Quantum
Aspects of
Life
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foreword by
SIR ROGER PENROSE
editors
Derek Abbott (University of Adelaide, Australia)
Paul C. W. Davies (Arizona State University, USAU
Arun K. Pati (Institute of Physics, Orissa, India)
ICP
Imperial College Press
Published by
Imperial College Press
57 Shelton Street
Covent Garden
London WC2H 9HE
Distributed by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Quantum aspects of life / editors, Derek Abbott, Paul C.W. Davies, Arun K. Pati ; foreword by
Sir Roger Penrose.
p. ; cm.
Includes bibliographical references and index.
ISBN-13: 978-1-84816-253-2 (hardcover : alk. paper)
ISBN-10: 1-84816-253-7 (hardcover : alk. paper)
ISBN-13: 978-1-84816-267-9 (pbk. : alk. paper)
ISBN-10: 1-84816-267-7 (pbk. : alk. paper)
1. Quantum biochemistry. I. Abbott, Derek, 1960– II. Davies, P. C. W. III. Pati,
Arun K.
[DNLM: 1. Biogenesis. 2. Quantum Theory. 3. Evolution, Molecular. QH 325 Q15 2008]
QP517.Q34.Q36 2008
576.8'3--dc22
2008029345
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Photo credit: Abigail P. Abbott for the photo on cover and title page.
Copyright © 2008 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
Printed in Singapore.
This book is dedicated to Arun’s baby daughter, Arshia, who
does not know what life is, yet she has a life.
Artwork credit: Arun K. Pati
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Foreword
When the remarkable book What is Life? was published in 1944, written
by the great quantum physicist Erwin Schrödinger and based on lectures
that he had given at Trinity College Dublin in February 1943, it had a very
considerable influence on several key figures in the development of molecular
biology. In particular, J. B. S. Haldane, James Watson, Francis Crick, and
Maurice Wilkins, have each expressed indebtedness to the penetrating ideas
that Schrödinger put forward. One of the basic questions that Schrödinger
raised was whether the ideas of classical physics, as normally employed by
biologists in their understanding of the behaviour of the physical world,
can be sufficient for explaining the basic features of life. He allowed that a
case could certainly be put forward that biological systems, being large as
compared with the atomic scale and containing vast numbers of constituent
atoms, would consequently have macroscopic actions determined essentially
by the statistical laws of large numbers. Together with some general overreaching principles of Newtonian mechanics such as conservation of energy,
he accepted that this could lead to an overall behaviour consistent with
classical Newtonian laws. However, he pointed out that a key feature of
the Darwinian/Mendelian nature of inheritance is its basis in discreteness,
which could only be explained through a quantum discreteness and stability,
in the basic carriers of genetic information. He argued that these carriers
had to be molecules of some nature—the molecules that we now know as
DNA.
Molecules, and their chemistry, are certainly governed by quantum laws,
according to our present understanding; nevertheless, chemists and biologists may not think of chemistry as very “quantum mechanical,” perhaps
because of the many ball-and-stick (or computer) models that they have
vii
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Quantum Aspects of Life
built their experience upon, and such a “hands-on” familiarity is not suggestive of the strange non-intuitive nature of quantum systems. In accordance
with such images, we may think of chemistry as being only rather “weakly”
quantum mechanical, where the more puzzling features of quantum mechanics in which distant entanglements and globally coherent behaviour
do not seem to feature significantly. Such coherent behaviour is witnessed
in the phenomena of superfluidity and superconductivity, and in the mysterious entanglements that one can find between the distantly separated
quantum particles of EPR (Einstein-Podolsky-Rosen) situations, where the
overall behaviour of the combined system cannot be understood simply in
terms of the individual nature of its constituent components.
A question of great interest, therefore, is whether or not such “strongly”
quantum-mechanical features of Nature might be playing significant roles in
the essential processes of life. An area where such a non-local role has been
argued for is in the operation of the brain, where the “binding problem”,
according to which widely separated areas of the brain, with very little in
the way of direct neuronal connection, are responsible for the processing of
different types of perception (such as colour, shape, and movement, in visual processing); nevertheless all come together in the formation of a single
conscious image. On the other side of the argument is the seemingly inhospitable environment that the warm and apparently “messy” living brain
provides for such delicate and subtle non-local quantum processes. Indeed,
there is no question that if the brain does make use of such “strongly”
quantum-mechanical phenomena, it must do so through the agency of some
very sophisticated organization. But the situation is certainly far from
hopeless as, on the one hand, there is indeed great subtlety in cell structure
and, on the other, the very existence of high-temperature superconductors
demonstrates that collective quantum phenomena can take place with a relatively small amount of sophistication and without the necessity of extreme
cold.
There is a further question that Schrödinger touched upon towards the
end of his book, in which he raised the more speculative issue of whether it
need actually be the case that even the physical laws provided by standard
20th century quantum mechanics are sufficient for a physical explanation
of life. He imagined the situation of an engineer, familiar only with Newtonian mechanics and general statistical principles, being presented with an
electric motor. Without any familiarity with the laws of electromagnetism
that Faraday and Maxwell have now presented us with, the engineer would
have no explanation for the motor’s behaviour, which might seem almost
Foreword
ix
like magic. But the Faraday-Maxwell laws are still mathematical laws of
physics, going beyond (but still consistent with) the overall scheme of things
laid down by the general framework of Newtonian mechanics and statistical physics. Likewise, Schrödinger argues, it is certainly possible that new
physical ingredients, going beyond those of 20th century physics, might be
needed for a full understanding of the physical underpinnings of life.
There are probably not many biologists today who would argue for the
necessity of such new physical ingredients in order to explain life. Yet, in an
Epilogue (On Determinism and Free Will) to his book, Schrödinger raises
the further conundrum of how the conscious mind, with its apparent free
will, can be accommodated within the “statistico-deterministic” framework
of our current quantum/classical pictures. The possible physical need for
going beyond this framework had already been raised by Schrödinger himself some eight years before his Dublin lectures, when he introduced his
famous “cat paradox”. Although he did not refer to this paradox explicitly in What is Life? (presumably because he had no desire to confuse
his lay audience by introducing such unsettling issues into his descriptions
of quantum mechanics), this unsatisfactory state of affairs in the foundations of quantum theory no doubt led him to be sceptical of the current
dogma that the rules of quantum mechanics must hold true at all levels
of physical description. (It may be pointed out that three others of the
key figures in the development of quantum mechanics, namely Einstein, de
Broglie, and Dirac, have also expressed the opinion that existing quantum
mechanics must be a provisional theory.) There is, indeed, a distinct possibility that the broadening of our picture of physical reality that may well
be demanded by these considerations is something that will play a central
role in any successful theory of the physics underlying the phenomenon of
consciousness.
These deep matters are still subject to much controversy, and the present
volume provides a multitude of closely argued opinions on the issues that
Schrödinger raised concerning the relation of biology to quantum physics.
Is it merely the complexity of biology that gives living systems their special
qualities and, if so, how does this complexity come about? Or are the special
features of strongly quantum-mechanical systems in some way essential?
If the latter, then how is the necessary isolation achieved, so that some
modes of large-scale quantum coherence can be maintained without their
being fatally corrupted by environmental decoherence? Does life in some
way make use of the potentiality for vast quantum superpositions, as would
be required for serious quantum computation? How important are the
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Quantum Aspects of Life
quantum aspects of DNA molecules? Are cellular microtubules performing
some essential quantum roles? Are the subtleties of quantum field theory
important to biology? Shall we gain needed insights from the study of
quantum toy models? Do we really need to move forward to radical new
theories of physical reality, as I myself believe, before the more subtle issues
of biology—most importantly conscious mentality—can be understood in
physical terms? How relevant, indeed, is our present lack of understanding
of physics at the quantum/classical boundary? Or is consciousness really
“no big deal,” as has sometimes been expressed?
It would be too optimistic to expect to find definitive answers to all these
questions, at our present state of knowledge, but there is much scope for
healthy debate, and this book provides a profound and very representative
measure of it.
Sir Roger Penrose, OM, FRS
The Mathematical Institute, University of Oxford
March 2007.
About the author
Sir Roger Penrose, OM, FRS was born on 8 August 1931 in Colchester, Essex, England. He is a mathematical physicist and Emeritus Rouse
Ball Professor of Mathematics at the Mathematical Institute, University of
Oxford and Emeritus Fellow of Wadham College. Penrose is concurrently
the Francis and Helen Pentz Distinguished Visiting Professor of Physics
and Mathematics at Penn State University. Penrose graduated with a first
class degree in mathematics from University College London. He obtained
his PhD at Cambridge (St John’s College) in 1958, writing a thesis on tensor methods in algebraic geometry under John Arthur Todd. In 1965 at
Cambridge, Penrose proved that black hole singularities could be formed
from the gravitational collapse of large dying stars. In 1967, Penrose invented twistor theory and in 1969 he conjectured the cosmic censorship
hypothesis—this form is now known as the weak censorship hypothesis. In
1979, Penrose formulated a stronger version called the strong censorship
hypothesis. He is also well-known for his 1974 discovery of Penrose tilings,
which are formed from two tiles that can surprisingly tile an infinite plane
aperiodically. Another noteworthy contribution is his 1971 invention of spin
Foreword
xi
networks, which later came to form the geometry of spacetime in loop quantum gravity. He was influential in popularizing what are commonly known
as Penrose diagrams. He has written 8 books, including The Emperor’s
New Mind (1989) and Shadows of the Mind (1994) that explore the lacunae between human consciousness and the known laws of physics. In 2004,
Penrose released his magnum opus The Road to Reality: A Complete Guide
to the Laws of the Universe. In 1975, Stephen Hawking and Roger Penrose were jointly awarded the Eddington Medal of the Royal Astronomical
Society. In 1985, Penrose was awarded the Royal Society Royal Medal. Together with Stephen Hawking, he was awarded the Wolf Foundation Prize
for Physics in 1988. In 1989, Penrose was awarded the Dirac Medal and
Prize of the British Institute of Physics. In 1990, he was awarded the Albert Einstein Medal and, in 1991, he was awarded the Naylor Prize of the
London Mathematical Society. In 1998, he was elected Foreign Associate
of the United States National Academy of Sciences and, in 2004, he was
awarded the De Morgan Medal.
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Preface
A landmark event in the history of science was the publication in 1944 of
Erwin Schrödinger’s book What is Life? Six decades later, the question
remains unanswered. Although biological processes are increasingly well
understood at the biochemical and molecular biological level, from the point
of view of fundamental physics, life remains deeply mysterious. Schrödinger
himself drew inspiration from his seminal work on quantum mechanics,
which had so spectacularly explained the nature of matter, believing it
was sufficiently powerful and remarkable to explain the nature of life too.
These dreams have not been realized. To be sure, quantum mechanics
is indispensable for explaining the shapes, sizes and chemical affinities of
biological molecules, but for almost all purposes scientists go on to treat
these molecules using classical ball-and-stick models. Life still seems an
almost magical state of matter to physicists; furthermore, its origin from
non-living chemicals is not understood at all.
In recent years, circumstantial evidence has accumulated that quantum
mechanics may indeed, as Schrödinger hoped, cast important light on life’s
origin and nature. In October 2003, the US space agency NASA convened
a workshop at the Ames Laboratory in California, the leading astrobiology
institution, devoted to quantum aspects of life. The workshop was hosted
by Ames astrobiologist Chris McKay and chaired by Paul Davies. In this
volume we solicit essays both from the participants in the workshop, and
from a wider range of physical scientists who have considered this theme,
including those who have expressed skepticism. The over-arching question
we address is whether quantum mechanics plays a non-trivial role in
biology.
xiii
xiv
Quantum Aspects of Life
We believe it is timely to set out a distinct quantum biology agenda.
The burgeoning fields of nanotechnology, biotechnology, quantum technology and quantum information processing are now strongly converging. The
acronym BINS, for Bio-Info-Nano-Systems, has been coined to describe the
synergetic interface of these several disciplines. The living cell is an information replicating and processing system that is replete with naturallyevolved nanomachines, which at some level require a quantum mechanical
description. As quantum engineering and nanotechnology meet, increasing
use will be made of biological structures, or hybrids of biological and fabricated systems, for producing novel devices for information storage and
processing, and other tasks. An understanding of these systems at a quantum mechanical level will be indispensable.
To broaden the discussion, we include chapters on “artificial quantum
life,” a rapidly-developing topic of interest in its own right, but also because
it may cast light on real biological systems. Related mathematical models include quantum replication and evolution, von Neumann’s universal
constructors for quantum systems, semi-quantum cellular automata, and
evolutionary quantum game theory.
Finally, we include the transcripts of two debates:
(1) “Dreams versus reality: quantum computing” hosted by the Fluctuations and Noise symposium held in Santa Fe, USA, 1–4 June 2003.
The panelists were Carlton M. Caves, Daniel Lidar, Howard Brandt,
Alex Hamilton (for) and David Ferry, Julio Gea-Banacloche, Sergey
Bezrukov and Laszlo Kish (against). The debate chair was Charles
Doering.
(2) “Quantum effects in biology: trivial or not?” hosted by the Fluctuations
and Noise symposium held in Gran Canaria, Spain, 25–28 May 2004.
The panelists were Paul Davies, Stuart Hameroff, Anton Zeilinger,
Derek Abbott (for) and Jens Eisert, Sergey Bezrukov, Hans Frauenfelder and Howard Wiseman (against). The debate Chair was Julio
Gea-Banacloche.
The second debate represents the topic of this book and a new reader to
the area may find it beneficial to jump directly to Chapter 16, as this will
help the reader navigate some of the competing arguments in an entertaining way. The first debate, in Chapter 15, is on whether useful man-made
quantum computers are possible at all. Placing these two debates side by
side exposes interesting conflicting viewpoints of relevance to this book:
(1) Those who would argue for quantum processing in various biological
Preface
xv
systems have to face the difficulty that useful man-made quantum computers are extremely hard to make, and if they are fodder for debate then the
biological proposition would appear to be on even weaker ground; (2) Those
physicists who are working towards realizing large scale man-made quantum
computers, when faced with skepticism, are on occasion tempted to appeal
to biology in their defence as can be seen in Chapter 15. This therefore
creates an exciting tension between the opposing viewpoints, namely, that
on one hand pessimistic experience with man-made quantum computers is
used to cast doubt on quantum effects in biology, whereas on the other
hand an optimistic view of quantum effects in biology is used to motivate
future man-made quantum computers. Physicists with a vested interest in
realizing quantum computers often find themselves in a strange superposition of these orthogonal viewpoints, which can only be finally resolved if
more detailed experiments on biomolecules are carried out.
Finally, it is our hope that at the very least this book will provoke further debate and help provide motivation for more experimental research
into nature’s nanostructures. If experiments can shed further light on our
understanding of decoherence in biomolecules, at scales where equillibrium
thermodynamics no longer applies, this may provide the required foundation for greatly accelerating our progress in man-made quantum computers.
Derek Abbott, University of Adelaide, Australia
Paul C. W. Davies, Arizona State University, USA
Arun K. Pati, Institute of Physics (IOP), Orissa, India.
November 2007
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Acknowledgments
We thank Rita Yordy, Stefanie Cross, Pauline Davies, and Megan Hunt of
Arizona State University (ASU) for administrative assistance, manuscript
proof reading, and correspondence.
A special thanks is due to Mathias Baumert, of the University of Adelaide,
who coordinated the LATEX formatting of the book.
DA
PCWD
AP
xvii
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Contents
Foreword
vii
Preface
xiii
Acknowledgments
xvii
Part 1: Emergence and Complexity
1
1. A Quantum Origin of Life?
3
Paul C. W. Davies
1.1. Chemistry and Information . . . . . . . . . . . . . . .
1.2. Q-life . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3. The Problem of Decoherence . . . . . . . . . . . . . .
1.4. Life as the “Solution” of a Quantum Search Algorithm
1.5. Quantum Choreography . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Quantum Mechanics and Emergence
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19
Seth Lloyd
2.1. Bits . . . . . . . . . . . . . .
2.2. Coin Flips . . . . . . . . . . .
2.3. The Computational Universe
2.4. Generating Complexity . . .
2.5. A Human Perspective . . . .
2.6. A Quantum Perspective . . .
References . . . . . . . . . . . . . .
xix
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Quantum Aspects of Life
Part 2: Quantum Mechanisms in Biology
31
3. Quantum Coherence and the Search for the First Replicator
33
Jim Al-Khalili and Johnjoe McFadden
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
When did Life Start? . . . . . . . . . . . . . . . .
Where did Life Start? . . . . . . . . . . . . . . .
Where did the Precursors Come From? . . . . . .
What was the Nature of the First Self-replicator?
The RNA World Hypothesis . . . . . . . . . . . .
A Quantum Mechanical Origin of Life . . . . . .
3.6.1. The dynamic combinatorial library . . .
3.6.2. The two-potential model . . . . . . . . .
3.6.3. Decoherence . . . . . . . . . . . . . . . .
3.6.4. Replication as measurement . . . . . . .
3.6.5. Avoiding decoherence . . . . . . . . . . .
3.7. Summary . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Ultrafast Quantum Dynamics in Photosynthesis
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51
Alexandra Olaya Castro, Francesca Fassioli Olsen,
Chiu Fan Lee, and Neil F. Johnson
4.1.
4.2.
4.3.
Introduction . . . . . . . . . . . . . . . . . . . . .
A Coherent Photosynthetic Unit (CPSU) . . . .
Toy Model: Interacting Qubits with a Spin-star
Configuration . . . . . . . . . . . . . . . . . . . .
4.4. A More Detailed Model: Photosynthetic Unit of
Purple Bacteria . . . . . . . . . . . . . . . . . . .
4.5. Experimental Considerations . . . . . . . . . . .
4.6. Outlook . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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5. Modelling Quantum Decoherence in Biomolecules
71
Jacques Bothma, Joel Gilmore, and Ross H. McKenzie
5.1.
5.2.
5.3.
Introduction . . . . . . . . . . .
Time and Energy Scales . . . .
Models for Quantum Baths and
5.3.1. The spin-boson model .
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Decoherence
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xxi
Contents
5.3.2. Caldeira-Leggett Hamiltonian . . . . . . .
5.3.3. The spectral density . . . . . . . . . . . . .
5.4. The Spectral Density for the Different Continuum
Models of the Environment . . . . . . . . . . . . .
5.5. Obtaining the Spectral Density from Experimental
Data . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6. Analytical Solution for the Time Evolution of the
Density Matrix . . . . . . . . . . . . . . . . . . . .
5.7. Nuclear Quantum Tunnelling in Enzymes and the
Crossover Temperature . . . . . . . . . . . . . . . .
5.8. Summary . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part 3: The Biological Evidence
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6. Molecular Evolution: A Role for Quantum Mechanics
in the Dynamics of Molecular Machines that Read and
Write DNA
97
Anita Goel
6.1.
6.2.
6.3.
Introduction . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . .
Approach . . . . . . . . . . . . . . . . . . . . .
6.3.1. The information processing power of a
molecular motor . . . . . . . . . . . . .
6.3.2. Estimation of decoherence times of the
motor-DNA complex . . . . . . . . . .
6.3.3. Implications and discussion . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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7. Memory Depends on the Cytoskeleton, but is it Quantum?
109
Andreas Mershin and Dimitri V. Nanopoulos
7.1.
7.2.
7.3.
7.4.
Introduction . . . . . . . . . . . . . . . . . . . . .
Motivation behind Connecting Quantum Physics
to the Brain . . . . . . . . . . . . . . . . . . . . .
Three Scales of Testing for Quantum Phenomena
in Consciousness . . . . . . . . . . . . . . . . . .
Testing the QCI at the 10 nm–10 µm Scale . . .
. . . . . 109
. . . . . 111
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xxii
Quantum Aspects of Life
7.5.
Testing for Quantum Effects in Biological Matter
Amplified from the 0.1 nm to the 10 nm Scale and
Beyond . . . . . . . . . . . . . . . . . . . . . . . .
7.6. Summary and Conclusions . . . . . . . . . . . . . .
7.7. Outlook . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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8. Quantum Metabolism and Allometric Scaling Relations
in Biology
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Lloyd Demetrius
8.1.
8.2.
Introduction . . . . . . . . . . . . . . . . . . . .
Quantum Metabolism: Historical Development
8.2.1. Quantization of radiation oscillators . .
8.2.2. Quantization of material oscillators . .
8.2.3. Quantization of molecular oscillators .
8.2.4. Material versus molecular oscillators . .
8.3. Metabolic Energy and Cycle Time . . . . . . .
8.3.1. The mean energy . . . . . . . . . . . .
8.3.2. The total metabolic energy . . . . . . .
8.4. The Scaling Relations . . . . . . . . . . . . . .
8.4.1. Metabolic rate and cell size . . . . . . .
8.4.2. Metabolic rate and body mass . . . . .
8.5. Empirical Considerations . . . . . . . . . . . .
8.5.1. Scaling exponents . . . . . . . . . . . .
8.5.2. The proportionality constant . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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9. Spectroscopy of the Genetic Code
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9.1.
9.2.
Background: Systematics of the Genetic Code . . . . .
9.1.1. RNA translation . . . . . . . . . . . . . . . . .
9.1.2. The nature of the code . . . . . . . . . . . . .
9.1.3. Information processing and the code . . . . . .
Symmetries and Supersymmetries in the Genetic Code
9.2.1. sl(6/1) model: UA+S scheme . . . . . . . . .
9.2.2. sl(6/1) model: 3CH scheme . . . . . . . . . .
9.2.3. Dynamical symmetry breaking and third base
wobble . . . . . . . . . . . . . . . . . . . . . .
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xxiii
Contents
9.3.
9.4.
Visualizing the Genetic Code . . . . . . . . . . . . . .
Quantum Aspects of Codon Recognition . . . . . . . .
9.4.1. N(34) conformational symmetry . . . . . . . .
9.4.2. Dynamical symmetry breaking and third base
wobble . . . . . . . . . . . . . . . . . . . . . .
9.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 168
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. . 181
10. Towards Understanding the Origin of Genetic Languages
187
Apoorva D. Patel
10.1. The Meaning of It All . . .
10.2. Lessons of Evolution . . . .
10.3. Genetic Languages . . . . .
10.4. Understanding Proteins . .
10.5. Understanding DNA . . . .
10.6. What Preceded the Optimal
10.7. Quantum Role? . . . . . . .
10.8. Outlook . . . . . . . . . . .
References . . . . . . . . . . . . .
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Languages?
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Part 4: Artificial Quantum Life
187
190
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195
201
204
211
215
217
221
11. Can Arbitrary Quantum Systems Undergo Self-replication?
223
Arun K. Pati and Samuel L. Braunstein
11.1. Introduction . . . . . . . . . . . . . . . .
11.2. Formalizing the Self-replicating Machine
11.3. Proof of No-self-replication . . . . . . .
11.4. Discussion . . . . . . . . . . . . . . . . .
11.5. Conclusion . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . .
12. A Semi-quantum Version of the Game of Life
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225
226
227
228
229
233
Adrian P. Flitney and Derek Abbott
12.1. Background and Motivation . . . . . . . . . . . . . . . . . 233
12.1.1. Classical cellular automata . . . . . . . . . . . . . 233
xxiv
Quantum Aspects of Life
12.1.2. Conway’s game of life . . . .
12.1.3. Quantum cellular automata
12.2. Semi-quantum Life . . . . . . . . . .
12.2.1. The idea . . . . . . . . . . .
12.2.2. A first model . . . . . . . . .
12.2.3. A semi-quantum model . . .
12.2.4. Discussion . . . . . . . . . .
12.3. Summary . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . .
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13. Evolutionary Stability in Quantum Games
234
237
238
238
239
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247
248
251
Azhar Iqbal and Taksu Cheon
13.1. Evolutionary Game Theory and Evolutionary Stability
13.1.1. Population setting of evolutionary game
theory . . . . . . . . . . . . . . . . . . . . . .
13.2. Quantum Games . . . . . . . . . . . . . . . . . . . . .
13.3. Evolutionary Stability in Quantum Games . . . . . .
13.3.1. Evolutionary stability in EWL scheme . . . .
13.3.2. Evolutionary stability in MW quantization
scheme . . . . . . . . . . . . . . . . . . . . . .
13.4. Concluding Remarks . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 253
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14. Quantum Transmemetic Intelligence
291
Edward W. Piotrowski and Jan Sladkowski
14.1.
14.2.
14.3.
14.4.
14.5.
14.6.
Introduction . . . . . . . . . . . . . . . . . . . . . . .
A Quantum Model of Free Will . . . . . . . . . . . .
Quantum Acquisition of Knowledge . . . . . . . . .
Thinking as a Quantum Algorithm . . . . . . . . . .
Counterfactual Measurement as a Model of Intuition
Quantum Modification of Freud’s Model of
Consciousness . . . . . . . . . . . . . . . . . . . . . .
14.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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291
294
298
300
301
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xxv
Contents
Part 5: The Debate
311
15. Dreams versus Reality: Plenary Debate Session on
Quantum Computing
313
16. Plenary Debate: Quantum Effects in Biology: Trivial or Not?
349
17. Nontrivial Quantum Effects in Biology: A Skeptical
Physicists’ View
381
Howard Wiseman and Jens Eisert
17.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2. A Quantum Life Principle . . . . . . . . . . . . . . . . . .
17.2.1. A quantum chemistry principle? . . . . . . . . . .
17.2.2. The anthropic principle . . . . . . . . . . . . . . .
17.3. Quantum Computing in the Brain . . . . . . . . . . . . .
17.3.1. Nature did everything first? . . . . . . . . . . . .
17.3.2. Decoherence as the make or break issue . . . . . .
17.3.3. Quantum error correction . . . . . . . . . . . . . .
17.3.4. Uselessness of quantum algorithms for organisms .
17.4. Quantum Computing in Genetics . . . . . . . . . . . . . .
17.4.1. Quantum search . . . . . . . . . . . . . . . . . . .
17.4.2. Teleological aspects and the fast-track to life . . .
17.5. Quantum Consciousness . . . . . . . . . . . . . . . . . . .
17.5.1. Computability and free will . . . . . . . . . . . . .
17.5.2. Time scales . . . . . . . . . . . . . . . . . . . . . .
17.6. Quantum Free Will . . . . . . . . . . . . . . . . . . . . . .
17.6.1. Predictability and free will . . . . . . . . . . . . .
17.6.2. Determinism and free will . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18. That’s Life!—The Geometry of π Electron Clouds
381
382
382
384
385
385
386
387
389
390
390
392
392
392
394
395
395
396
398
403
Stuart Hameroff
18.1.
18.2.
18.3.
18.4.
18.5.
What is Life? . . . . . . . . . . . . . . . . . .
Protoplasm: Water, Gels and Solid Non-polar
Van der Waals Forces . . . . . . . . . . . . .
Kekule’s Dream and π Electron Resonance .
Proteins—The Engines of Life . . . . . . . . .
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Regions
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403
405
407
409
413
xxvi
Quantum Aspects of Life
18.6. Anesthesia and Consciousness . . . . . . . . . . . . . . .
18.7. Cytoskeletal Geometry: Microtubules, Cilia and Flagella
18.8. Decoherence . . . . . . . . . . . . . . . . . . . . . . . . .
18.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1. Quantum Computing in DNA π Electron Stacks
Appendix 2. Penrose-Hameroff Orch OR Model . . . . . . . .
Index
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418
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425
427
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435
PART 1
Emergence and Complexity
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Chapter 1
A Quantum Origin of Life?
Paul C. W. Davies
The origin of life is one of the great unsolved problems of science. In the
nineteenth century, many scientists believed that life was some sort of magic
matter. The continued use of the term “organic chemistry” is a hangover
from that era. The assumption that there is a chemical recipe for life led
to the hope that, if only we knew the details, we could mix up the right
stuff in a test tube and make life in the lab.
Most research on biogenesis has followed that tradition, by assuming
chemistry was a bridge—albeit a long one—from matter to life. Elucidating the chemical pathway has been a tantalizing goal, spurred on by the
famous Miller-Urey experiment of 1952, in which amino acids were made by
sparking electricity through a mixture of water and common gases [Miller
(1953)]. But this concept turned out to be something of a blind alley, and
further progress with pre-biotic chemical synthesis has been frustratingly
slow.
In 1944, Erwin Schrödinger published his famous lectures under the title What is Life? [Schrödinger (1944)] and ushered in the age of molecular
biology. Schödinger argued that the stable transmission of genetic information from generation to generation in discrete bits implied a quantum
mechanical process, although he was unaware of the role of or the specifics
of genetic encoding. The other founders of quantum mechanics, including
Niels Bohr, Werner Heisenberg and Eugene Wigner shared Schrödinger’s
belief that quantum physics was the key to understanding the phenomenon
Received May 9, 2007
3
4
Quantum Aspects of Life
of life. This was a reasonable assumption at the time. Shortly before, quantum mechanics had solved the problem of matter, by explaining atomic and
molecular structure, chemical bonds and the nature of solids. It seemed
natural that quantum mechanics would soon also solve the riddle of the
living state of matter. To a physicist, life seems fundamentally weird, even
bizarre, in its properties, and bears almost no resemblance to any other
type of physical system. It is tempting to suppose that quantum mechanics
possesses enough weirdness to account for it.
These early musings about the place of quantum mechanics in life were
soon swept away in the rush. Molecular biology proved so successful that
rich pickings could be harvested merely from crude ball-and-stick models
of molecules. However, with the maturity of the subject, hints began to
surface that non-trivial quantum effects might be of crucial significance in
the functioning of certain biosystems. Some of these effects are reviewed
in other chapters in this volume. The question I wish to address in this
chapter is in what manner quantum mechanics played a role in the origin
of life. One point needs clarification. There is a trivial sense in which life is
quantum mechanical. Cellular function depends on the shapes of molecules
and their chemical affinities, properties that require quantum mechanics to
explain. However, what I have in mind are non-trivial quantum effects, for
example, the coherent wavelike nature of matter, tunnelling, entanglement,
intrinsic spin, Berry phase, environmental post-selection and the watchdog
effect.
Obviously at some level quantum mechanics cannot be ignored in the life
story, since by general consent, life somehow emerged from the molecular
realm, even if the specifics remain mysterious. The molecular road to life is
in contrast to the “magic matter” theories of the nineteenth century that
were essentially macroscopic in conception. Because the molecular realm
is unquestionably quantum mechanical in nature, the issue I am raising
is whether classicality emerged before life or whether life emerged before
classicality. My central hypothesis is that quantum mechanics enabled life
to emerge directly from the atomic world, without complex intermediate
chemistry. The orthodox view is that an extended period of increasingly
complex “ball-and-stick” chemistry preceded the transition to the first genuinely autonomous living system (which may not have been an individual
cell, more likely it was a cellular cooperative). The philosophical position
that underpins my hypothesis is that the secret of life lies not with its complexity per se, still less with the stuff of which it is composed, but with its
remarkable information processing and replicating abilities.
A Quantum Origin of Life?
1.1.
5
Chemistry and Information
Although there is no agreed definition of life, all living organisms are information processors: they store a genetic database and replicate it, with
occasional errors, thus providing the basis for natural selection. The direction of information flow is bottom up: the form of the organism and its
selective qualities can be traced back to molecular processes. The question then arises of whether, since this information flows from the quantum
realm, any vestige of its quantum nature, other than its inherent randomness, is manifested. Biological molecules serve the role of both specialized
chemicals and informational molecules, mirroring the underlying dualism of
phenotype/genotype. In computer terminology, chemistry is akin to hardware, information to software. A complete understanding of the origin of
life demands an explanation for both hardware and software. Most research in biogenesis focuses on the hardware aspect, by seeking a plausible
chemical pathway from non-life to life. Though this work has provided important insights into how and where the basic building blocks of life might
have formed, it has made little progress in the much bigger problem of
how those building blocks were assembled into the specific and immensely
elaborate organization associated with even the simplest autonomous organism [Davies (2003)]. But viewing life in terms of information processing transforms the entire conceptual basis of the problem of biogenesis.
Reproduction is one of the defining characteristics of life. Traditionally,
biologists regarded reproduction as the replication of material structures,
whether DNA molecules or entire cells. But to get life started all one
needs is to replicate information. In recent years our understanding of the
nature of information has undergone something of a revolution with the
development of the subjects of quantum computation and quantum information processing. The starting point of this enterprise is the replacement
of the classical “bit” by its quantum counterpart, the “qubit”. As a quantum system evolves, information is processed; significantly, the processing
efficiency is enhanced because quantum superposition and entanglement
represent a type of computational parallelism. In some circumstances this
enhancement factor can be exponential, implying a vast increase in computational speed and power over classical information processing. Many
researchers have spotted the sweeping consequences that would follow from
the discovery that living organisms might process information quantum mechanically, either at the bio-molecular level, or the cellular/neuronal level
[Penrose (1989); Beck and Eccles (1992); Hameroff (1994); Davies (2004);
6
Quantum Aspects of Life
Matsuno (1999); Patel (2001); Vedral (2003); Schempp (2003)]. Biological systems are quintessential information processors. The informational
molecules are RNA and DNA. Although quantum mechanics is crucial to
explain the structure of these molecules, it is normally disregarded when it
comes to their information processing role. That is, biological molecules are
assumed to store and process classical bits rather than qubits. In an earlier
paper [Davies (2004)] I speculated that, at least in some circumstances,
that assumption may be wrong. It is then helpful to distinguish between
three interesting possibilities:
(1) Quantum mechanics played a key role in the emergence of life, but
either ceased completely to be a significant factor when life became
established, or was relegated to a sporadic or subsidiary role in its
subsequent development. Nevertheless, there may be relics of ancient
quantum information processing systems in extant organisms, just as
there are biochemical remnants that give clues about ancient biological,
or even pre-biological, processes.
(2) Life began classically, but evolved some efficiency-enhancing “quantum
tricks.” For example, if biological systems were able to process information quantum mechanically, they would gain a distinct advantage in
speed and power, so it might be expected that natural selection would
discover and amplify such capabilities, if they are possible.
(3) Life started out as a classical complex system, but later evolved towards
“the quantum edge,” where quantum uncertainty places a bound on the
efficiency of bio-molecular processes.
As there is little doubt that some cellular machinery (e.g.
photosynthesis—see Chapter 4) does exploit quantum mechanics [Engel et
al. (2007)], the issue arises of whether quantum enhancement is a product
of evolution (as in 2), or a remnant of life’s quantum origin (as in 1).
1.2.
Q-life
The starting point of my hypothesis is the existence of a quantum replicator, a quantum system that can copy information with few errors [Wigner
(1961); Pati (2004)]. The information could be instantiated in the form
of qubits, but that is not necessary—the quantum replication of classical
bits is sufficient (see below). A quantum replicator need not be an atomic
system that clones itself. Indeed, there is a quantum no-cloning theorem
A Quantum Origin of Life?
7
that forbids the replication of wavefunctions [Wooters and Zurek (1982);
Pati (2004)]—see also Chapter 11 in this book. Rather, the information
content of an atomic system must be copied more or less intact—not necessarily in one step, maybe after a sequence of interactions. This information
might well be in binary form, making use of the spin orientation of an electron or atom for example. Quantum mechanics thus provides an automatic
discretization of genetic information. Quantum replicators certainly exist
in Nature. The simplest case is the stimulated emission of photons. Another is the atom-by-atom growth of a crystal lattice. But these examples
are not information-rich; they do not fulfill the additional requirement of
high algorithmic complexity demanded by biology, as neither the identical
photons not the identical crystal atoms store more than a very few bits of
information. So we seek a natural system in which high-fidelity replication of an information-rich assemblage of quantum objects takes place on
a short time scale. Henceforth I shall refer to this hypothetical system as
Q-life. Leaving aside wild speculations like life in neutron stars [Forward
(1980)], a venue for Q-life might plausibly be a condensed matter setting
at a low temperature, for example, the crust of an icy rogue planetesimal
in interstellar space.
Let me illustrate the basic idea of Q-life with a simple, and almost certainly unsatisfactory, example. Consider an array of atomic spins embedded
in a condensed matter system, defined relative to some fiducial direction.
The initial template A may be described by a ket vector such as
| ↑↑↓↑↓↓↓↑↓↑↑↑↓↑↓↓↑> .
This template then comes into interaction with an arbitrary system of
spins B, say,
| ↑↑↓↓↑↓↑↑↑↓↑↑↓↑↓↑↓> .
As a result of the interaction (which may entail many intermediate
steps), the following transition occurs
| ↑↑↓↑↓↓↓↑↓↑↑↑↓↑↓↓↑> | ↑↑↓↓↑↓↑↑↑↓↑↑↓↑↓↑↓>−→
| ↑↑↓↑↓↓↓↑↓↑↑↑↓↑↓↓↑> | ↑↑↓↑↓↓↓↑↓↑↑↑↓↑↓↓↑> .
8
Quantum Aspects of Life
Symbolically, the overall evolution of the state is AB −→ AA. Because
the transition has erased the information contained in state B, the replication process is asymmetric and irreversible, and accompanied by an increase
in entropy. The system thus requires an energy source to drive the reaction
forward. This could be in the form of an exciton that hops along the array
of atoms, flipping the B spins where necessary one-by-one but leaving the
A spins unchanged.
The foregoing model is very simplistic. A more realistic form of interaction, and a closer analogue of DNA replication, would be if the template
array A first created a complementary array
| ↓↓↑↓↑↑↑↓↑↓↓↓↑↓↑↑↓> ,
which then generated the original array by “base-pairing”. An additional
simplification is that the model described so far neglects interactions between neighbouring spins. Such interactions produce greater complexity,
and so increase the opportunity to encode algorithmically incompressible
information.
The replication rate of a spin array will depend on whether the sequence
is processed linearly, after the fashion of DNA, or all at once. It will also
depend on the availability of the necessary complementary structure. Once
the two structures are brought into interaction, each bit flip could occur
extremely fast (i.e. in less than a femtosecond). This can be compared to
the sluggish rate of only 100 base-pairs per second typical of DNA replication by polymerase enzymes, even when the system is not resource-limited
[Goel et al. (2003)]. Thus, in an appropriate quantum setting, Q-life could
replicate and evolve at least 12 orders of magnitude faster than familiar
life! However, in practice an ideal appropriate setting is unlikely to occur
in nature. More realistic is a model of replication in which the process is
managed by a catalytic structure, in analogy with the replicase enzymes of
DNA replication. The job of this conjectured structure would be to bring
the required components into interaction, perhaps by creating an “interaction centre” screened from the environment. The replication rate would
then be limited by the performance of this “Q-replicase.” In particular, the
Q-replicase is likely to be subject to the opportunities and limitations of
quantum mechanics. In the later category are fundamental limits of choreography set by the uncertainty principle, a topic that I shall defer to the
final section.
How, then, did organic life arise? Information can readily be passed
from one medium to another. At some stage Q-life could have co-opted large
A Quantum Origin of Life?
9
organic molecules for back-up memory, much as a computer uses a harddisk. The computer’s processor (analogous to Q-life) is much faster than
the hard disk drive (analogous to RNA and DNA), but more vulnerable and
in need of a continual input of energy. Robust computing systems require
something like a hard disk. Eventually the organic molecular system would
literally have taken on a life of its own. The loss in processing speed would
have been offset against the greater complexity, versatility and stability
of organic molecules, enabling organic life to invade many environments
off-limits to Q-life (e.g. high temperature).
Note that although the replicator I have used as a simple illustration
is fundamentally quantum mechanical in nature, the copying process as
described does not replicate any entanglement or phase information; i.e. the
process replicates bits rather than qubits. For that reason, decoherence
would not be an issue at the replication stage. Left out of the account so
far, however, is how the quantum replicator arises in the first place. The
nature of the transition from an arbitrary quantum system to a replicating
quantum system is far from clear, but the process is likely to be enormously
enhanced if it is at least partially coherent. Let me therefore make some
general remarks about decoherence in biosystems.
1.3.
The Problem of Decoherence
Coherence entails the preservation of delicate phase relationships between
different components of the wave function. Interactions between the quantum system and its environment will serve to decohere the wave function:
the noise of the environment effectively scrambles the phases. Once decohered, a quantum system behaves in most respects as a classical system
[Zurek (1982)]. The decoherence rate depends on the nature and temperature of the environment and the strength with which it couples to the quantum system of interest [Zurek (1982); Caldeira and Leggett (1985); Unruh
and Zurek (1989); Hu et al. (1992)]. The main burden in the development
of quantum computation, for example, is to screen out the decohering environment as efficiently as possible, e.g. by reducing the temperature. If
quantum mechanics is to play a role in the origin of life, typical decoherence rates must not be greater than the relevant transition rates. Simple
models of decoherence have been much studied over the past twenty years.
Typically, for a particle interacting with a heat bath at room temperature,
exceedingly short decoherence times result. Translated into the context of,
10
Quantum Aspects of Life
say, a nucleotide in the environment of a cell at room temperature, decoherence times of femtoseconds are typical, [Caldeira and Leggett (1985);
Unruh and Zurek (1989); Hu et al. (1992); Tegmark (2000)]. But on a second look, the situation is found to be more subtle. There are two ways in
which decoherence could be diminished for long enough to enable biologically important processes to occur. The first is screening: if the system
of interest can be quasi-isolated from the decohering environment then decoherence rates can be sharply reduced. According to Matsuno (1999),
organisms may exploit thermodynamic gradients by acting as heat engines
and thereby drastically reduce the effective temperature of certain molecular complexes. He cites the example of the slow release of energy from ATP
molecules at actomyosin complexes, which he claims implies an effective
temperature for the actomyosin of a mere 1.6 × 10−3 K. At any rate, the
lesson of high-temperature superconductivity reminds us that in complex
states of matter, simple “kT reasoning” can be misleading.
The second possibility involves decoherence-free subspaces. In the effort
to build a quantum computer, much attention has been given to identifying
subspaces of Hilbert space that are unaffected by the coupling of the system to its environment [Nielson and Chuang (2001)]. Paradoxically, when
a system couples very strongly to its environment through certain degrees
of freedom, it can effectively “freeze” other degrees of freedom by the quantum Zeno effect, enabling coherent superpositions and even entanglement to
persist. An explicit example is provided by a double-well one-dimensional
potential. A particle placed in the lowest energy state of one well will tunnel
back and forth through the intervening barrier, oscillating with a certain
frequency. If the particle is placed instead in an excited state of the well,
this flip-flop frequency will be different. Thus an initial state consisting of
a superposition of lowest energy and excited states will soon evolve into
a complicated muddle as the flip-flops get out of phase. However, if the
particle is now allowed to interact strongly with an external heat bath, the
environment has the effect of forcing the disparate oscillations into synchrony, thereby maintaining a limited form of quantum coherence, not in
spite of, but because of, environmental interactions [Davies (2003)]. Furthermore, if the system is placed in an entangled state of left and right
well-locations, this entanglement is also preserved by environmental interaction. The model was developed in the context of neutrino oscillations, but
has general applicability [Bell et al. (2002)]. It does, however, depend on
the interaction being “blind” between the two potential wells. It is unclear
how realistically this would translate into a biological scenario, or whether
A Quantum Origin of Life?
11
it has any relevance to the extended decoherence times reported recently
[Engel et al. (2007)].
1.4.
Life as the “Solution” of a Quantum Search Algorithm
The hypothesis I am proposing is that the transition from non-life to life was
a quantum-mediated process, and that the earliest form of life involved nontrivial quantum mechanical aspects. The power of quantum superpositions
is that the system can explore many alternative pathways simultaneously,
thereby potentially shortcutting the transition time by a large factor. Because life is a highly unusual state of matter, its formation from an arbitrary
initial state is presumably extremely improbable. Quantum mechanics provides a way to drastically shorten the odds and fast-track matter to life by
exploiting the parallel processing properties of superpositions. There is,
however, a deep philosophical issue that must be confronted. I am defining
“life” as a certain special state of low probability. Quantum mechanics enables the space of possibilities to be much more efficiently explored than a
stochastic classical system. Now, if there are branches of the wave function
“containing life” (e.g. a quantum replicator), they will, by assumption, have
very small amplitudes. We must therefore explain why the wave function
of the system “collapses” onto one of these states of such low intrinsic probability. Expressed differently, how does a quantum superposition recognize
that it has “discovered” life and initiate the said collapse? There seems to
be an unavoidable teleological component involved: the system somehow
“selects” life from the vastly greater number of states that are nonliving.
Actually, the way I have expressed it is an abuse of language. In the
standard formulation of quantum mechanics, a quantum system itself never
“initiates collapse.” The wavefunction collapses as a result of interaction
with the environment. One possibility is that replicators are the products
of environmental post-selection, perhaps amplified by a quantum feedback
loop. The importance of quantum post-selection has only recently been
recognized [Aharonov et al. (1996)]. The idea is this. The environment
serves as a sort of measuring device, and, by hypothesis, it somehow selects
for measurement a quantum variable relevant for life. Then even if the
amplitude is small, life will be “projected out” of the superposition by the
measurement-like interaction. It may even be “steered” towards life by
the inverse-Zeno effect. But this implies the environment somehow favours
life—that life is “built into” nature in a preordained manner. So an element
of teleology remains.
12
Quantum Aspects of Life
One way to envision the emergence of life by “state exploration” is in
terms of a vast decision tree of states (quantum or classical). The root of
the tree might correspond to a simple and easy-to-form initial state, which
might then evolve to any one of a huge range of possible subsequent states.
This can be represented by the tree of states splitting repeatedly into a
proliferating number of branches, each branch denoting a possible physical
path in state space leading away from the initial state. States of great complexity are represented by branches high up on the tree, and a subset of
these branches represents a quantum replicator, or some other state that we
may designate as life, or incipient life. The puzzle of life’s origin is how the
initial simple state “finds” one of the exceedingly rare branches associated
with life. Farhi and Gutmann (1998) have compared quantum and classical searches of decision trees, and they find that in some circumstances a
quantum search is exponentially faster than a classical search. Their model
cannot be immediately applied to the problem of biogenesis, however, because quantum coherence could not possibly be maintained through more
than a brief sequence of interactions in any likely prebiotic physical setting.
Nevertheless, as the example, of Engel et al. (2007) demonstrates, quantum
coherence over picosecond timescales is plausible, and leads to an enormous
speed-up in the transition to certain otherwise hard-to-attain states.
Our ignorance of the precise nature of the quantum replicator makes it
almost impossible to evaluate the probability that one will form as the end
product of a quantum search. However, some general points may be made
concerning quantum speed-up. If the replicator, or some other quantum
structure en route to it, is describable as a local minimum in an energy
landscape, with the formation of this unknown system being akin to a
phase transition, then quantum mechanics has the ability to enormously
enhance the probability of the transition by permitting tunnelling through
the relevant potential barrier in the energy landscape. So a possible model
of biogenesis is that of a phase transition analogous to bubble nucleation
in quantum field theory, where the nucleated lower-energy state is a community of interacting replicators—possibly a large community occupying
a mesoscopic region of a condensed matter system. This would constitute
a quantum version of Kauffman’s concept of an autocatalytic network of
molecules [Kauffman (1993)]. Secondly, if the “solution” of the quantum
“search” is defined to be a quantum replicator, and if the system does not
decohere faster than the replication time, then the replicator should act
in a manner similar to a quantum resonance (in view of the fact that the
wave function describing the replicator will be amplified by iteration), thus
greatly enhancing the probability for a transition to a replicator state.
A Quantum Origin of Life?
13
So far I have been describing the replicator as if it is a physical structure,
but the significant point about viewing life in terms of information is that, so
long as the information is replicated, the structures embodying that information need not be. In the case of familiar DNA based life, the information
represented by the base-pair sequence, and the base-pairs, are replicated together. Thus information replication is tied to structural replication. But
at the quantum level there are alternative possibilities. Consider, for example, a cellular automaton, such as the Game of Life—see, for example,
Gardner (1970). In this system a group of five clustered cells can form a
so-called glider. The glider moves across the array of cells as a coherent (in
the classical sense) object, and thus conserves information. However, individual cells are switched on and off, but in a way that preserves the overall
pattern. The origins of biological information could belong to this category
(perhaps constituting a quantum cellular automaton—see Chapter 12 by
Flitney and Abbott in this book). We can imagine a condensed matter system in which a pattern of excitation, or a pattern of spins, or some other
quantum variable, might induce transitions in neighbouring quantum states
in such a way as to conserve the pattern to high probability, but to “pass
on” the excitation, or spin, to adjacent atoms. The “information packet”
would thereby be preserved and propagate, until it encounters a suitable
quantum milieu in which it will replicate. Then two information packets
would propagate away from the interaction region, and so on. Quantum
fluctuations in the propagation and replication process would lead in a natural way to “mutations”, and to a Darwinian competition between rival
information packets.
1.5.
Quantum Choreography
An unresolved issue concerning replication is the matter of timing and
choreography. In the simplest templating arrangement one can imagine,
the formation of complementary base-pairs takes place by random access of
molecular components and will proceed at a rate determined by the slower
of two processes: the reaction time for pair bonding and the diffusion time
of the appropriate molecular or atomic building blocks. In real DNA replication, the base-pairing is incomparably more efficient and faster because
it is managed by a large and complex polymerase with complicated internal
states. Very little is known about the specifics of the replicase’s internal
activity, but it seems reasonable to conjecture in relation to its function
14
Quantum Aspects of Life
that in addition to the normal lowering of potential barriers to facilitate
quantum tunnelling (and thus accelerate the process), the replicase also
engages in a certain amount of choreography, making sure the right pieces
are in the right places at the right times. The concomitant speed-up over
the random access process would have a distinct evolutionary advantage.
Although the complexity of the replicase renders its internal workings obscure at this time, one may deploy general arguments to determine
whether quantum mechanics might be playing a non-trivial role in the hypothesized choreography, by appealing to the general analysis of quantum
time-keeping given by Wigner. As he pointed out, the energy-time uncertainty relation sets a fundamental limit to the operation of all quantum
clocks [Wigner (1957); Pesic (1993); Barrow (1996)]. For a clock of mass
m and size l, he found
T < ml2 / .
(1.1)
It is noteworthy that, for values of m and l of interest in molecular biology,
T also takes values of biological interest, suggesting that some biological
systems utilize quantum choreography. Let me give as an example the wellknown problem of protein folding, which is a major outstanding problem of
theoretical biology [Creighton (1993)]. Consider a peptide chain of N amino
acids, which folds into a specific three-dimensional structure. The number
of possible final configurations is astronomical, and it is something of a
mystery how the chaotically-moving chain “finds” the right configuration
in such a short time (typically microseconds). Quantum mechanics could
offer an explanation. If the average mass and length of an amino acid are
mo , and a respectively, then Eq. (1.1) yields
T < mo a2 N 3 / ,
(1.2)
suggesting a quantum scaling law for the maximum folding time of
T ∝ N3 .
(1.3)
T ∝ N 7/3
(1.4)
It is not clear that the linear dimension is the relevant size parameter
when it comes to large proteins. The assumption l ≡ N a in Eq. (1.2) may
be justified for small proteins (N = 80 to 100) that fold in one step, but
larger proteins do not remain “strung out” for a large fraction of the folding
process. Instead, they first fold into sub-domains. The opposite limit would
be to replace l by the diameter of the folded protein. Assuming it is roughly
spherical, this would imply T ∝ N 5/3 . The intermediate process of subdomain folding suggests a more realistic intermediate scaling law of, say,
A Quantum Origin of Life?
15
for large proteins. In fact, a power law of this form has been proposed on
empirical grounds [Gutlin et al. (1996); Cieplak and Hoang (2003)], with
the exponent in the range 2.5 to 3. Inserting typical numerical values from
Eq. (1.2), the limiting values of T for a 100 and 1000 amino acid protein
are 10−3 s and 0.3 s respectively. This is comfortably within the maximum
time for many protein folds (typically 10−6 s to 10−3 s for small proteins
in vitro), but near the limit for some, hinting that quantum choreography
may indeed be taking place in some cases.
Turning now to the polymerase enzyme, this is a molecular motor, or
ratchet, powered by ATP and using nucleotides as the raw material for the
base pairing. The physics of this system has been studied in some detail
for lambda-phage DNA [Goel et al. (2003)]. The Wigner inequality (1.1)
may be converted to a velocity bound
ν > /mL .
(1.5)
Using the parameters for the experimentally studied case, taking L to be
the length of the DNA (16 µm), and a polymerase mass of about 10−19 g,
Eq. (1.5) yields a minimum velocity of about 10−5 cm s−1 . The experimental results show the motor operates at about 100 base pairs per second,
which is indeed about 10−5 cm s−1 , suggesting that in normal operation the
motor could be limited by quantum synchronization uncertainty. Experiments demonstrate that applying tension to DNA using optical tweezers
decelerates the motor at a rate of about 3 bases per second per pN of applied tension [Goel et al. (2003)]. At a tension of about 40 pN the motor
stops altogether. (With further stretching of the DNA the motor runs backward). This suggests that the speed of the motor is not determined by the
availability of nucleotides or kT (which does not change as a function of
tension).
If quantum choreography underlies the efficiency of the polymerase motor, it seems reasonable to suppose that quantum choreography would be
even more important in the operation of Q-life. In the absence of a detailed
idea of the nature of the hypothetical Q-replicase, it is hard to know what
to choose for m and l, but by way of illustration if we take m to be 1000
proton masses and l to be 100 nm then the maximum running time of a
quantum clock is a few hundred femtoseconds. Quantum transitions that
take longer than about this limit could not be assisted in efficiency by such
a Q-replicase. For femtosecond transition rates, however, quantum choreography would seem, at least based on this crude estimate, to offer a good
mechanism for instantiating quantum replication.
16
Quantum Aspects of Life
Acknowledgements
I should like to thank Anita Goel, Gerard Milburn, Sandu Popescu, and
Jeff Tollaksen for helpful discussions.
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About the author
Paul C. W. Davies is a theoretical physicist, cosmologist, and astrobiologist. He received his PhD in 1970 from University College London, under
Michael Seaton and Sigurd Zienau. At Cambridge, he was a postdoc under Sir Fred Hoyle. He held academic appointments at the Universities of
Cambridge, London and Newcastle-upon-Tyne before moving to Australia
in 1990, first as Professor of Mathematical Physics at The University of
Adelaide, and later as Professor of Natural Philosophy at Macquarie University in Sydney, where he helped establish the NASA-affiliated Australian
18
Quantum Aspects of Life
Centre for Astrobiology. In September 2006, he joined Arizona State University as College Professor and Director of a new interdisciplinary research
institute called Beyond: Center for Fundamental Concepts in Science, devoted to exploring the “big questions” of science and philosophy. Davies’
research has been mainly in the theory of quantum fields in curved spacetime, with applications to the very early universe and the properties of
black holes, although he is also an expert on the nature of time. His astrobiology research has focused on the origin of life; he was a forerunner of
the theory that life on Earth may have originated on Mars.
Davies is the author of several hundred research papers and articles, as
well as 27 books, including The Physics of Time Asymmetry and Quantum
Fields in Curved Space, co-authored with his former PhD student Nicholas
Birrell. Among his recent popular books are How to Build a Time Machine
and The Goldilocks Enigma: Why is the Universe Just Right for Life? (US
edition entitled Cosmic Jackpot ). He writes frequently for newspapers,
journals and magazines in several countries. His television series “The Big
Questions”, filmed in the Australian outback, won national acclaim, while
his theories on astrobiology formed the subject of a specially commissioned
one-hour BBC 4 television production screened in 2003 entitled The Cradle
of Life. In addition, he has also devised and presented many BBC and
ABC radio documentaries on topics ranging from chaos theory to superstrings. Davies was awarded the 2001 Kelvin Medal and Prize by the UK
Institute of Physics and the 2002 Faraday Award by The Royal Society. In
Australia, he was the recipient of two Eureka Prizes and an Advance Australia award. Davies also won the 1995 Templeton Prize for his work on
the deeper meaning of science. The asteroid 1992 OG was renamed (6870)
Pauldavies in his honour.
Chapter 2
Quantum Mechanics and Emergence
Seth Lloyd
Quantum mechanics has two features that are important in guaranteeing
the emergence of complex systems such as life. The first is discreteness:
any finite quantum system with bounded energy has a finite number of
distinguishable states. Quantum mechanics is inherently digital. The second is chanciness: the outcomes of some quantum events are inherently
probabilistic [Peres (1995)].
This chapter shows that these two features combined, imply that quantum systems necessarily give rise to complex behaviour. Because it is digital, the universe can be thought of as a giant information processor: at
its most microscopic scales, it is flipping bits. The universe computes.
The computational nature of the universe is responsible for generating the
structures—galaxies, stars, planets, humans, bacteria—we see around us
[Lloyd (2006)]. Because it is probabilistic, the computing universe is effectively programmed by random quantum events called quantum fluctuations.
These events inject random bits into the operation of the universe: they
generate variation. It is this variation, processed by the universe’s ongoing
computation, which is responsible for the complexity of the structures we
see around us.
The chapter is organized as follows. A few paragraphs each will review
how quantum mechanics guarantees the digital and stochastic nature of the
universe. Then we shall delve more deeply into how those features combine
via the process of computation to produce complexity.
Received February 12, 2007
19
20
2.1.
Quantum Aspects of Life
Bits
First, let us examine the consequences of discreteness. The universe is
digital: the unit of digital information is the bit. A bit represents the
distinction between two possibilities—Yes or No, True or False, 0 or 1. The
digital nature of quantum mechanics is ubiquitous at the microscopic level.
Bits are everywhere: the spin of an electron or the polarization of a photon
registers a bit—strictly speaking, such bits are quantum bits, or “qubits”
[Nielsen and Chuang (2000)]. The interactions between those bits give rise
to a microscopic dynamics that is, in essence, a computation. As will be
shown in detail below, the computational capability of the universe at its
most microscopic level is a key ingredient in the spontaneous emergence
of complexity. Anything that computes can be programmed to produce
structure of any desired degree of complexity.
The digital nature of quantum mechanics implies that the universe is
effectively computing not only at its most microscopic level, but at larger
scales as well. The quantum nature of the universe implies that, at any given
energy scale, the spectrum of elementary particles is finite. Of these elementary particles, only a few—protons, neutrons, electrons, photons—are
sufficiently long-lived and interact sufficiently strongly to play a significant
role in the energy scales involved in everyday life. This small number of
elementary particles can combine to make a larger, but still finite set of
simple chemicals. Just as the nonlinear interactions between elementary
particles allow them to process information in a nontrivial way, the interactions between chemicals allow chemical reactions to perform the same type
of logical operations that are performed by digital computers.
As we shall see below, because they are computing, chemical reactions
spontaneously give rise to complex behaviour such as life. Exactly how and
where life arose, we do not know. Once something is computing, however,
we should not be surprised if it gives rise to the self-reproducing chemical
structures that presumably formed the basis for proto-life. Because chemistry is digital, this proto-life—whatever it was—inherited its digital nature
from chemistry. Later on, its descendants would inherit that beautiful digital structure, the genetic code: one base pair, two bits.
2.2.
Coin Flips
Now let us look at the probabilistic nature of the universe. Although many
processes in quantum mechanics are deterministic, like the great part of
Quantum Mechanics and Emergence
21
the computational processes described above, some quantum events are
intrinsically chancy. The best known of those events are the outcomes of
measurements. Let us look at some examples of probabilistic quantum
events.
As noted above, quantum systems are digital. The polarization of a
photon, for example, has two distinguishable states: it represents a bit. One
of those states could correspond to the electromagnetic fields of the photon
wiggling back and forth: such a photon is said to be horizontally polarized.
Regarding the photon as a bit, we can call this state, “0.” The second
state corresponds to the electromagnetic fields of the photon wiggling up
and down: such a photon is said to be vertically polarized. Regarding the
photon as a bit, we call this state, “1.”
Such polarized photons are easy to produce: just pass ordinary light,
which contains both types of photons, through the polarizing lenses of your
sunglasses. The vertical ones pass through, while the horizontal ones are
reflected. Since light reflected off water or the road consists primarily of
horizontally polarized photons, polarizing sunglasses filter out glare. (The
light of the sun consists of equal quantities of horizontally and vertically
polarized photons. When it bounces at an angle off the surface of the
water, many more horizontally polarized photons are reflected than are
vertically polarized photons. So glare consists mostly of horizontally polarized photons. By filtering out the glare’s horizontally polarized photons,
your sunglasses restore the balance of the light.)
So far, so good. But now quantum mechanics kicks in. Photons can also
be circularly polarized: their electromagnetic fields spiral around as they
move along at the speed of light. What happens if one takes a circularly
polarized photon and passes it through one’s sunglasses? The sunglasses
reflect horizontally polarized photons and transmit vertically polarized photons. But what about circularly polarized photons? It turns out that half
the time the circularly polarized photon is transmitted and half the time it
is reflected. When it is transmitted, its new polarization is vertical; when
it is reflected, its new polarization is horizontal.
Whether or not the circularly polarized photon is transmitted is a purely
chance event, like the toss of a fair coin. When you pass a string of circularly polarized photons through the sunglasses you are generating random,
probabilistic outcomes. If we take reflection (photon ends up in a horizontal polarization) to correspond to 0, and transmission (photon ends up
in a vertical polarization) to correspond to 1, then each time a circularly
polarized photon passes through the sunglasses, a brand new, random bit
is born.
22
Quantum Aspects of Life
The process by which quantum mechanics generates new, random bits
of information is called decoherence [Griffiths (2003)]. Measurement generates decoherence, and so do a host of other quantum mechanical processes,
notably quantum chaos. In particular, both measurement and chaos have
the tendency to amplify small and weak signals, so that a tiny quantum
fluctuation can end up having a substantial effect on the macroscopic world.
Similarly, if one traces back thermal and chemical fluctuations in energy,
temperature, pressure, particle number, etc., to their microscopic origins,
one finds that such statistical mechanical fluctuations are in fact quantum mechanical in nature. That is, the probabilistic nature of statistical
mechanics can, in the end, be traced back to the probabilistic nature of
quantum mechanics.
Decoherence is a rich topic of research, into which we will delve no
further here. For our purposes, what is important about decoherence is
that it is ubiquitous. Quantum mechanics is constantly injecting brand
new, random bits into the universe.
2.3.
The Computational Universe
Quantum mechanics makes the world digital and provides variation. Now
let us look more closely at how the universe is computing. We are all familiar with conventional digital electronic computers of the sort on which
I am writing these words. But computation is a more general and ubiquitous process than what occurs within a Mac or a PC. For our purposes,
computation can be thought of as a process that combines digital information with variation to produce complexity. This is a less familiar aspect of
computation than, say spreadsheets or word processing. But the generation of complexity is no less part of computation than computer games and
viruses. Indeed, the capacity of computers to generate complexity is one of
their most basic intrinsic abilities.
What is a computer? A computer is a device that processes information
in a systematic way. A digital computer operates on digital information,
i.e. on bits. When I play a computer game, my computer breaks up the
inputs that I give it into bits, and then processes and flips those bits one
or two at a time. At bottom, all that a PC or Mac is doing is flipping bits.
Now we see why it is natural to regard the universe itself as a computer. Quantum mechanics guarantees the digital nature of the universe:
at bottom, everything consists of quantum bits such as photon polarization.
Quantum Mechanics and Emergence
23
(Other famous microscopic qubits are electron spin, the presence or absence
of an electric charge, whether an atom is its ground or excited state, etc.)
These bits are constantly flipping due to the natural microscopic dynamics
of the universe. How they flip is governed by the laws of quantum mechanics. Some of this quantum bit flipping is deterministic, like the bit flipping
in a classical digital computer. Some of these quantum bit flips are random,
as when your sunglasses reflects or transmits a circularly polarized photon.
At first, the computational nature of the universe might seem like a radical, twenty first century discovery. In fact, the discovery that the universe
is, at bottom, registering and processing information, was made in the nineteenth century. In the second half of the nineteenth century, the great statistical mechanicians James Clerk Maxwell, Ludwig Boltzmann, and Josiah
Willard Gibbs, discovered the mathematical formula for entropy [Ehrenfest
and Ehrenfest (1990)]. Up until that point entropy was a somewhat mysterious quantity that gummed up the works of heat engines, preventing them
from performing as much work as they might otherwise. But what was
entropy in terms of the motions of individual atoms and molecules? The
energy of the molecules was simple to describe: it consisted of the kinetic
energy inherent in the molecules’ motion, and the potential energy stored
in their chemical bonds. Entropy was more mysterious.
Finally, over decades of painstaking analysis, Maxwell, Boltzmann, and
Gibbs discovered the mathematical formulation for entropy in terms of the
motion of atoms and molecules. Phrased in contemporary terms, their
formulae stated that entropy was proportional to the number of bits of information registered by the microsocopic state of the atoms and molecules.
Each atom carries with it a small number of bits of information. Every time
two atoms collide, those bits are flipped and processed. (The way in which
bits flip when two atoms collide is governed by the Boltzmann equation.)
The discovery that, at bottom, the universe computes was made more than
one hundred years ago.
Maxwell, Boltzmann, and Gibbs, together with their successors, used
the digital nature of the universe to analyze the behaviour of agglomerations
of atoms and elementary particles, with spectacular success. Here, we will
use the digital nature of the universe to analyse the creation of structure
and complexity in the universe.
In order to perform this analysis, we must go deeper into the notion of
digital computation. First, consider the concept of a program. A program
is a sequence of instructions that tells the computer what to do. In other
words, a program is a sequence of bits to which the computer attaches a
24
Quantum Aspects of Life
meaning. The meaning of the program need not be deep: for example,
at a particular point in the operation of a given computer, the instruction
“0” might mean “add 2 + 2,” while the instruction “1” might mean “add
4+0.” In a digital computer, each instruction on its own typically possesses
a simple, prosaic meaning. By stringing many simple instructions together,
however, arbitrarily complicated operations can be performed.
Above, it was noted that quantum fluctuations program the universe.
All that this means is that the random results of quantum events, such as
the reflection or transmission of a photon, can set into motion a sequence
of further bit flips. Just how quantum fluctuations program the universe
depends on the laws of quantum mechanics. Suppose, for example, that as
a result of a quantum fluctuation a vertically polarized photon penetrates
your sunglasses and is absorbed by your eye, causing a signal to propagate
down your optic nerve to your brain. Your brain interprets this signal as
indicating the presence of a fish just below the surface of the water, and
you throw your spear in its direction. What does this photon mean? In
this case, it means dinner!
Computers such as Macs and PCs have a special feature: they are universal digital computers. A universal digital computer is one that can be
programmed to simulate the operation of any other digital computer. That
is, a universal digital computer can be given a sequence of instructions that
allows it to perform the same sequence of operations that any other digital
computer performs. A familiar example of computational universality is
the fact that a program that runs on one computer can be translated into
a version of the program that runs on another computer. Microsoft Word
can run both on a Mac and on a PC.
The universe is also a universal computer. Matter can be configured to
compute. How so? Very simple. We “program” matter to imitate a Mac or
PC simply by constructing a Mac or PC! Over the last decade or so, it has
become clear that the universe itself is a universal computer at the most
microscopic levels. In 1993, I showed how atoms and elementary particles
could be programmed to perform any desired digital computation simply
by shining light on them in the right way [Lloyd (1993)]. Since that time,
my colleagues and I have constructed and demonstrated a wide variety
of quantum computers that store bits on the spins of individual electrons
and the polarizations of individual photons. Not only is the universe a
universal computer, it can perform such universal computations at the most
microscopic level.
Quantum Mechanics and Emergence
25
A system need not be complex to be a universal digital computer. For
example, the laws of physics are simple, and they support universal digital
computation. Many simple extended systems, such as uniform arrays of
bits with simple rules for interaction between them, are computationally
universal.
2.4.
Generating Complexity
In a way, the fact that the universe has generated huge complexity is highly
puzzling. Our observations of the cosmos indicate that the universe began
around 13.8 billion years ago. When it began, it was in a very simple,
uniform state, the physical analog of a bit string that is nothing but zeros.
As soon as the universe began, it began to evolve dynamically in time, governed by the laws of physics. But the known laws of physics are themselves
simple: the equations of the standard model for elementary particles fit on
the back of a T-shirt. Simple initial conditions and simple laws. So what
happened?
When I look out my window I see trees, people, dogs, cars, and buildings, all immensely varied and complex. When I look through a microscope,
I see the structure of materials and the behaviours of microorganisms, also
tremendously varied and complex. When I look up into the sky I see planets,
stars, and galaxies containing unimaginably greater quantities of complexity. Where did all this complexity come from?
The answer lies in the computational nature of the universe. Although
a universal computer need not be complex in and of itself, it is nonetheless
capable of exhibiting complex behaviour. A universal computer can be
programmed to do anything a Mac or PC can do. Indeed, because the
consequences of the laws of physics can be computed on a digital computer,
it can in principle be programmed to do anything the universe can do!
(Note, however, that a computer that was simulating the universe precisely
would have to be at least as large as the universe itself.)
Above, we saw that the universe itself is a universal computer that is
effectively programmed by random quantum fluctuations. Now we can ask
the question, “How likely is the universe, starting from a simple state,
to generate complex behavior?” There is an elegant branch of mathematics that deals with this very question. This branch of mathematics
is called algorithmic information theory [Solmonoff (1964); Kolmogorov
(1965); Chaitin (1987)]: one of the primary questions it asks and answers
26
Quantum Aspects of Life
is the question, “How likely is a universal computer, programmed with a
random program, to generate any given structure?”
The central quantity in algorithmic information theory is algorithmic
information content. The algorithmic content of some structure is equal
to the length of the shortest computer program, written in some suitable computer language such as Java or C, that instructs the computer
to produce that structure. Algorithmic information content was discovered
independently by (in chronological order) Ray Solmonoff (1964), Andrey
Kolmogorov (1965), and Gregory Chaitin (1987). It is Solomonoff’s interpretation of algorithmic information that concerns us most here.
Solomonoff was interested in formalizing the notion of Ockham’s razor.
William of Ockham was a medieval English Philosopher and Franciscan
monk. Ockham’s razor is not a shaving implement, but rather a philosophical principle for “cutting away” needless complexity. Ockham phrased his
razor in various ways: Pluralitas non est ponenda sine necessitate, (“Plurality should not be posited unless necessity”) and Frustra fit per plura quod
potest fieri per pauciora (“It is a mistake to make with more what can be
made with less”). A later paraphrase of Ockham’s razor is, Entia non sunt
multiplicand praeter necessitatem (“Beings should not be multiplied beyond
necessity”). Solomonoff made a mathematical paraphrase of the notion that
brief explanations are preferable. He identified an “explanation” of a structure or string of bits as a computer program that instructs a computer to
produce that structure or string. The briefest explanation is identified with
the shortest program that produces the structure. The length of this program is the algorithmic information content of the structure and is denoted
K(s), where s is the structure in question. (More precisely, we should write
KU (s) where and U denotes either the universal computer that is to be programmed to produce s, or the programming language that is to be used.
Because any universal computer can be programmed to simulate any other
computer, however, we have KU (s) = KV (s) + O(1), where the O(1) term
is no longer than the length of the program that allows universal computer
V to simulate U . In other words, algorithmic information content is to
some degree independent of the computer or programming language used.)
To make the connection with probability, Solomonoff invoked the notion
of a computer that has been programmed at random. A program is itself
nothing but a string of 0s and 1s. If these 0s and 1s are generated completely
randomly, as by tosses of a fair coin, then the probability that the first ℓ
random bits reproduce a particular program of length ℓ is 2−ℓ . Referring to
the concept of algorithmic information, we see that the probability that the
Quantum Mechanics and Emergence
27
randomly programmed computer produces the structure s is no less than
2−K(s) .
Suppose we take a random string of 0s and 1s and feed it into a computer
as a program. The computer interprets those 0s and 1s as instructions and
begins to execute them in sequence, reading and executing one bit after
another. Structures that can be generated by short programs are more likely
to arise than structures that can be generated only by long programs. Just
what sort of structures can be generated by short programs? The answer
is, all structures.
In particular, there exists a brief program that instructs the computer
to start computing all computable structures. Ironically, the length of this
program is much shorter than the program that causes the computer to
generate some particular complex structure: it is easier to generate all
structures than it is to generate any given one! Other structures that have
short programs include (a) the known laws of physics (printable on a
T-shirt), (b) the digits of π, (c) the spiral structure of galaxies, (d) . . . .
As noted above, many universal computers have very simple structure.
That is, only a short program is required for our computer to simulate
these other computers. A randomly programmed universal computer has a
high probability of generating all sorts of other universal computers. Those
computers in turn have a high probability of generating further computers,
etc.
This nested generation of computing structures is a familiar feature of
our universe. As noted above, the laws of physics support universal computation at their most microscopic scale, at the level of the laws of elementary
particles. Quantum computers are the expression of this most microscopic
computational power. As a consequence of this microscopic computational
universality, it is no surprise that the universe exhibits computational universality at the next higher level, given by the laws of chemistry. This
chemical computational universality is expressed by the coupled, nonlinear
equations of chemical reactions that can encode all sorts of computational
processes, including logic circuits, memory, switching processes, etc.
Similarly, it should be no surprise that the laws of chemistry give rise to
computational universality at the next higher level, in the form of life. Life
in its current form on earth is highly complex. But to get simple protolife, only a few features are required. Apparently, all that is required for
proto-life is the existence of physical systems that reproduce themselves
with variation. As much as one might like to set other requirements for the
origins of life, reproduction and variation seem to suffice.
28
Quantum Aspects of Life
Lots of chemical species manage to reproduce themselves within the
context of autocatalytic sets—networks of interacting chemical reactions in
which chemicals can catalyze, or enhance, their own production as part of
their reactions with other chemicals [Kauffman and Farmer (1986); Jain
and Krishna (1998)]. As part of the natural processes of chemical reactions, quantum fluctuations produce variation in the outputs of chemical
reactions. As soon as one of these self-catalyzing chemicals constructs variants of itself that are also auto-catalytic, the origins of life are off and
running. Variants that are more successful at producing copies of themselves become more prevalent in the chemical population; variants that are
more successful in their interactions with other chemicals also survive to
reproduce another day. These proto-living chemicals within their autocatalytic sets were presumably much simpler than life today. But if they were
not very complex, they were nonetheless ambitious. We are their lineal
descendants.
2.5.
A Human Perspective
As life continued to evolve more and more variants and more and more
complex forms, we should not be surprised that, amongst those forms,
new types of computational universality arose. Human language, together
with the mental apparatus that supports it, represents a remarkable and
wonderful type of computational universality. Human language can express,
well, anything that can be expressed in words. It is this universal capacity
for language that sets us apart from other species; until the time, of course,
that we discover such a capacity in them and begin to communicate with
them as equals. For the moment, however, it is not evident that other
species on Earth possess the rich linguistic ability that humans do.
Human beings, in turn, as they became more and more technically
adept, spawned a new type of computational universality in the form of
digital computation. Not only did we evolve the technologies to construct
computers, but we bequeathed to them a stripped-down form of universal
human language in the form of universal computer languages. Computers
do not yet possess the emotional depth that would make us treat them as
conscious beings. As time goes on, however, I suspect that we will regard
some computers as more and more human (some of my students at MIT
have already taken this step with their computers). There will be no single
moment of creation of a computer we regard as our intellectual, emotional,
Quantum Mechanics and Emergence
29
or even our spiritual equal. Rather, as computers become more and more
capable and sympathetic beings, we will gradually assign to them the rights
and capacities of more “conventional” humans, in just the same way that
as human society progressed over history, it became clear to most men that
women and slaves were, in fact, man’s equal.
2.6.
A Quantum Perspective
Quantum mechanics makes the world locally finite and discrete, allowing
it to compute. It is this computational ability that allows the universe
to generate complexity. But complexity does not arise from computation
alone: it requires variation.
There is no one quantum-mechanical perspective: it is in the nature
of quantum mechanics to reflect all possibilities at once. The chancy and
probabilistic nature of quantum mechanics comes about exactly because
quantum systems explore Yes and No, True and False, Horizontal and Vertical, or 0 and 1, simultaneously. Quantum computation, chemical reactions,
and life itself, are constantly exploring and searching out the consequences
of the possibilities allowed by the laws of physics.
At bottom, all the details that we see around us—the motions of air
and water, the microscopic pattern of cells in a leaf—arise from quantum
mechanical accidents that have been processed by the computing power of
the universe. Quantum mechanics makes nature digital: it supplies the bits
and the bit flips that are the substance of the the universe. Having supplied
the world with its substance, quantum mechanics then adds the variety that
is the proverbial spice of life. The result is what we see around us.
References
Chaitin, G. J. (1987). Algorithmic Information Theory (Cambridge University
Press).
Ehrenfest, P., and Ehrenfest, T. (1990, reprint of 1912 edition). The Conceptual
Foundations of the Statistical Approach in Mechanics (Dover).
Griffiths, R. (2003). Consistent Quantum Theory (Cambridge).
Jain, S. and Krishna, S. (1998). Autocatalytic Sets and the Growth of Complexity
in an Evolutionary Model, Phys. Rev. Lett. 81, pp. 5684–5687.
Kauffman, S. A., and Farmer, J. D. (1986). Autocatalytic sets of proteins, Origins
of Life and Evolution of Biospheres, 16, pp. 446–447.
30
Quantum Aspects of Life
Kolmogorov, A. N. (1965). Three approaches to the quantitative definition of information, Problems of Information Transmission, 1, pp. 1–11.
Lloyd, S. (1993). A potentially realizable quantum computer, Science 261,
pp. 1569–1571.
Lloyd, S. (2006). Programming the Universe (Knopf).
Nielsen, M. A., and Chuang, I. L. (2000). Quantum Computation and Quantum
Information (Cambridge University Press).
Peres, A. (1995). Quantum Theory: Concepts and Methods (Springer).
Solomonoff, R. J. (1964). A formal theory of inductive inference, Information and
Control 7, pp. 1–22.
About the author
Seth Lloyd was born on August 2, 1960. He received his AB from Harvard College in 1982, his MathCert and MPhil from Cambridge University
in 1983 and 1984, and his PhD from Rockefeller University in 1988, under Heinz Pagels, for a thesis entitled Black Holes, Demons, and the Loss
of Coherence: How Complex Systems Get Information, and What They
Do With It. After postdoctoral fellowships at the California Institute of
Technology and Los Alamos National Laboratory, he joined MIT in 1994
where he is currently a Professor of mechanical engineering, preferring to
call himself a “quantum mechanic.” His research area is the interplay of
information with complex systems, especially quantum systems. He has
made contributions to the field of quantum computation and proposed a
design for a quantum computer. In his book, Programming the Universe
(Knopf, 2006), Lloyd argues that the universe itself is one big quantum
computer producing what we see around us. Lloyd is principal investigator
at the MIT Research Laboratory of Electronics, and directs the Center for
Extreme Quantum Information Theory (xQIT) at MIT.
PART 2
Quantum Mechanisms in Biology
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Chapter 3
Quantum Coherence and the Search
for the First Replicator
Jim Al-Khalili and Johnjoe McFadden
3.1.
When did Life Start?
The Earth formed about 4 billion years ago out of the coalescing debris
of the nascent solar system. But the newly formed planet remained uninhabitable for several hundred million years as it continued to be showered
with massive rock fragments that would have vaporized the ocean. It was
only when this battering slowed down, about 3.8 billion years ago, that
liquid water became sufficiently stable to allow the formation of oceans. As
liquid water is an essential ingredient of life on this planet (and probably
on any other) it is generally agreed that life was not possible on the Earth
until about 3.8 billion years ago. The earliest undisputed evidence for life
arrives much later—about 3 billion years ago. There is however evidence—
much disputed—that life started much earlier than this. Microbe-like fossils
in Early Archaean rocks have been dated to 3.5 billion years ago [Schopf
(2006); Rasmussen (2000)] but other researchers have claimed that these
are of inorganic origin [Brasier et al. (2006, 2002)]. Chemical signatures
of life (carbon isotope enrichment) have also been found at this age, or
even a little earlier (some claim as early as 3.85 billion years ago), but once
again the evidence is disputed. Until the disputed evidence is resolved the
only thing we can say for sure is that life first emerged on Earth sometime
between 3.0 and 3.8 billion years ago.
Received March 29, 2007
33
34
3.2.
Quantum Aspects of Life
Where did Life Start?
Darwin famously mused on the possible origin of life in a warm little
pond. Alexander Oparin and J. B. S. Haldane independently put scientific flesh on Darwin’s speculation in the 1920s with what we now call the
“primordial soup theory”. The basic bones of the theory are that abiotic
processes—heat, lightning, wind, rain, impacts etc. churned up inorganic
material (ammonia, methane, water, hydrogen, etc.) on the early Earth
to form the basic chemical ingredients of life: amino acids, sugars, etc..
These ingredients were still a long way from life itself but, dissolved in the
early seas, oceans and ponds (the primordial soup), they could recombine,
through random thermodynamic processes, to form more and more complex
biomolecules. Eventually, and again through random thermodynamic processes, these primitive biomolecules would have associated with lipid membranes to enclose the biomolecules within some kind of “proto-cell”. These
proto-cells were not living in the conventional sense but they may have
possessed the ability to divide—perhaps by simple binary fission—to form
daughter proto-cells. Some of the daughter cells may have contained the
same mix of biomolecules as the original proto-cell, whereas others would
have inherited (again, through random processes) a slightly different mix
of ingredients generating a population of variant proto-cells with slightly
different properties. Some of the daughter proto-cells may have been incapable of self-replication, but others might have been more efficient than
the parent proto-cells or their siblings. Over many generations of replication the population would have tended to become dominated by the more
efficient self-replicating proto-cells and, as resources for replication became
limited, a kind of chemical selection process (analogous to natural selection) would have kicked in to drive the population towards faster and more
efficient replication. The more successful proto-cells would have gradually
captured the characteristics and capabilities of living systems cell membranes, DNA, RNA, enzymes etc. until eventually the first living cell was
born. There was obviously much that was vague and hand-waving about
the Oparin-Haldane primordial soup theory, particularly the precise nature
of the early self-replicators and how they formed, but the theory at least
provide a credible alternative to divine creation as the origin of life.
Quantum Coherence and the Search for the First Replicator
3.3.
35
Where did the Precursors Come From?
When Haldane and Oparin were speculating about primordial soups, the
mechanisms responsible for replication of actual living cells were still largely
unknown—the double helix was still nearly half a century away. But there
was a general appreciation that various biochemicals—proteins, nucleic
acids, sugars etc.—were probably necessary for the proto-cell to work. But
where did these simple biochemicals come from? Organic molecules found
on Earth today are nearly all the products of life; they are not formed in
significant quantities abiotically. They are also rather unstable and break
down pretty quickly once formed. For life to be initiated on the planet, it
needed some way of making biomolecules abiotically. To solve this problem Haldane proposed that the early Earth possessed an atmosphere that
was very different from today’s: He proposed that the primordial Earth
possessed a reducing atmosphere rich in compounds like methane, hydrogen, water and ammonia. In these conditions it is much easier to make
biomolecules such as protein or nucleic acids (as these are reduced forms of
carbon) and the biomolecules thus formed are more stable than they would
be today. It was thirty years before Oparin and Haldan’s speculations
were put to the test in a simple set of experiments performed by Stanley
Miller and Harold Urey in the 1950s. Miller and Urey constructed a kind
of laboratory primordial soup: the chemical constituents of the supposed
primordial atmosphere—ammonia, methane, hydrogen, water with electrical discharges to simulate lightning strike on the early Earth. Remarkably,
when Miller and Urey analyzed the chemical composition of their soup after it had been cooking for several days they found that 10-15% of the
carbon was now in the form of simple organic molecules such as amino
acids. At least some of the ingredients of life could indeed have been made
precisely as Haldane had predicted [Miller et al. (1976); Urey (1952)]. The
Miller-Urey experiments were so remarkable that many thought that the
problem of the origin of life had been more-or-less solved. When the work
was published there was widespread anticipation that it would not be long
before simple life forms were crawling out of origin of life experiments. But
it did not happen. Why not? The answer is that the situation has become more complicated. For a start, the atmosphere of the early Earth
is no longer thought to have been reducing. The present guess is that it
was probably at best redox neutral dominated by compounds like carbon
dioxide and nitrogen. Under these conditions it is much harder to form
biomolecules like amino acids. Another problem was that although amino
36
Quantum Aspects of Life
acids were made, no proteins (polymers of amino acids) were synthesized.
This is because the reactions were taking place in water, and since polymerization of biomolecules involves removal of water, life does not occur
spontaneously in aqueous solutions. With water present in such excess the
balance of the reaction is overwhelmingly towards hydrolysis rather than
polymerization. A third problem was the chirality of biomolecules—such as
(most) amino acids. All biomolecules come in either left-handed or righthanded forms, not both. But the Miller-Urey experiments (and all similar
primordial soup experiments) synthesized equal amounts of left and righthanded forms. These racemic mixtures simply do not polymerize to form
proteins. A fourth problem was that many essential biomolecules such as
nucleic acids—were not formed in the Miller-Urey experiments and have
since proven to be exceedingly difficult to form in any laboratory based
primordial soup experiment.
3.4.
What was the Nature of the First Self-replicator?
Even if all the biomolecules were present there was still a long way to go
before a self-replicating system was made. Somehow, the cooked-up ingredients of the soup would have to have formed a self-replicating structure,
such as the putative proto-cell. But how feasible is the abiotic synthesis of
the self-replicating proto-cell? To answer this question we must have some
idea of what the proto-cell was made of. Unfortunately none have survived
today and they have left no fossil records. The first undisputed evidence for
living cells are bacterial-like fossils within layered structures called stromatolites that existed about 3 billion years ago. These are similar to structures
that are still formed today in shallow seas that are generated by a photosynthetic group of bacteria called cyanobacteria (blue-green algae). Capturing
light from the sun and both carbon and nitrogen from the air, cyanobacteria form microbial mats in annual cycles that eventually build up into
stromatolites. But cyanobacteria are nothing like a proto-cell. Their cells
are rather complex for bacteria with outer and inner membranes, and include a set of internal organelles that perform the photosynthesis reaction
and (in many cyanobacteria) another organelle that captures and fixes nitrogen. Their cells contain DNA, RNA, proteins, sugars, fats and hundreds
of other biomolecules. The complete genome of several members of the
group have recently been sequenced and the size of their genome ranges
from the relatively small 1.6 megabases (1.6 million base pairs) genome of
Quantum Coherence and the Search for the First Replicator
37
Prochlorococcus marinus through to the hefty 9 megabase genome of Nostoc punctiforme. The genome of even the simplest of these organisms, the
diminutive P. marinus (with a cell size between 0.5-0.7 µm, these are the
smallest photosynthetic organisms known to date) encodes more than 1,800
genes. Each gene is composed of several hundred to several thousand bases
encoding a protein with many hundreds of amino acids folded into high specific and information rich structures—enzymes—that perform the complex
biochemical reactions of DNA replication, protein synthesis, photosynthesis
and energy generation. Prochlorococcus is clearly the product of a long evolutionary process and the same is presumably true for the organisms that
built the fossil stromatolites. They were not proto-cells but must have been
the descendants of simpler cellular life. But how simple is simple? What is
the minimal cell necessary for life? This question can at least be addressed
today and the answer is: not very simple at all. The simplest self-replicating
organisms alive today are, as far as we know, the mycoplasmas that have a
genome size of 580,000 base pairs, sufficient to code for nearly five hundred
proteins! But even mycoplasmas are not feasible primordial organisms because they are in fact descended from more complex bacteria by gene loss
and this reductive evolution has left them very enfeebled. They are parasites that grow best inside living cells requiring their host cell to make
many of their biomolecules. They are unlikely inhabitants of any primordial soup. Bacteria even simpler than mycoplasmas have been discovered,
such as the recently sequenced parasitic organism Nanoarchaeum equitans,
which, with a genome size of only 490,000 base pairs, is the smallest bacterial genome known. Viruses do of course have much simpler genomes,
but these (and N. equitans) are parasites of living cells. They rely wholly
(viruses) or partly (N. equitans) on their host cells to provide the essential
functions of life. They are not viable primordial soup organisms. So we are
left with a dilemma. The simplest self-replicating organisms alive today
are far from simple and unlikely to have formed spontaneously in the primordial oceans. The astronomer Fred Hoyle considered the probability of
assembling a structure like a bacterium from the random thermodynamic
processes available on the early Earth and likened its chances to that of a
tornado in a junkyard spontaneously assembling a Boeing 747.
3.5.
The RNA World Hypothesis
If cells are too complex to form spontaneously then perhaps some precellular self-replicator was not a cell but some kind of naked self-replicating
38
Quantum Aspects of Life
DNA or protein. But a problem with this scenario is that DNA or proteins
do not self-replicate today. DNA makes RNA makes proteins and proteins
make enzymes that replicate DNA. The genetic information is stored in
DNA, transcribed into the mobile messenger nucleic acid, RNA, and then
translated into strings of amino acids that fold into proteins. These folded
proteins then make enzymes capable of chemical catalysis that are responsible for all the dynamics of living cells: energy capture, mobility, DNA
replication etc. You need DNA and RNA to make enzymes. But you need
enzymes to make DNA or RNA! A possible way out of this chicken and egg
scenario was provided in the 1970s with the discovery that some forms of
RNA could also act as catalysts. Ribozymes, as they came to be known,
could close the loop between heredity (it can store genetic information)
and metabolism (it can catalyse biochemical reactions). Perhaps the first
self-replicator was a ribozyme? The idea of an early precellular phase of
life dominated by ribozymes has become probably the most popular origin
of life scenario [Orgel (2004)]. Ribozymes have been shown to be capable
of catalysing quite a wide range of biochemical reactions and can even act
as simple “replicases” capable of stitching together two fragments of themselves to generate a copy. There are however a number of problems with
the RNA world hypothesis. The first is that no one has yet found a feasible
way to generate RNA in any plausible primordial soup experiment. RNA is
a more complex biomolecule than an amino acid and is far harder to make!
The second problem is the fact that ribozymes are already rather complex
structures. A recent study by David Bartel’s group at MIT used an artificial evolutionary process to select for ribozymes capable of performing RNA
polymerization reactions (essential component of an RNA self-replicator).
The strategy did identify some novel ribozyme structures [Johnston et al.
(2001); Lawrence and Bartel (2005)] but the minimum size for any kind of
replicase activity was 165 bases, at least ten-fold bigger than anything that
might be synthesized in even the most optimistic primordial soup RNA
synthesis scenarios. Even if it were possible to generate structures as large
as 165 bases, the chances of generating Bartel’s ribozyme by random processes (in their experiments their starting point was a known RNA ligase)
are exceedingly small. There are 4165 (or 2 × 1099 ) possible 165 base long
RNA structures. If there was just one molecule of each of the possible 165
base long RNA molecules in the primordial soup then the combined mass
of all those RNA molecules would be 1.9 × 1077 kilograms. To put this
number in perspective, the entire mass of the observable universe is estimated as approximately 3 × 1052 kilograms. It clearly would have to have
Quantum Coherence and the Search for the First Replicator
39
been an astronomically big pond to have had any chance of generating a
ribozyme self-replicator by random processes alone. We thereby come to
the crux of the origin of life problem. It is not that it is difficult to form the
chemical precursors (although it is) or identify proteins or RNA molecules
capable of performing some of the necessary steps of self-replication. The
problem is a search problem. The self-replicator is likely to be only one
or a few structures in a vast space of possible structures. The problem
is that random searches (essentially thermodynamic processes) are far too
inefficient to find a self-replicator in any feasible period of time. An examination of the origin of life problem from an information science perspective
has recently reached the same conclusion [Trevors and Abel (2004)]. This
is probably true not only of carbon-based self-replication systems but also
of digital life. Although digital self-replicating programs, such as Tierra,
have been created in computers [Bedau et al. (2000)] digital life has not, so
far, emerged spontaneously out of the huge volume of digital traffic of the
internet. This is all the more surprising since we know that self-replicating
programs—able to infect and replicate in the primordial soup of the internet are surprisingly easy to make: we call them computer viruses. But,
as far as we are aware, all known computer viruses have been synthesized
by hackers; none has emerged spontaneously out of the digital primordial
soup of the internet. Why has not the internet generated its own digital
life? The answer is probably that self-replication, even in the protected
environment of digital computers, is too complex to emerge by chance.
3.6.
A Quantum Mechanical Origin of Life
It hardly needs stating that quantum mechanics plays an important role
in biology since it underlies the nature of atomic and molecular structure,
and therefore the nature of molecular shapes and bonding, and hence the
templating functions of nucleic acids and the specificity of proteins. It is
also crucial in explaining differential diffusion rates, membrane specificity,
and many other important biological functions. Thus quantum mechanics
underpins at the most fundamental level the machinery of life itself. However, what is emerging today is the notion that quantum mechanics may
play more than this basic role of determining molecular structure, bonding
and chemical affinity. After all, biology is based on chemistry, which in
turn is subject to quantum principles such as Pauli’s Exclusion Principle.
Thus, a number of the more counterintuitive features of the theory, such
40
Quantum Aspects of Life
as quantum superposition, entanglement, tunnelling and decoherence, may
also turn out to play a vital role in describing life itself. This is not so
speculative as it sounds; it is already well established that quantum tunnelling of protons may alter the structure of nucleotide bases and can be
responsible for certain types of mutations. Likewise, it plays a vital role
in many enzyme-driven reactions and enzyme catalysis. We propose here
that some of these more profound aspects of quantum mechanics may have
provided a helping hand in kick starting life within the primordial soup.
3.6.1.
The dynamic combinatorial library
Consider some small corner of the primordial soup—perhaps a drop of liquid trapped within a rock cavity. We propose that the drop contains some
proto-self replicator structures—perhaps RNA or amino acid polymers that
are large enough to form a self-replicator, but are in the wrong region of
sequence space so they are unable to self-replicate. We also imagine that
the proto self-replicators are subject to some kind of mutational process
that changes their structure by standard chemical reactions—breaking and
forming chemical bonds. Somewhere out there in the vast chemical structure space available to the proto self-replicators is the correct structure for
an actual self-replicating RNA or peptide but, as discussed above, for a
limited number of proto self-replicating molecules this become a massive
search problem that cannot be solved in any reasonable time period, at
least in a classical system. The problem for the classical system is that
the chemical mutational process of forming new configurations/structures
is very slow and limited by the number of molecules available to do the
searching. A proto-replicator may suffer a chemical change (a chemical
“mutation”) to form any new structure that may or may not be a selfreplicator. Usually of course the new structure will not be a self-replicator
so to try again the system must dismantle the newly-formed structure by
the same chemical processes—breaking and forming covalent bonds—before
another novel structure can be formed by another chemical mutation. With
a limited pool of proto self-replicators available in our primordial soup the
library of possible structures made by the system will be very tiny in comparison to the total space of structures that needs to be searched. The
time available for this search is of course tens, or hundreds, of millions of
years from the time that conditions on Earth first reached their “Goldilocks
values” to the time that the first simple self-replicators emerged. However,
given the shear improbability that the correct configuration is hit upon by
Quantum Coherence and the Search for the First Replicator
41
chance and the time taken for classical chemical mutation of breaking and
reforming covalent bonds, speeding up of the search mechanism would be
greatly desired. To make the search more tractable we propose that we
consider the library of structures in our soup as a dynamic combinatorial
library. Combinatorial libraries have been widely used in the pharmaceutical industry to identify novel drug compounds. Essentially a library of
related chemical compounds is synthesized by standard chemical synthesis
methods: the combinatorial library. The library can then be screened for
binding to a particular ligand say a viral protein—to identify a compound
that will bind to the virus and perhaps prevent it binding to host cells.
Dynamic combinatorial libraries are a recently introduced variation on the
combinatorial library approach [Ramstrom and Lehn (2002)] whereby reversible chemical self-assembly processes are used to generate the libraries
of chemical compounds. Dynamic combinatorial chemistry allows for the
generation of libraries based on the continuous interconversion between the
library constituents. Reversible chemical reactions are used for spontaneous
assembly and interconversion of the building blocks to continually generate novel structures. The dynamic combinatorial library is thereby able to
form all possible combinations of the building blocks, within the time and
chemical resources at its disposal. Addition of a target ligand or receptor can even be used to capture binding compounds and thereby drive the
synthetic reactions towards synthesis of chemical binders. It is easy to see
that the primordial soup may be considered to be a dynamic combinatorial
library capable of forming novel structures by reversible processes and some
of those novel structures could eventually become self-replicators. However,
the system still suffers from the search space problem: there are not enough
molecules within the dynamic combinatorial soup to find the self-replicator
within a feasible timescale. We thus add a further mechanism to the dynamic combinatorial library soup proposal: that the compounds within
the library are linked, not only via reversible reactions but via quantum
mechanics. Quantum tunnelling is a familiar process in chemistry where
it is often known by another name: chemical tautomerization. Many compounds, for instance nucleotide bases, are found in mixtures of related chemical structures, known as tautomers. For nucleotide bases the tautomeric
structures differ in the position of protons within the nucleotide bases to
form enol of keto forms of the bases with different base-pairing properties
(since these protons are involved in Watson and Crick base-pairing). The
alternative positions of the protons in the enol and keto forms are linked,
not by conventional chemical reactions, but by proton tunnelling [Douhal
42
Quantum Aspects of Life
et al. (1995)]. Each preparation of tautomeric compounds such as adenine
(a nucleotide base) contains a mixture of its tautomeric forms. The balance
between the alternative tautomeric forms depends on the relative stability
of each tautomer. Nucleotide bases usually exist predominantly in one of
the other keto or enol forms. But we must remember that the conversion
between the forms is quantum mechanical. Each molecule of a DNA base
is not in either the enol or keto form but must exist as a superposition of
both forms linked by proton tunnelling. We now return to our primordial
pool and imagine it to be a quantum dynamic combinatorial library with
many molecules of a single compound that can each exist in a quantum
superposition of many tautomeric states simultaneously. The issue here is
one of time scales; if such a quantum superposition of all possible states in
the combinatorial library can be built up before the onset of decoherence
then this becomes an extremely efficient way of searching for the correct
state: that of a simple replicator. Searches of sequence space or configuration space may proceed much faster quantum mechanically. In effect, a
quantum system can “feel out” a vast array of alternatives simultaneously.
So the question is: can quantum mechanics fast-track matter to life by “discovering” biologically potent molecular configurations much faster than one
might expect classically? This is, after all, just the principle that underlies the concept of a quantum computer. In effect, quantum computation
enables information processing to take place in a large number of states in
parallel, thus shortcutting the computation resources necessary to process
a given amount of information. There are two perceived problems with
this idea. Firstly, it has been argued that while it is easy to believe that
quantum superpositions might accelerate the “discovery” of the correct and
unique replicator state from amongst the myriad of other equally complex
but wrong structures, an element of teleology is required; namely that the
molecule must somehow know before hand what it is aiming for. We do
not believe this is necessary (as we argue below). The second problem is
more serious and less understood. It involves the estimate of how long
the delicate quantum superposition can be maintained before the onset of
decoherence, the process by which the delicate superposition is destroyed
through interactions with the surrounding environment. In order to quantify very roughly the timescales involved we employ a simple example.
3.6.2.
The two-potential model
We propose a 1-dimensional quantum mechanical model of a double potential well containing a single particle that must exist in a superposition of
Quantum Coherence and the Search for the First Replicator
43
being in both sides of the well until measured. There are plenty of examples
of such a situation, such as the inversion resonance in NH3 in chemistry or
strangeness oscillations in the neutral K-meson in physics. Consider a 1-D
double oscillator potential, V (x), symmetric about x = 0 and defined as
mω 2
(|x| − a)2 ,
(3.1)
2
where m is the mass of the particle, ω is the oscillator parameter and 2a
is the distance between the two sites. If the particle starts off on the right
hand side of the double well then it can be described by a superposition of
the lowest two eigenstates of the full Hamiltonian of the system
V (x) =
1
ψ(x, t = 0) = √ (ψ0 + ψ1 ) .
2
(3.2)
Standard textbook quantum mechanics shows the time evolution of this
state, which shuttles back and forth between the two wells. This “shuttling”
time that describes how long it takes for the particle to tunnel across from
one side of the double well to the other is given by
,
∆E = E1 − E0 ,
(3.3)
2∆E
where E0 and E1 are lowest two energy eigenvalues. However, as is wellknown with all quantum tunnelling, ts is extremely sensitive to the potential
parameters since
2V0
2V0
exp −
,
(3.4)
∆E = 2ω
πω
ω
ts =
where the height of the potential barrier between the two wells is
V0 =
1
mω 2 a2 .
2
(3.5)
Typically, a = 1 Å, m = 10−27 kg (proton mass) and ω = 1012 Hz (terahertz
vibration frequency typical in molecular physics). These give a value of
ts = 1012 seconds. Experimentally, coherent proton tunnelling in a hydrogen bonded network gives a timescale of ts = 10−7 seconds [Horsewill et al.
(2001)]. Of course, in order for our proto-replicator molecule to explore
all possibilities the quantum mechanical evolution of its wave function will
involve a large number of such tunnelling processes. However, they will all
take place simultaneously within a similar timescale ts from the moment
the molecule is left to its own devices as a quantum system isolated from
its environment.
44
3.6.3.
Quantum Aspects of Life
Decoherence
The crucial other timescale therefore is the decoherence time, tD —this is
the time that the full quantum superposition of all possible states in the
combinatorial library can be explored before the interaction with the surrounding environment destroys it. The noise of the environment effectively
scrambles the delicate phases between the many different terms in the full
wave function of the system. After collapse due to such thermal decoherence, the quantum state is reset and the wave function evolves again. Of
course if decoherence is rapid (tD < ts ) then we would essentially have a
quantum Zeno effect in which the molecule never gets a chance to explore
other configurations before collapsing back to its original state. However,
provided tD > ts then each time the complex quantum superposition collapses there is a (very tiny) chance that it will find itself in the correct
replicator configuration. The advantage this now has, of course, (over classical non-quantum searching) is that this search takes place far more rapidly
than it would if it had to explore (build-dismantle-rebuild) each possible
structure in the combinatorial library one by one through chemical reactions and random thermal collisions.
3.6.4.
Replication as measurement
But what if the correct structure for a replicator is hit upon following decoherence? Would not this unique state simply be lost as quickly as it is
found once the wave function evolves yet again? We argue not. Consider
if, once formed, the self-replicator then does what it is uniquely able to do:
replicate. Now, the dynamics of the system will be subtly but significantly
changed. Under these conditions, the thermal field is not the only source
of decoherence. Self-replication will incorporate precursor molecules into
newly-formed replicators. In these circumstances, self-replication will inevitably couple the system more strongly with the environment. In a sense,
the possibility of self-replication is constantly examining the system for
tunnelling events that can form the self-replicator. And, crucially, these
examinations do not merely look; they “capture” the replicator state by
virtue of their property of self-replication. Once a self-replicator is present
then it will be permanently coupled to its environment through the replication process. Thus, while decoherence takes place all the time and, with
overwhelming likelihood leads every time to the “wrong” structure, once
the replicator is formed, the process of replication becomes an “irreversible
Quantum Coherence and the Search for the First Replicator
45
act of amplification” (a classical measurement as defined by one of the
founders of quantum mechanics, Niels Bohr) of the system. That is, by
coupling more strongly to its environment, the replicator state announces
itself as being “macroscopically distinguishable” from all the other possible
structures the molecule could have. There is no teleology needed here since
we describe the measurement as a two-step process: the inevitable and
very rapid environment-induced decoherence process taking place all the
time followed, in the unique case of the replicator state being discovered,
by the irreversible dragging out of this state into the macroscopic world.
Of course, the biological process of replication is on a time scale far greater
than those discussed above, but we argue that it is the stronger coupling of
this state to its environment (such as its ability to utilize the chemicals in
the environment for replication) that marks it out as special. To reiterate
then: provided the search time needed for exploring all possible structures
within the quantum combinatorial library is shorter than the decoherence
time, which in turn is many orders of magnitude shorter than the time it
would take to explore all the possible structures “classically”, then quantum mechanics can provide a crucial advantage in the locating that special
replicator state.
3.6.5.
Avoiding decoherence
There remains, however, the issue of how long such a delicate and complex
quantum state can be maintained and decoherence kept at bay. Of course
we are not suggesting that every molecule of the requisite complexity will,
every time, explore all regions of the combinatorial library space, but rather
only those molecules that are already close enough to the replicator state
for them to be linked to it via quantum tunnelling of protons or even superpositions of different shapes. This “shape co-existence” is well known
in many areas of quantum physics. For instance, in nuclear physics, the
lead isotope 186 Pb has a nuclear ground state that looks like a superposition of three different shapes simultaneously: spherical, prolate and oblate.
Even in biology, the protein tubulin that makes up the microtubules within
neurons has been suggested to be in a superposition of two different shapes.
Keeping decoherence at bay is of course a tall order in the complex,
warm and wet conditions of a primordial pool that are not the kind of conditions where one would expect to find significant quantum coherence. However, there are many gaps in our understanding of decoherence in complex
systems. Recent demonstrations of dynamical tunnelling [Hensinger et al.
46
Quantum Aspects of Life
(2001)], indicates that our understanding of quantum coherence within dynamic systems is far from complete. Recent experiments [Margadonna and
Prassides (2002)] demonstrating superconductivity in doped fullerene (C60 )
molecules) at 117 K (with indications that higher temperatures may be
attainable) indicate that certain organic structures may indeed support
quantum superpositions. Transport of charges along the DNA double helix by hole transfer through quantum tunnelling has also been recently
demonstrated [Giese et al. (2001)], as has coherent proton tunnelling in a
hydrogen bonded network [Horsewill et al. (2001)]. It should also be remembered that intramolecular quantum tunnelling is of course responsible
for the room temperature chemical properties of conventional chemicals,
such as benzene (tunnelling of the three π electrons across all bonds in
the benzene ring) and the tautomeric forms of compounds such as nucleotide bases, as discussed above. Quantum tunnelling of electrons and
protons is also proposed to be involved in a number of enzyme reactions
[Scrutton (1999); Scrutton et al. (1999); Sutcliffe and Scrutton (2002)] and
proton tunnelling has recently been shown to be the dominant reaction
mechanism accounting for the rate acceleration performed by the enzyme
aromatic amine dehydrogenase [Masgrau et al. (2006)]. If our proposal
is correct then some way of sustaining quantum coherence, at least for
biochemically—if not biologically—significant time scales must be found.
It is already known there are two ways in which this can occur. The first is
screening: when the quantum system of interest can be kept isolated from
its surrounding (decohering) environment. Very little is known about the
screening properties of biological molecules. For example, a reaction region
enveloped in an enzyme molecule will be partially screened from van der
Waals-mediated thermal interactions from the rest of the cell. Similarly,
the histone-wrapped double helix might serve to shield coding protons in
DNA from decoherence. The second possibility concerns what are known
as decoherence-free subspaces. In the effort to build a quantum computer,
much attention has been given to identifying degrees of freedom (technically, subspaces of Hilbert space) that are unaffected by the coupling of
the system to its environment. Paradoxically, when a system couples very
strongly to its environment through certain degrees of freedom, it can effectively “freeze” other degrees of freedom by a sort of quantum Zeno effect,
enabling coherent superpositions and even entanglement to persist. A clear
example has been presented [Bell et al. (2002)] in the context of neutrino
oscillations in a medium, but their model serves to make the general point.
These authors consider a double-well one-dimensional potential—for further discussion see Section 1.3 in Chapter 1.
Quantum Coherence and the Search for the First Replicator
3.7.
47
Summary
There are of course many difficulties with this scenario, but, as described
above, there are many difficulties with all explanations of the origin of life.
If the emergence of life depended on an unlikely sequence of maintenance
of quantum coherence within some small primordial pool then it may yet
be the most plausible “origin of life” scenario. And the proposal has one
further merit: it could be explored experimentally. As stated earlier, our
proposal is of course tied closely to the feasibility of building a quantum
computer and we do not realistically see how the merits of the former can
be explored and tested before we fully understand the possibilities of the
latter.
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Ikegami, T., Kaneko, K., and Ray, T. (2000). Open problems in artificial
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Margadonna, S., and Prassides, K. (2002). Recent advances in fullerene superconductivity, J. Solid State Chem. 168, pp. 639–652.
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K., Mulholland, A., Sutcliffe, M., Scrutton, N., and Leys, D. (2006). Atomic
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primitive earth and in meteorites, J. Mol. Evol. 9, pp. 59–72.
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About the authors
Jim Al-Khalili obtained his PhD at the University of Surrey under Ronald
C. Johnson. He is presently a theoretical physicist at the University of
Surrey, UK, where he also holds a chair in the public engagement in science.
He has published widely in his specialist field of theoretical nuclear physics
where he has developed quantum scattering methods to model the structure
and properties of light exotic nuclei. He has published several popular
science books on a range of topics in physics and is a regular broadcaster
on radio and television. He is a fellow of the UK Institute of Physics and a
trustee of the British Association for the Advancement of Science.
Quantum Coherence and the Search for the First Replicator
49
Johnjoe McFadden obtained his PhD from Imperial College, London,
under Ken Buck. He is professor of molecular genetics at the University
of Surrey, UK. For more than a decade, he has specialised in examining
the genetics of microbes such as the agents of tuberculosis and meningitis
and he invented a test for the diagnosis of meningitis. He has published
on subjects as wide-ranging as bacterial genetics, tuberculosis, idiopathic
diseases and computer modelling of evolution and has edited a book on
the genetics of mycobacteria. He has produced a widely reported artificial
life computer model and is author of the popular science book, Quantum
Evolution.
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Chapter 4
Ultrafast Quantum Dynamics in
Photosynthesis
Alexandra Olaya Castro, Francesca Fassioli Olsen, Chiu Fan Lee,
and Neil F. Johnson
4.1.
Introduction
Photosynthesis lies at the heart of the process of life on Earth. After
photon absorption by a light-harvesting (LH) antenna, the central step
in photosynthesis is excitation-transfer to a molecular complex that serves
as a reaction centre (RC) where the excitation is trapped, thereby allowing
charge separation to take place [Ritz et al. (2002); van Amerongen et al.
(2002)] (see Fig. 4.1). This transfer takes only a few hundred picoseconds
and is performed with extraordinarily high efficiency: most of the absorbed
photons give rise to a charge separation event. These observations are all
the more remarkable when one considers the rather extreme environmental
conditions in which these organisms manage to survive, and they provide an
enormous motivation for studying and manipulating natural photosynthetic
systems, as well as building artificial nanostructures, which can emulate the
early steps of photosynthesis.
One of the most fundamental and long standing questions about these
early steps in the photosynthetic process, concerns the extent to which
quantum coherent phenomena might play a role in the high-efficiency transfer [Hu and Schulten (1997)]. The issue is tremendously controversial. Some
experimental works have claimed the observation of coherent delocalized excitations around the B850 ring of the LH-II complex of purple bacteria [van
Oijen et al. (1999)] at low temperature (1K). Others have suggested that
Received February 23, 2007
51
52
Quantum Aspects of Life
even at such low temperatures, coherence only extends around a few chromophores [Trinkunas et al. (2001)]. Although it has been widely accepted
that excitations in the B800 ring of LH-II are localized, recent discussions
suggest that coherence in such molecules does indeed affect the excitationtransfer dynamics [Cheng and Silbey (2006)].
In this chapter, we are interested in the dynamics of energy transfer in
the LH-I-RC subunit. Although the evidence for coherence in the LH-IRC complex is fairly sparse, Ketelaars et al. (2002) supports the view of
largely delocalized excited states in the LH1 assembly of pigments at low
temperatures. Remarkably, a recent report by Engel et al. (2007) presents
empirical evidence of quantum beats associated with electronic coherence
in the Fenna-Matthews-Olson (FMO) bacteriochlorophyll complex, which
connects a large peripheral LH antenna to the reaction centre. Theoretical support for quantum coherence in photosynthesis is ambiguous, with
some phenomenological works indicating that coherence will induce higher
excitation-transfer rates [Jang et al. (2004)] while others argue that this
may not necessarily be the case [Gaab and Bardeen (2004)]. Even in the
simplified limit where the LH network is replaced by a set of interacting twolevel systems, there is still no clear theoretical picture as to how quantum
coherence might affect the efficiency of an LH network [Gaab and Bardeen
(2004)].
Against this backdrop, the excitation transfer in organic dendrimers—
which are nanometre-size macromolecules with a regular tree-like array of
branch units [Fréchet (1994)]—has attracted significant attention recently
through the prospect of creating artificial photosynthetic systems [Ranasinghe et al. (2002); Varnavski et al. (2002); Lupton et al. (2002); Ranasinghe
et al. (2003); Andrews and Bradshaw (2004)]. One of the key elements in
these artificial systems is the evidence of coherent energy transfer mechanisms. These experiments therefore open up the possibility of exploring in
detail the interplay between quantum coherence and the efficiency of artificial light-harvesting units, and should in turn help understand the possible
roles of quantum coherence in natural photosynthesis.
Motivated by this experimental possibility, we wish to investigate how
quantum superposition and entanglement might be exploited in a prospective photosynthetic complex which exhibits such coherent energy transfer
mechanisms. Using a quantum jump approach [Carmichael (1993); Plenio
and Knight (1998)], we show a dual role for quantum superposition and
entanglement in the LH-I-RC subunit. It is shown that such quantum phenomena can be used to (1) increase the photosynthetic unit’s efficiency,
Ultrafast Quantum Dynamics in Photosynthesis
53
or (2) act as a “crowd control” mechanism that modifies the efficiency in
such a way that it reduces the possibility of burnout of the photosynthetic
machinery. These results therefore provide significant motivation for exploiting such quantum phenomena in the development of future artificial
photosynthetic complexes. They also give additional insight into the role
that quantum coherence could play in natural photosynthetic units.
We begin the chapter with a brief description of the main features that
define a photosynthetic unit in the coherent regime. With the photosynthetic apparatus of purple bacteria in mind (Fig. 4.1) we consider such a
unit to be made up of donors surrounding acceptors at a reaction centre,
as illustrated in Fig. 4.2(a). The fact that there has been no experimental
evidence presented to date for coherent transfer in the LH-I-RC of purple
bacteria, is mostly due to the lack of suitable ultrafast experiments so far.
At the same time, the justification for choosing it to investigate the role of
coherence rests on the large amount of experimental data that exists concerning its structure (see e.g. [Roszak et al. (2003)]), in addition to several
accompanying theoretical studies [Hu et al. (1997); Ritz et al. (2001); Hu
and Schulten (1998); Damjanović et al. (2000)]. As a first approximation,
we will model this unit as a collection of interacting two-level systems whose
interactions take the form of a star-like configuration (see Fig. 4.2(c)) and
we describe its dynamics following a quantum jump model [Carmichael
(1993); Plenio and Knight (1998)]. Given the lack of precise information
about interactions in the artificial systems, we will consider the effects of
various candidate interaction mechanisms (i.e. dipole-dipole, pairwise and
nearest-neighbour). In particular, we obtain analytical solutions for the
relationship between the efficiency, the symmetry of the initial state and
the number of initially entangled donors. We then consider a more detailed
interaction Hamiltonian for purple bacteria (see Fig. 4.2(b)). We finish
the chapter with a discussion of some of the open questions and relevant
experimental considerations.
4.2.
A Coherent Photosynthetic Unit (CPSU)
The photosynthetic efficiency depends primarily on the mechanism of energy transfer, the dissipation and charge separation rates, and the network
geometry [Sener and Schulten (2005)]. In photosynthetic systems operating at environmental temperatures [Ritz et al. (2002); van Amerongen et
al. (2002)], the transfer mechanism agrees with the picture of an electronic
54
Quantum Aspects of Life
Fig. 4.1. Schematic diagram of the photosynthetic apparatus of Purple Bacteria
Rhodobacter Sphaeroides. A photon is initially captured by the LH-II. This creates
an electronic excitation that is quickly transferred to the RC via the LH-I. For details
see Ritz et al. (2002).
excitation migrating between chromophores via a Förster interaction
[Förster (1965)]. Here we are interested however, in the efficiency of molecular aggregates that can mimic photosynthesis and which can operate in the
regime dominated by coherent excitation-transfer [Gilmore and MacKenzie
(2006)]. We will refer to such an aggregate as a coherent photosynthetic unit
(CPSU) and will show that quantum coherence and entanglement offer remarkable control over the efficiency of a CPSU. As mentioned before, such
systems could correspond to natural photosynthetic aggregates at very low
temperatures, or can be synthesized complexes [Ranasinghe et al. (2002);
Varnavski et al. (2002); Lupton et al. (2002); Ranasinghe et al. (2003);
Andrews and Bradshaw (2004)].
Motivated by the photosynthetic apparatus of purple bacteria [Hu et
al. (1997); Ritz et al. (2001); Hu and Schulten (1998); Damjanović et al.
(2000)], we consider the excitation transfer in a system of M donors in a
circular arrangement around a RC with n acceptors (see Fig. 4.2). We are
Ultrafast Quantum Dynamics in Photosynthesis
55
Fig. 4.2. Schematic diagram of the LH-I and Reaction Centre (RC) of purple bacteria
(a) Arrangement of the 32 Bactericlorophils (BChl) (light lines) or donors surrounding
the RC in photosynthetic Rodobacter Sphaeroides. The RC has two accessory BChl
(light lines) and two acceptors (shaded dark lines) forming a special pair responsible for
charge separation. The excitation (red cloud) is initially shared among several donors.
(b) Schematic diagram of the induced dipole moments in (a). The arrows indicate the
dipole moment directions corresponding to data taken from Hu and Schulten (1998). (c)
Toy model: The RC is assumed to be a single two-level system.
interested in the limit where electronic coherence is most relevant, hence
we consider the donors and acceptors to be on-resonance two-level systems.
The coherent interaction among donors and acceptors is described by the
Hamiltonian
HI =
M
n
j=1 c=1
γjc V̂jc +
M
j=1,k>j
Jjk V̂jk +
n
gcr V̂cr ,
(4.1)
c=1,r>c
with V̂ab = σa+ σb− + σb+ σa− . Here σ +(−) is the usual creation (annihilation)
operator for two-level systems, and γjc , Jjk and gcr are the donor-acceptor,
donor-donor, and acceptor-acceptor couplings, respectively.
56
Quantum Aspects of Life
To include the decoherence effects in the excitation transfer we employ
the quantum jump approach [Carmichael (1993); Plenio and Knight (1998)].
The irreversible dynamics of an open quantum system, i.e. a quantum system in contact with the environment, is often treated in terms of a master
equation for its density operator ρ(t). In the Markovian approximation
where all memory effects in the bath are neglected, the dynamics of the
open system is described by the Lindblad master equation ( = 1)
n
1
d
ρ = Lρ = −i[H, ρ] +
[2Aj ρA†j − A†j Aj ρ − ρA†j Aj ] ,
dt
2 j=1
(4.2)
where the commutator generates the coherent part of the evolution. The
second part on the right-hand side of this equation, represents the effect of
the reservoir on the dynamics of the system, where the action of each operator Aj accounts for a decohering process labelled j. In order to illustrate
how the system’s dynamics can be interpreted in terms of quantum trajectories we follow the approach given by Carmichael (1993). Let us define
L̃ = L − J and re-write Eq. (4.2) in the following way:
⎛
⎞
n
Lρ = (L̃ + J )ρ = ⎝L̃ +
Jj ⎠ ρ ,
(4.3)
j=1
where
†
L̃ρ = −i(Hcondρ − ρHcond
) and Jj ρ = Aj ρA†j
(4.4)
with
n
Hcond
i †
A Aj .
=H−
2 j=1 j
(4.5)
By re-writing the master equation as in Eq. (4.3) one can divide the dynamics of the system into two processes: A non-Hermitian evolution associated
with the generator L, and jump processes associated with the set of Aj
that are the operators describing the sources of decoherence of the open
system. One can then interpret the system’s dynamics as given by quantum trajectories that are defined by continuous evolutions interrupted by
stochastic collapses at the times the jumps occur. The dynamics given by
the master equation is recovered by averaging over all possible trajectories
[Carmichael (1993)].
The no-jump trajectory corresponds to the case in which no decay occurs. The evolution of a quantum state along this trajectory is governed
by the non-Hermitian Hamiltonian Hcond . For instance, if the initial state
Ultrafast Quantum Dynamics in Photosynthesis
57
is pure, i.e. |Ψ(0), the non-normalized state in the no-jump trajectory becomes |Ψcond(t) = exp(−iHcond t)|Ψ(0). For a wide variety of physical
situations, particularly in the context of quantum optics, it has been shown
that the no-jump trajectory can yield a good estimation of the system’s evolution in the presence of decoherence sources [Plenio et al. (1999); Nicolosi
et al. (2004); Beige (2003)]. In most of these schemes a single excitation is
present—hence if the excitation is “lost”, i.e. a photon is detected, the system collapses into its ground state. Therefore it is desirable to investigate
the system’s dynamics conditioned on no-excitation loss.
Most of the research in energy transfer in light-harvesting systems indicates that there is no more than a single excitation present in each complex
(LH-I or LH-II) [Hu et al. (1997)]. Also, the high efficiency in the transfer
of excitation to the reaction centre indicates that most likely no excitation
is lost in the transfer. Therefore, the dynamics along the no-jump trajectory provides a tractable description of the excitation-transfer dynamics in
a CPSU. The resulting non-unitary evolution, conditioned on there being
no loss of excitation, is interrupted by stochastic jumps that can be associated either with excitation dissipation by any of the donors or excitation
trapping in the RC. Between jumps, the dissipative dynamics is governed
by
Hcond = HI − iΓ
M
j=1
σj+ σj− − iκ
n
+ −
σci
σci ,
(4.6)
i=1
where the dissipation rate Γ is assumed to be identical for all the
donors, and the trapping rate κ is assumed to be identical for all
the acceptors at the RC. A basis is given by states in which one
of the two-level systems is excited and the rest are in their ground
state, i.e.
|dj = |01 02 . . . 1j . . . 0M ; 0c1 0c2 . . . 0cn union |Ci =
|01 02 . . . 0M ; 0c1 . . . 1ci . . . 0cn , j = 1, 2, . . . , M and i = 1, 2, . . . , n. The
labels after the semicolon in each ket denote the acceptors at the RC.
Since we are interested in describing the effects of initial entangled state
we will assume pure initial states Ψ0 . The non-unitary evolution is given
by U = exp[−iHcondt] and the unnormalized conditional state becomes
|Ψcond (t) =
M
j=1
bj (t)|dj +
n
i=1
bci (t)|Ci .
(4.7)
The monotonically decreasing norm of this conditional state gives the probability that the excitation is still in the system during the interval (0, t),
i.e. the probability of no-jump P (t; Ψ0 ) = |Ψcond 2 . The quantity of
58
Quantum Aspects of Life
interest is the probability density of having a jump between t and t + dt,
w(t; Ψ0 ) = −dP (t; Ψ0 )/dt which becomes
w(t; Ψ0 ) = Ψcond (t)|(−iH̃ † + iH̃)|Ψcond (t)
= 2Γ
M
j=1
|bj (t)|2 + 2κ
n
i=1
|bci (t)|2
= wD (t, Ψ0 ) + wRC (t, Ψ0 ) .
(4.8)
Here Ψ0 is the initial state, wRC (t; Ψ0 )dt is the probability that the excitation is used by the RC in (t, t + dt) while wD (t; Ψ0 )dt is the probability that
∞
it is dissipated by any of the donors. Notice that 0 w(t; Ψ0 )dt = 1 which
implies that the excitation will eventually either be dissipated or trapped
in the RC.
Within this framework we can now define the main features of our
CPSU: the excitation lifetime, the efficiency or quantum yield, and the
transfer times [Sener and Schulten (2005)]. Given an initial state Ψ0 , the
excitation lifetime (τ ) is the average waiting-time before a jump of any
kind occurs. The efficiency (η) is the total probability that the excitation
is used in charge separation. The forward-transfer time (tf ) is the average
waiting-time before a jump associated with charge-separation in the RC,
given that the excitation was initially in the donor subsystem:
∞
τ=
0
∞
dt t w(t; Ψ0 ) , η =
0
dt wRC (t; Ψ0 ) , tf =
∞
0 dt t wRC (t; Ψ0 )
∞
0 dt wRC (t; Ψ0 )
.
(4.9)
4.3.
Toy Model: Interacting Qubits with a Spin-star
Configuration
We start by considering the simplest model for which analytical solutions
can be obtained: the RC is a single two-level system on resonance with the
donors, i.e. n = 1, and all donor-RC couplings are identical, i.e. γjc ≡ γ;
see Fig. 4.2(c). Later we shall show that the main results obtained in
this situation also apply to a more realistic model featuring the detailed
structure of the RC as in purple bacteria.
First we establish a relationship between the efficiency and number of
initially entangled donors. We consider the excitation to be initially in
the donor subsystem, i.e. bc (0) = 0, and compare bc (t) for three different
mechanisms of interaction between the donors: (i) nearest neighbours with
59
Ultrafast Quantum Dynamics in Photosynthesis
−3
wRC(t)
1.5
x 10
1
0.5
0
0
0.01
0.02
0.04
0.05
x 10
0.03
0.04
0.05
D
w (t)
2
0.03
τ/τ*
−3
1.98
1.96
0
0.01
0.02
τ/τ*
Fig. 4.3. Short time behaviour of the waiting-time distribution for a jump associated
to the RC (top) or for a jump associated to any of the donors (bottom). Time in units
of τ ∗ = 100/γ.
Jjk = (J/2)δj,k−1 , (ii) pairwise interaction with Jjk ≡ J for all {j, k} pairs
3
where rjk is
and (iii) dipole-dipole interactions of the form Jjk = J/rjk
the relative position vector between the induced dipole moments of donors
j and k. Analytical solutions for bc (t) [Olaya-Castro et al. (2006)] and
therefore wRC (t), can be found in each of these cases:
wRC (t) = |B0 |2 F (t)
(4.10)
M
j=1 bj (0)
where B0 =
and F (t) = 8κγ 2 e−(κ+Γ)t/2 |sin(Ωt/2)|2 /|Ω|2 . Here
Ω is the complex collective frequency that determines
the timescale of coherent oscillations [Olaya-Castro et al. (2006)] i.e. Ω = 4M γ 2 − (δ + i∆)2
with δ = Γ − κ and
⎧
Nearest
⎨J
Pairwise .
(4.11)
∆ = J(M − 1)
⎩
M
J k=2 (1/r1k )3 Dipole
To illustrate the dynamics of wRC (t) and wD (t), we have plotted in Fig. 4.3
the short-time behaviour of these quantities for the case of dipole-dipole
interactions. In the case plotted, we take the excitation to be initially
localized on one of the donors. As we shall discuss below, the coherent
oscillations exhibited by wRC dominate the efficiency of our CPSU, while
60
Quantum Aspects of Life
wD (t) is a monotonically decreasing function of time that dominates the
dynamics of probability of no-jump P (t; Ψ0 ).
We consider initial entangled states in which the excitation is delocalM
ized among donors, i.e. Ψ0 = j=1 bj (0)|dj and we now proceed to show
how the symmetry and entanglement in this initial state can act as efficiency control mechanisms. Equation (4.10) shows that the efficiency η
becomes proportional to |B0 |2 . The latter quantity accounts for the relative phases between states |dj i.e. quantum coherence and therefore it is
clear that symmetric initial entangled states yield an increase in η, while
some asymmetric states could be used to limit or even prevent the transfer, i.e. η = 0. Unless otherwise stated, we henceforth
consider symmetric
√
N
initial entangled states of the form Ψ0 = (1/ N ) j=1 |dj where N is
the number of initially entangled donors, i.e. N ≤ M . For these states
|B0 |2 = N and hence the efficiency not only depends on the symmetry,
but becomes proportional to the number of initially entangled donors N
as shown in Fig. 4.4. These results can be understood in terms of one
striking feature of the entanglement, that is, entanglement sharing. This
feature refers to the fact that quantum correlations cannot be shared arbitrarily among several particles [Dawson et al. (2005)]. In our case, for any
initial state, the system evolves conditionally towards an entangled state
where the excitation is shared between donors and the RC. Since entanglement cannot be shared arbitrarily, the efficiency will therefore depend
on the dynamics of entanglement-sharing between donors and RC. When
the excitation is initially localized on only one of the donors, the average
donor-RC entanglement should be small since the excitation also has to be
shared directly among all interacting donors. By contrast when the excitation is already shared by several donors, there are fewer donors left to
be entangled and consequently the gain in entanglement between donors
and RC should be larger than in the previous situation. This feature is
illustrated in Fig. 4.4(a) which shows the long-time average entanglement
between donor and RC (AEDC) and the average entanglement between
donors (AEDD), as a function of N for the dipole-dipole case. As can be
seen, AEDC increases with N while AEDD does not change drastically.
Hence the efficiency can be seen as a monotonic function of AEDC as depicted in Fig. 4.4(b). The larger value of AEDC when N = M confirms the
strong damping effect of non-entangled donors on the AEDC.
In order to compute these averages, we have taken advantage of the
known results for W -class entangled states [Dawson et al. (2005)] as it is
the state given in Eq. (4.7). We quantify the pair entanglement using the
61
Ultrafast Quantum Dynamics in Photosynthesis
1
Dipole
Pairwise
Nearest
0.9
0.8
a
1
AEDC
AEDD
0.5
0.7
N
0
1 6 11 16 21 26 32
K
0.6
0.5
0.6
K
0.3
0.2
0.4
0.2
0.1
0
1
b
0.8
0.4
0
0
9
17
25
33
41
N
0.5
49
57
1
AEDC
65
Fig. 4.4. Efficiency (η) versus number of donors which are initially entangled (N ),
for the toy model. Main panel shows numerical results for three different interaction
mechanisms. For nearest-neighbour interactions (♦) the coupling is 100 meV, while for
the pairwise case (+) it is 10 meV and equals the average dipole-dipole coupling (•).
These values have been taken to be such that ∆dipole ≃ ∆pairwise > ∆nearest . In each
case, the donor-RC coupling equals 1 meV, Γ = 1 ns−1 and κ = 4 ps−1 . (a) The
average entanglement between donors and RC (solid line) and the average intra-donor
entanglement (dotted line) as a function of N , in the case of dipole-dipole interaction.
The same behaviour is observed for the pairwise and nearest-neighbour cases (not shown).
(b) η versus the average donor-RC entanglement for the three forms of interaction. In
(a) and (b), the average-entanglement values have been normalized to the maximum
value obtained in each case.
“tangle” [Coffman et al. (2000)] that equals |bj (t)bk (t)∗ |2 for the reduced
state of two two-level systems j and k in our CPSU. Here bj (t) are the
normalized versions of bj (t). The total intra-donor entanglement (EDD (t))
and donor-RC entanglement (EDC (t)) are each equal to the sum over all
distinct pair contributions, and their long-time averages are calculated for
the time when the probability of no-jump P (t; Ψ0 ) has decayed to 0.01 at
t = tmax . These averages are defined as follows:
AEDC =
AEDD =
lim
T →tmax
lim
T →tmax
1
T
1
T
T
EDC (t)dt
0
(4.12)
T
EDD (t)dt
0
62
Quantum Aspects of Life
400
Dipole
Pairwise
Nearest
1
P(0,t)
350
300
a
N=1
N=15
N=32
0.5
W [ps]
250
t [ns]
0
0
200
200
1
2
b
t [ps]
150
100
f
100
50
0
1
0
1 6 11 16 21 26 32
9
17
25
33
41
49
57
N
65
N
Fig. 4.5. Lifetime (τ ), transfer time (tf ), and probability of no-jump versus the number
of donors initially entangled (N ), for the toy model. Main panel shows numerical results
for τ , for the three forms of interaction: dipole-dipole (♦), pairwise (+) and nearestneighbour (•). Coupling strengths and parameter values are as in Fig. 4.4. Shorter
lifetimes are associated with faster decays in the probability of no-jump as shown in (a),
but also with shorter transfer times as shown in (b). (a) Probability of no-jump as a
function of time for different N values, for the case of dipole-dipole interactions. The
same behaviour is observed for pairwise and nearest-neighbour cases (not shown). (b)
tf as a function of N for the three mechanisms of interaction.
In Fig. 4.4, we note that the gradient in each case depends on the nature
of the interaction in the system. For a fixed N , η reaches higher values
in the case of nearest-neighbour interactions, while it reaches similar values for dipole-dipole and pairwise interactions. We have chosen γ to be
the same for all these situations and J has been taken to be such that
∆dipole ≃ ∆pairwise > ∆nearest . According to these results, the stronger the
effective interaction between one donor and the rest, the lower the efficiency.
This is due to the fact that a stronger interaction implies a larger value for
the average intra-donor entanglement, therefore limiting the donor-RC entanglement and hence the efficiency.
The decay-rate of P (t; ψ0 ) increases with the number of initially entangled donors, as shown in Fig. 4.5(a). Correspondingly, the excitation
lifetime τ decreases as shown in Fig. 4.5. As expected, situations for which
Ultrafast Quantum Dynamics in Photosynthesis
63
the efficiency reaches higher values imply lower lifetimes. Interestingly, for
these interaction mechanisms the transfer time tf is independent of N as
can be seen in Fig. 4.5(b). The three situations satisfy tf ≤ τ , where the
equality holds for the initial state in which all the donors are entangled.
Also, the lifetime and transfer times are less than the time at which probability of no-jump is no lower than 0.9 (see Fig. 4.5(a)), hence we can say
that the excitation transfer dynamics is indeed dominated by the short-time
behaviour illustrated in Fig. 4.3.
4.4.
A More Detailed Model: Photosynthetic Unit of
Purple Bacteria
We now consider an effective LH-I-RC interaction Hamiltonian given by
Hu et al. (1997) and Hu and Schulten (1998) which has been used to describe the excitation transfer in the photosynthetic unit of the purple bacteria Rhodobacter Spharoides [Hu et al. (1997); Hu and Schulten (1998);
Damjanović et al. (2000)]. The LH-I of these bacteria contain 32 identical
bacteriochlorophyll (BChl), or donors, surrounding the RC (see Fig. 4.2).
The RC is in turn made up of a special pair of BChl responsible for the
charge separation, and two more accessory BChl molecules whose function
is still under debate [Ritz et al. (2002)]. In general the BChl at the RC are
slightly off-resonance with the donors. Here, for the sake of simplicity, we
have assumed they are on-resonance such that the effective Hamiltonian is
of the form given in Eq. (4.1) but with certain particularities. First, the interactions between adjacent donors cannot be accounted for properly with
a dipole-dipole approximation. They should be quantified by two different
constants ν1 and ν2 , i.e. Jj,j+1 = ν1 and Jj,j−1 = ν2 whose difference reflects the dimeric structure of the LH-I ring—each BChl is bound to two
different proteins. Second, the coupling between non-neighbouring donors
corresponds to a dipole-dipole interaction of the form
Jjk =
3(rjk · µj )(rjk · µk )
µj · µk
−
,
3
5
rjk
rjk
|j − k| > 1 ;
(4.13)
where µj is the transition dipole moment of the j th donor and rjk is the
relative position vector between donors j and k. The directions of µj
have been taken from Hu and Schulten (1998). A top view of the dipole
representation of LH-I-RC is shown in Fig. 4.2(b).
64
Quantum Aspects of Life
We consider two cases for the RC: (i) the full structure, and (ii) the
structure without the two accessory BChls. Our numerical results suggest that a CPSU with an effective interaction as in the LH-I-RC
apparatus of purple
could exploit both the symmetrically en√ bacteria,
N
tangled Ψ0 = (1/ N ) j=1 |dj and the asymmetrically entangled Ψ0 =
√
N
(1/ N ) j=1 (−1)j |dj states in order to modify the efficiency. For the
symmetric states, η behaves very similarly to before: it increases with N
as shown in Figs. 4.6(a) and 4.6(b) which correspond to cases (i) and (ii)
respectively. The results indicate, however, that accessory BChls have a
strong damping effect for symmetric states which is seen not only in the
lower values for the efficiency in case (i), but in the fact that the transfer
time has an increasing trend as a function of N as shown in the inset in
Fig. 4.6(a). In the absence of the accessory BChls the transfer time presents
the opposite behaviour: it decreases with N (see inset in Fig. 4.6(b)) and
in consequence the efficiency values are larger. Conversely for the asymmetric states, the efficiency is a non-monotonic function of N , indicating
that there is an optimal number of entangled donors for which η has a maximum (Figs. 4.6(c) and 4.6(d)) and for which tf has a minimum (see insets
in Figs. 4(c) and 4(d)).
The slight differences between Figs. 4(c) and 4(d) indicate that for
asymmetric initial states, the presence of the accessory BChls in RC do
not significantly affect the efficiency of the transfer. Most importantly, the
non-monotonic profile for these states indicates that such a CPSU could use
the symmetry of the initial state as a “crowd-control” mechanism: it can
modify the efficiency, for example, in order to reduce the risk of burnout
on the RC. Interestingly, some recent experimental works have indeed indicated that the excitation in LH-II may be coherently delocalized over just
a few donors [van Oijen et al. (1999)]. Unfortunately, no such investigation
has been reported on the LH-I.
The above discussion raises several interesting questions, which hopefully justify and motivate further experimental work in these systems. First,
notice that the distinction between symmetric and asymmetric states is a
purely quantum coherent phenomena, i.e. a well defined phase-difference
between quantum states. Do these results suggest that quantum coherence
might be not just sufficient, but indeed necessary for the transfer of excitation to the RC? Second, even if entanglement is not necessary for the
transfer of excitation to the RC, might it be used to artificially enhance the
performance of natural photosynthetic units?
65
Ultrafast Quantum Dynamics in Photosynthesis
0.25
0.07
500
c
0.06
t [ps]
a
f
0.2
0.05
0
0.15
0 10 20 30
N
η
η
0.04
0.03
0.1
t [ps]
500
0.05
f
0.02
0.01
0
0
0
5
10
15
0
20
10
20
N
25
30
0
30
35
0
5
10
15
20
25
30
35
N
N
0.25
1
500
d
0.8
0.2
0.6
0.15
tf [ps]
b
0
0 10 20 30
N
η
η
200
tf [ps]
0.4
0.1
100
0.2
0
0.05
0
10
20
30
N
0
0
5
10
15
20
25
0
30
35
N
0
5
10
15
20
25
30
35
N
Fig. 4.6. Efficiency (η) and transfer time (tf ) versus number of initially entangled
donors (N ), for the photosynthetic bacterium Rhodobacter spharoides. Two situations
have been considered: (i) The RC with the special pair responsible for charge separation
and the two more accessory BChls, and (ii) the RC without the two accessory BChls.
The effective Hamiltonian has been taken following Hu et al. (1997). (a) and (b) show
numerical results for η as a function of N , for cases (i) and (ii) respectively, when the
initial state is a symmetric entangled state. For these initial states, the main difference
between situations (i) and (ii) is the behaviour of tf which is depicted in the insets in
(a) and (b). Numerical results for the situation where the initial state is an asymmetric
entangled state, are shown in (c) for case (i) and in (d) for case (ii). It turns out that
the transfer times (shown in the insets) are always less than or equal to the lifetimes
(not shown).
4.5.
Experimental Considerations
Experimental observation, and even manipulation, of the coherent excitation transfer in synthesized LH-I-RC [Davis et al. (1995)] should become
plausible as the temperature is lowered. An estimate of the temperature
below which robust coherent excitation transfer between donors and RC
66
Quantum Aspects of Life
should be expected is given by kB T ∗ = hγ/α where the coupling between the donor and the environment should satisfy α < 1/2 [Gilmore and
MacKenzie (2006)]. The α values for naturally occurring photosynthetic
structures are however unknown. Taking α = 0.1 yields an estimated temperature of T ∗ ≃ 10 K, which is larger than the temperatures at which
experiments have previously been performed (i.e. 1 K [van Oijen et al.
(1999)])—hence our belief that such quantum phenomena can be usefully
explored using current experimental expertise.
Most of the advances in our understanding of photobiological systems
have been due to recent improvements in instrumental techniques—for example, confocal microscopy which allows single-molecule experiments and
fluorescence correlation spectroscopy [Jung et al. (2002)]. We believe that
these techniques are ideal candidates for generating and manipulating entangled states within the chromophores of light-harvesting systems and to
probe the effects discussed in this work. The purely symmetric and antisymmetric entangled states discussed here will not, in the real system, have
these exact symmetries because of symmetry-breaking interactions within
the molecules and immediate environment. To the extent to which these
symmetries are broken, then the states will be either mostly symmetric
with some asymmetric component, or mostly asymmetric with some symmetric component. Both will now be allowed optically, and both will have
a greater or lesser character of the efficiency and transfer times for the pure
symmetric or antisymmetric states. In short, as with all symmetry breaking, one can expect the two resulting manifolds to still have predominantly
one of the two characters—hence the theoretical analysis of the present
chapter, while ideal, will still hold qualitatively. Given the current advances in nanotechnology, we also hope that the results in this chapter will
stimulate fabrication of novel nanoscale energy sources and devices built
around quantum coherent (or even mixed quantum-classical) dynamics. In
this direction we note the interesting theoretical possibility that natural
photosynthetic systems may one day be used as the basis of quantum logic
gates. An example of how this might be achieved using excitons, is given
by Hitchcock (2001).
4.6.
Outlook
In photosynthetic organisms such as purple bacteria, the LH-I-RC system
studied in this chapter is embedded in a network of other LH-I- RC systems
and LH-II rings. After sunlight is harvested by the LH-I and LH-II rings,
Ultrafast Quantum Dynamics in Photosynthesis
67
each excitation migrates from one ring to another until it either dissipates or
arrives to the RC where charge separation takes place. This process is characterized by the excitation transfer rates between the different complexes
(LH-I/LH- II, LH-I/LH-I, LH-II/LH-II and LHI-RC), by the probability
of an LH to be excited upon light and by the dissipation rate of excitations. The high efficiency of these organisms depend on these parameters
and in particular on the fact that dissipation occurs in a nanosecond scale
in contrast to the few tens picosecond scale of average lifetime excitation.
An important role in this process may be played by the different structures
of the various membranes themselves. We are currently investigating this
question, using both a classical walk simulation [Fassioli et al. (2007)] and
a quantum walk analysis (see Flitney et al. (2004) and references therein,
for discussions of quantum walks).
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About the authors
Alexandra Olaya-Castro is currently a Junior Research Fellow in Physics
at Trinity College, Oxford. She obtained her undergraduate degree in
Physics at Universidad Distrital Francisco José de Caldas and MSc at Universidad de Los Andes in Bogotá, Colombia. She obtained her DPhil at
Oxford under Neil F. Johnson. Her research interests currently focus on
entanglement in open systems and quantum coherence in biomolecular complexes.
Francesca Fassioli Olsen is currently a DPhil student under Neil
F. Johnson at Oxford University. She obtained her BSc in Physics at
Pontificia Universidad Católica de Chile. Her research interests are in entanglement in open quantum systems and excitation dynamics in photosynthetic membranes.
Chiu Fan Lee is currently a Glasstone Research Fellow at the physics
department and a Junior Research Fellow at Jesus College, Oxford. He
70
Quantum Aspects of Life
obtained his BSc at McGill University and DPhil at Oxford University
under Niel F. Johnson. His research interests lie in the areas of biophysics
and theoretical biology.
Neil F. Johnson until recently was a professor of physics at Oxford
University—he is now at the University of Miami, Florida. His training
in physics consisted of an undergraduate degree at Cambridge University
and a PhD at Harvard University under Henry Ehrenreich. His interests
focus around Complexity, and Complex Systems, in the physical, biological
and social sciences.
Chapter 5
Modelling Quantum Decoherence in
Biomolecules
Jacques Bothma, Joel Gilmore, and Ross H. McKenzie
5.1.
Introduction
Decoherence in the context of quantum mechanics is a concept that not
many people without a background in physics are familiar with. Anyone
who has carried out some general studies in science will be aware of the
strange manifestations of quantum mechanics that leads to counter intuitive effects like tunnelling and superpositions on the atomic scale. The
question of why these effects do not manifest on macroscopic scales is often
rationalized in terms of large objects having very small de Broglie wavelength. When this argument is applied to microscopic systems like biological chromophores or enzymes the relevant mass and length scales often
appear to be small enough to allow for the possibility of observing quantum
effects. It is this kind of thinking that has in some cases lead to sensationalist claims that quantum effects must be manifesting in biological systems
when anomalous observations are made. However this neglects the subtlety
of decoherence that tends to “wash out” quantum effects.
The phenomenon of decoherence comes about when different quantum
entities interact. An illustrative example of this is a beautiful experiment
that was performed by Anton Zeilinger’s group in 2003, which looked at
decoherence in C70 fullerenes [Hornberger et al. (2003)]. In this experiment
a beam of C70 molecules were fired at a series of slits. As a result the
Received April 7, 2007
71
72
Quantum Aspects of Life
wavefunctions of the fullerene molecules interfered to produce an interference pattern. When gas particles were introduced along the path, which
the C70 molecules needed to traverse, the interference pattern deteriorated
significantly. At high enough pressures no interference could be seen and
the C70 molecules behaved like classical particles. This arose simply from
the fact that the C70 and gas molecules were interacting, and as a result
the C70 molecule behaved like a classical entity.
In biological systems there is often a vast array of different sources of
decoherence. The cell is a “hot and wet” environment. A chromophore
on a protein can interact with the water molecules around the protein
and also the protein itself. It is of the utmost importance to be able to
model decoherence in biological molecules in order to understand whether
quantum mechanics plays a functional role in these systems.
A major problem in understanding biomolecules’ behaviour is that
the molecule itself may contain hundreds of atoms, while its surrounding environment might consist of thousands or even millions of water
molecules that all contribute to the biomolecule’s behaviour. To simulate the quantum dynamics of such systems on a computer is very time
intensive and, while it has been very informative for certain systems, the
general principles underlying the results are not always clear (similar to
using a pocket calculator to do complex sums—you can obtain the right
answer, but it may not be obvious why it is right). Most significantly, to
study even a slightly different molecule means the whole simulation must be
run again.
In this chapter we describe some minimal models that we and others
have developed to describe the interactions between biomolecules and their
surroundings. This includes looking at biological chromophores embedded
in a protein in solution and enzymes that catalyse hydrogen transfer reactions. These models capture the essential physics of the interactions but
are simple enough that very complex systems can be studied. They use
approximations like assuming the biomolecules are spherical, and treating
the water as a uniform fluid (not including individual molecules). Therefore, instead of describing every single atom, it is enough just to input the
size of the molecule and some information about its electric charge. It also
turns out that, mathematically, this same type of model has been used for
a number of other physical situations (such as in quantum computing), so
a lot of results for these models are already available, ready to be applied
to specific biological systems.
73
Modelling Quantum Decoherence in Biomolecules
These models let us separate out the effects on a biomolecule of the
nearby proteins and the solvent, and can tell us which part of the environment contributes most to the behaviour of different biomolecules. Ultimately, this could help in creating more efficient versions of, for example,
artificial photosynthesis. It also allows one to determine how valid the hypothesis that certain enzymes enhance their catalytic power by exploiting
tunnelling of hydrogen.
5.2.
Time and Energy Scales
Biology is remarkable in that the range of time and energy scales over which
biological processes occur spans seven orders of magnitude, ranging from
ultrafast solvation times in water on the order of femtoseconds to the slow
rotation of a protein which can take tens of nanoseconds. Figure 5.1 shows
some relevant processes and their corresponding time scales, Table 5.1 gives
some specific data for systems of relevance to this chapter.
In particular, we observe that within a given system, the various relevant processes often occur on widely separated time scales. For example,
a chromophore inside a protein has a radiative lifetime of 10 ns, the surrounding water can respond in 10-100 fs, while the protein exhibits internal
dynamics on the order of picoseconds, although in some circumstances much
longer relaxation times (up to 20 ns) have been observed [Pierce and Boxer
(1992)].
Elastic
vibrations of
Covalent bond vibrations globular regions Residence time of bound water
-15
10
Time (s)
-14
10
Ultrafast
solvation
in water
-13
10
-12
10
-11
10
-10
10
Bulk water
dielectric
relaxation
Fast
solvation
in water
ET in
PS RC
Surface
sidechain
rotation
-9
10
Whole
protein
rotation
-8
10
-7
10
Radiative lifetime of
chromophore
Protein dielectric
relaxation
Fig. 5.1. Schematic representation of the time scales of various processes in
biomolecules, proteins and solutions. ET stands for electron transfer, PS RC for photosynthetic reaction centre. See Table 5.1 for specific numbers.
74
Table 5.1. Timescales for various processes in biomolecules and solutions. The radiative lifetime of a
chromophore is order of magnitudes longer than all other timescales, except perhaps protein dielectric
relaxation. MD refers to results from molecular dynamics simulations. Of particular relevance to this work
is the separation of timescales, τs ≪ τb ≪ τp (compare Fig. 5.1).
Timescale
Ref.
Radiative lifetime
Bulk water dielectric relaxation
Protein dielectric relaxation (MD), τD,p
Ultrafast solvation in water
Fast solvation in water, τs
Solvation due to bound water, τb
Solvation due to protein, τp
Covalent bond vibrations
Elastic vibrations of globular regions
Rotation of surface sidechains
Reorientation of whole protein
10 ns
8 ps
1-10 ns
10’s fs
100’s fs
5-50 ps
1-10 ns
10-100 fs
1-10 ps
10-100 ps
4-15 ns
[van Holde et al. (1998)]
[Afsar and Hasted (1978)]
[Loffler et al. (1997); Boresch et al. (2000)]
[Lang et al. (1999)]
[Lang et al. (1999)]
[Peon et al. (2002)]
[Sen et al. (2003)]
[van Holde et al. (1998)]
[van Holde et al. (1998)]
[van Holde et al. (1998)]
[Boresch et al. (2000)]
Quantum Aspects of Life
Process
Modelling Quantum Decoherence in Biomolecules
5.3.
75
Models for Quantum Baths and Decoherence
As discussed above, systems are rarely completely isolated from their environment, and in many cases, particularly in biology, this system-bath
coupling may be very strong and in fact play an important role in their
functionality. This interaction may be through photons (i.e. light), phonons
(such as vibrational modes in the solvent or protein scaffold, or indeed the
molecule comprising the two-level system (TLS) itself) or specific localized
defects (such as local point charges).
In terms of dynamics, the presence of a strongly coupled environment
may “observe” the state of the TLS and destroy or weaken the Rabi quantum oscillations between the two states. At the most extreme limit, this
may lead to the quantum Zeno effect, where the system is completely localized [Sekatskii (2003); Joos (1984)] in one state or the other, even if an
alternative lower energy state is available.
Particularly with the current interest in creating a quantum computer,
it is necessary to have good models for the bath and how the resulting
decoherence will effect the system dynamics. In particular, the development of simple, minimal models allows for both physical insight and the
classifications of different parameter regimes exhibiting different dynamics.
A particularly interesting class of models focuses on interaction between a
two level system, described by a Pauli matrix σz , and its bath. It suffices
in many situations to represent this environment coupling by a term
Hint = σz · Ω̂ .
(5.1)
Here some, typically many-body, operator Ω̂ of the environment couples to
the state of the TLS via σz . Although situations exist where coupling to
the other operators σx or σy is relevant, in many cases these couplings will
be negligible [Leggett et al. (1987)].
Two specific forms for this interaction are worth discussion, as they can
be applied to a diverse range of physical systems. The first is the spin-bath
model (for a good review, see Prokof’ev and Stamp (2000)), where local
defects which are strongly coupled to the TLS of interest are themselves
modelled as two-level systems. This allows for strong coupling to localized
features of the environment.
The second is the spin-boson model [Weiss (1999); Leggett et al. (1987)].
This describes the interaction between a two-level system and the (typically) delocalized modes of the environment (phonons, photons, etc.), which
76
Quantum Aspects of Life
are treated as a bath of harmonic oscillators [Leggett et al. (1987)]. Provided the coupling to any single environment mode is sufficiently weak,
and intra-bath interactions can be neglected, a quite general environment
can be treated as a collection of harmonic oscillators [Caldeira and Leggett
(1983)]. It is distinct from the spin-bath model, however, because it cannot
describe any single, strongly coupled feature of the environment, so careful
consideration must be given to the applicability of each model.
5.3.1.
The spin-boson model
The spin-boson model is a powerful and widely used model for describing
decoherence. It exhibits rich quantum dynamics, and has found wide and
varied applications in modelling decoherence in qubits [Reina et al. (2002)]
and electron transfer reactions [Xu and Schulten (1994)], as well as deeper
questions about the role of quantum mechanics on the macroscopic level.
Most importantly, its dynamics have been widely studied [Weiss (1999);
Lesage and Saleur (1998); Costi and McKenzie (2003)] and its behaviour is
known through much of the parameter space.
In Leggett et al. (1987) the spin-boson Hamiltonian is defined as
1
1
1
1
(p2α /2mα + mα ωα2 x2α ) + σz
Cα xα . (5.2)
H = − ∆σx + ǫσz +
2
2
2
2
α
α
This Hamiltonian describes a two level system interacting with an infinite
bath of harmonic oscillators. Here, σz is a Pauli sigma matrix which describes the state of a two level system (TLS) with energy gap ǫ. Here, ∆
represents the bare tunnelling energy between the two levels (note that in
Leggett et al. (1987) a tunnelling frequency is used instead of energy, and
∆ appears in Eq. (5.2) instead). The TLS is coupled to a bath of harmonic oscillators identified by subscript α, described by frequency, mass,
position and momentum ωα , mα , xα and pα respectively. The third term
in (5.2) is the standard energy of a simple harmonic oscillator. The final
term describes the coupling to the position of the αth oscillator and has
units of force (energy per unit length, kg m s−2 ). Leggett et al. (1987) also
includes a coordinate q0 , representing a length scale derived from mapping
a continuous system (such as the double well potential) to a TLS that is
unnecessary for the intrinsically two-state system which we will be considering in this chapter, and is usually not included (although it should be
noted it would change the dimensions of the coupling constants Cα ).
77
Modelling Quantum Decoherence in Biomolecules
Table 5.2.
Parameters of the spin-boson model, and their units.
Symbol
Description
Units
ǫ
∆
ωα
Cα
J(ω)
α
Bias / TLS energy
Tunnelling element / coupling
Frequency of the αth oscillator
Coupling to the αth mode
Spectral density
Dimensionless coupling for Ohmic J(ω)
Energy (J)
Energy (J)
Frequency (s−1 )
Force (kg m s−2 )
Energy (J)
Unitless
It is also possible to rewrite the spin-boson model in terms of creation
and annihilation operators of the bath:
1
H = ǫσz + ∆σx
ωα a†α aα + σz
Mα (a†α + aα )
(5.3)
2
α
α
where the position and momentum are defined as
xα = 2mα ωα (aα + a†α )
and
pα = −i
mα ω α
mα ωα
(aα
2
− a†α )
i
√
or equivalently aα = xα
2 + pα 2mα ωα . In addition to allowing us
to use the framework of second quantization, we no longer need to specify
an effective mass mα for each oscillator; instead, it is sufficient to simply
specify the oscillator frequencies and their couplings.
5.3.1.1. Independent boson model
A closely related model is the independent boson model, where there is
no coupling between the two states of the TLS (i.e., ∆ = 0). This is
sometimes referred to as a polaron model [Mahan (1990)]. It is described
by the Hamiltonian
1
1
1
(p2α /2mα + mα ωα2 x2α ) + σz
Cα xα .
(5.4)
H = ǫσz +
2
2
2
α
α
This corresponds to two uncoupled energy levels interacting with an environment modelled by the harmonic oscillators. Now, the TLS operator σz commutes with the Hamiltonian and so is a constant of motion—
environment effects cannot act to change σz , but the off diagonal terms of
the reduced density matrix ρ(t), describing quantum coherence between the
two TLS states, will change over time. This system may be of particular
78
Quantum Aspects of Life
interest where transitions may be induced between the two states by an
external, “fast” influence, and the resulting changes and relaxation of the
environment are of interest. For example, solvent relaxation after a rapid,
laser induced transition in a chromophore is directly observable through
the gradual shift in the wavelength of the chromophore’s fluorescence peak.
5.3.2.
Caldeira-Leggett Hamiltonian
Another Hamiltonian that is highly analogous to the spin boson Hamiltonian can be used to model how coupling to the environment influences
hydrogen transfer reactions. This Hamiltonian is generally referred to as
the Caldeira-Leggett Hamiltonian and treats the position of the hydrogen
as a continuous one dimensional variable that is the subject of some position
dependent external potential V (x),
2
N
1 p2α
C
p2
α
+ mα ωα2 qα −
.
(5.5)
+ V (x) +
x
H=
2M
2 α=1 mα
mα ωα2
Figure 5.2 shows a double well potential that is the generic potential for an
arbitrary chemical reaction. There are two metastable positions located at
the reactant and product states and then an unstable position that corresponds to the transition state of the reaction.
Fig. 5.2. Potential energy as a function of the reaction coordinate, x, with the
metastable reaction state at A, the transition state at B and the final product state
at C. Escape occurs via the forward rate k and Eb is the corresponding activation
energy. The angular frequency of oscillations about the reactant state is ω0 , which depends on the curvature of the potential energy surface at the local minimum (x = xa )
and the mass of the particle. Similarly the barrier frequency ωb , depends on the curvature of the potential energy surface at the local maximum (x = xb ) and the mass of the
particle [Hänggi et al. (1990)].
Modelling Quantum Decoherence in Biomolecules
5.3.3.
79
The spectral density
In many situations, we are either unable to measure the state of the environment or have no interest in it beyond its effect on the chromophore
or hydrogen being transferred. A key result for the spin-boson model and
Caldeira-Leggett Hamiltonian is that the effect of the environment on the
dynamics of the subsystem of interest can be completely encapsulated in
the “spectral density” function J(ω) defined as:
4π 2
π Cα2
δ(ω − ωα ) =
Mα δ(ω − ωα ) ,
(5.6)
J(ω) =
2 α mα ω α
α
Here, δ(ω−ωα ) is the Dirac δ-function, and so J(ω) is effectively the density
of states of the environment, but weighted by the couplings Cα . It has units
of energy. The Caldeira-Leggett Hamiltonian is similarly characterized by a
spectral density that is similarly defined. In this context another function
known as the friction kernel is often used to characterise the interaction
with the environment. It is formally defined as:
1 Cα2
cos(ωα t) ,
(5.7)
γ(t) =
M α mα ωα2
where M is the effective mass of the particle involved in the chemical reaction. For most applications it is appropriate to assume that the spectrum of oscillator frequencies are sufficiently dense and the couplings Mα
are sufficiently smooth that J(ω) may be considered a smooth, continuous
function of ω. In this way, we remove the need to specify the couplings to
each individual oscillator (potentially requiring a large number of discrete
parameters) and can instead describe the functional form of the spectral
density.
In particular, for many (though by no means all) physical situations
J(ω) takes the form of a simple power law at low frequencies, but decays
to zero above some cut-off frequency ωc . We will later see that when the
dynamics occur on time scales much shorter than the environment response
times (specifically, quantum tunnelling with energy ∆ ≪ ωc ) that the
exact form of the cut-off, and the exact value of the cut-off frequency, will
be unimportant [Leggett et al. (1987); Weiss (1999)]. Clearly, though, if
both system and bath dynamics occur on comparable time scales, couplings
around ωc may be important.
Typically, ωc is introduced when mapping a continuous system to a discrete, two level system, but in many physical situations the high frequency
cut-off will occur naturally: intuitively, there is a minimum timescale over
80
Quantum Aspects of Life
which the TLS is capable of responding to the environment. For bath
events, particularly oscillatory events, on time scales shorter than this, the
TLS will only see the average behaviour of the environment. This corresponds to zero coupling at sufficiently high environment frequencies.
A general power law for the spectral density has been treated extensively
in Leggett et al. (1987):
J(ω) = Aω s e−ω/ωc .
(5.8)
The spectral density for s = 1 is referred to as Ohmic, and for s > 1 as
superohmic, and as subohmic for s < 1. The Ohmic spectral density is
important as J(ω) is roughly linear up to some cut-off frequency ωc :
J(ω) = hαω ,
ω < ωc
(5.9)
where α is a dimensionless measure of the strength of the systemenvironment coupling, independent of ωc , and will be critical in determining
the system dynamics. In many situations with an Ohmic spectral density
it is more convenient (or, indeed, more physically relevant) to consider a
Drude form for the spectral density [Weiss (1999)], as follows:
J(ω) =
5.4.
hαω/ωc
.
1 + (ω/ωc )2
(5.10)
The Spectral Density for the Different Continuum
Models of the Environment
In this section we consider dielectric continuum models of the environment
of a biological chromophore. For the different models it is possible to derive
an expression for the spectral density Eq. (5.6). This allows us to explore
how the relative importance of the dielectric relaxation of the solvent, bound
water, and protein depends on the relevant length scales (the relative size of
the chromophore, the protein and the thickness of the layer of bound water)
and time scales (the dielectric relaxation times of the protein, bound water
and the solvent). We find that even when the chromophore is completely
surrounded by a protein it is possible that the ultra-fast solvation (on the
psec timescale) is dominated by the bulk solvent surrounding the protein.
Many experimentally obtained spectral densities can be fitted to a sum
of Lorentzians of the form
α2 ω
α1 ω
+
+ ... .
(5.11)
J(ω) =
1 + (ωτ1 )2
1 + (ωτ2 )2
Modelling Quantum Decoherence in Biomolecules
81
Fig. 5.3. Schematic plot of the spectral density for a typical chromophore on a log-log
scale. We see three distinct peaks, which can be attributed to the relaxation of the
protein, bound water and bulk solvent, respectively, with corresponding relaxation times
τp ,τb and τe .
For a protein that is large compared to the size of the binding pocket of the
chromophore and the width of the bound water layer, we find the spectral
density is described as the sum of three Lorentzians which correspond to the
dynamics of the protein, bound water and bulk water dynamics respectively.
This is shown schematically in Fig. 5.3. Key to this is the separation of
time scales, associated with the solvation coming from each of the three
components of the environment.
We have shown that to a good approximation, the spectral density is
given by,
αb ω
αs ω
αp ω
+
+
(5.12)
J(ω) =
1 + (ωτp )2
1 + (ωτb )2
1 + (ωτs )2
where the relaxation times can be expressed as:
τp
τD,p
τs
τD,s
=
2ǫp,i + 1
2ǫp,s + 1
(5.13)
=
2ǫe,i + 1
2ǫe,s + 1
(5.14)
τb = τD,b .
(5.15)
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Quantum Aspects of Life
The subscripts x = p, s, b refer to the protein, solvent, and bound water
respectively; and ǫx,s , ǫx,i , τD,x are the static dielectric constant, high frequency dielectric constant, and relaxation times of a Debye model for each
medium, although we use ǫe (ω) for the dielectric of the bulk solvent to
avoid confusion between subscripts. The reorganization energies associated
with each part of the environment are given by αi /τi , where
6(ǫe,s − ǫe,i )
αp
(∆µ)2
=
3
τp
2πǫ0 a (2ǫp,s + 1)(2ǫe,i + 1)
(5.17)
(ǫ2b,s + 2ǫ2e,s )(ǫb,s − ǫb,i )
.
ǫ2b,s (2ǫe,s + 1)2
(5.18)
αs
6(ǫe,s − ǫe,i )
(∆µ)2
=
3
τs
2πǫ0 b (2ǫs + 1)(2ǫe,i + 1)
αb
3(∆µ)2
=
τb
2πǫ0 b3
c−b
b
(5.16)
9ǫp,i
(2ǫp,i + 1)2
In particular, for typical systems the above three quantities can be of the
same order of magnitude, i.e.
αs
αb
αp
∼
∼
.
(5.19)
τs
τb
τp
Hence, the peaks of the spectral density can be of the same order of magnitude. This is because although each contribution is due to different dielectrics constants, they only have a limited range of values. This is supported by experimental data (see Table 5.5) where several relaxation times
are observed which vary by several orders of magnitude, but whose relative contributions are comparable. Therefore, in many cases only a single
component of the environment (protein, bound water, bulk solvent) will
be relevant to a given process. These expressions allow us to predict the
ultrafast solvation times in the presence of a protein, which we find may
increase the solvation time by at most a factor of two. We also suggest
that at least some of the studies which have identified ultrafast dielectric
relaxation of proteins [Homoelle et al. (1998); Riter et al. (1996)] may in
fact be detecting the fast response of the distant solvent.
5.5.
Obtaining the Spectral Density from Experimental
Data
The spectral function J(ω) associated with optical transitions in chromophores can be extracted from ultra-fast laser spectroscopy [Fleming and
Cho (1996)]. The time dependence of the Stokes shift in the fluorescence
Modelling Quantum Decoherence in Biomolecules
83
spectrum, where ν(t) is the maximum (or the first frequency moment) of
the fluorescence spectrum at time t, can be normalized as
ν(t) − ν(∞)
C(t) =
(5.20)
ν(0) − ν(∞)
such that C(0) = 1, and C(∞) = 0 when the fluorescence maxima has
reached its equilibrium value. This is related to the spectral density by
∞
J(ω)
cos(ωt)
(5.21)
C(t) =
dω
ER 0
ω
where ER is the total reorganization energy given in Eq. (5.35), which also
equals half the total Stokes shift.
The function C(t) is sometimes referred to as the hydration correlation function and experimental results are often fitted to several decaying
exponentials,
C(t) = A1 exp(−t/τ1 ) + A2 exp(−t/τ2 ) + A3 exp(−t/τ3 ) + . . .
(5.22)
where A1 +A2 +. . . = 1. From (5.21), this corresponds to a spectral density
of the form
α2 ω
α1 ω
+
+ ... .
(5.23)
J(ω) =
1 + (ωτ1 )2
1 + (ωτ2 )2
The dimensionless couplings αj (j = 1, 2, . . .) are related to the total reorganization energy by
2ER Aj τj
ER τj
αj =
≃ 0.25Aj −1
.
(5.24)
π
cm psec
Table 5.5 gives values of the fitting parameters (ER , Aj , τj ) determined
by fast laser spectroscopy for a range of chromophores and different environments, both protein and solvent. We do not claim the list is exhaustive
of all the published values, but is meant to be indicative [Riter et al. (1996);
Kennis et al. (2002)]. We note the following general features:
(i) The Stokes shift varies significantly between different environments,
both solvent and protein. Generally, the presence of the protein reduces the total Stokes shift and the relative contribution of the ultrafast component, which can be assigned to the solvent. The less exposed
the chromophore is to the solvent the smaller is solvent contribution to
the spectral density. This is also seen in measurements of the dynamic
Stokes shift for a chromophore placed at three different sites in the B1
domain of protein G. (See Fig. 3c of Cohen et al. (2002)). Denaturing
the protein tends to expose the chromophore to more solvent and increase the total Stokes shift and increase the relative contribution of
the ultrafast component.
84
Table 5.3.
Protein
Eosin
Eosin
Trp
Trp
Trp
Trp
Dansyl
DCM
Prodan
Prodan
Acrylodan
Acrylodan
Acrylodan
Coumarin 153
C343-peptide
Coumarin 343
Phycocyanobilin
Phycocyanobilin
MPTS
MPTS
bis-ANS
bis-ANS
4-AP
4-AP
none
lysozyme
none
SC
Rube
Monellin
SC
HSA
none
HSA
HSA
HSA
HSA
none
Calmodulin
none
C-phycocyanin
C-phycocyanin
none
Ab6C8
GlnRS (native)
GlnRS (molten)
GlnRS (native)
GlnRS (molten)
Solvent
Ref.
water
[Lang et al. (1999)]
water
[Jordanides et al. (1999(@)]
water
[Zhong et al. (2002)]
water
[Pal et al. (2002)]
water
[Zhong et al. (2002)]
water
[Peon et al. (2002)]
water
[Pal et al. (2002)]
water
[Pal et al. (2001)]
water
[Kamal et al. (2004)]
water
[Kamal et al. (2004)]
water
[Kamal et al. (2004)]
0.2M Gdn.HCl
[Kamal et al. (2004)]
0.2M Gdn.HCl
[Kamal et al. (2004)]
acetonitrile
[Changenet-Barret et al. (2000)]
water
[Changenet-Barret et al. (2000)]
water
[Jimenez et al. (1994)]
water
[Homoelle et al. (1998)]
water
[Riter et al. (1996)]
water
[Jimenez et al. (2002)]
water
[Jimenez et al. (2002)]
water
[Sen et al. (2003)]
urea soln.
[Sen et al. (2003)]
water
[Sen et al. (2003)]
urea soln.
[Sen et al. (2003)]
ER
(cm−1 )
A1 , τ1
(fsec)
877
710
0.15, 400
0.1, 310
0.2, 180
0.6, 800
0.17, 1000
0.46,1300
0.94, 1500
1440
1180
515
2313
916
1680
2200
250
1953
372
372
2097
1910
750
500
1330
700
A2 , τ2
(psec)
0.12, 3
0.1, 7
0.8, 1
0.4, 38
0.26, 12
0.54, 16
0.06, 40
0.25, 600
0.47, 130
0.53, 0.770
0.19, 780
0.56, 2.6
0.23, 710
0.41, 3.7
0.16, 280
0.36, 5.4
0.2, 120
0.55, 2
0.8, 100
0.2, 700
0.9, 100
0.1, 2.4
0.2, 126
0.35, 0.880
0.2, 100 ± 30 0.2, 6 ± 5
0.1 , > 10
0.8, 20
0.2, 0.340
0.85, 33
0.1, 2
0.45, 170
0.63, 60
0.85, 40
0.77, 50
A3 , τ3
(nsec)
0.57, 0.320
0.75, 10
0.25, 0.032
0.36, 0.057
0.48, 0.061
0.25, 0.0135
0.05, 0.067
0.55, 2.4
0.37, 0.96
0.15, 0.580
0.23, 0.9
Quantum Aspects of Life
Chromophore
Solvation relaxation times for various chromophores in a range of environments.
Modelling Quantum Decoherence in Biomolecules
85
Fig. 5.4. Hydration correlation function C(t) for Trp (light) and Trp-3 in monellin
(dark) in aqueous solution at 300 K [Nilsson and Halle (2005)].
(ii) The different decay times observed for a particular system can vary by
as many as four orders of magnitude, ranging from 10’s fsec to a nsec.
(iii) The relative contributions of the ultrafast (100’s fsec) and slow (10’s
psec) response are often of the same order of magnitude, consistent
with Eq. (5.19).
(iv) Even when the chromophores are inside the protein, the coupling of
the chromophore to the solvent is large. For example, Prodan is in
a hydrophobic pocket of HSA, well away from the surface, and yet
αs ∼ 50. Even for the “buried” chromophores (Leu7 and Phe30 ) in
GB1, [Cohen et al. (2002)] the solvent contribution is As ER ∼ 100
cm−1 , τs ∼ 5 psec, and so αs ∼ 100. There are several proteins
for which a very slow (∼ 10’s nsec) dynamic Stokes shift has been
observed and has been assigned to dielectric relaxation of the protein
itself [Pierce and Boxer (1992); Pal et al. (2001)].
86
5.6.
Quantum Aspects of Life
Analytical Solution for the Time Evolution of the
Density Matrix
Of particular interest is the time evolution of the reduced density matrix for
the TLS of interest interacting with the oscillator bath. Reina et al. (2002)
have studied the decoherence of quantum registers through independent
boson models. They consider the 2 × 2 reduced density matrix ρ(0) with
elements
ρ11 ρ12
(5.25)
ρ(0) =
ρ21 ρ22
for the TLS, and assume that the bath is initially uncoupled to the TLS
and in thermal equilibrium so that the initial density matrix for the whole
system is
ρ̃(0) = ρ0 (0) exp(−βHb )
(5.26)
where Hb is the Hamiltonian representing the bath, for the independent
boson model given by Hb = α ωα a†α aα . The density matrix at a later
time t is then given by
ρ(t) = Tr eiHt ρ̃(0)e−iHt
(5.27)
where H is the total independent boson Hamiltonian for the system, given
by Eq. (5.4). They show that the time dependent density matrix has matrix
elements [Reina et al. (2002)]
ρ11 (t) = ρ11 (0)
ρ12 (t) =
ρ22 (t) = ρ22 (0) = 1 − ρ11 (0)
ρ∗21 (t)
(5.28)
= ρ12 (0) exp(iǫt + iθ(t) − Γ(t, T ))
where θ(t) is a phase shift given by
∞
θ(t, T ) =
0
J(ω)
[ωt − sin(ωt)]
dω
ω2
and
(5.29)
ω
(1 − cos ωt)
(5.30)
2kB T
ω2
0
describes the decoherence due to interaction with the environment.
Provided kB T ≪ ωc , for an Ohmic spectral density and for low temperatures compared to the cut-off frequency (ωc ≪ kB T ), the decoherence
rate is approximately
1 + ωc2 t2
1
Γ1 (t, T ) ≈ α1 2ωT t arctan(2ωT t) + ln
.
(5.31)
2
1 + 4ωT2 t2
Γ(t, T ) =
∞
dωJ(ω) coth
Modelling Quantum Decoherence in Biomolecules
87
This rate shows three different regimes of qualitative behaviour depending
on the relative size of the time t to the time scales defined by 1/ωc and
/kB T . For short times ωc t < 1,
Γ(t, T ) =
t2
2τg2
where
1
=
τg2
∞
dωJ(ω) coth
0
(5.32)
ω
2kB T
(5.33)
and so there is a Gaussian decay of decoherence. For kB T ≫ ωc , this
reduces to
(5.34)
= 2ER kB T /
τg
where ER is the reorganization energy given by
ER =
1
π
∞
0
J(ω)
dω .
ω
(5.35)
For an Ohmic spectral density of the form J(ω) = αω/[1 + (ω/ωc )2 ] one
obtains for intermediate times (the quantum regime [Unruh (1995)]),
Γ(t, T ) ≈ α ln(ωc t)
(5.36)
and for long times (t ≫ /kB T , the thermal regime) the decoherence is
linear in time,
Γ(t, T ) ≈ 2αkB T t/ .
(5.37)
as might be expected from a golden rule type calculation.
5.7.
Nuclear Quantum Tunnelling in Enzymes and the
Crossover Temperature
General tunnelling problems can be investigated by employing complextime path integrals [Hänggi et al. (1990); Weiss (1999)]. The functionalintegral representation of quantum mechanics pioneered by Feynman lends
itself to this treatment of the problem. Consider the partition function
Z = Tr{exp(−βH)} ,
(5.38)
here H denotes the full Hamiltonian operator corresponding to the system plus environment. Following [Feynman (1972)] this quantity can be
88
Quantum Aspects of Life
expressed in the form of a functional path integral over the tunnelling coordinate x(τ ), here τ = it is a real variable. This integral runs over all
paths that are periodic with period θ = β. Accordingly each trajectory
x(τ ) is weighted by the Euclidean action SE . In the Feynman path integral
formulation of quantum mechanics the transition probability between two
states involves the square of a transition amplitude which is the sum of all
possible paths joining those two states. At finite temperatures the path integral is dominated by the extrema of the imaginary time action. Pursuing
this analysis for the Caldeirra-Legget Hamiltonian shows that a non-trivial
periodic solution (which has been dubbed the bounce solution) only exists below a certain crossover temperature T0 [Hänggi et al. (1990)]. The
bounce solution is associated with tunnelling from the reactant to product
well. For temperatures T > T0 there is no oscillation of the particle in the
classically forbidden regime.
For temperatures T > T0 the role of the bounce solution is taken over by
the constant solution (xe (τ ) = xb ) where the particle sits at the barrier top.
In this kinetic regime there are still rate enhancements from quantum effects
but in this temperature regime the bounce solution which is associated with
conventional tunnelling does not exist. Figure 5.5 depicts the different
kinetic regimes.
Fig. 5.5. Different kinetic regimes as a function of temperature [Hänggi et al. (1990)].
At temperatures much greater than the crossover temperature T0 thermal activation
dominates as the means of getting over the energy barrier. As the temperature decreases
the dynamics become influenced by quantum effects which manifest themselves as a
correction to the classical rate expression. When the temperature is less than T0 the
dynamics becomes dominated by tunnelling since the effect of thermal hopping vanishes
[Hänggi et al. (1990)].
Modelling Quantum Decoherence in Biomolecules
89
The crossover temperature T0 is defined as
T0 = µ(2πkB )−1 = (1.216 × 10−12 secK)µ ,
(5.39)
0 = µ2 − ωb2 + µγ̂(µ) .
(5.40)
where µ is the effective barrier frequency which is be obtained by finding
the positive root of the following equation
Here γ̂(z) is simply the Laplace transform of the friction kernel defined in
Eq. (5.7). It is simply another means of representing the spectral density.
This equation shows that the effective barrier frequency depends on the
nature of the coupling to the environment. In the case of a Lorentzian
spectral density it takes the explicit form of
µωD γ
.
(5.41)
0 = µ2 − ωb2 +
ωD + µ
As T0 depends linearly on µ, one can gain an understanding of how T0 varies
as a function of the friction strength for different bath response frequencies
from Fig. 5.6. This figure shows how the positive root of Eq. (5.41) changes
as a function of the scaled friction and bath response frequency.
It is worthwhile to examine how the effective barrier frequency changes
in the different limits of the bath response. In the limit where ωD ≫ 2πT
the barrier frequency does not depend on the response frequency of the
bath and is essentially Ohmic. In this case the effective barrier frequency
takes the form of
µ=
ωb2 + γ 2 /4 − γ/2 .
(5.42)
In the other limit where µ ≫ ωD the effective barrier frequency becomes
µ = ωb2 − γωD .
(5.43)
In both cases an increase in the strength of the damping, γ, reduces
the effective barrier frequency and hence the crossover temperature. When
µ ≫ ωD increasing the response frequency of the bath also decreases the
crossover temperature. In the absence of any dissipative interaction with
the environment (γ = 0), the crossover temperature assumes the maximum
value of
T0 = ωb (2πkB )−1 .
(5.44)
This is an upper limit on the crossover temperature and any interaction
with the environment always tends to decrease it. This is a direct manifestation of the effect that decoherence has on the system showing that as the
environment more strongly interacts with the hydrogen being transferred
you need to go to lower and lower temperatures to see any quantum effects.
90
Quantum Aspects of Life
Fig. 5.6. The scaled crossover temperature as a function of the friction strength for a
Lorentzain spectral density. The scaled crossover temperature is the ratio of the actual
crossover temperature and the theoretical upper limit where there is no friction. The
friction strength is the dimensionless parameter γ/2ωb . The different plots show how
the relative size of the response frequency of the bath, ωD , and the barrier frequency, ωb
change the influence that friction strength has on the crossover temperature. These show
that in the limit where ωD /ωb → ∞ the crossover temperature becomes very sensitive
to the friction strength. However as the ratio is reduced the strength of the dependence
decreases.
5.8.
Summary
In the preceding sections we have given an illustrative guide to approaches
that we and others have taken to modelling decoherence in biomolecules. By
taking a minimalist approach one is able to capture a significant fraction of
the essential physics that describes how significant quantum effects can be
to the in vivo function of biomolecules. The results clearly show that interaction with the environment suppresses the significance of quantum effects
like interference and tunnelling. In the case of biological chromophores increasing the strength of interaction with the environment directly increases
the decoherence rate. Moreover, in the case of hydrogen transfer reactions
in enzymes significant interaction with the environment deceases the temperature at which quantum effects impinge on the kinetics to well outside
Modelling Quantum Decoherence in Biomolecules
91
the biologically relevant range. This shows that modelling the decoherence,
which a particular biomolecule is subject to, is of the utmost importance if
one is to determine whether quantum mechanics plays a non-trivial role in
its biological functionality.
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About the authors
Jacques Bothma is currently an MPhil student in the Physics department
at the University of Queensland. He completed his BSc in Physics at the
University of Queensland in 2006. His research interests are in the areas of
theoretical and experimental biological physics.
Joel Gilmore is a science communicator with the University of Queensland, and coordinator of the UQ Physics Demo Troupe. He completed
a PhD in Physics at the University of Queensland in 2007, under Ross
McKenzie, researching minimal models of quantum mechanics in biological
systems.
Ross McKenzie is a Professorial Research Fellow in Physics at the University of Queensland. He was an undergraduate at the Australian National
University and completed a PhD at Princeton University in 1988, under
Jim Sauls. His research interests are in the quantum many-body theory of
complex materials ranging from organic superconductors to biomolecules
to rare-earth oxide catalysts.
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PART 3
The Biological Evidence
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Chapter 6
Molecular Evolution: A Role for
Quantum Mechanics in the Dynamics
of Molecular Machines that Read &
Write DNA
Anita Goel
6.1.
Introduction
Are the dynamics of biological systems governed solely by classical physics
or could they somehow be influenced by quantum effects? New developments from the convergence of physics, biology, and nanotechnology are
leading us to critically reexamine our conventional assumptions about the
role of quantum physics in life and living systems. Schrödinger in his 1944
book What is Life? [Schrödinger (1967)] questioned whether the laws of
classical physics were really sufficient to understand life. Schrödinger also
wondered whether life, at the most fundamental level, could somehow be a
quantum phenomena or at least be influenced by quantum effects? In recent times, such lines of inquiry have, for the most part, been dismissed by
mainstream scientists because biological systems are wet and swampy, complex macroscopic systems, where it is presumed that quantum coherences
would be destroyed much before their effects become relevant to biological
processes. It is widely accepted, however, that quantum mechanics must
play some role, albeit a trivial one, in life; namely the electronic structures
of biomolecules are determined as per the laws of quantum chemistry.
My own quest to understand the physics of living systems is driven,
in part, by an inner, intuitive conviction that twentieth century physics
developed in the context of inanimate matter has not yet adequately come
to terms with life and living systems. Living systems provide an excellent
Received May 1, 2007
97
98
Quantum Aspects of Life
laboratory to probe the interplay of matter, energy, and information. I
have for many years been fascinated with molecular machines that read
and write information into molecules of DNA. These nanomotors (matter )
transduce chemical free energy into mechanical work as they copy biological
information stored in a DNA molecule. These motors can be thought of as
information processing machines that use information in their environment
to evolve or adapt the way they read out DNA. In ways yet unknown to
us, information from the environment can couple into and modulate the
dynamics of these nanomachines as they replicate or transcribe genetic
information.
For the past several years, I have been seeking, with the aid of fundamental physics concepts and emerging experimental tools, to identify and
elucidate the various “knobs” in a motor’s environment that can exert control on its dynamics as it replicates or transcribes the genetic code. Here,
I heuristically examine the role that quantum mechanics may play in influencing the dynamics of the motors as they read/write bits of DNA. I
begin by discussing how Wigner’s inequalities for a quantum clock impose
fundamental constraints on the accuracy and precision with which these
nanomotors can read or write DNA. Contrary to implicit assumptions, I
discuss how the relaxation times of DNA polymer molecules can be quite
long, and hence lead to decoherence times that are long compared to the
timescale of the internal state transitions in the motor and relevant compared to the timescale associated with the motor reading a DNA base.
Thus, we argue that it is entirely plausible for quantum effects to influence not only the structure but also the dynamics of biomolecular motors.
Lastly, some broader implications of this work are discussed.
6.2.
Background
Nature packs information into DNA molecules with remarkable efficiency.
Nanometre-sized molecular engines replicate, transcribe, and otherwise process this information. New tools to detect and manipulate single molecules
have made it possible to elicit how various parameters in the motor’s microenvironment can control the dynamics of these nano-motors (i.e. enzymes). At small length scales, noise plays a non-negligible role in the
motor’s movement along DNA.
Biological information in DNA is replicated, transcribed, or otherwise processed by molecular machines called polymerases. This process of
Molecular Evolution: A Role for Quantum Mechanics in Molecular Machines
99
Fig. 6.1. Single molecule experiments reveal that mechanical tensions above 35 pN
on the DNA induce a velocity reversal in the T7 DNA polymerase (DNAp) motor. A
single DNA molecule is stretched between two plastic beads, while the motor catalyzes
the conversion of ss to dsDNA. The speed of polymerization or formation of dsDNA is
denoted by Vpoly , while the reverse excision of dsDNA (n → n − 1) is denoted by Vexo .
Inset: Crystal structure of T7 complex indicates that the polymerase and exonuclease
activities (forward and reverse “gears”) of the motor arise from structurally distinct
catalytic domains.
reading and writing genetic information can be tightly coupled or regulated
by the motor’s environment. Environmental parameters (like temperature,
nucleoside concentrations, mechanical tension of template, etc.) [Goel et
al. (2003, 2002); Wuite et al. (2000)] can directly couple into the conformational dynamics of the motor. Theoretical concepts in concert with
emerging nanotools to probe and manipulate single molecules are elucidating how various “knobs” in a motor’s environment can control its real-time
dynamics. Recent single molecule experiments have shown, for example,
that increasing the mechanical tension applied to a DNA template can appreciably “tune” the speed at which the motor enzyme DNA polymerase
(DNAp) replicates DNA. In addition to the tuning effect, a tension-induced
reversal in the motor’s velocity has been found to occur at high stretching
forces (i.e. above ∼ 35 pN). See Fig. 6.1.
100
Quantum Aspects of Life
We have been working to understand how mechanical tension on a
DNA polymer can control both the “tuning” and “switching” behaviour
in the molecular motor. The tension-induced switching observed in single
molecule experiments is similar to the natural reversal that occurs after
a mistake in DNA incorporation, whereby the reaction pathways of the
biochemical network are kinetically partitioned to favour the exonuclease
pathway over the polymerase one. We seek to develop a framework to
understand how environmental parameters (like tension, torsion, external
acoustic or electromagnetic signals) can directly couple into the conformational dynamics of the motor. By understanding how these various perturbations affect the molecular motor’s dynamics, we can develop a more
holistic picture of their context-dependent function. These motors are fundamentally open systems and very little is understood today about how
their (local and global) environment couples into their function. Viewing
the motor as a complex adaptive system that is capable of utilizing information in its environment to evolve or learn may shed new light on how
information processing and computation can be realized at the molecular
level.
As it becomes possible to probe the dynamics of these motors at increasingly smaller length and time scales, quantum effects, if relevant, are more
likely to become experimentally detectable. Paul Davies (2004) has very elegantly posed the question “Does quantum mechanics play a non-trivial role
in life?” and whether quantum mechanics could somehow enhance the information processing capabilities of biological systems. Here we revisit such
fundamental questions in the context of examining the information processing capabilities of motors that read DNA. We use Wigner’s relations for a
quantum clock to derive constraints on the information processing accuracy
and precision of a molecular motor reading DNA. In order for this motor to
process information quantum mechanically, it must have long decoherence
times. Here we also calculate the decoherence time for our motor-DNA
system.
6.3.
Approach
In the late 1950’s Eugene Wigner showed how quantum mechanics can limit
both the accuracy and precision with which a clock can measure distances
between events in space-time [Wigner (1957)]. Wigner’s clock inequalities
can be written as constraints on the accuracy (maximum running time
Molecular Evolution: A Role for Quantum Mechanics in Molecular Machines
101
Tmax ) and precision (smallest time interval Tmin) achievable by a quantum
clock as a function of its mass M , uncertainty in position λ, and Planck’s
constant :
M λ2
(6.1)
Tmax <
Tmax
.
(6.2)
Tmin >
M c2 Tmin
> 1) than
Wigner’s second constraint is more severe (by a factor TTmax
min
the Heisenberg uncertainty principle, which requires that only one single
simultaneous measurement of both energy (E = M c2 ) and the time Tmin
be accurate. Wigner’s constraints require that repeated measurements not
disrupt the clock and that it must accurately register time over the total
running period Tmax . An intuitive way of saying this is that a Wigner
clock must have a minimum mass so that its interaction with a quantum
of light (during the measurement of a space-time interval) does not significantly perturb the clock itself. Wigner suggested that these inequalities
(Eqs. (6.1) and (6.2)) should fundamentally limit the performance of any
clock or information processing device [Barrow (1996)], even “when the
most liberal definition of a clock” is applied [Wigner (1957)].
These inequalities have elegantly been applied by John Barrow (1996) to
describe the quantum constraints on a black hole (a rather “liberal definition for a clock”). He shows that the maximum running time for a black hole
(Tmax in Wigner’s relations) corresponds to its Hawking lifetime and that
Wigner’s size constraints are equivalent to the black hole’s Schwarzchild
radius. Furthermore, Barrow demonstrates that the information processing power of a black hole is equivalent to its emitted Hawking radiation.
Wigner inequalities should likewise provide nontrivial constraints on the
performance of any information processing nanomachine or time-registering
device.
Here we heuristically examine the ramifications of these limits on the
capability of a nanomachine to read DNA. We assume that λ is the uncertainty in the motor’s position along the DNA (linear span over which
it processes information) and can be estimated by the length of the DNA
molecule (e.g. ∼ 16 µm for lambda–phage DNA used in typical single
molecule experiments). Then Eq. (6.1) gives Tmax < 387 sec as the maximum running time for which the motor can reliably run and still be accurate. For comparison, the error rate for a polymerase motor from the
species Thermus aquaticus (TAQ) is about 1 error every 100 sec. Likewise,
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Quantum Aspects of Life
Eq. (6.2) gives Tmin > 5 × 10−14 sec as the motor’s precision, or the minimum time interval that it can measure. A plausible interpretation of this
value is that Tmin corresponds not to the motor incorporating one base but
to the motor undergoing an internal state transition. Note this time corresponds well to the timescale for the lifetime of a transition state [Peon and
Zewail (2001)].
These Wigner constraints can also be written in terms of Lmax , the
maximum readout length over which the motor is accurate, and Lmin, the
minimum effective step size of the motor. If the motor’s speed vmotor ∼ 100
bases/sec, then Lmax = vmotor x Tmax ∼ 4 x 104 bases. This compares
reasonably well with known error rates of the DNA polymerase motor. For
example, a TAQ polymerase is known to make about one mistake once for
every 104 bases it reads. Likewise, Lmin = vmotor x Tmin ∼ 5 x 10−12 bases,
which corresponds to about 2 x 10−21 m. This is the effective step size or
the minimum “distance” interval that can be accurately registered by the
motor. This linear coordinate corresponds to the time associated with the
fastest internal state transition within the motor and indicates a minimum
lengthscale over which the motor can register information.
6.3.1.
The information processing power of a
molecular motor
E
required by any information processor can be calcuThe power P ≡ Tmax
lated using Wigner’s second clock inequality [Barrow (1996)]. Analogous to
Barrow’s treatment for a quantum black hole, we can estimate the motor’s
information processing power as
E
−2
2
= (Tmin ) = (ν) ,
(6.3)
Tmax
−1
where ν = Tmin
is the fastest possible information processing frequency of
the motor. For a theoretical estimate of the precision Tmin ∼ 5 × 10−14 sec,
this corresponds to a power of Pinf ∼ 4 × 10−8 J/ sec.
The maximum number of computational steps carried out by the motor
∼ 7 × 1015 . Note the number of computacan be estimated as m = TTmax
min
tional steps the motor performs are much larger than the number of bases
(4 × 104 ) that the motor actually reads in the running time Tmax . Thus
each base in the DNA molecule corresponds to roughly about 2 × 1011
computational (information processing) steps carried out by the motor.
Molecular Evolution: A Role for Quantum Mechanics in Molecular Machines
103
In comparison, the power actually generated by the motor Pout can be
estimated using experimental force vs velocity data [Wuite et al. (2000);
Goel et al. (2003)] as
Pout = f × v ∼ 5pN × 100bases/ sec ∼ 1.5 × 10−19 J/ sec .
(6.4)
If a motor molecule consumes 100 NTP (i.e. nucleotide triphosphate
fuel) molecules per second, this corresponds to an input power Pin ∼ 8 ×
∼ 20%. From the
10−19 J/ sec and a thermodynamic efficiency ǫ = PPout
in
actual power Pout generated, we can better estimate the actual precision
of our motor Tmin measured to be about 26 n sec. This suggests that the
actual number of computational steps taken by our motor during its longest
running period is about
mmeasured =
Tmax
∼ 1010 ,
Tmin measured
(6.5)
which means about 3 × 105 computational steps are taken (or information
bits processed) for every DNA base that the motor reads. Each of the
internal microscopic states of the motor or clock can store information.
This leads to dramatically higher information storage densities than if the
information were stored solely in the DNA molecule itself.
As discussed above, the first Wigner relation imposes constraints on the
maximum timescales (and length scales) over which DNA replication is accurate or, in other words, remains coherent. The second Wigner inequality
sets limits on the motor’s precision and its information processing power.
By viewing the motor as an information processing system, we also calculated the number of computational steps or bits required to specify the
information content of the motor-DNA system. However, in order for quantum mechanics to play a more proactive role in the dynamics of these motors
(i.e. beyond just imposing the above constraints on their performance), it
is critical that the decoherence time (τD ) of the motor-DNA complex be
much longer than the motor’s base reading time (τbase reading ∼ 10 milliseconds). This decoherence time (τD ) denotes a time scale over which
quantum coherence is lost.
6.3.2.
Estimation of decoherence times of the
motor-DNA complex
We consider the motor-DNA complex as a quantum system moving in one
dimension where the environment is a heat bath comprised of a set of n
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Quantum Aspects of Life
harmonic oscillators, each vibrating with a given frequency and a given
coupling strength between oscillations. Then, using an expression derived
by Zurek (1991), we can estimate the decoherence time of this system as
2
λT
(6.6)
t D = tr
∆x
where the motor of mass M is in a superposition of two position states that
are separated spatially by ∆x. This effective interstate spacing ∆x can be
estimated as L/mmeasured ∼ 10−15 metres, for a motor that takes roughly
1010 computational steps while reading a 16 micron long molecule of DNA.
The thermal de Broglie wavelength λT for a motor in equilibrium with its
surrounding heat bath is estimated as
2π
∼ 3 × 10−13 m .
(6.7)
λT =
mkB T
When the thermal de Broglie wavelength is much smaller than the distance
∆x, the motor-DNA system will behave as a classical system. On the other
hand, when the thermal de Broglie wavelength is on the order of, or larger
than the spacing ∆x, quantum effects will predominate. Thus, for our
motor-DNA complex, Eq. (6.6) gives tD ∼ 8.4 × 104 tr .
The spectrum of relaxation times for a DNA polymer vary with the
length of the molecule and can be estimated from the Zimm model [Grosberg and Khokhlov (1994)] and have been experimentally verified [Perkins
et al. (1994)] to range from microseconds to milliseconds. For instance,
the slowest relaxation time for a DNA polymer chain of length L and
persistence length P can be approximated via the Zimm model as
3
6 (LP ) 2 η
.
(6.8)
tr = 2
π kB T
This corresponds, for a typical ∼16 µm long lambda-phage DNA
molecule, to the longest relaxation time being about 500 milliseconds for
double-stranded DNA and about 3 milliseconds for single stranded DNA.
With the longest DNA relaxation times being in the milliseconds, the corresponding longest decoherence times, Eq. (6.8), of the motor-DNA complex
will range from several minutes to several hours. This easily satisfies the
condition that tD ≫ τbase reading . Thus, it is indeed quite possible that
quantum mechanical effects play a proactive role in influencing the dynamics of motors reading DNA.
Molecular Evolution: A Role for Quantum Mechanics in Molecular Machines
6.3.3.
105
Implications and discussion
The heuristic exercise above leads to a few intriguing implications.
(1) The first Wigner inequality sets fundamental constraints on the motor’s
accuracy for reading DNA. Our numerical estimates are comparable to
known error rates of the polymerase motor.
(2) The second Wigner inequality sets fundamental constraints on the motor’s precision and information processing power. This exercise suggests
that the information content or the number of bits stored in a DNAmotor system is much larger than that typically assumed (1 bit per
base). This increase in information storage density results from the
motor itself having several internal microscopic states. Conventionally,
information in DNA is seen as being stored in the DNA bases itself.
This work, in contrast, suggests that DNA, the replicating motors,
and their environment comprise a dynamic and complex informationprocessing network with dramatically higher information storage and
processing capabilities.
(3) The power of information processing was compared to the actual power
generated by the motor as it consumes energy to mechanically move
along a DNA track. Molecular motors provide an excellent laboratory
for probing the interplay of matter, energy, and information. These
molecules (matter) transduce chemical free energy into mechanical work
with remarkably high efficiency; and (in ways as of yet unknown to us)
information from their environment plays a critical role in controlling
or modulating their dynamics. What is needed is a more rigourous
conceptual framework, where the molecular motor’s dynamics can be
intrinsically and strongly coupled to its exchange of information and
energy with its environment.
(4) The decoherence times for the motor-DNA system was found to be on
the order of minutes to hours, paving the way for quantum mechanics
to play a non-trivial role. In order for quantum effects on the motor
dynamics along DNA to enter the realm of experimental detection, a
few prerequisites must be met: i) the decoherence times must indeed
be sufficiently long; ii) single molecule experiments must be carefully
designed so they do not destroy coherences; and iii) we should look
more seriously for emergent macroscopic quantum effects, including for
instance evidence for quantum information processing occurring within
these molecular systems.
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Quantum Aspects of Life
There is fervent interest in developing technologies that can store, retrieve, and process information at the nanoscale. Biological systems have
already evolved the ability to efficiently process remarkable amounts of information at the nanoscale. Perhaps futuristic quantum information technologies could find their best realization yet in the context of biomolecular
motors.
In his classic book, What is Life, Schrödinger first speculated that quantum mechanical fluctuations could give rise to mutations. In more recent
times, McFadden and Al-Khalili (1999) describe how quantum mechanics may provide a mechanism for understanding “adaptive mutations”—
i.e. mutations that are not purely random but are driven by environmental
pressures. We have discussed how Wigner relations limit the accuracy with
which polymerase motors can copy DNA. This suggests that mutations are
fundamentally built into the replication machinery. We have also argued
that these complex macromolecular systems can have decoherence times
that are long compared to the timescale associated with the motor reading
a DNA base, suggesting that quantum features are not really destroyed
in these systems. We can dare to speculate and ask some provocative
questions: Could quantum noise or fluctuations perhaps give rise to mistakes made during the motor’s copying of the DNA? Since these motors
propagate genetic information, such molecular mistakes made during DNA
replication lead to mutations in the organism. Could the environment be
somehow deeply entangled with the dynamics of these molecular motors
as they move along DNA? Could information embedded in the motor’s environment somehow modulate or influence its information processing, and
hence how it reads the DNA bases? Could the environment somehow be
selectively driving evolution and if so could it be that evolution, at least at
the molecular level, is more Lamarckian than it is Darwinian?
As fields like nanotech, biotech, and quantum information processesing
come together and new fields like quantum biology are born, it will become more fashionable to ask such questions and increasingly possible to
experimentally address them.
References
Anglin, J. R., Paz, J. P., and Zurek W. H. (1996). Deconstructing Decoherence.
quant-ph/9611045v1.
Barrow, J. D. (1996). Wigner inequalities for a black hole. Physical Review D 54,
pp. 6563–6564.
Molecular Evolution: A Role for Quantum Mechanics in Molecular Machines
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Davies, P. C. W. (2000). The Fifth Miracle: The Search for the Origin and Meaning of Life (Touchstone).
Davies, P. C. W. (2004). Does quantum mechanics play a non-trivial role in life?,
BioSystems 78, pp. 69–79.
Goel, A., Ellenberger, T., Frank-Kamenetskii, M. D., and Herschbach, D. R.
(2002). Unifying themes in DNA replication: reconciling single molecule
kinetic studies with structrual data on DNA polymerases, J. Biolmolecular
Structure and Dynamics 19.
Goel, A., and Herschbach, D. R., 2–4 June 2003. Controlling the speed and direction of molecular motors that replicate DNA, Proc. SPIE Fluctuations
and Noise in Biological, Biophysical, and Biomedical Systems, Eds. Sergey
M. Bezrukov, Hans Frauenfelder, and Frank Moss, Santa Fe, New Mexico,
USA 5110 pp. 63–68 (Santa Fe, New Mexico).
Goel, A., Astumian, R. D., and Herschbach, D. R. (2003). Tuning and switching
a DNA polymerase motor with mechanical tension, Proc. of the National
Academy of Science, 100 pp. 9699–9704.
Grosberg, A. Y., and Khokhlov, A. R. (1994). Statistical Physics of Macromolecules eds. Larson, R. and Pincus, P.A. (AIP Press).
McFadden, J., and Al-Khalili, J. (1999). A quantum mechanical model of adaptive
mutation, BioSystems 50, pp. 203–211.
Peon, J., and Zewail, A. H. (2001). DNA/RNA nucleotides and nucleosides: direct measurement of excited-state lifetimes by femtosecond fluorescence upconversion, Chem. Phys. Lett. 348, p. 255.
Perkins, T. T., Quake, S. R., Smith, D. E., and Chu, S. (1994). Relaxation
of a single DNA molecule observed by optical microscopy, Science 264,
pp. 822–826.
Salecker, H., and Wigner, E. P. (1958). Quantum limitations of the measurement
of space-time dimensions, Physical Review 109, pp. 571–577.
Schrödinger, E. (1967). What is Life? (Cambridge University Press).
Scully, M. O., and Zubairy, M. S. (1997). Quantum Optics (Cambridge University
Press).
Wigner, E. P. (1957). Relativistic invariance and quantum phenomena. Reviews
of Modern Physics 29, pp. 255–268.
Wuite, G. J. L., Smith, S. B., Young, M., Keller, D., and Bustamante, C. (2000).
Single-molecule studies of the effect of template tension on T7 DNA polymerase activity, Nature 404, pp. 103–106.
Zurek, W. H. (1991). Decoherence and the transition from quantum to classical,
Phys. Today 43, pp. 36–44.
About the author
Anita Goel holds both a PhD in Physics from Harvard University under
Dudley Herschbach, and an MD from the Harvard-MIT Joint Division of
Health Sciences and Technology (HST) and BS in Physics from Stanford
108
Quantum Aspects of Life
University. She was named in 2005 as one of the world’s “top 35 science and
technology innovators under the age of 35” by the MIT Technology Review
Magazine. Goel is the Chairman, and Scientific Director of Nanobiosym
Labs and the President and CEO of Nanobiosym Diagnostics, Inc. Goel
also serves as an Associate of the Harvard Physics Department, an Adjunct
Professor at Beyond: Center for Fundamental Concepts in Science, Arizona
State University; a Fellow of the World Technology Network; a Fellow-atLarge of the Santa Fe Institute; and a Member of the Board of Trustees
and Scientific Advisory Board of India-Nano, an organization devoted to
bridging breakthrough advances in nanotechnology with the burgeoning
Indian nanotech sector.
Chapter 7
Memory Depends on the
Cytoskeleton, but is it Quantum?
Andreas Mershin and Dimitri V. Nanopoulos
7.1.
Introduction
One of the questions the book you are reading debates is whether there
is a non-trivial place for quantum physics in explaining life phenomena.
This chapter will shed light on the possible role quantum mechanics may
play in the cognitive processes of life, in particular memory encoding, storage, and retrieval that represents a subsection of the nearly thirty-five year
old “Quantum Mind” theory or, more accurately, the loosely-connected
collection of numerous and generally experimentally unsupported notions
regarding the importance of quantum effects in consciousness.
From its inception, this space of human enquiry has been populated by
bright and open minded folk, to whom it usually represented a secondary
field to their areas of expertise, most famously mathematician Sir Roger
Penrose, who along with anesthesiologist Stuart Hameroff are responsible
for the most widely known incarnation of the Quantum Consciousness Idea
(QCI) they title “Orch-OR” to stand for orchestrated objective reduction
(by gravity) of the wavefunction of microtubules (MTs) inside neural cells
[Hameroff (1998)]. This presents perhaps the most far-reaching conjecture
about consciousness to date as it bridges the quantum realm of atoms and
molecules to the cosmic scale of gravity through the human brain. It is
suspiciously anthropocentric and brings to mind times when the Earth was
considered the centre of the universe (here, human consciousness is at the
Received May 9, 2007
109
110
Quantum Aspects of Life
centre of the infamously difficult to bridge gap between quantum theory
and general relativity). At the same time, one can argue that this is just
the anthropic principle in action and it is only natural for consciousness,
which occupies itself with the study of both the atomic and the cosmic to
be the very centre of the two [Nanopoulos (1995)].
This space has also been highly attractive to people who are eager to
believe that the currently mysterious physical phenomenon of consciousness
can be somehow automatically tamed when coupled to the famously mysterious and counterintuitive quantum theory. This is similar to the “god
of the gaps” strategy of assigning a deity to poorly understood phenomena until they are adequately explained by science—from Zeus the god of
lighting thousands of years ago to the “intelligent designer” heavily and
implausibly marketed by some as responsible for the origin and complexity
of life even now.
Before continuing, we will put forth two necessary and we feel entirely
reasonable assumptions: 1) the phenomenon of consciousness depends on
(brain) cellular processes [Crick (1994)] and 2) any “quantum effects” such
as non-locality, coherence and entanglement at the macromolecular or cellular scales must originate at the atomic and molecular scales (where quantum
effects are commonplace) and be somehow “amplified” in both size and time
span.
Instances of such amplification of quantum behaviour to the macroscopic world (in non-biological systems) have started to appear in recent
years albeit using very intricate apparata including either very low temperatures [Corbitt et al. (2007)] or very large collection of particles at room
temperature that are put into a superposition of states such that decoherence takes a long time [Julsgaard et al. (2001)]. While these experiments
are very far from being biologically relevant they are a clear indication that
science and technology are moving in the direction of amplifying quantum
behaviour with all its weirdness in tow to the meso- and macro- worlds.
One of the key objections to all manners of QCIs by skeptics such as
Max Tegmark (2000) has been that the brain is a “wet, warm and noisy
environment” where any amplification of atomic scale quantum effects is
washed away and his QCI-damning quotation in an article in Science [Seife
(2000)] still ranks amongst the first hits of a Google search for “quantum
mind” and is featured prominently in the Wikipedia lemma for the same
keyword. These two measures accurately reflect the current scientific consensus on QCI: it is worth mentioning but it cannot yet be taken seriously.
Memory Depends on the Cytoskeleton, but is it Quantum?
111
In fact, consciousness research even without any controversial quantum
hypotheses is mired in taboo. In some learned neuroscience circles, consciousness is referred to as the “unmentionable C-word” indicating there is
something deeply wrong with its study. Clearly, the reason for the taboo
has to do with the fact that it is a difficult and unexplored subject with fundamental philosophical implications (e.g. free will) and is so very personal
to many. Additionally, we think that part of the blame for this sad state
lies with the unchecked proliferation of quantum brain hypotheses, and
their adoption by pseudoscientists, new age enthusiasts, and others seeking
to validate their shaky reasoning by association with the well-established
quantum theory. Sometimes, the most prolific of these receive well-deserved
prizes for books with titles such as Quantum Healing [IgNobel (1998)].
We feel that the QCI is worth more than a passing mention in current
neuroscience and in some of its more realistic incarnations, exhibits bold
and interesting, experimentally testable predictions. But the field is sick
and the best cure for it is not to avoid it completely [Dennett (1991)] but
to perform well-controlled experiments and cull this space of the gratuitous assumptions, while at the same time illuminate the possibilities of
non-trivial quantum effects playing a non-trivial role in life’s processes. To
clarify, there are certainly a slew of quantum phenomena important to biology, for instance the quantum nature of photon absorption by rhodopsin
in the retina that leads to the—classical as far as we know—cascade of
G-protein release and subsequent neural membrane depolarization and action potential propagation. While this type of phenomenon starts rooted
in the quantum realm, it quickly becomes classical and does not amplify
any of the quantum weirdness and for that reason we consider it “trivial.”
An example of a non-trivial quantum phenomenon would be the experimental observation of quantum coherence, non-locality or—the clearest
case—entanglement occurring at a biologically relevant time and size scale
as we describe in more detail below. But first, we would like to touch upon
the motivation behind looking at quantum physics to explain the brain.
7.2.
Motivation behind Connecting Quantum Physics
to the Brain
While different scientists have different motivations behind connecting
quantum physics to consciousness [Penrose (1989)], it seems that a unifying trend in the community is the argument that “qualia” [Koch (2004)]
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Quantum Aspects of Life
and the “binding problem” (i.e. the problem of how the unity felt while
perceiving the results of highly distributed activities of the nervous system is achieved), cannot be explained by classical physics alone and the
weirdness of quantum mechanics (usually superposition of states and quantum entanglement) is needed [Penrose (1994)]. Additionally, the biological
brain’s unsurpassed ability at pattern recognition and other massively parallel processes is also sometimes alluded to in comparison to the algorithms
of Shor (1995) and Grover (1996) that afford “super powers” to quantum
computers.
As far as the authors can discern, the basic reasoning behind such assertions stems from the observation that just like Schrödinger’s cat is both
alive and dead until measured, so is a mental state undetermined until a
collapse of the superposed wavefunctions (presumably each representing a
different mental state). Quantum entanglement is sometimes employed to
explain the “differentiated yet integrated” [Edelman et al. (1987)] aspect of
experience in general and memory in particular where different parts of a
memory are encoded in macroscopically separated parts of the brain while
when one remembers, everything is integrated into a coherent whole. Both
quantum superposition and entanglement are usually alluded to when arguing that the brain is too ill-equipped and regular neurotransmission too slow
to achieve these spectacular results unless it works as a quantum computer.
Qualia and binding are fascinating philosophical and scientific problems
and approaching them from a quantum mechanical perspective is provocative. Amongst others, we have in the past conjectured a possible mechanism
for quantum entanglement to lead to correlated neural firing [Mavromatos
et al. (2002)] thus addressing aspects of the binding problem. To do this,
we approximated the internal cavity of microtubules (MTs) as quantum
electrodynamic (QED) cavities and proposed that if this turns out to be
an experimentally verified assumption, nature has provided us with the
necessary MT structures to operate as the basic substrate for quantum
computation either in vivo, e.g. intracellularly, or in vitro, e.g. in fabricated
bioqubit circuits. This would mean that we could in principle construct
quantum computers by using MTs as building blocks, in much the same
way as QED cavities in quantum optics are currently being used in successful attempts at implementing qubits and quantum logic gates [Song and
Zhang (2001)]. Detecting quantum behaviour in biological matter at this
level would undoubtedly advance attempts at implicating quantum physics
in consciousness while at the same time uncover intriguing new possibilities
at the interface of bio-nanotechnology and quantum information science.
Memory Depends on the Cytoskeleton, but is it Quantum?
113
However, to our knowledge, there is currently no complete and experimentally testable quantum mechanical model of neural function or consciousness. In this chapter we show that there are ways to phenomenologically start addressing this issue and perform experiments that can falsify
large swaths of the QCI space thus narrowing the possibilities down to a
manageable few while moving the field towards scientific credibility.
7.3.
Three Scales of Testing for Quantum Phenomena
in Consciousness
There are three broad kinds of experiments that one can devise to test
hypotheses involving the relevance of quantum effects to the phenomenon
of consciousness. The three kinds address three different scale ranges associated roughly with tissue-to-cell (1 cm–10 µm), cell-to-protein (10 µm–
10 nm) and protein-to-atom (10 nm–0.1 nm) sizes. Note that we are excluding experiments that aim to detect quantum effects at the “whole human”
or “society” level as these have consistently given either negative results or
been embarrassingly irreproducible when attempted under properly controlled conditions (e.g. the various extra sensory perception and remote
viewing experiments [Lilienfeld (1999)]).
The consciousness experiments belonging to the tissue-cell scale frequently utilize apparatus such as electro-encephalographs (EEG) or magnetic resonance imaging (MRI) to track responses of brains to stimuli. A
first-class example of consciousness-research at the tissue-to-cell scale [Crick
and Koch (1990); Crick et al. (2004)] is that by Christoff Koch and his group
at Caltech sometimes in collaboration with the late Francis Crick; tracking the activity of living, conscious human brain neurons involved in visual
recognition. While any quantum phenomena found at these time and size
scales would shake the world of science and lead to profound breakthroughs
in the study of both consciousness and quantum physics, we feel that the
wet/warm/noisy-type objections have their strongest footing here as any
quantum effects will most likely be washed-out by the measurement techniques used (unless of course the provocative thought experiment proposed
in Koch and Hepp (2006) is, somehow, performed).
The second scale in the quest to understand consciousness is that covering sizes between a cell and a protein. Inspired by QCI, seminal experimental work has been carried out by Nancy Woolf et al. (1994, 1999) on
dendritic expression of microtubule associated protein-2 (MAP-2) in rats
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Quantum Aspects of Life
and has been followed by experiments performed by members of our group
on the effects of MAP-TAU overexpression on the learning and memory of
transgenic Drosophila. Working at this scale leads to an understanding of
the intracellular processes that undoubtedly plays a significant role in the
emergence of consciousness but it is hard to see how experiments involving
tracking the memory phenotypes and intracellular redistribution of proteins
can show a direct quantum connection. It seems clear that experimentation at this size scale can invalidate many flavours of QCI if they predict
phenomena that are found to not occur so it is a fruitful field if one is
aiming to weed out bad assumptions and hypotheses. On the other hand,
if one is trying to find support for a QCI at this scale, one can at best
expect evidence that is “not inconsistent with” and perhaps “suggestive
of” a QCI [Mershin et al. (2004a, 2006)]. In Sec. 7.4 we summarize experimental evidence obtained by our group that the correct stochiometry, and
therefore local electrostatic and electrodynamic properties of MTs are of
paramount importance to memory storage and retrieval as first reported
in a 2004 publication in Learning & Memory titled “Learning and memory
deficits upon TAU accumulation in Drosophila mushroom body neurons.”
While this experimental in vivo study provides clear and direct proof of
MTs’ involvement in memory, these data are merely consistent with—but
not proof of—a quantum mechanical role. The significance of these experiments to the QCI is twofold: firstly, it has been discovered that there is
a cytoskeletal pathway underlying the very first steps towards associative
olfactory memory encoding in Drosophila showing that QCI-motivated research can have direct impact on conventional neurobiology. Secondly, it is
shown for the first time that it is possible to at least indirectly test some
aspects of the various QCIs, i.e. they are experimentally falsifiable. Had
that data shown that MTs are not involved in memory we could then confidently write that the entire MT-consciousness connection would need to
be discarded.
The third scale regime is that of protein-to-atom sizes. It is well understood that at the low end of this scale, quantum effects play a significant role in both chemistry and biology and it is slowly being recognized
that even at the level of whole-protein function, quantum-mechanical effects such as quantum tunneling may be key to biological processes such
as for instance enzymatic action [Ball (2004)] or photosynthesis [Ritz et
al. (2002)] or, more controversially, signal transduction [Kell (1992); Turin
(1996)]. Showing that a quantum effect is significant at this level, and is
also somehow propagated up in size to the protein-cell scale would be a
tremendous discovery across many fields.
Memory Depends on the Cytoskeleton, but is it Quantum?
115
So far however, no-one has demonstrated direct evidence for either superposition of biologically relevant states in biomolecules (e.g. a superposition of two conformational states for a membrane protein such as an ion
channel) or larger structures such as for instance neural cells. Similarly,
no-one has ever shown, and very few have suggested tangible ways to show,
in living matter, the gold-standard of quantum behaviour: entanglement.
In Sec. 7.5 we will suggest a direct experimental path to possibly measuring
quantum-entangled electric dipole moment states amongst biomolecules.
7.4.
Testing the QCI at the 10 nm–10 µm Scale
This section summarizes and condenses original research published in
[Mershin et al. (2004a)].
We asked whether it is possible to directly test one key prediction of the
microtubular QCI, namely that memory must be affected by perturbations
in the microtubular (MT) cytoskeleton.
We induced the expression of vertebrate (human and bovine) tau genes,
producing microtubule-associated protein TAU in specific tissues and at
specific times in Drosophila using directed gene expression. We disturbed
the fly MTs as little as possible, avoiding perturbation of the cytoskeleton by formation of such large protein aggregates as neurofibrillary tangles
(NFTs) that could effectively “strangle” the neuron disrupting or even stopping intracellular (axonal) transport. In addition, NFTs and/or amyloid
or senile plaques have been unequivocally shown to contribute to neurodegeneration and eventual neuronal death and it is reasonable to expect a
dying neuron to dysfunction, regardless of the state of its MTs. We also
carefully selected gene promoters that were activated in the adult fly only
to avoid any developmental problems that would be difficult to discern from
a cytoskeleton-specific dysfunction.
It was established that the bovine and human TAU protein expressed
in our transgenic flies bound to the appropriate (mushroom body) neurons responsible for olfactory associative memory in Drosophila. The flies
were then conditioned using two standard negatively reinforced associative
learning paradigms that essentially generalize the Pavlovian conditioning
protocol by coupling aversive odours as conditioned stimuli to electric shock
as the unconditioned stimulus. This way, olfactory cues are coupled with
electric shock to condition the flies to avoid the odourant associated with
the negative reinforcer. These conditioning protocols for Drosophila were
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Quantum Aspects of Life
initially developed by Tully and Quinn (1985) and modified by Skoulakis
et al. (1996); Philip (2001).
Following these protocols it is routinely observed that post-conditioning
(or “training”), a large percentage of wild type flies choose to avoid the
smell that was present when they received the electric shocks. The memory “score” is calculated as a normalized performance index (PI) where
× 100. Typical PI values for various genotype wildPI = trained−untrained
total
type flies averaged to 75% when tested 180 s post training and 52% at 1.5
hours post training while the transgenic flies, whose neuronal cytoskeleton
was burdened with excess microtubule associated protein TAU only scored
52% and 30% respectively (averaged over various genotypes). These results
are summarized in Fig. 7.1.
These data, when coupled to the numerous controls that eliminated the
possibility of any residual effects due to: non-MT specific overexpression
of protein, changes in: pre-exposure sensitivity, mechanosensory ability,
overall viability and virility, neurodegeneration, decreased olfactory acuity
to both attractive and aversive odors and finally histology, compel us to
conclude that MTs and their stochiometry to MAPs are in fact intimately
involved in memory. Taken together, the results strongly suggest that excess TAU binding to the neuronal microtubular cytoskeleton causes mushroom body neuron dysfunction exhibited as learning and memory deficits
while the cells and flies are normal in every other way. This also indicates
that although excessive TAU may not result in (immediate or mediumterm) neurodegeneration, it is sufficient to cause significant decrements in
associative learning and memory that may underlie the cognitive deficits
observed early in human tauopathies such as Alzheimer’s.
As an intriguing aside, looking at the cytoskeletal diagram of Fig. 7.1 it
is difficult not to speculate that the abacus-like binding pattern of MAPs
represents information coding which is disrupted when extra MAPs are
introduced and this is indeed alluded to in [Woolf et al. (1994, 1999)].
We have conjectured [Mershin et al. (2006)] this as the “Guitar String
Model” of the memory engram where the MAPs play the roles of fingers
clamping vibrating strings (played by MTs) at various positions. While
purely speculative at this point, this model is too provocative to ignore
as it automatically addresses the issue of both information storage (the
pattern of MAPs), information loss (MAPs break loose), long-term synaptic
potentiation (pattern of MAPs guides kinesins during axonal transport to
synapses in need of reinforcement) etc.
Memory Depends on the Cytoskeleton, but is it Quantum?
117
Wild Type & Trangenic
cytoskeletons exhibit
Identical
Viability
Pre-exposure
Olfactory Acuity
Mechanosensory
Neuroanatomy
Histology
Native MAP
PERFORMANCE INDEX
Significantly different
Olfactory associative memory
Foreign MAP
Memory retention
52% after 180s
30% after 1.5h
100% x ((number of trained)-(number of untrained))/ total
Memory retention
75% after 180s
52% after 1.5h
Yet
Fig. 7.1. Wild-type and transgenic cytoskeleton of neurons involved in olfactory associative memory representing the expected distribution of MAP tau. Summary of results
presented in [Mershin et al. (2004a)]. Wild type and transgenic flies’ average performance index (PI) for short term (180 s) and long term (1.5 h). PI suffers in transgenic
flies expressing excess MAP-tau in their mushroom body neurons indicating a direct
involvement of microtubular cytoskeleton in short and long term olfactory associative
memory.
7.5.
Testing for Quantum Effects in Biological Matter
Amplified from the 0.1 nm to the 10 nm Scale and
Beyond
Objections to a quantum role in biological processes of the “warm/wet/
noisy” kind usually come from the application of equilibrium principles to
the constituent particles. We hope to investigate deeper, as although—
for instance—the tubulin molecule consists of some 17,000 atoms that are
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Quantum Aspects of Life
subject to considerable thermal noise, the mesoscopically relevant electric
dipole moment state of this protein depends crucially on only a few electrons that can be in two sets of orbitals and the use of the conventional
notion of ambient temperature at the atomic scale is unwarranted since
tubulin is not an equilibrium system, rather it is a dynamic dissipative
system. Theoretical work has suggested that for a certain set of parameters (such as the value of the local dielectric constant, dipole moment, the
pH etc.) tubulin could indeed sustain a quantum mechanically coherent
state of its electric dipole moment for times of the order of microseconds
[Mavromatos and Nanopoulos (1998)]. Clearly, the best way to settle this
is to experimentally determine whether in fact a biomolecule can store a
qubit and the most profound way of doing so would be to exhibit quantum
“entanglement” amongst biomolecules.
Erwin Schrödinger, one of the fathers of quantum theory, coined the
word “entanglement” in 1935 to refer to a state where the wavefunction describing a system is unfactorizable. This peculiar phenomenon has turned
out to be very useful in quantum information science, quantum cryptography and quantum teleportation. Entanglement is the “gold standard” of
non-classical behavior and has been experimentally realized in light [Ou et
al. (1992); Togerson et al. (1995)], in matter [Sackett (2000)] and in combinations of those [Julsgaard et al. (2001); Rauschenbeutel (2000)]. Here
we suggest one way to experimentally test for the survival of entanglement
in biological matter by using entangled states of light to couple to surface
plasmons (mesoscopic quantum objects) that can couple, in turn, to the
electric dipole moment of proteins.
One way to produce entangled states in light is via type-II phasematching parametric downconversion, which is a process occurring when
ultraviolet (UV) laser light is incident on a non-linear beta-barium borate
(BBO) crystal at specific angles. A UV photon incident on a BBO crystal
can sometimes spontaneously split into two correlated infrared (IR) photons. The infrared photons are emitted on opposite sides of the UV pump
beam, along two cones, one of which is horizontally polarized and the other
vertically. The photon pairs that are emitted along the intersections of
the two cones have their polarization states entangled. This means the
photons of each pair necessarily have perpendicular polarizations to each
other. The state Ψ of the outgoing entangled photons can be written as:
|Ψ = | ↔, + eiα | , ↔ where the arrows indicate polarizations for the
(first,second) IR photon and can be controlled by inserting appropriate half
wave plates, while the phase factor eiα can be controlled by tilting the
Memory Depends on the Cytoskeleton, but is it Quantum?
119
crystal or using an additional BBO. Measuring the polarization state of
one of the outgoing photons—say IR1, immediately determines the state of
the other (IR2) regardless of their separation in space. This counterintuitive
phenomenon is referred to as the Einstein Podolsky Rosen (EPR)-paradox
and such pairs are called EPR pairs.
To use EPR-pair photons to check for the ability of biological matter to
carry quantum entangled states, we propose to follow a protocol similar to
one developed by Kurtsiefer et al. (2001) capable of producing brightness
in excess of 360,000 entangled photon pairs per second, coupled to a setup
similar to the one developed by Altwischer (2002) where entangled photons
are transduced into (entangled) surface plasmons and re-radiated back as
(surprisingly still entangled) photons.
The essential difference would be that the insides of the perforations in
the gold film of Altwischer (2002) would be covered with a monolayer of
tubulin dimers or microtubules (there exist numerous protocols for doing
this e.g. [Schuessler et al. (2003)]). The evanescent wave of the (entangled)
surface plasmon generated at resonance will interact with the electric dipole
moment of the immobilized protein complexes and presumably transfer the
entanglement (and quantum bit) to a dipole state in a manner similar to
the transfer of the photon polarization entanglement to surface plasmons.
At the end of the tunnel, the surface plasmons would be reradiated having undergone the interaction with the protein electric dipole moment. If
partial entanglement with the partner photon (that underwent a subset or
even none of these transductions) is found, then this would suggest that
the protein is capable of “storing” the entanglement in its electric dipole
moment state and characteristic decoherence times could be measured by
varying the length of the tunnels from 200 nm up.
We have previously preformed numerous experiments immobilizing
monomolecular layers of tubulin on gold substrates and probing them with
surface plasmons and have measured the changes of the refractive index
and dielectric constant with tubulin concentration: ∆n
∆c = (1.85 ± 0.20) ×
∆ǫ
10−3 (mg/ml)−1 ⇒ ∆c
= (5.0 ± 0.5) × 10−3 (mg/ml)−1 as reported in
[Schuessler et al. (2003)] where n and ∆ǫ are the changes in the refractive index n and dielectric constant ǫ and ∆c is the change in concentration
c. We also checked using direct refractometry (a method based on the
same underlying physical principle as surface plasmon resonance but very
different in implementation), and found a ∆n/∆c of 1.8 × 10−3 (mg/ml)−1 .
These methods alone cannot provide the permanent dipole moment of
the molecule since they address only the high frequency region where the
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Quantum Aspects of Life
Au sieve with sub-IR wavelength
diameter tunnels (~200nm)
MTs immobilized on tunnel walls
IR1 and IR2 are re-radiated and
for the empty tunnels case are found
to still be entangled (Altewischer et al.
Nature 418, 304-306, 2002)
BB
O
op
ti c
al
ax
is
m
~200n
BBO
UV
IR beam 1
Entangled to IR2
&?
IR beam 2
Entangled to IR1
IR1 and IR2 transduce to SPs
(increasing transmission tenfold)
IR2 SPs couple to Electric Dipole of MTs
If they are found to still be entangled
after coupling to MTs’ electric dipole
moment, it will suggest that MTs
can sustain a quantum entangled state
Not to scale
Fig. 7.2. For certain orientations, a UV photon is absorbed by the BBO crystal and reemitted as two entangled IR photons (IR1, IR2). One of the EPR-paired beams can be
allowed to undergo the quadruple transduction of photon → surface plasmon → electric
dipole moment → surface plasmon → photon while the other beam can either be left
undisturbed or undergo any subset of transductions determining whether proteins can
sustain entanglement.
permanent dipole is “frozen out” so to calculate it we resorted to supercomputer molecular dynamic simulations [Mershin et al. (2004b)] and arrived
at a value of 552D and 1193D for the α- and β- monomers and 1740D for
the αβ-dimer with a refractive index for the protein at 2.90 and the highfrequency dielectric constant at 8.41 and polarizability 2.1 × 10−33 Cm2 /V.
Knowing these parameters paves the way towards implementing the experiment described in Fig. 7.2 as the high values of the dielectric constant
and dipole moments are particularly encouraging suggesting strong interaction between the entangled surface plasmons and the biomolecules’ electric
dipole moment.
7.6.
Summary and Conclusions
By performing carefully controlled in-vivo experiments it has been shown
that olfactory associative memory encoding, storage and retrieval is intimately tied to the neuronal microtubular cytoskeleton in Drosophila. This
finding undoubtedly applies to the microtubular cytoskeleton of many other
animals including humans. Since memory is a necessary ingredient of consciousness, our finding is consistent with a microtubular involvement in
consciousness but at this stage does not lean toward a classical or quantum
Memory Depends on the Cytoskeleton, but is it Quantum?
121
role. We have proposed an experiment to test for the ability of microtubules to sustain entangled states of their electric dipole moments, set up
by EPR-pair photons transduced to surface plasmons. A positive result to
this experiment would revolutionize the way we look at proteins and will
undeniably create a more positive climate for the QCI.
7.7.
Outlook
Fabrication of novel biomaterials through molecular self-assembly is going
to play a significant role in material science [Zhang (2003)] and possibly
the information technology of the future [Ou et al. (1992)]. Tubulin, microtubules and the dynamic cytoskeleton are fascinating self-assembling
systems and we have here asked whether they underlie the possibly quantum nature of consciousness. Whatever the consciousness case may be, our
work with the neurobiology of transgenic Drosophila [Mershin et al. (2004a)]
compels us to recognize that the cytoskeleton is very near the “front lines”
of intracellular information manipulation and storage. Our work with surface plasmon resonance and tubulin biophysics suggests that cytoskeletal
structure and function contains clues on how to fabricate biomolecular information processing devices whether they are fundamentally quantum in
nature or not.
So perhaps one way to proceed is to accept Richard Feynman’s adage
“What I cannot create, I do not understand” and try to create biomolecular
information-processing circuits using nature’s cytoskeletal building blocks.
If we discover that they are in fact harnessing quantum effects, then it will
be that much easier to take the leap to quantum consciousness, but for
now, we must hedge our bets.
Acknowledgements
The authors wish to thank E. M. C. Skoulakis, H. A. Schuessler,
A. A. Kolomenski, J. H. Miller, H. Sanabria, R. F. Luduena,
D. Nawarathna, and N. E. Mavromatos.
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About the authors
Andreas Mershin received his MSci in Physics from Imperial College
London (1997) and his PhD in Physics from Texas A&M University (2003),
under D. V. Nanopoulos, where he studied the theoretical and experimental
biophysics of the cytoskeleton. He performed molecular dynamic simulations on tubulin and after winning an NSF grant initiated wide-reaching,
cross-disciplinary collaborations performing experiments utilizing surface
plasmon resonance, dielectric spectroscopy and molecular neurobiology to
successfully test the hypothesis that the neuronal microtubular cytoskeleton
is involved in memory encoding, storage and retrieval in Drosophila. Currently, he is a postdoctoral associate at the Center for Biomedical Engineering of the Massachusetts Institute of Technology developing bioelectronic
photovoltaic and chemical sensing applications using membrane proteins
integrated onto semiconductors. A patent holder and entrepreneur in the
field of biosensors, he is also a co-founder of the Royal Swedish Academy of
Sciences’ international annual “Molecular Frontiers Inquiry Prize” for the
best scientific question posed by children (www.molecularfrontiers.org).
Dimitri V. Nanopoulos received his BSc in Physics from the University
of Athens (1971) and his PhD in High Energy Physics from the University
of Sussex (1973), under Norman Dombey. He has been a Research Fellow
at CERN, at Ecole Normale Superieure and at Harvard University. In 1989,
he was elected professor (Department of Physics, Texas A&M University)
where since 1992 he is a Distinguished Professor of Physics and since 2002
Memory Depends on the Cytoskeleton, but is it Quantum?
125
holds the Mitchell/Heep Chair in High Energy Physics. He is also Head of
the Astroparticle Physics Group at the Houston Advanced Research Center. He has made several contributions to particle physics and cosmology
working on string unified theories, fundamentals of quantum theory, astroparticle physics and quantum-inspired models of brain function. With
over 550 original papers and 31,000 citations he has been ranked 4th most
cited high energy physicist of all time by the Stanford University Census.
A fellow of the American Physical Society since 1988 he was elected a member of the Academy of Athens in 1997 and became President of the Greek
National Council for Research and Technology as well as the National representative of Greece to CERN in 2005. He has received numerous awards
and honors including the Onassis International prize.
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Chapter 8
Quantum Metabolism and Allometric
Scaling Relations in Biology
Lloyd Demetrius
The basal metabolic rate of an organism is its steady-state rate of heat
production under a set of standard conditions. There exists a class of
empirical rules relating this physiological property with the body size of
an organism. The molecular mechanism that underlies these rules can be
understood in terms of quantum metabolism, an analytical theory which
deals with the dynamics of energy transduction within the membranes of
the energy transducing organelles. This chapter delineates the origin and
analytical basis of quantum metabolism and illustrates its predictive power
with examples drawn from the empirical literature.
8.1.
Introduction
The metabolic rate of an organism—the rate of energy expenditure,—and
body size—an organism’s total metabolic mass—are highly interrelated
characteristics. Elucidating the rules that define this dependency and delineating the mechanisms that underlie these rules are central problems
in bioenergetics. Since body size is correlated with many physiological,
ecological and life-history traits, an understanding of the relation between
metabolic rate and body size has important implications in many areas of
biology.
Body size is a highly variable property: size changes during an organism’s ontogeny. Hence, most efforts to determine the relation between
Received May 8, 2007
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Quantum Aspects of Life
metabolic rate and body size have focused on adult size, which is a relatively
constant morphological index. Metabolic rate is also highly variable—even
for an adult individual. Three situations are usually distinguished: standard metabolic rate, maximal metabolic rate, and field metabolic rate [Rolfe
and Brown (1997)].
Standard metabolic rate is the steady-state rate of heat production under a set of standard conditions; in mammals, these conditions are that
the individual is an adult, resting and maintained at a temperature that
generates no thermoregulatory effect. The maximal metabolic rate is the
maximum steady-state metabolic rate attainable during hard exercise over
a defined period of time. Field metabolic rate represents the average rate
over an extended period of time when the organism is living in its natural
environment.
The distinction between the different measures of metabolic rate and
its relevance in understanding the constraints which body size imposes on
metabolic activity was recognized as early as 1781 by Lavoisier and Laplace.
They carried out calorimetry experiments on adult guinea pigs who were
maintained in a resting state. One of the main achievements of these two
pioneers in the field of bioenergetics was the discovery that the standard
metabolic rate is quantitatively related to the caloric nutrient requirements
of the organism. More than a century later, in 1804, Rubner, the German
physiologist, repeated the experiments of Laviosier and Laplace on several
species of domesticated animals and proposed a series of scaling rules relating the standard metabolic rate (SMR) to body size. A systematic approach
to the study of bioenergetics and allometry, however, only began with the
investigations of Kleiber (1961). He expanded on the work of Rubner by
including non-domesticated species in his studies, and subjecting his data
to statistical analysis.
During the last 50 years, Kleiber’s work has been extended to include
uni-cellular organisms, various species of invertebrates, birds and plants.
Much of this work has now been reviewed and analyzed in various texts
[Brody (1946); Peters (1983); Calder (1984)].
The allometric rules that have emerged from these investigations can be
expressed in the form
P = αW β ,
here W denotes the body size, and P , the standard metabolic rate.
(8.1)
Quantum Metabolism and Allometric Scaling Relations in Biology
129
Equation (8.1) involves two critical parameters:
(i) a scaling exponent, β, which refers to the fractional change in metabolic
rate to a change in body size. Typically, β is found to be 2/3 for small
animals, 3/4 for large animals, and 1 for perennial plants.
(ii) A normalizing coefficient α, which describes the rate of energy expenditure in an organism of unit mass. This parameter also shows a large
interspecific variation. For example, endotherms have a much higher
metabolic rate than ectotherms of the same body size.
The problem of explaining the allometric rules in mechanistic terms
has generated considerable controversy; see for example McMahon (1973);
West et al. (1997); Banavar et al. (2002). These studies have been largely
concerned with explaining the incidence of a 3/4 scaling exponent for the
metabolic rate. Such models have largely ignored the significance of proportionality constants, and the deviations from the 3/4 rule that characterizes
small animals (β = 2/3), and perennial plants (β = 1).
The difficulties in explaining the interspecific variation in scaling exponents and proportionality constants derive in large part from the fact
that the models proposed are framed in terms of macroscopic organismic
properties. However, empirical studies show that variations in scaling exponents and proportionality constants are largely contingent on cellular level
properties, in particular, on the molecular dynamics of metabolic activity
[Hochachka and Somero (2000)], Chap. 4).
The mechanism underlying the allometric rules was recently addressed
in [Demetrius (2003, 2006)], in terms of a cellular-level model based on
the molecular dynamics of metabolic regulation. The model rests on the
observation that metabolic activity within organisms has its origin in the
processes of energy transduction localized in biomembranes: the plasma
membrane in uni-cells, the inner membrane of mitochondria in animals,
and the thylakoid membrane in plants, see [Harold (1986)].
Energy transduction in biomembranes can be understood in terms of
the chemiosmotic coupling of two molecular motors [Mitchell (1966)].
(a) The redox chain which describes the transfer of electrons between redox
centers within the biomembrane.
(b) The ATP-ase motor, which is involved in the phosphorylation of ADP
to ATP.
The molecular dynamics model, [Demetrius (2003, 2006)], integrates
the chemiosmotic theory of energy transduction [Mitchell (1966)], with the
130
Quantum Aspects of Life
proposition, due to Fröhlich (1968), that the energy generated by the redoxreactions can be stored in terms of the coherent vibrational modes of the
molecular oscillators embedded in biomembranes.
This new synthesis, which we call quantum metabolism, postulates that
the collective modes of vibration of the molecular oscillators is quantized.
Quantum metabolism exploited certain methods, integral to the methods
developed in the quantum theory of solids, to derive a set of analytic relations between cellular metabolic rate and cycle time—that is, the mean
transit time of the redox reactions within the cell. These relations were
then invoked to establish an allometric rule relating cycle time, τ , with cell
size, Wc , namely,
τ = αWcβ .
(8.2)
The distinct values assumed by the scaling exponent are contingent on the
magnitude of the metabolic energy stored in the molecular oscillators. Typically, β = 1, when the metabolic energy associated with the vibrational
modes is much greater than the vibrational spacing. In this case the energy
transduction process is continuous and the total metabolic energy generated is proportional to the cycle time. The condition β = 3/4 is obtained
when the metabolic energy per mole is smaller than the vibrational spacing.
Under this constraint energy transduction becomes discrete, and the total
metabolic energy generated is now the fourth power of the cycle time.
The proportional constant α was shown to be determined by (i) the
capacity of the system to transport protons across the membrane, (ii) the
extent to which the dynamics of the electron transport chain, the energy
donating process, is coupled to ADP phosphorylation, the energy accepting
process. The first property depends on the bioenergetic parameters, the
proton conductance, denoted C, and the proton motive force, denoted ∆p.
The second property defines the metabolic efficiency, denoted µ.
The relation between the metabolic cycle time τ , and cell size Wc , given
by Eq. (8.2) can be extended to higher levels of biological organization, such
as organs and multicellular organisms, by appealing to a multi-level scaling
argument, [Demetrius (2006)]. These extensions provide the basis for a
series of allometric rules relating the metabolic rate, denoted P , with the
body size, denoted W , of multicellular organisms. The rules are given by
P = γC∆pW β .
(8.3)
The parameter C (dimension: nmol H + per unit time, per mg protein, per
mV); and ∆p (dimension: mV) represent bioenergetic variables averaged
Quantum Metabolism and Allometric Scaling Relations in Biology
131
over the different cells and organ systems. The parameter γ is the product,
γ = aµ, where µ is the metabolic efficiency, and a the capacity of the
organism to transport nutrients from the external environment to cellular
organelles.
This chapter aims to provide a conceptual overview and a non-technical
account of the mathematical ideas that constitute quantum metabolism.
We pursue this aim by
(i) providing a historical account of the ideas of quantum theory and
the implications of the theory to energy transduction in sub-cellular
processes in living systems;
(ii) delineating the main mathematical ideas that underlie the derivation
of the allometric rules;
(iii) illustrating, with empirical examples, the predictive power of the allometric laws.
8.2.
Quantum Metabolism: Historical Development
The law of conservation of energy asserts that an object in a mechanically
stable state and isolated from its surroundings has a definite energy E. In
classical mechanics, E is a continuous variable and can assume any value
consistent with the stability of the object. In quantum mechanics, the
possible values of the energy are discrete,
E = Ei ,
i = 0, 1, 2, . . .
Here E0 denotes the ground state energy, E1 , the energy in the first excited
state, and so on.
We will describe the pertinence of this quantization postulate in the
context of Planck’s derivation of the black body radiation spectrum. We
will then discuss its application to the study of the heat capacity of solids
and, finally describe its application to the analysis of the metabolic rate
of subcellular processes in living organisms. Our historical overview draws
extensively from [Longair (2004)].
8.2.1.
Quantization of radiation oscillators
The quantization concept was invoked by Planck in 1900 to explain the
Stephan-Boltzmann law of electromagnetic radiation. This empirical law
132
Quantum Aspects of Life
asserts that the energy density of radiation, denoted e, is given by
e = σT 4 .
(8.4)
Here T denotes the absolute temperature, and σ a proportionality constant,
with an empirically precise value.
The first attempt to explain the radiation law was made by Boltzmann
who exploited classical thermodyamical arguments to account for Eq. (8.4).
These methods were able to determine the value for the scaling exponent.
However, in Boltzmann’s model, the proportionality constant appeared as
a constant of integration without any physical meaning.
An explanation of both the scaling exponent and the proportionality
constant was achieved by Planck in 1900. The crucial insight in Planck’s
model was to analyze radiation as a photon gas. The oscillations of these
photons were assumed to be the mechanism underlying radiation in a metallic cavity. The model resides on the so called quantization principle: The
energy that can be stored by an oscillator with frequency ω can only be
integral multiples of a basic energy unit which is proportional to the characteristic frequency of the oscillator.
Analytically, we write
En = nhω ,
n = 1, 2, . . . ,
(8.5)
where h is Planck’s constant and n an integer.
An important achievement of Planck’s model was his expression for the
spectral density of radiation at temperature T ,
u(ω) =
hω
.
)−1
exp( khω
BT
(8.6)
Here kB denotes Boltzmann’s constant.
For small hω/kB T , the exponential term can be expanded and retention
of only the first terms gives u(ω) = kB T , which is consistent with the
Rayleigh-Jeans law. For large hω/kB T , the expression in Eq. (8.6) tends to
hω exp(−hω/kB T ). This is consistent with Wien’s law. The significance of
the quantization rule resides in the fact that Eq. (8.6), provides an excellent
agreement with experimental data at all temperatures.
8.2.2.
Quantization of material oscillators
The fundamental nature of Planck’s radiation law was evidently recognized
by Einstein, who in 1907 appealed to quantum theory to study the thermal
properties of solids. The characteristic property of a solid is that its atoms
Quantum Metabolism and Allometric Scaling Relations in Biology
133
execute small vibrations about their equilibrium positions. When in thermal equilibrium the atoms are arranged on a regular lattice—a condition
called the crystalline state.
Einstein treated the atoms in a crystalline solid as vibrating independently of each other about fixed lattice sites. The vibrations are assumed
to be simple harmonic. In contrast to the radiation oscillators that can assume all possible frequencies, the material oscillators are assumed to have
a single frequency.
By invoking Planck’s quantization rule, the mean energy associated with
a given frequency will now be given by
E(ω) =
hω
.
exp( khω
)−1
BT
(8.7)
This expression yielded an analytical framework for predicting the specific
heat per molecule at high temperatures. The model thus yielded a molecular explanation of the law of Dulong and Petit. However the argument was
not able to explain the heat capacity at low temperatures.
The reason for this failure is due to the independence assumption made
regarding the vibrations of the atoms at the lattice sites. This is now recognized as a highly restrictive condition. A crystal does not consist of atoms
vibrating totally independently of each other about fixed lattice sites: there
is a strong coupling between the atoms. On account of this assumption,
Einstein’s model was inconsistent with the empirically observed values of
heat capacity at low temperatures.
The discrepancy at low temperatures was elucidated in 1912 by Debye
who formulated a new model in which the atoms are assumed to execute
coupled vibrations about the fixed sites. Hence for a crystal consisting of
N atoms, the system can be described by 3N normal modes of vibration of
the whole crystal, each with its own characteristic frequency. Hence if we
know the characteristic frequencies, we can write down the total energy of
the crystal.
8.2.3.
Quantization of molecular oscillators
Quantum metabolism is a molecular biological application of Planck’s quantization concept. Quantum metabolism is concerned with explaining certain
empirical relations between the size of an organism and its metabolic rate.
The model addresses this problem by first investigating at the cellular
level the mechanisms underlying the scaling rules relating cycle time and cell
134
Quantum Aspects of Life
size. The fundamental idea invoked draws from the chemiosmotic theory
proposed by Mitchell (1966). According to this theory, metabolic activity
is localized in energy-transducing membranes and is determined by the
coupling of an energy donating process (oxidation-reduction reaction) and
an energy-accepting process (ADP phosphorylation).
According to Fröhlich (1968), a characteristic property of an energytransducing membrane is that the phospholipid head groups, which constitute an integral component of the membrane, can execute small vibrations
due to their oscillating dipole moments. When the system is subject to a
continuous supply of energy, coherent elastic vibrations will be generated—
a consequence of the strong coupling between the molecular elements and
the long-range Coulomb interaction between the molecular groups.
Quantum metabolism rests on the postulate that Planck’s quantization
principle, which was invoked by Einstein and Debye in their studies of the
heat capacity of solids, can also be applied to the vibration of molecular
groups embedded in the biomembrane.
There exists, however, a fundamental difference between the EinsteinDebye models and quantum metabolism. In the quantum theory of solids,
the fundamental unit of energy is given by
E(T ) = kB T ;
(8.8)
the typical thermal energy per molecule.
In a crystal lattice each atom is in equilibrium when it occupies its
designated position in the lattice; and if perturbed the atom undergoes oscillations about the equilibrium state with a dynamic that is approximately
simple harmonic. The quantization of the material oscillators ensures that
the mean energy per atom will now be dependent on the ratio hω/kB T .
Here hω denotes the vibrational spacing of the harmonic oscillator. The
mean energy will now be given by Eq. (8.7), the expression derived by
Planck in the context of radiation oscillators.
In quantum metabolism, the fundamental unit of energy is
E(τ ) = gτ .
(8.9)
Here τ denote the metabolic cycle time, the mean turnover time of the
oxidation-reduction reaction. The quantity g = (g̃w)/NA , where w denotes
the mean protein mass, and g̃ = (C∆p), and NA denotes Avogadro’s number. The parameter C, we recall describes the proton conductance and ∆p,
the proton motive force—both bioenergetic parameters. The quantity E(τ )
is the total metabolic energy generated per cycle per mole.
Quantum Metabolism and Allometric Scaling Relations in Biology
135
The metabolic energy of the cell is derived from the vibration of the
molecular groups embedded in the biomembrane. Each molecular group
embedded in the membrane undergoes oscillations about its steady state
with a motion which is simple harmonic for small oscillations. The frequency of the oscillations can also be computed in terms of the size of
molecular group and elastic constants of the membrane.
The mean metabolic energy can be computed using methods analogous
to the derivation of Eq. (8.7). We have
Ẽ(ω) =
hω
.
exp( hω
gτ ) − 1
(8.10)
The fundamental difference between Eq. (8.7) and Eq. (8.10) is the replacement of thermal energy, E(T ) = kB T , by the metabolic energy per cycle,
E(τ ) = gτ .
The difference between the components of energy invoked in the two
classes of models resides in the fact that energy transformation in living
organisms, in contrast to energy transformations in physical systems, occur under isothermal conditions. There are no significant differences in
temperature between the parts of a cell or between cells in a tissue. Cells
cannot function as heat engines. Energy transformation in cells proceeds
through differences in ion-gradients, hence processes of energy transformation in cells are not described by equilibrium conditions. Living systems, in
sharp contrast to physical systems, are in a dynamic steady state where the
notion of a cycle time now replaces temperature as the critical organizing
parameter.
8.2.4.
Material versus molecular oscillators
Quantum metabolism rests on the recognition that the molecular oscillators in biomembranes and the material oscillators in crystalline solids can
be analyzed in terms of the same mathematical formalism. This realization
derives from a formal correspondence between the thermodynamic variables that describe material systems and the metabolic parameters that
define certain biological processes [Demetrius (1983)]. We summarize this
correspondence in Table 8.1.
This correspondence between thermodynamic variables and metabolic
parameters is a consequence of the following analytical fact: The growth rate
parameter in metabolic processes satisfies a variational principle which is
formally analogous to the minimization of the free energy in thermodynamic
systems [Demetrius (1983); Arnold et al. (1994)].
136
Quantum Aspects of Life
Table 8.1. Relation between thermodynamic variables and
metabolic parameters.
Thermodynamic Variables
Metabolic Parameters
Temperature
Heat capacity
Thermodynamic entropy
Cycle time
Metabolic rate
Entropy production rate
The conceptual framework and the analytical methods invoked in quantum metabolism is an elaboration of this mathematical principle.
8.3.
Metabolic Energy and Cycle Time
The fundamental equation in quantum metabolism, Eq. (8.2), relates the
metabolic cycle time with cell size. This relation derives from the analysis
of energy transduction within biomembranes. The basic information on
the structure of biomembranes is due to Singer and Nicholson (1972). This
study, which led to the fluid-mosaic model, describes the membrane as a
sheet-like structure with a thickness of about 10−6 cm. This structure,
which consist of lipid-protein complexes, are non-covalent aggregates. The
constituent proteins, which are embedded in the phospholipid layer, are
held together by many cooperative non-covalent interactions.
According to the chemiosmotic theory, the energy released in oxidations is coupled by proton translocation across the biomembrane to ADP
phosphorylation.
The energy transformation involves the interconversion of three forms
of energy; see [Nicholls and Ferguson (2002); Harris (1995)].
(1) The redox potential difference, that is, the actual redox potential between the donor and acceptor couples in the electron transfer chain.
(2) The proton motive force which describes the free energy stored in the
membrane electrochemical proton gradients
(3) The phosphorylation potential for ATP synthesis.
Let g̃ denote the proton current induced by the electromotive force. We
can now apply Ohm’s law to the proton circuit and obtain g̃ = C∆p.
The proton circuit which describes the coupling of the electron transport
chain with ADP phosphorylation by means of the proton flux, denoted g̃,
is schematically represented by Fig. 8.1.
Quantum Metabolism and Allometric Scaling Relations in Biology
High-energy
electrons
~
g
137
ADP
+P1
Low-energy
electrons
ATP
~
g
Fig. 8.1.
Proton circuit linking the electron transport chain with ADP phosphorylation.
The energy generated per cycle, per mole, is given by Ẽ = gτ . Here
g = (g̃w)/NA , where w denotes protein mass, and NA , Avogadro’s constant.
We let N denote the number of molecular groups in the membrane. The
system has 3N degrees of freedom corresponding to the 3N coordinates,
which are necessary to specify the location of the molecular groups. The
molecular oscillations of the system can be described in terms of 3N normal modes of vibration; each with characteristic frequency ω1 , ω2 . . . , ω3N .
We will consider the molecular oscillators to be collective properties of the
membrane as a whole and we will compute the metabolic energy, a property which is generated by the vibrations of the membrane into which the
molecular groups are bound.
8.3.1.
The mean energy
We will assume that the vibrational modes of the molecular groups are
quantized and then apply a statistical argument to obtain an average energy
for each independent mode of oscillation.
For a given energy En = nhω, the probability Wn that the oscillation
has an energy corresponding to its nth allowed value is
Wn ∼ exp[−
where Ẽ = gτ .
En
]
Ẽ
(8.11)
138
Quantum Aspects of Life
To normalize this expression, we write
nhω
]
gτ
Wn = ∞
nhω
]
exp[−
gτ
n=0
exp[−
which reduces to
nhω
]
gτ
.
Wn =
1 − exp[ −hω
gτ ]
exp[−
The mean energy associated with an independent mode of oscillation is now
given by
E∗ =
En Wn .
In view of the expression for Wn given by Eq. (8.11) we obtain the mean
metabolic energy Ẽ(ω) given by Eq. (8.10).
8.3.2.
The total metabolic energy
The total metabolic energy generated by the redox reactions and stored in
the membrane will be given by
u=
3N
k=1
hωk
.
k
exp[ hω
gτ ] − 1
(8.12)
We will derive an approximate value for this energy by ignoring the discrete
structure of the membrane and considering the system as a homogeneous
elastic medium. In order to determine the total metabolic energy, we need
to calculate the different standing wave patterns generated by the vibrations
of the molecular groups.
Now, the number of standing waves in an enclosure with wave vectors
in the range ω to ω + dω is determined by the geometry of the system.
It is proportional to the volume of the enclosure and to ω 2 dω. Hence for
elastic waves, the density of modes will be given by, see for example [Mandl
(1988)],
3vω 2
dω .
(8.13)
2π 2 c3
Here v denotes the volume of the wave medium. We can now exploit this
expression to derive an approximate value for the total metabolic energy, u.
f (ω) =
Quantum Metabolism and Allometric Scaling Relations in Biology
In view of Eq. (8.13), we can approximate Eq. (8.12) by
ωmax
3hv
w3 dω
.
u=
2π 2 c3 exp( hω
gτ ) − 1
139
(8.14)
0
Here ωmax is the maximum frequency.
We will now evaluate Eq. (8.14) under certain particular constraints on
the energy function, gτ . We first introduce the notion of a characteristic
cycle time, denoted τ ∗ , by writing
gτ ∗ = hωmax .
Hence
hωmax
.
τ∗ =
g
The characteristic cycle time thus depends critically on the bioenergetic
parameters C and ∆p, variables which define the proton current, g.
We consider two limiting regimes of cycle time:
(a) τ ≫ τ ∗ : When this condition holds, gτ ≫ hω and the metabolic energy
is thus much larger than the vibrational spacing.
(b) τ ≪ τ ∗ : When this condition is obtained, we have gτ ≪ hω and the
metabolic energy is now much smaller than the vibrational spacing.
We will observe that these two constraints will lend to distinct characterizations of the metabolic energy as a function of cycle time.
(I) τ ≫ τ ∗ .
When this condition holds, the integral given by Eq. (8.14) can now be
simplified and we obtain
ωmax
3v
ω 2 dω .
(8.15)
u = gτ ×
2π 2 c3
0
By expressing N , the total number of oscillators in terms of ωmax , we obtain
the following expression for the total metabolic energy,
u = 3N gτ .
(8.16)
∗
(II) τ ≪ τ .
When this condition prevails, the Eq. (8.14) can be approximated by the
integral,
3v
(gτ )4
u=
2
2π h3 c3
∞
x3
dx
−1
ex
0
which reduces to
u = aτ 4 ,
4
where a = kg and k =
π2 v
h3 c 3 .
(8.17)
140
8.4.
Quantum Aspects of Life
The Scaling Relations
The scaling relationship of metabolic rates with body size, given by Eq. (8.3)
can now be derived from Eq. (8.16) and Eq. (8.17) by appealing to thermodynamic arguments.
8.4.1.
Metabolic rate and cell size
The relations given by Eqs. (8.16) and (8.17) can be exploited to determine
scaling relations between metabolic rate and cell size. Thermodynamic
arguments entail that the total metabolic energy u generated by the redox
reactions can be expressed in the form
u = ρµWc .
(8.18)
Here ρ is a proportionality constant with dimension [energy/mass], and
µ the metabolic efficiency, which is defined by the ratio: rate of ADP
phosphorylation / rate of electron transport.
We now return to the two limiting cases and appeal to Eqs. (8.16),
(8.17) and (8.18).
(I) τ ≫ τ ∗ :
This describes the classical regime where the energy transduction process is continuous. The cycle time and metabolic rate now becomes
τ=
(ρµ)
;
g
P = gµWc .
(II) τ ≪ τ ∗ :
When this constraint on cycle time is obtained, the energy transduction process is discrete. The scaling exponents for the cycle time and
metabolic rate now assume the values 1/4 and 3/4, respectively. The
allometric relations become
ρµ
τ = ( )1/4 Wc1/4 ; P = a1/4 (ρµ)3/4 Wc3/4 .
a
8.4.2.
Metabolic rate and body mass
The analytic relation between metabolic rate and cell mass can be extended
to yield an allometric relation between organism metabolic rate and body
mass. Extension to allometry at the whole organism level rests on a multilevel scale hypothesis, see [Suarez and Darveau (2005)]. This hypothesis
entails that the scaling exponent (β) of a multicellular organism is similar to
Quantum Metabolism and Allometric Scaling Relations in Biology
141
that of its constituent cells, while the proportionality constant (α) depends
on the efficiency with which nutrients are transported within the organism
to the energy-transducing membranes in the cell.
According to the multi-level scale hypothesis, the metabolic rate of the
whole organism, P , is now given by
P ∼ C∆pW β
(8.19)
where W denotes organism body mass.
The scaling exponent β depends on the cycle time τ . For systems described by a large cycle time, τ ≫ τ ∗ , we have β = 1, for systems defined
by a small cycle time, τ ≪ τ ∗ , we obtain the characterization β = 3/4.
The large majority of organisms are characterized by either the exponent
β = 1, or β = 3/4. This observation is consistent with the fact that when
the limiting conditions hold, the expressions for metabolic energy given
by Eq. (8.16) and Eq. (8.17) are exact. The expression for the metabolic
energy is an interpolation formula between two correct limits.
Beyond these two limiting conditions for the cycle time, the scaling
exponent may assume values distinct from 1 or 3/4. However, a minimal
value for β can be derived by invoking an energetic argument.
We assume that the metabolic rate P , scales with body size; that is
P ∼ W β . Now, let Q denote the rate of assimilation of energy. By appealing
to surface-area relations, we have, Q ∼ W 2/3 . Since the minimal metabolic
rate, the rate needed for homeostasis, must exceed the rate of assimilation
of energy, we have P ≥ Q, and we conclude that β ≥ 2/3.
8.5.
Empirical Considerations
Quantum metabolism is a mathematical model of energy transduction
which provides a quantitative explanation of the empirical rules relating
body size with metabolic rate. The model shows that the dependency on
body size involves three main factors:
(a) Metabolic efficiency: The extent to which the electron transfer process
is coupled to ADP phosphorylation.
(b) Membrane composition: The phospholipid composition of the biomembrane. This property determines the bioenergetic parameters such as
proton conductance and the proton motive force.
(c) Network geometry: The complexity of the circulatory network by which
nutrients are transported within the cells and tissues of the organism.
142
Quantum Aspects of Life
The analytic relation between metabolic rate and body size is given by
Eq. (8.19). This relation yields explicit predictions regarding both the
scaling exponent and the proportionality constant. We now describe the
nature of these predictions and discuss their empirical support.
8.5.1.
Scaling exponents
The model predicts that the scaling exponent β will satisfy the condition,
2/3 ≤ β ≤ 1.
Different classes of organisms will be defined by different range of values
of β — this range will be contingent on the metabolic cycle time.
Quantum metabolism distinguishes between two classes of organisms.
Type (I): defined by the condition τ > τ ∗ . Chloroplasts the energy transducing organelles in plants are relatively large and described by
a large cycle time. Hence plants are typical members of Type
(I).
Type (II): defined by the condition τ < τ ∗ . Mitochondria the energy transducing organelles in animals are described by a relatively short
cycle time. Hence animals are typical members of Type (II).
This classification according to cycle time indicates that in plants, the scaling exponents will satisfy the condition 2/3 < β < 1, whereas in animals,
we have 2/3 < β < 3/4.
However, the actual value of β will depend on the ecological constraints
experienced by the population during its evolutionary history. The effect
of these constraints on the scaling exponent can be understood in terms
of directionality theory [Demetrius (1997)]. This model of the evolutionary
process distinguishes between equilibrium and opportunistic species.
(i) Equilibrium species. This condition characterizes populations that
spend most of their time in the stationary phase or undergoing small
fluctuations in population numbers around some constant value. (Examples: perennial plants, large mammals.)
(ii) Opportunistic species, This property defines species subject to large
irregular fluctuations in population size. (Examples: annual plants,
small mammals, birds.)
Quantum Metabolism and Allometric Scaling Relations in Biology
143
The central parameter in directionality theory is demographic entropy, S,
which is given by
∞
S=−
p(x) log p(x)dx .
(8.20)
0
Here, p(x) is the probability that the mother of a randomly-chosen newborn
belongs to the age-class (x, x + dx).
Directionality theory predicts that evolution will result in a unidirectional increase in entropy in equilibrium species. In opportunistic species,
evolution results in a unidirectional decrease in entropy for large populations, and random, non-directional change in entropy for small populations
[Demetrius (1997)].
In view of the allometric rules relating body size to cycle time and
metabolic rate, we can derive an expression relating the entropy, S to the
mass-specific metabolic rate, denoted P ∗ . The relation is given by
S = a − c log P ∗ .
(8.21)
Equation (8.21) provides a basis for predicting evolutionary trends in
metabolic rate by appealing to the directionality principles for evolutionary
entropy. These principles imply the following patterns: (a) in equilibrium
species, evolution will act to decrease P ∗ ; (b) in opportunistic species, evolution will act to increase P ∗ , when population size is large, and result in
random non-directional changes in P ∗ when population size is small.
Now, the mass-specific metabolic rate, P ∗ , is given by
P ∗ ∼ C∆pW β−1 .
(8.22)
We can therefore exploit directionality theory to infer the following characterization of the scaling exponents for perennial plants and large mammals
(equilibrium species), and annual plants and small mammals (opportunistic
species).
(I) The scaling exponent for plants will range between 3/4 and 1. Perennial
angiosperms are described by β = 1, annual ones by β = 3/4.
(II) The scaling exponent for mammals will range between 2/3 and 3/4.
Large mammals are described by β = 3/4, small mammals by β = 2/3.
Empirical observations on mammals [Dodds (2001)]; plants [Reich et al.
(2006)] and a range of other taxa [Glazier (2005)] broadly corroborate these
two classes of predictions.
144
8.5.2.
Quantum Aspects of Life
The proportionality constant
The proportionality constant α is determined principally by the microlevel
variables proton conductance and proton motive force and one macrolevel
variable, nutrient supply.
Proton conductance is highly dependent on the degree of polyunsaturation of membrane phospholipids: the more polyunsaturated the mitochondrial membranes, the larger the proton conductance.
There exists a large variation in phospholipid composition between
species. However, certain distinct patterns of variation exist [Hulbert
(2005); Brand et al. (1994)]:
(a) Membrane bilayers of endotherms are more polyunsaturated than those
of similar-size ectotherms;
(b) Membrane bilayers of tissues of small mammals are highly polyunsaturated, while in large mammals, membrane polyunsaaturation decreases
with increased body size.
Since proton conductance is positively correlated with proton current, we
can invoke (a), (b), and Eq. (8.19) to predict the following patterns.
(III) The metabolic rate of an endotherm will be greater than that of an
equivalently-sized ectotherm at the same body temperature.
(IV) Tissues from larger mammals should have lower in vitro metabolic
rate than homologous tissues from small mammals
Experimental studies of metabolic rate, described for example by
Hulbert (2005) and Hochachka and Somero (2000), are consistent with these
predictions.
References
Arnold, L., Demetrius, L., and Gundlach, M. (1994). Evolutionary formalism and
products of random matrices. Ann. Appl. Prob. 4, pp. 859–901.
Banavar, J. R., Damuth, J., Martan, A., and Rinaldo, A. (2002). Supply-demand
balance and metabolic scaling, Proc. Natl. Acad. Sci. 99, pp. 1056–1059.
Brand, M. D., Chien, L. F., and Diolez, P. (1994). Experimental discrimination
between proton leak and redox slip during mitochondrial electron transport.
Biochem. J. 297, pp. 27–29.
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Calder, W. A. (1984). Size, Function and Life History (Harvard University Press).
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Demetrius, L. (1983). Statistical mechanics and population biology, Jour. Stat.
Phys. 30, pp. 709–783.
Demetrius, L. (1997). Directionality principles in thermodynamics and evolution.
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Demetrius, L. (2003). Quantum statistics and allometric scaling of organisms.
Physica A. 322, pp. 477–480.
Demetrius, L. (2006). The origin of allometric scaling laws in biology. Jour. Theor.
Bio. 243, pp. 455–467.
Dodds, P. S., Rothmann, D. H., and Weitz, J. S. (2001). Re-examination of the
3/4 law of metabolism. Jour. Theor. Biol. 209, pp. 2–27.
Fröhlich, H. (1968). Long range coherence and energy storage in biological systems. Int. Jour. Quantum Chem. 11, pp. 641–649.
Glazier D. S. (2005). Beyond the 3/4 power law: variation in the intra- and
interspecific scaling of metabolic rate in animals. Biol. Rev. 80, pp. 611–
662.
Harold, F. M. (1986). The Vital Force. A Study of Bioenergetics (Freeman, New
York).
Harris, D. A. (1995). Bioenergetics at a Glance (Blackwell Scientific Publications).
Hochachka, P. W., and Somero, G. W. (2000). Biochemical Adaptation (Oxford
University Press).
Hulbert, A. H. (2005). On the importance of fatty acid composition of membranes
for aging. J. Theor. Biol. 234, pp. 277–288.
Hulbert, A H., and Else, P. (2000). Mechanisms underlying the cost of living in
animals. Ann. Rev. Phys. 62, pp. 207–253.
Kleiber, M. (1961). The Fire of Life. An Introduction to Animal Energetics
(Wiley, New York).
Linsteadt, S. L., and Calder III, W. A. (1981). Body size, physiological time and
longevity of homeothermic animals. Quart. Review of Biology 56, pp. 1–15.
Longair, M. S. (2004). Theoretical Concepts in Physics, 2nd Ed. (Cambridge
University Press).
McMahon, T. A. (1973). Size and Shape in Biology. Science 171, pp. 1201–1204.
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Mitchell, P. (1966). Chemiosmotic coupling in oxidative phosphorylation. Bio.
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Peters, R. H. (1983). The Ecological Implication of Body Size (Cambridge University Press).
Reich, P. B., Tjoelker, M. G., Machado, T. L., and Oleksyn, J. (2006): Universal
scaling of respiratory metabolism, size and nitrogen in plants, Nature 439,
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Suarez, R. K., and Darveau, C. A. (2005). Multi-level regulation and metabolic
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About the author
Lloyd Demetrius was educated at Cambridge University, England and
the University of Chicago, USA. His present research interests include the
application of ergodic theory and dynamical systems to the study of evolutionary processes in biology. He has been on the mathematics faculty
at the University of California, Berkeley, Brown University and Rutgers.
His current affiliations are the Max Planck Institute for molecular genetics,
Berlin, and Harvard University.
Chapter 9
Spectroscopy of the Genetic Code
Jim D. Bashford and Peter D. Jarvis
Discussions of the nature of the genetic code cannot be divorced from the
biological context of its origin and evolution. We briefly review some of the
main arguments that have been put forward for the evolution of the genetic code, together with the salient biological background. Longstanding
observations of genetic code regularities have led to combinatorially-based
assertions about its structure. However, it is also possible to extend such
“symmetry” descriptions to continuous symmetries or supersymmetries, especially in relation to the pattern of redundancy (degeneracy) of the genetic
code. We give an account of some recent work along these lines. This is
supported by graphical presentations, and some data fits, of samples of
measured physico-chemical properties of codons and amino acids across
the genetic code table. Finally, we review codon-anticodon recognition in
terms of conformational degrees of freedom, and structural, stereochemical
and kinetic considerations. Based on this, we suggest a possible role for
quantum processes at important stages of codon reading and translation.
9.1.
Background: Systematics of the Genetic Code
Discussion of the nature, and organization, of the genetic code dates from
almost before the early work on its detailed elucidation, and has spawned
a great variety of ingenious suggestions and insights.1
Received April 3, 2007
of the first such speculations was the so-called “diamond code” proposed by the
physicist Gamow (1954).
1 One
147
148
Quantum Aspects of Life
The dual aims of this chapter are firstly, to review physics-inspired approaches for describing and analysing the patterns of codon-amino acid
assignments that characterize the nature of the genetic code, almost universally across all life forms, and secondly, to discuss possible roles for
quantum processes within the genetic code recognition system.
The simplest abstraction of the “genetic code” is as a mapping of genetic
information, encoded by one type of biological macromolecule, the nucleic
acids, into another family, the amino acids, which constitute the building
blocks of proteins.2 The scheme is simple enough to state: a dictionary
of 64 possible code-words (codons), is associated with 20 amino acids plus
a “stop” signal. The vast bulk of built-in redundancy in this mapping is
conserved within all living organisms, and this strongly suggests that the
code in its present form is the result of an evolutionary process at the
molecular level, whereby the code derived from some more primitive form.
A common first step in attempting to describe the evolution of the genetic
code is simply to explain the two numbers, “64” and “20”.3
The remainder of this section is devoted to a rapid survey of the salient
biological and biochemical facts relating to the biomolecules and processes
involved in the genetic coding system, followed by a brief commentary on
the traditional “explanations” of the origins of the code. More recent,
information-theory perspectives, of potential relevance to a role for quantum processes, are also mentioned. In Sec. 9.2 the “systematics” of the
genetic code in terms of the observed patterns and regularities of the genetic code are developed in the group-theoretical language of dynamical
Lie algebras and superalgebras. In particular, we show that an sl(6/1)
supersymmetry model proposed by us [Bashford et al. (1998)], is able to
give an interpretation in this language, of two main models of code evolution developed quite independently in the biological literature, [Jiménez
Sanchez (1995); Jiménez-Montaño (1999)]. The most subtle stage of code
evolution relates to the third codon base (see below), and this is taken up in
Sec. 9.4, with a detailed discussion of codon-anticodon recognition taking
into account ribonucleotide conformational degrees of freedom. On this basis we suggest a possible role for quantum processes at important stages of
codon-anticodon reading. Meanwhile in Sec. 9.3, the dynamical symmetry
2 A recent review of the origin of the genetic code is, for example, Szathmáry (1999);
see also articles in the special issue Vol. 33, 2003 of Origins of Life and Evolution of the
Biosphere.
3 An entertaining discussion on how such numbers might relate to the Tower of Hanoi
Problem can be found in Berryman, Matthew J. (2006). Mathematic principles underlying genetic structures, http://www.manningclark/events/stars.Berryman.pdf.
149
Spectroscopy of the Genetic Code
description is corroborated by giving some simple numerical fits between
various measured biological and physico-chemical codon and amino acid
properties, and appropriate polynomials in the group labels, or “quantum”
numbers of the codons.
9.1.1.
RNA translation
The machinery of gene translation is dependent upon four major kinds
of biomolecule [Woese et al. (1966)]: mRNA—the gene transcript, which
contains sequences of codons; amino acids (a.a.)—the building blocks of
polypeptides; tRNA—the intermediary molecules, which carry amino acids
and recognise specific codons on mRNA via base-pairing with the anticodons which they present: and the amino-acyl (-tRNA-) synthetases,
aaRS —enzymes that bind specific amino acids to their cognate tRNA.
The basic reactions connecting these biomolecules are sketched in Fig. 9.1.
Other kinds of molecule also participate: the ribosome complex, at which
translation occurs, and elongation factor EF-Tu: a protein that transports
amino-acylated tRNAs to the ribosome. However these latter pathway components have more generic roles, independent of codon or amino acid properties, and will not be discussed in great detail. The basic unit of the
genetic code is the codon: a triplet of nucleotide bases. Four such bases,
Guanine, C ytosine, Adenosine and U racil (T hymine for DNA) occur naturally in mRNA. Therefore there exist 64 = 43 possible triplet combinations which are distributed, unequally, amongst 20 amino acids as shown in
Table 9.1. Even a cursory inspection of this, mitochondrial, genetic code
Amino acids
(20)
tRNA (anticodons)
(23< n <45)
aaRS
amino−acyl
tRNA
EF−Tu
GTP
mRNA (codons)
(n=64)
Fig. 9.1.
Ribosome
Codon
Reading
Organisational chart of key steps in codon reading.
150
Quantum Aspects of Life
Table 9.1.
a. a.a codon a.c.b
Mitochondrial genetic code.
a. a.a codon a.c.b
a. a.a codon a.c.b
a. a.a codon a.c.b
Phe
Phe
Leu
Leu
UUU GAA
UUC GAA
UUA UAA
UUG UAA
Ser
Ser
Ser
Ser
UCU
UCC
UCA
UCG
UGA
UGA
UGA
UGA
Tyr
Tyr
Ter
Ter
UAU
UAG
UAA
UAG
GUA
GUA
-
Cys
Cys
Trp
Trp
UGU
UGC
UGA
UGG
GCA
GCA
UCA
UCA
Leu
Leu
Leu
Leu
CUU
CUC
CUA
CUG
UAG
UAG
UAG
UAG
Pro
Pro
Pro
Pro
CCU
CCC
CCA
CCG
UGG
UGG
UGG
UGG
His
His
Gln
Gln
CAU
CAC
CAA
CAG
GUG
GUG
UUG
UUG
Arg
Arg
Arg
Arg
CGU
CGC
CGA
CGC
UCG
UCG
UCG
UCG
Ile
Ile
Met
Met
AUU
AUC
AUA
AUG
GAU
GAU
UAU
UAU
Thr
Thr
Thr
Thr
ACU
ACC
ACA
ACG
UGU
UGU
UGU
UGU
Asn
Asn
Lys
Lys
AAU
AAC
AAA
AAG
GUU
GUU
UUU
UUU
Ser
Ser
Ter
Ter
AGU
AGC
AGA
AGG
GCC
GCC
-
Val
Val
Val
Val
GUU
GUC
GUA
GUG
UAC
UAC
UAC
UAC
Ala
Ala
Ala
Ala
GCU
GCC
GCA
GCG
UGC
UGC
UGC
UGC
Asp
Asp
Glu
Glu
GAU
GAC
GAA
GAG
GUC
GUC
UUC
UUC
Gly
Gly
Gly
Gly
GGU
GGC
GGA
GGG
UCC
UCC
UCC
UCC
a
b
a.a. = amino acid.
a.c. = anticodon. Anticodon base modifications not shown.
reveals systematic patterns in the observed degeneracy of amino acids to
codons. Specifically, changes in the first and second codon letters always
change the coded amino acid, whilst changes in the third position often do
not. Furthermore, codons with the purine (R)-derivative bases (A and G)
in the third position always code for the same amino acid, as do those with
pyrimidine-derivatives (U and C )4 . The overall effect seen in Table 9.1 is
that codons cluster in families, with either 4-fold (“family boxes”) or 2fold degeneracies (“mixed boxes”), which all code for the same amino acid;
moreover there are two cases of “hexanumerate” codons in which both a
family box and a mixed box contribute. This structure is a direct consequence of the tRNA-mRNA pairing: bases in the first5 two positions of
the codon always bind to their complements (G with C, A with U ) on
the anticodon (a.c., as shown in Fig. 9.2). The binding in the third position is less precise. The nature of this ambiguous or “wobble” pairing,
first postulated by Crick (1966), is still not completely understood, and we
4 The
code for eukaryotic organisms is more involved, as will be discussed below.
to the standard (5’ carbon→3’ carbon) orientation of the sugar-phosphate
backbone; see Sec. 9.4.
5 Relative
Spectroscopy of the Genetic Code
151
will review current knowledge in a subsequent section (see Sec. 9.4), as it
forms the cornerstone of our suggestions for quantum processes in codon
reading.
Finally let us mention the fourth class of molecule, the aaRS . Each
enzyme contains a receptor for a specific amino acid, and binds to the
anticodon-containing region of the cognate tRNA. Detailed comparison
of aaRS structures [Eriani et al. (1990)] led to the discovery that two
structurally-distinct families of molecule exist. Furthermore, these structural motifs are strongly conserved amongst organisms with only one, primitive, exception amongst archaebacteria, discovered to date [Fabrega et al.
(2001)]. Remarkably, each aaRS class contains species cognate for 10 amino
acids, with the resulting families being “complete” in the sense that each
contains physicochemically-distinct (in terms of hydrophobicity and acidity) amino acids [Cavalcanti et al. (2004)], capable of producing key protein
structural motifs. Moreover the so-called class II aaRS are associated with
smaller, polar a.a.’s, commonly believed to have been incorporated in the
genetic code earlier than the bulkier residues of class I. This observation
has led to speculation that the modern genetic code formed via a “doublet”
predecessor.
9.1.2.
The nature of the code
During the 1960s and 1970s organisms as diverse as certain bacteria, viruses
and vertebrates were all found to have the same genetic code, leading to
the concept of a “universal” genetic code, present at least since the Last
Universal Common Ancestor (LUCA). This observed universality was the
motivation for the “frozen accident” hypothesis [Crick (1968)], which stated
that as evolution progressed, the increased complexity of proteins made
incorporation of new amino acids unlikely to be beneficial.
Although the “universal” genetic code incorporates 20 amino acids, recognized by the procedure in Fig. 9.1, several recently-discovered exceptions exist, whereby “new” amino acids are encoded by tRNA simultaneously binding to a “stop” codon and recognizing a secondary structural
motif. Examples include selenocysteine [Bock et al. (1991)] in eukaryotes and pyrrolysine [Srinivasan et al. (2002)] in archaebacteria. Differences in amino acid-codon assignments have also been discovered (for a
review see the paper by Osawa et al. (1992)) and currently 16 variants
on the “universal” code are catalogued on the NCBI Taxonomy webpage
http://www.ncbi.nlm.nih.gov/Taxonomy/Utils/wprintgc.cgi.
152
Quantum Aspects of Life
Table 9.2.
Wobble pairing rulesa .
First a.c. positionb
Third codon position
U
G
C
I
U, C, A, G
U, C
G
U, C, A
a
b
Adapted from [Osawa et al. (1992)].
Nucleoside modifications not shown.
In this paper we shall be concerned with two codes: the (vertebrate)
mitochondrial code (VMC), posited to be related to an ancestor of the universal or eukaryotic code (EC). The EC has qualitatively similar 2- and 4fold degeneracies, and codon assignments to amino acids, to those observed
for the VMC. As mentioned, these degeneracies are due to the ambiguous
or “wobble” nature of pairing between third codon, and first anticodon,
bases [Crick (1966); Agris (1991)]. Here it suffices to state the wobble rules
(Table 9.2); we shall discuss them in greater detail in Sec. 9.4. As seen from
Table 9.1, in the VMC only first a.c. position U and G are present, leading
to the characteristic “4” and “2+2” box degeneracies. While a.a.—codon
assignments are very similar in the EC, the anticodon usage is different.
Firstly the purine-derived base Inosine (I) replaces A (in all but a few exceptions) in the first a.c. position, while C may compete with I and U for
codons ending with G. We shall comment further on this competition in
Sec. 9.2.
Regularities inherent in the nucleobase “alphabet” allow discussion of
codon-amino acid relationships to be abstracted from a biochemical setting to a mathematical/logical one. Nucleobases are commonly classified in
terms of three dichotomous indices [Saenger (1984)]: Strong (G, C) versus
Weak (A, U/T ) pertaining to the number of H-bonds formed in canonical
pairs (Fig. 9.2); puRine-derived bases (A, G) contain two heterocyclic rings,
while pYrimidine bases (C, U/T ) have one. Thirdly one can distinguish
the proton acceptor/donor nature of the functional group attached to the
C1 atom: aMino (A, C) versus Keto (G, U/T ). Of course any two of these
indices are sufficient to establish the identity of any given nucleobase. Finally there exists the common notation aNy base, that is N = U/T, C, A, G.
In terms of this language, regularities in the code are easily expressed. Arguably the best known is Rumer’s rule [Rumer (1966)]: replacement of
the bases in codon positions I and II by their M/K counterparts changes
the nature of the 4-codon box. For example, the box U U N is split, with
153
Spectroscopy of the Genetic Code
H
N
O
H
N
Cytosine
N
H
N
N
N
N
Guanine
N
H
O
H
H
H
O
N
N
Uridine
N
N
Adenosine
H
N
N
N
O
Fig. 9.2.
Watson-Crick pairing of RNA bases.
codons U U Y and U U R coding for Phe(nylalanine) and Leu(cine) respectively. Replacing U (I) and U (II) by the other K-type base (G) changes the
structure to GGN , which is a family (unsplit) box, coding for Gly(cine).
Another, more recent example is the observed correlation [Biro et al. (2003);
Chechetkin (2006)] between class I and class II aaRS and anticodon families
of the forms
(W W W, W W S, SW W, SW S) , (SSS, SSW, W SS, W SW ) ,
(M M M, M M K, KM M, KM K) , (KKK, KKM, M M K, M KM ) .
Other regularities apparent in the code, in particular relating amino acid
and codon physico-chemical properties will be discussed in more detail in
subsequent sections.
Theories on the evolution of the code fall into one of three broad categories. The co-evolution theory posits that the genetic code evolved in
parallel with the emergence of increasingly complex amino acids [Wong
(1976); Weberndorfer et al. (2003)]. Thus similar amino acids would be
154
Quantum Aspects of Life
coded by similar codons because more recent a.a.’s “captured” codons from
their precursors. The physico-chemical hypothesis [Di Giulio (2003)] suggests that, at an early stage of evolution, direct contacts between amino
acids and codons/anticodons facilitated translation, dictating patterns of
physico-chemical regularities observed within the modern, universal code.
Finally the selection theory suggests that the code evolved to minimize phenotypic errors [Freeland et al. (2003); Ronneberg et al. (2000)] and, indeed,
secondary structure of mRN A transcripts [Shabalina et al. (2006)]. The
three streams of thought are not, however, mutually exclusive and each
mechanism may have influenced different stages of evolution.
9.1.3.
Information processing and the code
On an abstract level, the flow of genetic information from DNA to polypeptide can readily be viewed in terms of a digital code. Generally the Y /R
and S/W characteristics of each nucleobase are represented as “bits” and
regularities within patterns of codon and amino acid assignments are investigated [Freeland et al. (2003); Mac Dónaill and Manktelow (2004)]. For
example in a 2-bit scheme (1 each for Y /R and S/W ) Rumer’s conjugate
rule (G ↔ U , A ↔ C) can be implemented as the negation operation
[Négadi (2003)]. Informational aspects of coding evolution, including adaptor enzymes (aaRS) have been discussed by Nieselt Struwe and Wills (1997).
A fuller discussion of such coding labelling is given in Sec. 9.2 and Sec. 9.3.
From the evolutionary point of view however Gray, or error-checking
codes [Freeland et al. (2003)] are of particular interest. For example Mac
Dónaill (2003) proposed a 4-bit scheme, where 3 bits indicated proton
donor/acceptor sidegroups on the bases and 1 labelled the base Y /R nature,
viz,
G = (0, 1, 1; 0) , C = (1, 0, 0; 1)
aA = (1, 0, 1; 0) , U/T = (0, 1, 0; 1) .
Here “aA” (amino-Adenosine) was considered on theoretical grounds. With
base parity defined as the sum of all 1’s appearing in the corresponding
vector, it is clear that the fourth, Y /R bit acts as a parity check upon
permissible H-bonding patterns (i.e. those which do not disrupt the regular
helical geometries of DNA or RNA)6 . Using this scheme an extended set of
24 candidate bases was considered for inclusion in the nucleotide alphabet,
6 Changing from aA to A (deletion of the third “1” of aA) does not affect the even
parity of the resulting alphabet.
Spectroscopy of the Genetic Code
155
whereupon it was argued that same-parity alphabets have high recognition
fidelity, in contrast to those of mixed parity. Further, alphabets with even
parity, such as the natural one, are likely to be favoured over odd ones as
the occurrence of tautomers is generally less likely.
A different kind of information-processing hypothesis was proposed by
Patel (2001a,b,c) whereby DNA or protein assembly is reduced to a computational task. The problem is to determine a maximally-sized “database”
of items (nucleobases), which are readily distinguishable, with a minimal
number of search queries. For a database containing N randomly-ordered
items denote the number of binary (“yes/no”) queries, Q, required to locate
the desired item. In a classical ensemble, where rejected items are returned
to the database, the expected mean number of questions, Q = N/2.
However in a quantum-mechanical system, superpositions of items are
permitted, and Grover’s algorithm exploits this feature [Grover (1997)].
Starting with the symmetric initial state N −1/2 (|1 + |2 + . . . |N ), the
number of queries needed is determined by
(9.1)
(2Q + 1)ArcsinN −1/2 = π/2 .
The potential significance of several solutions of this equation for DNA
replication and expression have been pointed out by Patel (2001a,b,c). For
Q = 1 the database has N = 4 items which, if the “items” are nucleobases,
and (Watson-Crick) base-pairing the “quantum oracle”, has implications
for DNA replication. For Q = 3 (one question per base pair in a codon)
N ≃ 20.2, while the number of “letters” in the genetic code is 21 (20 amino
acids plus a stop signal). Lastly when Q = 2 one finds N ≃ 10.5, which is
potentially of interest in regard to the two classes of aaRS and amino acids
[Patel (2005)].
In order for quantum genetic information-processing to occur, as outlined above, there are two important considerations. Firstly quantum decoherence, which occurs far more quickly than proton tunnelling, at room
temperature needs to be mitigated. Indeed particular enzymes have been
suggested to facilitate proton tunnelling—for a recent review see the paper by Knapp and Klinman (2002)—in other reactions, via the exclusion
of H2 O, although whether DNA polymerases have such capabilities is unknown. Secondly there is the inference that, somehow a quantum superposition of base molecules is set up in the vicinity of the assembly site. It
is unclear precisely how an enzyme might achieve this. However, as has
recently been pointed out [Shapira et al. (2005)], the Grover algorithm can
produce advantageous searches under a variety of initial conditions including mixed states.
156
9.2.
Quantum Aspects of Life
Symmetries and Supersymmetries in the Genetic Code
Attempts to understand the non-random nature of the genetic code invite
a description in a more abstract setting. Combinatorial symmetry considerations amount to statements about certain transformations amongst the
basic ingredients, like bases and codons, or groupings thereof, which display
or predict regularities.7 A natural extension of this language is to formal
continuous linear transformations amongst the physical objects themselves.
This mathematical viewpoint thereby allows access to the rich theory of Lie
groups and their representations. In group theoretical language, the existence of the genetic code itself entails a simple counting problem—find the
semisimple Lie groups8 that have irreducible representations of dimension
64, the number of codons in the genetic code. Secondly, such a genetic code
group should have a “reading subgroup”, for which the dimension and multiplicity of its representations in the decomposition of the 64-dimensional
codon representation, coincides with, or is a refinement of, the known pattern of codon redundancies in the amino acid assignments. Such groups
and subgroups are candidates for “symmetries of the genetic code”.
The first work along these lines was carried out in a pioneering paper
by Hornos and Hornos (1993), and elaborated in Forger et al. (1997)—for
a review see Hornos et al. (1999). A comprehensive sort by rank and dimension of representations led to the almost unique identification of the
64-dimensional representation of the group Sp(6). It was shown that standard symmetry-breaking scenarios using eigenvalues of Casimir operators
in group-subgroup branching chains, for plausible assignments of codons
to basis vectors, could provide a quantitative and statistically significant
match to a certain major composite index of amino acid functionality, the
so-called Grantham polarity index. The group-subgroup branching chain
also gave support to the interpretation of the code as a “frozen accident”:
generically unequal polarity eigenvalues of certain distinct representations
of the “reading subgroup” are constrained by the numerical fit to be degenerate, because in practice they still code for the same amino acid.
7 There have been many studies attempting to unlock the “secret of the genetic code” by
careful examination of code patterns. See Sec. 9.1 for historical remarks, and Szathmáry
(1999) for a modern account. A recent orthodox study attempting to establish objective
support for code trends is Biro et al. (2003). Intriguing geometrical insights have been
developed by Yang (2005) (also see references therein).
8 A restriction to simple groups would proscribe the reasonable candidate SL(4) ×
SL(4) × SL(4) for example (see below); on the other hand, weakening the search criteria
(for example to non-classical groups or even reducible representations) would considerably complicate the discussion, and so is avoided just on technical grounds.
Spectroscopy of the Genetic Code
157
Subsequent work has exploited the connection between algebraic structures and allied transformation groups. In Bashford et al. (1997, 1998),
and Forger and Sachse (2000a,b), the notion of symmetry transformations
is generalized to that of supersymmetries, which make allowance for the
fact that objects in the underlying space may possess a grading, that is
an assigned “even”, “odd”, or in physics language “bosonic”, “fermionic”,
character. In Bashford et al. (1997, 1998) a specific type of scheme based
on the Lie superalgebra sl(6/1) is developed, and in Forger and Sachse
(2000a,a) more general possibilities are identified. The work of Frappat
et al. (1998), reviewed by Frappat et al. (2001) on the other hand, deviates
further from the Lie group—Lie algebra connection in exploiting a specific
type of “quantum”, or q-deformed algebra, here slq (2) × slq (2), in which
tensor products of representations reduce in a simple way, as desirable if
unique assignments of abstract state vectors to objects in the genetic code
system are to be maintained. An important paper which critically reviews
all group-based and related attempts at a description of the genetic code,
especially the Sp(6) models and an alternative SO(13) version, is the article
by Kent et al. (1998).
As reviewed in Sec. 9.1 above, current thinking is that the origin of the
genetic code is distinct from the origin of life itself, and that its present-day,
near-universal structure is the result of some evolutionary process of refinement from earlier, primitive versions. The potential of group-theory based
accounts is that, in contrast to combinatorial schemes, which merely serve
to express genetic code regularities via succinct statements about various
discrete transformations, the code structure as a whole can be described
in terms of a succession of group-subgroup steps, in a chain starting with
the initial codon group, and ending with the final “reading” group. In the
remainder of this section we describe the sl(6/1) model in some detail, in
this context. The focus is not so much on quantitative predictions (but
see Sec. 9.3 below), but rather to demonstrate that a group-theory based
account can indeed be broadly compatible with established biochemicallyand biologically-based understandings about the origin and evolution of the
code. The claim would be that an eventual, “dynamical code model” will
bear out the group-theoretical steps in detail.
A useful starting point for code degeneracy, compatible with both
physico-chemical and coevolution views of the code origin, is to regard the
outputs of the early translation system as “stochastic proteins”. Possibly,
early proto-amino acid/nucleic acid associations were useful in the context
of optimizing replication, and only incrementally acquired sufficient specificity for the synthesis of functional oligopeptides to emerge as an end in
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Quantum Aspects of Life
itself. It is reasonable to suggest that early coding for such primitive enzymes was quite non-specific and error-prone, but also, that the system
as a whole was error-tolerant. The group-theoretical counterpart of this is
that the degeneracies of the early code should be associated with the decomposition of the 64-dimensional representation of the codon group into
irreducible representations (irreps) of intermediate subgroup(s), such that
codon assignments within, or between, such subgroup irreps, respectively
minimize, or maximize, variations in amino acid properties at the largest
possible levels of functional synonymy.
As was pointed out in Sec. 9.1 above, a major determinant of amino
acid type is the character of the second codon base. Specifically weak bases
W = {A, U } are associated with hydrophilicity/hydrophobicity extremes
respectively—see Weber and Lacey (1978). Also, there is an argument from
biosynthetic complexity [Jiménez Sanchez (1995)] that the earliest utilized
bases should be the simplest chemically, namely A, U again (there are
of course other arguments, for example thermodynamic stability of codonanticodon pairs, [Baumann and Oro (1993)] for the strong bases S = {C, G}
to have been the earliest coding bases). Finally, it can be argued [Woese
et al. (1966)] that a minimal requirement for useful oligopeptides, should
be the existence of tunable hydrophobic/hydrophilic regions in the primary
structure, so as to allow the possibility of folding and the presentation of
stereochemically specific, enzymatically active contact regions.
9.2.1.
sl(6/1) model: UA+S scheme
The representation-theoretical equivalent is thus that there should be an
assignment of codons to a basis for the 64-dimensional representation of
the genetic code algebra, which is adapted to a subalgebra decomposition
which distinguishes the second base letter and assigns codons N AN and
N U N to different representations (necessarily of dimension 16). It turns
out that the class of superalgebras sl(n/1) possesses a family of so-called
typical irreducible representations of dimension 2n , which moreover branch
to members of the corresponding family under restriction to smaller subalgebras sl(n′ /1), sl(n′′ /1) with n′′ < n′ < n. This property, shared by
spinor representations of the orthogonal groups, singles out for attention in
the genetic code context the superalgebra sl(6/1), and its typical irreducible
representations of dimension 26 = 64, with Dynkin label (0, 0, 0, 0, 0; b) for
appropriate values of b > 5 (denoted hereafter by 64b ). The branching rule
Spectroscopy of the Genetic Code
159
(for n′ = n − 2 = 4) reads
sl(6/1) → sl(2)(2) × sl(4/1)(1,3) × gl(1) ,
64b → 1 × 16b+2 + 2 × 16b+1 + 1 × 16b ,
where the Dynkin label of the 24 = 16 dimensional typical irreducible representation 16 of sl(4/1) is given as a subscript,9 and the superscripts on
superalgebra labels (or multiplets as needed) refer to the codon positions
on which the subalgebra factors act. Making the natural identification of
the 1’s with the A and U codons as suggested by the above discussion, and
assigning the strong bases S to the doublet 2, a more descriptive form of
the branching rule is thus
64b → 1A × 16b+2 + 2S × 16b+1 + 1U × 16b
with the understanding that the codon groups being assigned to the symmetry adapted bases for the subalgebra representations are N AN , N SN ,
and N U N , respectively.
From the standpoint that redundancy in codon reading and amino acid
translation, equates with degeneracy in codon assignments to irreducible
representations in the group theoretical schemes, it can be suggested that
this stage of code evolution would have corresponded to the existence of
three proto-amino acids, or possibly three groups of amino acids with shared
functional uses within each group. Alternatively, in the earliest stages the
middle N SN group could have simply been unassigned to a definite amino
acid coding role. Code elaboration became possible once the developing
translation system had achieved a requisite degree of accuracy and reliability. Further major determinants of amino acid assignments to codons
are once again the precise identity of the second codon base (thus, not
just a coding role for the N SN group, but perhaps separately for N CN
and N GN ), but also the modulation of codon assignments afforded by the
identity of the first codon base. Both options are plausible, and lead to
different group branching scenarios.
Consider, for example, the second option. It is natural to repeat, at the
level of the first base letter, the previous branching pattern, this time at
the level of sl(4/1)(1,3) → sl(2)(1) × sl(2/1)(3) × gl(1),
16b′ → 1 × 4b′ +2 + 2 × 4b′ +1 + 1 × 4b′ ,
9 This label is closely related to the weight of the irreducible representations of gl(1)
which occur in the decomposition; however these are not given explicitly.
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Quantum Aspects of Life
where again the gl(1) label has been omitted in favour of the related nonzero
Dynkin index of the sl(2/1)(3) typical irreps 4 (given as a subscript). At
this stage the full list of sl(2)(1) × sl(2)(2) × sl(2/1)(3) irreps (again omitting
gl(1) factors but including the nonzero Dynkin label of the sl(2/1) third
base letter quartets) in the decomposition of the codon representation is
64b → (1 × 1 × 4b+2 + 1 × 2 × 4b+3 + 1 × 1 × 4b+4 )
+ (2 × 1 × 4b+3 + 2 × 2 × 4b+2 + 2 × 1 × 4b+1 )
+ (1 × 1 × 4b + 1 × 2 × 4b+1 + 1 × 1 × 4b+2 ) ,
corresponding to the codon groups
(AU N + ASN + AAN )
+ (SU N + SSN + SAN )
+ (U U N + U SN + U AN )
(9.2)
respectively. Once again, depending on whether the codons with middle letter S are translated or unassigned (or ambiguous), this code stage suggests
5 or 6, or possibly as many as 8 or 9, active groups of mutually exchangeable
proto-amino acids. This group-theoretical description closely matches the
scheme for code evolution proposed by Jiménez Sanchez (1995) where the
weak U, A bases are argued to be the first informative parts of primordial
(three-letter) codons (with the strong bases merely providing stability for
the codon-anticodon association). Van den Elsen and coworkers have argued for an intermediate expansion stage of evolution of the genetic triplet
code via two types of doublet codons, namely both “prefix codons” in which
both the middle and first bases are read (as in the above scenario), but also
“suffix codons” involving reading of the middle and third codon bases [Wu
et al. (2005)]. Conflicts are resolved by allowing certain amino acids to
possess both prefix and suffix codons, which are still visible in the present
eukaryotic code in the form of the six-fold codon degeneracies for Arg, Leu
and Ser. It should be noted that the above scheme involving A, U and
S in both first and second bases, which has been introduced as a partial
doublet prefix codon genetic code, could also develop suffix codon reading;
for example SW N → SW W , and SSN → SSW wherein the third base
position is read and the remaining S base positions confer stability.
The final step in this scheme is the breaking of the strong base sl(2)
symmetries which hold C, G bases in the first and second codon positions
degenerate (or unassigned). If the second codon position is the major
determinant of amino acid differentiation, then the sl(2)(2) breaking step
Spectroscopy of the Genetic Code
161
proceeds first, yielding (referring to (9.2) above)
four W W N quartets ;
ASN and U SN → four quartets ACN, AGN, U CN, U GN ;
two unbroken octets SU N , SAN ;
SSN → two octets SCN, SGN .
The code has thus expanded to eight degenerate quartets and four octets,
for up to twelve readable amino acids.10 The final step is first base position
symmetry breaking leading to 16 family boxes, with both first and second
base positions being read. This proposal for code evolution, with or without
the variation of prefix and suffix doublet codons, can be referred to as the
“UA+S” scheme, to distinguish it from the following alternative model.
9.2.2.
sl(6/1) model: 3CH scheme
A somewhat different proposal for the origin and organization of the genetic code has been developed by Jiménez-Montaño et al. (1996); JiménezMontaño (1999) under the motto “protein evolution drives the evolution
of the genetic code, and vice-versa”. According to this scenario, the code
has evolved by sequential full elaboration of the second codon base letter, followed by the first (and lastly the third); however at each stage
pyrimidine-purine reading occurs, before further strong-weak base reading within the Y, R types. This is argued by strictly applying a systematic
criterion of code evolution via incremental, minimum change coding pathways, whereby the hierarchical order of codon-anticodon Gibbs free energy
of interaction, C2 > H2 > C1 > H1 (followed by · · · > C3 > H3 ), which
can be established in vitro, is adopted to infer a temporal sequence of code
expansion. This means that the chemical type C = {Y, R} (that is, whether
bases are pyrimidines Y or purines R), and then the hydrogen bonding type
H = {S, W } (that is, whether bases are strong, S, or weak, W ) of the
second, first (and finally third) codon base are successively able to be read
by the evolving translation system.
A group-theoretical branching scheme reflecting this scenario would entail symmetry breaking of transformations on successively the second, first
(and lastly the third) base letters, in contrast to the hierarchical UA + S
scheme’s adoption of the partial A, U and S breaking scheme on the second
10 In the account of Jiménez Sanchez (1995), the original code used triplet codons entirely
of the W W W form, which later acquired S codons in all positions.
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Quantum Aspects of Life
and first base letters before differentiation of the S bases into C, G. Thus
corresponding to the chemical type Y, R, within each of which is in turn a
strong and a weak base (Y includes C, U and R includes G, A respectively),
each base quartet is assigned two dichotomic labels, which serve to distinguish subgroup transformations. Remarkably group-theoretical branching
rules incorporating these steps are once again natural within the class of
64-dimensional typical representations of the sl(6/1) superalgebra. In this
scheme the required labels are eigenvalues of gl(1) generators, and the relevant sl(n′ /1), sl(n′′ /1) subalgebras are now successively n′ = n − 1 = 5,
n′′ = n′ − 1 = 4, rather than n′ = n − 2 = 4, n′′ = n − 4 = 2 as in the
UA + S scheme. The branching rules read finally (with the gl(1)’s being
tagged according to whether they refer to chemical or hydrogen bonding
type),
sl(6/1) → sl(5/1) × gl(1)C2 ,
64 → 32Y + 32R ∼
= N Y N + N RN ;
sl(5/1) × gl(1)C2 → sl(4/1)(1,3) × gl(1)H2 × gl(1)C2 ,
32Y → 16YS + 16YW , 32R → 16RS + 16RW ,
N Y N → N CN + N U N, N RN → N GN + N AN .
This pattern is repeated for the first codon base letter giving eventually
sl(6/1) → sl(2/1)(3) × gl(1)H1 × gl(1)C1 × gl(1)H2 × gl(1)C2
with 16 codon quartets in which the first two bases are read. This scenario
can be referred to as the “3CH” scheme.
A variant of this picture was in fact proposed earlier by Swanson (1984).
That version considered code elaboration based on a C2 > C1 > H2 > H1
hierarchy. In the sl(6/1) model, the corresponding subalgebra branching
pattern would be
sl(6/1) → sl(4/1)(1,3) × sl(2)(3) × gl′ (1)
→ sl(4/1)(1,3) × gl(1)(3) × gl(1)′
→ sl(2/1)(3) × sl(2)(1) × gl′′ (1) × gl(1)(3) × gl(1)′
→ sl(2/1)(3) × gl(1)(1) × gl′′ (1) × gl(1)(3) × gl(1)′ .
However, in a quantitative study of the effect of base changes on amino
acid similarity across the code (using amino acid correlation matrices from
alignment methodologies), it was shown by Mac Dónaill and Manktelow
Spectroscopy of the Genetic Code
163
Fig. 9.3. Code evolution according to Jiménez Sanchez (1995) transcribed into a group
branching scheme in the sl(6/1) chain. Dynamical symmetry breaking stages: I:
sl(1,3) (4/1)×gl(2) (2)S ×gl(2) (1). II: sl(2/1)(3) ×gl(1) (2)S ×gl(1) (1)×gl(2) (2)S ×gl(2) (1).
III: sl(2/1)(3) × gl(1) (2)S × gl(1) (1)× gl(2) (1)m × gl(2) (1)d . IV : sl(2/1)(3) × gl(1) (1)m ×
gl(1) (1)d ×gl(2) (1)m ×gl(2) (1)d .
(2004) that this scheme is less supported than the standard 3CH version
above.
Thus far, both group-theoretical branching scenarios based on the
sl(6/1) scheme have arrived at the 16 codon boxes (quartets) of the standard genetic code, regarded as 4-dimensional irreps of the residual third
letter sl(2/1)(3) dynamical symmetry to which both branching chains reduce. These are shown in Figs. 9.3 and 9.4 for the UA+S and 3CH schemes
respectively (compare Fig. 9.3 with (Jiménez Sanchez, 1995) Table 2 and
Fig. 2, and Fig. 9.4 with (Jiménez-Montaño, 1999), Figs. 1a, 1b and 1c).
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Quantum Aspects of Life
The SO(13) model mentioned above, would have similar counterparts because of the intimate relation between the typical irreducible representations of the gl(n/1) superalgebras and the spinor representations of SO(2n)
or SO(2n + 1) groups.11
9.2.3.
Dynamical symmetry breaking and third base wobble
The final stage in code evolution is the expansion of the amino acid repertoire via reading of the third codon letter. The relevant feature of the
genetic code in this respect, is that the canonical Crick-Watson pairing between the codon base letters on the mRN A strand, and the tRN A base
anticodon (recognition) letters, breaks down. Namely, the “first”, 5’-3’ base
of the anticodon triplet (which structurally occurs at base position 34 in
the so-called anticodon loop of 7 bases in each tRNA), admits the so-called
Crick wobble pairing with respect to the third codon base, which is more
flexible than canonical pairing.12 The “degeneracy of the genetic code” as
a whole, is in fact a convolution of the association between the 45 or so used
anticodons and amino acids (and the 20-strong aaRS enzyme system), and
the wobble pairing. Indeed, pairing at the third codon position determines
almost all of the degeneracy in the genetic code and, as such, is least correlated with amino acid properties. On the other hand, as will be seen in
detail below, the pattern of such pairing depends upon genomic G+C content, and also post-transcriptional modification of tRNA bases; usually at
bases 34 (a.c. position 1) and also at base 37 (downstream of a.c. position
3). It is reasonable to contend then that reading at this codon position is
associated with the latter stages of evolution of the genetic code, the basic
translation apparatus necessarily already having been established.
The viewpoint adopted here is that the dynamical symmetry description
must relate to codon-anticodon binding and amino acid recognition as a
whole; in the UA + S scheme stereochemical or other considerations are
dominant in organizing coding according to hydrophobicity; in the 3CH
scheme the free energy of formation is the major determinant of coding.
11 In the first scenario the UA + S split is natural within spinor reductions SO(n) →
SO(n − 4) × SO(4), wherein a four dimensional spinor of SO(4) ≃ SU (2) × SU (2)
decomposes into a direct sum 1 + 1 + 2 with respect to one of the SU (2) factors. The
second scenario is compatible with successive branchings of the form SO(n) → SO(n −
2) × SO(2) wherein a spinor representation of a certain dimension reduces to a pair of
spinors of the subgroup.
12 The explicit notation N (34) · N ′ (III) can be used to denote this anticodon-codon base
(wobble) pairing, or simply N · N ′ where no confusion arises.
Spectroscopy of the Genetic Code
165
Fig. 9.4. Code evolution steps according to Jiménez-Montaño (1999) transcribed
into a group branching scheme in the sl(6/1) chain. Dynamical symmetry breaking stages: I: sl(5/1) × gl(2) (1)m−d . II: sl(4/1)(3,1) × gl(2) (1)m−d × gl(2) (1)m+d . III:
sl(3/1)×gl(1) (1)m−d×gl(2) (1)m−d×gl(2) (1)m+d . IV : sl(2/1)(3)×gl(1) (1)m−d×gl(1) (1)m+d×
gl(2) (1)m−d ×gl(2) (1)m+d .
For the third codon base, attention is naturally focussed on the patterns
not only of amino acid assignments within codon boxes, but also, in view
of the wobble pairing, of anticodon usage.
We now take up in detail these issues of the codon-anticodon degeneracy
and the wobble rules. The mitochondrial codes, believed to show similar
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Quantum Aspects of Life
structural simplifications to a hypothetical ancestral code,13 are especially
simple: in each family box, one tRNA codes; Uridine U (34) in the first a.c.
position can pair with U , C, A, or G—in the case of U · A by canonical
pairing, and in U · U , U · C and U · G by wobble pairing. Mixed boxes on
the other hand have two tRNA species; Guanine G(34) recognizes U and
C—via wobble pairing in G · U and canonical pairing in G · C, while U (34)
binds A and G—again via canonical pairing in U ·A, and in U ·G by wobble
pairing. Uridine U (34) in these mixed boxes is commonly prevented from
misreading U and C by chemical modification (and in the family boxes, may
indeed undergo different modification to facilitate the U · N wobbles).14
The dynamical symmetry description of code elaboration for the third
codon position in the mitochondrial codes is therefore rather straightforward. Codon boxes (quartets) either lead to family boxes (“intact”
sl(2/1)(3) irreps), or reduce for mixed boxes to two doublets of some subalgebra, N1 N2 N → N1 N2 Y + N1 N2 R. It follows that the symmetry breaking
at the third position is partial (8 family, and 8 mixed, boxes); moreover the
pattern of breaking turns out to be specified completely by the identity
of bases occupying the first two codon positions. We shall return to these
points presently, along with a specific choice for the unbroken subalgebra.
In higher organisms, which use the “universal” genetic code (for example, the eukaryotic code), codon usage is strongly linked with genomic
base content. In particular, within the open reading frames in a genome,
the G+C content of the third codon position correlates well with genomic
G + C content, in contrast to the other positions. In genomes with high
A+T content, G+C-rich codons are seldom used, and can even be deleted
from the code (for example CGG in Mycoplasma capricolum [Andachi et al.
(1989)]). Conversely, if G + C content is high, A + U -rich codons become
rare, and may also disappear (for example N N A within family boxes in
Micrococcus luteus [Kano et al. (1991)]). In between these extremes, different tRNA species can “compete” for the same set of synonymous codons.
Representative patterns of codon degeneracy within a box are shown in
Table 9.3. The 2 + 2 and 2 + 1 + 1 patterns on the left-hand side of Table 9.3 are utilized in seven of the eight mixed boxes, while the 2 + 1 + 1
patterns on the right (the triplet of anticodons involving Inosine is actually
2 + 1 arising from N N Y + N N A) determine seven of the family boxes.
13 Code variations across biota have been intensively studied in recent years; see for
example Osawa et al. (1992) for a comprehensive early review.
14 We shall discuss further the effects of post-transcriptional modification in a quantal
model of codon recognition in Sec. 9.4.
Spectroscopy of the Genetic Code
Table 9.3.
otes.
codon
NNU
NNC
NNA
NNG
167
Anticodon usage patterns in eukary-
mixed boxes
a.c
GN N
GN N
UNN
(CN N, U N N )
family boxes
codon
a.c.
NNU
NNC
NNA
NNG
IN N
IN N
(IN N, U N N )
(CN N, U N N )
The exceptions are boxes AU N (in which Met has the single codon AU G,
and Ile is coded for by three codons AU Y and AU C) and GGN (the Gly
family box) in which these patterns are reversed. However, as emphasized
already, we also need to consider how codon usage shifts with genomic base
content. For A+T -rich genomes, codons N N G are selected against, resulting in the probable disappearance of anticodons CN N . In this instance,
the codon/anticodon box pattern coincides with that of the mitochondrial
code 2 + 2 mixed boxes, N1 N2 N → N1 N2 Y + N1 N2 R. Conversely, in
G+C-rich organisms codon N N A is relatively rare, and consequently there
is little need for tRNA species U N N : recognition of codon N N G is predominantly due to anticodon CN N . Thus, in this case there tends to be
a further “breaking” of the N N R codon doublet assignment leading to
N1 N2 N → N1 N2 Y + N1 N2 R → N1 N2 Y + N1 N2 A + N1 N2 G.
It remains to tie the third base reading patterns to a dynamical symmetry breaking account relating to the proposed sl(2/1)(3) codon quartet
superalgebra. We suggest [Bashford et al. (1998)] that the basic pattern
of sl(2/1)(3) breaking for the primitive and mitochondrial codes is to a
gl(1/1)(3) superalgebra, which is further reduced to an appropriate gl(1)(3)
label for the eukaryotic code. gl(1/1) is the well-known superalgebra of
supersymmetric quantum mechanics , with supercharge generators Q± satisfying {Q− , Q+ } = H, [F, Q± ] = ±Q± where H is the “Hamiltonian”
and F (with eigenvalues = 0, 1) labels fermion number. Irreducible representations are generically two-dimensional, so that quartets of sl(2/1)(3)
decompose under gl(1/1)(3) to two degenerate doublets as 4 → 2 + 2
(the mixed box codon-antiocodon pattern). Partial symmetry breaking—
the fact that 8 family boxes remain intact, and do not show this codonanticodon splitting—must be attributed to the varying strength of this
breaking across the code. The same applies to the final 2 + 2 → 2 + 1 + 1
decomposition manifested in the eukaryotic code, which can be attributed
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Quantum Aspects of Life
in turn to partial gl(1/1)(3) breaking, this time to the generator F of its
Abelian gl(1)(3) subalgebra. The codon-anticodon doublet degeneracy is
potentially completely lifted by the additional fermion number-dependent
shift; however again this is realized only in certain of the mixed boxes for
N1 N2 R, and never for N1 N2 Y .
As mentioned already, the existence of partial symmetry breaking was
argued by Hornos and Hornos (1993) to support the “frozen accident” account of the structure of the genetic code (see also Sec. 9.1). In terms
of dynamical symmetry breaking, appropriate breaking parameters are to
be fine-tuned, so that otherwise non-degenerate codons remain degenerate,
and can consistently be assigned the same amino acids. The mechanism
operates similarly in our present sl(6/1) scheme, except that we have been
discussing codon-anticodon pairings rather than codon-amino acid assignments (which are not the same if different tRNA’s can be charged with the
same amino acid). Moreover, we have linked the emergence of 2 + 1 + 1
pairing to genomic G+C content, and so effectively injected an organismdependence into the breaking patterns. Further numerical aspects of the
partial breaking, and of the related issue of codon-amino acid assignments,
are given in Sec. 9.3. A more refined discussion of the codon-anticodon
recognition process is given in Sec. 9.4.
9.3.
Visualizing the Genetic Code
The discussion of the genetic code so far has centred on qualitative aspects
of its systematics. These include both longstanding trends, noticed almost
as soon as the code was fully elucidated in the 1960s (Sec. 9.1), as well as
more elaborate Lie symmetry and supersymmetry-based schemes (Sec. 9.2),
which served to transcribe selected accounts grounded in biological understanding, into a mathematical language.
The utility of broken dynamical symmetry schemes in physics is specifically, in helping to quantify the hierarchy of symmetry and symmetrybreaking in the spectroscopy of complex quantum systems such as atoms,
molecules and nuclei. Schematically, suppose that a Hamiltonian operator
can be constructed as a series of the form
H = H0 + H1 + H2 + · · ·
where the terms are successively “smaller” in the appropriate sense. Also,
on the Hilbert space of quantum-mechanical states of the system, suppose there are operators representing the transformations of a hierarchy of
Spectroscopy of the Genetic Code
169
symmetry groups G0 ⊃ G1 ⊃ G2 ⊃ · · · such that G0 is a symmetry of
(commutes with) H0 , G1 commutes with H1 , · · · , and so on. Then, by
general theorems, the energy eigenfunctions of the system (the energy levels of physical states) are organized into unitary irreducible representations
of the successive subgroups. The spectra of the partial Hamiltonians H0 ,
H0 + H1 , H0 + H1 + H2 can be labelled by these irreducible representations, each of which corresponds to states with degenerate energy levels.
Moreover, as the corrections introduced become smaller, this labelling thus
provides a hierarchical “symmetry breaking” scheme for understanding the
structure of the system. In ideal cases the contributions to H are moreover appropriate combinations of so-called Casimir operators of the various
subgroups, such that when the states are accorded their correct ancestry in
terms of the descending hierarchy of subgroups and respective irreducible
representations, their energies (eigenvalues of H) are the corresponding sum
of Casimir eigenvalues (polynomials in the labels, or quantum numbers, of
the respective representations, for example the highest weight labels).
This methodological approach has indeed been taken, with some success, for the genetic code problem in the work of Hornos and Hornos (1993).
As mentioned in Sec. 9.2, the symmetry groups were taken to be a chain of
subgroups of Sp(6), with codons assigned to its 64-dimensional representation, with the role of the energy being played by a composite measure of
codon and amino acid organization, the Grantham polarity index. An attractive feature of the argument was that although the symmetry breaking
chain taken implied complete degeneracy in the generic case, the “frozen
accident” visible in the instances of synonymous codon assignments in the
real genetic code could be explained by particular parameter constraints
between the strengths with which Casimirs belonging to the partial Hamiltonians appeared in the total Hamiltonian H.
In our work we have taken a somewhat weaker approach to quantifying
the structure of the genetic code. Within the group branching scenario, it is
often sufficient15 to distinguish states within an irreducible representation
of a starting group G by their so-called weights, which are labels for (one
dimensional) representations of the smallest available continuous subgroup,
the Cartan (maximal Abelian) subgroup. Thus a parametrization of physical properties via fitted polynomials in these labels, can be regarded as a
kind of general proxy for the more specific approach sketched above, where
a definite subgroup chain is declared, and specific Casimir operators are
15 Technically
the irreducible representation of G must have no weight multiplicities.
170
Quantum Aspects of Life
included at the outset. It is this more flexible method that we have used
as an attempted confirmation of the sl(6/1)-based supersymmetric schemes
for the structure of the genetic code in Sec. 9.2 above. It is apparent from
the discussion in Sec. 9.2 that the subgroup and state labelling required is
closely matched to the four base letter alphabet, three letter word lexicon
of the genetic code. Mention has already been made of the fact that the
nucleic acid bases stand in very symmetrical relationships with respect to
each other, and it is natural to reflect this in the state labelling appropriate to the 64-dimensional codon “space” (see the introductory discussion
in Sec. 9.2 above). Indeed, any bipartite labelling system which identifies
each of the four bases A, C, G, U , extends naturally to a composite labelling
for codons, and hence amino acids. We choose for bases two coordinates
d, m = 0, ±1 as A = (−1, 0), C = (0, −1), G = (0, 1), U = (1, 0), so that
codons are labelled as ordered sextuplets, N N N = (d1 , m1 , d2 , m2 , d3 , m3 );
for example ACG = (−1, 0, 0, −1, 0, 1). Our choice of dichotomic base
labels is of course equivalent to 0, 1 binary labelling and the geometrical picture of the code as a 6 dimensional hypercube16 as has been noted
by several authors (see the discussion above). Fitting polynomial functions in these labels to code properties is furthermore compatible with any
group labelling scheme for which the 64-dimensional codon representation
is equivalent to a hypercube in weight space; as discussed earlier, candidate
groups and algebras include sl(6/1) but also so(13), and non-simple groups
such as so(4) × so(4) × so(4). With these preliminaries it only remains to
present sample genetic code (codon and or amino acid), physico-chemical or
biological, data, and compare this data to polynomials in the codon labels.
Figure 9.5 gives a two-dimensional presentation of the genetic code
whereby each of the m, d paired labels for each base letter are plotted or
projected onto the plane. In the case of the first two base letters this occurs
by showing four d2 , m2 diamonds separated by their different d1 , m1 coordinates; for the third base letter instead a linear rank ordering U, C, A, G
of the bases is used (corresponding to a one-dimensional projection of the
diamond to a line skew to the sides of the basic diamond). Remarkably,
essentially this organization of the code was discussed some time ago by
Siemion (1994) in connection with so-called “mutation rings”, designed to
present a rank ordering of codons reflecting their relative interconvertability or functional similarity (so that near neighbours in the mutation ring
are also likely to be correlated in their occurrences in nucleic acid coding).
16 The d, m labels give a diamond rather than a square orientation to the fundamental
base quartet.
Spectroscopy of the Genetic Code
Fig. 9.5.
171
Diamond presentation of codon labelling N N N = (d1 ,m1 ,d2 ,m2 ,d3 ,m3) .
Fig. 9.6. Genetic code “mutation rings” according to Siemion. The “mutation number”
0 ≤ k ≤ 63, labels the looping closed path around the main second base letter rings,
with excursions into and out of the G ring starting with GAU (Asp).
Figure 9.6 shows Siemion’s rings, with the linear rank ordering (related by
Siemion to a “mutation number” 0 ≤ k ≤ 63), following a looping closed
path around the main second base letter rings with excursions into and out
of the G ring starting with GAU (Asp) (compare Fig. 9.5, which has had
the G2 = {d2 = 0, m2 = 1} ring shifted downwards, with Fig. 9.6).
Important quantitative indicators of coding functions are the so-called
Chou-Fasman parameters, which give log frequency measures of the presence of each amino acid in protein tertiary structures such as β sheets and
172
Quantum Aspects of Life
α helices and associated turns. We have fitted several of these parameters to the codon weight labels as described above, and we present here
representative fits—taken from (Bashford and Jarvis, 2000). In Figs. 9.8
and 9.9 are plotted histograms of the P α and P β parameters against the
Siemion number, together with least-squares polynomial fits to functions
F α (d1 ,m1 ,d2 ,m2 ,d3 ,m3), F β (d1 ,m1 ,d2 ,m2 ,d3 ,m3) given by
F α = 0.86 + 0.24d22 + 0.21m1m2 (m2 − 1) − 0.02(d3 − m3 )
− 0.075d22 (d3 − m3 ) ,
F β = 1.02 + 0.26d2 + 0.09d21 − 0.19d2 (d1 − m1 ) − 0.1d1 m2 (m2 − 1)
− 0.16m21 m2 (m2 − 1) ,
respectively.
In Figs. 9.7(a) to 9.7(f) measured values for a selection of further experimental parameters are plotted (without any fitting), not against Siemion
number, but as histograms over the rings themselves.17 It is clear from
the fitted plots Figs. 9.8 and 9.9 and from these further plots that simple
numerical fitting of the type given in (9.3) can capture the major trends
in such genetic code data. For example in Bashford and Jarvis (2000), it
was found that for the Grantham polarity itself, the important terms were
simply d2 (second codon base hydrophobicity) and d3 − m3 (third base
chemical Y /R type) with appropriate coefficients, modulo some first base
dependence. Similar numerical considerations also support the “partial
symmetry breaking” scenarios. For example, the polynomial
1 2 2
(9.3)
d m (1 + m2 ) + m21 d22 (1 − d2 )
(d1 d2 )2 +
2 1 2
takes the value 1 on W W N , W GN and SAN and 0 on SSN , W CN and
SU N and so serves to “turn on” the partial codon-anticodon sl(2/1)(3)
breaking leading to mixed versus family boxes (the key parameter underlying Rumer’s rule [Rumer (1966)]). Finally it can be noted that the difference between codon-anticodon pairing degeneracy and codon-amino acid
assignment synonymy also has numerical support: periodicity or symmetry
patterns of codon-amino acid properties over the Siemion rings is consistent
with repeated amino acid assignments (belonging to different codon boxes)
occurring on certain symmetrical ring locations—see Bashford and Jarvis
(2000); Siemion (1994).
17 Recently entire databases of physico-chemical and biological codon and amino acid
properties have become available; see for example Kawashima and Kanehisa (2000).
173
Spectroscopy of the Genetic Code
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 9.7. Histograms of physicochemical parameters superimposed upon Siemion’s
rings: (a) aaRS synthetase class I=0, II=1; Chou-Fasman parameters relating (b) to
beta sheets and (c) coils [Jiang et al. (1998)]; (d) pKb (a measure of codon polarity);
[Sober (1970)] (e) hydrophobicity [Bull and Breese (1974)] and (f) isoelectronic potential
[Sober (1970)].
174
Quantum Aspects of Life
Fig. 9.8. P α versus k. Histogram: data; solid and dashed curves: polynomial fits (four
parameters); dots: preferred codon positions.
Fig. 9.9. P β versus k. Histogram: data; solid curve: polynomial least squares fit (five
parameters).
9.4.
Quantum Aspects of Codon Recognition
In this section we present the proposition that the codon-anticodon recognition process has an initial, quantum-mechanical step. Previously, Patel
(2001a,b) discussed the genetic code in terms of quantum information processing, however despite the striking numerical predictions stemming from
Grover’s search algorithm, the model required some unlikely properties of
enzymes.
Spectroscopy of the Genetic Code
175
Our basic assertion rests on the observation that the first anticodon base
(labelled henceforth as N (34)) is conformationally flexible, whereas a.c.
sites 35, 36 are constrained by the geometry of the tRNA anticodon loop
(in addition to modifications to base 37). In an unpaired tRNA, N (34) could
therefore be expected to be in a superposition of conformational states. In
proximity to the complementary codon base, one such state becomes increasingly favoured, facilitating the “collapse” to the classical, paired state.
Thus, in contrast to the Patel picture, the superposition of nucleobase states
occurs at a structural, rather than chemical level. There is still the issue
of thermal effects; however in this regard, we note that aminoacyl-tRNA
is transported to the ribosome by elongation factor (EF-Tu). There are
thus two distinct tRNA-protein environments, in either of which quantum
coherence could be maintained.
9.4.1.
N(34) conformational symmetry
In order to develop this quantal hypothesis it is necessary to first discuss nucleobase conformational states. The RNA oligomer is formed of repeated
ribonucleotide-phosphate units, one of which is sketched in Fig. 9.10(a).
The conformer degrees of freedom fall into three broad categories—for a
full discussion see Yokoyama and Nishimura (1995). First are the torsion angles between ribonucleotide and phosphate groups: there are respectively three C4’-C5’ (gg, gt and tg) and two C3’-O3’ (G± ) bond rotamers. Secondly there is a twofold degree of freedom (anti/syn) describing the relative orientation of the base to the ribose ring. Only R-type
bases can form two H-bonds (commonly argued to be the minimum required for recognition) in the syn conformation, however such R · R pairings are not observed in vivo [Yokoyama and Nishimura (1995)]. Finally
there is a nonplanar deformation of the ribose ring (Figure 9.10(b)), commonly described by the pseudorotation parameter τ . Typically one of two
conformers: C2’-endo (τ ≃ 180◦) or C3’-endo (τ ≃ 0◦ ) is favoured, as
sketched in Fig. 9.10(b). Note however that other states, such as O4’-exo,
may also become favourable under special circumstances. By correlating
these degrees of freedom with stereochemical considerations arising from
linking nucleobase units, it is possible to identify likely, low-energy conformer states. For example, according to Altona and Sundralingam (1972),
for mononucleosides in the solid state favoured low-energy conformers of R
and Y bases can be summarized as in Table 9.4. Any extrapolation to duplex RNA is likely to restrict the number of favourable states even further.
176
Quantum Aspects of Life
0
1
0
1
0
1
0
1
0
1
0
1
P
−
0
1
O 0000
O
00
11
1111
00
11
00
11
00
11
00
11
00
11
00
11
00O5’
11
00
11
00
11
00
11
X=U, C, A, T
00
11
11
00
00
11
11
00
11
00
11
00
11
00
0000
1111
00
11
0000
1111
00
11
O4’
(anti,syn)
C5’
0000
1111
00000011
111111
0000
1111
000000
111111
00000000
111111
00
11
0000111111
1111
000000
111111
00
11
0000000000
1111
000000
111111
000000
111111
00
11
0000
1111
000000
111111
000000
111111
00
11
0000
1111
111
000
111
000
000000
111111
000000
111111
00
11
(gg, gt, tg)
111
000
111
000
111
000 C1’
111
000
C4’ 111
111
000
000
111
000
000 111111
111
000
111
000 000000
111
000 C3’
111
C2’
000
111
000
111
000
111
−
+
000
111
( G , G )111
000
111
000
111
000
O3’
C1’
C4’
C3’−endo
C2’
C1’
C4’
C2’−endo
C3’
(a)
(b)
Fig. 9.10. (a) RNA backbone showing rotamer degrees of freedom. (b) Preferred ribose
buckling conformations in A-form RNA.
Table 9.4.
Base
a
Mononucleoside conformations.
Conformer
R
C2’-endo
R
C3’-endo
Y
C3’-endo
Y
C2’-endo
Rotamera
syn
gg
×
anti
gt
gg
(anti) ×
gt
(anti) ×⎛(gg)⎞
gg
(anti) × ⎝ gt ⎠
tg
States
4
2
1
3
Rotamer states G± have been neglected.
For example, the above classification is modulo G± rotamer states. Within
a duplex the combination of G− and C2’-endo ribose places an oxygen (O2’)
group in close proximity to a (backbone) P unit, with the resulting repulsion making such conformers highly unfavourable. In fact (C3’-endo, G− )
and (C2’-endo, G+ ) are the stable combinations [Yokoyama and Nishimura
(1995)] of these degrees of freedom. Additional constraints upon allowable
states may arise since the binding occurs with the codon as a ribosomal
substrate, rather than in solution. To date only RC3 and YC2 , YC3 conformers have been observed in vivo [Takai (2006)] and on these grounds we
may neglect the RC2 states in a first approximation.
From the rules in Table 9.4 it is easy to see how anticodon GN N might
accommodate codons N N C and N N U : G(34) is predominantly in the
Spectroscopy of the Genetic Code
177
C3’-endo form. The WC pairing geometry (G · C) requires the gg rotamer,
while G · U requires a Guanine deformation towards the major groove,
possibly facilitated by transition to the gt form. In the present picture the
flexible G(34) base would be in a superposition of conformer states, until
it encounters the third codon base, whereupon it is required to collapse to
either an optimal or suboptimal state (in the contexts of G · C and G · U
respectively).
The case of U (34) is more complex. From the rules above, one identifies
the UC3 (34) gg singlet, which participates in WC pairing, and a triplet
of UC2 rotamers. Empirical evidence strongly suggests U · G and U · U
wobble pairs occur in the C2’-endo form, hence can be placed in the triplet.
However little is known about the U ·C pair. Note that U ·Y mismatches are
physically impossible in the C3’-endo form; on the other hand the C2’-endo
conformer theoretically suffers from steric hindrance. Other proposals for
the U · C pairing geometry include water-mediated H-bonds [Agris (2004)]
and protonation of C or, possibly a different ribose conformer. Based upon
current knowledge, the U · C pair is not inconsistent as the third member
of the UC2 triplet. Uridine is unique amongst the bases, in that the C2’endo and C3’- endo forms are almost equally favoured: ∆G∗ as defined in
Fig. 9.11(b) is of the order of -0.1 kcal mol−1 [Yokoyama et al. (1985)] and
it therefore readily forms wobble pairs. However U (34) is almost invariably
modified post-transcriptionally, presumably to enhance recognition fidelity
in one of several ways. The 5-hydroxyuridine derivatives18 (xo5 U ∗ ) almost
always participate in 4-way wobbles [Takai (2006)]. This modification shifts
the pseudorotation double well in Fig. 9.11(a) in favour of the C2-endo
form (∆G∗ = 0.7 kcal mol−1 ), thereby enhancing recognition of the U · U ,
U · G (and presumably U · C) wobble pairs. Conversely, 5-methyl-2-thiouridine derivatives (m5 s2 U ∗ ) strongly stabilize the C3’ form (∆G∗ = −1.1
kcal mol−1 ) [Takai and Yokoyama (2003)]. These modifications appear in
the split boxes, where misreading of U - and C- ending codons would be
potentially lethal. Note that such misreadings still occur, albeit several
orders of magnitude less frequently than the “correct” G · C and G · U
pairings [Inagaki et al. (1995)].
9.4.2.
Dynamical symmetry breaking and third base wobble
In the “modified wobble hypothesis”, [Agris (1991, 2004)] patterns of nucleotide modification are proposed to modify anticodon loop dynamics so
18 The
“*” superscript denotes possible further modification.
178
Quantum Aspects of Life
∆G
∆G
C3’−endo
C2’−endo
C3’−endo
C2’−endo
∆ G*
0
π/2
π
τ
0
π/2
(b)
∆G
π
τ
0
π/2
(d)
π
τ
(a)
∆G
∆ G*
0
π/2
(c)
π
τ
Fig. 9.11. (a) Sketch of Uridine ribose pseudorotation potential, showing equally stable
C2’- and C3’-endo conformers. (b) Effect of xo5 modification to Uridine on potential.
(c) Same as (a) but with hypothetical (conformer) bound states imposed. (d) Same as
(b) showing hypothetical bound states.
as to be compatible with the codon-ribosome complex. In addition to the
effects of post-transcriptional modification upon N (34) conformations, as
discussed above, bases 32 and 38 (which demarcate the anticodon loop) are
commonly modified to enhance H-bonding, thereby facilitating an “open”
loop. Further, modifications to R(37) (just downstream of a.c. position 3),
the so-called “universal purine”, generally correlate with the base content
of position 36.
Structural studies lend support to kinetic models of codon reading
[Takai (2006); Ninio (2006)] describing multiple-stage processes. Initial
contacts between ribosome and canonical A-form duplex RNA (for pairs
N (35) · N (II) and N (36) · N (I)) have been observed to promote conformational changes [Ogle et al. (2003)] in the ribosome which facilitate the
release of the amino acid from tRNA. In fact the conservation of A-form
structure is more important than stability conferred by base-pairing. For example, thermodynamically, the difference between contributions of canonical, U (36) · A(I), and a potential first codon position wobble, U (36) · G(I),
Spectroscopy of the Genetic Code
179
pairs is of the order of 10%. Yet when bound to the ribosome, reading of
the “correct” Watson-Crick pair proceeds 3-4 orders of magnitude faster
[Kurland et al. (1996)].
With the codon bound at the ribosomal A-site, it is reasonable to assume that these three bases are intrinsically rigid, and amenable to A-form
duplex formation. Moreover, it can be argued [Lim and Curran (2001)]
that anticodon nucleotides 35, 36 are inflexible, whether by way of proximity to the 5’ end of the anticodon loop, or due to R(37) modifications.
With these considerations it is straightforward to envisage a simple, lattice
Hamiltonian containing three sites with one, corresponding to position (34),
carrying an internal conformer degree of freedom which feels a double-well
potential of the sort sketched in Fig. 9.11(a) above.
Using the “low energy” conformer states of Table 9.4 (and tentatively
assuming the xo5 U ∗ (34) base adopts the C2’-endo tg form in the context of
U (34) · C(III)) we can write down correlated anticodon states associated
with third letter codon recognition. In the mitochondrial code, for example,
in the mixed boxes the possible states for the first anticodon letter are
G(34) → (|C3′ > ⊗|G− > ⊗(α1 |gg > +α2 |gt >) ,
m5 s2 U ∗ (34) → β1 |C3′ > ⊗|G− > ⊗|gg > +β2 |C2′ > ⊗|G+ > ⊗|gg > ,
while the family boxes have
xo5 U ∗ (34) → γ1 |C3′ > ⊗|G− > ⊗|gg >
+|C2′ > ⊗|G+ > ⊗(γ2 |gg > +γ3 |gt > +γ4 |tg >) .
In their model of permitted wobble-rules, Lim and Curran (2001) predicted
that, given certain wobble pairs U (34) · N (III), a second position wobble,
U (35)·G(II) was not forbidden, but prevented from occurring by the use of
anticodons GN N . This kind of observation naturally connects the above,
quantum picture of codon reading with our original discussion of broken dynamical symmetries. Second position wobbles are likely to be suppressed
by the local rigidity of the 7-member anticodon loop, in addition to ribosomal contacts with the bound codon-anticodon complex. It is therefore
plausible that higher, dynamically-broken symmetries could describe some
ancestral translation system with simpler structural features (and lower fidelity). In this manner the pattern of dynamical symmetry breaking may
be indicative of the evolution of the translation apparatus (see, for example,
Seligmann and Amzallag (2002); Poole et al. (1996); Beuning and Musier
Forsyth (1999)).
180
Quantum Aspects of Life
The possibility of a graded symmetry underlying the scenario just described is left open. Whenever a 4-way wobble (xo5 U (34)∗ ) is present, the
bound states described in Fig. 9.11(d) are in one-to-one correspondence
with the “physical” codon-anticodon reading complexes. In a mixed box
(or indeed in any eukaryotic box) multiple tRNA species exist, and several
analogues of Fig. 9.11(d) are required, with the lowest-lying states of each
comprising the set of reading complexes. One possibility is to postulate
that certain kinds of pairing geometry have even or odd grading, the motivation being the gl(1/1) dynamical symmetry analogue of supersymmetric
quantum mechanics, whereby an even Watson-Crick (ground) state would
lie below an odd (excited) wobble state in the case of U (34) · R(III) or
G(34) · Y (III) pairs. The following, speculative grading of pairs
{G · C, C · G, A · U, U · A, U · C, I · U } even
{G · U, U · G, U · U, I · C, U · C, I · A} odd
within the VMC and EC is compatible with the supermultiplet structure
described previously in Sec. 9.2.
Finally we wish to emphasize the following point: the differences in
codon-anticodon binding energies are several orders of magnitude less than
differences in codon reading rates. The connection with the “spectroscopic”
theme of earlier sections does not lie directly within the bound states
sketched in the potentials of Fig. 9.11(c)-(d), which are indicative of only
the first stage of a multi-step kinetic pathway. Rather the “spectrum”, if
there is one, is in terms of total reading reaction rates, analogously to the
“potentiation” concept of Takai (2006).
9.5.
Conclusions
Regularities inherent in the genetic code “alphabets” allow discussion of
codon-amino acid relationships to be abstracted from a biochemical setting
to a mathematical/logical one. In this review we have attempted to present
insights into the form (and possible evolution) of the genetic code, borrowing group theory concepts from spectroscopy. In Sec. 9.2 an argument for
an evolutionary role of continuous and/or graded symmetries was made, in
comparison with different models in the (biological) literature. Section 9.3
provided some numerical support for this view, via fits of physico-chemical
properties of amino acids to those of codons. Finally in Sec. 9.4 we proposed a possible role for quantum processes in codon reading. Our hope is
Spectroscopy of the Genetic Code
181
that an eventual, dynamical code model will bear out the preliminary steps
taken here in this direction.
Acknowledgements
This research was partly funded by Australian Research Council grant
DP0344996. Collaboration with Ioannis Tsohantjis in earlier stages of
this work is also acknowledged. PDJ thanks I. Z. Siemion and his group
at Wroclaw for hospitality and useful discussions. JDB wishes to thank
K. Takai for helpful discussions. The assistance of Elizabeth Chelkowska
in formatting the 3D plots of genetic code data is gratefully acknowledged.
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About the authors
Jim Bashford is presently an ice sheet data analyst at the Australian
Government Antarctic Division. He graduated with a PhD in theoretical physics from the University of Adelaide in 2003, under Anthony
W. Thomas. Recent research interests have included modelling of codonamino acid degeneracy, oligomer thermodynamics, nonlinear models of
DNA dynamics and phylogenetic entanglement.
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Peter Jarvis obtained his PhD from Imperial College, London, 1976, under Robert Delbourgo. He is presently at the School of Mathematics and
Physics, University of Tasmania. His main interests are in algebraic structures in mathematical physics and their applications, especially combinatorial Hopf algebras in integrable systems and quantum field theory. In
applications of group theory to physical problems, aside from the work
on supersymmetry in the genetic code, recent papers have included applications of classical invariant theory to problems of quantum physics (entanglement measures for mixed state systems), and also to phylogenetic
reconstruction (entanglement measures, including distance measures, for
taxonomic pattern frequencies).
Chapter 10
Towards Understanding
the Origin of Genetic Languages
Apoorva D. Patel
“. . . four and twenty blackbirds baked in a pie . . . ”
Molecular biology is a nanotechnology that works—it has worked for billions of years and in an amazing variety of circumstances. At its core is
a system for acquiring, processing and communicating information that is
universal, from viruses and bacteria to human beings. Advances in genetics
and experience in designing computers have taken us to a stage where we
can understand the optimization principles at the root of this system, from
the availability of basic building blocks to the execution of tasks. The languages of DNA and proteins are argued to be the optimal solutions to the
information processing tasks they carry out. The analysis also suggests simpler predecessors to these languages, and provides fascinating clues about
their origin. Obviously, a comprehensive unraveling of the puzzle of life
would have a lot to say about what we may design or convert ourselves
into.
10.1.
The Meaning of It All
I am going to write about some of the defining characteristics of life. Philosophical issues always arise in discussions regarding life, and I cannot avoid
that. But let me state at the outset that such issues are not the purpose
of my presentation. I am going to look at life as an exercise in information
theory, and extend the analysis as far as possible.
Received April 13, 2007
187
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Quantum Aspects of Life
Let me begin with the textbook answer to the question made famous by
Schrödinger (1944): What is life? Life is fundamentally a non-equilibrium
process, commonly characterized in terms of two basic phenomena. One is
“metabolism”. Many biochemical processes are needed to sustain a living
organism. Running these processes requires a continuous supply of free
energy, which is extracted from the environment. (Typically this energy is
in electromagnetic or chemical form, but its ultimate source is gravity—
the only interaction in the universe that is not in equilibrium.) The other
is “reproduction”. A particular physical structure cannot survive forever,
because of continuous environmental disturbances and consequent damages.
So life perpetuates itself by a succession of generations.
It is obvious that both these phenomena are sustaining and protecting
and improving something, often against the odds. So let us figure out what
is it that is being sustained and protected and improved.
All living organisms are made up of atoms. These atoms are fantastically
indestructible. In all the biochemical processes, they just get rearranged in
different ways. Each of us would have a billion atoms that once belonged
to the Buddha, or Genghis Khan, or Isaac Newton—a sobering or exciting
realization depending on one’s frame of mind! We easily see that it is not
the atoms themselves but their arrangements in complex molecules, which
carry biochemical information. In the flow of biochemical processes, living
organisms synthesize and break up various molecules, by altering atomic
arrangements. The biochemical information resides in what molecules to
use where, when and how. Characterization of this information is rather
abstract, but central to the understanding of life. To put succinctly:
Hardware is recycled, while software is refined!
At the physical level, atoms are shuffled, molecules keep on changing, and
life goes on. At the abstract level, it is the manipulation and preservation of
information that requires construction of complex structures. Information
is not merely “a” property of life—it is “the” basis of life.
Now information is routinely quantified as entropy of the possible forms
a message could take [Shannon (1948)]. What the living organisms require,
however, is not mere information but information with meaning. A random
arrangement of components (e.g. a gas) can have large information, but it
is not at all clear how that can be put to any use. The molecules of life
are destined to carrying out specific functions, and they have to last long
enough to execute their tasks. The meaning of biological information is
carried by the chemical properties of the molecules, and a reasonably stable
Towards Understanding the Origin of Genetic Languages
189
cellular environment helps in controlling the chemical reactions. What the
living organisms use is “knowledge”,
Knowledge = Information + Interpretation.
Knowledge has to be communicated using a language. A language uses a
set of building blocks (e.g. letters of an alphabet) whose meaning is fixed,
and whose variety of arrangements (invariably aperiodic) compose different
messages. It is the combination of information and interpretation that
makes languages useful in practice.
Thus to understand how living organisms function, we need to focus
on the corresponding languages whose interpretation remains fixed, while
all manipulations of information processing go on. A practical language
is never constructed arbitrarily—criteria of efficiency are always involved.
These criteria are necessarily linked to the tasks to be implemented using
the language, and fall into two broad categories. One is the stability of the
meaning, i.e. protection against error causing fluctuations. And the other is
the efficient use of physical resources, i.e. avoidance of unnecessary waste of
space, time, energy etc. while conveying a message. The two often impose
conflicting demands on the language, and the question to investigate is: Is
there an optimal language for a given task, and if so how can we find it?
From the point of view of a computer designer, the question has two parts:
Software: What are the tasks? What are the algorithms?
Hardware: How are the operations physically implemented?
It goes without saying that the efficiency of a language depends both on
the software and the hardware.
In the computational complexity analysis, space and time resources are
often traded off against each other, and algorithms are categorized as polynomial or non-polynomial (usually exponential). In the biological context,
however, the efficiency considerations are not quite the same. Time is
highly precious, while space is fairly expendable. Biological systems can
sense small differences in population growth rates, and even an advantage
of a fraction of a percent is sufficient for one species to overwhelm another
over many generations. Spatial resources are frequently wasted, that too
on purpose. For instance, how many seeds does a plant produce, when just
a single one can ensure continuity of its lineage? It must not be missed
that this wastefulness leads to competition and Darwinian selection.
Before going on to the details of the genetic languages, here is a quick
summary of the components making up the biochemical machinery of
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Quantum Aspects of Life
living organisms, at different scales. A framework for understanding genetic languages must incorporate this hierarchical structure.
Atoms
Nucleotide bases and amino acids
Peptides and drugs
Proteins
Genomes
Size
H,C,N,O, and infrequently P,S
10-20 atoms
40-100 atoms
100-1000 amino acids
103 -109 nucleotide base pairs
1 nm (molecules)-104 nm (cells)
Gene and protein databases have been accumulating a lot of data, which can
be used to test hypotheses and consequences of specific choice of languages.
To summarize, the aim of this chapter is to understand the physical and
the evolutionary reasons for (a) the specific genetic languages, and (b) their
specific realizations. A tiny footnote is that such an understanding would
have a bearing on the probability of finding life elsewhere in the universe
and then characterizing it.
10.2.
Lessons of Evolution
Evolution is the centrepiece of biology. It has been the cause of many
controversies, mainly because it is almost imperceptible—the evolutionary
timescales are orders of magnitude larger than the lifetimes of individual
living organisms. But it is the only scientific principle that provides a
unifying framework encompassing all forms of life, from the simple origin
to an amazing variety. We need to understand the forces governing the
direction of evolution, in order to comprehend where we came from as well
as what the future may have in store for us.
Genetic information forms the quantitative underpinning of evolution.
Certain biological facts regarding genetic languages are well-established:
(1) Languages of genes and proteins are universal. The same 4 nucleotide
bases and 20 amino acids are used in DNA, RNA and proteins, all the
way from viruses and bacteria to human beings. This is despite the fact
that other nucleotide bases and amino acids exist in living cells. This
clearly implies that selection of specific languages has taken place.
Towards Understanding the Origin of Genetic Languages
191
(2) Genetic information is encoded close to data compression limit and
maximal packing. This indicates that optimization of information storage has taken place.
(3) Evolution occurs through random mutations, which are local changes in
the genetic sequence. In the long run, however, only a small fraction of
the mutations survive—those proving advantageous to the organisms.
This optimizing mechanism is labelled Darwinian selection, i.e. competition for limited resources leading to survival of the fittest.
Over the years, many attempts have been made to construct evolutionary scenarios that can explain the universality of genetic languages. They
can be broadly classified into two categories. One category is the “frozen
accident” hypothesis [Crick (1968)], i.e. the language somehow came into
existence, and became such a vital part of life’s machinery that any change
in it would be highly deleterious to living organisms. This requires the birth
of the genetic machinery to be an extremely rare event, without sufficient
time to explore other possibilities. There is not much room for analysis in
this ready-made solution. I do not subscribe to it, and instead argue for
the other category. That is the “optimal solution” end-point [Patel (2003)],
i.e. the language arrived at its best form by trial and error, and it did not
change thereafter, because any change in it would make the information
processing less competitive. This requires the evolution of genetic machinery to have sufficient scope to generate many possibilities, and subsequent
competition amongst them whence the optimal solution wins over the rest.
It should be noted that the existence of an optimizing mechanism does
not make a choice between the two categories clear-cut. The reason is that
a multi-parameter optimization manifold generically has a large number
of minima and maxima, and an optimization process relying on only local changes often gets trapped in local minima of the undulating manifold
without reaching the global optimum. In such situations, the initial conditions and history of evolution become crucial in deciding the outcome of
the process, and typically there arise several isolated surviving candidates.
The globally optimal solution is certainly easier to reach, when the number
of local minima is small and/or the range of exploratory changes is large.
The extent of optimization is therefore critically controlled by the ratio of
time available for exploration of various possibilities to the transition time
amongst them. For the genetic machinery to have reached its optimal form,
the variety of possibilities thrown up by the primordial soup must have had
a simple and quick winner.
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Quantum Aspects of Life
The procedure of optimization needs a process of change, and a process of selection. The former is intrinsic, the latter is extrinsic, and the
two take place at different levels in biology. Indeed the difference between
the two provides much ammunition for debates involving choice vs. environment, or nature vs. nurture. The changes are provided by mutations,
which occur essentially randomly at the genetic level. That describes the
genotype. The selection takes place by the environmental pressure at the
level of whole organisms. It is not at all random, rather it is biased towards short-term survival (till reproduction). That describes the phenotype. We have good reasons to believe that the primitive living organisms
were unicellular, without a nucleus, with small genomes, and having a simple cellular machinery. In such systems, the genotype and phenotype levels
are quite close, and the early evolution can easily be considered a direct
optimization problem.
Before exploring what could have happened in the early stages of evolution, let us also briefly look at the direction in which it has continued.
The following table summarizes how the primitive unicellular organisms
progressed to the level of humans (certainly the most developed form of
life in our own point of view), using different physical resources to process
information at different levels.
Organism
Messages
Physical Means
Single cell
Molecular
(DNA, Proteins)
Chemical bonds,
Diffusion
Multicellular
Electrochemical
(Nervous system)
Convection,
Conduction
Families,
Societies
Imitation, Teaching,
Languages
Light, Sound
Humans
Books, Computers,
Telecommunication
Storage devices,
Electromagnetic
waves
Gizmos or
Cyborgs ?
Databases
Merger of brain
and computer
Towards Understanding the Origin of Genetic Languages
193
It is clear that evolution has progressively discovered higher levels of
communication mechanisms, whereby the communication range has expanded (both in space and time), the physical contact has reduced, abstraction has increased, succinct language forms have arisen and complex
translation machinery has been developed. More interesting is the manner
in which all this has been achieved, with cooperation (often with division
of labour) gradually replacing competition. This does not contradict Darwinian selection—it is just that the phenotype level has moved up, and components of a phenotype are far more likely to cooperate than compete. The
mathematical formulation underlying this behaviour is “repeated games”,
with no foresight but with certain amount of memory [Aumann (2006)].
The evolutionary features useful for the purpose of this article are:
• The older and lower information processing levels are far better optimized
than the more recent higher levels. This is a consequence of the fact that
in the optimization process the lower levels had less options to deal with
and more time to settle on a solution.
• The capacity of gathering, using and communicating knowledge has
grown by orders of magnitude in the course of evolution. Indeed one
can surmise that, in the long run, the reach of knowledge overwhelms
physical features in deciding survival fitness.
Knowledge is the essential driving force behind evolution,
providing a clear direction even when the goal remains unclear.
10.3.
Genetic Languages
Let us now return to analysing the lowest level of information processing,
i.e. the genetic languages. There are two of them—the language of DNA
and RNA with an alphabet of four nucleotide bases, and the language of
proteins with an alphabet of twenty amino acids. The tasks carried out by
both of them are quite specific and easy to identify.
(1) The essential job of DNA and RNA is to sequentially assemble a chain of
building blocks on top of a pre-existing master template. One can call
DNA the read-only-memory of living organisms. When not involved
in the replication process, the information in DNA remains idle in a
secluded and protected state.
(2) Proteins are structurally stable molecules of various shapes and
sizes, with precise locations of active chemical groups. They carry out
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Quantum Aspects of Life
various functions of life by highly selective binding to other molecules.
Molecular interactions are weak and extremely short-ranged, and so
the binding necessitates matching of complementary shapes, i.e. lockand-key mechanism in three dimensions. Proteins are created whenever
needed, based on the information present in DNA, and disintegrated
once their function is over.
The identification of these tasks makes it easy to see why there are two
languages and not just one. Memory needs long term stability, on the other
hand fast execution of functions is desirable, and the two make different
demands on the hardware involved. (The accuracy of a single language performing both the tasks would be limited, which is the likely reason why the
RNA world, described later, did not last very long.) Indeed, our electronic
computers compute using electrical signals, but store the results on the disk
using magnetic signals. The former encoding is suitable for fast processing, while the latter is suitable for long term storage. The two hardware
languages fortunately correspond to the same binary software language,
and are conveniently translated into each other by the laws of electromagnetism. In case of genetic information, the two hardware languages work
in different dimensions—DNA is a linear chain while proteins are three dimensional structures—forcing the corresponding software languages also to
be different and the translation machinery fairly complex.
We want to find the optimal languages for implementing the tasks of
DNA/RNA and proteins. So we have to study what constraints are imposed
on a language for minimization of errors and minimization of resources.
Minimization of errors inevitably leads to a digital language, having a set
of clearly distinguishable building blocks with discrete operations. With
non-overlapping signals, small fluctuations (say less than half the separation between the discrete values) are interpreted as noise and eliminated
from the message by resetting the values, while large changes represent
genuine change in meaning. The loss of intermediate values is not a drawback, as long as actual applications need only results with bounded errors.
Minimization of resources is achieved by using a small number of building
blocks, with simple and quick operations. A versatile language is then obtained by arranging the building blocks together in as many different ways
as possible.
In this optimization exercise, the “minimal language”, i.e. the language
with the smallest set of building blocks for a given task, has a unique status
[Patel (2006a)]:
Towards Understanding the Origin of Genetic Languages
195
• It has the largest tolerance against errors, since the discrete variables are
spread as far apart as possible in the available range of physical hardware
properties.
• It has the smallest instruction set, since the number of possible transformations is automatically limited.
• It can function with high density of packing and quick operations, which
more than make up for the increased depth of computation.
• It can avoid the need for translation, by using simple physical responses
of the hardware.
The genetic languages are undoubtedly digital, and that has been crucial
in producing evolution as we know it. Some tell-tale signatures are:
• Digital language helps in maintaining variation, while continuous variables would average out fluctuations.
• It is a curious fact that evolution is a consequence of a tiny error rate.
With too many errors the organism will not be able to survive, but
without mutations there will be no evolution.
• Even minimal changes in discrete genetic variables generate sizeable disruptions in the system, and they will be futile unless the system can
tolerate them. Often a large number of trial variations are needed to find
the right combinations, and having only a small number of discrete possibilities helps. Continuous variables produce gradual evolution, which
appears on larger phenotypic scales when multiple sources contributing
to a particular feature average out.
• With most of the trial variations getting rejected as being unproductive, digital variables give rise to punctuated evolution—sudden changes
interspersed amongst long periods of stasis.
In the following sections, we investigate to what extent the digital genetic languages are minimal, i.e. we first deduce the minimal languages for
the tasks of DNA/RNA and proteins, and then compare them to what the
living organisms have opted for. A worthwhile bonus is that we gain useful
clues about the simpler predecessors of the modern genetic languages.
10.4.
Understanding Proteins
Finding the minimal language for proteins is a straightforward problem in
classical geometry [Patel (2002)]. The following is a rapid-fire summary of
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Quantum Aspects of Life
the analysis:
• What is the purpose of the language of amino acids?
To form protein molecules of different shapes and sizes in three dimensions, and containing different chemical groups.
• What is the minimal discrete geometry for designing three dimensional
structures?
Simplicial tetrahedral geometry and the diamond lattice. Secondary protein structures, i.e. α-helices, β-bends and β-sheets, fit quite well on the
diamond lattice.
• What are the best physical components to realize this geometry?
Covalently bonded carbon atoms, also N+ and H2 O. Silicon is far more
abundant, but it cannot form aperiodic structures needed to encode a
language. (In the graphite sheet arrangement, carbon also provides the
simplicial geometry for two dimensional membrane patterns.)
• What is a convenient way to assemble these components in the desired
three dimensional structures?
Synthesize one dimensional polypeptide chains, which carry knowledge
about how to fold into three dimensional structures. The problem then
simplifies to assembling one dimensional chains. (Note that images in
our electronic computers are stored as folded sequences.)
• What are the elementary operations needed to fold a polypeptide chain on
a diamond lattice, in any desired manner?
Nine discrete rotations, represented as a 3×3 array on the Ramachandran
map (see Fig. 10.2). Additional folding operations are trans-cis flip and
long distance bonds.
• What can the side groups of polypeptide chains do?
They favour particular orientations of the polypeptide chain by interactions amongst themselves. They also fill up cavities in the structure by
variations in their size.
To put the above statements in biological perspective, and to illustrate
the minimalistic choices made by the living organisms (in the context of
what was available), here are some facts about the polypeptide chains:
(a) Amino acids are easily produced in primordial chemical soup. They
even exist in interstellar clouds.
(b) Amino acids are the smallest organic molecules with both an acid group
(−COOH) and a base group (−NH2 ). They differ from each other in
terms of distinct R-groups, which become the side groups of polypeptide
chains.
Towards Understanding the Origin of Genetic Languages
H
COOH
H2 N
C
R
(a)
Fig. 10.1.
H
197
O
H
H❍ ✟ R
❍ ✟
C ♣ ♣ ♣ ♣ ♣ ✟ Cα❍ ✟
N ❍φ ψ
N♣ ♣ ♣ ♣
♣
♣
✟
✟
♣
✟
❍
C✟
♣♣♣♣
❍
❍ N ✟ ❍ C ♣✟
♣ ♣ ♣ ♣ C♣ ♣
α
✟ ❍
✟
❍
H
χ R
O
H
O
(b)
Chemical structures of (a) an amino acid, (b) a polypeptide chain.
Fig. 10.2. The allowed orientation angles for the Cα bonds in real polypeptide chains
for chiral L-type amino acids, taking into account hard core repulsion between atoms
[Ramachandran et al. (1963)]. Stars mark the nine discrete possibilities for the angles,
uniformly separated by 120◦ intervals, when the polypeptide chain is folded on a diamond
lattice.
(c) Polypeptide chains are produced by polymerisation of amino acids by
acid-base neutralisation (see Fig. 10.1).
(d) Folded ↔ unfolded transition of polypeptide chains requires flexible
joints and weak non-local interactions (close to critical behaviour).
198
Quantum Aspects of Life
(e) Transport of polypeptides across membranes is efficient in the unfolded
state than in the folded one, preventing leakage of other molecules at
the same time. (A chain can slide through a small hole.)
The structural language of polypeptide chains would be the most versatile when all possible orientations can be generated by every amino acid
segment. This cannot be achieved by just a single property of the R-groups
(e.g. hydrophobic to hydrophilic variation). The table below lists the amino
acids used by the universal language of proteins. They are subdivided into
several categories according to the chemical properties of the R-groups, and
their molecular weights provide an indication of the size of the R-groups
[Lehninger et al. (1993)]. The language of bends and folds of the polypeptide chains is non-local, i.e. the orientation of an amino acid is not determined by its own R-group alone, rather the orientation is decided by the
interactions of the amino acid with all its neighbours. Still, by analysing
protein databases, one can find probabilities for every amino acid to participate in specific secondary structures, and the dominant propensities are
listed in the table below as well [Creighton (1993)].
Deciphering the actual orientations of amino acids in proteins is an outstanding open problem—the protein folding problem. Even then a rough
count of the number of amino acids present can be obtained with one additional input. This is the division of the amino acids into two classes, according to the properties of the corresponding aminoacyl-tRNA synthetases
(aaRS). In the synthesis of polypeptide chains, tRNA molecules are the
adaptors with one end matching with a genetic codon and the other end
attached to an amino acid. The aaRS are the truly bilingual molecules
in the translation machinery, that attach an appropriate amino acid to
the tRNA corresponding to its anticodon. There is a unique aaRS for every amino acid, even though several different tRNA molecules can carry
the same amino acid (the genetic code is degenerate). It has been discovered that the aaRS are clearly divided in two classes, according to their
sequence and structural motifs, active sites and the location where they
attach the amino acids to the tRNA molecules [Arnez and Moras (1997);
Lewin (2000)]. The classes of amino acids are also listed in the table below,
and here is what we find:
(a) The 20 amino acids are divided into two classes of 10 each.
(b) The two classes divide amino acids with each R-group property equally,
in such a way that for every R-group property the larger R-groups
correspond to class I and the smaller ones to class II.
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Towards Understanding the Origin of Genetic Languages
Amino acid
R-group
Mol. wt.
Class
G Gly (Glycine)
A Ala (Alanine)
P Pro (Proline)
V Val (Valine)
L Leu (Leucine)
I Ile (Isoleucine)
Propensity
Non-polar
aliphatic
75
89
115
117
131
131
II
II
II
I
I
I
turn
α
turn
β
α
β
S Ser (Serine)
T Thr (Threonine)
N Asn (Asparagine)
C Cys (Cysteine)
M Met (Methionine)
Q Gln (Glutamine)
Polar
uncharged
105
119
132
121
149
146
II
II
II
I
I
I
turn
β
turn
β
α
α
D Asp (Aspartate)
E Glu (Glutamate)
Negative
charge
133
147
II
I
turn
α
K Lys (Lysine)
R Arg (Arginine)
Positive
charge
146
174
II
I
α
α
H His (Histidine)
F Phe (Phenylalanine)
Y Tyr (Tyrosine)
W Trp (Tryptophan)
Ring/
aromatic
155
165
181
204
II
II
I
I
α
β
β
β
(c) The class label of an amino acid can be interpreted as a binary code for
its R-group size, in addition to the categorization in terms of chemical
properties.
(d) This binary code has unambiguous structural significance for packing
of proteins. Folding of an aperiodic chain into a compact structure
invariably leaves behind cavities of different shapes and sizes. The use
of large R-groups to fill big cavities and small R-groups to fill small
ones can produce dense compact structures.
200
Quantum Aspects of Life
(e) Each class contains a special amino acid, involved in operations other
than local folding of polypeptide chains—Cys in class I can make long
distance disulfide bonds, and Pro in class II can induce a trans-cis flip.
We thus arrive at a structural explanation for the 20 amino acids as
building blocks of proteins. Local orientations of the polypeptide chains
have to cover the nine discrete points on the Ramachandran map. They
are governed by the chemical properties of the amino acid R-groups, and
an efficient encoding can do the job with nine amino acids. The binary
code for the R-group sizes fills up the cavities nicely without disturbing
the folds. And then two more non-local operations increase the stability of
protein molecules.
The above counting does not tell which sequence of amino acids will lead
to which conformation of the polypeptide chain. That remains an unsolved
exercise in coding as well as chemical properties. On the other hand, it is
known that amino acids located at the active sites and at the end-points
of secondary structures determine the domains and activity of proteins,
while the amino acids in the intervening regions more or less act like spacefillers. Among the space-fillers, many substitutions can be carried out that
hardly affect the protein function—indeed protein database analyses have
produced probabilistic substitution tables for the amino acids. We need to
somehow incorporate this feature into our understanding of the structural
language of proteins, so that we can progress beyond individual letters to
words and sentences [see for example, Socolich et al. (2005); Russ et al.
(2005)]. A new perspective is necessary, and perhaps the following selfexplanatory paragraph is a clue [Rawlinson (1976)]. Surprise yourself by
reading it at full speed, even if you are not familiar with crossword puzzles!
You arne’t ginog to blveiee taht you can aulaclty uesdnatnrd waht I am
wirtnig. Beuacse of the phaonmneal pweor of the hmuan mnid, aoccdrnig
to a rscheearch at Cmabrigde Uinervtisy, it deosn’t mttaer in waht oredr
the ltteers in a wrod are, the olny iprmoatnt tihng is taht the frist and
lsat ltteer be in the rghit pclae. The rset can be a taotl mses and you
can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid
deos not raed ervey lteter by istlef, but the wrod as a wlohe. Amzanig
huh? Yaeh and you awlyas tghuhot slpeling was ipmorantt!
Written English and proteins are both non-local languages. Evolution,
after all, is no stranger to using a worthwhile idea—here a certain amount
of parallel and distributed processing—over and over again.
Towards Understanding the Origin of Genetic Languages
10.5.
201
Understanding DNA
Now let us move on to finding the minimal language for DNA and RNA.
Once again, here is a quick-fire summary of the analysis [Patel (2001a)].
• What is the information processing task carried out by DNA?
Sequential assembly of a complementary copy on top of the pre-existing
template by picking up single nucleotide bases from an unsorted ensemble. The same task is carried out by mRNA in the assembly of polypeptide chains, but proceeding in steps of three nucleotide bases (triplet
codons).
• What is the optimal way of carrying out this task?
Lov Grover’s database search algorithm [Grover (1996)], which uses binary queries and requires wave dynamics. It optimizes the number of
queries, providing a quadratic speed up over any Boolean algorithm, irrespective of the size of spatial resources the Boolean algorithm may use.
In a classical wave implementation the database is encoded as N distinct
wave modes, while in a quantum setting the database is labeled by log2 N
qubits.
• What is the characteristic signature of this algorithm?
The number of queries Q required to pick the desired object from an
unsorted database of size N are given by:
⎧
⎪
⎪
⎨Q = 1, N = 4
π
−1 1
(10.1)
=⇒
(2Q + 1) sin √ =
Q = 2, N = 10.5
⎪
2
N
⎪
⎩Q = 3, N = 20.2
(Non-integral values of N imply small errors in object identification,
about 1 part in 700 and 1050 for Q = 2 and Q = 3 respectively.)
• What are the physical ingredients needed to implement this algorithm?
A system of coupled wave modes whose superposition maintains phase
coherence, and two reflection operations (phase changes of π).
Again to clarify the biological perspective, and to illustrate the minimalistic choices made by the living organisms, here are some facts about
the biochemical assembly process:
(a) Instead of waiting for a desired complex biomolecule to come along, it
is far more efficient to synthesize it from common, simple ingredients.
(b) There should be a sufficient number of clearly distinguishable building
blocks to create the wide variety of required biomolecules.
202
Quantum Aspects of Life
(c) The building blocks are randomly floating around in the cellular environment. They get picked one by one and added to a linearly growing
polymer chain.
(d) Complementary nucleotide base-pairing decides the correct building
block to be added at each step of the assembly process.
(e) The base-pairings are binary questions; either they form or they do not
form. The molecular bonds involved are hydrogen bonds.
With these features, the optimal classical algorithm based on Boolean
logic would be a binary tree search. But the observed numbers do not fit
that pattern (of powers of two). On the other hand, the optimal search
solutions of Grover’s algorithm are clearly different from and superior to
the Boolean ones, and they do produce the right numbers. The crucial difference between the two is that wave mechanics works with amplitudes and
not probabilities, which allows constructive as well as destructive interference. Grover’s algorithm manages the interference of amplitudes cleverly,
and the individual steps are depicted in Fig. 10.3 for the simplest case of
four items in the database.
Now note that classically the binary alphabet is the minimal one for
encoding information in a linear chain, and two nucleotide bases (one
complementary pair) are sufficient to encode the genetic information. As a
matter of fact, our digital computers encode all types of information using
Amplitudes
0.5
(1)
0
U
❄b
0.25
0
(2)
−Us
❄
(3)
0.25
♣
♣
♣ 0
(4) Projection
Algorithmic Steps
Physical Implementation
Uniform
distribution
Equilibrium
configuration
Binary oracle
Yes/No query
Amplitude of
desired state
flipped in sign
Sudden
impulse
Reflection
about average
Overrelaxation
Desired state
reached
Opposite end
of oscillation
Algorithm
is stopped
Measurement
Fig. 10.3. The steps of Grover’s database search algorithm for the simplest case of
four items, when the first item is desired by the oracle. The left column depicts the
amplitudes of the four states, with the dashed lines showing their average values. The
middle column describes the algorithmic steps, and the right column mentions their
physical implementation.
Towards Understanding the Origin of Genetic Languages
203
only 0’s and 1’s. The binary alphabet is the simplest system, and so would
have preceded (during evolution) the four nucleotide base system found in
nature. Then, was the speed-up provided by the wave algorithm the real
incentive for nature to complicate the genetic alphabet? Certainly, if we
have to design the optimal system for linear assembly, knowing all the physical laws that we do, we would opt for something like what is present in
nature. But what did nature really do? We have no choice but to face the
following questions:
• Does the genetic machinery have the ingredients to implement Grover’s
algorithm?
The physical components are definitely present, and it is not too difficult
to construct scenarios based on quantum dynamics [Patel (2001a)] as
well as vibrational motion [Patel (2006b)]. Although Grover’s algorithm
was discovered in the context of quantum computation, it is much more
general, and does not need all the properties of quantum dynamics. In
particular, highly fragile entanglement is unnecessary, while much more
stable superposition of states is a must. The issue of concern then is
whether coherent superposition of wave modes can survive long enough
for the algorithm to execute. This superposition may be quantum (i.e. for
the wavefunction) or may be classical (as in case of vibrations). It need
not be exactly synchronous either—if the system transits through all the
possible states at a rate much faster than the time scale of the selection
oracle, that would simulate superposition, averaging out high frequency
components (e.g. the appearance of spokes of a rapidly spinning wheel).
Provided that the superposition is achieved somehow, the mathematical
signature, i.e. Eq. (10.1), follows. Explicit formulation of a testable scenario, based on physical properties of the available molecules and capable
of avoiding fast decoherence, is an open challenge.
• Did nature actually exploit Grover’s algorithm when the genetic machinery evolved billions of years ago?
Unfortunately there is no direct answer, since evolution of life cannot be
repeated.
• Do the living organisms use Grover’s algorithm even today?
In principle, this is experimentally testable. Our technology is yet to
reach a stage where we can directly observe molecular dynamics in a liquid environment. But indirect tests of optimality are plausible, e.g. constructing artificial genetic texts containing a different number of letters
and letting it compete with the supposedly optimal natural language
[Patel (2001b)].
204
Quantum Aspects of Life
This is not the end of the road, and I return to a deeper analysis later on.
But prior to that let us look at what the above described understanding
of the languages of proteins and DNA has to say about the translation
mechanism between the two, i.e. the genetic code. That investigation does
offer non-trivial rewards, regarding how the complex genetic machinery
could have arisen from simpler predecessors.
10.6.
What Preceded the Optimal Languages?
Languages of twenty amino acids and four nucleotide bases are too complex
to be established in one go, and evolution must have arrived at them from
simpler predecessors. On the other hand, continuity of knowledge has to be
maintained in evolution from simpler to complex languages, because sudden
drastic changes lead to misinterpretations that kill living organisms. Two
evolutionary routes obeying this restriction, and still capable of producing
large jumps, are known:
(1) Duplication of information, which allows one copy to carry on the required function while the other is free to mutate and give rise to a new
function.
(2) Wholesale import of fully functional components by a living organism,
distinct from their own and developed by a different living organism.
In what follows, we study the genetic languages within this framework.
The two classes of amino acids and the Q = 2 solution of Grover’s algorithm, described in preceding sections, suggest a duplication event, i.e. the
universal non-overlapping triplet genetic code arose from a more primitive
doublet genetic code labelling ten amino acids [Patel (2005); Wu et al.
(2005); Rodin and Rodin (2006)]. To justify this hypothesis, we have to
identify evolutionary remnants of (a) a genetic language where only two
nucleotide bases of a codon carry information while the third one is a punctuation mark, (b) a set of amino acids that can produce all the orientations
of polypeptide chains but without efficiently filling up the cavities, and (c) a
reasonable association between these codons and amino acids. Amazingly,
biochemical signals for all of these features have been observed.
The central players in this event are the tRNA molecules. They are older
than the DNA and the proteins in evolutionary history, and are believed to
link the modern genetic machinery with the earlier RNA world [Gesteland
et al. (2006)]. It has been discovered that RNA polymers called ribozymes
Towards Understanding the Origin of Genetic Languages
205
can both store information and function as catalytic enzymes, although
not very accurately. The hypothesis is that when more accurate DNA and
proteins took over these tasks from ribozymes, tRNA molecules survived
as adaptors from the preceding era.
As illustrated in Fig. 10.4, the tRNAs are L-shaped molecules with the
amino acid acceptor arm at one end and the anticodon arm at the other.
The two arms are separated by a distance of about 75 Å, too far apart
for any direct interaction. The aaRS molecules are much larger than the
tRNAs, and they attach an amino acid to the acceptor stem corresponding
to the anticodon by interacting with both the arms. The two classes of aaRS
perform this attachment from opposite sides, in a mirror image fashion as
shown in Fig. 10.4. Class I attachment is from the minor groove side of
the acceptor arm helix, and class II attachment is from the major groove
side. It has been observed that the tRNA acceptor stem sequence, which
directly interacts with the R-group of the amino acid being attached, plays
a dominant role in the amino acid recognition and the anticodon does not
matter much. This behaviour characterizes the operational RNA code,
formed by the first four base pairs and the unpaired base N73 of the acceptor
stem [Schimmel et al. (1993)]. The operational code relies on stereochemical
atomic recognition between amino acid R-groups and nucleotide bases; it
is argued to be older than the genetic code and a key to understanding the
goings on in the RNA world.
We now look at the amino acid class pattern in the genetic code. The
universal triplet genetic code has considerable and non-uniform degeneracy,
Amino
acid arm
1 5'
Tψ C arm
64
54
56
DHU
arm
(residues
10-25)
72
7
20
3'
69
12
44
26
38
Anticodon
arm
32
Anticodon
Fig. 10.4. The structure of tRNA [Lehninger et al. (1993)] (left), and the tRNA-AARS
interaction from opposite sides for the two classes [Arnez and Moras (1997)] (right).
206
Quantum Aspects of Life
Table 10.1. The universal genetic code. Boldface letters
indicate class II amino acids.
UUU Phe
UUC Phe
UUA Leu
UUG Leu
UCU
UCC
UCA
UCG
Ser
Ser
Ser
Ser
UAU Tyr
UAC Tyr
UAA Stop
UAG Stop
UGU Cys
UGC Cys
UGA Stop
UGG Trp
CUU Leu
CUC Leu
CUA Leu
CUG Leu
CCU
CCC
CCA
CCG
Pro
Pro
Pro
Pro
CAU His
CAC His
CAA Gln
CAG Gln
CGU Arg
CGC Arg
CGA Arg
CGG Arg
AUU Ile
AUC Ile
AUA Ile
AUG Met
ACU
ACC
ACA
ACG
Thr
Thr
Thr
Thr
AAU
AAC
AAA
AAG
AGU Ser
AGC Ser
AGA Arg
AGG Arg
GUU Val
GUC Val
GUA Val
GUG Val
GCU
GCC
GCA
GCG
Ala
Ala
Ala
Ala
GAU Asp
GAC Asp
GAA Glu
GAG Glu
Asn
Asn
Lys
Lys
GGU
GGC
GGA
GGG
Gly
Gly
Gly
Gly
with 64 codons carrying 21 signals (including Stop) as shown. Although
there is a rough rule of similar codons for similar amino acids, no clear
pattern is obvious.
By analysing genomes of living organisms, it has been found that during the translation process 61 mRNA codons (excluding Stop) pair with a
smaller number of tRNA anticodons. The smaller degeneracy of the anticodons is due to wobble pairing of nucleotide bases, where the third base
carries only a limited meaning (either binary or none) instead of four-fold
possibilities [Crick (1966)]. The wobble rules are exact for the mitochondrial code—all that matters is whether the third base is a purine or a
pyrimidine, and the number of possibilities reduces to 32 as shown. (Note
that the mitochondrial code works with rather small genomes and evolves
faster than the universal code, and so is likely to have simpler optimization
criteria.)
The departures exhibited by the mitochondrial genetic code, as well as
the genetic codes of some living organisms, from the universal genetic code
are rather minor, and only occur in some of the positions occupied by class
I amino acids. It can be seen that all the class II amino acids, except Lys,
←−−−
can be coded by codons NNY and anticodons GNN (wobble rules allow
Towards Understanding the Origin of Genetic Languages
207
Table 10.2. The (vertebrate) mitochondrial genetic
code. Pyrimidines Y=U or C, Purines R=A or G.
UUY Phe
UUR Leu
UCY Ser
UCR Ser
UAY Tyr
UAR Stop
UGY Cys
UGR Trp
CUY Leu
CUR Leu
CCY Pro
CCR Pro
CAY His
CAR Gln
CGY Arg
CGR Arg
AUY Ile
AUR Met
ACY Thr
ACR Thr
AAY Asn
AAR Lys
AGY Ser
AGR Stop
GUY Val
GUR Val
GCY Ala
GCR Ala
GAY Asp
GAR Glu
GGY Gly
GGR Gly
pairing of G with both U and C) [Patel (2005)]. This pattern suggests
that the structurally more complex class I amino acids entered the genetic
machinery later, and a doublet code for the class II amino acids (with
the third base acting only as a punctuation mark) preceded the universal
genetic code.
The class pattern becomes especially clear with two more inputs:
(1) According to the sequence and structural motifs of their aaRS, Phe is
assigned to class I and Tyr to class II. But if one looks at the stereochemistry of how the aaRS attach the amino acid to tRNA, then Phe
belongs to class II and Tyr to class I [Goldgur et al. (1997); Yaremchuk
et al. (2002)]. Thus from the operational RNA code point of view the
two need to be swapped.
(2) Lys has two distinct aaRS, one belonging to class I (in most archaea)
and the other belonging to class II (in most bacteria and all eukaryotes)
[Woese et al. (2000)]. On the other hand, the assignment of AGR
codons varies from Arg to Stop, Ser and Gly. This feature is indicative
of an exchange of class roles between AAR and AGR codons (models
swapping Lys and Arg through ornithine have been proposed).
These two swaps of class labels do not alter the earlier observation that
the two amino acid classes divide each R-group property equally. We thus
arrive at the predecessor genetic code shown below. The binary division of
the codons according to the class label is now not only unmistakable but
produces a perfect complementary pattern [Rodin and Rodin (2006)].
208
Quantum Aspects of Life
Table 10.3. The predecessor genetic code. Pyrimidines
Y=U or C, Purines R=A or G.
UUY Phe
UUR Leu
UCY Ser
UCR Ser
UAY Tyr
UAR Stop
UGY Cys
UGR Trp
CUY Leu
CUR Leu
CCY Pro
CCR Pro
CAY His
CAR Gln
CGY Arg
CGR Arg
AUY Ile
AUR Met
ACY Thr
ACR Thr
AAY Asn
AAR Lys*
AGY Ser
AGR Arg*
GUY Val
GUR Val
GCY Ala
GCR Ala
GAY Asp
GAR Glu
GGY Gly
GGR Gly
When the middle base is Y (the first two columns), it indicates the class
on its own—U for class I and C for class II. When the middle base is R (the
last two columns), the class is denoted by an additional Y or R, in the third
position when the middle base is A and in the first position when the middle
base is G. (Explicitly the class I codons are NUN, NAR and YGN, while
the class II codons are NCN, NAY and RGN.) The feature that after the
middle base, the first or the third base determines the amino acid class in a
complementary pattern, has led to the hypothesis that the amino acid class
doubling occurred in a strand symmetric RNA world, with complementary
tRNAs providing complementary anticodons [Rodin and Rodin (2006)].
The complementary pattern has an echo in the operational code of the
tRNA acceptor stem too. When the aaRS attach the amino acid to the
−CCA tail of the tRNA acceptor arm, the tail bends back scorpion-like,
and the R-group of the amino acid gets sandwiched between the tRNA
acceptor stem groove (bases 1-3 and 70-73) and the aaRS. Analysis of tRNA
consensus sequences from many living organisms reveals [Rodin and Rodin
(2006)] that (a) the first base-pair in the acceptor stem groove is almost
invariably G1 -C72 and is mapped to the wobble position of the codon,
(b) the second base-pair is mostly G2 -C71 or C2 -G71 , which correlate well
respectively with Y and R in the middle position of the codon, and (c) the
other bases do not show any class complementarity pattern.
The involvement of both the operational RNA code and the anticodon
in the selection of appropriate amino acid, and the above mentioned correlations between the two, make it very likely that the two had a common
Towards Understanding the Origin of Genetic Languages
209
origin. Then piecing together all the observed features, the following scenario emerges for the evolution of the genetic code:
(1) Ribozymes of the RNA world could replicate, but their functional capability was limited—a small alphabet (quite likely four nucleotide bases)
and restricted conformations could only produce certain types of structures. Polypeptide chains, even with a small repertoire of amino acids,
provided a much more accurate and versatile structural language, and
they took over the functional tasks from ribozymes. This takeover required close stereochemical matching between ribozymes and polypeptide chains, in order to retain the functionalities already developed.
(2) The class II amino acids provided (or at least dominated) the initial
structural language of proteins. With smaller R-groups, they are easier to synthesize, and so are likely to have appeared earlier in evolution. They can fold polypeptide chains in all possible conformations,
although some of the cavities may remain incompletely filled. They
also fit snugly into the major groove of the tRNA acceptor stem, with
the bases 1-3 and 70-74 essentially forming a mould for the R-group, for
precise stereochemical recognition. Indeed, this stereochemical identification of an R-group by three base-pairs, necessitated by actual sizes
of molecules, would be the reason for the triplet genetic code, even in
a situation where all the bases do not carry information.
(3) The modern tRNA molecules arose from repetitive extensions and complementary pairing of short acceptor stem sequences. In the process,
the 1-2-3 bases became the forerunners of the 34-35-36 anticodons.
With different structural features identifying the amino acids, paired
bases in the acceptor stem and unpaired bases in the anticodon, the
evolution of the operational code and the genetic code diverged. The
two are now different in exact base sequences, but the purine-pyrimidine
label (i.e. R vs. Y) still shows high degree of correlation between the
two.
(4) In the earlier era of class II amino acid language, the wobble base was
a punctuation mark (likely to be G in the anticodon, as descendant of
the 1-72 pair), the central base was the dominant identifier (descendant
of the 2-71 pair), and the last anticodon base provided additional specification (equivalent to the 3-70 pair and the unpaired base 73). During
←−−−
subsequent evolution, these GNN anticodons have retained their meaning, and all minor variations observed between genetic codes are in the
other anticodons corresponding to class I amino acids.
210
Quantum Aspects of Life
(5) Class I amino acids got drafted into the structural language, because
they could increase stability of proteins by improved packing of large
cavities without disrupting established structures. The required binary
label for the R-group size, appeared differently in the operational code
and the genetic code. For the operational code, the minor groove of the
acceptor stem was used, and utilization of the same paired bases from
the opposite side led to a complementary pattern. The class I amino
acids fit loosely in the minor groove, and subsequent proof-reading is
necessary at times to remove incorrectly attached amino acids. For
the genetic code, several of the unassigned anticodons were used for
the class I amino acids, introducing a binary meaning to the wobble
position whenever needed. The Darwinian selection constraint that the
operational code and the genetic code serve a common purpose ensured
a rough complementary strand symmetry for the anticodons as well.
(6) The structural language reached its optimal stage, once both classes
of amino acids were incorporated. With 32 anticodons (counting only
a binary meaning for the wobble position) and 20 amino acid signals,
enough anticodons may have remained unassigned. Most of them were
taken over by amino acids with close chemical affinities (wobble position
did not assume any meaning), and a few left over ones mapped to the
Stop signal.
(7) All this could have happened when each gene was a separate molecule,
coding for a single polypeptide chain. Additional selection pressures
must have arisen when the genes combined into a genome. To take
care of the increased complexity, some juggling of codons happened
and the Start signal appeared. The present analysis is not detailed
enough to explain this later optimization. Nevertheless, interpretation
of similar codons for similar amino acids and the wobble rules, as relics
of the doubling of the genetic code—indicative but not perfect—is a
significant achievement.
At the heart of the class duplication mechanism described above is (a)
the mirror image pattern of the amino acid R-group fit with the tRNA acceptor stem, and (b) the complementary pattern of the anticodons. More
detailed checks for these are certainly possible. The amino acids have been
tested for direct chemical affinities with either their codons or their anticodons (but not both together), and most results have been lukewarm
[Yarus et al. (2005)]. Instead, chemical affinities of amino acids with paired
codon-anticodon grooves should be tested, both by stereochemical models
Towards Understanding the Origin of Genetic Languages
211
and actual experiments. It should be also possible to identify which amino
acid paired with which one when the genetic code doubled. Some pairs
can be easily inferred from biochemical properties [Ribas de Pouplana
and Schimmel (2001); Patel (2005)]—(Asp,Glu), (Asn,Gln), (Lys,Arg),
(Pro,Cys), (Phe,Tyr), (Ser&Thr,Val&Ile)—while the others would be revealed by stereochemical modeling.
The next interesting exercise, further back in time and therefore more
speculative, is to identify how a single class 10 amino acid language took
over the functional tasks of 4 nucleotide base ribozymes. This is the stage
where Grover’s algorithm might have played a crucial role, and so we go
back and look into it more inquisitively.
10.7.
Quantum Role?
The arguments of the preceding section reduce the amino acid identification
problem by a triplet code, to the identification problem within a class by a
doublet code plus a binary class label. It is an accidental degeneracy that
the Q = 3 solution of Grover’s algorithm, Eq. (10.1), can be obtained as the
Q = 2 solution plus a classical binary query. To assert that the sequential
assembly process reached its optimal solution, we still need to resolve how
the Q = 1, 2 solutions of Eq. (10.1) were realized by the primordial living
organisms.
Clearly, the assembly processes occur at the molecular scale. We know
the physical laws applicable there—classical dynamics is relevant, but quantum dynamics cannot be bypassed. Discrete atomic structure provided by
quantum mechanics is the basis of digital genetic languages. Molecular
bonds are generally given a classical description, but they cannot take place
without appropriate quantum correlations among the electron wavefunctions. Especially, hydrogen bonds are critical to the genetic identification
process, and they are inherently quantum—typical examples of tunnelling
in a double well potential. The assemblers, i.e. the polymerase enzymes
and the aaRS molecules, are much larger than the nucleotide bases and the
amino acids, and completely enclose the active regions where identification
of nucleotide bases and amino acids occurs. They provide a well-shielded
environment for the assembly process, but the cover-up also makes it difficult to figure out what exactly goes on inside.
Chemical reactions are typically described in terms of specific initial
and final states, and transition matrix elements between the two that
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Quantum Aspects of Life
characterize the reaction rates. That is a fully classical description, and
it works well for most practical purposes. But to the best of our understanding, the fundamental laws of physics are quantum and not classical—
the classical behaviour arises from the quantum world as an “averaged
out” description. Quantum steps are thus necessarily present inside averaged out chemical reaction rates, and would be revealed if we can locate
their characteristic signatures. In the present context, such a fingerprint is
superposition.
The initial and final states of Grover’s algorithm are classical, but the
execution in between is not. In order to be stable, the initial and final
states have to be based on a relaxation towards equilibrium process. For the
execution of the algorithm in between, the minimal physical requirement is
a system that allows superposition of states, in particular a set of coupled
wave modes. As illustrated earlier in Fig. 10.3, the algorithm needs two
reflection operations. Provided that the necessary superposition is achieved
somehow, it is straightforward to map these operations to: (i) the impulse
interaction during molecular bond formation which has the right properties
to realize the selection oracle as a fairly stable geometric phase, and (ii) the
(damped) oscillations of the subsequent relaxation, which when stopped at
the right instant by release of the binding energy to the environment can
make up the other reflection phase.
Beyond this generic description, the specific wave modes to be superposed can come from a variety of physical resources, e.g. quantum evolution,
vibrations and rotations. With properly tuned couplings, resonant transfer
of amplitudes occurs amongst the wave modes (the phenomenon of beats),
and that is the dynamics of Grover’s algorithm. When the waves remain
coherent, their amplitudes add and subtract, and we have superposition.
But when the waves lose their coherence, we get an averaged out result—a
classical mixture. Thus the bottom line of the problem is:
Can the genetic machinery maintain coherence of appropriate wave
modes on a time scale required by the transition matrix elements?
Explicitly, let tb be the time for molecular identification by bond formation, tcoh be the time over which coherent superposition holds, and trel
be the time scale for relaxation to equilibrium. Then, Grover’s algorithm
can be executed when the time scales satisfy the hierarchy
tb ≪ tcoh ≪ trel .
(10.2)
Towards Understanding the Origin of Genetic Languages
213
Other than this constraint, the algorithm is quite robust and does not rely
on fine-tuned parameters. (Damping is the dominant source of error; other
effects produce errors which are quadratic in perturbation parameters.)
Wave modes inevitably decohere due to their interaction with environment, essentially through molecular collisions and long range forces. Decoherence always produces a cross-over leading to irreversible loss of information [Guilini et al. (1996)]—collapse of the wavefunction in the quantum
case and damped oscillations for classical waves. The time scales of decoherence depend on the dynamics involved, but a generic feature is that no
wave motion can be damped faster than its natural undamped frequency
of oscillation. For an oscillator,
ẍ + 2γ ẋ + ω02 x = 0 , x ∼ eiωt =⇒ γcrit = max(Im(ω)) = ω0 .
(10.3)
Too much damping freezes the wave amplitude instead of making it decay.
Thus ω0−1 is both an estimate of tb and a lower bound on tcoh . Molecular
properties yield ω0 = ∆E/ = O(1014 ) sec−1 , for the transition frequencies
of weak bonds as well as for the vibration frequencies of covalent bonds.
Decoherence must be controlled in order to observe wave dynamics,
irrespective of any other (undiscovered) physical phenomena that may be
involved. In case of vibrational and rotational modes of molecules, the fact
that we can experimentally measure the excitation spectra implies that the
decoherence times are much longer than tb . In case of quantum dynamics,
the decoherence rate is often estimated from the scattering cross-sections of
environmental interactions, in dilute gas approximation using conventional
thermodynamics and Fermi’s golden rule. For molecular processes, these
times are usually minuscule, orders of magnitude below ω0−1 . In view of
Eq. (10.3), such minuscule estimates are wrong—the reason being that
Fermi’s golden rule is an approximation, not valid at times smaller than
the natural oscillation period. A more careful analysis is necessary.
According to Fermi’s golden rule, the environmental decoherence rate
is inversely proportional to three factors: the initial flux, the interaction
strength and the final density of states. We know specific situations, where
quantum states are long-lived due to suppression of one or more of these
factors. The initial flux is typically reduced by low temperatures and shielding, the interaction strength is small for lasers and nuclear spins, and the
final density of states is suppressed due to energy gap for superconductors
and hydrogen bonds. We need to investigate whether or not these features
are exploited by the genetic machinery, and if so to what extent.
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Quantum Aspects of Life
Large catalytic enzymes (e.g. polymerases, aaRS, ribosomes) have an
indispensable role in biomolecular assembly processes. These processes do
not take place in thermal equilibrium, rather the enzymes provide an environment that supplies free energy (using ATP molecules) as well as shields.
The assembly then proceeds along the chain linearly in time. In a free
solution without the enzymes, the assembly just does not take place, even
though such a free assembly would have the advantage of parallel processing (i.e. simultaneous assembly all along the chain). The enzymes certainly
reduce the external disturbances and decrease the final density of states by
limiting possible configurations. But much more than that, they stabilize
the intermediate reaction states, called the transition states. The traditional description is that the free energy barrier between the reactants and
the products is too high to cross with just the thermal fluctuations, and
the enzymes take the process forward by lowering the barrier and supplying free energy. The transition states are generally depicted using distorted
electron clouds, somewhere in between the configurations of the reactants
and the products, and they are unstable when not assisted by the enzymes.
They can only be interpreted as superpositions, and not as mixtures—we
have to accept that the enzymes stabilize such intermediate superposition
states while driving biomolecular processes. Thus we arrive at the heart of
the inquiry:
Grover’s algorithm needs certain type of superpositions, and catalytic
enzymes can stabilize certain type of superpositions. Do the two match,
and if so, what is the nature of this superposition?
The specific details of the answer depend on the dynamical mechanism
involved. The requisite superposition is of molecules that have a largely
common structure while differing from each other by about 5-10 atoms. I
have proposed two possibilities [Patel (2001a, 2006b)]:
(1) In a quantum scenario, wavefunctions get superposed and the algorithm enhances the probability of finding the desired state. Chemically
distinct molecules cannot be directly superposed, but they can be effectively superposed by a rapid cut-and-paste job of chemical groups
(enzymes are known to perform such cut-and-paste jobs). Whether
this really occurs, faster than the identification time scale tb and with
the decoherence time scale significantly longer than /ω0 , is a question
that should be experimentally addressed. It is a tough proposition, and
most theoretical estimates are pessimistic.
Towards Understanding the Origin of Genetic Languages
215
(2) In a classical wave scenario, all the candidate molecules need to be
present simultaneously and coupled together in a specific manner. The
algorithm concentrates mechanical energy of the system into the desired
molecule by coherent oscillations, helping it cross the energy barrier
and complete the chemical reaction. Enzymes are required to couple
the components together with specific normal modes of oscillation, and
long enough coherence times are achievable. This scenario provides the
same speed up in the number of queries Q as the quantum one, but
involves extra spatial costs. The extra cost is not insurmountable in the
small N solutions relevant to genetic languages, and the extra stability
against decoherence makes the classical wave scenario preferable. (Once
again note that time optimization is far more important in biology than
space optimization.)
Twists and turns can be added to these scenarios while constructing
a detailed picture. But in any implementation of Grover’s algorithm, the
requirement of superposition would manifest itself as simultaneous presence of all the candidate molecules during the selection process, in contrast
to the one by one trials of a Boolean algorithm. This particular aspect
can be experimentally tested by the available techniques of isotope substitution, NMR spectroscopy and resonance frequency measurements. The
algorithm also requires the enzymes to play a central role in driving the nonequilibrium selection process, but direct observation of that would have to
await breakthroughs in technologies at nanometre and femtosecond scales.
10.8.
Outlook
Information theory provides a powerful framework for extracting essential
features of complicated processes of life, and then analysing them in a systematic manner. The easiest processes to study are no doubt the ones at
the lowest level. We have learned a lot, both in computer science and in
molecular biology, since their early days [Schrödinger (1944); von Neumann
(1958); Crick (1968)], and so we can now perform a much more detailed
study. Physical theories often start out as effective theories, where predictions of the theories depend on certain parameters. The values of the
parameters have to be either assumed or taken from experiments; the effective theory cannot predict them. To understand why the parameters have
the values they do, we have to go one level deeper—typically to smaller
216
Quantum Aspects of Life
scales. When the deeper level reduces the number of unknown parameters, we consider the theory to be more complete and satisfactory. The
level below conventional molecular biology is spanned by atomic structure
and quantum dynamics, and that is the natural place to look for reasons
behind life’s “frozen accident”. It is indeed wonderful that sufficient ingredients exist at this deeper level to explain the frozen accident as the optimal
solution. The first reward of this analysis has been a glimpse of how the
optimal solution was arrived at.
Evolution of life occurs through random events (i.e. mutations), without
any foresight or precise rules of logic. It is the powerful criterion of survival,
in a usually uncomfortable and at times hostile environment, that provides
evolution a direction. Even though we do not really understand why living
organisms want to perpetuate themselves, we have enough evidence to show
that they use all available means for this purpose [Dawkins (1989)]. This
struggle for fitness allows us to assign underlying patterns to evolution—not
always perfect, frequently with variations, and yet very much practical. By
understanding these patterns, we can narrow down the search for a likely
evolutionary route among a multitude of possibilities. Such an insight is
invaluable when we want to extrapolate in the unknown past with scant
direct evidence. That is certainly the case in trying to understand the
origin of life as we know it. Of course, the inferences become stronger
when supported by simulated experiments, and worthwhile tests of every
hypothesis presented have been pointed out in the course of this article.
Counting the number of building blocks in the languages of DNA and
proteins, and finding patterns in them, is only the beginning of a long exercise to master these languages. Natural criteria for the selection of particular building blocks would be chemical simplicity (for easy availability
and quick synthesis) and functional ability (for implementing the desired
tasks). Life can be considered to have originated, not with just complex
chemical interactions in a primordial soup, but only when the knowledge
of functions of biomolecules started getting passed from one generation to
the next. This logic puts the RNA world before the modern genetic machinery; ribozymes provide both function and memory, to a limited extent
but with simpler ingredients. During evolution, the structurally more versatile polypeptides—they have been observed to successfully mimic DNA
[Walkinshaw et al. (2002)] as well as tRNA [Nakamura (2001)]—took over
the task of creating complex biochemistry, while leaving the memory storage job to DNA. The work described in this chapter definitely reinforces this
point of view, with simpler predecessors of the modern genetic languages
Towards Understanding the Origin of Genetic Languages
217
to be found in the stereochemical interaction between the tRNA acceptor
stem and simple class II amino acids. Experimental verification of this hypothesis would by and large solve the translation mystery, i.e. which amino
acid corresponds to which codon/anticodon? Then we can push the analysis further back in time, to the still simpler language of ribozymes, and try
to figure out what went on in the RNA world.
The opposite direction of investigation, of constructing words and sentences from the letters of alphabets, is much more than a theoretical adventure and closely tied to what the future holds for us. We want to design
biomolecules that carry out specific tasks, and that needs unraveling how
the functions are encoded in the three dimensional protein assembly process.
This is a tedious and difficult exercise, involving hierarchical structures and
subjective variety. But some clues have appeared, and they should be built
on to understand more and more complicated processes of life. We may feel
uneasy and scared about consequences of redesigning ourselves, but that
after all would also be an inevitable part of evolution!
“. . . when the pie was opened, the birds began to sing . . . ”
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About the author
Apoorva D. Patel is a Professor at the Centre for High Energy Physics,
Indian Institute of Science, Bangalore. He obtained his MSc in Physics
from the Indian Institute of Technology, Mumbai, and PhD in Physics
from the California Institute of Technology (Caltech) under Geoffrey Fox
(1984). His major field of work has been the theory of QCD, where he
has used lattice gauge theory techniques to investigate spectral properties,
phase transitions and matrix elements. In recent years, he has worked on
quantum algorithms, and used information theory concepts to understand
the structure of genetic languages.
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PART 4
Artificial Quantum Life
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Chapter 11
Can Arbitrary Quantum Systems
Undergo Self-replication?
Arun K. Pati and Samuel L. Braunstein
Arbitrary quantum states cannot be copied. In fact, to make a copy we must
provide complete information about the system. However, can a quantum
system self-replicate? This is not answered by the no-cloning theorem. In
the classical context, von Neumann showed that a “universal constructor”
can exist that can self-replicate an arbitrary system, provided that it had
access to instructions for making a copy of the system. We question the
existence of a universal constructor that may allow for the self-replication
of an arbitrary quantum system. We prove that there is no deterministic
universal quantum constructor that can operate with finite resources. Further, we delineate conditions under which such a universal constructor can
be designed to operate deterministically and probabilistically.
11.1.
Introduction
The basis of classical computation is the Church-Turing thesis [Church
(1936); Turing (1936)], which says that every recursive function can be
computed algorithmically provided the algorithm can be executed by a
physical process. However, fundamental physical processes are not governed by classical mechanics, rather by quantum mechanical laws. The
possibility of performing reversible computation [Bennett (1973)] and the
fact that classical computers cannot efficiently simulate quantum systems
[Feynman (1982); Benioff (1982)] gave birth to the concept of the quantum
Received April 26, 2007
223
224
Quantum Aspects of Life
Turing machine [Deutsch (1985)]. This led to a flurry of discoveries in quantum computation [Feynman (1986)], quantum algorithms [Bernstein and
Vazirani (1993); Deutsch and Jozsa (1992); Shor (1997); Grover (1997)],
quantum simulators [Lloyd (1996)], quantum automata [Albert (1983)] and
programmable gate arrays [Nielsen and Chuang (1997)]. In another development, von Neumann (1966) thought of an extension of the logical
concept of a universal computing machine that might mimic a living system. One of the hall-mark properties of a living system is its capability
of self-reproduction. He asked the question: Is it possible to design a machine that could be programmed to produce a copy of itself, in the same
spirit that a Turing machine can be programmed to compute any function
allowed by physical law? More precisely, he defined a universal constructor as a machine that can reproduce itself if it is provided with a program
containing its own description. The process of self-reproduction requires
two steps: first, the constructor has to produce a copy of itself and second, it has to produce the program of how to copy itself. The second step
is important in order that the self-reproduction continues, otherwise, the
child copy cannot self-reproduce. When the constructor produces a copy of
the program, then it attaches it to the child copy and the process repeats.
Unexpectedly, working with classical cellular automata it was found that
there is indeed a universal constructor capable of self-reproducing.
In a sense, von Neumann’s universal constructor is a “Turing test of life”
[Adami (1995)] if we attribute the above unique property to a living system,
though there are other complex properties such as the ability to self-repair,
grow and evolve. From this perspective, the universal constructor is a very
useful model to explore and understand under what conditions a system is
capable of self-reproducing (either artificially or in reality). If one attempts
to understand elementary living systems as quantum mechanical systems in
an information theoretic sense, then one must first try to find out whether
a universal quantum constructor exists. In a simple and decisive manner,
we find that an all-purpose quantum mechanical constructor operating in
a closed universe with finite resources cannot exist.
Wigner was probably the first to address the question of replicating
machines in the quantum world and found that it is infinitely unlikely that
such machines can exist [Wigner (1961)]. It is now well known that the
information content of a quantum state has two remarkable properties: first,
it cannot be copied exactly [Wootters and Zurek (1982); Dieks (1982)] and
second, given several copies of an unknown state we cannot delete a copy
[Pati and Braunstein (2000)]. In addition, non-orthogonal quantum states
Can Arbitrary Quantum Systems Undergo Self-replication?
225
cannot be perfectly copied [Yuen (1986)]. Indeed, the extra information
needed to make a copy must be as large as possible—a recent result known
as the stronger no-cloning theorem [Jozsa (2002)]. The no-cloning and the
no-deleting principles taken together reveal some kind of “permanence” of
quantum information.
11.2.
Formalizing the Self-replicating Machine
First, we observe that merely the copying of information is not selfreplication. Therefore, along with the usual quantum mechanical toolkit, we
must formalize the question of a self-replicating machine. A quantum mechanical universal constructor may be completely specified by a quadruple
UC = (|Ψ, |PU , |C, |Σ), where |Ψ ∈ HN is the state of the (artificial or
real) living system that contains quantum information to be self-replicated,
|PU ∈ HK is the program state that contains instructions to copy the original information, i.e. the unitary operator U needed to copy the state |Ψ
via U (|Ψ|0) = |Ψ|Ψ is encoded in the program state, |C is the state of
the control unit, and |Σ = |0|0 · · · |0 ∈ HM is a collection of blank states
onto which the information and the program will be copied. Let there be
n blank states each with |0 ∈ HN , then the dimension of the blank state
Hilbert space is M = N n . Without loss of generality we may assume that
an individual blank state |0 may belong to a Hilbert space of dimension
equal to N . It is assumed that a finite string of blank states are available
in the environment in which the universal constructor is operating (they
are analogous to the low-entropy nutrient states that are required by a real
living system). The justification for a finite number of such states comes
from the fact that in the universe the total energy and negative entropy
available at any time is always finite [Wigner (1961)]. To copy the program
state the machine uses m blank states in one generation, so K = N m .
Thus M is finite but M ≫ N, K. The initial state of the universal constructor is |Ψ|PU |C|Σ. A universal constructor will be said to exist if
it can implement copying of the original and the stored program by a fixed
linear unitary operator L acting on the combined Hilbert space of the input, program, control and (m + 1) blank states that allows the following
transformation
L(|Ψ|0|PU |0m |C)|0n−(m+1)
= |Ψ|PU L(|Ψ|0|PU |0m |C ′ )|0n−2(m+1) ,
(11.1)
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Quantum Aspects of Life
where |C ′ is the final state of the control unit. It is worth emphasizing that
Eq. (11.1) is not a cloning transformation. It is a recursively defined transformation where the fixed unitary operator L acts on the initial (parent)
configuration and the same operator acts on the final (child) configuration
after the copies have been produced. This definition is required in order
that the self-replication proceeds in an autonomous manner until the blank
states are exhausted. The fixed unitary operator will not act on the child
configuration unless (m + 1) nutrient states are available in the universe.
Once the transformation is complete, the control unit separates the original information from the program states (parent information) so that the
off-spring exists independently. It then continues to self-reproduce.
If such a universal constructor exists, then when it is fed with another
state |Φ and a suitable program |PV to create it via V (|Φ|0) = |Φ|Φ
then it will allow the transformation
L(|Φ|0|PV |0m |C)|0n−(m+1)
= |Φ|PV L(|Φ|0|PV |0m |C ′′ )|0n−2(m+1) .
11.3.
(11.2)
Proof of No-self-replication
If such a machine can make a copy of any state along with its program in
a unitary manner, then it must preserve the inner product. This implies
that we must have
Ψ|Φ PU |PV = Ψ|Φ2 PU |PV 2 C ′ |C ′′ ,
(11.3)
holds true. However, the above equation tells us that the universal constructor can exist only under two conditions, namely, (i) either Ψ|Φ = 0
0 or (ii) Ψ|Φ =
0 and PU |PV = 0. The first conand PU |PV =
dition suggests that for orthogonal states as the carrier of information,
there is no restriction on the program state. This means that with a finite
dimensional program state and a finite number of blank states orthogonal states can self-replicate. Such a universal constructor can exist with
finite resources. This corresponds to the realization of a classical universal constructor, and is consistent with von Neumann’s thesis, that a selfreproducing general purpose machine can exist, in principle, in a deterministic universe [von Neumann (1966)]. However, the second condition tells us
that for non-orthogonal states, the program states have to be orthogonal.
This means that to perfectly self-replicate a collection of non-orthogonal
states {|Ψi } together with their program states {|PUi , i = 1, 2, . . .} one
Can Arbitrary Quantum Systems Undergo Self-replication?
227
requires that the states |PUi should be orthogonal. Since an arbitrary
state such as |Ψ = i αi |i with the complex numbers αi ’s varying continuously can be viewed as an infinite collection of non-orthogonal states
(or equivalently the set of non-orthogonal states for a single quantum system is infinite, even for a simplest two-state system such as a qubit), one
requires an infinite-dimensional program state to copy it. In one generation of the self-replication the number of blank states used to copy the
program state is m = log2 K/ log2 N and when K → ∞ the nutrient resource needed also becomes infinite. As a consequence, to copy an infinitedimensional Hilbert space program state one needs an infinite collection
of blank states to start with. Furthermore, the number of generations g
for which the self-reproduction can occur with a finite nutrient resource is
g = log2 M/(log2 KN ). When K becomes infinite, then there can be no
generations supporting self-reproduction. Therefore, we surmise that with
a finite-dimensional program state and a finite nutrient resource there is no
deterministic universal constructor for arbitrary quantum states.
11.4.
Discussion
One may ask is it not possible to rule out the nonexistence of deterministic
universal constructor from the no-cloning principle? The answer is “no” for
two reasons. First, a simple universal cloner is not a universal constructor.
Second, in a universal constructor we provide the complete specification
about the input state, hence it should have been possible to self-reproduce,
thus reaching an opposite conclusion! The surprising and remarkable result
is that when we ask a constructor to self-replicate any arbitrary living
species, then it cannot. The perplexity of the problem lies where we attempt
to copy the program. If it has to self-replicate then it violates the unitarity
of quantum theory.
This result may have immense bearing on explaining life based on quantum theory. One may argue that after all if everything comes to the molecular scale then there are a variety of physical actions and chemical reactions
that might be explained by the basic laws of quantum mechanics. However, if one applies quantum theory, then as we have proved an arbitrary
quantum mechanical living organism cannot self-replicate. Interpreting this
differently, we might say that the present structure of quantum theory cannot model a living system as it fails to mimic a minimal living system.
Quantum mechanizing a living system seems to be an impossible task. If
228
Quantum Aspects of Life
that holds true, then this conclusion is going to have rather deep implications for our present search for the ultimate laws of nature encompassing
both the physical and biological world. On the other hand, because the
self-reproducible information must be “classical” the replication of DNA
in a living cell can be understood purely by classical means. Having said
this, our result does not preclude the possibility that quantum theory might
play a role in explaining other features of living systems [Penrose (1994);
McFadden (2000)]. For example, there is a recent proposal that quantum
mechanics may explain why living organisms have four nucleotide bases
and twenty amino acids [Patel (2001)]. It has been also reported that the
game of life can emerge in the semi-quantum mechanical context [Flitney
and Abbott (2002)]—see Chapter 12.
Implications of our results are multifold for physical and biological sciences. It is beyond doubt that progress in the burgeoning area of quantum
information technology can lead to revolutions in the machines that one
cannot think of at present. If a quantum mechanical universal constructor
would have been possible, future technology would have allowed quantum
computers to self-replicate themselves with little or no human input. That
would have been a completely autonomous device—a truly marvelous thing.
However, as we have shown, a deterministic universal constructor with finite resources is impossible in principle. One may have to look instead
to probabilistic universal constructors that could self-replicate with only a
limited probability of success, similar to a probabilistic cloner [Duan and
Guo (1998)]. This could still have great implications for the future. With
a complete specification such a machine could construct copies based on its
own quantum information processing devices. Future lines of exploration
may lead to the design of approximate universal constructors in analogy
with approximate universal quantum cloners [Buzek and Hillery (1996)].
11.5.
Conclusion
How life emerges from inanimate quantum objects has been a conundrum
[Schrödinger (1944); Elsasser (1958); Chaitin (1970); Davies (1995)]. What
we have shown here is that quantum mechanics fails to mimic a selfreproducing unit in an autonomous way. Nevertheless, if one allows for
errors in self-replication, which actually do occur in real living systems,
then an approximate universal constructor should exist. Such a machine
would constitute a quantum mechanical mutation machine. It would be
Can Arbitrary Quantum Systems Undergo Self-replication?
229
important to see how variations in “life” emerge due to the errors in selfreplication. From this perspective, if quantum mechanics is the final theory
of Nature, our result indicates that the information stored in a living organism is copied imperfectly and the error rate may be just right in order
for mutation to occur to drive Darwinian evolution. In addition, one could
study how the quantum evolution of species leads to an increase in the level
of complexity in living systems. Since understanding these basic features
of life from quantum mechanical principles is a fundamental task, we hope
that the present result is a first step in that direction, and will be important
in the areas of quantum information, artificial life, cellular automata, and
last but not least in biophysical science.
Acknowledgments
We thank C. Fuchs for bringing the paper of E. P. Wigner to our attention.
AKP wishes to thank C. H. Bennett, S. Lloyd, and I. L. Chuang for useful
discussions. SLB currently holds a Wolfson–Royal Society Research Merit
Award.
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Dieks, D. (1982). Communication by EPR devices, Phys. Lett. A 92, pp. 271–272.
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McFadden, J. (2000). Quantum Evolution (Harper Collins).
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pp. 467–488.
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pp. 507–531.
Flitney, A. P., and Abbott, D. (2002). A semi-quantum version of the game of
life, arXiv quant-ph 0208149.
Grover, L. K. (1997). Quantum mechanics helps in searching a needle in a
haystack, Phys. Rev. Lett. 79, pp. 325–328.
Jozsa, R. (2002). A stronger no-cloning theorem, arXiv quant-ph 0204153.
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von Neumann, J. (1966). The Theory of Self-Replicating Automata (University of
Illinois Press).
Patel, A. (2001). Why genetic information processing could have a quantum basis,
J. of Bioscience 26, pp. 145–151.
Pati, A. K., and Braunstein, S. L. (2000). Impossibility of deleting an unknown
quantum state, Nature 404, p. 164.
Penrose, R. (1994). Shadows of the Mind (Oxford University Press).
Schrödinger, E. (1944). What is life? (Cambridge University Press).
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About the authors
Arun K. Pati obtained his PhD from the University of Bombay, Mumbai, India. He has been a theoretical physicist in the Theoretical Physics
Division, BARC, Mumbai, India since 1989, and is currently a visiting scientist at the Institute of Physics, Bhubaneswar, India. His research is in
Can Arbitrary Quantum Systems Undergo Self-replication?
231
quantum information and computation, the theory of geometric phases and
its applications, and the foundations of quantum mechanics. He is also interested in the quantum mechanics of bio-systems. He has published over
60 papers on these topics and has edited a book on quantum information
theory. Pati is the recipient of the India Physics Association Award for
Young Physicist of the Year (2000) and the Indian Physical Society Award
for Young Scientists (1996). His research has been featured in news items
in Nature, Science and many national and international newspapers.
Samuel L. Braunstein obtained his PhD from Caltech, USA, under Carlton Caves. He joined the University of York, in 2003 and is heading a
group in non-standard computation. He is a recipient of the prestigious
Royal Society-Wolfson Research Merit Award. He was recently awarded
the honorary title of 2001 Lord Kelvin Lecturer. He is editor of three
books Quantum Computing, Scalable Quantum Computing and Quantum
Information with Continuous Variables and serves on the editorial board of
the journal Fortschritte der Physik. He is a Founding Managing Editor of
Quantum Information and Computation. He has over 90 papers published
in refereed journals. His work has received extensive coverage in prestigious
scientific venues such as Science, Nature, Physics Today, New Scientist and
daily newspapers around the world.
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Chapter 12
A Semi-quantum Version of the
Game of Life
Adrian P. Flitney and Derek Abbott
Cellular automata provide a means of obtaining complex behaviour from a
simple array of cells and a deterministic transition function. They supply
a method of computation that dispenses with the need for manipulation of
individual cells and they are computationally universal. Classical cellular
automata have proved of great interest to computer scientists but the construction of quantum cellular automata pose particular difficulties. This
chapter presents a version of John Conway’s famous two-dimensional classical cellular automata Life that has some quantum-like features, including
interference effects. Some basic structures in the new automata are given
and comparisons are made with Conway’s game.
12.1.
12.1.1.
Background and Motivation
Classical cellular automata
A cellular automaton (CA) consists of an infinite array of identical cells, the
states of which are simultaneously updated in discrete time steps according
to a deterministic rule. Formally, they consist of a quadruple (d, Q, N, f ),
where d ∈ Z+ is the dimensionality of the array, Q is a finite set of possible
states for a cell, N ⊂ Zd is a finite neighbourhood, and f : Q|N | → Q is
a local mapping that specifies the transition rule of the automaton. The
simplest cellular automata are constructed from a one-dimensional array
Received February 10, 2006
233
234
Quantum Aspects of Life
...
...
✒
cells ∈ {0, 1}
❅❅
❘
time
rule
❄
...
❄
...
Fig. 12.1. A schematic of a one-dimensional, nearest neighbour, classical cellular automaton showing the updating of one cell in an infinite array.
of cells taking binary values, with a nearest neighbour transition function,
as indicated in Fig. 12.1. Such CA were studied intensely by Wolfram
(1983) in a publication that lead to a resurgence of interest in the field.
Wolfram classified cellular automata into four classes. The classes showed
increasingly complex behaviour, culminating in class four automata that
exhibited self-organization, that is, the appearance of order from a random
initial state.
In general, information is lost during the evolution of a CA. Knowledge
of the state at a given time is not sufficient to determine the complete
history of the system. However, reversible CA are of particular importance, for example, in the modeling of reversible phenomena. Furthermore,
it has been shown that there exists a one-dimensional reversible CA that
is computationally universal [Morita (1989)]. Toffoli (1977) demonstrated
that any d-dimensional CA could be simulated by a (d + 1)-dimensional
reversible CA and later Morita (1995) found a method using partitioning
(see Fig. 12.2) whereby any one-dimensional CA can be simulated by a
reversible one-dimensional CA. There is an algorithm for deciding on the
reversibility of a one-dimensional CA [Amoroso and Patt (1972)], but in
dimensions greater than one, the reversibility of a CA is, in general, undecidable [Kari (1990)].
12.1.2.
Conway’s game of life
John Conway’s game of Life [Gardner (1970)] is a well known twodimensional CA, where cells are arranged in a square grid and have binary
values generally known as “dead” or “alive.” The status of the cells change
in discrete time steps known as “generations.” The new value depends
235
A Semi-quantum Version of the Game of Life
...
...
✏
PP
✏✏
PP
✮✏
q ❄✏
P
rule
❄
(a)
...
time
❄
...
original cell
...
(b)
...
...
❄
❄
...
❄
❄
❄
❄
❄
rule one
...
rule two
time
❄
...
Fig. 12.2. A schematic of a one-dimensional, nearest neighbour, classical (a) partitioned
cellular automaton [Morita (1995)] and (b) block (or Margolus) partitioned cellular automata. In (a), each cell is initially duplicated across three cells and a new transition
rule f : Q3 → Q3 is used. In (b), a single step of the automata is carried out over two
clock cycles, each with its own rule f : Q2 → Q2 .
upon the number of living neighbours, the general idea being that a cell
dies if there is either overcrowding or isolation. There are many different rules that can be applied for birth or survival of a cell and a number
of these give rise to interesting properties such as still lives (stable patterns), oscillators (patterns that periodically repeat), spaceships or gliders
(fixed shapes that move across the Life universe), glider guns, and so on
236
Quantum Aspects of Life
❅ = alive
(a) still lives
(b) blinker
❅❅
❅❅
❅
❅
❅
♣
❅
❅
♣
❅
❅
❅
(i) block
(ii) tub
(iii) boat
♣
❅
♣
❅
❅
initial
(c) beacon
♣ = empty or dead
❅
♣
❅
♣
❅
♣
❅
❅
❅❅❅
♣
♣
1st gen.
2nd gen.
❅❅ ♣ ♣
❅ ♣ ♣ ♣
♣ ♣ ♣ ❅
♣ ♣
❅❅
❅❅ ♣ ♣
❅❅ ♣ ♣
♣ ♣ ❅❅
♣ ♣
❅❅
❅❅ ♣ ♣
❅ ♣ ♣ ♣
♣ ♣ ♣ ❅
♣ ♣
❅❅
initial
1st gen.
2nd gen.
Fig. 12.3. A small sample of the simplest structures in Conway’s Life: (a) the simplest
still-lives (stable patterns) and (b)–(c) the simplest period two oscillators (periodic patterns). A number of these forms will normally evolve from any moderate sized random
collection of alive and dead cells.
[Gardner (1971, 1983); Berlekamp et al. (1982)]. Conway’s original rules
are one of the few that are balanced between survival and extinction of the
Life “organisms.” In this version a dead (or empty) cell becomes alive if it
has exactly three living neighbours, while an alive cell survives if and only
if it has two or three living neighbours. Much literature on the game of
Life and its implications exists and a search on the world wide web reveals
numerous resources. For a discussion on the possibilities of this and other
CA the interested reader is referred to Wolfram (2002).
The simplest still lives and oscillators are given in Fig. 12.3, while
Fig. 12.4 shows a glider, the simplest and most common moving form.
237
A Semi-quantum Version of the Game of Life
♣
♣ ♣
♣
❅
♣ ♣
❅❅
♣
♣
❅❅
♣ ♣ ♣ ♣
♣ ♣
♣
❅
♣ ♣ ♣
❅
♣
❅❅❅
♣ ♣ ♣ ♣
♣ ♣ ♣ ♣
♣
♣
❅ ❅
♣ ♣
❅❅
♣ ♣ ❅ ♣
initial
1st gen.
2nd gen.
3rd gen.
♣
♣
❅
♣
♣
❅
❅
♣
♣
❅❅
♣ ♣ ♣ ♣
Fig. 12.4. In Conway’s Life, the simplest spaceship (a pattern that moves continuously
through the Life universe), the glider. The figure shows how the glider moves one cell
diagonally over a period of four generations, the fourth generation (not shown) is the
same as the first moved diagonally down and to the right.
A large enough random collection of alive and dead cells will, after a period
of time, usually decay into a collection of still lives and oscillators like those
shown here, while firing a number of gliders off toward the outer fringes of
the Life universe.
12.1.3.
Quantum cellular automata
The idea of generalizing classical cellular automata to the quantum domain
was already considered by Feynman (1982). Grössing and Zeilinger made
the first serious attempts to consider quantum cellular automata (QCA)
[Grössing and Zeilinger (1988a,b)], though their ideas are considerably different from modern approaches. Quantum cellular automata are a natural
model of quantum computation where the well developed theory of classical CA might be exploited. Quantum computation using optical lattices
[Mandel et al. (2003)] or with arrays of microtraps [Dumke et al. (2002)] are
possible candidates for the experimental implementation of useful quantum
computing. It is typical of such systems that the addressing of individual
cells is more difficult than a global change made to the environment of
all cells [Benjamin (2000)] and thus they become natural candidates for
the construction of QCA. An accessible discussion of QCA is provided by
Gruska (1999). The simple idea of quantizing existing classical CA by making the local translation rule unitary is problematic: the global rule on an
infinite array of cells is rarely described by a well defined unitary operator.
One must decide whether a given local unitary rule leads to “well-formed”
unitary QCA [Dürr and Santha (2002)] that properly transform probabilities by preserving their sum squared to one. One construction method to
achieve the necessary reversibility of a QCA is to partition the system into
238
Quantum Aspects of Life
...
Û
Û
Û
Û
Û
❍
❍
❍
❍
❍❍
❍❍
❍❍
❍❍
❍
❍
❍
❍
❍
...
φ
φ
. . . one qubit unitaries
right shift
time
❄
. . . control-phase gates
Fig. 12.5. A schematic of a one-dimensional nearest neighbour quantum cellular automaton according to the scheme of Schumacher and Werner (2004) (from Fig. 10 of
that publication). The right-shift may be replaced by a left-shift or no shift.
blocks of cells and apply blockwise unitary transformations. This is the
quantum generalization to the scheme shown in Fig. 12.2(b)—indeed, all
QCA, even those with local irreversible rules, can be obtained in such a
manner [Schumacher and Werner (2004)]. Formal rules for the realization
of QCA using a transition rule based on a quasi-local algebra on the lattice
sites is described by Schumacher and Werner (2004). In this formalism,
a unitary operator for the time evolution is not necessary. The authors
demonstrate that all nearest neighbour one-dimensional QCA arise by a
combination of a single qubit unitary a possible left- or right-shift, and a
control-phase gate,1 as indicated in Fig. 12.5.
Reversible one-dimensional nearest neighbour classical CA are a subset
of the quantum ones. In the classical case, the single qubit unitary can only
be the identity or a bit-flip, while the control-phase gate is absent. This
leaves just six classical CA, all of which are trivial.
12.2.
12.2.1.
Semi-quantum Life
The idea
Conway’s Life is irreversible while, in the absence of a measurement, quantum mechanics is reversible. In particular, operators that represent measurable quantities must be unitary. A full quantum Life on an infinite array
would be impossible given the known difficulties of constructing unitary
QCA [Meyer (1996)]. Interesting behaviour is still obtained in a version
0
1 A control-phase gate is a two-qubit gate that multiplies the target qubit by ( 1
0 exp(iφ) )
if the control qubit is 1.
A Semi-quantum Version of the Game of Life
239
of Life that has some quantum mechanical features. Cells are represented
by classical sine-wave oscillators with a period equal to one generation, an
amplitude between zero and one, and a variable phase. The amplitude of
the oscillation represents the coefficient of the alive state so that the square
of the amplitude gives the probability of finding the cell in the alive state
when a measurement of the “health” of the cell is taken. If the initial
state of the system contains at least one cell that is in a superposition of
eigenstates the neighbouring cells will be influenced according to the coefficients of the respective eigenstates, propagating the superposition to the
surrounding region.
If the coefficients of the superpositions are restricted to positive real
numbers, qualitatively new phenomena are not expected. By allowing the
coefficients to be complex, that is, by allowing phase differences between the
oscillators, qualitatively new phenomena such as interference effects, may
arise. The interference effects seen are those due to an array of classical
oscillators with phase shifts and are not fully quantum mechanical.
12.2.2.
A first model
To represent the state of a cell introduce the following notation:
|ψ = a|alive + b|dead,
(12.1)
subject to the normalization condition
|a|2 + |b|2 = 1.
(12.2)
The probability of measuring the cell as alive or dead is |a|2 or |b|2 , respectively. If the values of a and b are restricted to non-negative real numbers,
destructive interference does not occur. The model still differs from a classical probabilistic mixture, since here it is the amplitudes that are added
and not the probabilities. In our model |a| is the amplitude of the oscillator.
Restricting a to non-negative real numbers corresponds to the oscillators
all being in phase.
The birth, death and survival operators have the following effects:
B̂|ψ = |alive ,
D̂|ψ = |dead ,
Ŝ|ψ = |ψ .
(12.3)
240
Quantum Aspects of Life
A cell can be represented by the vector ab . The B̂ and D̂ operators are
not unitary. Indeed they can be represented in matrix form by
1
0
0
D̂ ∝
1
B̂ ∝
1
,
0
0
,
1
(12.4)
where the proportionality constant is not relevant for our purposes. After applying B̂ or D̂ (or some mixture) the new state will require (re-)
normalization so that the probabilities of being dead or alive still sum to
unity.
A new generation is obtained by determining the number of living neighbours each cell has and then applying the appropriate operator to that cell.
The number of living neighbours in our model is the amplitude of the superposition of the oscillators representing the surrounding eight cells. This
process is carried out on all cells effectively simultaneously. When the cells
are permitted to take a superposition of states, the number of living neighbours need not be an integer. Thus a mixture of the B̂, D̂ and Ŝ operators
may need to be applied. For consistency with standard Life the following conditions will be imposed upon the operators that produce the next
generation:
• If there are an integer number of living neighbours the operator applied
must be the same as that in standard Life.
• The operator that is applied to a cell must continuously change from
one of the basic forms to another as the sum of the a coefficients from
the neighbouring cells changes from one integer to another.
• The operators can only depend upon this sum and not on the individual
coefficients.
If the sum of the a coefficients of the surrounding eight cells is
A=
8
i=1
ai
(12.5)
A Semi-quantum Version of the Game of Life
241
then the following set of operators, depending upon the value of A, is the
simplest that has the required properties
0 ≤ A ≤ 1; Ĝ0 = D̂ ,
√
1 < A ≤ 2; Ĝ1 = ( 2 + 1)(2 − A)D̂ + (A − 1)Ŝ ,
√
2 < A ≤ 3; Ĝ2 = ( 2 + 1)(3 − A)Ŝ + (A − 2)B̂ ,
√
3 < A < 4; Ĝ3 = ( 2 + 1)(4 − A)B̂ + (A − 3)D̂ ,
(12.6)
A ≥ 4; Ĝ4 = D̂ .
For integer values of A, the Ĝ operators are the same as the basic operators
of standard Life, as required. For non-integer values in the range (1, 4), the
operators
are a linear combination of the standard operators. The factors
√
the
of 2 + 1 have been inserted to give more appropriate behaviour in √
middle of each range. For example, consider the case where A = 3 + 1/ 2,
a value that may represent three neighbouring cells that are alive and one
the has a probability of one-half of being alive. The operator in this case is
1
1 11
1
Ĝ = √ B̂ + √ D̂, ∝ √
.
(12.7)
2
2
2 11
Applying this to either a cell in the alive, 10 or dead, 01 states will
produce the state
1
1
|ψ = √ |alive + √ |dead
2
2
(12.8)
which represents a cell with a 50% probability of being alive. That is, Ĝ
is an equal combination of the birth and death operators, as might have
been expected given the possibility that A represents an equal probability
of three or four living neighbours. Of course the same value of A may have
been obtained by other combinations of neighbours that do not lie half way
between three and four living neighbours, but one of our requirements is
that the operators can only depend on the sum of the a coefficients of the
neighbouring cells and not on how the sum was obtained.
In general the new state of a cell is obtained by calculating A, applying
the appropriate operator Ĝ:
′
a
a
= Ĝ
,
(12.9)
b′
b
242
Quantum Aspects of Life
and then normalizing the resulting state so that |a′ |2 + |b′ |2 = 1. It is this
process of normalization that means that multiplying the operator by a
constant has no effect. Hence, for example, Ĝ2 for A = 3 has the same
effect
√ as Ĝ3 in the limit as A → 3, despite differing by the constant factor
( 2 + 1).
12.2.3.
A semi-quantum model
To get qualitatively different behaviour from classical Life we need to introduce a phase associated with the coefficients, that is, a phase difference
between the oscillators. We require the following features from this version
of Life:
• It must smoothly approach the classical mixture of states as all the
phases are taken to zero.
• Interference, that is, partial or complete cancellation between cells of
different phases, must be possible.
• The overall phase of the Life universe must not be measurable, that
is, multiplying all cells by eiφ for some real φ will have no measurable
consequences.
• The symmetry between (B̂, |alive) and (D̂, |dead) that is a feature
of the original game of Life should be retained. This means that if the
state of all cells is reversed (|alive ←→ |dead) and the operation of
the B̂ and D̂ operators is reversed the system will behave in the same
manner.
In order to incorporate complex coefficients, while keeping the above properties, the basic operators are modified in the following way:
B̂|dead = eiφ |alive ,
B̂|alive = |alive ,
D̂|alive = eiφ |dead ,
(12.10)
D̂|dead = |dead ,
Ŝ|ψ = |ψ ,
where the superposition of the surrounding oscillators is
α=
8
i=1
ai = Aeiφ ,
(12.11)
A Semi-quantum Version of the Game of Life
243
A and φ being real positive numbers. That is, the birth and death operators
are modified so that the new alive or dead state has the phase of the sum
of the surrounding
cells. The operation of the B̂ and D̂ operators on the
a
state b can be written as
a
a + |b|eiφ
=
,
b
0
a
0
,
D̂
=
b
|a|eiφ + b
B̂
(12.12)
with Ŝ leaving the cell unchanged. The modulus of the sum of the neighbouring cells A determines which operators apply, in the same way as before
[see Eq. (12.6)]. The addition of the phase factors for the cells allows for
interference effects since the coefficients of alive cells may not always reinforce in taking the sum, α =
ai . A cell with a = −1 still has a unit
probability of being measured in the alive state but its effect on the sum
will cancel that of a cell with a = 1. A phase for the dead cell is retained in
order to maintain the alive ←→ dead symmetry, however, it has no effect.
Such an effect would conflict with the physical model presented earlier and
would be inconsistent with Conway’s Life, where the empty cells have no
influence.
A useful notation to represent semi-quantum Life is to use an arrow
whose length represents the amplitude of the a coefficient and whose angle
with the horizontal is a measure of the phase of a. That is, the arrow
represents the phasor of the oscillator at the beginning of the generation.
For example
1
−→ =
,
0
1/2
i/2
iπ/2 √
√
=
,
↑=e
(12.13)
3/2
i 3/2
√
1/√2
(1 + i)/2
ր = eiπ/4
=
,
1/ 2
(1 + i)/2
etc. In this picture α is the vector sum of the arrows. This notation includes
no information about the b coefficient. The magnitude of this coefficient
can be determined from a and the normalization condition. The phase of
the b coefficient has no effect on the evolution of the game state so it is not
necessary to represent this.
244
12.2.4.
Quantum Aspects of Life
Discussion
The above rules have been implemented in the computer algebra language
Mathematica. All the structures of standard Life can be recovered by making the phase of all the alive cells equal. The interest lies in whether there
are new effects in the semi-quantum model or whether existing effects can
be reproduced in simpler or more generalized structures. The most important aspect not present in standard Life is interference. Two live cells can
work against each other as indicated in Fig. 12.6 that shows an elementary
example in a block still life with one cell out of phase with its neighbours.
In standard Life there are linear structures called wicks that die or “burn”
at a constant rate. The simplest such structure is a diagonal line of live
cells as indicated in Fig. 12.7(a). In this, it is not possible to stabilize an
end without introducing other effects. In the new model a line of cells of
alternating phase (. . . −→←− . . .) is a generalization of this effect since
(a)
✲ ✲
q
q
q
q
✲✛
q
✛
q
q
initial
(b)
1st gen.
✲ ✲
✲ ✲
✲
✲ ❏
❪
✻
(i)
(c)
2nd gen.
2π/3
(ii)
✲ ✲
✲ ✲
✲ ✲
✲ ■
❅
❅ 3π/4
✲ ■
❅
❅
✲
initial
1st gen.
❅
■
❅
2nd gen.
Fig. 12.6. (a) A simple example of destructive interference in semi-quantum Life: a
block with one cell out of phase by π dies in two generations. (b) Blocks where the phase
difference of the fourth cell is insufficient to cause complete destructive interference; each
cell maintains a net of at least two living neighbours and so the patterns are stable. In
the second of these, the fourth cell is at a critical angle. Any greater phase difference
causes instability resulting in eventual death as seen in (c), which dies in the fourth
generation.
A Semi-quantum Version of the Game of Life
245
♣♣♣
❅
❅
❅
❅
(a)
❅
❅
❅
❅
♣♣♣
✲ ✲
(b)
(c)
✲ ✲✛
✲✛
✲✛
✲ ...
✻ ✲✛
✲✛
✲✛
✲ ...
✻ ✲✛
✲✛
✲✛
✲ ...
Fig. 12.7. (a) A wick (an extended structure that dies, or “burns”, at a constant rate)
in standard Life that burns at the speed of light (one cell per generation), in this case
from both ends. It is impossible to stabilize one end without giving rise to other effects.
(b) In semi-quantum Life an analogous wick can be in any orientation. The block on
the left-hand end stabilizes that end; a block on both ends would give a stable line;
the absence of the block would give a wick that burns from both ends. (c) Another
example of a light-speed wick in semi-quantum Life showing one method of stabilizing
the left-hand end.
it can be in any orientation and the ends can be stabilized easily. Figures 12.7(b)–(c) show some examples. A line of alternating phase live cells
can be used to create other structures such as the loop in Fig. 12.8. This
is a generalization of the boat still life, Fig. 12.3(a)(iii), in the standard
model that is of a fixed size and shape. The stability of the line of −→←−’s
results from the fact that while each cell in the line has exactly two living
neighbours, the cells above or below this line have a net of zero (or one at a
corner) living neighbours due to the canceling effect of the opposite phases.
No new births around the line will occur, unlike the case where all the cells
are in phase.
Oscillators (Fig. 12.3) and spaceships (Fig. 12.4) cannot be made simpler
than the minimal examples presented for standard Life. Figure 12.9 shows a
stable boundary that results from the appropriate adjustment of the phase
246
Quantum Aspects of Life
✛
✲✛
✲
✲
✛
✛
✲
✲
✛
✛
✲✛
✲✛
✲
Fig. 12.8. An example of a stable loop made from cells of alternating phase. Above
a certain minimum, such structures can be made of arbitrary size and shape compared
with a fixed size and limited orientations in Conway’s scheme.
...
✡
✣
✡
✲ ✲
❏
❪
❏
❏
✡
✣
❏
❫
✡
✲ ✲ ✲ ✲ ✲ ...
✡
✡
✢
❏
❪
❏
Fig. 12.9. A boundary utilizing appropriate phase differences to produce stability. The
upper cells are out of phase by ±π/3 and the lower by ±2π/3 with the central line.
differences between the cells. The angles have been chosen so that each cell
in the line has between two and three living neighbours, while the empty
cells above and below the line have either two or four living neighbours and
so remain life-less. Such boundaries are known in standard Life but require
a more complex structure.
In Conway’s Life interesting effects can be obtained by colliding gliders.
In the semi-quantum model additional effects can be obtained from colliding
gliders and “anti-gliders,” where all the cells have a phase difference of π
with those of the original glider. For example, a head-on collision between
a glider and an anti-glider, as indicated in Fig. 12.10, causes annihilation,
where as the same collision between two gliders leaves a block. However,
there is no consistency with this effect since other glider-antiglider collisions
produce alternative effects, sometimes being the same as those from the
collision of two gliders.
A Semi-quantum Version of the Game of Life
247
✲
✲
✲
✲ ✲
✛ ✛
✛
✛
✛
Fig. 12.10. A head on collision between a glider and its phase reversed counter part,
an anti-glider, produces annihilation in six generations.
12.3.
Summary
John Conway’s game of Life is a two-dimensional cellular automaton where
the new state of a cell is determined by the sum of neighbouring states that
are in one particular state generally referred to as “alive.” A modification
to this model is proposed where the cells may be in a superposition of the
alive and dead states with the coefficient of the alive state being represented
by an oscillator having a phase and amplitude. The equivalent of evaluating
the number of living neighbours of a cell is to take the superposition of the
oscillators of the surrounding states. The amplitude of this superposition
will determine which operator(s) to apply to the central cell to determine
its new state, while the phase gives the phase of any new state produced.
Such a system show some quantum-like aspects such as interference.
Some of the results that can be obtained with this new scheme have
been touched on in this chapter. New effects and structures occur and
some of the known effects in Conway’s Life can occur in a simpler manner.
However, the scheme described should not be taken to be a full quantum
analogue of Conway’s Life and does not satisfy the definition of a QCA.
The field of quantum cellular automata is still in its infancy. The protocol of Schumacher and Werner (2004) provides a construction method
for the simplest QCA. Exploration and classification of these automata is
an important unsolved task and may lead to developments in the quantum
domain comparable to those in the classical field that followed the exploration of classical CA. Quantum cellular automata are a viable candidate
for achieving useful quantum computing.
248
Quantum Aspects of Life
References
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injectivity of parallel maps for tessellation structures, J. Comput. Syst. Sci.
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Benjamin, S. C. (2000). Schemes for parallel quantum computation without local
control of qubits, Phys. Rev. A 61, art. no. 020301.
Berlekamp, E. R., Conway, J. H., and Guy, R. K. (1982). Winning Ways for Your
Mathematical Plays Vol. 2, (Academic Press).
Dumke, R., Volk, M., Müther, T., Buchkremer, F. B. J., Birkl, G., and Ertmer, W. (2002). Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with
atomic qubits, Phys. Rev. Lett. 89, art. no. 097903.
Dürr, C., and Santha M. (2002). A decision procedure for well-formed unitary
linear quantum cellular automata, SIAM J. Comput. 31, pp. 1076–1089.
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Conway’s new solitaire game of “Life”, Sci. Am. 223, 10, p. 120.
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2, p. 116.
Gardner, M. (1983). Wheels, Life and Other Mathematical Amusements (W. H.
Freeman)
Grössing G., and Zeilinger, A. (1988a). Quantum cellular automata, Complex
Systems 2, pp. 197–208.
Grössing G., and Zeilinger, A. (1988b). Structures in quantum cellular automata,
Physica B 151, pp. 366–370.
Gruska, J. (1999). Quantum Computing ch. 4, pp. 181–192 (McGraw Hill, Maidenhead, Berkshire, UK).
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(2003). Coherent transport of neutral atoms in spin-dependent optical lattice potentials, Phys. Rev. Lett. 91, art. no. 010407.
Meyer, D. A. (1996). From quantum cellular automata to quantum lattice gases,
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Morita, K. (1989). Computation universality of one-dimensional reversible (injective) cellular automata, Trans. IEICE 72, pp. 758–762.
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Schumacher, B., and Werner, R. F. (2004). Reversible quantum cellular automata,
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Toffoli, T. (1977). Cellular Automata Mechanics, PhD thesis, University of Michigan.
A Semi-quantum Version of the Game of Life
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Wolfram, S. (1983). Statistical mechanics of cellular automata, Rev. Mod. Phys.
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About the authors
Adrian P. Flitney completed a Bachelor of Science with first class honours
in theoretical physics at the University of Tasmania in 1983. He worked in
the field of ionospheric physics and high frequency radio communications
for the Department of Science for two years and later for Andrew Antennas
Corporation. He was a researcher in quantum field theory in the Physics
Department at the University of Adelaide for some time, and worked as a
tutor in physics and mathematics both within the University and privately.
In 2001 Flitney finally saw the light and began a PhD in the field of quantum
information with the Department of Electrical and Electronic Engineering
at the University of Adelaide. At the beginning of 2005 he submitted a
thesis entitled “Aspects of Quantum Game Theory,” under Derek Abbott,
and was awarded his doctorate with highest commendation. He is currently
working in the School of Physics, University of Melbourne on quantum
games and decoherence under an Australian Research Council Postdoctoral
Fellowship. Currently Flitney’s major non-academic interests are chess,
where he has achieved considerable success in the past, and kayaking.
Derek Abbott was born in South Kensington, London, UK. He received
his BSc (Hons) in Physics from Loughborough University of Technology. He
obtained his PhD in Electrical and Electronic Engineering from the University of Adelaide, under Kamran Eshraghian and Bruce R. Davis. He is
with The University of Adelaide, Australia, where he is presently a full professor and the Director of the Centre for Biomedical Engineering (CBME).
He has served as an editor and/or guest editor for a number of journals
including Chaos (AIP), Smart Structures and Materials (IOP), Journal
of Optics B (IOP), Microelectronics Journal (Elsevier), Proceedings of the
IEEE, and Fluctuation Noise Letters (World Scientific). He is a life Fellow
of the Institute of Physics (IOP) and Fellow of the Institution of Electrical
& Electronic Engineers (IEEE). He has won a number of awards including
a 2004 Tall Poppy Award for Science. He holds over 300 publications and
250
Quantum Aspects of Life
is a co-author of the book Stochastic Resonance published by Cambridge
University Press. Prof Abbott is co-founder of two international conference series: Microelectronics in the New Millennium (with J. F. Lopez)
and Fluctuations and Noise (with L. B. Kish).
Chapter 13
Evolutionary Stability in
Quantum Games
Azhar Iqbal and Taksu Cheon
In evolutionary game theory an Evolutionarily Stable Strategy (ESS) is a
refinement of the Nash equilibrium (NE) concept that is sometimes also recognized as evolutionary stability. It is a game-theoretic model, well known
to mathematical biologists, that was found quite useful in the understanding of evolutionary dynamics of a population. This chapter presents an
analysis of evolutionary stability in the emerging field of quantum games.
Games such as chess, warfare and politics have been played throughout history. Whenever individuals meet who have conflicting desires, and
priorities, then strategic games are likely to be played. Analysis and understanding of games has existed for long a time but the emergence of
game theory as a formal study of games is widely believed to have taken
place when Neumann and Morgenstern [Neumann and Morgenstern (1953)]
published their pioneering book “The Theory of Games and Economic Behaviour” in 1944. Game theory [Rasmusen (1989)] is now an established
discipline of mathematics that is a vast subject having a rich history and
content. Roughly speaking, game theory is the analysis of the actions made
by rational players when these actions are strategically interdependent.
The 1970s saw game theory being successfully applied to problems of
evolutionary biology and a new branch of game theory, recognized as evolutionary game theory [Maynard Smith (1982); Hofbrauer and Sigmund
(1998); Weibull (1995)], came into existence. The concept of utility from
Received April 8, 2007
251
252
Quantum Aspects of Life
economics was given an interpretation in terms of Darwinian fitness. Originally, evolutionary game theory considered animal conflicts occurring in
the macro-world. Later, research in biology [Turner and Chao (1999)] suggested that nature also plays classical games at the micro-level. Bacterial
infections by viruses are classical game-like situations where nature prefers
dominant strategies. Economics and biology are not the only areas that
have benefited from game theory as there is recent interest in the application of game theory to problems in physics [Abbott et al. (2002)].
In game theory [Neumann and Morgenstern (1953); Rasmusen (1989)]
one finds many examples where multiple Nash equilibria (NE) [Nash (1950,
1951)] emerge as solutions of a game. To select one (or possibly more)
out of these requires some refinement of the equilibrium concept [Myerson
(1978)]. A refinement is a rule/criterion that describes the preference of one
(in some cases more than one) equilibrium out of many. Numerous refinements are found in game theory, for example, perfect equilibrium (used for
extensive- and normal-form games), sequential equilibrium (a fundamental
non-cooperative solution concept for extensive-form games), and correlated
equilibrium (used for modelling communication among players). The ESS
concept is known to be a refinement on the set of symmetric NE and is
generally considered central to evolutionary game theory.
During recent years quantum game theory [Meyer (1999); Eisert et
al. (1999); Eisert and Wilkens (2000); Flitney and Abbott (2002a)] has
emerged as a new research field within quantum information and computation [Nielsen and Chuang (2000)]. A significant portion of research in
quantum games deals with the question asking how quantization of a game
affects/changes the existence/location of a NE. This question has been addressed in a number of publications [Flitney and Abbott (2002a,b); Iqbal
(2004)] in this area and now it seems that it is generally agreed that quantization of a game indeed affects/changes the existence/location of a NE.
In this chapter we argue that, like asking how quantization of a game
affects/changes the existence/location of a NE, an equally important question for quantum games is to ask how quantization of a game can affect a
refinement of the NE concept. While focussing on the refinement of the notion of an ESS, we motivate those quantum games in which a NE persists1
in both of its classical and quantum versions while its property of being
an ESS survives in either classical or its quantum version, but not in both.
1 By saying that a NE persists in both the classical and quantum version of a game
we mean that there exists a NE consisting of quantum strategies that rewards both the
players exactly the same as the corresponding NE does in the classical version of the
game.
Evolutionary Stability in Quantum Games
253
We argue that, the quantum games offering such situations, along with
their quantization procedures, can justifiably be said to extend the boundary of investigations in quantum games from existence/location of NE to
existence/location of one (or more) of its refinements.
13.1.
Evolutionary Game Theory and Evolutionary
Stability
The roots of evolutionary game theory [Weibull (1995)] can be traced to the
puzzle of the approximate equality of the sex ratio in mammals. In 1930
Fisher ([Stanford (2003)]) noticed that if individual fitness is defined in
terms of the expected number of grandchildren, then it becomes dependent
on how males and females are distributed in a population. Fisher showed
that the evolutionary dynamics then leads to the sex ratio becoming fixed
at equal numbers of males and females. Although Fisher’s argument can be
recast in game-theoretic language, it was not originally presented in those
terms. Perhaps it was due to the fact that until that time modern game
theory had not yet emerged as a formal study.
Modern game theory was used, for the first time, to understand evolution when in 1972 Maynard Smith and G. R. Price introduced the concept of an Evolutionarily Stable Strategy (ESS) [Maynard Smith and Price
(1973); Maynard Smith (1982)]. Presently, this concept is widely believed
to be the cornerstone of evolutionary game theory [Hofbrauer and Sigmund
(1998)] and has been found quite useful to explain and understand animal
behaviour.
Traditionally, game theory had concerned analyzing interactions among
hyperrational players and the idea that it can be applied to animals seemed
strange at the time. The ESS concept made three important changes in
the traditional meaning of the concepts of a) strategy, b) equilibrium, and
c) players’ interactions.
a) Strategy: In traditional game theory, players have strategy set from
which they choose their strategies. In biology, animals belonging to a
species have strategy sets, which are genotypic variants that may be mutated, of which individuals inherit one or another variant, which they then
play in their strategic interactions. A mixed strategy in game theory means
a convex linear combination (with real and normalized coefficients) of pure
strategies. Because genotypic variants are taken as pure strategies, the evolutionary game theory interprets a mixed strategy in terms of proportion
of the population that is playing that strategy.
254
Quantum Aspects of Life
b) Equilibrium: An ESS represents an equilibrium and it is a strategy
having the property that if a whole population plays it, it cannot be invaded
under the influence of natural selection, by a small group of players playing
mutant strategies. Because strategies of evolutionary games are genotypes
the ESS definition takes the following form: If adapted by a whole population an ESS is a genotype that cannot be invaded by another genotype
when it appears in a small fraction of the total population.
c) Player interactions: The ESS concept is about repeated and random
pairing of players who play strategies based on their genome and not on the
previous history of play. This concept presented a new approach to analyze
repeated games that are played in evolutionary settings.
Consider a large population [Weibull (1995); Hofbrauer and Sigmund
(1998)] in which members are matched repeatedly and randomly in pairs
to play a bi-matrix game. The players are anonymous, that is, any pair
of players plays the same symmetric bi-matrix game. The symmetry of a
bi-matrix game means that for a strategy set S Alice’s payoff when she
plays S1 ∈ S and Bob plays S2 ∈ S is the same as Bob’s payoff when he
plays S1 and Alice plays S2 . Hence, a player’s payoff is defined by his/her
strategy and not by his/her identity and an exchange of strategies by the
two players also exchanges their respective payoffs. A symmetric bi-matrix
game is represented by an expression G = (M, M T ) where M is the first
player’s payoff matrix and M T , being its transpose, is the second players’
payoff matrix. In a symmetric pair-wise contest one can write P (x, y) as
being the payoff to a x-player against a y-player.
To be precise [Hofbrauer and Sigmund (1998); Bomze (1996); Bomze
and Pötscher (1989)] a strategy x is said to be an ESS if:
a) for each mutant strategy y there exists a positive invasion barrier.
b) if the population share of individuals playing the mutant strategy
y falls below the invasion barrier, then x earns a higher expected payoff
than y.
Mathematically speaking [Weibull (1995); Hofbrauer and Sigmund
(1998)] x is an ESS when for each strategy y = x the inequality
P [x, (1 − ǫ)x + ǫy] > P [y, (1 − ǫ)x + ǫy]
(13.1)
holds for all sufficiently small ǫ > 0. In Eq. (13.1) the expression on the
left-hand side is payoff to the strategy x when played against the mixed
strategy (1 − ǫ)x + ǫy. This condition for an ESS is shown [Maynard Smith
and Price (1973); Maynard Smith (1982); Weibull (1995)] equivalent to the
Evolutionary Stability in Quantum Games
255
following requirements:
a) P (x, x) > P (y, x)
b) If P (x, x) = P (y, x) then P (x, y) > P (y, y) .
(13.2)
It turns out [Maynard Smith (1982); Weibull (1995)] that an ESS is a
symmetric NE that is stable against small mutations. Condition a) in the
definition (13.2) shows (x, x) is a NE for the bi-matrix game G = (M, M T )
if x is an ESS. However, the converse is not true. That is, if (x, x) is a NE
then x is an ESS only if x satisfies condition b) in Definition (13.2).
Evolutionary game theory defines the concept of fitness [Prestwich
(1999)] of a strategy as follows. Suppose x and y are pure strategies played
by a population of players that is engaged in a two-player game. Their
fitnesses are
W (x) = P (x, x)Fx + P (x, y)Fy ;
W (y) = P (y, x)Fx + P (y, y)Fy (13.3)
where Fx and Fy are frequencies (the relative proportions) of the pure
strategies x and y respectively.
It turned out that an ESS is a refinement on the set of symmetric Nash
equilibria [Weibull (1995); Cressman (1992)]. For symmetric bi-matrix
games this relationship is described [van der Laan and Tiemen (1996)] as
△ESS ⊂ △P E ⊂ △N E where △P E = Φ and △N E , △P E , △ESS are the sets
of symmetric NE, symmetric proper equilibrium, and ESSs respectively.
The property of an ESS of being robust against small mutations is also
referred to as evolutionary stability [Bomze (1996); Bomze and Pötscher
(1989)]. In evolutionary game theory, the Darwinian natural selection is
formulated as an algorithm called replicator dynamics [Hofbrauer and Sigmund (1998); Weibull (1995)], which is a mathematical statement saying
that in a population the proportion of players playing better strategies increases with time. Mathematically, ESSs come out as the rest points of
replicator dynamics [Hofbrauer and Sigmund (1998); Weibull (1995)].
The concept of evolutionary stability has provided significant part of
the motivation for later developments in evolutionary game theory and was
found to be a useful concept because it says something about the dynamic
properties of a system without being committed to a particular dynamic
model. Sometimes, it is also described as a model of rationality that is
physically grounded in natural selection.
256
13.1.1.
Quantum Aspects of Life
Population setting of evolutionary game theory
Evolutionary game theory introduces the so-called population setting
[Weibull (1995); Hofbrauer and Sigmund (1998)] that is also known as a
population-statistical setting. This setting assumes a) an infinite population
of players who are engaged in random pair-wise contests, b) each player being programmed to play only one strategy, and c) an evolutionary pressure
ensuring that better-performing strategies have better chances of survival
at the expense of other competing strategies. Because of b) one can refer
to better-performing players as better-performing strategies.
Although it may give such an impression, the population setting of evolutionary game theory is not alien to the concept of the NE. In fact, as
was found later, John Nash himself had this setting in his mind when he
introduced this concept in game theory. In his unpublished PhD thesis
[Nash (1950); Hofbrauer and Sigmund (1998)] he wrote “it is unnecessary
to assume that the participants have...the ability to go through any complex
reasoning process. But the participants are supposed to accumulate empirical information on the various pure strategies at their disposal...We assume
that there is a population...of participants...and that there is a stable average
frequency with which a pure strategy is employed by the ‘average member’
of the appropriate population.”
That is, Nash had suggested a population interpretation of the NE concept in which players are randomly drawn from large populations. Nash
assumed that these players were not aware of the total structure of the
game and did not have either the ability nor inclination to go through any
complex reasoning process.
13.2.
Quantum Games
This chapter considers the game-theoretic concept of evolutionary stability
in quantum games that are played in the two quantization schemes: the
Eisert, Wilkens, Lewenstein (EWL) scheme [Eisert et al. (1999); Eisert
and Wilkens (2000)] for playing quantum Prisoners’ Dilemma (PD) and
Marinatto and the Weber (MW) scheme [Marinatto and Weber (2000a)]
for playing the quantum Battle of the Sexes (BoS) game.
The EWL quantization scheme appeared soon after Meyer’s publication
[Meyer (1999)] of the PQ penny-flip—a quantum game that generated significant interest and is widely believed to have led to the creation of the new
research field of quantum games. The MW scheme derives from the EWL
Evolutionary Stability in Quantum Games
257
scheme, but it gives a different meaning to the term “strategy” [Benjamin
(2000); Marinatto and Weber (2000b)].
EWL quantum PD assigns two basis vectors |C and |D in the Hilbert
space of a qubit, where C and D refer to the strategies of Cooperation
and Defection, respectively, in PD. States of the two qubits belong to twodimensional Hilbert spaces HA and HB , respectively. The state of the game
is defined as being a vector residing in the tensor-product space HA ⊗ HB ,
spanned by the bases |CC , |CD , |DC and |DD. The initial state of
the game is |ψini = Jˆ |CC where Jˆ is a unitary operator known to both
the players. Alice’s and Bob’s strategies are unitary operations ÛA and
ÛB , respectively, chosen from a strategic space Ş. After players’ actions
the state of the game changes to ÛA ⊗ ÛB Jˆ |CC. Finally, the state is
measured and it consists of applying reverse unitary operator Jˆ† followed
by a pair of Stern-Gerlach type detectors. Before detection the final state
of the game is |ψfin = Jˆ† ÛA ⊗ ÛB Jˆ |CC. Players’ expected payoffs are
the projections of the state |ψfin onto the basis vectors of tensor-product
space HA ⊗ HB , weighed by the constants appearing in the following Game
Matrix,
Bob
CD
C (r, r) (s, t)
.
Alice
D (t, s) (u, u)
(13.4)
The first and the second entry in small braces correspond to Alice’s and
Bob’s (classical, pure strategy) payoffs, respectively. When s < u < r < t
the Matrix (13.4) represents PD. In EWL quantum PD Alice’s payoff, for
example, reads
2
2
2
2
PA = r | CC | ψfin | + s | CD | ψfin | + t | DC | ψfin | + u | DD | ψfin | .
(13.5)
With reference to the Matrix (13.4) Bob’s payoff is, then, obtained by
the transformation s ⇄ t in Eq. (13.5). Eisert and Wilkens (2000) used
following matrix representations of unitary operators of their one- and twoparameter strategies, respectively:
cos(θ/2) sin(θ/2)
U (θ) =
(13.6)
- sin(θ/2) cos(θ/2)
iφ
e cos(θ/2) sin(θ/2)
(13.7)
U (θ, φ) =
- sin(θ/2)
e−iφ cos(θ/2)
258
Quantum Aspects of Life
where
0 ≤ θ ≤ π and 0 ≤ φ ≤ π/2 .
(13.8)
To ensure that the classical game is faithfully represented in its quantum
ˆ
version, EWL imposed an additional conditions on J:
!
!
!
ˆ Ĉ ⊗ D̂ = 0
ˆ D̂ ⊗ Ĉ = 0, J,
ˆ D̂ ⊗ D̂ = 0, J,
J,
(13.9)
with Ĉ and D̂ being the operators corresponding to the classical strategies
C and D, respectively. A unitary operator satisfying the condition (13.9) is
"
#
Jˆ = exp iγ D̂ ⊗ D̂/2
(13.10)
where γ ∈ [0, π/2] and Jˆ represents a measure of the game’s entanglement.
At γ = 0 the game can be interpreted as a mixed-strategy classical game.
For a maximally entangled game γ = π/2 the classical NE of D̂ ⊗ D̂ is replaced by a different unique equilibrium Q̂ ⊗ Q̂ where Q̂ ∼ Û (0, π/2). This
new equilibrium is found also to be Pareto optimal [Rasmusen (1989)],
that is, a player cannot increase his/her payoff by deviating from this pair
of strategies without reducing the other player’s payoff. Classically (C, C)
is Pareto optimal, but is not an equilibrium [Rasmusen (1989)], thus resulting in the “dilemma” in the game. It is argued [Benjamin and Hayden
(2001); Eisert et al. (2001)] that in its quantum version the dilemma disappears from the game and quantum strategies give a superior performance
if entanglement is present.
The MW quantization scheme [Marinatto and Weber (2000a); Benjamin
(2000); Marinatto and Weber (2000b)] for BoS identifies a state in 2 ⊗ 2
dimensional Hilbert space as a strategy. At the start of the game the players
are supplied with this strategy and the players manipulate the strategy
in the next phase by playing their tactics. The state is finally measured
and payoffs are rewarded depending on the results of the measurement. A
player can carry out actions within a two-dimensional subspace. Tactics are
therefore local actions on a player’s qubit. The final measurement, made
independently on each qubit, takes into consideration the local nature of
players’ manipulations. This is done by selecting a measurement basis that
respects the division of Hilbert space into two equal parts.
259
Evolutionary Stability in Quantum Games
Set of unitary operators
A maximally entangled
two-qubit quantum
state
Uˆ (T ) or
Uˆ (T , I )
Alice
Final
quantum
state
Uˆ A
Inverse gate
Alice’s payoff
Jˆ
Entangling
gate
Two
qubits
Jˆ †
Measurement
Bob’s payoff
Uˆ B
Bob
Uˆ (T ) or
Uˆ (T , I )
Fig. 13.1.
The EWL scheme for playing a quantum game.
Essentially the MW scheme differs from the EWL scheme [Eisert et al.
(1999); Eisert and Wilkens (2000)] in the absence of the reverse gate2 J † .
Finally, the quantum state is measured and it is found that the classical
game remains a subset of the quantum game if the players’ tactics are limited to a convex linear combination, with real and normalized coefficients, of
applying the identity Iˆ and the Pauli spin-flip operator σ̂x . Classical game
results when the players are forwarded an initial strategy |ψin = |00.
Let ρin be the initial strategy the players Alice and Bob receive at the
start of the game. Assume Alice acts with identity Iˆ on ρin with probability
p and with σ̂x with probability (1 − p). Similarly, Bob act with identity Iˆ
with probability q and with σ̂x with probability (1 − q). After the players’
actions the state changes to
†
†
†
†
⊗ σ̂xB
⊗ IˆB
+ p(1 − q)IˆA ⊗ σ̂xB ρin IˆA
ρfin = pq IˆA ⊗ IˆB ρin IˆA
†
†
+ q(1 − p)σ̂xA ⊗ IˆB ρin σ̂xA
⊗ IˆB
†
†
+ (1 − p)(1 − q)σ̂xA ⊗ σ̂xB ρin σ̂xA
⊗ σ̂xB
.
(13.11)
2 EWL introduced the gate J † before measurement takes place that makes sure that
the classical game remains a subset of its quantum version.
260
Quantum Aspects of Life
Hermitian and unitary
operators
Preparation of a
two-qubit
quantum state
Vˆ x
Iˆ
Alice
p
(1-p)
Alice’s payoff
Two
qubits
Measurement
Bob’s payoff
q
(1-q)
Final quantum
state
Bob
Initial quantum
state
Vˆ x
Iˆ
\ ini
Fig. 13.2.
The MW scheme for playing a quantum game.
When the game is given by the bi-matrix:
Bob
S1 S2
S1 (αA , αB ) (βA , βB )
Alice
S2 (γA , γB ) (δA , δB )
(13.12)
the payoff operators are:
(PA )oper = αA |00 00| + βA |01 01| + γA |10 10| + δA |11 11|
(PB )oper = αB |00 00| + βB |01 01| + γB |10 10| + δB |11 11| ,
(13.13)
and payoff functions are then obtained as mean values of these operators:
PA,B = Tr {(PA,B )oper ρfin } .
(13.14)
It is to be pointed out that in the EWL set-up a quantum game results
when the entanglement parameter γ of the initial quantum state is different
from zero. Also, when γ is non-zero the players have strategies available to
them that result in the classical game. The general idea is to allow a range
of values to the parameter γ and then to find how it leads to a different,
i.e. non-classical, equilibrium in the game.
In the MW scheme [Marinatto and Weber (2000a); Benjamin (2000);
Marinatto and Weber (2000b)] an initial strategy is forwarded to two players who then apply their tactics on it and the classical game corresponds to
Evolutionary Stability in Quantum Games
261
the initial state |00. Assume now that the players receive pure two-qubit
states, different from |00, while the measurement remains the same. A
quantum form of the game then corresponds if initial states different from
the product state |00 are used. This translates finding the quantum form of
a matrix game to finding appropriate initial state(s). This can be justified
on the ground that the only reasonable restriction [Marinatto and Weber
(2000b)] on a quantum form of a game is that the corresponding classical
game must be reproducible as its special case. As the product initial state
|00 always results in the classical game, this approach towards obtaining
a quantum game remains within the mentioned restriction.
In the EWL scheme one looks for new equilibria in games in relation to
the parameter γ. In the above approach, however, one finds equilibria in
relation to different initial states. In this chapter, we restrict ourselves to
pure states only.
13.3.
Evolutionary Stability in Quantum Games
The concept of a NE from noncooperative game theory was addressed in
the earliest research publications in quantum games [Eisert et al. (1999);
Eisert and Wilkens (2000)]. Analysis of this solution concept generated significant interest thereby motivating an emerging new research field. These
publications do not explicitly refer to the population interpretation of the
NE concept, which was behind the development of the ESS concept in
evolutionary game theory. When this interpretation is brought within the
domain of quantum games it becomes natural to consider ESSs in this domain.
One may ask how and where the population setting may be relevant to
quantum games. How can a setting, originally developed to model population biology problems, be relevant and useful to quantum games? One can
sharpen this argument further given the fact that, to date, almost all of
the experimental realizations of quantum games are artificially constructed
in laboratories using quantum computational circuits [Nielsen and Chuang
(2000)].
It seems that reasonable replies can be made to this question. For example, that the population setting, which, in fact, was behind the NE—the
concept that was addressed in the earliest constructions of quantum games.
It, then, justifies consideration of this setting within quantum games. One
also finds that evolutionary stability has very rich literature in game theory, mathematical biology and in evolutionary economics [Friedman (1998);
262
Quantum Aspects of Life
Witt (2006)], thus making it almost natural to explore how this concept
comes into play when games are quantized. In quantum games the NE
has been discussed in relation to quantum entanglement [Nielsen and
Chuang (2000)] and the possibility that the same can be done with evolutionary stability clearly opens a new interesting role for this quantum
phenomenon. It is conjectured that the possibility of this extended role for
entanglement may perhaps be helpful to better understand entanglement
itself.
Evolutionary stability presents a game-theoretic model to understand
evolutionary dynamics. Quantum games motivate us to ask how this gametheoretic solution concept adapts/shapes/changes itself when players are
given access to quantum strategies. It appears that this question is clearly
related to a bigger question: Can quantum mechanics have a role in directing, or possibly dictating, the dynamics of evolution? We believe that, for
an analysis along this direction, evolutionary stability offers an interesting
situation because, it is a simple and a beautiful concept that is supported
by extensive literature [Bomze (1996); Hofbrauer and Sigmund (1998)].
Discussing evolutionary stability in quantum games may appear as if a
concept originally developed for population biology problems is arbitrarily
being placed within the domain of quantum games. In reply we notice that
population biology is not the only relevant domain for evolutionary stability.
This concept can also be interpreted using infinitely repeated two-player
games and without referring to a population of players. Secondly, as the
Nash’s thesis [Nash (1950); Hofbrauer and Sigmund (1998)] showed, it is not
the population biology alone that motivates a population setting for game
theory—responsible for the concept of evolutionary stability. Surprisingly,
the concept of NE also does the same, although it may not be recognized
generally.
The usual approach in game theory consists of analyzing games among
hyper-rational players who are always found both ready and engaged in
their selfish interests to optimize their payoffs or utilities. Evolutionary
stability has roots in the efforts to get rid of this usual approach that
game theory had followed. The lesson it teaches is that playing games can
be disassociated from players’ capacity to make rational decisions. This
disassociation seems equally valid in those possible situations where nature
plays quantum games.3 It is because associating rationality to quantuminteracting entities is of even a more remote possibility than it is the case
3 Although, no evidence showing nature playing quantum games has been found to date,
the idea itself does not seem far-fetched.
Evolutionary Stability in Quantum Games
263
when this association is made to bacteria and viruses, whose behaviour
evolutionary game theory explains.
In the following we will try to address the questions: How ESSs are
affected when a classical game, played by a population, changes itself to a
quantum form? How pure and mixed ESSs are distinguished from one another when such a change in the form of a game takes place? Can quantum
games provide a route that can relate evolutionary dynamics, for example,
to quantum entanglement? Considering a population of players in which
a classical strategy has established itself as an ESS, we would like to ask:
a) What happens when “mutants” of ESS theory come up with quantum
strategies and try to invade the classical ESS? b) What happens if such an
invasion is successful and a new ESS is established—an ESS that is quantum
in nature? c) Suppose afterwards another small group of mutants appears
which is equipped with some other quantum strategy. Will it successfully
invade the quantum ESS?
13.3.1.
Evolutionary stability in EWL scheme
EWL used the matrix (13.4) with r = 3, s = 0, t = 5, and u = 1 in
their proposal for quantum PD. Assume a population setting where in each
pair-wise encounter the players play PD with the same matrix and each
contest is symmetric. Which strategies will then be likely to be stable?
Straightforward analysis [Prestwich (1999)] shows that D will be the pure
classical strategy prevalent in the population and hence the classical ESS.
We consider following three cases:
• Case (a) A small group of mutants appear equipped with one-parameter
quantum strategy Û (θ) when D exists as a classical ESS;
• Case (b) Mutants are equipped with two-parameter quantum strategy
Û(θ, φ) against the classical ESS;
• Case (c) Mutants have successfully invaded and a two-parameter quantum strategy Q̂ ∼ Û (0, π/2) has established itself as a new quantum
ESS. Again another small group of mutants appear, using some other
two-parameter quantum strategy, and tries to invade the quantum ESS,
which is Q̂.
Case (a): Because players are anonymous one can represent P (Û (θ), D)
as the payoff to Û (θ)-player against the D-player. Here Û (θ) is the Eisert
and Wilkens’ one-parameter quantum strategy set as in Eq. (13.6). Players’ payoffs read P (Û (θ), D) = sin2 (θ/2); P (Û (θ), Û (θ)) = 2 cos2 (θ/2) +
264
Quantum Aspects of Life
5 cos2 (θ/2) sin2 (θ/2) + 1; P (D, Û(θ)) = 5 cos2 (θ/2) + sin2 (θ/2); and
P (D, D) = 1. Now P (D, D) > P (Û(θ), D) for all θ ∈ [0, π). Hence the
first condition for an ESS holds and D ∼ Û(π) is an ESS. The case θ = π
corresponds to one-parameter mutant strategy coinciding with the ESS,
which is ruled out. If D ∼ Û (π) is played by almost all the members of
the population—which corresponds to high frequency FD for D—we then
have W (D) > W (θ) for all θ ∈ [0, π) using the definition in Eq. (13.3). The
fitness of a one-parameter quantum strategy4 , therefore, cannot exceed the
fitness of a classical ESS. And a one-parameter quantum strategy cannot
invade a classical ESS.
Case (b): Let Û (θ, φ) be a two-parameter strategy from the set
(13.7). The expected payoffs read P (D, D) = 1; P (D, Û (θ, φ)) =
5 cos2 (φ) cos2 (θ/2) + sin2 (θ/2); P (Û (θ, φ), D) = 5 sin2 (φ) cos2 (θ/2) +
sin2 (θ/2); and
$2
$
P (Û (θ, φ), Û (θ, φ)) = 3 $cos(2φ) cos2 (θ/2)$
2
+ 5 cos2 (θ/2) sin2 (θ/2) |sin(φ) − cos(φ)|
$
$2
+ $sin(2φ) cos2 (θ/2) + sin2 (θ/2)$ .
(13.15)
√
Here P (D, D) > P (Û (θ, φ), D) if φ < arcsin(1/ 5) and if P (D, D) =
P (Û (θ, φ), D) then P√
(D, Û(θ, φ)) > P (Û(θ, φ), Û (θ, φ)). Therefore, D is an
ESS if φ < arcsin(1/ 5) otherwise the strategy Û (θ, φ) will be in position
to invade D. Alternatively, if most of the members of the population play
D ∼ Û(π, 0)—which means a high frequency FD for D—then the fitness
√
W (D) will remain√greater than the fitness W [Û(θ, φ)] if φ < arcsin(1/ 5).
For φ > arcsin(1/ 5) the strategy Û (θ, φ) can invade the strategy D, which
is the classical ESS.
√
In this analysis mutants are able to invade D when φ > arcsin(1/ 5)
and the invasion may seem not so unusual given the fact that they exploit
richer strategies. But it leads to the third case i.e. when “quantum mutants”
have successfully invaded and a two-parameter strategy Û has established
itself. Can now some new mutants coming up with Q̂ ∼ Û (0, π/2) and
invade the “quantum ESS”?
Case (c): EWL [Eisert et al. (1999); Eisert and Wilkens (2000)] showed
that in their quantum PD the quantum strategy Q̂, played by both the
4 In the EWL set-up one-parameter quantum strategies correspond to mixed (randomized) classical strategies.
Evolutionary Stability in Quantum Games
265
players, is the unique NE. How mutants playing Q̂ come up against Û (θ, φ)
which already exists as an ESS? To find it the following payoffs are obtained. P (Q̂, Q̂) = 3; P (Û (θ, φ), Q̂) = [3 − 2 cos2 (φ)] cos2 (θ/2); and
P (Q̂, Û(θ, φ)) = [3 − 2 cos2 (φ)] cos2 (θ/2) + 5 sin2 (θ/2). Now the inequality P (Q̂, Q̂) > P (Û (θ, φ), Q̂) holds for all θ ∈ [0, π] and φ ∈ [0, π/2]
except when θ = 0 and φ = π/2, which is the case when the mutant
strategy Û (θ, φ) is the same as Q̂. This case is obviously ruled out.
The first condition for Q̂ to be an ESS, therefore, holds. The condition
P (Q̂, Q̂) = P (Û (θ, φ), Q̂) implies θ = 0 and φ = π/2. Again we have the
situation of mutant strategy same as Q̂ and the case is neglected. If Q̂
is played by most of the players, meaning high frequency FQ̂ for Q̂, then
W (Q̂) > W [Û (θ, φ)] for all θ ∈ (0, π] and φ ∈ [0, π/2). A two-parameter
quantum strategy Û(θ, φ), therefore, cannot invade the quantum ESS (i.e.
the strategy Q̂ ∼ Û (0, π/2)). Mutants’ access to richer strategies, as it
happens in the case (B), does not continue to be an advantage as most of
the population also have access to it. Hence Q̂ comes out as the unique NE
and ESS of the game.
13.3.1.1. Evolutionary stability and entanglement
The above analysis motivates us to obtain a direct relationship between a
measure of entanglement and the mathematical concept of evolutionary stability for two-player games. The following example shows this relationship.
Consider the two-player game given by the Matrix (13.16):
Bob
S1 S2
S1 (r, r) (s, t)
Alice
S2 (t, s) (u, u)
(13.16)
and suppose Alice and Bob play the strategy S1 with probabilities p and q,
respectively. The strategy S2 is then played with probabilities (1 − p) and
(1 − q) by Alice and Bob, respectively. We denote Alice’s payoff by PA (p, q)
when she plays p and Bob plays q. That is, Alice’s and Bob’s strategies are
now identified by the numbers p, q ∈ [0, 1], without referring to S1 and S2 .
For the Matrix (13.16) Alice’s payoff PA (p, q), for example, reads
PA (p, q) = rpq + sp(1 − q) + t(1 − p)q + u(1 − p)(1 − q) .
(13.17)
Similarly, Bob’s payoff PB (p, q) can be written. In this symmetric game
we have PA (p, q) = PB (q, p) and, without using subscripts, P (p, q), for
266
Quantum Aspects of Life
example, describes the payoff to p-player against q-player. In this game the
inequality
P (p∗ , p∗ ) − P (p, p∗ ) 0
(13.18)
says that the strategy p∗ , played by both the players, is a NE. We consider
the case when
s = t,
r = u , and (r − t) > 0
(13.19)
in the Matrix (13.16). In this case the Inequality (13.18) along with the
Definition (13.17) gives
P (p∗ , p∗ ) − P (p, p∗ ) = (p∗ − p)(r − t)(2p∗ − 1) ,
(13.20)
and the strategy p∗ = 1/2 comes out as a mixed NE. From the ESS definition (13.2) we get P (1/2, 1/2) − P (p, 1/2) = 0 and the part a) of the
definition does not apply. Part b) of the definition (13.2), then, gives
P (1/2, p) − P (p, p) = (r − t) {2p(1 − p) − 1/2} ,
(13.21)
which can not be strictly greater than zero given (r − t) > 0. For example,
at p = 0 it becomes a negative quantity. Therefore, for the matrix game
defined by (13.16) and (13.19) the strategy p∗ = 1/2 is a symmetric NE,
but it is not evolutionarily stable. Also, at this equilibrium both players
get (r + t)/2 as their payoffs.
Now consider the same game, defined by (13.16) and (13.19), when
it is played by the set-up proposed by EWL. We set sA ≡ (θA , φA ) and
sB ≡ (θB , φB ) to denote Alice’s and Bob’s strategies, respectively. Because
the quantum game is symmetric i.e. PA (sA , sB ) = PB (sB , sA ) we can write,
as before, P (sA , sB ) for the payoff to sA -player against sB -player. For the
quantum form of the game defined by (13.16) and (13.19) one finds
P (sA , sB ) = (1/2)(r − t)
{1 + cos θA cos θB + sin θA sin θB sin γ sin(φA + φB )} + t .
(13.22)
The definition of a NE gives P (s∗ , s∗ ) − P (s, s∗ ) 0 where s = (θ, φ) and
s∗ = (θ∗ , φ∗ ). This definition can be written as
{∂θ P |θ∗ ,φ∗ (θ∗ − θ) + ∂φ P |θ∗ ,φ∗ (φ∗ − φ)} ≥ 0 .
(13.23)
Evolutionary Stability in Quantum Games
267
We search for a quantum strategy s∗ = (θ∗ , φ∗ ) for which both ∂θ P |θ∗ ,φ∗
and ∂φ P |θ∗ ,φ∗ vanish at γ = 0 and which, at some other value of γ, is not
zero. For the payoffs in Eq. (13.22) the strategy s∗ = (π/2, π/4) satisfies
these conditions. For this strategy Eq. (13.22) gives
P (s∗ , s∗ ) − P (s, s∗ ) = (1/2)(r − t) sin γ {1 − sin(φ + π/4) sin θ} . (13.24)
At γ = 0 the strategy s∗ = (π/2, π/4), when played by both the players,
is a NE and it rewards the players same as does the strategy p∗ = 1/2
in the classical version of the game i.e. (r + t)/2. Also, then we have
P (s∗ , s∗ ) − P (s, s∗ ) = 0 from Eq. (13.24) and the ESS’s second condition
in (13.2) applies. Using Eq. (13.22) to evaluate
P (s∗ , s) − P (s, s) = −(r − t) cos2 (θ)
+ (1/2)(r − t) sin γ sin θ {sin(φ + π/4) − sin θ sin(2φ)} ,
(13.25)
which at γ = 0 reduces to P (s∗ , s) − P (s, s) = −(r − t) cos2 (θ), that can assume negative values. The game’s definition (13.19) and the ESS’s second
condition in (13.2) show that the strategy s∗ = (π/2, π/4) is not evolutionarily stable at γ = 0.
Now consider the case when γ = 0 in order to know about the evolutionary stability of the same quantum strategy. From (13.8) we have both
sin θ, sin(φ + π/4) ∈ [0, 1] and Eq. (13.24) indicates that s∗ = (π/2, π/4)
remains a NE for all γ ∈ [0, π/2]. The product sin(φ + π/4) sin θ attains
a value of 1 only at s∗ = (π/2, π/4) and remains less than 1 otherwise.
Equation (13.24) shows that for γ = 0 the strategy s∗ = (π/2, π/4) becomes a strict NE for which the ESS’s first condition in (13.2) applies.
Therefore, for the game defined in (13.19) the strategy s∗ = (π/2, π/4) is
evolutionarily stable for a non-zero measure of entanglement γ. That is,
entanglement gives evolutionary stability to a symmetric NE by making it
a strict NE, that is, it is achieved by using in (13.2) the ESS’s first condition only. Perhaps a more interesting example would be the case when
entanglement gives evolutionary stability via the ESS’s second condition.
In that case, entanglement will make P (s∗ , s) strictly greater than P (s, s)
when P (s∗ , s∗ ) and P (s, s∗ ) are equal.
268
Quantum Aspects of Life
It is to be pointed out here that in literature there exists an approach [Demitrius and Gundlach (2000)] which characterizes ESSs in terms
of extremal states of a function known as evolutionary entropy that is
defined by
(13.26)
E = − µi log µi
i
where µi represents the relative contribution of the i-th strategy to the
total payoff. A possible extension of the present approach may be the
case when quantum entanglement decides extremal states of evolutionary
entropy. Extension along similar lines can be proposed for another quantity called relative negentropy [Bomze (1996)] that is optimized during the
course of evolution.
13.3.2.
Evolutionary stability in MW quantization scheme
Another interesting route that allows us to consider evolutionary stability in
relation to quantization of a game is provided by MW scheme [Marinatto
and Weber (2000a)]. In this scheme a transition between classical and
quantum game is achieved by the initial state: classical payoffs are obtained
when the initial state is a product state |ψin = |00. In this scheme one
can consider evolutionary stability in a quantum game by asking whether
it possible that a particular symmetric NE switches-over between being an
ESS and not being an ESS when the initial state (initial strategy) changes
from being |ψin = |00 to another state. MW scheme offers the possibility
to make transition from classical to quantum version of a game by using
different initial states and it appears to be a more suitable quantization
scheme to analyze evolutionary stability in quantum games. It is because:
a) In a symmetric bi-matrix game, played in a population setting, players
have access to two pure strategies and a mixed strategy is interpreted as
a convex linear combination of pure strategies. Similar is the case with
the players’ strategies in MW scheme where a mixed strategy consists
of a convex linear combination of the players’ actions with two unitary
operators.
b) Fitness of a pure strategy can be given a straightforward extension in
MW scheme. It corresponds to a situation when, for example, in the
quantum game, a player uses only one unitary operator out of the two.
Evolutionary Stability in Quantum Games
269
c) Theory of ESSs, in the classical domain, deals with anonymous players
possessing discrete number of pure strategies. EWL scheme involves a
continuum of pure quantum strategies. The ESS concept is known to
encounter problems [Oechssler and Riedel (2000)] when players possess
a continuum of pure strategies.
13.3.2.1. 2 × 2 asymmetric games
An ESS is defined as a strict NE [Weibull (1995)] for an asymmetric bimatrix game, i.e. the game G = (M, N ) for which N = M T . That is, a
⋆ ⋆
⋆ ⋆
⋆
strategy pair (x, y) ∈ S is an ESS of the game G if PA (x, y) > PA (x, y) and
⋆ ⋆
⋆
⋆
⋆
PB (x, y) > PB (x, y) for all x = x and y = y. For example, the BoS:
(α, β) (γ, γ)
(γ, γ) (β, α)
(13.27)
where α > β > γ is a asymmetric game with three classical NE [Marinatto
⋆
⋆
⋆
⋆
and Weber (2000a)] given as 1) p1 = q1 = 0 2) p2 = q2 = 1 and 3)
⋆
⋆
α−γ
β−γ
p3 = α+β−2γ
, q3 = α+β−2γ
. Here the NE 1) and 2) are also ESS’s but
3) is not because of not being a strict NE. When the asymmetric game
(13.27) is played with the initial state |ψin = a |S1 S1 + b |S2 S2 , where
S1 and S2 are players’ pure classical strategies, the following three NE
⋆
⋆
⋆
⋆
[Marinatto and Weber (2000a)] emerge 1) p1 = q1 = 1 2) p2 = q2 = 0
2
2
⋆
+(β−γ)|a|2
+(β−γ)|b|2 ⋆
and 3) p3 = (α−γ)|a|
, q3 = (α−γ)|b|
. It turns out that,
α+β−2γ
α+β−2γ
similar to the classical case, the quantum NE 1) and 2) are ESSs while 3)
is not. Now, play this game with a different initial state:
|ψin = a |S1 S2 + b |S2 S1
for which players’ payoffs are:
"
#
2
2
PA (p, q) = p −q(α + β − 2γ) + α |a| + β |b| − γ
#
"
2
2
+ q α |b| + β |a| − γ + γ
"
#
PB (p, q) = q −p(α + β − 2γ) + β |a|2 + α |b|2 − γ
#
"
2
2
+ p β |b| + α |a| − γ + γ
(13.28)
(13.29)
270
Quantum Aspects of Life
⋆
2
2
α2 β2
γ2 σ2
⋆
2
2
+β|b| −γ
+α|b| −γ
, q3 = α|a|α+β−γ
, which
and there is only one NE, i.e. p = β|a|α+β−γ
is not an ESS. So that, no ESS exists when BoS is played with the state
(13.28).
Consider now another game:
(α1 , α2 ) (β1 , β2 )
(13.30)
(γ1 , γ2 ) (σ1 , σ2 )
for which
α1 β1
γ1 σ1
=
T
(13.31)
and that it is played by using initial state |ψin = a |S1 S1 + b |S2 S2 with
2
2
|a| + |b| = 1. Players’ payoffs are:
"
#
PA,B (p, q) = α1,2 pq |a|2 + (1 − p)(1 − q) |b|2
"
#
2
2
+ β1,2 p(1 − q) |a| + q(1 − p) |b|
"
#
2
2
+ γ1,2 p(1 − q) |b| + q(1 − p) |a|
"
#
2
2
(13.32)
+ σ1,2 pq |b| + (1 − p)(1 − q) |a| .
The NE conditions are
⋆ ⋆
⋆
PA (p, q) − PA (p, q)
⋆
2
2
⋆
= (p − p) |a| (β1 − σ1 ) + |b| (γ1 − α1 ) − q {(β1 − σ1 ) + (γ1 − α1 )}
≥0
(13.33)
⋆ ⋆
⋆
PB (p, q) − PB (p, q)
⋆
2
2
⋆
= (q − q) |a| (γ2 − σ2 ) + |b| (β2 − α2 ) − p {(γ2 − σ2 ) + (β2 − α2 )}
≥ 0.
!
!
(13.34)
⋆
⋆
So that, for p = q = 0 to be a NE we have
!
PA (0, 0) − PA (p, 0) = −p (β1 − σ1 ) + |b|2 {(γ1 − α1 ) − (β1 − σ1 )} ≥ 0
!
2
PB (0, 0) − PB (0, q) = −q (γ2 − σ2 ) + |b| {(β2 − α2 ) − (γ2 − σ2 )} ≥ 0
(13.35)
271
Evolutionary Stability in Quantum Games
and for the strategy pair (0, 0) to be an ESS in the classical game5 we
require PA (0, 0) − PA (p, 0) = −p(β1 − σ1 ) > 0 and PB (0, 0) − PB (0, q) =
−q(γ2 − σ2 ) > 0 for all p, q = 0. That is, (β1 − σ1 ) < 0 and (γ2 − σ2 ) < 0.
2
For the pair (0, 0) not to be an ESS for some |b| = 0, let take γ1 = α1 and
β2 = α2 and we have
#
"
2
PA (0, 0) − PA (p, 0) = −p(β1 − σ1 ) 1 − |b|
(13.36)
#
"
2
PB (0, 0) − PB (0, q) = −q(γ2 − σ2 ) 1 − |b|
i.e. the pair (0, 0) does not remain an ESS at |b|2 = 1. A game having this
property is given by the matrix:
(1, 1) (1, 2)
(2, 1) (3, 2)
.
(13.37)
2
For this game the strategy pair (0, 0) is an ESS when |b| = 0 (classical
2
game) but it is not when for example |b| = 12 , though it remains a NE in
both the cases. The example shows a NE switches between ESS and “not
ESS” by using different initial state. In contrast to the last case, one can
also find initial states—different from the one corresponding to the classical
game—that turn a NE strategy pair into an ESS. An example of a game
for which it happens is
Bob
S1 S2
S1 (2, 1) (1, 0)
.
Alice
S2 (1, 0) (1, 0)
(13.38)
Playing this game again via the state |ψin = a |S1 S1 + b |S2 S2 gives the
following payoff differences for the strategy pair (0, 0):
2
PA (0, 0) − PA (p, 0) = p |b|
2
and PB (0, 0) − PB (0, q) = q |b|
(13.39)
for Alice and Bob respectively. Therefore, (13.38) is an example of a game
for which the pair (0, 0) is not an ESS when the initial state corresponds
to the classical game. But the same pair is an ESS for other initial states
2
for which 0 < |b| < 1.
5 which
corresponds when |b|2 = 0.
272
Quantum Aspects of Life
13.3.2.2. 2 × 2 symmetric games
Consider now a symmetric bi-matrix game:
Bob
S1 S2
S1 (α, α) (β, γ)
Alice
S2 (γ, β) (δ, δ)
(13.40)
that is played by an initial state:
2
2
|ψin = a |S1 S1 + b |S2 S2 , with |a| + |b| = 1 .
(13.41)
Let Alice’s strategy consists of applying the identity operator Iˆ with probability p and the operator σ̂x with probability (1 − p), on the initial state
written ρin in density matrix notation. Similarly Bob applies the operators
Iˆ and σ̂x with the probabilities q and (1 − q) respectively. The final state is
(13.42)
Pr(ÛA ) Pr(ÛB )[ÛA ⊗ ÛB ρin ÛA† ⊗ ÛB† ]
ρfin =
ˆ x
Û=I,σ̂
where unitary and Hermitian operator Û is either Iˆ or σ̂x . Here, Pr(ÛA ),
Pr(ÛB ) are the probabilities, for Alice and Bob, respectively, to apply the
operator on the initial state. The matrix ρfin is obtained from ρin by making
a convex linear combination of players’ possible quantum operations. Payoff
operators for Alice and Bob are [Marinatto and Weber (2000a)]
(PA,B )oper = α, α |S1 S1 S1 S1 | + β, γ |S1 S2 S1 S2 |
+ γ, β |S2 S1 S2 S1 | + δ, δ |S2 S2 S2 S2 | .
(13.43)
The payoffs are then obtained as mean values of these operators i.e. PA,B =
T r [(PA,B )oper ρfin ]. Because the quantum game is symmetric with the initial
state (13.41) and the payoff matrix (13.40), there is no need for subscripts.
We can , then, write the payoff to a p-player against a q-player as P (p, q),
⋆
where the first number is the focal player’s move. When p is a NE we find
the following payoff difference:
⋆ ⋆
⋆
⋆
2
P (p, p) − P (p, p) = (p − p)[ |a| (β − δ)
⋆
+ |b|2 (γ − α) − p {(β − δ) + (γ − α)}] . (13.44)
273
Evolutionary Stability in Quantum Games
Now the ESS conditions for the pure strategy p = 0 are given as
1.
2
|b| {(β − δ) − (γ − α)} > (β − δ)
2
2. If |b| {(β − δ) − (γ − α)} = (β − δ)
then q 2 {(β − δ) + (γ − α)} > 0
(13.45)
where 1 is the NE condition. Similarly the ESS conditions for the pure
strategy p = 1 are
1.
2
|b| {(γ − α) − (β − δ)} > (γ − α)
2. If |b|2 {(γ − α) − (β − δ)} = (γ − α)
then (1 − q)2 {(β − δ) + (γ − α)} > 0 .
(13.46)
Because these conditions, for both the pure strategies p = 1 and p = 0,
depend on |b|2 , therefore, there can be examples of two-player symmetric
games for which the evolutionary stability of pure strategies can be changed
while playing the game using initial state in the form |ψin = a |S1 S1 +
2
⋆
(β−δ)+|b|2 (γ−α)
b |S2 S2 . However, for the mixed NE, given as p = |a| (β−δ)+(γ−α)
, the
corresponding payoff difference (13.44) becomes identically zero. From the
⋆
second condition of an ESS we find for the mixed NE p the difference
⋆
P (p, q) − P (q, q)
=
1
(β − δ) + (γ − α)
2
2
× [(β − δ) − q {(β − δ) + (γ − α)} − |b| {(β − δ) − (γ − α)} ] . (13.47)
⋆
Therefore, the mixed strategy p is an ESS when {(β − δ) + (γ − α)} > 0.
⋆
This condition, making the mixed NE p an ESS, is independent 6 of |b|2 . So
that, in this symmetric two-player quantum game, evolutionary stability of
⋆
the mixed NE p can not be changed when the game is played using initial
quantum states of the form of Eq. (13.41).
However, evolutionary stability of pure strategies can be affected, with
this form of the initial states, for two-player symmetric games. Examples
of the games with this property are easy to find. The class of games for
which γ = α and (β − δ) < 0 the strategies p = 0 and p = 1 remain NE for
(β−δ)−q{(β−δ)+(γ−α)}
alternative possibility is to adjust |b|2 =
which makes the
{(β−δ)−(γ−α)}
" ⋆
#
⋆
difference P (p, q) − P (q, q) identically zero. The mixed strategy p then does not
6 An
remain an ESS. However such “mutant dependent” adjustment of |b|2 is not reasonable
because the mutant strategy q can be anything in the range [0, 1].
274
Quantum Aspects of Life
2
2
all |b| ∈ [0, 1]; but the strategy p = 1 is not an ESS when |b| = 0 and the
2
strategy p = 0 is not an ESS when |b| = 1.
Consider the symmetric bi-matrix game (13.40) with the constants
α, β, γ, δ satisfying the conditions:
α, β, γ, δ ≥ 0; (δ − β) > 0; (γ − α) ≥ 0; (γ − α) < (δ − β) .
(13.48)
The condition making (p⋆ , p⋆ ) a NE is given by (13.44). For this game three
Nash equilibria arise i.e. two pure strategies p∗ = 0, p∗ = 1, and one mixed
2
−(γ−α)|b|2
. These three cases are considered below.
strategy p∗ = (δ−β)|a|
(δ−β)−(γ−α)
For the strategy p⋆ = 0 to be a NE one requires
p
(γ − α)
P (0, 0) − P (p, 0) =
|a|2 −
≥0
(γ − α) + (δ − β)
(γ − α) + (δ − β)
Case p⋆ = 0 :
(13.49)
2
and the difference {P (0, 0) − P (p, 0)} > 0 when 1 ≥ |a| >
(γ−α)
(γ−α)+(δ−β) .
2
In this range of |a| the equilibrium p⋆ = 0 is a pure ESS. However, when
2
(γ−α)
we have the difference {P (0, 0) − P (p, 0)} identically
|a| = (γ−α)+(δ−β)
zero. The strategy p⋆ = 0 can be an ESS if
P (0, p) − P (p, p)
%
&
(1 − p)(γ − α) + p(δ − β)
2
= p {(γ − α) + (δ − β)} |a| −
(γ − α) + (δ − β)
>0
(13.50)
that can be written as
#
"
2
P (0, p) − P (p, p) = p {(γ − α) + (δ − β)} |a| − ̥ > 0
where
(γ−α)
(γ−α)+(δ−β)
⋆
≤ ̥ ≤
(δ−β)
(γ−α)+(δ−β)
2
p = 0 can be an ESS only when |a| >
2
because |a|
ESS for 1 ≥
(13.51)
when 0 ≤ p ≤ 1. The strategy
(δ−β)
(γ−α)+(δ−β) ,
which is not possible
(γ−α)
. Therefore the strategy p⋆ = 0 is an
is fixed at (γ−α)+(δ−β)
2
2
(γ−α)
(γ−α)
|a| > (γ−α)+(δ−β) and for |a| = (γ−α)+(δ−β)
this NE becomes
2
⋆
unstable. The classical game is obtained by taking |a| = 1 for which p = 0
is an ESS or a stable NE. However this NE does not remain stable for
2
(γ−α)
which corresponds to an entangled initial state; though
|a| = (γ−α)+(δ−β)
the NE remains intact in both forms of the game.
275
Evolutionary Stability in Quantum Games
Case p⋆ = 1 : Similar to the last case the NE condition for the strategy
p⋆ = 1 can be written as
(1 − p)
(δ − β)
2
P (1, 1) − P (p, 1) =
− |a| +
≥ 0.
(γ − α) + (δ − β)
(γ − α) + (δ − β)
(13.52)
2
Now p⋆ = 1 is a pure ESS for 0 ≤ |a|
<
(δ−β)
(γ−α)+(δ−β) .
(δ−β)
(γ−α)+(δ−β)
2
For |a|
=
the difference {P (1, 1) − P (p, 1)} becomes identically zero.
The strategy p⋆ = 1 is an ESS when
P (1, p) − P (p, p)
%
&
(1 − p)(γ − α) + p(δ − β)
2
= (1 − p) {(γ − α) + (δ − β)} − |a| +
(γ − α) + (δ − β)
> 0.
(13.53)
2
(γ−α)
⋆
(γ−α)+(δ−β) . Therefore the strategy p = 1 is a
2
(δ−β)
|a| < (γ−α)+(δ−β)
. It is not stable classically (i.e.
It is possible only if |a| <
stable NE (ESS) for 0 ≤
2
for |a| = 1) but becomes stable for an entangled initial state.
Case p⋆ =
(δ−β)|a|2 −(γ−α)|b|2
(δ−β)−(γ−α)
:
In case of the mixed strategy:
2
p⋆ =
2
(δ − β) |a| − (γ − α) |b|
(δ − β) − (γ − α)
(13.54)
the NE condition (13.44) turns into P (p⋆ , p⋆ ) − P (p, p⋆ ) = 0. The mixed
strategy (13.54) can be an ESS if
P (p⋆ , p) − P (p, p)
!
= (p⋆ − p) − |a|2 (δ − β) + |b|2 (γ − α) + p {(δ − β) − (γ − α)} > 0
(13.55)
for all p = p⋆ . Write now the strategy p as p = p⋆ + △. For the mixed
strategy (13.54) the payoff difference of the Eq. (13.55) is reduced to
P (p⋆ , p) − P (p, p) = − △2 {(δ − β) − (γ − α)} .
(13.56)
Hence, for the game defined in the conditions (13.48), the mixed strategy
2
−(γ−α)|b|2
p⋆ = (δ−β)|a|
cannot be an ESS, though it can be a NE of the
(δ−β)−(γ−α)
symmetric game.
276
Quantum Aspects of Life
It is to be pointed out that above considerations apply when the game
is played with the initial state given by Eq. (13.41).
To find examples of symmetric quantum games, where evolutionary stability of the mixed strategies may also be affected by controlling the initial
states, the number of players is now increased from two to three.
13.3.2.3. 2 × 2 × 2 symmetric games
In extending the two-player scheme to a three-player case, we assume that
three players A, B, and C play their strategies by applying the identity
operator Iˆ with the probabilities p, q and r respectively on the initial state
|ψin . Therefore, they apply the operator σ̂x with the probabilities (1 −
p), (1 − q) and (1 − r) respectively. The final state becomes
!
Pr(ÛA ) Pr(ÛB ) Pr(ÛC ) ÛA ⊗ ÛB ⊗ ÛC ρin ÛA† ⊗ ÛB† ⊗ ÛC†
ρfin =
ˆ x
Û=I,σ̂
(13.57)
where the 8 basis vectors are |Si Sj Sk , for i, j, k = 1, 2. Again we use
initial quantum state in the form |ψin = a |S1 S1 S1 + b |S2 S2 S2 , where
|a|2 + |b|2 = 1. It is a quantum state in 2 ⊗ 2 ⊗ 2 dimensional Hilbert space
that can be prepared from a system of three two-state quantum systems or
qubits. Similar to the two-player case, the payoff operators for the players
A, B, and C can be defined as
(PA,B,C )oper
= α1 , β1 , η1 |S1 S1 S1 S1 S1 S1 | + α2 , β2 , η2 |S2 S1 S1 S2 S1 S1 |
+ α3 , β3 , η3 |S1 S2 S1 S1 S2 S1 | + α4 , β4 , η4 |S1 S1 S2 S1 S1 S2 |
+ α5 , β5 , η5 |S1 S2 S2 S1 S2 S2 | + α6 , β6 , η6 |S2 S1 S2 S2 S1 S2 |
+ α7 , β7 , η7 |S2 S2 S1 S2 S2 S1 | + α8 , β8 , η8 |S2 S2 S2 S2 S2 S2 | (13.58)
where αl , βl , ηl for 1 ≤ l ≤ 8 are 24 constants of the matrix of this threeplayer game. Payoffs to the players A, B, and C are then obtained as mean
values of these operators i.e. PA,B,C (p, q, r) =Tr[(PA,B,C )oper ρfin ].
Here, similar to the two-player case, the classical payoffs can be obtained
2
when |b| = 0. To get a symmetric game we define PA (x, y, z) as the payoff
to player A when players A, B, and C play the strategies x, y, and z
respectively. With following relations the players’ payoffs become identityindependent.
PA (x, y, z) = PA (x, z, y) = PB (y, x, z)
= PB (z, x, y) = PC (y, z, x) = PC (z, y, x) .
(13.59)
277
Evolutionary Stability in Quantum Games
The players in the game then become anonymous and their payoffs depend
only on their strategies. The relations in (13.59) hold with the following
replacements for βi and ηi :
β2 → α3
β3 → α2
β4 → α3
β1 → α1
β5 → α6
β6 → α5
β7 → α6
β8 → α8
(13.60)
η1 → α1
η2 → α3
η3 → α3
η4 → α2
η5 → α6
η6 → α6
η7 → α5
η8 → α8 .
Also, it is now necessary that we should have α6 = α7 , α3 = α4 .
A symmetric game between three players, therefore, can be defined by
only six constants of the payoff matrix . These constants can be taken as
α1 , α2 , α3 , α5 , α6 , and α8 . Payoff to a p-player, when other two players play
⋆
q and r, can now be written as P (p, q, r). A symmetric NE p is now found
⋆ ⋆ ⋆
⋆ ⋆
from the Nash condition P (p, p, p) − P (p, p, p) ≥ 0, i.e.
⋆ ⋆ ⋆
⋆ ⋆
⋆
⋆2
2
P (p, p, p) − P (p, p, p) = (p − p) p (1 − 2 |b| )(σ + ω − 2η)
"
#
⋆
2
+ 2p |b| (σ + ω − 2η) − ω + η
#!
"
2
+ ω − |b| (σ + ω)
≥0
(13.61)
where (α1 − α2 ) = σ, (α3 − α6 ) = η, and (α5 − α8 ) = ω.
Three possible NE are found as
{(ω−η)−|b|2 (σ+ω−2η)}±
√
⎫
{(σ+ω)2 −(2η)2 }|b|2 (1−|b|2 )+(η 2 −σω) ⎪
⎪
p1 =
⎬
(1−2|b|2 )(σ+ω−2η)
⋆
. (13.62)
p2 = 0
⎪
⎪
⋆
⎭
p3 = 1
⋆
It
observed that# the mixed NE p1 makes the differences
" is
⋆ ⋆ ⋆
⋆ ⋆
⋆
P (p, p, p) − P (p, p, p) identically zero and two values for p1 can be found
⋆
⋆
⋆
2
for a given |b| . Apart from p1 the other two NE (i.e. p2 and p3 ) are pure
⋆
strategies. Also now p1 comes out a NE without imposing further restric⋆
tions on the matrix of the symmetric three-player game. However, the pure
⋆
⋆
strategies p2 and p3 can be NE when further restriction are imposed on the
⋆
2
matrix of the game. For example, p3 can be a NE provided σ ≥ (ω + σ) |b|
⋆
2
2
for all |b| ∈ [0, 1]. Similarly p2 can be NE when ω ≤ (ω + σ) |b| .
Now we address the question: How evolutionary stability of these three
NE can be affected while playing the game via initial quantum states given
in the following form?
|ψin = a |S1 S1 S1 + b |S2 S2 S2 .
(13.63)
278
Quantum Aspects of Life
For the two-player asymmetric game of BoS we showed that out of three
NE only two can be evolutionarily stable. In classical evolutionary game
theory the concept of an ESS is well-known [Broom et al. (2000, 1997)] to
be extendable to multi-player case. When mutants are allowed to play only
one strategy the definition of an ESS for the three-player case is written as
[Broom et al. (2000)]
1.
P (p, p, p) > P (q, p, p)
2. If P (p, p, p) = P (q, p, p) then P (p, q, p) > P (q, q, p) .
(13.64)
Here p is a NE if it satisfies the condition 1 against all q = p. For our
⋆
⋆
case the ESS conditions for the pure strategies p2 and p3 can be written as
⋆
follows. For example p2 = 0 is an ESS when
2
2
2
2
σ |b| > ω |a|
1.
2
2
2. If σ |b| = ω |a| then − ηq 2 (|a| − |b| ) > 0
⋆
(13.65)
⋆
where 1 is NE condition for the strategy p2 = 0. Similarly, p3 = 1 is an
ESS when
1.
2
2
2
2
σ |a| > ω |b|
2
2
2. If σ |a| = ω |b| then η(1 − q)2 (|a| − |b| ) > 0
⋆
⋆
(13.66)
2
2
and both the pure strategies p2 and p3 are ESSs when |a| = |b| . The
conditions (13.66) can also be written as
1.
σ > (ω + σ) |b|2
2. If σ = |b|2 (ω + σ) then
γ(ω − σ)
> 0.
(ω + σ)
(13.67)
⋆
For the strategy p2 = 0 the ESS conditions (13.65) reduce to
1.
2
ω < (ω + σ) |b|
2
2. If ω = |b| (ω + σ) then
γ(ω − σ)
> 0.
(ω + σ)
(13.68)
Examples of three-player symmetric games are easy to find for which a pure
strategy is a NE for the whole range |b|2 ∈ [0, 1], but it is not an ESS for
some particular value of |b|2 . An example of a class of such games is for
⋆
which σ = 0, ω < 0, and η ≤ 0. In this class the strategy p2 = 0 is a NE
2
2
for all |b| ∈ [0, 1] but not an ESS when |b| = 1.
Apart from the pure strategies, the mixed strategy equilibrium
⋆
p1 forms the most interesting case. It makes the payoff difference
279
Evolutionary Stability in Quantum Games
" ⋆ ⋆ ⋆
#
⋆ ⋆
P (p1 , p1 , p1 ) − P (p, p1 , p1 ) identically zero for every strategy p. The
" ⋆
#
⋆
⋆
⋆
strategy p1 is an ESS when P (p1 , q, p1 ) − P (q, q, p1 ) > 0 but
⋆
⋆
⋆
P (p1 , q, p1 ) − P (q, q, p1 )
⋆
2
2
= ±(p1 − q)2 {(σ + ω)2 − (2η)2 } |b| (1 − |b| ) + (η 2 − σω) . (13.69)
⋆
⋆
Therefore, out of the two possible roots (p1 )1 and (p1 )2 of the quadratic
equation7 :
⋆2
p1 (1 − 2 |b|2 )(σ + ω − 2η)
"
# "
#
⋆
+ 2p1 |b|2 (σ + ω − 2η) − ω + η + ω − |b|2 (σ + ω) = 0
(13.70)
⋆
only (p1 )1 can exist as an ESS. When the square root term in Eq. (13.69)
becomes zero we have only one mixed NE, which is not an ESS. Hence, out
of four possible NE in this three-player game only three can be ESSs.
An interesting class of three-player games is the one for which η 2 = σω.
For these games the mixed NE are
"
#
2
(w
−
η)
−
|b|
(σ
+
ω
−
2η)
± |a| |b| |σ − ω|
⋆
p1 =
(13.71)
2
(1 − 2 |b| )(σ + ω − 2η)
and, when played classically, we can get only one mixed NE that is not an
ESS. However for all |b|2 , different from zero, we generally obtain two NE
out of which one can be an ESS.
Similar to the two-player case, the equilibria in a three-player symmetric
game where quantization affects evolutionary stability, are the ones that
survive for two initial states, one of which is a product state and corresponds
⋆
2
to the classical game. Suppose p1 remains a NE for |b| = 0 and some other
⋆
2
non-zero |b| . It is possible when (σ − ω)(2p1 − 1) = 0. Alternatively, the
⋆
2
strategy p = 12 remains a NE for all |b| ∈ [0, 1]. It reduces the defining
⋆
quadratic
equation (13.70)#for p1 to σ + ω + 2η = 0 and makes the difference
"
⋆
⋆
⋆
⋆
2
P (p1 , q, p1 ) − P (q, q, p1 ) independent of |b| . Therefore the strategy p =
1
2,
2
even when it is retained as an equilibrium for all |b| ∈ [0, 1], cannot be
7 These
roots make the difference
zero, respectively.
"
#
⋆
⋆
⋆
P (p1 , q, p1 ) − P (q, q, p1 ) greater than and less than
280
Quantum Aspects of Life
an ESS in only one version of the symmetric three-player game. For the
⋆
second possibility σ = ω the defining equation for p1 is reduced to
(η − σ) + η 2 − σ 2
(η − σ) − η 2 − σ 2
⋆
⋆
2
=0
p1 −
(1 − 2 |b| ) p1 −
2(η − σ)
2(η − σ)
(13.72)
for which
2
⋆
⋆
⋆
⋆
P (p1 , q, p1 ) − P (q, q, p1 ) = ±2(p1 − q)2 |b| −
Here the difference
2
1 2
η − σ2 .
2
(13.73)
⋆
⋆
⋆
P (p1 , q, p1 ) − P (q, q, p1 ) still depends on |b|2 and
becomes zero for |b| = 1/2.
Hence, for the class of games for which σ = ω and η > σ, one of the
⋆
⋆
2
mixed strategies (p1 )1 , (p1 )2 remains a NE for all |b| ∈ [0, 1] but not an
2
ESS when |b| = 1/2. In this class of three-player quantum games the
evolutionary stability of a mixed strategy can, therefore, be changed while
the game is played using initial quantum states in the form of Eq. (13.63).
13.3.2.4. Rock-Scissors-Paper game
Rock-Scissors-Paper (RSP) is a game for two players that is typically played
using the players’ hands. This game has been played for long as a children’s
pastime or as an odd-man-out selection process. The players opposite each
others, tap their fist in their open palm three times (saying rock, scissors,
paper) and then show one of three possible gestures. The rock wins against
the scissors (crushes it) but looses against the paper (is wrapped into it).
The scissors wins against the paper (cuts it) but looses against the rock
(is crushed by it). The paper wins against the rock (wraps it) but looses
against the scissors (is cut by it). The game is also played in nature like
many other games. Lizards in the Coast Range of California play this
game [Peterson (1996)] using three alternative male strategies locked in an
ecological never ending process from which there seems little escape.
In a slightly modified version of the RSP game both players get a small
premium ǫ for a draw. This game can be represented by the payoff matrix:
RSP
⎞
R −ǫ 1 −1
S ⎝ −1 −ǫ 1 ⎠
P
1 −1 −ǫ
⎛
(13.74)
Evolutionary Stability in Quantum Games
281
where −1 < ǫ ≤ 0. The matrix of the usual game is obtained when ǫ is
zero.
One cannot win if one’s opponent knew which strategy was going to
be picked. For example, when picking rock consistently, all the opponent
needs to do is pick paper and s/he would win. Players find soon that in case
predicting opponent’s strategy is not possible the best strategy is to pick
rock, scissors, or paper at random. In other words, the player selects rock,
scissors, or paper with a probability of 1/3. In case opponent’s strategy is
predictable picking a strategy at random with a probability of 1/3 is not
the best thing to do unless the opponent does the same [Prestwich (1999)].
Analysis [Weibull (1995)] of the modified RSP game of Matrix (13.74)
shows that its NE consists of playing each of the three different pure strategies with a fixed equilibrial probability 1/3. However it is not an ESS
because ǫ is negative.
Here we want to study the effects of quantization on evolutionary stability for the modified RSP game. The game is different, from others considered earlier, because classically each player now possesses three pure
strategies instead of two. A classical mixed NE exists which is not an ESS.
Our motivation is to explore the possibility that the classical mixed NE
becomes an ESS for some quantum form of the game.
Quantization of Rock-Scissors-Paper game: Using simpler notation:
R ∼ 1, S ∼ 2, P ∼ 3 we quantize this game via MW scheme [Marinatto
and Weber (2000a)]. We assume the two players are in possession of three
ˆ Ĉ and D̂ defined as follows.
unitary and Hermitian operators I,
Iˆ |1 = |1 ,
Iˆ |2 = |2 ,
Iˆ |3 = |3 ,
Ĉ |1 = |3 ,
D̂ |1 = |2
Ĉ |2 = |2 ,
D̂ |2 = |1
Ĉ |3 = |1 ,
D̂ |3 = |3
(13.75)
where Ĉ † = Ĉ = Ĉ −1 and D̂† = D̂ = D̂−1 and Iˆ is the identity operator.
Consider a general two-player payoff matrix when each player has three
strategies:
1 2 3
⎛
⎞
1 (α11 , β11 ) (α12 , β12 ) (α13 , β13 )
2 ⎝ (α21 , β21 ) (α22 , β22 ) (α23 , β23 ) ⎠
3 (α31 , β31 ) (α32 , β32 ) (α33 , β33 )
(13.76)
282
Quantum Aspects of Life
where αij and βij are the payoffs to Alice and Bob, respectively, when Alice
plays i and Bob plays j and 1 ≤ i, j ≤ 3. Suppose Alice and Bob apply the
operators Ĉ, D̂, and Iˆ with the probabilities p, p1 , (1 − p − p1 ) and q, q1 ,
(1 − q − q1 ) respectively. The initial state of the game is ρin . Alice’s move
changes the state changes to
A
†
†
†
.
+ p1 D̂A ρin D̂A
+ pĈA ρin ĈA
ρin = (1 − p − p1 )IˆA ρin IˆA
(13.77)
The final state, after Bob too has played his move, is
A,B
ρf
A
A
A
†
†
†
= (1 − q − q1 )IˆB ρin IˆB
+ q ĈB ρin ĈB
+ q1 D̂B ρin D̂B
.
(13.78)
This state can be written as
A,B
ρf = (1
#
"
†
†
+ p(1 − q − q1 )
⊗ IˆB
− p − p1 )(1 − q − q1 ) IˆA ⊗ IˆB ρin IˆA
"
#
"
#
†
†
†
†
⊗ IˆB
⊗ IˆB
× ĈA ⊗ IˆB ρin ĈA
+ p1 (1 − q − q1 ) D̂A ⊗ IˆB ρin D̂A
#
"
#
"
†
†
†
†
+ pq ĈA ⊗ ĈB ρin ĈA
⊗ ĈB
⊗ ĈB
+ (1 − p − p1 )q IˆA ⊗ ĈB ρin IˆA
"
#
"
#
†
†
†
†
+ p1 q D̂A ⊗ ĈB ρin D̂A
⊗ ĈB
⊗ D̂B
+ (1 − p − p1 )q1 IˆA ⊗ D̂B ρin IˆA
"
#
"
#
†
†
†
†
+ p1 q1 D̂A ⊗ D̂B ρin D̂A
.
⊗ D̂B
+ pq1 ĈA ⊗ D̂B ρin ĈA
⊗ D̂B
(13.79)
The nine basis vectors of initial quantum state with three pure classical
strategies are |ij for i, j = 1, 2, 3. We consider the initial state to be a
pure quantum state of two qutrits, i.e.
2
|cij | = 1 .
(13.80)
cij |ij ,
where
|ψin =
i,j=1,2,3
i,j=1,2,3
The payoff operators for Alice and Bob are [Marinatto and Weber (2000a)]
(PA,B )oper = (α, β)11 |11 11| + (α, β)12 |12 12| + (α, β)13 |13 13|
+ (α, β)21 |21 21| + (α, β)22 |22 22| + (α, β)23 |23 23|
+ (α, β)31 |31 31| + (α, β)32 |32 32| + (α, β)33 |33 33| .
(13.81)
The players’ payoffs are then
A,B
PA,B = Tr[{(PA,B )oper } ρf ]
(13.82)
Payoff to Alice, for example, can be written as
PA = ΦΩΥT
(13.83)
283
Evolutionary Stability in Quantum Games
where T is for transpose, and the matrices Φ, Ω, and Υ are
Φ = [ (1 − p − p1 )(1 − q − q1 ) p(1 − q − q1 ) p1 (1 − q − q1 )
(1 − p − p1 )q pq p1 q (1 − p − p1 )q1 pq1 p1 q1 ]
Υ = [ α11 α12 α13 α21 α22 α23
⎡
|c11 |2 |c12 |2 |c13 |2 |c21 |2
⎢ |c |2 |c |2 |c |2 |c |2
⎢ 31
32
33
21
⎢
2
2
2
2
⎢ |c21 | |c22 | |c23 | |c11 |
⎢
2
2
2
⎢ |c13 | |c12 | |c11 | |c23 |2
⎢
2
2
2
2
Ω=⎢
⎢ |c33 | |c32 | |c31 | |c23 |
⎢ |c |2 |c |2 |c |2 |c |2
22
21
13
⎢ 23
⎢
2
2
2
2
⎢ |c12 | |c11 | |c13 | |c22 |
⎢
2
2
2
⎣ |c32 | |c31 | |c33 | |c22 |2
2
2
2
2
|c22 | |c21 | |c23 | |c12 |
α31 α32 α33 ]
|c22 |2
2
|c22 |
|c12 |2
2
|c22 |
2
|c22 |
2
|c12 |
2
|c21 |
|c21 |2
2
|c11 |
|c23 |2
2
|c23 |
|c13 |2
2
|c21 |
2
|c21 |
2
|c11 |
2
|c23 |
|c23 |2
2
|c13 |
|c31 |2
2
|c11 |
|c31 |2
2
|c33 |
2
|c13 |
2
|c33 |
2
|c32 |
|c12 |2
2
|c32 |
|c32 |2
2
|c12 |
|c32 |2
2
|c32 |
2
|c12 |
2
|c32 |
2
|c31 |
|c11 |2
2
|c31 |
⎤
|c33 |2
2
|c13 | ⎥
⎥
⎥
|c33 |2 ⎥
2⎥
|c31 | ⎥
⎥
2
|c11 | ⎥
⎥.
2⎥
|c31 | ⎥
2⎥
|c33 | ⎥
⎥
|c13 |2 ⎦
2
|c33 |
(13.84)
The payoff (13.83) corresponds to the Matrix (13.76). Payoffs in classical
mixed strategy game can be obtained from Eq. (13.82) for the initial state
|ψin = |11. The game is symmetric when αij = βji in the Matrix (13.76).
The quantum game played using the quantum state (13.80) is symmetric
2
2
when |cij | = |cji | for all constants cij in the state (13.80). These two
conditions together guarantee a symmetric quantum game. The players’
payoffs PA , PB then do not need a subscript and we can simply use P (p, q)
to denote the payoff to the p-player against the q-player.
The question of evolutionary stability in quantized RSP game is addressed below.
Analysis of evolutionary stability: Assume a strategy is defined by a
pair of numbers (p, p1 ) for players playing the quantized RSP game. These
numbers are the probabilities with which the player applies the operators Ĉ
and D̂. The identity operator Iˆ is, then, applied with probability (1−p−p1).
Similar to the conditions a) and b) in Eq. (13.2), the conditions making a
strategy (p⋆ , p⋆1 ) an ESS can be written as [Maynard Smith and Price (1973);
Weibull (1995)]
1.
P {(p⋆ , p⋆1 ), (p⋆ , p⋆1 )} > P {(p, p1 ), (p⋆ , p⋆1 )}
2. if P {(p⋆ , p⋆1 ), (p⋆ , p⋆1 )} = P {(p, p1 ), (p⋆ , p⋆1 )} then
P {(p⋆ , p⋆1 ), (p, p1 )} > P {(p, p1 ), (p, p1 )} .
(13.85)
284
Quantum Aspects of Life
Suppose (p⋆ , p⋆1 ) is a mixed NE then
*
+
∂P
∂P
⋆
⋆
|
|
(p⋆1 − p1 ) ≥ 0 .
(p⋆ − p) +
⋆
⋆
∂p pp=q=p
∂p1 pp=q=p
1 =q1 =p1
1 =q1 =p1
(13.86)
Using substitutions
|c11 |2 − |c31 |2 = △1 , |c21 |2 − |c11 |2 = △´1
2
2
2
2
|c13 | − |c33 | = △2 , |c22 | − |c12 | = △´2
2
2
2
2
|c12 | − |c32 | = △3 , |c23 | − |c13 | = △´3
(13.87)
we get
∂P
⋆
|
= p⋆ (△1 − △2 ) {(α11 + α33 ) − (α13 + α31 )}
⋆
∂p pp=q=p
=q
=p
1
1
1
+ p⋆1 (△1 − △3 ) {(α11 + α32 ) − (α12 + α31 )}
− △1 (α11 − α31 ) − △2 (α13 − α33 ) − △3 (α12 − α32 ) ,
(13.88)
∂P
⋆
|
= p⋆ (△´3 − △´1 ) {(α11 + α23 ) − (α13 + α21 )}
⋆
∂p1 pp=q=p
=q
=p
1
1
1
+ p⋆1 (△´2 − △´1 ) {(α11 + α22 ) − (α12 + α21 )}
+ △´1 (α11 − α21 ) + △´2 (α12 − α22 ) + △´3 (α13 − α23 ) .
(13.89)
For the Matrix (13.74) the Eqs. (13.88) and (13.89) can be written as
∂P
⋆
|
= △1 {−2ǫp⋆ − (3 + ǫ)p⋆1 + (1 + ǫ)}
⋆
∂p pp=q=p
1 =q1 =p1
+ △2 {2ǫp⋆ + (1 − ǫ)} + △3 {(3 + ǫ)p⋆1 − 2}
(13.90)
∂P
⋆
|
= △´1 {−p⋆ (3 − ǫ) + 2ǫp⋆1 + (1 − ǫ)}
⋆
∂p1 pp=q=p
1 =q1 =p1
− △´2 {2ǫp⋆1 − (1 + ǫ)} + △´3 {(3 − ǫ)p⋆ − 2} . (13.91)
Evolutionary Stability in Quantum Games
285
The payoff difference in the second condition of an ESS given in the
Eq. (13.85) reduces to
P {(p⋆ , p⋆1 ), (p, p1 )} − P {(p, p1 ), (p, p1 )}
= (p⋆ − p)[− △1 {2ǫp + (3 + ǫ)p1 − (1 + ǫ)}
+ △2 {2ǫp + (1 − ǫ)} + △3 {(3 + ǫ)p1 − 2}]
+ (p⋆1 − p1 )[− △´1 {(3 − ǫ)p − 2ǫp1 − (1 − ǫ)}
− △´2 {2ǫp1 − (1 + ǫ)} + △´3 {(3 − ǫ)p − 2}] .
(13.92)
With the substitutions (p⋆ − p) = x and (p⋆1 − p1 ) = y the above payoff
difference is
P {(p⋆ , p⋆1 ), (p, p1 )} − P {(p, p1 ), (p, p1 )}
= △1 x {2ǫx + (3 + ǫ)y} − △2 (2ǫx2 ) − △3 xy(3 + ǫ)
− △´1 y {2ǫy − (3 − ǫ)x} + △´2 (2ǫy 2 ) − △´3 xy(3 − ǫ)
(13.93)
provided that
∂P
⋆ = 0
|
⋆
∂p pp=q=p
1 =q1 =p1
∂P
⋆ = 0.
|
⋆
∂p1 pp=q=p
1 =q1 =p1
(13.94)
The conditions in Eq. (13.94) together define the mixed NE (p⋆ , p⋆1 ). Consider now the modified RSP game in classical form obtained by setting
2
|c11 | = 1. Equation (13.94) becomes
−2ǫp⋆ − (ǫ + 3)p⋆1 + (ǫ + 1) = 0
(−ǫ + 3)p⋆ − 2ǫp⋆1 + (ǫ − 1) = 0
(13.95)
and p⋆ = p⋆1 = 31 is obtained as a mixed NE for all the range −1 < ǫ < 0.
From Eq. (13.93) we get
P {(p⋆ , p⋆1 ), (p, p1 )} − P {(p, p1 ), (p, p1 )}
2
3
= 2ǫ(x2 + y 2 + xy) = ǫ (x + y)2 + (x2 + y 2 ) ≤ 0 .
(13.96)
In the classical RSP game, therefore, the mixed NE p⋆ = p⋆1 = 31 is a
NE but not an ESS, because the second condition of an ESS given in the
Eq. (13.85) does not hold.
Now define a new initial state as
1
(13.97)
|ψin = {|12 + |21 + |13 + |31}
2
286
Quantum Aspects of Life
and use it to play the game, instead of the classical game obtained from
|ψin = |11. The strategy p⋆ = p⋆1 = 13 still forms a mixed NE because
the conditions (13.94) hold true for it. However the payoff difference of
Eq. (13.93) is now given below, when −1 < ǫ < 0 and x, y = 0:
P {(p⋆ , p⋆1 ), (p, p1 )} − P {(p, p1 ), (p, p1 )}
2
3
= −ǫ (x + y)2 + (x2 + y 2 ) > 0 .
(13.98)
Therefore, the mixed NE p⋆ = p⋆1 = 13 , not existing as an ESS in the classical
form of the RSP game, becomes an ESS when the game is quantized and
played using an initial (entangled) quantum state given by the Eq. (13.97).
Note that from Eq. (13.82) the sum of the payoffs to Alice and Bob
(PA + PB ) can be obtained for both the classical mixed strategy game
(i.e. when |ψin = |11) and the quantum game played using the quantum
state of Eq. (13.97). For the Matrix (13.74) we write these sums as (PA +
PB )cl and (PA + PB )qu for classical mixed strategy and quantum games,
respectively. We obtain
(PA + PB )cl = −2ǫ {(1 − p − p1 )(1 − q − q1 ) + p1 q1 + pq}
(13.99)
and
(PA + PB )qu = −
%
1
(PA + PB )cl + ǫ
2
&
.
(13.100)
In case ǫ = 0 both the classical and quantum games are clearly zero sum.
For the slightly modified version of the RSP game we have −1 < ǫ < 0 and
both versions of the game become non zero-sum.
13.4.
Concluding Remarks
Evolutionary stability is a game-theoretic solution concept that tells which
strategies are going to establish themselves in a population of players engaged in symmetric contests. By establishing itself it means that the strategy becomes resistant to invasion by mutant strategies when played by a
small number of players. Our analysis of evolutionary stability in quantum games shows that quantization of games, played by a population of
players, can lead to new stable states of the population in which, for example, a quantum strategy establishes itself. Our results show that quantum
strategies can indeed change the dynamics of evolution as described by the
concept of evolutionary stability. Quantum strategies being able to decide evolutionary outcomes clearly gives a new role to quantum mechanics
Evolutionary Stability in Quantum Games
287
which is higher than just keeping the atoms together. Consideration of this
role also provides a mathematically tractable method of analysis for studying multi-player quantum games [Benjamin and Hayden (2001)] played in
evolutionary arrangements.
Using EWL and MW quantization schemes, we explored how quantization can change evolutionary stability of Nash equilibria in certain
asymmetric bi-matrix games. We showed that quantization can change evolutionary stability of a NE in certain types of symmetric bi-matrix games
to which the ESS concept refers. We identified the classes of games, both
symmetric and asymmetric, for which, within the EWL and MW schemes,
the quantization of games becomes related to evolutionary stability of NE.
For example, in the case of Prisoners’ Dilemma we found that when a population is engaged in playing this symmetric bi-matrix game, a small number
of mutant players can invade the classical ESS8 when they exploit Eisert
and Wilken’s two-parameter set of quantum strategies. As another example we studied the well-known children’s two-player three-strategy game of
Rock-Scissors-Paper. In its classical form a mixed NE exists that is not evolutionarily stable. We found that in a quantum form of this game, played
using MW quantization scheme, the classical NE becomes evolutionarily
stable when the players share an entangled state.
We speculate that evolutionary stability in quantum games can potentially provide a new approach towards the understanding of rise of complexity and self-organization in groups of quantum-interacting entities, although
this opinion, at the present stage of development in evolutionary quantum
game theory, remains without any supportive evidence, either empirical or
experimental. However, it seems that the work presented in this chapter
provides a theoretical support in favour of this opinion. Secondly, evolutionary quantum game theory benefits from the methods and concepts of
quantum mechanics and evolutionary game theory, the second of which is
well known to facilitate better understanding of complex interactions taking place in communities of animals as well as that of the bacteria and
viruses. Combining together the techniques and approaches of these two,
seemingly separate, disciplines appears to provide an ideal arrangement to
understand the rise of complexity and self-organization at molecular level.
Although it is true that evolutionary stability and evolutionary computation provide two different perspectives on the dynamics of evolution,
it appears to us that an evolutionary quantum game-theoretic approach
8 Consisting
of a Defection-Defection strategy pair.
288
Quantum Aspects of Life
can potentially provide an alternative viewpoint in finding evolutionary
quantum search algorithms that may combine the advantages of quantum
and evolutionary computing [Greenwood (2001)]. This will then also provide the opportunity to combine the two different philosophies representing these approaches towards computing: evolutionary search and quantum
computing.
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Flitney, A. P., and Abbott, D., (2002b). Proc. Roy. Society London A, 459,
pp. 2463–2474.
Friedman, D. (1998). J. Evol. Econ. 8, p. 15 and references within.
Greenwood, G. (2001). Congress on Evol. Comp., pp. 815–822.
Harsanyi, J. C., and Selten, R. (1988). A General Theory of Equilibrium Selection
in Games (The MIT Press).
Hofbauer, J., and Sigmund, K. (1998). Evolutionary Games and Population Dynamics (Cambridge University Press).
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http://arxiv.org/abs/quant-ph/0503176.
Marinatto, L., and Weber, T. (2000a). Phys. Lett. A 272, p. 291.
Marinatto, L., and Weber, T. (2000b). Phys. Lett. A 277, p. 183.
Maynard Smith, J. (1982). Evolution and the Theory of Games (Cambridge University Press).
Maynard Smith, J., and Price, G. R. (1973). Nature 246, pp. 15–18.
Meyer, D. A. (1999). Phys. Rev. Lett. 82, p. 1052.
Myerson, R. B. (1978). Int. J. Game Theo. 7, p. 73.
Nash, J. (1950). Proc. of the National Academy of Sciences 36, p. 48.
Nash, J. (1950). Non-cooperative games, PhD dissertation, Princeton University.
Nash, J. (1951). Ann. Math. 54, pp. 287–295.
Nielsen, M. A., and Chuang, I. L. (2000). Quantum Computation and Quantum
Information (Cambridge University Press).
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Behaviour (Princeton).
Oechssler, J., and Riedel, F. (2000). Discussion paper 7/2000, Bonn Graduate
School of Economics, University of Bonn, Adenauerallee 24-42, D-53113
Bonn, http://www.bgse.uni-bonn.de/papers/liste.html#2000.
Peterson, I. (1996). Lizard Game, Available at
http://www.maa.org/ mathland/mathland 4 15.html.
Prestwich, K. (1999). Game Theory, A report submitted to the Department of Biology, College of the Holy Cross, Worcester, MA, USA 01610. Available at
http://www.holycross.edu/departments/biology/kprestwi/behavior/
ESS/pdf/games.pdf.
Rasmusen, E. (1989). Games and Information (Blackwell).
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“Competition and Cooperation” of the Faculty of Economics and Econometrics, Free University, Amsterdam, November 6, 1996. Available online
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Weibull, J. W. (1995). Evolutionary Game Theory (The MIT Press).
Williams, C. P., and Clearwater, S. H. (1998). Explorations in Quantum Computing (Springer-Verlag).
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https://papers.econ.mpg.de/evo/discussionpapers/2006-05.pdf.
About the authors
Azhar Iqbal graduated in Physics in 1995 from the University of Sheffield,
UK. From 1995 to 2002 he was associated with the Pakistan Institute of
Lasers & Optics. He earned his first PhD from Quaid-i-Azam University,
Pakistan, under Abdul Hameel Toor. He earned his second PhD from
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the University of Hull, UK, in 2006 in the area of quantum games under
Tim Scott. He is Assistant Professor (on leave) at the National University
of Sciences and Technology, Pakistan, and has been a Visiting Associate
Professor at the Kochi University of Technology, Japan. In 2007, he won
the prestigious ARC Australian Postdoctoral (APD) Fellowship to carry out
further research on quantum games at the University of Adelaide, under
Derek Abbott.
Taksu Cheon graduated in Physics in 1980 from the University of Tokyo,
Japan. He earned his PhD from the University of Tokyo in 1985 in the
area of theoretical nuclear physics under Akito Arima. He is Professor of
Theoretical Physics at the Kochi University of Technology, Japan.
Chapter 14
Quantum Transmemetic Intelligence
Edward W. Piotrowski and Jan Sladkowski
Congregados los sentidos, surge el alma. Haba que esperarla.
Madeleine estaba para la vista, Madeleine estaba para el odo,
Madeleine estaba para el sabor, Madeleine estaba para el olfato,
Madeleine estaba para el tacto: Ya estaba Madeleine.
Adolfo Bioy Casares, La Invencin de Morel1
14.1.
Introduction
Richard Dawkins put forward the fascinating idea of a meme—a selfreplicating unit of evolution of human behaviour2 , that is analogous to
a gene, the fundamental unit of biological evolution [Dawkins (1989)]. Although the memetic model of human consciousness and intelligence is not
widely accepted by scientists investigating the phenomenon of human beings it seems to be unrivalled with respect to the evolutionary paradigm
so successful in biology. In some sense, it passes the Ockham’s razor test
of efficiency and holds out hope of unification of knowledge. It certainly
deserves a thorough analysis from the point of view of the qualitatively
new perspective opened by quantum information processing [Nielsen and
Chuang (2000)]. Restrictions, such as no-cloning theorems, imposed by
the unitarity of quantum evolution would certainly shed new light on the
Received June 26, 2006
all the senses are synchronized, the soul emerges... When Madeleine existed
for the senses of sight, hearing, taste, smell, and touch, Madeleine herself was actually
there.
2 e.g. ideas, tunes, fashions, habits etc.
1 When
291
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Quantum Aspects of Life
otherwise interesting and bewildering aspects of Darwin’s ideas.3 The idea
of a quantum meme (qumeme) offers a unique opportunity of interpretation
of human consciousness as an element of material evolution. This holds out
hope of overcoming the soul-matter dychotomy that has been dominating
research since Descartes, and perhaps even Plato.
Although very interesting, the problem of whether the memetic structures are abstract ideas or could possibly be identified with some substructures of individual human brains is of secondary significance.4 Whatever
the answer is, it might be that while observing the complex ceremonial of
everyday human behaviour we are in fact observing quantum games eluding
classical description. If human decisions can be traced to microscopic quantum events one would expect that nature would have taken advantage of
quantum computation in evolving complex brains. In that sense one could
indeed say that sorts of quantum computers are already playing games according to quantum rules. Even if this is not true, the investigation into
the quantum aspects of information processing opens new chapters in information science the quantum mechanism might have the power to overcome
complexity barriers stemming from the classical Turing theory. What that
science will look like is currently unclear, and it is difficult to predict which
results will turn out to be fruitful and which will have only marginal effect. The results of the research will probably influence the development of
cryptography, social sciences, biology, and economics.
The emergent quantum game theory [Meyer (1999); Eisert (1999); Flitney and Abbott (2002); Piotrowski (2004a, 2002)] is, from the information
theory point of view, a proposal of a new language game [Wittgenstein
(1961)] describing empirical facts that, although a having precise mathematical model, resist classical analysis.5 It forms a promising tool because
quantum theory is up to now the only scientific theory that requires the
observer to take into consideration the usually neglected influence of the
method of observation on the result of observation and strategies can be
3 The reversed process can also be fruitful: Quantum Darwinism—the process by which
the fittest information is propagated at the expense of incompatible information can be
useful in the quantum measurement theory. The fittest information becomes objective
and the incompatible redundant [Zurek (2004)].
4 It is very difficult, if not impossible, to identify the algorithm being executed by a
computer by, say, microscopic analysis of its hardware, especially if one notices that
often the computer in question might be only a minute part of a network performing
parallel computation.
5 Full and absolutely objective information about the investigated phenomenon is impossible and this is a fundamental principle of nature and does not result from deficiency
in our technology or knowledge.
Quantum Transmemetic Intelligence
293
intertwined in a more complicated way than probabilistic mixtures. In
this chapter we discuss several simple quantum systems that resist classical
(non-quantum) description. They form information processing units that
can “proliferate” via scientific publications and experiments. We propose to
call them qumemes. We will neither consider here “technological” realization nor replication mechanisms of qumemes6 [Iqbal (2001)]. New artificial
sensors might result in development analogous to that caused by transgenic
plants in agriculture. But this time the revolutionary changes are brought
about in human intelligence/mind theory. Since the first implementations
of algorithms as computer programs, the information content has became
an abstract notion separated from its actual (physical) realization—all such
realizations (representations) are equivalent. Moreover, a way of division
into substructures can be quite arbitrary, dictated only by conventions or
point of view. Engineers commonly use analogies with natural evolution
to optimize technical devices. If sciences, techniques, human organizations,
and more generally all complex systems, obey evolutionary rules that have
a good genetic model, even if genes and chromosomes are only “virtual”
entities [Krähenbühl (2005)]. Thus, the genetic representation is not only
a powerful tool in the design of technological solutions, but also a global
and dynamic model for the action of human behaviour. Let us have a closer
look at such as yet virtual objects. Following examples from classical logical circuits, David Deutsch put forward the idea of quantum logical circuits
made up from quantum gates. Quantum gates seem to be too elementary to
represent quantum operations that could be referred to as memes—rather
they play the roles of RNA (DNA) bases in genetics. The qumeme functionality (as an analogue of a gene) can be attained only at the level of
a circuit made up from several quantum gates representing, for example,
tactics in a quantum game7 —examples would be discussed below. The due
ceremonial of everyday performance of quantum physicists and, possibly
not yet discovered, natural phenomena outside the area of human activities might already be the theater of activity of qumemes that cannot be
replaced by classical ones—they might participate in evolutionary struggle
6 Quantum states cannot be cloned, but such no-go theorems do not concern evolution
and measurements of quantum systems. The no-cloning theorem is not so restricting
to our model as the reader might expect. The solution is coding the information in
the statistics of a set of observables [Ferraro (2005)]. The concepts of both exact and
approximate cloning of classes of observables can be introduced. Explicit implementations for cloning machines for classes of commuting observables based on quantum
non-demolition measurements have already been proposed [Ferraro (2005)].
7 From the information theory point of view (qu-)memes correspond to algorithms.
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Quantum Aspects of Life
for survival with themselves, genes or memes.8 Is the notion of a qumeme,
a replicable quantum tactics or unit of quantum information living in a
kind of quantum information soup that is being detected, a newly recognized autonomous class of replicators? In the light of recent speculations
[Patel (2005)] a fascinating relationship between qumemes and mechanisms
for functioning of the genetic code emerged. Does the chain of replicators
driving the evolution end at the qumemes stage or shall we look for a more
fundamental modules? The theory of evolution can, to some extent, be perceived as decision making in conflict situations.9 We will restrict ourselves
to simple cases when memes can be perceived as strategies or tactics or,
more precisely, self-replicating strategies/tactics. Details of the formalism
can be found in [Piotrowski (2004a)]. Game theory considers strategies
that are probabilistic mixtures of pure strategies. Why cannot they be intertwined in a more complicated way, for example interfered or entangled?
Are there situations in which quantum theory can enlarge the set of possible strategies? Can quantum memes-strategies be more successful than
classical ones? Do they replicate in the way we suspect?
14.2.
A Quantum Model of Free Will
The idea of human free will is one of most infectious memes. It can be illustrated in game theoretical terms as was shown by Newcomb [Levi (1982)].10
Martin Gardner proposed the following fabulous description of the game
with pay-off given by the Matrix (14.1) [Gardner (1982)]. An alien Omega
being a representative of alien civilization (player 2) offers a human (player
1) a choice between two boxes:
$1000 $1 001 000
M :=
(14.1)
0
$1 000 000
Player 1 can take the content of both boxes or only the content of the second
one. The first one is transparent and contains $1,000. Omega declares to
8 The possibility that human consciousness explores quantum phenomena, although it
seems to be at least as mysterious as the quantum world, is often berated. Nevertheless,
one cannot reject the the idea that the axioms of probability theory are too restrictive and
one, for example, should take quantum-like models into consideration. Such a possibility
removes some paradoxes in game theory.
9 For example, games against nature [Milnor (1954)]. These include those for which
Nature is quantum mechanical.
10 In 1960 William Newcomb, a physicist, intrigued the philosopher Robert Nozick with
the parable of faith, decision-making, and free will [Nozik (1969)].
Quantum Transmemetic Intelligence
295
have put into the second box that is opaque $1,000,000 (strategy |12 ) but
only if Omega foresaw that player 1 decided to take only the content of
that box (|11 ). A male player 1 thinks: If Omega knows what I am going
to do then I have the choice between $1,000 and $1,000,000. Therefore I
take the $1,000,000 (strategy |11 ). A female player 1 thinks: It is obvious
that I want to take the only the content of the second box therefore Omega
foresaw it and put the $1,000,000 into the box. So the one million dollars
is in the second box. Why should I not take more?—I take the content
of both boxes (strategy |01 ). The question is whose strategy, male’s or
female’s, is better? If between deciding what to do and actually doing it
the male player was to bet on the outcome he would certainly bet that if
he takes both boxes he will get $1,000 and if he takes the opaque box only
he will get $1,000,000. Why should he act in a way that he would bet will
have a worse result? But suppose you are observing the game and that
you know the content of the boxes. From your point of view the player
should always choose both boxes because in this case the player will get
better of the game. Does the prediction blur the distinction between past
and future and therefore between what can and what cannot be affected
by one’s actions? One cannot give unambiguous answer to this question,
without precise definition of the measures of the events relevant for the
pay-off. Quantum theory offers a solution to this paradox.
Suppose that Omega, as representative of an advanced alien civilization, is aware of quantum properties of the Universe that are still obscure
or mysterious to humans. The boxes containing pay-offs are probably coupled. One can suspect that because the human cannot take the content of
the transparent box alone ($1,000). The female player is sceptical about the
possibility of realization of the Omega’s scenario for the game. She thinks
that the choice of the male strategy results in Omega putting one million
dollars in the second box, and after this being done no one can prevent her
from taking the content of both boxes in question (i.e. $1,001,000). But
Meyer proposed recently the use of quantum tactics [Meyer (1999)] that,
if adopted by Omega, allows Omega to accomplish his scenario. Omega
may not be able to foresee the future [Gardner (1982)]. It is sufficient
that Omega is able to discern human intentions regardless of their will or
feelings on the matter. This can be accomplished by means of teleportation [Milburn (1999)]: Omega must intercept and then return the human’s
strategies. The manipulations presented below leading to thwarting humans are feasible with contemporary technologies. The game may take the
following course. At the starting-point, the density operator W acting on
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Quantum Aspects of Life
the Hilbert space of both players (1 and 2) H1⊗H2 describes the human’s
intended strategy and the Omega’s strategy based on its prediction of the
human’s intentions. The game must be carried on according to quantum
rules, that is, the players are allowed to change the state of the game by
unitary actions on W [Eisert (1999)]. The human player can only act on
her/his q-bit Hilbert space H1 . Omega’s tactics must not depend on the
actual move performed by the human player (it may not be aware of the
human strategy): its moves are performed by automatic device that couples
the boxes. Meyer’s recipe leads to:
(1) Just before the human’s move, Omega set the automatic devise according to its knowledge of human’s intention. The device executes the
tactics F ⊗ I, where I is the identity transform (Omega cannot change
its decision) and F is the well known Hadamard
transform frequently
1 1
1
used in quantum algorithms: F := √2
.
1 −1
(2) The human player uses with probability w the female tactics N ⊗ I ,
where N is the negation operator11 and with probability 1−w the male
tactics I ⊗ I.
(3) At the final step the boxes are being opened and the built-in coupling
mechanism performs once more the transform F ⊗ I and the game is
settled.
Players’ tactics, by definition, could have resulted in changes in the (sub-)
space H1 only. Therefore it suffices to analyze the human’s strategies. In
a general case the human can use a mixed strategy: the female one with
the probability v and the male one with probability 1 − v. Let us begin
with the extreme values of v (pure strategies). If the human decided to
use the female strategy (v = 1) or the male one (v = 0) then the matrices
Wi , i = 0, 1 corresponding to the density operators [Nielsen and Chuang
(2000)]
W0 =
2
W0 rs |r−11 |02 1 s−1| 2 0|
(14.2)
2
W1rs |r−11 |12 1 s−1| 2 1|
(14.3)
r,s=1
and
W1 =
11 N |0
= |1, N |1 = |0.
r,s=1
Quantum Transmemetic Intelligence
are calculated as follows:
1 v 0
v 0
1 1
1 1
−→ 2
1 −1 0 1−v 1 −1
0 1−v
1 2v−1
01
1 2v−1 0 1
−→ w2
= 12
2v−1 1
1 0 2v−1 1
10
1 2v−1
1 2v−1
1
=
+ 1−w
2
2 2v−1
2v−1 1
1
1
1 2v−1 1 1
v 0
1 1
−→ 4
=
.
1 −1 2v−1 1
1 −1
0 1−v
297
(14.4)
It is obvious that independently of the employed tactics, the human’s strategy takes the starting form. For the mixed strategy the course of the game
is described by the density operator
W = v W0 + (1−v) W1 .
(14.5)
which also has the same diagonal form at the beginning and at the end of
the game [Piotrowski (2003)].
Therefore the change of mind resulting from the female strategy cannot
lead to any additional profits. If the human using the female tactics (that
is changes his/her mind) begins the game with the female strategy then at
the end the opaque box will be empty and he/she will not get the content
of the transparent box: the pay-off will be minimal (0). If the human acts
in the opposite way the transparent box must not be opened but nevertheless the pay-off will be maximal ($1,000,000). Only if the human begins
with the female strategy and then applies the male tactics the content of the
transparent box is accessible. If restricted to classical game theory, Omega
would have to prevent humans from changing their minds. In the quantum
domain the pay-off M21 (female strategy and tactics) is possible: humans
regain their free will but they have to remember that Omega has (quantum)
means to prevent humans from profiting out of altering their decisions. In
this way the quantum approach allows us to remove the paradox from the
classical dilemma. One can also consider games with more alternatives for
the human player. The respective larger pay-off matrices would offer even
more sophisticated versions of the Newcomb’s observation. But even then
there is a quantum protocol that guarantees that Omega keeps its promises
(threats) [Wang (2000)]. Thus, even if there exists nothing like a quantum
meme, the meme of quantum theory is likely to replicate using human hosts
and to influence their behaviour so to promote its replication.
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Quantum Aspects of Life
Fig. 14.1.
14.3.
The game Master and Pupil (dense coding).
Quantum Acquisition of Knowledge
Acquisition of knowledge certainly belongs to the class of behaviours that
can be interpreted in terms of memes replication. Let us consider a collective game that has no classical counterpart and can shed some light on
qumemes replication. We call it Master and Pupil . Suppose Alice (A)
is ready to sell some asset G at low price and Bob (B) wants to buy G
even at high price. But Bob, instead of making the deal (according to
the measured strategies), enters into an alliance12 with Alice. In the aftermath, Alice changes her strategy and enters into an alliance with Bob.
As a result an entangled quantum state13 |z, αAB ∈ RP 3 ⊂ CP 3 is formed,
cf. Fig. 14.1:
|z, αAB := C (Uz,α ⊗ I) C ′ |0A |0′ B = cos(α) |0′ A |0B
+ i sin(α) Ez (X ) |0′ A |IB + Ez (X ′ ) |I′ A |0B
+ Ez (X X ′ ) |I′ A |IB .
(14.6)
Although Bob cannot imitate Alice’s tactics Uz,α by cloning of the state, he
can gather substantial knowledge about her strategy when she is buying (he
is able to measure proportions among the components I, X , X ′ and X X ′ ).
The game is interesting also from the Alice’s point of view because it allows
her to form convenient correlations of her strategy with the Bob’s one. Such
a procedure is called dense coding in quantum information theory [Rieffel
(2000)]. If Alice and Bob are separated from each other and have formed
12 Alliances are represented by controlled NOT gates denoted here by C [Nielsen and
Chuang (2000)].
13 We call any unitary transformation that changes agent’s (player’s) strategy a tactics. We follow the notation introduced in [Piotrowski (2004b)]: SU (2) ∋ U z,α =
→
−
→
−
→
−
−
−
−
eiα σ ·Ez ( σ ) = I cos α + i →
σ ·E (→
σ ) sin α , where the vector E (→
σ ) = z| σ |z represents
z
z
z|z
−
the expectation value of the vector of Pauli matrices →
σ := (σ1 , σ2 , σ3 ) for a given strategy |z. The family {|z},z ∈ C of complex vectors (states) |z := |0 + z |I (|±∞ := |I)
represents all trader’s strategies in the linear subspace spanned by the vectors |0 and |I.
Quantum Transmemetic Intelligence
299
Fig. 14.2. Teleportation of the strategy |z consisting in measurement of the tactic
′ ′
Um′ n := X [n=I] X ′[m =I ] (the notation [true] := 1 and [false] := 0 is used).
the entangled state |0A |0B + |IA |IB (this is the collective strategy before
the execution of Uz,α ⊗ I) then Alice is able to communicate her choice of
tactic (I, X , X ′, X X ′ ) to Bob (bits of information) by sending to him a
single qubit. Bob can perform a joint measurement of his and Alice’s qubits.
Only one of four orthogonal projections on the states |0′ A |0B , |0′ A |IB ,
|I′ A |0B and |I′ A |IB will give a positive result forming the message.14 Such
concise communication is impossible for classical communication channels
and any attempt at eavesdropping would irreversibly destroy the quantum
coherence (and would be detected).
If one player forms an alliance with another that has already formed
another alliance with a third player then the later can actually perform
measurements that will allow him to transform his strategy to a strategy
that is identical to the first player’s primary strategy (teleportation [Bennet
(1993)]). This is possible due to the identity (remember that X , X ′, X X ′
are involutive maps)
2 (C ⊗ I) (I ⊗ C) |z|0′ |0 = |0′ |0|z + |0′ |IX |z
+ |I′ |0X ′ |z + |I′ |IX X ′ |z .
(14.7)
Recall that quantum strategies cannot be cloned (no-cloning theorem) and
if there are several identical strategies their number cannot be reduced by
classical means (no-reducing theorem). A possible working mechanism for
replication is coding the information in the statistics of a set of observables [Ferraro (2005)]. Both exact and approximate cloning of classes of
observables can be considered as a quantum replication of (qu-)memes.
14 Answers to the questions Would Alice buy at high price? and Would Bob sell at low
price? would decode the message.
300
14.4.
Quantum Aspects of Life
Thinking as a Quantum Algorithm
Let us recall the anecdote popularized by John Archibald Wheeler [Davies
(1993)]. The plot concerns the game of 20 questions: the player has to
guess an unknown word by asking up to 20 questions (the answers could be
only yes or no and are always true). In the version presented by Wheeler,
the answers are given by a “quantum agent” who attempts to assign the
task the highest level of difficulty without breaking the rules. Any quantum
algorithm (including classical algorithms as a special cases) can be implemented as a sequence of appropriately constructed questions-measurements.
The results of the measurements (i.e. answers) that are not satisfactory
cause further “interrogation” about selected elementary ingredients of the
reality (qubits). If Quantum Intelligence (QI) is perceived in such a way
(as quantum game) then it can be simulated by a deterministic automaton that follows a chain of test bits built on a quantum tenor [Deutsch
(1998)]. The automaton completes the chain with afore prepared additional questions at any time that an unexpected answer is produced. Although the results of the test will be random (and actually meaningless—
they are instrumental), the kind and the topology of tests that examine
various layers multi-qubit reality and the working scheme of the automaton are fixed prior to the test. The remarkability of performance of such
an automaton in a game against nature is by the final measurement that
could reveal knowledge that is out of reach of classical information processing, cf. the already known Grover and Shor quantum algorithms and the
Elitzur-Vaidman “bomb tester”. Needless to say, such an implementation
of a game against quantum nature leaves some room for perfection. The
tactics CN OT and H belong to the normalizer of the n-qubit Pauli group
Gn [Nielsen and Chuang (2000)], hence their adoption allows to restrict
oneself to single corrections of “errors” made by nature that precede the
final measurement. It is worth noting that a variant of implementation of
the tactics T makes it possible to postpone the correction provided the respective measurements methods concern the current state of the cumulated
errors [Jorrand (2003)]. Therefore in this setting of the game some answers
given by nature, though being instrumental, have a significance because of
the influence of the following tests. There is no need for the final error
correction—a modification of the measuring method is sufficient. In that
way the course of the game is fast and the length of the game is not a
random variable. This example shows that in some sense the randomness
in the game against quantum nature can result from awkwardness of agents
Quantum Transmemetic Intelligence
301
and erroneous misinterpretation of answers that are purely instrumental. If
only one error (lie) in the two-person framework is allowed, fast quantum
algorithms solving the problem exist (Ulams’s problem) [Mancini (2005)].
There is a wide class of human behaviours that are adopted during the
process of education (classical memes!) that manifests quantum-like character. If realization of own or some else behaviour is to be perceived as
a measurement, then, contrary to the classical approach, there are restrictions on conscious transfer of emotions [Ferraro (2005)] but appropriate
measurement can help to became aware of emotions. In that way qumemes
(replicated via education process quantum strategies) might represent forming of emotions that would be unique individual features. Moreover, the
process of realization (measurement) of such quantum behaviourism would
itself form a class of qumemes.
14.5.
Counterfactual Measurement as a Model of Intuition
Mauritius Renninger has discovered a fascinating qumeme that allows to
identify events that are possible but did not occur and distinguish them
from events that are impossible [Renninger (1953)].
Let us now consider a modification of the method of jamming the strategy measuring game in which the circuit-breaker gate I/N OT 15 is implemented as a part in a separate switching-off strategy, cf. Fig. 14.3. To
this end, the alliance CN OT was replaced by the Toffoli gate (controlledcontrolled-NOT). Contrary to the former case we are now interested in an
effective accomplishment of the measurement. Therefore, we assume that
there are no correlations between the state of the gate I/N OT and the
strategy |1/0. The role of the gate N OT that comes before the measurement of the central qubit is to guarantee that the measurement of the state
|1 stands for the switching-off the subsystem consisting of the two bottom qubits. To quantize this game we will follow Elitzur and Vaidman
[Vaidman (1996)] who explored Mauritius Renninger’s idea of the negative
measurement [Renninger (1953)], see Fig. 14.4. The method
is based on
√
gradual unblocking the switching-off gate (n steps of n N OT ) and giving
up the whole measurement at any step, if only the change of the third
qubit is observed (measuring the first qubit). Hence, the game is stopped
15 The
gate I/N OT is defined as a randomly chosen gate from the set {I, N OT } and is
used to switching-off the circuit in a random way. It can be generalized to have some
additional control qubits [Miakisz (2006)].
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Quantum Aspects of Life
Fig. 14.3.
Fig. 14.4.
Modification of the system by adding a switching-off strategy.
The Elitzur–Vaidman tactics of gradual unblocking the switching-off strategy.
by the “exploding bomb” in circumstances when at some step the value of
the auxiliary strategy measured after the alliance CN OT is measured to
be |1, see Fig. 14.4.
√
The tactics n N OT of gradual unblocking is represented by the operator:
√
π
n
π
π
+ N OT sin 2n
= eN OT 2n ∈ SU (2) .
N OT := I cos 2n
(14.8)
The probability of continuation of the game after one step is equal to
$
$ √
$ 0| n N OT |0$2 = cos2 π
2n
π
=
and all steps are successfully accomplished with the probability cos2n 2n
2
4
π
π
−3
1 − 4n + 32n2 + O(n ). Therefore, in the limit n → ∞ the probability
of stopping the game tends to zero.16 The inspection of the value of the
first qubit with help of the third qubit acquires a transcendental dimension
because if |1/0 = |1 the measuring system is switched-off and if |1/0 = |0
the switching-off strategy cannot be unblocked. The bomb plays the key
role in the game because it freezes the second qubit in the state |0—this
is the famous quantum Zeno effect [Facchi (2000)]. However, the information about the state of the first qubit (|0 or |1) can only be acquired
via the effectiveness of the unblocking the second qubit. The presented
16 The
limit can be found by application of the de L’Hospital rule to ln cos2n
π
.
2n
Quantum Transmemetic Intelligence
Fig. 14.5.
Fig. 14.6.
303
Safe Elitzur–Vaidman bomb tester.
A bomb tester constructed on the basis of the quantum anti-Zeno effect.
implementation and analysis of the Elitzur-Vaidman circuit-breaker paves
the way for a completely new class of technologies that might be shocking for those unacquainted with quantum effects. For example, if the first
qubit represents a result of quantum computation, then such a breaker allows the access in that part of the Deutsch Multiversum [Deutsch (1998)]
where this computer is turned off [Mitchison (2001)]. If the first qubit
of the circuit represented in Fig. 14.4 is fixed in the state |1, then this
machinery can be used to nondestructive testing, for example, to select
bombs with damaged fuse. The respective measuring system is presented
in Fig. 14.5 (the shaded-in qubits in Fig. 14.4 are absent because they are
redundant). The breaker controlled − (I/N OT ) that replaces the alliance
CN OT is in the state I/N OT = I if the bomb fuse is damaged and in the
state I/N OT = N OT if the fuse is working. The result |1 of the measurement of the first qubit informs us that the bomb is in working order.
This is due to the fact that the √working bomb always reduces this qubit
to |0 after the transformation n N OT (quantum Zeno effect). Without
doubt such a bomb tester (and the Elitzur–Vaidman circuit–breaker) can
be constructed on the basis of the quantum anti-Zeno effect [Facci (2001)].
In this case the working but unexploded bomb accelerates the evolution
of the system instead of “freezing” it. Such an alternative tester is represented in Fig. 14.6, where the working bomb causes at any of the n stages
π
in the phase ϕ of the cumulative tactics eN OT ϕ . Let us
the increase of 2n
304
Quantum Aspects of Life
Fig. 14.7.
Supply-demand switch.
define V (β) := N OT cos β + (I cos α + H · N OT · H sin α) sin β. It is not
difficult to show that V (β2 ) · N OT 3 · V (β1 ) = V (β1 + β2 ). Therefore, we
can replace the gate N OT
N OT cos
π
2n
n−1
n
with any of the gates
π
+ (I cos α + H · N OT · H sin α) sin 2n
,
where α ∈ [0, 2π). But only for α = 0, π such gate belongs to the class
eN OT ϕ and we can claim that the transformation N OT results from the
acceleration or freezing of the evolution of the system. For α = 0, π
we observe a kind of para-Zeno effect because the measurement of the
qubit entangled with the qubit in question stops the free evolution corresponding to a damaged bomb. Consider a slight modification of the
π
π
circuit presented in Fig. 14.7, where now exp πH
2n = I cos 2n + H sin 2n .
Again, there is a strong likelihood that we can avoid explosion because
π
π 2 n
π
+ √i2 sin 2n
| ) > cos2n 2n
. In this case the information revealed
( | cos 2n
by the breaker is more subtle because the “bomb” can only cause transition to a corresponding state in the conjugated basis [Wiesner (1983)].
Nevertheless, the bomb being in the working order causes strategy change.
14.6.
Quantum Modification of Freud’s Model of
Consciousness
In the former section we have put great emphasis on distinction between
measuring qubits and qubits being measured. The later were shaded in
figures. Analogously to the terminology used in the computer science, we
can distinguish the shell (the measuring part) and the kernel (the part being measured) in a quantum game that is perceived as an algorithm implemented by a specific quantum process. Note that this distinction was introduced on the basis of abstract properties of the game (quantum algorithm,
quantum software) and not properties of the specific physical implementation. Quantum hardware would certainly require a great deal of additional
Quantum Transmemetic Intelligence
305
measurements that are nor specific to the game (or software), cf. the process of starting a one-way quantum computer. For example, consider a
Quantum Game Model of Mind (QGMM) exploring the confrontation of
quantum dichotomy between kernel and shell with the principal assumption
of psychoanalysis of dichotomy between consciousness and unconsciousness
[Freud (1923)]. The relation is as follows.
• Kernel represents the Ego, that is the conscious or more precisely, that
level of the psyche that is aware of its existence (it is measured by the
Id). This level is measured due to its coupling to the Id via the actual or
latent (not yet measured) carriers of consciousness (in our case qubits
representing strategies)
• Shell represents the Id that is not self-conscious. Its task is monitoring (that is measuring) the kernel. Memes, the AI viruses [Dawkins
(1989)], can be nesting in that part of the psyche.
Memes being qutrojans, that is quantum parasitic gates (not qubits!) can
replicate themselves (qubits cannot—no-cloning theorem). There is a limited knowledge of the possible threat posed by qutrojans to the future
of quantum networks. In quantum cryptography teleportation of qubits
might be helpful in overcoming potential threats posed by qutrojans therefore, we should only be concerned about attacks by conventional trojans
[Lo (1999)]. If the qutrojan is able to replicate itself it certainly deserves
the name quvirus. A consistent quantum mechanism of such replication is
especially welcome if quantum computers and cryptography are to become
a successful technology. Measuring apparatus and “bombs” reducing (projecting) quantum states of the game play the role of the nervous system
providing the “organism” with contact with the environment that sets the
rules of the game defined in terms of supplies and admissible methods of using of tactics and pay-offs [Piotrowski (2004b)]. Contrary to the quantum
automaton put forward by Albert (1983), there is no self-consciousness—
only the Ego is conscious (partially) via alliances with the Id and is infallible
only if the Id is not infected with memes. Alliances between the kernel and
the Id (shell) form kind of states of consciousness of quantum artificial intelligence (QAI) and can be neutralized (suppressed) in a way analogous
to the quantum solution to the Newcomb’s paradox [Piotrowski (2003)].
In the context of unique properties of the quantum algorithms and their
potential applications, the problem of deciding which model of artificial
intelligence (AI) (if any) faithfully describes human mind is regarded as
fascinating, though less important. The discussed earlier variants of the
306
Quantum Aspects of Life
Elitzur-Vaidman breaker suggests that the addition of the third qubit to
the kernel could be useful in modelling the process of forming the psyche
by successive decoupling qubits from the direct measurement domain (and
thus becoming independent of the shell functions). For example dreams
and hypnosis could take place in shell domains that are temporary coupled
to the kernel in this way. The example discussed in the previous section
illustrates what QAI intuition resulting in a classically unconveyable belief
might be like. It is important that QAI reveals more subtle properties than
its classical counterparts because it can deal with counterfactual situations
[Mitchison (2001); Vaidman (1996)] and in that sense analyze hypothetical situations (imagination). Therefore QAI is anti-Jourdainian: Molier’s
Jourdain speaks in prose without having knowledge of it; QAI might be
unable to speak but QAI knows that it would have spoken in prose if it
were able to speak.
14.7.
Conclusion
Quantum intertwining of tactics creates unique possibilities of parallel actions on practically unlimited number of strategies. Therefore quantum
systems can adopt various types of ambivalent tactics [Makowski (2006)].
In probabilistic models life is kind of gambling scheme. Quantum tactics,
being deterministic from the theoretical point of view, can represent fascinating and yet fully understood wealth of behaviours and the probabilistic
nature emerges only after brutal interactions with classical environment17 —
measurements that extort information from the system. Not only God does
not play dice! Morel, brought to existence by Casares’ vivid imagination18 ,
neglected the fact that Madeleine is a being of intelligence that is not representable by classically computable functions. Does a quantum mathematics
that, among others, investigates quantum-computable functions wait for its
discovery? Will the paradoxes following from the Gödel and Chaitin theorems survive? The specific character of quantum models of consciousness
and thinking that consists in information barrier between conscious and unconscious activities (e.g. computing) suggests a possibility for a complete
understanding of the physical world.19 Would the dream of the Theory of
Everything come true via a Quantum Metatheory of Everything? Quantum
17 One
can say, a brutal invasion of privacy of an isolated quantum system.
Bioy Casares, La Invencin de Morel, we do recommend reading this novel.
19 The world is not reduced to abstract idea such that the axiom of intelligibility is
satisfied [Barrow (1992)].
18 Adolfo
Quantum Transmemetic Intelligence
307
(artificial) sensors are already being used, mostly in physical laboratories.
Humans have already overcome several natural limitations with the help of
artificial tools. Would quantum artificial intelligence/life ever come to existence? Adherents of artificial intelligence should welcome a great number
of new possibilities offered by quantum approach to AI.
Acknowledgements
This paper has been supported by the Polish Ministry of Scientific Research and Information Technology under the (solicited) grant No PBZMIN-008/P03/2003.
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About the authors
Edward W. Piotrowski is presently the head of the Applied Mathematics Group at The University of Bialystok, Poland. He was born in Rybnik
(Poland) in 1955, graduated in theoretical physics from the University of
Silesia (Katowice) and earned his PhD and habilitation from the University
of Silesia, under Andrzej Pawlikowski. He has published about 40 papers in
leading international journals on statistical physics, quantum game theory,
and econophysics. His most important results are the analysis of quantum
strategies, showing connections between the Kelly criterion, thermodynamics, and special theory of relativity. He discovered extremal properties of
fixed points profit intensities of elementary merchant tactics.
Jan Sladkowski was born in Świćetochlowice, Poland in 1958. He earned
his PhD, under Marek Zralek, and habilitation in theoretical physics from
the University of Silesia (Katowice, Poland). He has published over 50 papers on quantum field theory, mathematical physics, quantum information
processing, and econophysics. He has held visting posts at the Bielefeld
University and at the State University of Wisconsin in Madison. During
the past 15 years he has been working on the role of exotic smoothness in
cosmology, quantum games, and applications of thermodynamics in finance
theory. He presently holds the Chair of Astrophysics and Cosmology at the
University of Silesia.
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PART 5
The Debate
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Chapter 15
Dreams versus Reality: Plenary
Debate Session on Quantum
Computing
6:00pm, Wednesday, 4th June 2003, La Fonda Hotel, Santa Fe, USA, The
International Symposium on Fluctuations and Noise (FaN’03).
The Panel - Dramatis Personae
Chair/Moderator: Charles R. Doering, Univ. of Michigan (USA); Editor
of Physics Letters A
Pro Team (assertion “reality”):
Carlton M. Caves, Univ. of New Mexico (USA);
Daniel Lidar, Univ. of Toronto (Canada); Editor of Quantum Information
Processing;
Howard Brandt, Army Research Lab. (USA);
Alex Hamilton, Univ. of New South Wales (Australia).
Con Team (assertion “dream”):
David Ferry, Arizona State University (USA); Editor of Journal of Computational Electronics; Journal of Applied Physics/Applied Physics Letters;
Solid State Electronics; Superlattices and Microstructures;
Julio Gea-Banacloche, Univ. of Arkansas (USA); Editor of Physical Review A;
Sergey Bezrukov, National Institutes of Health (USA); Editor of Fluctuation Noise Lett.;
First published in abridged form in Quantum Information Processing, 2 2003,
pp. 449–472.
313
314
Quantum Aspects of Life
Laszlo Kish, Texas A&M (USA); Editor-in-Chief of Fluctuation Noise
Lett.
Transcript Editor :
Derek Abbott, The University of Adelaide (Australia); Editor of Fluctuation Noise Lett.; Smart Materials and Structures.
Disclaimer: The views expressed by all the participants were for the purpose of lively debate and do not necessarily express their actual views.
Transcript conventions: Square brackets [...] containing a short phrase
indicate that these words were not actually spoken, but were editorial insertions for clarity. Square brackets [...] containing a long section indicate
that the recording was unclear and the speaker has tried to faithfully reconstruct what was actually said; [sic] indicates the transcript has been faithful
to what was spoken, even though grammatically incorrect. Where acoustic
emphasis was deemed to occur in the recording, the transcript reflects this
with italics.
The Debate
Charlie Doering (Chair): [Welcome everybody to the Plenary Debate
on quantum computing “Dream or Reality.” We are going to start with
the Pro team and then the Con team. Each speaker will strictly have 3
minutes. Before we start, I would like to remind everybody about the
dangers of trying to make future predictions about computers with the
following quote:]
“I think there is a world market for maybe five computers.”
—Thomas Watson, Chairman of IBM, 1943.
Audience laughter
Charlie Doering (Chair): [OK, now let’s move straight to our first panellist. Carl.]
Carl Caves (Pro Team): [I’m going to declare that the subject of this
debate is the question: “Is it possible to build a quantum computer?”
With this question in hand, we still have to define our terms. What does
“possible” mean? It could mean, “Are quantum computers allowed by
physical law?” Since we think they are, and since small numbers of qubits
Dreams versus Reality: Plenary Debate Session on Quantum Computing
315
have been demonstrated, I’m going to define “possible” to mean, “Can it
be done in n years?” And then we have the further question of the value of
n. Does n = 1,000, n = 100, n = 30 or n = 10? Finally, we need to define
what we mean by a “quantum computer.” Do we mean a rudimentary, but
scalable device that can, say, factor 15? Do we mean a useful quantum
simulator? Or do we mean a scalable, general-purpose quantum computer
(e.g., one that factors interestingly large numbers)? Before proceeding,
I will issue a warning about physicists’ estimates of the time needed to
accomplish some task:
• n = 1 year: This is a reliable guess, but it will probably take 2 to 5
years.
• n = 10 years: I have a clue how to proceed, but this is a guess that I’m
hoping the funders will forget before the 10 years are out.
• n > 30 years: I don’t have a clue, but someone put a gun to my head
and made me guess.]
Audience laughter
Carl Caves (Pro Team): [So n is going to have a different answer depending on what we mean. Here are my estimates for the three cases:
• Rudimentary, but scalable 10-15 years device that can, say, factor 15?
Motivation: High.
• Useful quantum simulator? 20-30 years Motivation: Medium.
• General-purpose quantum computer? 50 years Motivation: Need more
algorithms.
The really important question for discussion is not whether we can build a
quantum computer, but rather, “Is quantum information science a worthwhile interdisciplinary research field?” Yes! It puts physical law at the core
of information-processing questions. It prompts us to ask what can be accomplished in a quantum-mechanical world that can’t be accomplished in
a classical world. And it prompts us to investigate how to make quantum
systems do what we want instead of what comes naturally.]
Charlie Doering (Chair): [OK, good timing. Next, Daniel.]
Second Pro Panelist (Daniel Lidar): I think it was in ’95, [a paper in
Physics Today by Haroche and Raimond,]1 and the title of the paper was
1 Transcript editor’s note: It was actually 1996, see: Haroche, S. and Raimond, J.-M.,
(1996) Quantum computing: Dream or Nightmare?, Physics Today, 49, pp. 51–52.
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Quantum Aspects of Life
“Quantum computing: dream or nightmare,”—so we’re making progress
by making this [debate] “dream or reality.”
Audience laughter
Daniel Lidar (Pro Team): I believe that there’s no question that quantum computers will be built and my reasoning for this is very simple. There
is simply no law of nature, which prevents a quantum computer from being
built and there are some damn good reasons for building one. Now, why is
it that there is no law of nature? Well, in that “dream or nightmare” paper
and as well as some other papers by Bill Unruh and Landauer around the
same time, it was argued that decoherence was going to kill quantum computers and therefore there was no chance. And there was a quick reaction to
that, which astonished a lot of people because it seemed to somehow violate
the second law of thermodynamics; and this was the discovery of quantum
error correcting codes. So while it was believed naively that quantum computers would never work because of decoherence, quantum error correcting
theory shows that this belief was false and that in fact it is possible to overcome the decoherence problem, at least in principle. And this theory has
been refined to the level where we now know that there exists a threshold,
which is measured in terms of a number that’s rather small—about 10−4 or
so. It’s basically something like the ratio between the decoherence time and
the time it takes to apply an elementary logic gate. So you have to be able
to squeeze in 104 logic gates within a unit of decoherence time. If you can
do that, we know from the theory of quantum error correction that there is
nothing in principle preventing quantum computers from being built. They
will be robust; they will be able to resist decoherence absolutely. So this
disproves the old skepticism and I believe it reduces the problem of constructing a quantum computer to a very interesting one, but it has basically
now become a problem of finding the optimal system and fine-tuning the
ways that we’re going to implement quantum error correction, quantum
logic gates, and measurements.
Charlie Doering (Chair): Excellent. Thank you. Twenty seconds to
spare.
Audience applause
Charlie Doering (Chair): Now can we have Howard?
Howard Brandt (Pro Team): Yes.
Charlie Doering (Chair): OK, go!
Dreams versus Reality: Plenary Debate Session on Quantum Computing
317
Howard Brandt (Pro Team): Quantum information processors are certainly viable: quantum crypto systems, operational quantum crypto systems, have already been demonstrated. That’s small-scale quantum information processing. Quantum teleportation has been demonstrated. Some
of the basic ingredients of quantum repeaters have been demonstrated.
Quantum copiers are certainly feasible. Grover’s algorithm has had a proofof-principle—Grover’s algorithm has been demonstrated for a database of
8 entries, doing it faster than a classical computer could have. So the algorithm has been proof-of-principle demonstrated. Shor’s algorithm has been
proof-of-principle demonstrated in factoring the number 15 faster than a
classical computer could. Quantum error correction has also been demonstrated on a small scale—proof-of-principle. As Lidar has pointed out, the
laws of physics do not prohibit even large-scale quantum computers capable of exercising Shor’s algorithm, or Grover’s algorithm to search a large
database. There’s potentially a big pay-off in solving problems not possible
to solve classically, and breaking unbreakable codes. Speaking for the viability of quantum information processors is the worldwide effort including
many elements, many disciplines, special sections of our most prestigious
journals on quantum information, and a number of entirely new journals on
this subject. The real feasibility of developing a robust largescale quantum
computer by any of the current approaches remains in question. It will
likely take a lot of time.
Charlie Doering (Chair): 30 seconds.
Howard Brandt (Pro Team): Luv Grover has warned us against erring
on the side of pessimism. Witness the pessimism at the end of the ENIAC
in 1949 in terms of projected size and number of vacuum tubes.
Charlie Doering (Chair): One second.
Howard Brandt (Pro Team): My time is up, is it? All right, well...the
Army Research Lab is tasked with assessing the viability of quantum information science and technology for possible transition to the Army. Quantum crypto systems are ready and that’s being pursued—the other systems
are not ready but they will be.
Charlie Doering (Chair): OK, thank you very much.
Audience applause
Charlie Doering (Chair): OK, I’m pretty excited now! Alex.
Alex Hamilton (Pro Team): OK, so to follow Carl’s theme; the first
thing we’ve got to do is look at the question: Is it a dream or reality?
318
Quantum Aspects of Life
And the answer is we best not follow that path, actually, because it is
an entanglement of both. The dream is really to—as system engineers—
to understand nature and to try to control nature. What’s the simplest
quantum mechanical thing we can understand? [It is] the quantum two-level
system. What could be simpler? Let’s get one and control it. That will be a
beautiful thing to do. Understanding even what quantum mechanics means
at the most fundamental level—this is all part of the dream. Well, what do
we mean by doing a quantum measurement? We teach our high school and
undergraduate students that you have a quantum system, you come along,
do a measurement and that collapses the wave function—but we’re not
really sure how it collapses the wave function—that’s never really discussed.
It just comes in, it collapses and you get a 1 or a 0. The cat is either
alive or dead. So, we’re having to think very hard about what a quantum
measurement means. This seemingly esoteric and irrelevant question now
has a very real physical meaning in terms of doing a measurement on a
quantum bit. And then, how do you couple these two-level systems? What
does it mean to entangle them? Do you actually need entanglement for
quantum computing?2 So these are, physically, very important questions to
answer. The other breakthrough is that it does bring together people from
all sorts of different disciplines. In the solid-state area there are people from
superconductivity, semiconductors, surface science, other variant schools of
physics, all talking about the same thing and for the first time, in a long
time, speaking the same language. So there are 3 level quantum systems,
decoherence, T1 , T2 . In liquid-state NMR, same thing is happening. Every
phase of matter is being represented: solid-state, through liquid, through
gas, even the others: Bose-Einstein condensates, fractional quantum Hall
liquids. We’re all coming together and talking the same language, so that’s
the dream, who knows? The reality—can it be done? Well, there’s good
evidence that we can make one-qubit systems. There’s evidence we can
couple n small numbers of qubits. So on a very small scale, yes, it looks
like it can be done. Can it be scaled to a usefully large quantum computer?
That really is a very difficult question to answer. I would say that it is
perhaps too early to say because it’s a big engineering problem but there’s
no law of physics that says that it simply cannot be done. And, again, if we
look at the history of electronics the first vacuum valves were in operation
in the early twentieth century but it wasn’t until about the 1960s that it
was possible to really build a useful [classical] computer.
2 Transcript editor’s note: See, for example, D. A. Meyer (2002) Sophisticated quantum
search without entanglement, Phys Rev Lett, 85, pp. 2014-2017.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
319
Charlie Doering (Chair): Twenty seconds.
Alex Hamilton (Pro Team): Will it actually be useful? Will I be able
to go down to Walmart and buy one for my grandmother for Christmas?
Audience laughter
Alex Hamilton (Pro Team): This is a question that one of my students
asked. Maybe it will never be one per household, but perhaps we don’t need
it to. Supercomputers—my grandmother doesn’t have a supercomputer at
home—she’s quite happy, in fact she doesn’t have a computer at home.
And perhaps you can say, “Look, it doesn’t matter. It’s just never going to
work.” How do we know? If you look at the history of computers, people
said that it would never work. They were weighing no more than 1.5 tonnes
and they’ll consume no more than the power of a small city; and look what
we’ve got today, so I think it is possible and it’s just... you better go and
see what happens.
Charlie Doering (Chair): OK, thank you very much.
Audience applause
Charlie Doering (Chair): OK. I’m glad you addressed the issue of how
much they’re going to weigh. That’s certainly something on a lot of people’s
minds. At least a while ago,
“Computers in the future may weigh no more than 1.5 tonnes.”
—Popular Mechanics, forecasting the relentless march of science, 1949.
Audience laughter
Charlie Doering (Chair): Now we’re going to go over to the Con side,
which I believe is some kind of Republican view of [quantum computing].
Hysterical audience laughter
Charlie Doering (Chair): We’ll start off with David Ferry—please.
David Ferry (Con Team): Just a simple, little 2-level system. It would
be the easiest thing in the world to make, all right? We’ve had twenty
years working on quantum computers, more than two decades in fact and
we haven’t got it going yet. The problem is that in those two decades—more
than two decades—they’ve only got two algorithms. Although I heard in a
rumour, today, that a third algorithm may have been followed up. Without
having the architecture and an algorithm making the system work, you can’t
make a system. So you really have to have more than just a device—the
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Quantum Aspects of Life
world is littered with devices and it takes more than just a theory. When
I was young, last century or just before ...
Audience laughter
David Ferry (Con Team): [Many years ago,] one of the first conferences I
went to was on superconductivity, and there was a god of superconductivity
who made the statement that all the theory in the world, integrated over
time had not raised the transition temperature one milliKelvin. So it takes
more than just ideas about where science goes—you have to have practical
working examples from the laboratory. You have to see the results. If
you haven’t seen it yet, it’s quite a difficult problem. We’ve been arguing
about [quantum] measurement since the 1927 Solvay Conference, and even
the idea of wave-function collapse depends upon your view of quantum
mechanics. [Bob] Griffiths doesn’t believe in wave-function collapse. So
you have to be careful now about your interpretation. This makes it a very
difficult problem both intellectually and practically, but it’s a dream with
a shift in emphasis over there by Caves and he probably should work on
quantum information.
Chair taps to indicate time.
OK. Great.
Audience applause
Charlie Doering (Chair): Excellent, excellent. The economy of your
presentation was perfect. Julio?
Julio Gea-Banacloche (Con Team): Alright, well ...I ...um ...I’m surprised, actually, that I’m sitting here on the Con side.
Audience laughter
Julio Gea-Banacloche (Con Team): ... because I just realized that I’m
actually more optimistic than Carl is.
Audience laughter
Julio Gea-Banacloche (Con Team): I would like to mention, nonetheless, that the reason I’m here, I think, is because I understood the question
to mean the last of these options, that is to say, the general purpose, huge,
big, million physical qubit factoring machine of strategic importance and
so on. And that actually ...personally, I don’t think that we will ever see
that, for the reason that it’s basically—even though we may call it a universal quantum computer, and that seems to confuse some people—we really
don’t mean that this is a computer that will replace current computers in
Dreams versus Reality: Plenary Debate Session on Quantum Computing
321
any sense. We’re not building this so that we can run Microsoft Office on
it.
Audience laughter
Julio Gea-Banacloche (Con Team): In fact, there is no reason to build
anything to run Microsoft Office on it [sic].
Audience laughter
Julio Gea-Banacloche (Con Team): But this is obviously going to be—
even if it is built—a special purpose machine and in fact, as Carl also has
pointed out, so far we have only one reason to build it... and that is to
break certain encryption systems, which are currently very popular. But
the thing is that this device is not going to be built, if at all, for some
20 years, 30 years or something like that. And I find it very hard to believe
that in 20-30 years people are still going to be relying for the encryption
of sensitive data on the same encryption algorithm that today a quantum
computer can break. Given that, my personal prediction is that this idea
is going to go basically the way of some other technologies that looked very
promising at one time, but they turned out to be so extremely challenging
that they failed to deliver on their promise in a timely fashion and they
simply fell by the wayside—mostly we found ways around them, and that’s
basically what I think is going to happen with quantum computers; I mean
the large scale quantum computers. Like Carl and like everybody else, I
think that this is very valuable scientific research and having a small, say
100 qubit, quantum simulator in 10 or 15 years will be a big accomplishment
and not out of the realm of possibility.
Charlie Doering (Chair): Thank you.
Audience applause
Charlie Doering (Chair): Sergey.
Sergey Bezrukov (Con Team): The organizers have asked me to say
something about quantum computing and biology.3 This is something of
a very short message, which is “there is no place for quantum computing
in our brain.” The main function [of the brain] is based on nerve pulse
propagation and this process has been studied in great detail. What I
mean is that most of you in this audience do not have any idea of how
many people [are working on these problems] and how much effort is put
into this investigation. It is well understood that this [pulse propagation] is
3 Transcript editor’s note: The possible implication being that if nature hasn’t somehow
made use of quantum computing, itself, then there probably isn’t much hope for it.
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Quantum Aspects of Life
a dissipative macroscopic process. The next in line is synaptic transduction.
This is how nerve cells talk to teach other. Again, this is a macroscopic
dissipative process, which is understood right down to molecular detail.
Next, there are—and these are necessary for “computation” in our brain—
short-term and long-term memory. Well, these things are not as well studied
as the previous two, but one can say that the short-term memory is related
to the short-term changes in the chemical composition of interacting cells.
For example, if I say something to you right now, you are able to recall it
within time intervals of several seconds, because of the transient chemical
changes in the right places. And, finally, our long-term memory is definitely
related to, again, macroscopic dissipative processes leading to structural
changes in the brain. This is all.
Charlie Doering (Chair): Thank you.
Audience applause
Charlie Doering (Chair): OK, everybody take a deep breath because—
next—Laszlo is going to give us his view on the subject.
Laszlo Kish (Con Team): I don’t have much to say... yes, it’s really marvellous that the quantum field has found new effects. This is really great.
My problem is with—just like Julio—general-purpose quantum computing, it seems, is like analog computing:4 we have to build a system that
is special purpose. The error space is analog. What we have to see is
that quantum parallelism is a consequence of Hilbert space. But classical
systems also can inhabit Hilbert space. So to save time, you can also try
classical systems. When we compare classical and quantum computing, it
is very important to use the same temperature and the same speed <clock
frequency> and then compare a classical hardware version with a quantum version, with the same number of elements, and ask what is the power
dissipation. Another question is where are the general-purpose quantum
algorithms? It is important to note that a classical Hilbert space computer
4 Transcript editor’s note: The fact a state in a quantum computer (QC) can be described by a vector in a Hilbert space that is complex and real valued has led some to
believe that QCs are analog machines. This is not quite true, as it is not the full picture.
QCs are digital because it has been rigorously proven, for a general quantum Turing
machine, that for n computational steps only O(log n) bits of precision is required. So
when constructing a quantum computer, the measurement of, say, a rotation of a nuclear spin by arbitrary real-valued angles is not needed as the required precision can
still be maintained by throwing away real-valued information in the sense of a digital
computer. However, what Kish probably really means is that QCs might display some
“analog-like” disadvantages in terms of power dissipation and the requirement for special
purpose circuitry.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
323
is already working in Japan! A 15-qubit classical quantum computer was
built by Fujishima5 —we saw this in the talks. Thanks.
Charlie Doering (Chair): OK, thank you very much.
Audience applause
Charlie Doering (Chair): What we’re going to do now is... I’m going to
show you something to the Con side here: the Pro side is very busy taking
notes while you were all speaking. So, now, I’d like to have a reality check
every once in a while about telling the future of computer science:
“There is no reason anyone would want a computer in their home.”
—Ken Olson, president, chairman and founder of Digital Equipment Corp.,
1977.
Audience laughter
Charlie Doering (Chair): That’s right, that’s why DEC doesn’t exist
anymore, OK!
Audience laughter
Charlie Doering (Chair): Now what we’re going to do is we’re going to
go back. Quickly, one minute, each person, same direction; any comments
they want to make, any ridiculing they want to do. Then we’re going to
open it up to a free discussion to take comments and questions from the
audience and so on, OK, so, we’ll start off right now with Carl.
Carl Caves (Pro Team): I’m going to try and make four quick points.
First, it is good to know that the editor of PRA, i.e. for quantum information views research in this field as useful.
Audience laughter
Carl Caves (Pro Team): I think we might learn that... this is in response to some comments by Dave Ferry... I think we might learn some
things about how to interpret quantum mechanics by thinking in terms
of quantum information processing. I think quantum mechanics is partly
information theory, partly physical theory, and we’ve never understood exactly how these two go together. We might learn something in this regard,
but I don’t think we have to know anything about the interpretation of
quantum mechanics to know how physicists will interpret and make predictions for what a computer will do. I guess I want to get to my third point:
I agree that we need more algorithms. Let me say that the only reason
5 Transcript
editor’s note: Fujishima M., and Hoh K., (2003) High speed quantum
computing emulator utilizing a dedicated processor, Proc. SPIE Noise and Information
in Nanoelectronics, Sensors, and Standards, 5115, pp. 281–287.
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Quantum Aspects of Life
I’m optimistic about that is—because I don’t know much about that and I
don’t think there’s anyone here in this room who’s a real expert on that—
Umesh Vazirani told me that after probabilistic algorithms came out, it
took people 15 years before they realised what could be done with probabilistic algorithms. Maybe something like that will happen with quantum
algorithms.
Charlie Doering (Chair): OK, thank you. No applause for this [round].
Too much time, too much time. Daniel.
Daniel Lidar (Pro Team): OK, let me take these guys on one by one.
Audience laughter
Daniel Lidar (Pro Team): David Ferry says that “20 years and we have
no qubits yet, no [new] algorithms, no practical devices”—but he neglects
the amazing results in trapped ions, 4-qubit entanglement, Josephson qubits
have already shown entanglement, and in quantum dots single qubit operations have already been performed. [This] all happened in the last three
or four years—no reason that it won’t continue. Julio says “only reason is
to break crypto” but he forgets that quantum computers will be to simulate quantum mechanics exponentially faster than we can do on classical
devices. Now, Sergey: “no role for quantum computers in the brain”—I
agree.
Audience laughter
Daniel Lidar (Pro Team): Laszlo: “quantum computers are like analog
systems, that are special purpose,” well, they are not analog. Actually
they are digital. That is a subtle point. “Classical computers can be
described in Hilbert space.” Yes, but there’s no entanglement, no tensor
product structure. The whole speed up issue just breaks down for classical
computers, even if you use Hilbert spaces.
Charlie Doering (Chair): Perfect. OK. Howard.
Howard Brandt (Pro Team): I agree with Dan that David is not up to
date. I’m not surprised, because when I looked at his paper, he speaks of
entanglement as being a hidden variable... enough for David.
David Ferry (Con Team): Clutches his chest as if he’s been shot.
Howard Brandt (Pro Team): Julio, well, I think that you have to realize
that a universal quantum computer is a mathematical artifice, as was the
Turing machine. It’s an idealisation—something that will be approached—
it does not deal with decoherence, it doesn’t deal properly with a halt bit.
There are certain operations and certain unitary transformations that are
Dreams versus Reality: Plenary Debate Session on Quantum Computing
325
suspect. However, related to the universal quantum computer—we now
have a generalized Church-Turing thesis...
Charlie Doering (Chair): 10 seconds.
Howard Brandt (Pro Team): The original Church-Turing thesis is not
true because of quantum computers. Also I heard that factoring might
not be that important. But Grover’s search will [be important], and there
will be other NP incomplete algorithms (such as the travelling salesman
problem) that may happen. And Laszlo, sure you can use Hilbert space
for some classical systems, but that’s an entirely different ballgame than in
quantum mechanics.
Charlie Doering (Chair): OK, right, let’s move on. You’ll get another
chance.
Alex Hamilton (Pro Team): I think everything has been said, so let
me add just two quick points. One is, well perhaps, we don’t have many
[quantum] algorithms, but that’s OK, we don’t have that many [quantum]
computers to run them on just yet...
Audience laughter
Alex Hamilton (Pro Team): ... so, you know, algorithms—it helps if
we have something to do to run them with and that will probably come in
time. Second thing is that, does it have to be a general-purpose quantum
computer?6 The floating-point unit in my laptop is not general purpose.
All it does is crunches numbers but it makes my games so much better,
and I think what we really need to do is quantum gaming and that’s what’s
really driven the microprocessor industry and that’s what will drive the
quantum gaming industry.
Audience laughter
Alex Hamilton (Pro Team): And finally, the classical representation of
quantum computing. If you want to represent 300 qubits for a quantum
computer classically, you can, but there won’t be much left of the universe
once you’ve done that.
Charlie Doering (Chair): Excellent. And in time. David? Hold the
mike closer.
6 Transcript
editor’s note: An important point that is often missed in such debates is
that general-purpose quantum computing is out of the question in the first place. This
is because an arbitrary unitary operation on n qubits takes a number of gates that is
exponential in n, see Nielsen, M. A., and Chuang, I. L. (2000). Quantum Computation
and Quantum Information (Cambridge University Press), pp. 198–200.
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Quantum Aspects of Life
David Ferry (Con Team): Alright, I believe, Dan, I used the word
“practical” but there’s a big difference in “practical” and the number of
the qubits you need out there. I spent a great deal of time working on
quantum dots and I know how practical they are for this purpose. And
there are other examples of some massively parallel analog systems, which
factor the number 15 really fast—it’s called the [human] brain.
Audience laughter
Charlie Doering (Chair): Onward. Julio.
Julio Gea-Banacloche (Con Team): Ummm ... ummm
Audience laughter
Julio Gea-Banacloche (Con Team): Ummm ... ummm
Audience laughter
Charlie Doering (Chair): Fourty seconds.
Audience laughter
Julio Gea-Banacloche (Con Team): Ummm ... ummm
Charlie Doering (Chair): Okaaaay. Now, Sergey.
Sergey Bezrukov (Con Team): My only point is that the solutions
adopted by nature are very, very good. For example, the other day, Laszlo
Kish and I discussed the dissipation issue of 1-bit processing in our brain
and in a conventional computer... and it turned out [that] our brain is 10
times more efficient in power dissipation. Why is that? For two reasons.
[Firstly,] because our brain uses ten times smaller voltages. The computer
uses about 1 V and the brain only about 0.1 V. The second reason is that our
brain is a massively parallel computer, so that mistakes are not prohibited
but, to a degree, are welcome for our spontaneity and ability to think.
Charlie Doering (Chair): Excellent. Thank you. Last one.
Laszo Kish (Con Team): The brain is using noise to communicate,
which is important. Concerning Hilbert space: yes classical and quantum
is different. Classical is better because it is not statistical like quantum.
But finally to you Charlie [Doering], your quotes are against us—I mean
they are Pro! How about the moon base? In the 1970s, we expected that
we would have a base on the moon at the end of the century, [which did
not eventuate.] That’s all.
Charlie Doering (Chair): It’s no good taking shots at me! I just work
here.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
327
Audience laughter
Charlie Doering (Chair): What we’re going to do now is ... I’d like to
open it up to the audience here. If people have questions, you can direct
questions toward either a particular side or particular person, but we’ll keep
it short and then we may allow some rebuttal from the other side, whatever.
So, we have a question right here.
Audience member (Unknown): I would like to ask a question to just
anybody who’s most motivated to, to one or two of you who might answer.
I would like to stick to the moon. What do you think is harder—to build
a 10,000-qubit-quantum computer right now, say in the next years—some
big effort—or to decide in 1960 to go and put a man on the Moon within
10 years?
Charlie Doering (Chair): Who would like to take that? Carlton, it
looks like you’re reaching for that.
Carl Caves (Pro Team): No question. It’s easier to put a man on
the moon. That’s basically engineering. There’s a huge amount of basic
research that has to be done to make a quantum computer work.
Charlie Doering (Chair): Anybody else? Everybody agrees.
The whole Pro team nods affirmatively.
Charlie Doering (Chair): That’s an interesting take-home message.
Derek Abbott (The University of Adelaide, Australia): It seems to
me, without a doubt, that small numbers of qubits have been demonstrated.
So the real question for this debate should be: “Is it possible to scale
quantum computers?” I think that’s your real question and if you look at
the most sensible way of scaling, which is on silicon, in my opinion—because
it’s a mature scaleable technology—you have then got to ask, “What is the
decoherence time in silicon?” And all the papers say, “If you use pure silicon
and blah, blah, blah, it’s all very good.” But putting my Con hat on, to
help the Con team a bit...
Charlie Doering (Chair): They need it.
Derek Abbott, The University of Adelaide, Australia (Audience
member): ... they need it, so I’m going to help them a bit. What the
papers don’t address is that, “OK, I’ve got this...”
Charlie Doering (Chair): Questions cannot last longer than three minutes.
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Quantum Aspects of Life
Audience laughter
Derek Abbott, The University of Adelaide, Australia (Audience
member): [What the papers don’t address is that,] “OK, I’ve got this
scaleable quantum computer; I’ve got zillions of qubits on here; I’ve got
all these A and J gates switching like crazy. That is a coupling into the
environment. What’s going to happen to that decoherence time when they
are all switching like crazy? That is my question to this [Pro] side. Thank
you.
Daniel Lidar (Pro Team): Well, the answer once again is in quantum
error correction. Provided that you can get the single qubit decoherence
rate below a certain threshold, the theory of quantum error correction guarantees that you can scale-up a quantum computer.
Alex Hamilton (Pro Team): Just to finish, to go back to your point
about scalability. Although silicon is one of the things I’m working in—I
don’t think it’s the only one that’s scaleable—superconductive technology
is equally scaleable. It’s very good—you can go out right now and buy
RSFQ <Rapid Single Flux Quantum> electronics that’s basically a superconducting electronics that’s been scaled-up and there’s no reason that
other systems can’t be scaled-up. Ion traps can be put on-chip and so
on. So, there’s no reason that semiconductors are the only ones that are
scaleable.
Charlie Doering (Chair): Julio.
Julio Gea-Banacloche (Con Team): I think that it’s always a big jump
to say that just because you have demonstrated something for, say, 100
qubits that you’re going to be able to scale that up 4 orders of magnitude,
without encountering any unexpected problems. I don’t think there are
any engineers here that will support such a point of view. And, there are
constraints that we can already begin to imagine, as you just mentioned.
If you’re going to address your qubits by frequency, for instance, it’s not
the same thing to have a hundred different frequencies, as it is to have a
million different frequencies. And that’s not all. There are constraints on
the amount of energy that you need in order to perform the gates, and it’s
not the same to operate a hundred qubits as to operate a million of them.
So, the scaling is not by any means trivial. I am willing to grant that once
we have demonstrated, say, 5 qubits—with some effort in a 5-10 years time
frame, we may be able to do 50 to 100 qubits.
Minor audience applause
Charlie Doering (Chair): David, did you want to...?
Dreams versus Reality: Plenary Debate Session on Quantum Computing
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David Ferry (Con Team): Scaling is not all it’s cracked up to be. You
can go to the Intel website. You can find there a view graph, which predicts
that in about 6 or 7 years from now, the power dissipation figure for your
Pentium will about that of a nuclear reactor.7
Audience laughter
Laszlo Kish (Con Team): Yeah, Alex Hamilton said in his talk yesterday,
that if you use error correction you need an error rate of 10−6 or less.
A 10−6 error rate—this is a huge thing, because this is just like analog
circuits, which can achieve [a] 10−6 error [rate] by using very strong negative
feedback. You know, [a] 10−6 error rate [for quantum computation] seems
to be hopeless. Anyway, this is very difficult.
Carl Caves (Pro Team): I think it’s generally 10−4 , which is also incredibly small. But there’s a lot of work in getting error correction worked out
and in some systems based on dits instead of bits—that is higher dimensional quantum systems—or systems based on topological quantum computing, there’s some indication that the error threshold might get up to
one per cent, and then you’re in the ballgame, I think. So we’re just at
the start of this and to dismiss the whole thing because the first results say
fault tolerance is going to be extremely difficult to achieve, seems to be a
mistake. Let’s do some further work and see what the error threshold can
get up to in other kinds of architectures and designs.
Charlie Doering (Chair): OK. All right, let’s move on to a different...
Another question.
Howard Wiseman, Griffith University, Australia (Audience member): This is addressing Carl’s observation comparing probabilistic computing with quantum computing. The genuine question is, “Was probabilistic computing as sexy an area as quantum computing is now?” Because
it is sort of worrying that there are so many smart people working on quantum algorithms and there hasn’t been another reasonable one since ’97. It
does indicate a genuine concern.
Charlie Doering (Chair): Anyone know what the response is? David?
You have the mike.
David Ferry (Con Team): In the beginning, Poppelbaum (University
of Illinois) was working on probabilistic computing back in the mid-70s,
around ’73 or ’74, and it was not a big area like this. He was kind of
7 Transcript editor’s note: ftp://download.intel.com/research/silicon/
TeraHertzshort.pdf, slide number 9.
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Quantum Aspects of Life
trudging on alone with a small, dedicated group working in it, but I don’t
think it grabbed the attention of big research groups around the world like
quantum computing did.
Daniel Lidar (Pro Team): There is a misconception that there are no
good quantum algorithms out there. For problems in number theory and
pure computer science, yes—there are very few. But let’s not forget that
quantum computers are exponentially faster at simulating quantum mechanics. Every university in the world has people in chemistry and physics
departments working on trying to find fast algorithms to solve problems in
quantum mechanics. A quantum computer would be enormously helpful
there, so that’s a huge benefit.
Charlie Doering (Chair): OK.
Carl Caves (Pro Team): I think that’s a good point Howard [Wiseman].
I’m just relying on the fact that Umesh Vazirani, who has worked on both,
suggested that given the current scale of effort in computer science among
people who think about this, you might expect to make a big breakthrough
in quantum algorithms any time or you might expect it to be in another
decade. My direct response to you is that quantum mechanics is a much
richer theory than classical probability theory, so you might think it is
harder to come up with quantum algorithms, and it might take longer even
with more people working on it.
Charlie Doering (Chair): Another question here.
Audience member (Unknown): Just like to make a quick comment.
Doing all those is fine but as general-purpose computers, I’m just wondering
in the ’60s, ’70s and ’80s people doing optical computing. Except for certain
special purpose optical computing, there isn’t any general purpose optical
computing.
Charlie Doering (Chair): Anybody?
Alex Hamilton (Pro Team): My understanding is that for optical computing that one of the great things that it would be good for would be for
Fourier transforms, and with the invention of the fast Fourier transform
algorithm, there really wasn’t any more need for optical computing.
Audience member (Unknown): That’s not quite true because you’ve
got optical parallelism and your Fourier transform [is traditionally computed in] series.
Kotik Lee, BAH, USA (Audience member): Optical computers are
used extensively with defence systems, for special purpose processes.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
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Charlie Doering (Chair): Another question. Up the back.
Fred Green, University of New South Wales, Australia (Audience
member): Well, it’s really a comment. It’s another take on the relative
lack of algorithms. One of the things that happens when you make a
machine, that’s enormously complex, is that it may well become something
that uses its emergent behaviour—to copy a buzzword. The thing is, that
in a sense we are thinking in a reductionist way about machines, we’re
thinking of a specification and rules and designs to make an enormous
machine, but it’s equally likely that the machine will go and do things that
you simply cannot predict from its underlying equations. That is just an
open question. For example, you cannot—just by having a set of equations
and putting it on a computer—you cannot get superconductivity out of
that. Something has to make it all complete and all I’m doing is actually
repeating what Laughlin said some years ago now. It’s quite conceivable
that a machine in all its complexity will be able to do things like that. It’s
something that human brains are quite good at.
Charlie Doering (Chair): Any response? Julio?
Julio Gea-Banacloche (Con Team): I’ll venture a response. That’s
certainly a possibility but it’s not currently an envisioned possibility, the
way people envision this huge fault-tolerant quantum computer. Most of
its time—99.99% of its time in every clock cycle it will not be doing anything except error correction. Emergent behaviour would be, you know,
remarkable—almost anything could show emergent behavior more likely
than such a machine.
Charlie Doering (Chair): You Carl, you have to respond to that.
Carl Caves (Pro Team): I’m not really directly responding to that. I
want to say something that popped into my head that has something to do
with that. Now what if there were a fundamental decoherence mechanism
in the universe that couldn’t be explained by coupling to external systems.
You could error correct that. Wouldn’t that be pretty neat? You could restore linear quantum mechanics even though the universe is fundamentally
[might possibly not be] linear quantum mechanics.
Audience laughter
Charlie Doering (Chair): Whoa, whoa, whoa. OK.
Daniel Lidar (Pro Team): I just wanted to say something about the
[observation that] 99% of the time is spent doing error correction. This is
true, but it does in no way contradict the fact that a quantum computer
offers a speed up.
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Quantum Aspects of Life
Michael Weissman, Univ. Illinois Urbana Champagne, USA (Audience member): I did not understand those last remarks [of Caves on
restoring linearity], but while we are on the same topic: you mentioned
earlier that the interpretation of quantum mechanics does not affect the
operation of a quantum computer. That is certainly true up to an extent.
However, if there were an intrinsic non-unitary operator involved, at some
point, for example, you would find decoherence in cases where [you] do not
expect decoherence if there are only unitary operators. If you made a quantum computer more or less of the same physical scale as your head and with
similar amounts of mass and current involved in its thoughts as in yours
and it did not show unknown nonunitary operations that would be very important for understanding quantum mechanics. If it did, it would be even
more important because it would support the idea that some modification
is needed. Either way, conceivably, it would have something to do with the
experimental realisation of tests of modification-type interpretations.
Carl Caves (Pro Team): Yeah, I certainly agree with you that one thing
you might find out is that there are fundamental non-unitary processes
when you get a sufficiently large system, and those are processes responsible for making the world classical and they would represent a barrier to
making a quantum computer of sufficient size.8 Those are important issues.
I don’t call them interpretational because they’re changing quantum mechanics, whereas when I refer to the interpretation of quantum mechanics I
mean keeping what we’ve got and figuring out what it means—not making
changes.
Charlie Doering (Chair): As the Chair of this session, I’m going to
declare we’re going to keep quantum mechanics the same way, as it’s not
fair to try to change quantum mechanics for either the Pro side or the Con
side.
Audience laughter
Charlie Doering (Chair): OK, yes, absolutely, absolutely.
Howard Brandt (Pro Team): We’ve got some affirmation of the worth
of the pursuit of quantum information science and technology by one of the
editors of Physical Review A, Julio here. But Charles, one of the things I
hear is you’re an editor of Physics Letters A. I’m very concerned about one
8 Transcript
editor’s note: In fact, the holographic principle indeed sets an upper bound,
the “Davies Limit”, to the size of a quantum computer, as there is a cosmological information bound for quantum information. See, P. C. W. Davies in quant-ph/0703041.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
333
of the things you said previously and that the editor of Physics Letters A
may be ignorant about quantum mechanics. You stated that yourself!
Audience laughter
Charlie Doering (Chair): Very good. I apologize.
Audience laughter
Charlie Doering (Chair): The great thing about Physics Letters A is
that each editor has their own area of expertise and mine is explicitly not
quantum information, [which means I have no bias]. So, anyway, quantum
mechanics shall not be changed in remaining discussion and we’ll move on
from there. Peter Hänggi.
Peter Hänggi, University of Augsburg, Germany (Audience member): I like quantum computing because you can see all this knowledge
being brought together from different areas of science and great progress
in understanding quantum mechanics. But I also believe in human nature,
you know. After five, ten years I get a bit tired because I’ve seen enough
of it. See, there are those things that can be done quickly, and I’m not so
sure this momentum carries on when it comes to do the very hard work.
Most of the people here don’t want to do the nitty-gritty work, the core
and the details about this stuff, and so on. I think the excitement, which is
so high up with tackling all these problems on quantum information, will
eventually slow down. [If] the problems are not [solved in] three, four, five
years maximum, and then of course we need something else and we don’t
know what the next excitement in science will be; but most likely we physicists don’t want to do for two or even more years [the] nitty-gritty work
on a detail. Moreover, we also need to talk about the engineering, so that
explaining this heightened expectation [sic]; we also need practical things
from this whole exciting quantum computer and computation [area].
Charlie Doering (Chair): OK, so that sounded like a ... ? Is that a... ?
Could we have a response to that? Is that a Pro or a Con?
Peter Hänggi (Audience member): I don’t know what it is.
Charlie Doering (Chair): Yeah, that’s OK.
Audience laughter
Daniel Lidar (Pro Team): I think it would be great [if people got tired],
because there are way too many papers in this field right now.
Audience laughter. Julio nods affirmatively.
Charlie Doering (Chair): OK, the editor of Physical Rev. A seconds the
motion.
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Quantum Aspects of Life
Audience laughter
Carl Caves (Pro Team): I think the example of quantum cryptography
shows that people are willing to do very sophisticated, higher mathematical
and physical work on a system that is closer to the point of transference
into something useful. And I think that’s a good example of quantum cryptography inspiring extremely useful work—detailed work about improving
security in quantum crypto systems—for real systems. So I think that as
long as the experimental work in the field is moving forward to increasing
numbers of qubits, there are going to be important theoretical problems to
address, and we have plenty of theorists to work on them, and I think they
will.
Charlie Doering (Chair): Yup.
Julio Gea-Banacloche (Con Team): I think that the concern is more
with the funding agencies losing interest and... clearly, if this machine is
20 or 30 years in the future, I think that it doesn’t take a prophet to predict, that they are not going to continue... the current level of funding
for the next 20 or 30 years. Moreover, as I said, without any more algorithms there is the possibility that they will lose interest much earlier
because all we need is basically an easy, convenient alternative to RSA encryption and you’re in business, and there are already encryption—public
key encryption—algorithms that nobody knows whether they are equivalent to factoring or not. Which means that even if you had the big quantum
computer today, you would not know how to use it to crack those forms of
encryption. So, it’s really only a matter of time. So...
Charlie Doering (Chair): Howard? Comment?
Howard Brandt (Pro Team): [Regarding] the business about, you know,
if it’s going to take thirty years to build a quantum computer, that the government agencies aren’t going to wait that long and continue to fund it.
I don’t believe that, because the imperative is still considerable. Witness
thermonuclear fusion. Sakharov came up with the invention of the Tokomak. Now, that was a long, long time ago. That has continued to be funded.
Also, newly, and nicely, inertial confinement fusion. And you know, I remember in the ’70s I was asked to predict when we would have controlled
thermonuclear fusion, and I said, “At the earliest 2030”, and certainly after
all the major participants are dead. And that is true of large-scale quantum
computers too. The government will still have this imperative and it will
be supported at some level, I believe.
Charlie Doering (Chair): Now let Julio talk.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
335
Julio Gea-Banacloche (Con Team): Now I think there’s a big difference
between physical controlled fusion and quantum computing, as we know it
now. I mean, once you get a fusion reactor going, then you can do a lot of
things with the energy. Once you get these huge quantum computers going
you can do exactly one thing ...
Audience laughter
Julio Gea-Banacloche (Con Team): No, sorry, apologies to Daniel,
actually-one thing of strategic importance, OK? Which is to break the RSA
code. How much longer is RSA encryption going to be of strategic importance? My guess is not 30 years, OK. Now, I completely agree with Daniel,
this is of extremely high [scientific value] and I hope the NSF will continue
to support the development of quantum computers at the medium-sized
scale for all the universities that will want to have a quantum computer.
Charlie Doering (Chair): Alex?
Alex Hamilton (Pro Team): Well, I think my point has been said, actually. It’s not just the one algorithm you want to include—there’s a whole
raft of fundamental science reasons, there’s a whole raft of computational
reasons that you [want a quantum computer for] as well for simulating physical systems. I mean, it’s crazy that we have a transistor with 50 electrons
in it and we still can’t calculate, properly, what its properties should be
from a fully quantum mechanical viewpoint. So that would be kind of nice,
and ... the second thing is about, going back to RSA: if everyone switched
to quantum hard codes there’d be no need for this computer, but wouldn’t
you love to know what Clinton really said about Lewinsky? I mean, you
could go back and ...
Audience laughter
Charlie Doering (Chair): OK, OK, let’s keep this clean!
Audience laughter
Charlie Doering (Chair): Let’s move on here. Anybody have a comment
or complaint, or a ... you know. Gottfried.
Gottfried Mayer-Kress, Penn State University, USA (Audience
member): Yes, just a question or comment on the statement about the
brain and it was so sweeping a rejection of any kind of possibility of quantum computation in the brain, and you gave the impression [that] everything was known how the brain works and, you know, there’s really no
open questions—so do you really know how we make a decision? How you
make a choice between different alternatives and the speed at which this is
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Quantum Aspects of Life
happening? So, it seems to me like ... just from the problem solving point
of view: if you think about it, how fast a human brain, can select from a
huge database of visual or sensory inputs and make a very rapid decision.
I mean, that sounds very much like a quantum computation to me, and
if you go down to the biochemical processes of how ion-channels open and
close I think, you know, that quantum processes certainly play a role. So, I
don’t understand why you just completely reject the possibility of quantum
computation occurring.
Sergey Bezrukov (Con Team): I agree with you that we don’t understand how our brain operates in... concerning what you just said. My only
point is that according to the current knowledge of the “elemental base” of
the brain, responsible for logical operations, there is no place for quantum
computing.
Charlie Doering (Chair): Carlton?
Carl Caves (Pro Team): I used to have the conventional view—and I still
have it—that the probability is about [point] 50 nines in a row that there
aren’t any coherent quantum processes going on in the brain that are of any
value. You can do simple calculations that show that decoherence removes
any coherent quantum information processes in the brain. But we now
know that in complex quantum systems there are these decoherence free
subspaces just sitting around that are free of certain kinds of decoherence
and it’s not out of the question that maybe something’s going on there
and, you know, evolution by natural selection is awfully good at figuring
out how to do stuff. I’ll give it a probability of epsilon, where epsilon is
smaller than the error threshold, but I wouldn’t rule it out.
Charlie Doering (Chair): Interesting point. Anybody else? Yeah, let
me see your hand.
Howard Wiseman, Griffith University, Australia (Audience member): I’ll keep supporting the Con side, just to be fair. Daniel, you keep
bringing up this simulating quantum systems thing, but how big—given
that classical computers will probably keep going faster for the next, say,
20 years—how big a quantum computer do you actually need to make it
useful? To make it definitely useful? And, you know, is there anything that
can bridge the gap between, you know, the next 5 to 10 years, and that
sort of level?
Daniel Lidar (Pro Team): Well, there are several papers, which have
looked at this question in detail, and not taking into account the error
correction overhead, it turns out that at about 100 qubits you can solve
Dreams versus Reality: Plenary Debate Session on Quantum Computing
337
problems in mesoscopic quantum physics, which are not possible on any
reasonable classical computer. So, 100 qubits is my answer but you’d have
to multiply that probably by a factor of like at least 15 if you want to take
error correction into account.
Howard Wiseman (Audience member): Is there something that can
take us from where we will be in the foreseeable future to that level of some
1500 qubits?
Charlie Doering (Chair): Did you hear the question?
Howard Wiseman (Audience member): What is going to motivate us
to go from the level of having 10 or 100 qubits—where we can do interesting
things from the point of communications and distillation and stuff like
that—to that level, which is considerably harder?
Dan Lidar (Pro Team): Well, one problem, for example, is understanding superconductivity in metallic grains. So, if that is a problem that is of
considerable interest, which I believe it probably is, I can see that motivating going to that number of qubits that’s required, and there are plenty of
other problems in this class of highly-correlated-electron systems that are
mesoscopic, for which you would need a quantum computer on the order of
100-1000 qubits.
Charlie Doering (Chair): Another question here.
Carl Caves (Pro Team): Can I say one more thing about that?
Charlie Doering (Chair): Certainly.
Carl Caves (Pro Team): Let me say something pretty quickly. The systems that are proposed for quantum computing and quantum information
processing are the cleanest we know of. The best records for quantum coherence are the atomic physics systems, now using trapped ions and trapped
neutrals. Those are pristine systems for which decoherence is very low, but
it’s not so clear how you scale those. The condensed systems are easier to
see how to scale because they rely on more conventional technology, but
their record for decoherence isn’t as good. All the superconducting qubits
are getting there now in terms of decoherence, but we still have to see how
they do when they’re coupled together. We don’t yet know which one of
these systems, if any, is going to be one that ultimately works out. We don’t
know what the architecture is going to look like for a 1500 qubit quantum
computer.
Audience member (Unknown): Yes, I would like to ask Carlton a question. You seem to be hopping around a bit together with other members in
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the Pro team, appeasing the two editors of PRA and PLA. One [statement]
is that in principle you can do [quantum computing] operations with error
correction... Simply, it sounds like it’s just an engineering problem; get
enough engineers together and enough money together and [quantum error
correction] will work. And on the other hand the suggestion is that you
actually need some more basic research—you may find that, you know, you
run into something like a mental limitation. So like, the question is: Is it
just an engineering problem or not?
Carl Caves (Pro Team): You might have exposed a rift in the Pro team,
I don’t know. We’re sitting awfully close together up here, so if you can see
any rift between us ... but I think there’s a lot of basic research to be done.
It’s not an engineering problem yet. I think a fairer comparison, when I was
asked about the space program, would have been the Manhattan Project. If
you put in an amount of money comparable to that in today’s dollars, would
we get a quantum computer a lot faster? Which would be several billion
dollars a year, I reckon. Oh, I don’t think so. I think we wouldn’t know
what to do with it, because it’s not yet an engineering problem. There’s a
lot of basic science yet to be done before we know which physical system is
the best one.
Charlie Doering (Chair): So, we have something from the Pro side?
Carl Caves (Pro Team): Yeah.
Audience laughter
Dan Lidar (Pro Team): Alright, I think you are probably referring to—
or extrapolating from a comment that I made—that, well, we have error
correction, therefore problem solved. No, that’s not the case. The fact that
we have an existence proof or a viability proof, if you wish, that quantum
computers are possible, does not in any sense imply that there’s no basic
research left to be done. I mean, it’s like—what’s a good analog?—maybe:
an existence proof is like saying that we have an axiomatic system for doing
mathematics and now, that’s it, we’re done. Of course, a lot of theorems
remain to be discovered. There’s a lot of basic research to be done on
how we can actually construct a device and there’ll also be lots of spin-offs
in terms of just interesting fundamental questions that are not necessarily
related to how you construct a device.
Howard Brandt (Pro Team): I agree with Carl. I think Derek’s comment was very appropriate and it sort of addresses this question of the
fundamental nature of current research. [Regarding] Derek’s comment,
well you know, the hybrid Kane-type quantum computer in silicon, and
Dreams versus Reality: Plenary Debate Session on Quantum Computing
339
other solid-state approaches that we include here, like quantum dots, well,
they’re scalable. What does that mean in practice? It means that there’s
a giga-dollar industry in semiconductors and solid-state, and frankly, you
know, I think that the funding agencies sort of translate that into scalability. I mean, after all, you know, the classical widgets scaled, so we make
one quantum widget and put the widgets together, but as Derek sensibly
questioned, you know, right now—be it Josephson junctions, quantum dots
or a Kane-type of quantum computer—the study of decoherence is at a very
primitive level. The study of how to produce controlled entanglement is at
a very primitive level. Gates and solid-state approaches are at a very, very
primitive level. People do not know how to do this. They’re doing research
to hopefully, you know, be able to do this, but it’s a big question mark. It’s
a basic research issue. And so Derek, you know, is justified in questioning
the scalability, of the solid-state semiconductor approaches anyway. So, it’s
a basic research issue. The answers are not there. If they were, we’d hear
about it in program reviews. I mean, I’ve heard tens and tens of program
reviews, and nobody is coming close yet, but that doesn’t mean they won’t.
Basic research is needed in fact.
Charlie Doering (Chair): Any other comments or questions from the
audience? Derek wants to rebuff.
Derek Abbott, The University of Adelaide, Australia (Audience
member): [No, I don’t, but] I just thought I’d give the Con team some
more help because they need it.
Audience laughter
Julio Gea-Banacloche (Con Team): I wish you would stop saying that.
Audience laughter
Derek Abbott (Audience member): OK, so I think we’ve established
that the real question is: scaling. To make it practical we need a scaleable
quantum computer. To make it scaleable you’re talking about chip technology for a number of reasons because it’s the only way we know how to
make scaleable things—and we’ve got millions of dollars of backing behind
that. Now, as soon as you put qubits on a chip and line them up in a nice
little pretty order, I find it very hard to believe that you can make a useful
quantum computer with that, because on top of those qubits you’re going
to need classical control registers to control the gates and you are going to
need read-out circuitry. So there’s going to a number of post-processing
steps on top of that. Now, I know some clever guys have put phosphorous
ions [on chip] nicely in a neat little row and it works. But once you do all
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Quantum Aspects of Life
that post-processing, are they [the qubits] really going to stay still? So,
this is my question to the Pro team.
Charlie Doering (Chair): Yes, and it’s a good question. Alex?
Alex Hamilton (Pro Team): OK, so for the specific case of phosphorous and silicon it actually looks like they do stay put, during the postprocessing—but that’s just a specific answer. But the more general answer
is, you do need control chip circuitry, absolutely, and so there’s no reason
that that has to be on the same chip. There’s no reason that we can’t build
a [separate] complete high-frequency silicon germanium control electronics
[chip] and match it up.
Derek Abbott (Audience member): You have to use the same chip
because of noise.
Alex Hamilton (Pro Team): No, no, it doesn’t have to be on the same
chip, Derek, because they [only] have to be physically close to each other.
They don’t have to be on the same chip.
Charlie Doering (Chair): OK. He says no and you say yes.
Audience laughter
Julio Gea-Banacloche (Con Team): Yes, I wanted to say something
about that too, regarding the same sort of thing: the control systems. I
actually gave a talk yesterday9 on this subject and there are constraints
there: how large the control systems have to be in some sense. So in some
sense, some of these amusing quotes referring to the famous quotes recalled
by Charlie Doering are a little misleading because ...
Charlie Doering (Chair): We don’t know.
Julio Gea-Banacloche (Con Team): ...they suggest that, you know,
quantum computing might follow a path similar to classical computers
where you start with something huge, like vacuum tubes, and then slowly
and over time, sometimes fast, you start making things smaller and more
efficient and so forth. And the indications are that it’s not going to be
like that. I mean, when we get the quantum computer we’re stuck at the
vacuum tube level. The control systems have to be large because they have
to be classical, so there is going to be no pocket quantum computer that
somebody will be able to carry around, and there are minimum energy requirements and so another question to ask, you know, is: how are you going
to deal with the heating and so on? How are you going to extract it?
9 Transcript editor’s note: J. Gea-Banacloche (2003), Energy requirements for quantum computation, Proc. SPIE Noise and Information in Nanoelectronics, Sensors, and
Standards, 5115, pp. 154–166.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
341
Charlie Doering (Chair): OK. Laszlo’s going to make one more comment and then we’re going to move into the next stage, the final stage of
this panel.
Laszlo Kish (Con Team): [Regarding] the comment of Julio’s: the calculations I showed yesterday—at the same temperature, same speed and
same number of elements—the quantum computer dissipates more energy
when processing the same information. So, again, the dissipation and noise
is the key [as to why classical computers are better].
Charlie Doering (Chair): What I would like to...you’re going to talk? I
think we need to move on ...
Carl Caves (Pro Team): Well, we’re at about the last stage ...
Charlie Doering (Chair): OK.
Carl Caves (Pro Team): We might have to run the gauntlet to get out
of here.
Charlie Doering (Chair): OK. That’s right. Let me organize the last
stage as follows. Let me have each one talking. Think about a two-sentence
summary of your view, a two-sentence summary of your view, and we’ll run
down the road here and then we’ll take a vote on how the audience feels
what the panel says. A forum—OK? Carl.
Carl Caves (Pro Team): Well, my consistent view has been that it would
be extremely difficult to build a general-purpose quantum computer though
it might be somewhat easier to build quantum simulators, but that’s not
the point of why information science—quantum information science—is a
discipline worth pursuing.
Charlie Doering (Chair): That’s one sentence. OK, good.
Audience laughter
Charlie Doering (Chair): Daniel.
Dan Lidar (Pro Team): I agree.
Howard Brandt (Pro Team): Well, again, no one has demonstrated that
a large-scale quantum computer is, you know, physically impossible, and
certainly small scale quantum information processors are possible and have
already been demonstrated. It’s a worthwhile enterprise and will continue.
Alex Hamilton (Pro Team): We’re scientists. Our job is to try and
understand nature and if we want to—and we’re humans—and we try to
control nature. And we’ve been given this amazing curiosity and we want
to do things. Why does one climb Mount Everest? Because it’s there.
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Quantum Aspects of Life
The Con Team all purse their lips in a thoughtful pose, gently nodding,
apparently conceding this point.
If we build this thing [a quantum computer], let’s have a go or let’s prove
that it’s simply not physically possible.
Charlie Doering (Chair): Very good. Julio?
Julio Gea-Banacloche (Con Team): I really would like to say that I
am also very impressed ...
Charlie Doering (Chair): Julio, also let me just say, that the Con team
does not have the volume of the Pro team...
Julio Gea-Banacloche (Con Team): OK.
Charlie Doering (Chair): ...we would like to hear how the argument’s
going. Hold the mike closer!
Julio Gea-Banacloche (Con Team): Well, I would really like to say
that I find it incredibly amazing and very, very impressive. [A lot of] good
science has come out of quantum information, for the past seven years. And
if quantum computing is responsible for this, then it’s a good thing. The
dream, at least, [is] of a quantum computer.
Sergey Bezrukov (Con Team): While [many] functional processes in
the brain are not understood, the “elemental base” [of the brain] is very
well studied. Also, main interactions between the elements are already well
understood. So, as I said, there is no any [sic] place for quantum computing
in the human brain. But, the concepts, which are being developed by the
scientists working in this field, will find their way into the brain studies and
will be very useful there.
Charlie Doering (Chair): Yes.
Laszlo Kish (Con Team): Quantum compared to classical. Quantum
means more noise, statistical in nature, more dissipation, and higher price.
Audience laughter
Charlie Doering (Chair): OK, that was enough. Now we’ve got first ...
for the record ... I guess the question is a “pro-con/dream-reality” thing. I
would like to take a show of hands for ... first, question number one. How
many people think that quantum computing is really a dream and it’s just
going to fall by the wayside and our attention will go some place else, and
... can I have a show of hands?
A few hands show
Dreams versus Reality: Plenary Debate Session on Quantum Computing
343
Charlie Doering (Chair): OK, how many people think that it’s possible
that quantum computers as—people envision it as a tool—is simply not
going to happen? The way we’re visioning it now?
A few hands show, some people holding up two hands
Charlie Doering (Chair): You are not allowed to hold up two hands.
But you can attempt to have your hands in a superposition of “for” and
“against.”
Audience laughter
Charlie Doering (Chair): That’s good, that’s good. I’m impressed!
Now, now ... but that does not mean that the complement to that set will
be reality, OK. So, how many people think that there’s a possibility that it
may be a useful tool, based on the ideas that we’re now tossing around in
the year 2003? That it’s going to emerge—and some of us in this room are
[still] alive to realise it? Anybody agree with that?
A unanimous majority of hands show
Charlie Doering (Chair): OK. I think the conclusion is clear. I would
just like to reinforce the whole idea of predicting the future in computer
science [is dangerous]:
“640 K ought to be enough for anybody.”
—Bill Gates, 1981.
Let’s thank everyone on the panel here.
Audience applause
End of transcript.
Acknowledgements
The assistance of Phil Thomas, as the audio transcript typist and amanuensis, and Gottfried Mayer-Kress, for the mp3 recording, are gratefully
acknowledged. The mp3 recording can be downloaded from the Complexity
Digest at: http://www.comdig2.de/Conf/SPIE2003/. Thanks are due to
the many people who proof read this transcript—especially Howard Wiseman whose remarkable aural abilities miraculously decoded otherwise inaudible parts of the recording; any remaining errors are of course mine
(Derek Abbott).
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Quantum Aspects of Life
About the Panelists
Charles R. Doering is professor of mathematics at the University of
Michigan, Ann Arbor. He received his BS from Antioch College, 1977; his
MS from the University of Cincinnati, 1978; and his PhD from The University of Texas at Austin under Cécile DeWitt-Morette, 1985, in the area of
applying stochastic differential equations to statistical mechanics and field
theory. In 1986–87, he was a Director’s Postdoctoral Fellow 1986–87, Center for Nonlinear Studies, Los Alamos National Laboratory; in 1987–96, he
rose to Professor of Physics, 1987–96, Clarkson University; in 1994–96, he
was Deputy Director of Los Alamos’ Center for Nonlinear Studies. Doering
has received a number of honours including the NSF Presidential Young
Investigator, 1989–94; Fellow of the American Physical Society, 2000; and
the Humboldt Research Award, 2003. His research is generally focused
on the analysis stochastic dynamical systems arising in biology, chemistry
and physics, to systems of nonlinear partial differential equations. Recently
he has been focusing on fundamental questions in fluid dynamics as part
of the $1M Clay Institute millennium challenge concerning the regularity of solutions to the equations of fluid dynamics. With J. D. Gibbon,
he co-authored the book Applied Analysis of the Navier-Stokes Equations,
published by Cambridge University Press. Doering is an editor of Physics
Letters A.
Carlton M. Caves holds the position of Distinguished Professor in physics
at the University of New Mexico. He received his BA in physics and mathematics from Rice University, 1972; and his PhD from Caltech under Kip
Thorne, 1979, in the area of gravitation. In 1979–81, he was a Research
Fellow, Caltech; in 1982-87, he was a Senior Research Fellow, Caltech; in
1987–92, he was an associate professor at the University of Southern California; and in 1992–2006, he was Professor of Physics and Astronomy,
University of New Mexico. He has received a number of honours including Fellow of the American Physical Society; National Science Foundation
(NSF) Predoctoral Fellow, 1972–75; Richard P. Feynman Fellow, Caltech,
1976–77; first Öcsi Bácsi Fellow, Caltech, 1976–77; and the Einstein Prize
for Laser Science, Society for Optical and Quantum Electronics, 1990. His
research focus is in the area of physics of information; information, entropy,
and complexity; quantum information theory; quantum chaos quantum optics; theory of nonclassical light; theory of quantum noise; and the quantum
theory of measurement.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
345
Daniel Lidar is the Director and co-founding member of the USC Center
for Quantum Information Science & Technology (CQIST). He received his
BS from the Hebrew University of Jerusalem, 1989; and obtained his PhD
under Robert Benny Gerber and Ofer Biham, 1997, in the area of disordered
surfaces also from the Hebrew University of Jerusalem. In 1997-2000, he
was a postdoc at UC Berkeley; in 2000–05, he was an assistant and then
later an associate professor of Chemistry at the University of Toronto. He
moved to Southern California in 2005 as an associate professor of Chemistry
and Electrical Engineering, with a cross-appointment in Physics. Lidar’s
research interests lie primarily in the theory and control of open quantum
systems, with a special emphasis on quantum information processing. His
past interests include scattering theory and disordered systems.
Howard E. Brandt was born in Emerado, North Dakota, and is presently
with the US Army Research Laboratory in Maryland. He received his BS
in physics from MIT as a National Sloan Scholar, 1962; he received his MS
in physics from the University of Washington, 1963; and obtained his PhD
at the University of Washington under Marshall Baker, 1970, calculating
the divergent part of the charge renormalization constant in quantum electrodynamics to sixth order in perturbation theory in Feynman gauge to
verify the gauge invariance of the calculation. In 1970-76, he was a postdoc in general relativity at the University of Maryland. In 1976, he joined
the Army Research Laboratory in 1976 (then the Harry Diamond Laboratory). From 1986 to 1995, he technically directed three major programs
for the Office of Innovative Science and Technology of the Strategic Defense Initiative Organization, involving nation-wide research on high-power
microwave source development, sensors for interactive discrimination, and
electromagnetic missiles and directed energy concepts. Brandt has received
a number of honours including the Siple Medal, Hinman Award, and Ulrich
Award. He is an elected Fellow of the US Army Research Laboratory. He
received a major achievement award from the US Army Research Laboratory for his publications and research on quantum information processing.
He also received the ARL 2004 Science Award. He is inventor of the Turbutron, a high power millimeter-wave source (US Patent 4,553,068), and
co-inventor of a quantum key receiver based on a positive operator valued
measure (US Patent 5,999,285). His broad research interests include quantum field theory, quantum computation, quantum cryptography, quantum
optics, general relativity, and non-neutral plasma physics. Most recently,
his research has concentrated on quantum information processing includ-
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Quantum Aspects of Life
ing quantum computing and quantum cryptography. Brandt’s extramural
interests include philosophy, theology, art, and classical music.
Alex Hamilton is an Associate Professor in the School of Physics at the
University of New South Wales (UNSW), and manager of the quantum measurement program in the ARC Centre of Excellence for Quantum Computer
Technology. He obtained his BSc in Physics from the University of London in 1988, and a PhD, under Michael Pepper and Michael Kelly, from
the University of Cambridge in 1993. He was awarded a highly competitive
EPSRC postdoctoral fellowship to continue his work at the Cavendish Laboratory, which led to new understandings of electrical conduction in highly
correlated low-dimensional quantum systems. Alex moved to UNSW in
1999, where his team is developing techniques for controlling and reading
out quantum information in silicon quantum computer devices. His expertise lies in the field of Experimental Condensed Matter Physics, having
worked on semiconductor nanofabrication and the study of quantum effects in nanometer scale electronic devices at ultra-low temperatures. He
has published over 50 research papers, and is Australasian editor of the
international journal Solid State Communications.
David K. Ferry is the Regents’ Professor of Electrical Engineering at
Arizona State University (ASU). He received his BSEE, 1962, and MSEE,
1963, both from Texas Technical College. Ferry obtained his PhD under
Arwin A. Dougal, 1996, from the University of Texas at Austin. Following a postdoctoral year in Vienna (1966–67) under Karl-Heinz Seeger, he
spent time at Texas Tech University (1967–73), the Office of Naval Research (1973–77), and Colorado State University (1977–83), and the joined
ASU in 1983. He has received a number of honours including the IEEE
Cledo Brunetti Award, 1999; IEEE (Phoenix) Engineer of Year, 1990; Fellow of the IEEE Fellow, 1987; and Fellow of the American Physical Society,
1974. His research involves the physics and simulation of semiconductor
devices and quantum effects and transport in mesoscopic device structures.
His books include Quantum Mechanics (Adam Hilger, Bristol, 1995), 2nd
Edition (Inst. Physics Publ., London, 2000); Quantum Transport in Ultrasmall Devices (Plenum, New York, 1995), Edited with Hal Grubin, Carlo
Jacoboni, and Antti-Pekka Jauho; Transport in Nanostructures (Cambridge
University Press, Cambridge, 1997), with Steve Goodnick; Semiconductor
Transport (Taylor and Francis, London, 2001); and Electronic Materials
and Devices (Academic Press, San Diego, 2001), with Jon Bird.
Dreams versus Reality: Plenary Debate Session on Quantum Computing
347
Julio Gea-Banacloche was born in 1957, Seville, Spain and is presently
professor of physics at the University of Arkansas. He received his BS from
Universidad Autonoma de Madrid, 1979; and obtained his PhD under Marlan O. Scully, 1985, on quantum theory of the free-electron laser, from the
University of New Mexico. In 1985–87, he served as a Research Associate,
Max Planck Institute for Quantum Optics; in 1988–90 he was a Staff Scientist, Instituto de Optica, Madrid, Spain; and in 1990 he joined the Universisty of Arkansas initially as an assistant professor. He is an Associate editor
of Physical Review A and Fellow of the American Physical Society. He has
carried out theoretical work in laser physics, quantum optics, and quantum
information. His main contribution to the field of quantum information has
been the observation that the quantum mechanical nature of the fields used
to manipulate the quantum information carriers themselves—often called
“qubits”, or “quantum bits”—might lead to unpredictable errors in the performance of the quantum logical operations. The lower bound on the size
of these errors can be made smaller by increasing the energy of the control
system. This has led Banacloche to predict a minimum energy requirement
for quantum computation, which has given rise to some controversy.
Sergey Bezrukov is a Section Chief in the Laboratory of Physical and
Structural Biology, National Institutes of Health (NIH), Bethesda, Maryland. He received his MS in Electronics and Theoretical Physics from
St. Petersburg Polytechnic University, 1973; and he obtained his PhD under Giliary Moiseevich Drabkin in Physics and Mathematics from Moscow
State University, Russia, 1981. In 1981–87, he was a Research Scientist,
St. Petersburg Nuclear Physics Institute, Laboratory of Condensed Matter
Physics; in 1987–90 a Senior Research Scientist, St. Petersburg Nuclear
Physics Institute, Laboratory of Condensed Matter Physics; in 1990–92
he was a Visiting Research Associate, University of Maryland and Special
Volunteer, National Institutes of Health, LBM, NIDDK; in 1992–98 he was
Visiting Scientist, National Institutes of Health, LSB, DCRT and LPSB,
NICHD; in 1998-02 he was an Investigator, Head of Unit, National Institutes of Health, LPSB, NICHD; and he took up his present position in
2002. Bezrukov was elected Member of Executive Council of the Division
of Biological Physics of the American Physical Society in 2002. One of his
key research interests is in the physics of ion channels.
Laszlo B. Kish is a professor of Electrical Engineering at Texas A&M
University. He received his MS in physics from Attila József University
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Quantum Aspects of Life
(JATE), Hungary, 1980; and a PhD in Solid-State Physics, at JATE in
1984. He had no official PhD advisor, though his mentors were Laszlo Vize
and Miklos Torok. He received a Docent in Solid State Physics (habilitation) from Uppsala University, Sweden in 1994. He received a Doctor of
Science (Physics), from the Hungarian Academy of Science in 2001. He won
the 2001 Benzelius Prize of the Royal Society of Science of Sweden for his
activities in chemical sensing. He is the foundation Editor-in-Chief of Fluctuation and Noise Letters and serves on the Editorial Board of the Journal
of Nanoscience and Nanotechnology. He holds 8 patents and over 200 international publications. He coauthored the innovative HTML document
available for download from Amazon, The Dancer and the Piper: Resolving
Problems with Government Research Contracting. His research interests lie
in the physics of noise and fluctuations. Professor Kish is co-founder of the
international conference series Fluctuations and Noise (with D. Abbott).
Chapter 16
Plenary Debate: Quantum Effects in
Biology: Trivial or Not?
6:00pm, Friday, May 28th, 2004, Costa Meloneras Hotel, Canary Islands, Spain Second International Symposium on Fluctuations and Noise
(FaN’04).
The Panel - Dramatis Personae
Chair/Moderator: Julio Gea-Banacloche, Univ. of Arkansas (USA); Editor of Physical Review A.
No Team (assertion “not trivial”):
Paul C. W. Davies, Macquarie University (Australia);
Stuart Hameroff, Univ. of Arizona (USA);
Anton Zeilinger, Univ. of Vienna (Austria);
Derek Abbott, Univ. of Adelaide (Australia); Editor of Fluctuation Noise
Lett. and Smart Structures and Materials.
Yes Team (assertion “trivial”):
Jens Eisert, Imperial College (UK) and Potsdam University (Germany);
Howard Wiseman, Griffith University (Australia);
Sergey Bezrukov, National Institutes of Health (USA); Editor of Fluctuation Noise Lett.;
Hans Frauenfelder, Los Alamos National Laboratories (USA).
Transcript Editor :
Derek Abbott, The University of Adelaide (Australia);
349
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Quantum Aspects of Life
Disclaimer: The views expressed by all the participants were for the purpose of lively debate and do not necessarily express their actual views.
Transcript conventions: Square brackets [...] containing a short phrase
indicate that these words were not actually spoken, but were editorial insertions for clarity. Square brackets [...] containing a long section indicate that
the recording was unclear and the speaker has tried to faithfully reconstruct
what was actually said. [sic] indicates the transcript has been faithful to
what was spoken, even though grammatically incorrect. Where acoustic
emphasis was deemed to occur in the recording, the transcript reflects this
with italics.
The Debate
Julio Gea-Banacloche (Chair): [I am pleased to introduce] this special debate, Quantum effects in biology: trivial or not? I am Julio GeaBanacloche, originally from here: a native of Spain. As Laszlo Kish pointed
out yesterday at the banquet that there was an intense pre-debate debate
concerning what the debate should be about—and such topics as whether
the title of the debate should phrased as a “yes” or “no” question or not.
That’s something some people were very adamant about and in some sense
it is kind of a miracle that we are having this debate at all. I am really
glad about it. So the question is: “Quantum effects in biology: trivial or
not?” and on the Yes Team we have Professor Paul Davies from Macquarie
University...
Stuart Hameroff (No Team): [Interrupts] We’re the non-trivial team...
we’re the No Team!
Audience laughter
Julio Gea-Banacloche (Chair): See: sorry, I got it wrong already. With
apologies to the teams, so, actually the No Team are the good guys, eh...
Audience laughter
Julio Gea-Banacloche (Chair): We must not get this bit confused. The
No Team claim that quantum effects in biology are not trivial and we have
Professor Paul Davies from Macquarie University, and Professor Stuart
Hameroff from the University of Arizona, and Professor Anton Zeilinger,
who is currently in a superposition of states and thus invisible1 ...
1 Transcript editor’s note: At the beginning of the debate Zeilinger’s chair was vacant—
but he did turn up later.
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
351
Audience laughter
Julio Gea-Banacloche (Chair): And masquerading as Jack Tuszynski
is Professor Derek Abbott,2 our organizer from The University of Adelaide.
On the Yes Team we have Professor Jens Eisert, from the University of
Potsdam; Professor Howard Wiseman, from Griffith University; Dr Sergey
Bezrukov, from the National Institutes of Health; and Dr Hans Frauenfelder from Los Alamos National Labs. There are a couple of things about
the way I see this today, and of course the audience—and they represent
themselves—are welcome to think any way they want to. But way, way
back in the early, early stages of the pre-debate debate, somebody had suggested the title of Quantum mechanical effects on the mind: important or
not? I actually thought that was a very exciting question, not having any
idea about what the possible answers might have been and what anybody
might have to say for or against it. So at the time, when I was asked
whether I would chair the debate, I thought that I would ask the panelists
about their quantum mechanical states of mind and, if so, what drugs we
need to take in order to experience them...
Audience laughter
Julio Gea-Banacloche (Chair): I would still like somebody to address
that, except for the drugs part—which of course is just a joke. The other
thing—that I am probably responsible for—is the final wording of the question: putting the word “trivial” in there. What I am personally expecting
to get from this [is for] the No Team to try to provide some surprising
facts—things that we would not have expected. In a sense, of course, we
believe the entire world is described by quantum mechanics, so you can
always say quantum mechanics is responsible for this or that. So what?
Of course it would be responsible, because ultimately everything can be
described by quantum mechanics. What I would like to see [are things that
make me say] “I would not have expected that. Wow, this is unexpected.
That is a surprise.” That’s the sort of thing that I would like to see come
out of this. That is the sort of thing that would lead me to vote, if I were
to vote, for the No Team as opposed to the Yes Team, and this is perhaps a
suggestion to the audience that maybe you should look for the wow-factor
at the end of the debate when I will ask for an informal vote on whose side
was the most persuasive. If you have been wowed by something that the
No Team said that the Yes Team did not manage to make sound totally
mundane, and so forth, then [consider this in your voting decision].
2 Transcript editor’s note: Jack Tuszynski was invited to be on the panel but unfortunately called in sick, so Derek Abbott substituted him at the last minute.
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Quantum Aspects of Life
So, the format that we are going to adopt is the following. First, we
will start with the No Team presenting their case, each panelist will speak
in turn only for three minutes, and then we move to the Yes Team, again
three minutes each. Then [in order to] reply we return again to the same
order—limited to two minutes for panelists, and then at the end of that
I would like to go one more round to give each one of the panelists a
chance to summarize his position in two sentences. After that we will take
questions from the audience for the panelists and then I will ask for an
informal vote. I hope the panelists summarize their points from their talks
yesterday. I apologize to the panelists for having missed the presentations
of Wednesday, because I haven’t mastered the quantum mechanical art of
being in two places at the same time. Unfortunately, I just couldn’t. I hope
they will be kind enough to repeat the things that they might have said in
their talks that might be relevant to [today’s] subject as I would very much
like to learn myself. So without further ado, let’s start with the No Team:
Professor Davies.
Paul Davies (No Team): Thank you. Well I’m sorry to disappoint you
but I was particularly not going to repeat the content of my lecture on
Wednesday morning, where I set out what I felt were the most persuasive
arguments on the grounds of science for the non-trivial effects in biology.
I thought I would restrict my remarks, this evening, to addressing a more
philosophical point as befits a professor of natural philosophy. The problem
of the origin of life remains one of the great unsolved problems of science,
and that in itself is a highly non-trivial problem. The simplest living thing
is already so immensely complex—[when you consider] the first living thing
that arose just from the random shuffling of building blocks of molecules—
that it is quite clear that the odds against that happening are so stupendous
that it would have happened only once in the observable universe. It will in
fact be a near miracle. The alternative is that life is a natural and more or
less automatic part of the outworkings of the laws of nature; in which case
there must be some sort of life-principle, or what the Christian de Duve calls
a “cosmic imperative.” And if there is such a principle then this principle
must exist within physical theory. It must be a part of physics and, though
we haven’t deduced this principle yet, we imagine that it is something that
is incorporated within physics; and then the question arises: Does that life
principle come from quantum physics or is it part of classical physics? I can
give three reasons at least why I think it will be part of quantum physics.
The first is that quantum mechanics is, of course, a fundamental theory;
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
353
the idea that you can choose the world as classical or quantum mechanical
is nonsense. The world is quantum mechanical. We live in a quantum
mechanical universe, so quantum mechanics is the default position. If we
are looking for a new principle in physics—by default it belongs in quantum
mechanics, or else quantum mechanics is not the correct description of the
world.
The second point is that life, of course, proceeds from the quantum
world. Life is ultimately molecular and it must have begun in the molecular domain. And so life came out of the quantum domain, and to insist that
the quantum domain somehow had to stop or that one had to move beyond
it before life came to exist, seems to me to be completely unreasonable.
The third point of view—I’ll mention it very briefly, famously introduced
by Eugene Wigner—is that the connection between life and quantum mechanics is there all along in the role of the observer. Of course, we are
all persuaded that wave functions don’t collapse [due to conscious agents],
[rather] they decohere by the environment—but, as Anton Zeilinger said
so clearly in his lecture,3 we’re still left with the issue that quantum mechanics is incomplete inasmuch as it gives a probabilistic description of the
world and the actual outcome of any given observation clearly depends on
the observer. So, I think that the other team had better state, first of all,
whether they think life is a miracle—and if it’s not a miracle, why is it that
the life-principle, which must be there, is disconnected from fundamental
quantum physics? What sort of principle emerges only when wave functions
decohere? That’s what I’d like them to answer. Thank you.
Gea-Banacloche (Chair): I will try to stand up and signal that it is 15
seconds before the end.
Stuart Hameroff (No Team): Thank you. Real-time activities of living systems are all performed by conformational fluctuations of proteins,
changing shape on the order of nanoseconds. How do proteins control their
shape? Within proteins high energy charge interactions tend to cancel
out. So functional dynamics are governed by collective actions of numerous
quantum mechanical van der Waals-London forces, influenced by external
factors. London forces are instantaneous dipole couplings of electron clouds
in non-polar amino-acids, which form non-polar hydrophobic pockets within
proteins. These tiny pockets—where quantum forces rule—are essentially
the “brain” of each protein. Thus proteins—at least certain proteins—are
3 Transcript
editor’s note: Jennewein, T. and Zeilinger, A. (2004), Quantum noise and
quantum comunication, Proc. SPIE Fluctuations and Noise in Photonics and Quantum
Optics II, 5468, pp. 1–9.
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Quantum Aspects of Life
quantum switches, which should exist for some time in a superposition
of conformational states. Protein qubits: why would nature need qubits?
Quantum computing is the answer, I believe. Geometric lattice assemblies
of protein qubits, for example, microtubules, are well-suited to act as quantum computers. The evolution of organisms utilizing quantum computation
would have enormous advantages.
Is there evidence of quantum computation in biology? If any of us
needed surgery, say our appendix ruptured, we would be anaesthetized,
rendered non-conscious by breathing a gas mixture containing roughly one
per cent of an anaesthetic gas molecule, which is inert. These molecules
float from our lungs into our blood and brains selectively and act at the
same hydrophobic pockets, which govern the dynamics of certain proteins in
our brain. They form no chemical bonds or ionic interactions, acting merely
by the same quantum mechanical van der Waals-London forces, occurring
naturally in these proteins, thereby preventing the normally occurring London forces, inhibiting protein quantum switching in the brain and erasing
consciousness. [Other brain activities continue.] The surgeon takes his
knife and we feel nothing. So consciousness—the most elegant biological
process—utilizes quantum processes quite possibly quantum computation.
Other functions like immune cell recognition, cell division, etc. may also
utilize the unitary oneness of quantum coherence and entanglement.
Ah! The decoherence! Biology is warm and wet—it’s very warm.
Yes, but bio-systems may utilize laser-like coherent phonon-pumping from
a heat-bath of biochemical metabolism. Several months ago a paper in
Science by Ouyang and Awschalom4 showed quantum spin transfer between quantum dots connected by benzene rings—the same type of [electron cloud] ring found in hydrophobic pockets. The quantum spin transfer
was more efficient at higher temperature. Benzene rings are identical to the
electron resonance rings of the amino-acids found in hydrophobic pockets
in proteins.
Wet—yes, we are mostly water, but cell interiors exist much of the time
in a gel state in which all water is ordered, coupled to protein dynamics.
Additionally structures like microtubules can use Debye layer screening
and topological quantum error correction—that is, utilizing the AharanovBohm effect as suggested for quantum computers by Kitaev. In conclusion,
whether quantum processes evolved as an adaptation of biology, or whether
biology and consciousness [evolved as] classical adaptations of pre-existing
4 Transcript editor’s note: Ouyang, M. and Awschalom, D.D. (2003). Coherent spin
transfer between molecularly bridged quantum dots, Science, 301, pp. 1074–1078.
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
355
quantum information, they [may be] like artificial neural networks, [which
were copied after] brain systems in the 80s. Quantum information technology can learn a great deal from the study of certain biomolecules.
Julio Gea-Banacloche (Chair): Thank you, and that was a little long,
but I don’t mind perhaps giving this team a little more time since they are
slightly handicapped here...
Audience laughter
Julio Gea-Banacloche (Chair): ...by the absence of one person [Anton
Zeilinger]. Thank you for addressing the drugs question too.
Audience laughter
Julio Gea-Banacloche (Chair): Moving on...Derek.
Derek Abbott (No Team): OK, the position I’m going to take in this
debate is one of cautious optimism and my question I’m going to put to the
Con Team is: Why not explore the relationship between quantum mechanics and biology? Why not have a go? It might be fruitful because, when
you think about it, nature has been around for 3.5 billion years and it’s
produced some marvelous, fascinating things that we don’t understand. It’s
often said that nature is the world’s best engineering text book. Anything
man sets out to go and do technologically—if he looks at nature first—he
will always find examples beforehand carried out by nature [often] even
better than we can do. So my challenge to the Con Team is to think of
a counterexample and actually tell me something that Man has invented
that nature hasn’t. So, if this is true, then it’s hard to believe that nature
hasn’t [already] made clever use of quantum effects.
Now, there are two people that have famously made a counter-example
to [the standard] argument that I’ve made, and one is John Maynard Smith,
and the other is Charles Bennett. And I’m going to take on these guys
one by one now. So, Maynard Smith says that nature has never used the
wheel—man has invented the wheel, but nature hasn’t. Now of course he’s
wrong because bacteria have flagella, which have true 360 degree axes of
rotation! Maynard Smith visited [my lab at] The University of Adelaide
3-4 years ago and actually admitted this oversight. Also, Charles Bennett
famously commented that nature has never made explosives and has never
made the radio, but this of course is wrong as well. For example, there is the
bombardier beetle that excretes a gas that explodes when it hits the air, to
keep off its predators. Also, nature invented the radio as well. The example
is fireflies; when they’re [about to] mate they modulate an electromagnetic
signal—in other words they flash on and off—and the female receives that
message. And so that’s a radio—it’s [albeit] a slightly different wavelength.
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Quantum Aspects of Life
Audience laughter
So, anything that you can think of biology has already thought of it. So, if
these team members, over there, really believe in quantum computation—
and I assume they do because they’ve actually made quantum computers to
a few qubits, so they must believe in it—then nature must have “thought”
of it or used it in some way first.
Just finally, to finish off, I want to remind the Con Team that Carlton Caves—their quantum information guru—is on record as saying that
because nature is so clever, there possibly is a very small chance that nature, through natural selection, may have found some use for quantum
decoherence-free subspaces.5 And so I would just like to leave it at that
point.
Julio Gea-Banacloche (Chair): Thank you, Derek. I just want to remind the No Team that this team is actually a Yes Team, not the Con
Team. I realize that is confusing but...
Audience laughter
Anyway, thank you very much to all of you and now we are going to hear
what the Yes Team has to say ...
Jens Eisert (Yes Team): Well then, I suppose it’s up to me to open the
case for the Yes-No Team...
Audience laughter
Jens Eisert (Yes Team): ...well, unless someone expects me to volunteer to fill in for Anton Zeilinger on the other side. Well, let me start
with some numbers and science, rather than the philosophical issues raised
before. It seems that in the whole debate the “make-or-break” issue is
whether coherence can be preserved over the timescales relevant for nontrivial biological processes. And of course, one has to say what one means
by non-trivial here. Needless to say, everything in biology is built via quantum mechanics—bonds, hydrogen tunnelling, and so on. But, however,
if one regards living entities as information processing devices, one could
make a definition of “non-trivial” that this kind of information processing
is in some form quantum information processing. There I would doubt
whether coherence can be preserved over timescales that would be relevant
to have efficient quantum computation—in the sense that it is discussed in
[the field of] quantum information. In a nice paper by Tegmark6 —where
5 Transcript
editor’s note: see Chapter 15, p. 384.
editor’s note: See Tegmark, M. (2000). The importance of quantum decoherence in brain processes, Phys. Rev. E 61, pp. 4194–4206.
6 Transcript
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
357
he looks at the decoherence timescale in a simple model of neurons firing
or not—for a neuron to fire or not means that 106 sodium atoms are on
one side of the brain or the other side of the brain. If one just looks at
the neighbouring sodium atoms then the decoherence timescale just from
simple high-temperature Brownian motion gives a lower bound of 10−20
seconds, which is very, very short and even if one [considers] very low levels for decoherence timescales in microtubules inside the cytoskeleton, then
the most conservative bounded scheme is 10−30 seconds for a typical decoherence timescale. So, I mean in the light of this, it seems fairly unlikely
that something non-trivial as quantum information processing [appears in]
biological systems.
Let me remark on Paul Davies’ [point] directly. What he would suggest
is that [a quantum process in] life is some sort of fast-track to the early
emergence of life. There are two points [I can make]. If one thinks of
searching—well, if it’s a quantum search there must be some sort of oracle
that says, “OK, you have life, or you do not have life,” so—the point is also
that everything must be coherent all the time. One thing is, what must
be coherent over all generations so that we have the superpositions of us,
our kids and our kids’ kids, and so on, and they do not cohere in order
to have a proper fast-track to life—and then, my final point: there’s the
teleological point that, hey, we search for something—but first point—one
has to specify is what one looks for when we’re searching. So, it doesn’t
seem entirely clear what the search is for life, because the pay-off is not
clear. It is not clear what one in fact is looking for.
Julio Gea-Banacloche (Chair): Thanks, Jens. Howard.
Howard Wiseman (Yes Team): Thanks Julio. Okay, so to begin I would
just like to say that the Yes Team in this debate obviously wins by default.
The topic is debating “Quantum effects in biology: trivial or not?”—well,
clearly the answer is yes, they are trivial or not ...
Audience laughter
Howard Wiseman (Yes Team): ... No, seriously, I think the point
is that, yes, you do need to look at the title of this debate carefully and
the keyword in my mind is “trivial.” I’m glad that people see that people
are taking this in a good spirit, and indeed claiming things to be nontrivial that I would agree are non-trivial. There are a number of people
that have said that obviously quantum mechanics is behind everything,
but Julio said that what he is really looking for is for the No Team to
come up with something surprising. I guess I think a word or a note of
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Quantum Aspects of Life
warning is necessary here: that what physicists like Julio and many of us
here are surprised or interested by here are notoriously obscure: the fact
that we’re all here at a conference called Fluctuations and Noise would be
very surprising to most members of the public.
Audience laughter
Howard Wiseman (Yes Team): So, I think a better definition, of what
is non-trivial, is not what will surprise and interest physicists but, rather,
something that would convince a biologist that they need to learn quantum
mechanics. A non-trivial quantum effect in biology is something that will
make a biologist want to go out and, you know, take a second year quantum
mechanics course and learn about Hilbert space and operators, so that they
understand what’s going on. That’s my concept. Now, again, we all seem
to be in agreement that an obvious example of this is quantum information
processing: that this is somehow the heart of what we’re getting at here,
because what most people involved in biology are interested in these days
is information processing in some form—in control, in genetics, in cells,
in thoughts, in consciousness, and all of that is information processing.
So something non-trivial would be whether there’s quantum information
processing in there. So, needless to say, I don’t believe there is.
To respond to some of the points of the No Team so far, Paul Davies said
he wants the No Team...er...I mean...Yes Team to respond to his challenge
of explaining where the life-principle comes from—that is supposed to be
a new principle of physics. So, I have two responses to that: (a) I don’t
believe there is one, and (b) I think that it’s outside this debate, so it
shouldn’t be talked about at all. Firstly, I don’t believe it because I think
it comes from the fallacy that “I don’t understand A, I don’t understand
B. Therefore, A and B must be the same thing.” And secondly, I think
it’s outside the realms of debate. This debate is about quantum effects,
not about effects that result from some new theory—that we don’t even
understand yet—but which has something to do with quantum mechanics.
I think this remark is also directed at Stuart Hameroff’s ideas that involve
quantum gravity and things that don’t even exist yet in theory. I guess I’ll
leave it at that.
Anton Zeilinger enters the room
Julio Gea-Banacloche (Chair): Right, I’ll take a break to welcome
Anton Zeilinger. I’ll probably let him have some four or five minutes in the
next round. But now, to keep going with the flow, the third member of the
Yes Team: Sergey.
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
359
Sergey Bezrukov (Yes Team): Thank you. I think that everybody
is too serious for this late hour, so I decided, instead of sticking to my
introduction and answering questions, to tell you an anecdote. This anecdote is a real true story, which happened about twenty five years ago at the
NIH—National Institutes of Health—campus in Bethesda. The story is that
one young post-doc—actually a candidate who looked for a post-doctoral
position and who is now a well-established professor at the University of
Maryland and, by the way, came to be a very famous scientist (I would tell
you all the names if I did not know that Derek is going to publish all this).
Audience laughter
Sergey Bezrukov (Yes Team): Anyway, [this young man] came to the
famous scientist (anyone of you, who ever did anything in rate theory knows
his name for sure) [looking for a job]. And he was told “Okay, I like what
you’re saying, I like you, so I think am offering you the job, but please
remember that in Building 5 is zero”.
Audience laughter
Sergey Bezrukov (Yes Team): And all I have to add here is that after
25 years, in Building 5 (where most of the physicists at the NIH sit) is still
zero. Now, I agree completely with what Paul just said. Of course, life is a
non-miracle, it follows from physical laws, but if quantum mechanics plays
a decisive role in the explanation of the life phenomenon I’m not sure about
this; [I say “not sure” rather than “no”] because it is very dangerous to say
“no” when talking about science. As for Stuart’s short speech, I can say
all those fluctuations that he was talking about are pretty well understood
from the point of view of classical physics; there is no problem in all of
this. In his talk [at this conference] Prof Frauenfelder [was discussing the]
timescales of these fluctuations and the timescales are understood [in the
framework of classical physics]. Thank you.
Julio Gea-Banacloche (Chair): Thank you.
Hans Frauenfelder (Yes Team): I think as an experimentalist I would
only start looking for non-trivial quantum effects if I find something that
I cannot explain in any other way—but since we have started in telling
stories, I have a story about what is trivial. Frank Yang, the Nobel Prize
winner, gave a talk in Seattle many years ago and he said, when discussing
a particular point, “It is trivial.” He was challenged by another physicist,
Boris Jacobson who asked, “Is it really trivial?” After the talk they went
back to the office of Boris, worked for two days and they came out agreeing
that it was trivial.
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Quantum Aspects of Life
Audience laughter
Hans Frauenfelder (Yes Team): Some of Paul’s statements sound to
me like religion and it’s very hard to discuss religion. Two statements
about the fluctuations—I probably know as much about fluctuations as
most people—and I’ve never yet seen something that looks like a miracle.
The fluctuations that we observe are explainable using standard physics.
At the moment, I see no reason to invoke something that’s supernatural or
quantum mechanical in the understanding of biology.
Julio Gea-Banacloche (Chair): Thank you and that finishes the first
round, so everybody has now two minutes to reply, except for Professor
Zeilinger who can have 5 minutes. Thank you.
Paul Davies (No Team): Let me just respond to a few of these points.
First of all, is my position religious? I would say that it is exactly the
opposite—[it is the person] who says that life originates because of some
stupendously improbable set of events [who] is effectively appealing to religion by calling [life’s origin] a miracle. There’s no real difference between
a miracle and an event that is so improbable that it’s going to happen
only once in the universe. So I think by saying that there are principles in
physics, which encourage matter to organize itself into life, is the scientific
position. We don’t know [this hypothetical principle] yet, but to [claim]
perversely that this principle does not belong within quantum mechanics
seems to me a very peculiar position. As far as the teleological aspects are
concerned, I think that’s very easily dealt with because the physicist will
define life as a system that would replicate, which is a well-defined physical
criterion. The pay-off for the system is that it gets to replicate—so that
takes care of that point. And I’ve probably used up my two minutes—well,
I think I’ve got nothing else to say!
Audience laughter
Paul Davies (No Team): ... because I’ll defer to Stuart on the question
of timescales for decoherence.
Stuart Hameroff (No Team): First of all, Jens, I’m glad you read
Tegmark’s paper; unfortunately you didn’t read our reply to Tegmark.
Audience laughter
Howard Wiseman (Yes Team): [You mean in] Phys. Rev. E, the same
journal as your paper?
Stuart Hameroff (No Team): Yes, Phys. Rev. E, the same journal
that he published in. Tegmark successfully disproved his own theory about
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361
microtubules [implying] that he was disproving ours. However, his decoherence time of 10−13 [seconds] was based on a superposition separation
distance of 24 nm, whereas our [superposition separation distance] was
a Fermi length—so that’s seven orders [of magnitude lengthening of the
decoherence time] right there. He also neglected things like dielectric, [permittivity], Debye layers, and [decoherence free sub-spaces] and some other
things. We corrected for those, we used his same formula, and we got [the
calculated decoherence time] to 10-100 milliseconds. If you add the potential effect of topological quantum error correction you get an indefinite
extension. As far as the comments about the fluctuations of the proteins
not needing any miracle, quantum effects are not supernatural. It may seem
that that protein dynamics is [straightforward and classical], however, as
I mentioned, the biochemistry text, by Voet and Voet, clearly states that
the strong forces cancel out, and the weak London forces rule on timescales
relevant to conformational fluctuations. Also, [in] Professor Frauenfelder’s
lecture in his wonderful work on myoglobin, he showed a xenon molecule,
which is an anaesthetic, exactly like [the other inert anesthetic gases we
use clinically]. [Xenon is] a perfectly good anaesthetic, we [could use for
patients] to go to sleep for surgery. It’s expensive but it works just fine.
It’s completely inert and neutral; gets into hydrophobic pockets and does
exactly what the other anaesthetics do [binding only by quantum London
forces]. Professor Frauenfelder has said that xenon prevents the dynamics
of myoglobin, so while he was looking at the other effects and attributing [protein conformational control] to external classical fluctuations rather
than internal control. [Control of myoglobin may be mediated through
quantum London forces in a hydrophobic pocket blocked by xenon]. Thank
you.
Anton Zeilinger (No Team): OK, first I would like to apologize it’s my
fault that I didn’t realize that the debate was moved down by half an hour. I
had the old schedule, but this is my fault and I apologize for this happening.
Now, let me make one or two statements. I have no idea whether quantum
states play a trivial or non-trivial role in biological systems; otherwise I
might not be sitting here. But I feel that there is no reason why they should
not. And as an experimentalist I view—I see a challenge—to prove that
quantum systems basically no matter how complex can exist in quantum
superposition. This is independent of the question of whether this plays a
role in living systems or not but my fervent approach is, that if we are able
to prove that quantum superpositions can be shown in the laboratory and
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Quantum Aspects of Life
later on for very complex systems including living systems, it might change
the way biologists view their own business, because biologists’ essential
paradigm is that we are essentially classical machines.
Now I am saying—in including living systems—I do not see any reason
whatsoever why we should not be able to see a quantum superposition of
bacteria, for example some day. I know that all the papers which talk about,
you know, decoherence, about coupling linking systems with environment,
and why this would not allow quantum interference—I read such papers
as instructions about what not to do and how to avoid decoherence. For
example, this specific thing, which I am sure will be done in the future—I
don’t know when—is that you take a small living system, say a bacterium
provided with other technology with its own living shell around it, so that
the whole system does not couple to the environment, and then I’m sure
you can put the whole thing through a double slit or whatever. There
is no reason not to have quantum superpositions of living systems. As
I said this might change the viewpoint of people. It might really lead to
something new. The question is certainly interesting and whether [quantum
effects are] trivial or non-trivial in living systems and what is called for is
something like Bell’s theorem. It is also under-appreciated, how gigantic
the achievement of John Bell was. Namely that he was able to give a general
set of quantified criteria, which tells you whether in a given situation, you
can explain what you observe, by a local, realistic viewpoint. We need to
see exactly the same kind of thing for a living system. So, if somebody
would be able to provide criteria which tell you if a certain condition is
met, we know that quantum phenomena play no role in living systems that
would be an equally important achievement as the opposite [viewpoint]. It
might be possible, it might not be possible, I have no idea whatsoever...
Julio Gea-Banacloche (Chair): Not much time left for you.
Anton Zeilinger (No Team): Alright I should give the rest of the time to
others, but I just would like to point out one more thing. Suppose we have
some dynamics, which release two coherent superpositions in living systems.
No matter for how short a time, then my claim is that decoherence does
not [matter]—because decoherence gets rid of non-diagonal terms in the
density matrix, but it does not explain to you why a specific event the
living system happens and we are dealing with living systems in relation
to each other. Finally I want to share with you something I learned from
yesterday, and in the quick discussions we had about today’s debate. It
would be extremely non-trivial if quantum mechanics did not play a role
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363
in living systems, it would be the only area in which we know quantum
mechanics is not at work. Thank you very much.
Derek Abbott (No Team): Just in rebuttal to the Pro Team...I get
confused...so I’ll call them the Fundamentalist Team. So I find it surprising
that they take their particular position. I’d like to remind Jens Eisert of
one of his own papers, where he motivated the idea of quantum games
as perhaps helping to explain how nature might play quantum games at
the molecular level.7 So this leads to an interesting point that perhaps
quantum mechanics might—if it doesn’t [already play a non-trivial] part
at the biological level—then what about the [evolutionary] pre-biotic level?
So that is another thing I’d like to throw out to the Con Team... er...
Fundamentalist Team. And I’d also like to remind them that they haven’t
yet found a counterexample to my little conundrum I made the first time.
Thank you.
Julio Gea-Banacloche (Chair): That was short. Alright, Jens you may
as well address your guilty past.
Audience laughter
Jens Eisert (Yes Team): I don’t want to: that’s my game. OK, we’d
better start with Anton—there seem to be three things that need to be
addressed and need to be done, in order to have a meaningful debate. And
that’s, in my point of view: experiments, experiments, and experiments.
What I would like to see, for example, is [based on] some of the arguments
that we were discussing are in principle explainable. So I would like to
see, say, an experiment where you prepare entangled photons, let them be
absorbed in a photosynthesis reaction, the energy is transferred to internal
degrees of freedom, and then you have a Bell inequality violated. The point
is that, some of the statements about coherence are testable in principle. It
only becomes difficult if one thinks of processes that are underlying other
processes that are so much larger than the things you’re talking about. To
Stuart, and if you look for, say, the effects of gravitational objective state
reduction, it looks beautiful to do experiments to confirm or to falsify the
hypothesis made in this context, but I mean, if the effect you’re looking for
is orders of magnitudes smaller than the environment induced decoherence
that you have, then I would rather like to see an experiment that first deals
with the numbers that are there and are accessible by experiment. And
7 Transcript editor’s note: See Eisert, J., and Wilkens, M., and Lewenstein M. (1999).
Quantum games and quantum strategies, Phys Rev Lett. 83, pp. 3077–3080.
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Quantum Aspects of Life
then, what about timescales? I liked Howard’s comment on the A and B.
One needs to be careful of the fallacy that, well, if A happens on timescale
something and B happens on timescale something, therefore A must be
B. You have decoherence processes happening on, say, one second—which
happens to be decoherent—and our thought processes happening on the
level of one, then we should not be really tempted to say it is the thought
process.
Three, concerns the search for life, I mean, if one is really searching
for life and the goal is to have a successful species in the end of maybe if
you want to, you can have a quantum strategy or whatever. I mean, this
concludes that there are so many rounds going on with the generations, this
has to be coherent. I find it hard to imagine that the power to replicate or
to have a powerful species again can play any role in the search for life on
the quantum level. My time is up.
Howard Wiseman (Yes Team): So yeah, in terms of counter-examples
to Derek’s claim, it seems to me a ludicrous suggestion that nature has
developed every technology that you can think of. And yes, I just thought
of four things here, which to my knowledge, Nature hasn’t done: an internal
combustion engine, a television, a refrigerator, an inferometric measurement
of gravity.
Audience laughter and applause
Howard Wiseman (Yes Team): So, to move on—oh, Anton—Anton
said that in his experiments that there is no reason that quantum effects
couldn’t play a role in biology. I would say if that’s the case, Anton, why
don’t you do your interference experiments in a saline solution? Warm,
wet... my point was, why do you do yours in a high vacuum rather than in
a warm, wet environment? OK, so to go back again to what Julio began
with by asking: he not only asks to be surprised, he asks for the No Team
to provide surprising facts. I think that’s one thing that’s been lacking.
You have surprising speculations ...
Audience laughter
I’d like to make a comment specifically for Stuart: the word “coherent”
and the word “quantum” are not synonyms. So you can have coherence
without having quantum effects. I think that’s a very important point. OK,
in terms of whether there’s quantum computation going on in the brain, I
think it’s an enormous leap of faith to believe something like that. To begin with, in Stuart’s scheme involving tubulin proteins, as far as I know—I
would like to know if I’m wrong—there’s no evidence that tubulin even
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
365
does classical computations, still less that it does quantum computations,
and even less than it does quantum computation that involves general relativistic effects. So I think this theory is so far ahead of the experiments
that it is fairly pointless.
OK, another point about why we are skeptical about quantum computation happening in the brain is [the following]. An important point about
quantum computation is that it is useless unless you have a real large-scale
computer—because quantum computers are very hard to build and they
tend to be slower than classical computers. So you only get an advantage
out of them if you can actually get to a very large scale, and then finally
quantum computers are going to be faster than a classical computer. So
how a quantum computer could ever have evolved, as it’s completely useless
until it’s extremely large? And my time’s up.
Sergey Bezrukov (Yes Team): I would like to comment on Prof
Zeilinger’s last statement where he proposed that “it would be very surprising if quantum mechanics has nothing to do with biology.”
Anton Zeilinger (No Team) [Interjecting]: I did not say “surprising,” I
said “non-trivial.”
Sergey Bezrukov (Yes Team): Okay, I meant “non-trivial.” Anyway,
quantum mechanics has actually a lot to do with biology and it was a problem for me [to decide] on which side of the debate to join. [This is] because
my only point is that quantum mechanics as of this moment hasn’t provided
any qualitatively new insights into the mechanisms [studied by] biological
physics. At the same time, it has great importance in the quantification of
many phenomena at the elementary level. And the best example to discuss
is probably van der Waal’s forces, because to explain van der Waal’s forces
one doesn’t need any quantum mechanics. If you know the history of the
subject, the first explanation, the first qualitative explanation of van der
Waal’s forces, belongs to [the Russian physicist] Lebedev [who worked it
out] more than hundred years ago. There was not any quantum mechanics
at that time. However, to calculate with reasonable accuracy the constants
of interaction, one has to have the quantum mechanical input. So what I’m
saying is that up to now—and I don’t know what is going to happen in the
future—but up to now, “ is zero” for the reason that quantum mechanics
did not supply us with any qualitatively new insights about the molecular mechanisms in biology. At the same time [quantum mechanics] is of
greatest importance for quantifying [the] dynamic interactions, including
interactions [involved] in protein folding [and functioning].
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Quantum Aspects of Life
Hans Frauenfelder (Yes Team): First of all I have to remind the audience that the roles we play in here have been pre-assigned, so we cannot
be accused of being a believer or non-believer.
Audience laughter
Hans Frauenfelder (Yes Team): The second point is quantum mechanics is absolutely essential for biology. There is no question. For instance
electron transfer occurs in quantum mechanical tunneling—it is essential.
I think that the question which we’re really discussing hasn’t been clearly
stated is: “If we find a general law that determines or explains life, is it
quantum mechanical or classical?” Here I have no opinion and I will wait
till it is discovered.
Julio Gea-Banacloche (Chair): Thanks to all the panelists. I think
that they have raised lots of interesting points, on both sides, and I’m sure
that the audience will have many questions. Yes, we’ll have a two sentence
summary and so, Paul.
Paul Davies (No Team): I just want to respond to Jen’s point about
remaining coherent, generation after generation. That’s certainly not what
I had in mind. Just getting the first replicator seems to me is very hard.
Quantum mechanics would potentially be very useful for [acting over fractions of a second to avoid] hanging in there for thousands of millions of
years. To sum up, I would say that quantum mechanics is the default [option] and until somebody persuades me that it’s not [part of] the origin of
life, I think that’s the only reasonable thing to assume.
Stuart Hameroff (No Team): OK, well, in the first of two very long
sentences I want to respond to the chairman’s point about (a) drugs, and
(b) consciousness, which were brought into the discussion. In the ’70s it
was shown that the potency of the psychedelic drugs was proportional to
their ability to donate electron resonance energy to their receptors, suggesting that receptors are more prone to go in the quantum state, suggesting
further that the psychedelic experience might use quantum information,
and I might add that psychoactive neurotransmitters like serotonin and
dopamine also have electron resonance effects. The second sentence is that
aspects of the subconscious including [dreams] are increasingly being expressed in terms of quantum information, that the subconscious mind is to
consciousness what the quantum world is to the classical world.
Anton Zeilinger (No Team): First, one little remark on Howard’s remark on my remark: “Why don’t we do our experiments in saline solution?”
And I guess I pointed out that someday we hope we can do these exper-
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
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iments, in quantum superposition, with living things including the saline
solution. So it might simply be that the picture is a little bit too narrow.
If we dissect a living system to the centre of the saline solution to the
brain and it is clear that information goes back and forth to the between
all kinds of components in there, and may be we should do that and we
should include those kind of considerations somehow. I know there are all
kinds of reasons why this is stupid, but there might be a reason—it may
work some time. The second point, the second sentence, I would like to
mention—I think Howard’s argument against some of these that they are
far ahead of experiments that some scientists have been discouraged about
what us experimentalists can do. I mean I saw it happen in the field of
quantum information. People simply didn’t think about ideas or writing
them down, because they thought the experiment was not possible or the
experiment too far ahead. You never know how fast we can move and in
which direction—this might be true in this area too, I hope.
Derek Abbott (No Team): Howard’s8 refrigerator: the principles are
there and animals cool themselves down by sweating, by evaporation. The
internal combustion engine: Howard’s stomach when he eats and consumes
food—the principles are there. Television: the principles of encoding information and transmitting it through electromagnetic radiation are there—
that’s the firefly example. As for detection of gravity waves: wrong scale
in biology. That one doesn’t count.
Audience laughter
Derek Abbott (No Team): I would like to make an observation now,
in my summary. I would like to make an observation that, you know, our
team has been pro-active whereas the other team has been reactive. They
haven’t come up with any interesting stuff.
Audience laughter
Derek Abbott (No Team): ... we’re the interesting team!
Audience laughter
Julio Gea-Banacloche (Chair): Your two sentences are over.
Derek Abbott (No Team): The question of quantum mechanics in biology: it’s a fascinating question—so in summary, I want to say: let’s stop
being armchair scientists, let’s apply the scientific method, let’s go out,
and let’s do some experiments in the spirit of what Anton is saying. We
8 Transcript
editor’s note: This refers to Howard Wiseman.
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Quantum Aspects of Life
may—if we do this—we may just find something that nature can teach us
about quantum computation.
Julio Gea-Banacloche (Chair): Well, these have turned out to be rather
long sentences, but...
Audience laughter
Jens Eisert (Yes Team): To start with, nature has television, but does
nature also have game shows?
Audience laughter
Jens Eisert (Yes Team): OK so we are on the boring side, but something
seems to be buried. There seems to be some sort of psychological belief that
the mystery of life sounds cool if some sort of quantum effect is involved. I
must say that I’m not very happy if the brain just works as a classical Turing
machine—admittedly—and not that happy that the brain is a quantum
Turing machine either.
Howard Wiseman (Yes Team): OK, I just want to say, quantum information processing is extremely hard to do and as far as we know, it’s only
used for solving obscure number-theoretical problems.
Audience laughter
Howard Wiseman (Yes Team): I’d be happy for the experimentalists on
the other team to find evidence for [non-trivial quantum effects] in biology,
but I don’t believe it so far.
Sergey Bezrukov (Yes Team): Just in two sentences. Quantum mechanics will be very important for the future development of the biological
physics, or “physics of biology”, but up to now the situation is very simple.
As I told you already, [quantum mechanics is of] great importance for the
quantitative understanding of the [basic interactions and] processes’ rates,
but, unfortunately, [of little importance] for the qualitatively new insights
and new concepts. Thank you.
Hans Frauenfelder (Yes Team): I think essentially everything has been
said, and I am concerned with summarizing. The answer is we need experiments to answer the question, but first we have to find out what the real
question is, and we haven’t done that.
Audience laughter
Julio Gea-Banacloche (Chair): Well, I would like to thank all the panelists again, and now we have questions from the audience. Ah, there is
one!
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
369
Laszlo Kish (Texas A&M): Just a short comment: Paul didn’t take
the “trivial” part seriously. The question was about non-trivial quantum
effects. Anything that refers to material properties, dead material properties: chemistry, tunneling, van der Waals forces, even if it is quantum, is
trivial because still occurs in dead non-living material.
Julio Gea-Banacloche (Chair): Does anybody want to address this
point?
Paul Davies (No Team): I dread to reopen the whole debate about what
is trivial and what is non-trivial, as it is a non-trivial topic.
Audience laughter
Peter Heszler (University of Uppsala): I just have a comment that, as
far as I know, if you take high quantum numbers, quantum theory converges
to classical theory. My question is: “Are [humans made up of materials]
with high quantum numbers or low quantum numbers?” Because if we are
high quantum numbers—apparently according to Bohr—we are classical.
If we are low quantum numbers, then we are quantum mechanical. That’s
my point.
Anton Zeilinger (No Team): I would simply disagree with your starting
point. The limit of high quantum numbers is not always classical.
Juan Parrondo (Universidad Complutense): It looks like when we’re
talking about the brain here, the only way to escape from this picture of the
brain as a classical Turing machine is going to the quantum world. I would
like to say that there are other ways of escaping from the Turing machine:
for instance [by appealing to] chaotic classical systems or [by appealing to]
recent studies of treating the mind-body relationship as a whole. [Thus] I
think there are other ways of escaping from this narrow aspect. This is a
kind of an off-side comment.
Stuart Hameroff (No Team): I’ll cover that now. I got interested in
Roger Penrose’s argument using [Gödel’s] theorem that human consciousness uses something that is non-computable. Still deterministic, but not
algorithmic and chaotic systems and everything you said are still basically
deterministic and algorithmic. The only way out of that is something that
he called non-computable and he brought in his hypothesis of quantum
gravity, which is the only way out of us being helpless spectators and conscious automata. And [Jens Eisert] mentioned how seemingly ludicrous it is
to bring in quantum gravity because [the energy] is 24 orders of magnitude
lower than [the environment], however, in Roger [Penrose’s] scheme the re-
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Quantum Aspects of Life
duction is instantaneous so the power is actually calculated as a kilowatt
per tubulin protein.
Unknown (Audience Member): You say that it’s instantaneous?
Stuart Hameroff (No team): If you approximate “instantaneous” to
one Planck time, you take the very low energy divided by the [even smaller]
Planck time of 10−43 seconds, giving a kind of a karate chop.
Anton Zeilinger (No Team): Why don’t we all boil if it is a kilowatt?
Stuart Hameroff (No Team): Because [the energy is delivered] only
over a Planck time of 10−43 seconds.
Jens Eisert (Yes Team): In the mid-19th century as far as I know there
was this debate whether there’s something special about this kind of energy
of life or not...er...I only know the German word—very sorry—a kind of
spirit of life specific for...
Derek Abbott (No Team): “Vitalism” [is the word you are looking for.]
Jens Eisert (Yes Team): Ah, vitalism. Okay, if one pushes that picture
so far, then one can just speak of consciousness particles that are in the
brain and then we are done. If you go beyond the theories that are open
to empirical verification [then you have gone too far].
Stuart Hameroff (No Team): You’re putting words in our mouths, we
didn’t say that—although I will admit to being a quantum vitalist—but
it’s not that we have quantum particles, it’s that [biology utilizes quantum
superposition, entanglement and computation. Why wouldn’t it?]
Unknown (Audience Member): What quantum information processing
could be useful in biology?
Paul Davies (No Team): If quantum information processing is not useful, why is so much money being spent on trying to improve it?
Audience laughter
Paul Davies (No Team): Quantum information processing bestows upon
nature awesome information processing power and I would find it totally
extraordinary if in the entire history of the universe this has never been put
to use until now.
Unknown (Audience Member): What is the one example? One possible example?
Paul Davies (No Team): Well, we naturally look to nature’s great information processing system which is called life. Really, for me, that is the
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
371
most persuasive point. Where else would we expect to see this happening in nature, except life, which is just so wonderfully adept at processing
information? And [according to orthodoxy] we are to suppose that life
hasn’t discovered the awesome information processing power that quantum
mechanics provides—but that’s all gone to waste over the 13.7 billion year
history of the universe and it’s only now that human beings have discovered
[quantum information processing]. That seems extraordinary hubris.
Stuart Hameroff (No Team): Just think of your subconscious mind as
being quantum information, which 40 times a second collapses or reduces
into your conscious mind.
Unknown (Audience Member): In order to live you’ve got to have
very large redundancy because you are constantly attacked by microbes,
viruses with all kinds of things—if I use my glasses it is because I see not
so well. Quantum information seems to be very, very fragile. Whatever
you do, a small perturbation and the whole computation fails. If quantum
computation has anything to do with living [systems], it seems to me that
nature should have found the optimal quantum error correction scheme. So
maybe this is what we should be [looking for.]
Paul Davies (No Team): Yes, they may indeed to have done so and it
seems to me very sensible to look at nature’s nanostructures within cells to
see if they are deploying any tricks that we could make use of.
Stuart Hameroff (No Team): I mentioned [yesterday] that microtubules
seem to have used the Fibonacci series in terms of their helical winding
and it has been suggested that they utilize a topological quantum error
correction code that could be emulated in [man-made] technology. As far
as redundancy, there is a lot of parallelism in the brain and memory seems
to be represented holographically, so redundancy is not a problem.
Jens Eisert (Yes Team): My concern with quantum error correction is
that some care is probably appropriate. I mean it’s not known what the
actual thresholds are for quantum computation against arbitrary errors and
if you take off your glasses it is not so clear what errors are going to come
against you. But the best known bounds on the “market” are 10−4 for
quantum computation. And so we need to see the perspective that really
small errors can be corrected with error correction.
Stuart Hameroff (No Team): That’s why you should look at biology.
It might be better.
Julio Gea-Banacloche (Chair): I’m seriously trying to bite my tongue.
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Quantum Aspects of Life
Audience laughter
Julio Gea-Banacloche (Chair): Any other questions. I’m sure there
must be some comments. Yes.
Michael Hoffman (University of Ulm): It’s just a comment. I would
like to recall why quantum mechanics have been invented. They have been
invented because experiments were done that could not possibly be explained by classical models. So, I’m waiting for the experiment that you
are doing that cannot possibly be explained by classical models, and now
we should do another one.
Paul Davies (No Team): I’d like to respond to that because it’s the point
I was going to make, that ... everyone here seems to assume that biologists
seem to have a wonderful understanding of what is going on inside a living
cell, and the biologists I talk to are continually baffled. For example I
spoke in my lecture about the polymerase motor. You may think the basic
physical principles must be understood. Absolutely not. The people who
work on that say they really haven’t a clue what is going on there in terms
of basic physics. Somebody made the comment, “Biologists, you know,
why don’t they go out and learn some physics?” Mostly, biologists don’t
know quantum physics so, and that doesn’t mean they fully understand
everything in terms of classical physics. That doesn’t follow. They are
continually troubled and do not understand most of what is going on inside
a living cell, except at the level of individual molecular interactions. Once
you start getting something that’s complicated as the polymerase motor
[they are in trouble].
Michael Hoffman (Unversity of Ulm): But have they not tried to
explain it by classical models?
Paul Davies (No Team): Yes, I mean there are all sorts of hand waving
models around, but the classical models aren’t terribly satisfactory either. I
think it’s a real mystery what’s going on inside a cell, and the fact that biologists say, “Well, you know, it isn’t a problem” doesn’t mean that physicists
can just sit on the sidelines saying, “Oh, it will all be explained in terms of
classical physics.” I think [scientists] do not understand what’s happening.
There are lots of mysteries and we do need experiments. It seems to me
an entirely open question as to whether these experiments will reveal that
quantum mechanics is going to be really essential for understanding some
of these molecular biological properties—at least at the level of smaller
components.
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
373
Sergey Bezurkov (Yes Team): I would like to disagree with Paul and
support Michael. From my point of view, it depends on what you call
“experiment.” We probably do not actually understand what is going on in
a single cell. Absolutely, I am with Paul [in this respect]. But the moment
you do your experiments properly, the moment you dissect the cells into
the parts that you can study with a real control over the [experimental]
parameters, they [the parts] are “golden” [i.e. can be described by laws of
physics]. So, [here] I’m with Michael.
Stuart Hameroff (No Team): Can I respond to that? I think what you
just said, Sergey, is that if you cut the cell up into small enough pieces,
eventually you get an experiment you can perform.
Sergey Bezrukov (Yes Team): Absolutely, but I’m not ashamed by the
lack of understanding [of complex biological systems]; this is where we are
now, with our physics, and I am not talking only about Building 5 at the
NIH, [I am talking about] everybody here and in the world.
Stuart Hameroff (No Team): So when you reduce it too much you
throw away the baby with the bathwater, you’ve lost the essential feature
of life. And the other thing is consciousness is completely unexplained by
classical means.
Howard Wiseman (Yes Team): To [Stuart], about the possibility of
doing the experiment: could you show one tubulin protein being in a superposition? I mean that’s pretty small. It seems to me that, you know,
you criticize people for saying that you can’t take things apart but it would
bolster your position enormously if you could demonstrate that just one
protein like this could be in a superposition state. So, that’s why I think,
really, this isn’t going forward until an experiment like that is done.
Anton Zeilinger (No Team): Suppose what I believe will come true,
namely that we will find superpositions of all kinds of things including
tubulin. Would that have any impact for today’s debate?
Stuart Hameroff (No Team): Addressing Wiseman and referring to
Zeilinger. He showed it for Porphyrin so why not Tubulin?
Anton Zeilinger (No Team): Addressing Wiseman. If so, what impact
would it have on your opinion?
Howard Wiseman (Yes Team): I think it would have an impact, it
would advance your side enormously. At the moment there is no evidence
whatsoever that there is anything quantum going on or even potentially
quantum going on. So, I’m not saying it would win the debate, but it
would certainly do a lot for you.
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Quantum Aspects of Life
Audience laughter
Howard Wiseman (Yes Team): And a comment on the difference between porphyrin and tubulin—it’s enormously difficult to show a macromolecule being in a superposition of two different configurations. That is
like a miniature Schrödinger’s cat. A porphyrin molecule being in a superposition in slightly different positions, is a completely different thing and
it’s much easier to do and to conceive of being true and to demonstrate
experimentally.
Stuart Hameroff (No Team): The difference between the two tubulin
states in our model: Professor Frauenfelder mentioned that the spatial difference in functional states is very small in a protein—is only the diameter
of an atomic nucleus—so you couldn’t really tell by looking at it. Of course
if you looked at it you’d collapse it anyway.
Most view the brain as a hundred billion neurons with synapses acting
as simple switches making up a computer. But if you look at one cell
organism like the paramecium, it swims around and finds food, it avoids
predators and obstacles. If it gets sucked up into a capillary tube, it gets
out faster and faster each time—it can learn! It finds a mate and has sex.
[Paramecium] doesn’t have any synapses to process information, [but does
so] very efficiently using its microtubules. If you try to develop a machine
to do that, you would need a hundred million dollars.
Paul Davies (No Team): And a sense of humour?
Audience laughter
Stuart Hameroff (No Team): I’d like to think so!
Juan Parrondo (Universidad Complutense): We know that life
[forms] prefer to live at a relatively high temperature, say around 300
Kelvin. If there would be quantum effects in biological processes, wouldn’t
there be [a preference for] life to develop at a colder temperature? Is it
possible to say something general about this?
Stuart Hameroff (No Team): Biology has apparently adapted to utilize
heat to promote rather than destroy quantum states, somewhat like a laser
pumps quantum coherence. For example increased temperature enhances
the quantum [spin] transfer through benzene, a perfect example of an organic molecule [found in protein hydrophobic pockets], so there must be
some [kind of] electronic resonance that harnesses heat to promote quantum [processes].
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
375
Julio Gea-Banacloche (Chair): Well, it is probably about time to begin
winding down. I think I would like to informally pose the question to the
audience. Basically, I’m not sure that I can pose the question in a way that
everybody here will approve of, so I’m not going to try. Let me just say by
every individual’s definition of trivial then, how many here—we can have a
show of hands—would think that it’s possible that there may be non-trivial
quantum effects in biology, [that is], they tend to believe that there really
may be non-trivial [quantum] effects in biology?
A little under 50% of the audience raise their hands
Julio Gea-Banacloche (Chair): OK. Now, how many would tend to
believe that there are really no non-trivial quantum effects in biology?
A little over 50% of the audience raise their hands
Julio Gea-Banacloche (Chair): What do you say? I think there are
probably more on the non-trivial side... I mean the trivial side—sorry!
Audience laughter
Derek Abbott (No Team): I’d like to see a show of hands whether
people think it’s worth going out and doing some experiments to check
the relationship between quantum mechanics and biology. Hands up who
thinks that’s worth doing.
Julio Gea-Banacloche (Chair): It’s a fair question.
>70% of the audience raise their hands
Derek Abbott (No Team): Hands up if you think you’d be wasting your
time.
No hands are raised
Derek Abbott (No Team) Ah, nobody!
Then slowly one hand is raised—that of Laszlo Kish. Audience laughter
Julio-Gea Banacloche: Well, I would like to thank you all again and the
participants for taking part in this debate. Thank you.
Audience applause
End of transcript.
Acknowledgements
The assistance of Phil Thomas, as the audio transcript typist, and Matthew
J. Berryman for the sound recording are gratefully acknowledged. Thanks
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Quantum Aspects of Life
are due to the many people who proof read the manuscript and helped decode the sound recording—any remaining errors are mine (Derek Abbott).
About the Panelists
Julio Gea-Banacloche was born in 1957, Seville, Spain and is presently
professor of physics at the University of Arkansas. He received his BS from
Universidad Autonoma de Madrid, 1979; and obtained his PhD under Marlan O. Scully, 1985, on quantum theory of the free-electron laser, from the
University of New Mexico. In 1985–87, he served as a Research Associate,
Max Planck Institute for Quantum Optics; in 1988–90 he was a Staff Scientist, Instituto de Optica, Madrid, Spain; and in 1990 he joined the Universisty of Arkansas initially as an assistant professor. He is an associate editor
of Physical Review A and Fellow of the American Physical Society. He has
carried out theoretical work in laser physics, quantum optics, and quantum
information. His main contribution to the field of quantum information has
been the observation that the quantum mechanical nature of the fields used
to manipulate the quantum information carriers themselves—often called
“qubits”, or “quantum bits”—might lead to unpredictable errors in the performance of the quantum logical operations. The lower bound on the size
of these errors can be made smaller by increasing the energy of the control
system. This has led Banacloche to predict a minimum energy requirement
for quantum computation, which has given rise to some controversy.
Paul C. W. Davies is a theoretical physicist, cosmologist, and astrobiologist. He received his PhD in 1970 from University College London,
under Michael Seaton and Sigurd Zienau. At Cambridge, he was a postdoc under Sir Fred Hoyle. He held academic appointments at the Universities of Cambridge, London and Newcastle upon Tyne before moving
to Australia in 1990, first as Professor of Mathematical Physics at The
University of Adelaide, and later as Professor of Natural Philosophy at
Macquarie University in Sydney, where he helped establish the NASAaffiliated Australian Centre for Astrobiology. In September 2006, he joined
Arizona State University as College Professor and Director of a new interdisciplinary research institute called Beyond: Center for Fundamental
Concepts in Science, devoted to exploring the “big questions” of science
and philosophy. Davies’s research has been mainly in the theory of quantum fields in curved spacetime, with applications to the very early universe
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
377
and the properties of black holes, although he is also an expert on the
nature of time. His astrobiology research has focused on the origin of life;
he was a forerunner of the theory that life on Earth may have originated
on Mars.
Davies is the author of several hundred research papers and articles, as
well as 27 books, including The Physics of Time Asymmetry and Quantum
Fields in Curved Space, co-authored with his former PhD student Nicholas
Birrell. Among his recent popular books are How to Build a Time Machine
and The Goldilocks Enigma: Why is the universe just right for life? (U.S.
edition entitled Cosmic Jackpot ). He writes frequently for newspapers,
journals and magazines in several countries. His television series “The Big
Questions”, filmed in the Australian outback, won national acclaim, while
his theories on astrobiology formed the subject of a specially commissioned
one-hour BBC 4 television production screened in 2003 entitled The Cradle
of Life. In addition, he has also devised and presented many BBC and
ABC radio documentaries on topics ranging from chaos theory to superstrings. Davies was awarded the 2001 Kelvin Medal and Prize by the UK
Institute of Physics and the 2002 Faraday Award by The Royal Society. In
Australia, he was the recipient of two Eureka Prizes and an Advance Australia award. Davies also won the 1995 Templeton Prize for his work on
the deeper meaning of science. The asteroid 1992 OG was renamed (6870)
Pauldavies in his honour.
Stuart Hameroff is an anesthesiologist and Professor of Anesthesiology
and Psychology at the University of Arizona in Tucson, Arizona. He received his MD from Hahnemann College, Philadelphia, Pennsylvenia, in
1973. He has teamed with Sir Roger Penrose to develop the “Orch OR”
(orchestrated objective reduction) model of consciousness based on quantum computation in brain microtubules, and has also researched the action
of anesthetic gases. As Director of the University of Arizona’s Center for
Consciousness Studies, Hameroff organizes the biennial “Tucson conferences” Toward a Science of Consciousness, among other Centre activities.
His website is www.quantumconsciousness.org.
Anton Zeilinger was born on 20 May 1945, Ried im Innkreis, Austria.
His family moved to Vienna, where Zeilinger went on to study physics
and mathematics at the University of Vienna. In 1971, he completed
his PhD Neutron Depolarization in Dyprosium Single Crystals under Helmut Rauch. In 1979 he completed his Habilitation in neutron physics at
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Quantum Aspects of Life
the Vienna Technical University. From 1972–81, he worked as a research
assistant at the University of Vienna under Helmut Rauch, followed by two
years as an associate professor at the Massachusetts Institute of Technology (MIT). He then taught at numerous universities in Austria and abroad,
such as Melbourne, Munich, Paris, Innsbruck and Oxford. He is currently
a professor of physics at the University of Vienna, previously the University of Innsbruck. He is director of the Vienna branch of the Institute of
Quantum Optics and Quantum Information (IQOQI). Zeilinger is known
for multiple experiments in the realm of quantum interferometry, which
include the first demonstration of quantum teleportation between two separately emitted photons. With Daniel Greenberger and Michael Horne he
demonstrated an extension of the Einstein-Podolsky-Rosen paradox, where
one considers three, not just two entangled particles. In 1999, his Innsbruck
group demonstrated experimentally that one can indeed observe three particle Greenberger-Horne-Zeilinger correlations. In Vienna his molecular interferometry team, co-led by Markus Arndt, discovered the possibility of
observing quantum interference of heavy chemical molecules, including the
C60 molecule (fullerene). Zeilinger received many awards for his scientific
work, one of the most recent being the Isaac Newton Medal (2007) of the
Institute of Physics (IOP). He is a fan of the Hitchhiker’s Guide To The
Galaxy by Douglas Adams, and has named his yacht “42.”
Derek Abbott was born in South Kensington, London, UK. He received
his BSc(Hons) in Physics from Loughborough University of Technology. He
obtained his PhD in Electrical and Electronic Engineering from the University of Adelaide, under Kamran Eshraghian and Bruce R. Davis. He is
with The University of Adelaide, Australia, where he is presently a full professor and the Director of the Centre for Biomedical Engineering (CBME).
He has served as an editor and/or guest editor for a number of journals
including Chaos (AIP), Smart Structures and Materials (IOP), Journal
of Optics B (IOP), Microelectronics Journal (Elsevier), Proceedings of the
IEEE, and Fluctuation Noise Letters (World Scientific). He is a life Fellow
of the Institute of Physics (IOP) and Fellow of the Institution of Electrical
& Electronic Engineers (IEEE). He has won a number of awards including
a 2004 Tall Poppy Award for Science. He holds over 300 publications and
is a co-author of the book Stochastic Resonance published by Cambridge
University Press. Professor Abbott is co-founder of two international conference series: Microelectronics in the New Millennium (with J. F. Lopez)
and Fluctuations and Noise (with L. B. Kish).
Plenary Debate: Quantum Effects in Biology: Trivial or Not?
379
Jens Eisert is a lecturer and holder of the European Young Investigator Award at Imperial College London in the UK (Diploma, University of
Freiburg, Germany; MSc, University of Connecticut, USA; PhD, University
of Potsdam, Germany). His research interests are in quantum information
science and related fields. This includes formal aspects of entanglement theory and computational models, as well as quantum optical implementations
and the study of complex quantum systems.
Howard Wiseman is a theoretical quantum physicist. His principle interests are quantum measurements, quantum feedback control, quantum information, fundamental questions in quantum mechanics, and open quantum
systems. He completed his PhD under Gerard J. Milburn at the University
of Queensland in 1994, and then undertook a postdoc under Dan Walls at
the University of Auckland. Since 1996 he has held Australian Research
Council research fellowships. He is currently Professor and Federation Fellow at Griffith University, where he is the Director of the Centre Quantum
Dynamics. He is also a Program Manager in the ARC Centre for Quantum Computer Technology. He has over 120 refereed journal papers, and
his awards include the Bragg Medal of the Australian Institute of Physics,
the Pawsey Medal of the Australian Academy of Science and the Malcolm
Macintosh Medal of the Federal Science Ministry.
Sergey Bezrukov is a Section Chief in the Laboratory of Physical and
Structural Biology, National Institutes of Health (NIH), Bethesda, Maryland. He received his MS in Electronics and Theoretical Physics from
St. Petersburg Polytechnic University, 1973; and he obtained his PhD under Giliary Moiseevich Drabkin in Physics and Mathematics from Moscow
State University, Russia, 1981. In 1981–87, he was a Research Scientist,
St. Petersburg Nuclear Physics Institute, Laboratory of Condensed Matter
Physics; in 1987–90 a Senior Research Scientist, St. Petersburg Nuclear
Physics Institute, Laboratory of Condensed Matter Physics; in 1990–92
he was a Visiting Research Associate, University of Maryland and Special
Volunteer, National Institutes of Health, LBM, NIDDK; in 1992–98 he was
Visiting Scientist, National Institutes of Health, LSB, DCRT and LPSB,
NICHD; in 1998–02 he was an Investigator, Head of Unit, National Institutes of Health, LPSB, NICHD; and he took up his present position in
2002. Bezrukov was elected Member of Executive Council of the Division
of Biological Physics of the American Physical Society in 2002. One of his
key research interests is in the physics of ion channels. He still works at the
NIH in a building where = 0.
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Quantum Aspects of Life
Hans Frauenfelder was born 28th June 1922 in Schaffhausen, Switzerland. He received his Dr. sc. nat. in physics in 1950 at the Swiss Federal
Institute of Technology (ETH) in Zurich under Paul Scherrer. His thesis
concerned the study of surfaces with radioactivity. In 1951, he discovered perturbed angular correlation. At ETH he was also taught by Gregor
Wentzel and Wolfgang Pauli. Through Pauli, he also got to know many of
the leading scientists such as Kramers, Heisenberg, Hans Jensen, and Wolfgang Paul (“Pauli’s real part”). Frauenfelder migrated to the US in 1952,
joining the Department of Physics at the University of Illinois in Urbana
Champaign as a research associate. Despite the absence of mountains, he
stayed at the UIUC till 1992, ultimately as Center for Advanced Study Professor of Physics, Chemistry, and Biophysics. His research interests included
nuclear physics, particle physics, conservation laws, Mössbauer effect, and
biological physics. In 1992, Frauenfelder moved to the Los Alamos National
Laboratory where he is currently the director of the Center for Nonlinear
Studies and continues research in biological physics. He wrote three books,
Mössbauer Effect, and together with Ernest Henley, Nuclear and Particle
Physics, and Subatomic Physics. Frauenfelder is a member of the National
Academy of Sciences, the American Philosophical Society, and a Foreign
Member of the Royal Swedish Academy of Sciences.
Chapter 17
Nontrivial Quantum Effects in
Biology: A Skeptical Physicists’ View
Howard Wiseman and Jens Eisert
When you have excluded the trivial, whatever remains, however
improbable, must be a good topic for a debate.1
17.1.
Introduction
This chapter is somewhat of an anomaly in this book. Firstly, its authors profess no particular knowledge of any effects in biology, (whether
quantum or non-quantum, trivial or non-trivial), both being theoretical
quantum physicists by trade. Secondly, we adopt here a skeptical view on
the existence of such effects if they fall in the non-trivial class. That two
such skeptical non-experts have been invited to contribute to this volume
came about as a result of the public debate (reproduced earlier in this volume) at the Second International Symposium on Fluctuations and Noise,
held in the Canary Islands in 2004 (see Chapter 16). We were invited by
Derek Abbott to affirm the statement that “Quantum effects in biology are
trivial.”
This chapter will reproduce many of the arguments that we put in that
debate, although hopefully somewhat more coherently than we communicated them at the time. It also contains some arguments that were not
covered in the debate. Obviously the debate would have been pointless unless both sides had agreed on what counts as a non-trivial quantum effect in
biology. Thankfully, all participants in the debate did agree, more or less,
Received April 17, 2007
to A. C. Doyle.
1 Apologies
381
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Quantum Aspects of Life
although only one (HMW) offered a formal definition: that a non-trivial
quantum effect in biology is one that would convince a biologist that they
needed to take an advanced quantum mechanics course and learn about
Hilbert space and operators etc., so that they could understand the effect.
To use the word “trivial” to characterize all quantum effects in biology that do not increase enrollments of biologists in advanced quantum
physics courses is unfortunate. Neither we, nor, we imagine, any of the
debate participants, wish to denigrate the interesting and challenging research into quantum effects relevant to biology such as coherent excitations
of biomolecules [Helms (2002); Gilmore and McKenzie (2005)], quantum
tunneling of protons [Kohen and Klinman (1999)], van der Waals forces
[Parsegian (2005)], ultrafast dynamics through conical intersections [Cederbaum et al. (2005)], and phonon-assisted electron tunneling as the basis for
our sense of smell [Brookes et al. (2007)]. But here we are concerned not
with these real (or at least plausible) quantum effects, but rather with more
exotic, unproven (and, we believe, implausible) effects.
What might these non-trivial sorts of quantum effects be? Several have
been suggested in the literature (see other papers in this volume), but we
will concentrate upon four: A quantum life principle; quantum computing
in the brain; quantum computing in genetics; and quantum consciousness.
These intriguing topics provide the structure of our chapter. We devote one
section each to briefly explaining, and then arguing the implausibility of,
these hypothetical effects. It is hence the purpose of the present chapter to
be cautionary: to warn of ideas that are more appealing at first sight than
they are realistic.
We end, however, on a more constructive note in our final section, by
pointing out that there is one sense in which it seems likely that quantum
effects introduce a non-trivial difference between brains and digital computers. This section (quantum free will) is of interest philosophically rather
than scientifically, so we do not see it as an exception to our claim that
biologists should not want to enroll in advanced quantum physics courses.2
17.2.
17.2.1.
A Quantum Life Principle
A quantum chemistry principle?
It is a widely held belief that the origin of life is extremely unlikely according
to established science. This has led some to argue that there exists a natural
2 Philosophers,
on the other hand, should!
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
383
principle, in addition to existing laws, that guarantees that life must arise
in the universe—see Davies (2004a). In this review, Davies points out
difficulties with this argument, but apparently he gives it some credibility
since he used it in the 2004 Canary Island debate (Chapter 16). There
he stated that, unless life is miraculous, there must be a life principle,
and that since it is a fundamental physical principle, it must be related
to our most fundamental physical theory: quantum mechanics. In Davies
(2004a) he suggests that the origin of life may be related to quantum search
algorithms, an idea we discuss in Sec. 17.4.
That a belief is widely held does not make it correct. Indeed, we claim
that the origin of life is entirely plausible according to established physical
theories. Moreover, the relevant physical theory, chemistry, has no deep relation to quantum physics. To understand chemical structure and reactions
at a fundamental level it is, of course, necessary to use quantum physics.
But chemistry is usually regarded as emerging from physics in a straightforward (upwardly causal, not downwardly causal [Davies (2004b)]) way.
If this were not the case, it would be necessary to postulate not merely a
“quantum life principle”, but also a “quantum chemistry principle” (along
with, presumably, a “quantum condensed matter principle”, a “quantum
atom principle”, and so on).
That life is an epiphenomenon of chemistry, and one whose appearance
on earth is unsurprising, even expected, is well argued by Dawkins in his
most recent popular book on evolution [Dawkins (2004)]. First, he stresses
(pp. 575-81) that the essence of life, the aspect of life that must precede all
others, is heredity. Heredity means the existence of a varied population of
replicators in which the variation is (at least partially) inherited in the act
of replication. To quote Dawkins,
“[S]ome writers . . . have sought a theory of metabolism’s spontaneous origin, and somehow hoped that heredity would follow, like
other useful devices. But heredity . . . is not to be thought of as a
useful device. Heredity has to be on the scene first because, before
heredity, usefulness itself has no meaning. Without heredity, and
hence natural selection, there would have been nothing to be useful
for.”
Accepting Dawkin’s imperative, the origin of life can be illuminated by
seeking the simplest carrier of hereditary information in the natural world.
A well publicized example [Davies (1999); Dawkins (2004)] is Spiegelman’s
Monster, named after its developer [Kacian et al. (1972)]. It is far simpler
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Quantum Aspects of Life
than the viruses from which it was derived. It is an RNA strand a mere
218 nucleotides long; that is, it is a large molecule containing less than 104
atoms. The environment in which it replicates is an aqueous solution of
activated nucleotides, plus an enzyme Qβ-replicase. As shown by Spiegelman and Orgel, the monster carries hereditary information, and undergoes
natural selection.3 Most remarkably, self-replicating monsters appear spontaneously in the environment described above [Sumper and Luce (1975)].
The point of these investigations is not that Spiegelman’s monster is the
first life—that probably developed in quite different environments [Davies
(1999); Dawkins (2004)]. Rather, in the present context, the points are: (i)
that the beginnings of life need not be nearly so complicated as is imagined by those who stress its implausibility; and (ii) that nothing in these
experiments suggest that anything other than chemistry is involved in the
self-assembly and replication of these largish molecules. Indeed, it is likely
that the chemical reactions involved could be reproduced, in the not too
distant future, by simulations based at the atomic level. Such a simulation
would be a definitive refutation of the idea of a quantum life principle.
17.2.2.
The anthropic principle
It could be argued that, even if life is an almost inevitable consequence
of chemistry in suitable environment, this fact itself requires explanation.
That is, does it not seem miraculous that the physical world enables life to
arise? Specifically, it has been argued that the fundamental constants of
physics are “fine-tuned” so as to allow the existence of long-lasting stars,
planets, liquid water etc. that are apparently necessary for life to arise
[Barrow and Tipler (1986)]. Such an argument is known as the strong anthropic principle.4 According to the standard model of particle physics,
there are some 20 fundamental constants whose values are arbitrary, and
according to theories like string theory, these “constants” are in fact quantum variables [Susskind (2005)]. Thus it might seem plausible to claim that
life is somehow linked to quantum cosmology [Davies (2004a)].
Leaving aside the possible lack of imagination of physicists with regard
to the sorts of universes in which life may exist, it seems unnecessary to
invoke the strong anthropic principle to argue for a quantum life principle,
3 Indeed, later work [Eigen and Oehlenschlager (1997)] showed that, through natural
selection, the monster reduced even further in size, down to a mere 50 nucleotides—a
few thousand atoms!
4 Martin Gardiner has suggested the name Completely Ridiculous Anthropic Principle
for the more extreme versions of this principle [Gardiner (1986)].
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
385
when the weak anthropic principle has just as much explanatory power.
The weak anthropic principle simply states that we should condition all
our predictions on the undeniable fact that we are here to ask the question
[Barrow and Tipler (1986)]. Thus, if asked, what is the chance that the
fundamental constants will be found to have values that enable life to evolve,
we would have to say that the chance is essentially unity, since life evidently
has evolved. That is, invoking some special principle to explain that life
must have appeared and evolved intelligent observers is as unnecessary as
invoking a special principle to explain that the language we call English
must have developed and become widespread.
17.3.
17.3.1.
Quantum Computing in the Brain
Nature did everything first?
In the past decade or so, the field of quantum information (theory and experiment) has exploded [Nielsen and Chuang (2000)]. This is driven largely
by the prospect of building a large-scale quantum computer, that could
compute much faster than any conceivable classical computer by existing
in a superposition of different computational states. This leads naturally
to the conjecture that the brain itself may be a quantum computer [Hagan,
Hameroff, and Tuszyński (2002)].
When looking at the wealth of existing life forms, the following observation becomes apparent: nature had the idea first. Indeed, in nature we
can find parachutes and explosives, surfaces reminiscent of the most sophisticated nanostructured materials used in aeronautics today to reduce the
aerodynamic resistance. Many effects and concepts of physics can indeed
be found to be exploited by some life form to its benefit. So, after all, why
should this not apply to the brain being a quantum computer?
We would argue that this is not a legitimate argument. While it is
striking that some features have been “invented” by nature, the argument
as such is a “postselected argument”, based on case studies of anecdotal
character. It is equally (if not more) easy to collect counterexamples of
the same character; that is, inventions for which no counterpart in nature
is known. For example, there are no metal skeletons, despite metal being
much stronger than bone. There is no radio (long distance) communication,
albeit this certainly being a useful and feasible means of communication.
No closed-cycle refrigeration based on gas expansion is known. There is no
use of interferometry to measure distances. Also, the eye is as such a really
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Quantum Aspects of Life
lousy camera, corrected by the “software” of the brain. This last example
illustrates a general point: nature makes do with things that are good
enough; it does not do precision engineering. If there is one thing a quantum
computer requires, it is precision, as we discuss below in Sec. 17.3.3.
17.3.2.
Decoherence as the make or break issue
The case for the brain being a quantum computer, or indeed for quantum
mechanics playing any key role at a macroscopic level in the nervous system,
is weakest because of one effect: decoherence [Zurek (2003, 1998)].
A quantum system is never entirely isolated from its environment, which
is always “monitoring” its dynamics. That is, information is transferred
into the environment, where it is diluted into a larger and larger number
of degrees of freedom. As a result, superposition states become, for all
practical purposes, indistinguishable from classical mixtures of alternatives
on a time scale known as the decoherence time [Zurek (2003, 1998); Eisert
(2004)]. In short, quantum coherence is lost, as an effect of the environment
monitoring the system.
This effect of decoherence is one of the main concerns in research on
quantum computation [Nielsen and Chuang (2000)], where ingenious ways
are being explored of shielding engineered and strongly cooled quantum
systems from their respective environments. In fact, decoherence is the key
challenge in the realization of a full-scale quantum computer. In large scale
biological systems, like the brain, decoherence renders large scale coherence
(as necessary for quantum computation) very implausible. Even the most
optimistic researchers cannot deny the fact that the brain is a warm and
wet environment. This is in contrast to the high-vacuum environment used
in the beautiful experiments on spatial superpositions of organic molecules
from Anton Zeilinger’s group in Vienna [Hackermüller et al. (2003)]. In
the realistic biological setting, even the most conservative upper bounds to
realistic decoherence times are dauntingly small [Tegmark (2000)].
It is essential to keep in mind that large-scale quantum computation
does not mean merely computing with a large number of systems, each of
which behaves quantum mechanically. If coherence prevails only in subsystems of a quantum computer, but not over wide parts of the whole system,
the computation would be no more powerful than its classical counterpart.
Simply putting subsystems together operating on quantum rules with no
coherence between them cannot give rise to a quantum computer [Nielsen
and Chuang (2000)]. To create the large scale superpositions necessary for
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
387
quantum computation requires preserving coherence for a long time, long
enough to enable all the different subsystems to interact.
Tegmark’s article [Tegmark (2000)] is a careful discussion of the plausibility of preserving coherence over long times under the conditions in the
brain. He focuses on two situations where it has been suggested that quantum superpositions could be maintained: a superposition of a neuron firing
or not [Schadé and Ford (1973)]; and a superposition of kink-like polarization excitations in microtubules, playing a central role in the proposal
of Hameroff and Penrose (1996). The firing of a neuron is a complex dynamical process of a chain reaction, involving Na+ and K+ ions to quickly
flow across a membrane. Tegmark provides a conservative estimate of the
relevant decoherence times for a coherent superposition of a neuron firing including only the most relevant contributions, arriving at a number of 10−20
seconds. Similarly, he discusses decoherence processes in microtubules, hollow cylinders of long polymers forming the cytoskeleton of a cell. Again,
a conservative estimate gives rise to an estimated time of 10−13 seconds
on which superpositions decohere to mere mixtures.5 The general picture
from a discussion of decoherence times that emerges is the following: Even
if superposition states were to appear in the processes relevant for brain
functioning, they would persist for times that fall short (by many orders of
magnitude!) of the time scales necessary for the proposed quantum effects
to become relevant for any thought processes.
17.3.3.
Quantum error correction
The theory of quantum computation offers a number of strategies for preserving coherence of quantum evolution in the presence of a decohering
environment. To be sure, the idea of classical error correction of simply
storing information redundantly and measuring the full system to decide
with a majority rule whether an error has occurred does not work; in that
case the measurement itself necessarily destroys the coherence in the system, as it acquires information about the encoded quantum state. It was
one of the major breakthroughs in the field of quantum computation that
quantum error correction could nevertheless be realized. One indeed encodes quantum information in several physical systems, but in a way that
in later partial measurements, one can only infer whether an error has oc5 To
be fair, it should be noted that Hagan et al. [Hagan, Hameroff, and Tuszyński
(2002)] themselves argue that decoherence times may be significantly shorter than this
[Rosa and Faber (2004)].
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Quantum Aspects of Life
curred or not, but without being able to gather any information about the
encoded state itself [Shor (1996); Steane (1996)]. Based on this knowledge,
the error can then be corrected.
The idea of quantum error correction has been further developed into
the theory of fault-tolerance [Aharonov and Ben-Or (1998); Aliferis, Gottesman and Preskill (2006)]. Even using faulty devices, an arbitrarily long
quantum computation can be executed reliably. In topological quantum
memories, systems are arranged in a two-dimensional array on a surface
of nontrivial topology [Dennis et al. (2002)]. In physical systems, all these
ideas may further be enhanced with ideas of trying to stay within physical decoherence-free subspaces [Zanardi and Rasetti (1997)], or bang-bang
control. In the debate, Stuart Hameroff said:
“I mentioned [yesterday] that microtubules seem to have used
the Fibonacci series in terms of their helical winding and it has been
suggested that they utilize topological quantum error correction
codes that could be emulated in [man-made] technology. As far as
redundancy there’s a lot of parallelism in the brain and memory
seems to be representable holographically, so redundancy is not a
problem.”
So why should, after all, nature not operate like the brain as a fault tolerant
quantum computer?
Although this is a tempting idea it is by far more appealing than it
is a realistic option. Beautiful as the idea is, it only works if the basic
operations (called gates) are not too faulty. In realistic terms, they have
to be very, very good. Specifically, quantum fault tolerance, employing
complicated concatenated encoding schemes [Aharonov and Ben-Or (1998);
Aliferis, Gottesman and Preskill (2006)], works if the performance of logic
operations is better than a certain finite threshold. If the probability of
failure of a basic logic operation is below this threshold, then a computation
can indeed be performed as if perfect quantum gates were available. To
obtain good bounds to the exact value of this threshold is a topic of intense
research, but values of about 10−3 are realistic. Presently, we are a long
way from achieving such low probability of error experimentally, even in
sophisticated systems of laser cooled ions in traps, or in optical systems.
To say, as Hameroff did in the public debate, that
“[...] if you add the potential effect of topological quantum error
correction you get an indefinite extension,”
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
389
misses the point that such quantum error correction is only possible once
you have already reached the regime of very small errors. The required
accuracy is in very sharp contrast to any accuracy that seems plausibly
to be available in the slightly above room temperature environment of the
brain. To think of performing reliable arbitrarily long quantum computation under these conditions is frankly unrealistic. Thus while the appeal of
fault tolerance as an argument in favour of large scale coherence is indeed
enormous, the numbers very strongly argue against that.
17.3.4.
Uselessness of quantum algorithms for organisms
A final objection to the idea that quantum computing in the brain would
have evolved through natural selection is that it would not be useful. Quantum computing has no advantage over classical computing unless it is done
on a large scale [Nielsen and Chuang (2000)]. It is difficult to make statements about the time scales for quantum operations in the brain because
there is zero evidence for their existence, and because existing platforms on
which quantum computing is being explored are immensely different from
any known biological system. But for no other reason than the difficulty
in doing quantum error correction compared to classical error correction, it
can only be expected that the time required to do a quantum logic operation
would be greater than the corresponding time for classical logic operations.
Because of this, quantum computing to solve any given problem would actually be slower than classical computing until the problem reaches some
threshold size.
History is littered with case studies of organs and attributes that seem
to defy Darwinian evolution because any intermediate stage on the path
towards their full development would be useless or even harmful. But none
have stood up to scrutiny [Dawkins (2004)]. So perhaps the hypothetical
quantum computer in the brain could have come into existence despite
the above arguments. After all, quantum computers are generally thought
to provide an exponential speed up in solving certain problems [Nielsen
and Chuang (2000)], so the threshold problem size needed to overtake the
limitations of intrinsically slow quantum logic operations is not so large.
Unfortunately, the sort of problems for which such a speed up exists have
no obvious application to a biological organism. Basically, the problems
quantum computers are really good at are number theoretic in nature.
Instances of these problems, such as factoring large semi-prime numbers,
form the basis of modern cryptography as used millions of times a day on
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Quantum Aspects of Life
the internet (RSA encryption). If it were not for this fact, such problems
would be regarded as mathematical curiosities. Do enthusiasts for biological
quantum computing imagine that animals evolved the ability to send RSAencrypted messages to one another, and subsequently evolved the means to
eavesdrop by quantum computing?
To be fair, there are problems of more general use that quantum computers can attack using the Grover search algorithm [Grover (1996)] and
its relatives [Nielsen and Chuang (2000)]. Grover’s algorithm is sometimes
described as being useful for “searching a database”, suggesting that, for
example, it would help one find a person in an (alphabetically ordered)
phonebook if all one had was their phone number. This is a misconception. The Grover algorithm is an important quantum algorithm—indeed
it was one of the breakthrough results—but it cannot search a classical
database. What it requires is a quantum database: a fixed, fully hardwired database-“oracle”, a black box that is “called” in the process of the
quantum algorithm. Nevertheless, Grover’s algorithm and its relations may
be applied to hard problems, such as finding good routes in a network, that
would conceivably be useful to an animal. Unfortunately, the speed-up
offered by Grover’s algorithm on such problems is at best quadratic. Moreover, it has been proven that no algorithm can do better than Grover’s
algorithm. Thus quantum computers make no difference to the complexity
class of these problems. The lack of an exponential speed-up means that the
threshold problem size for any speed-up at all is very large. This makes it
exceedingly unlikely that evolution could have made the leap to large-scale
quantum computing.
17.4.
17.4.1.
Quantum Computing in Genetics
Quantum search
If not in the brain, then perhaps coherent quantum effects, or even fully
fledged quantum computations, are operating behind the scenes at the microscopic level of our genes [Davies (2004a)]. It has been argued that the
genetic code contains evidence for optimization of a quantum search algorithm. Again, this is intriguing idea, and it may not be possible at the
present stage to definitively rule it out. Here we argue, however, that the
case for such an idea is, if anything, weaker than that for quantum computing in the brain.
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
391
The argument formulated, albeit cautiously, in Patel (2001) in favour
of quantum effects to play a role in genetics, is to a large extent based on
suggestive numbers that are involved: On the one hand, the genetic code is
based on triplets of nucleotides of 4 varieties that code for 20 or 21 amino
acids. On the other hand, the optimal number Q of sampling operations in
Grover’s algorithm on an unsorted database of N objects is given by Q = 1
for N = 4 and Q = 3 for N = 20 or N = 21. This might appear indeed as
a remarkable coincidence of numbers.
But then, some caution is appropriate: To start with, the role of Q and
N is very different. More convincing as an argument against a connection,
however, is probably the observation that 3, 4, 20, 21 also appear, say, in
the sequence of numbers which appear the same6 when written in base 5
and base 10/2. This is easily revealed by using the On-Line Encyclopedia
of Integer Sequences of AT&T Research [AT&T (2006)]. It is an interesting and educational pastime to see how essentially every finite sequence of
integer numbers that one can possibly come up with appears in, for example, the “number of isolated-pentagon fullerenes with a certain number of
vertices”, or the “decimal expansion of Buffon’s constant”. The sequence
2, 4, 6, 9 in this order, to consider a different random example, appears in
no fewer than 165 (!) listed integer sequences, each of which is equipped
with a different construction or operational meaning. The lesson to learn is
that one should probably be not too surprised about coincidences of small
tuples of integers.
Moreover, as has been emphasized above, Grover’s search is not an
algorithm that sorts a database given as a number of objects following the
laws of classical mechanics: One needs a hard-wired oracle, following the
rules of quantum mechanics between all involved objects throughout the
computation [Grover (1996)]. It is difficult to conceive how such a hardwired coherent oracle would be realized at the genome level. The optimal
improvement in the sampling efficiency, in turn, would be of the order of
the square root of N . It does seem unlikely that the overhead needed in a
reliable quantum computation, possibly even enhanced by error correction
requiring again an enormous overhead, would by any figure of merit be
more economical than, say, a simple doubling of the waiting time in case of
N = 4.
6 To represent a given number in base b, one proceeds as follows: If a digit exceeds b,
one has to subtract b and carry 1. In a fractional base b/c, one subtracts b and carries c.
392
17.4.2.
Quantum Aspects of Life
Teleological aspects and the fast-track to life
One of the most interesting open questions at the interface of the biological and physical sciences is the exact mechanism that led to the leap from
complex molecules to living entities. The path from a non-living complex
structure to one of the possible living structures may in some way be a
regarded as a search procedure, the number of potential living structures
being likely a tiny subset of all possible ones consisting of the same constituents [Davies (2004a)]. Now, how has nature found its way to this tiny
subset? Needless to say, we have very little to say about this key question.
In this subsection, we merely cautiously warn that whatever the mechanism,
the involvement of quantum superpositions to “fast-track” this search again
in the sense of a quantum search appears implausible.
When assessing the possibility of quantum search here one has to keep
in mind that quantum search is, once again, not just a quantum way of
having a look in a classical database of options: Necessarily, the coherence
must be preserved. This means that in the search, the figure of merit, the
oracle, needs to be hard-wired. This oracle has to couple to all subspaces
corresponding to all different options of developments. What is more, there
is a teleological issue here: It is not entirely clear what the search algorithm
would be searching for. The figure of merit is not well defined. If a search
is successful, life has been created, but what features does life have? Arguably, this might be linked to the structure being able to reproduce. But
again, this figure of merit could only be evaluated by considering subsequent generations. Thus it seems that it would be necessary to preserve
a coherent superposition through multiple generations of such structures,
which we would argue is particularly implausible.
17.5.
17.5.1.
Quantum Consciousness
Computability and free will
Recent years have seen significant advances in the understanding of neural
correlates of consciousness [Koch (2004)]. Yet, needless to say, the understanding of consciousness on the biological elementary level is not sufficiently advanced to decide the case of quantum mechanics playing a role or
not in consciousness, beyond the obvious involvement of ruling the underlying physical laws. Hence, any discussion on the role of quantum mechanics in form of long-range entanglement in the brain or in actual realistic
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
393
collapses of wave-functions is necessarily of highly speculative character.
Here, we limit ourselves to addressing arguments put forward in the public
debate that triggered the publication of this book, and warn of the possibility of fallacies in some of these arguments.
Where could quantum mechanics play a key role in consciousness?
Hameroff argued in the debate, based on an earlier proposal put forth in
Hameroff and Penrose (1996), that the gravitational induced collapse of the
wave-function is eventually responsible for conscious acts. Moreover, microtubules forming the cytoskeleton of neurons should be the right place to
look for such state reductions. These reductions should be realistic, actually happening state reductions, in what is called an orchestrated objective
reduction (Orch-OR).
This is interesting, but also dangerous territory. To start with, it does
not refer to the established physical theory of quantum mechanics as such
[Grush and Churchland (1995); Penrose and Hameroff (1995)]. The motivation for this approach is to seek a way for human consciousness to
be noncomputable, in order to differentiate it from mere computation as
performed by artificial intelligence machines (see also Sec. 17.6). But computability and noncomputability are the same in quantum computer science
as in classical computer science. Thus Penrose and Hameroff must appeal
to a new theory of nature that may allow for noncomputable physical effects. They speculate that the key feature of this new theory would result
from unifying quantum mechanics with general relativity (i.e. gravity).
There is no generally accepted theory of quantum gravity. Hence, to
invoke a realistic collapse in this sense bears the risk that the debate is
pushed into a dark corner where everybody simply has to admit that he or
she has no proper understanding what is happening there: “Ha, I told you
that you do not know the answer!” In the debate, Hameroff invoked the
“[...] hypothesis of quantum gravity, which is the only way out
of us being helpless spectators,”
(that is, the only way to prevent our thoughts from being computable). The
mere wish that gravity could leave a loophole for free will does not seem to
us to be a very strong argument for this hypothesis. Finally, it should be
pointed out that there is essentially no experimental evidence for any sort
of information processing in microtubules, still less quantum information
processing, and yet less again for noncomputable quantum gravitational
information processing.
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Quantum Aspects of Life
17.5.2.
Time scales
Invoking quantum gravity also leads to confusions in the comparison of
time scales relevant for coherent quantum effects. In the debate, Hameroff
said:
“One of these guys [on the affirmative team] mentioned that
how seemingly ludicruous it is to bring in quantum gravity because
it is 24 orders of magnitude lower than decoherence.7 However,
in Roger’s scheme the reduction is instantaneous so the power is
actually calculated as a kilowatt per tubulin protein.”
To this Zeilinger (also on the negative team) asked
“But why don’t we all boil if it is a kilowatt?”
to which the response was
“Because it is only over a Planck time 10−42 seconds.”
These statements refer to the postulated Orch-OR time scale of state vector reduction. The relevant decoherence time scales are given in Hagan,
Hameroff, and Tuszyński (2002); this collection of numbers contains on the
one hand estimates for environment-induced decoherence times, for example of a superposition of neural firing (10−20 seconds). On the other hand,
it gives the time scale of superposition decay in Orch-OR, 10−4 –10−5 seconds. Based on these numbers, the obvious conclusion would be that, since
the gravitationally induced Orch-OR time scale is so much slower than
decoherence, the former process will be basically irrelevant.
What is more, the status of these two numbers is very different: The
environment-induced decoherence time scale is calculated with the help
of standard quantum mechanics as could be taught in any second year
quantum mechanics course. In contrast, the number on Orch-OR derives
from a speculative reading of what effects quantum gravity could possibly
play here. In this figure in Hagan, Hameroff, and Tuszyński (2002), these
two numbers are put together on the same footing, written in the same font
size. There is nothing wrong with openly speculating, and the presented
approach is not necessarily wrong or uninteresting. But it can become
problematic when the right disclaimers are not put in the right places, where
speculation on time scales of a potential theory of gravity are discussed
7 For an actual comparison of the relevant time scales, see Hagan, Hameroff, and
Tuszyński (2002).
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
395
with the same words and on the same footing as an elementary standard
quantum mechanics calculation. Regarding the status of the 10−4 –10−5
seconds it is not even entirely clear what object it refers to. Also, the fact
that the conscious thinking process occurs on similar time scales to this
hypothetical Orch-OR, does not make the processes causally linked. To
make that link is to risk introducing a rather postmodern tone into the
debate, where “anything goes”.
17.6.
17.6.1.
Quantum Free Will
Predictability and free will
As mooted in the Introduction, there is a relation between life and quantum
physics that may motivate a philosopher, if not a biologist, to try to understand advanced quantum physics. This is the fact that quantum physics
implies an in-principle distinction between (classical) digital computers and
human brains: the behaviour of the former is predictable, while that of the
latter is not. Note that we are not just making the obvious observation that
in practice the actions of human beings are unpredictable; we are making
the stronger statement that no matter how well you observed your neighbour (and your neighbour’s surroundings), with the help of any level of
technology, and how well you understood them, with the help of any level
of computing power (including quantum computers!), you could not predict
precisely how they would respond to a given stimulus (such as your kicking
a ball into their yard) at some point in the sufficiently distant future.
Digital computers are designed to have deterministic outputs for a given
input. Apart from hardware errors, which happen very infrequently, the
output of computer is completely predictable simply by feeding the input
into an identically designed machine. Human brains are not designed at
all, but more to the point they are analog devices. Moreover, they are extremely complicated systems, comprising roughly 1011 neurons, each electrically connected with varying strength to many other neurons. And each
neuron is a non-trivial element in itself, with complex biochemical reactions
determining how it responds to its stimuli. Thus there is every reason to
expect the brain to be a chaotic system, in that a small difference to the
initial microscopic conditions of the brain would be amplified over time so
as to lead to macroscopically different behaviour (such as kicking the ball
back, or throwing it back).
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Quantum Aspects of Life
The above argument does not yet establish an in-principle difference
between brain and computer, because in principle it would seem that a
sufficiently advanced technology would allow you to know the microscopic
state of your neighbour’s brain (and the microscopic state of their body
and other surroundings) to any degree of accuracy, so that in principle
its state at some fixed future time could be be predicted to any degree of
accuracy. What prevents this is of course quantum mechanics: it is impossible to know precisely the position and momentum of a particle. Under
chaotic dynamics, this microscopic quantum uncertainty will be amplified
up to macroscopic uncertainty. Even for a huge system with few degrees of
freedom—Saturn’s moon Hyperion —the time taken for its orientation to
become completely unpredictable according to quantum mechanics is only
20 years [Zurek (1998)]. For a far smaller and far more complex system such
as the human brain, we would expect this time to be far, far smaller—see
also Dennett (1984).
Thus quantum mechanics implies that, even if artificial intelligence were
realized on a classical digital computer, it would remain different from human intelligence in being predictable. Of course this does not mean artificial
intelligence would be deficient in any aspect of human intelligence that we
value, such as empathy or the ability to write poetry. However, such an
artificial intelligence would lack free will, at least in the following operational sense: If it thought that it had free will, then it would make the
wrong decision in Newcomb’s problem (see Chapter 14), by thinking that
it could outwit a Predictor of its behaviour [Nozik (1969)]. For humans,
by contrast, the above arguments imply that such a Predictor cannot exist,
except as a supernatural being (a case we have no call to address).
17.6.2.
Determinism and free will
Having made this distinction between human brains and deterministic digital computers, it is important to note that the above arguments do not
mean that human brains are non-deterministic (still less that they are uncomputable, as Penrose feels they must be [Penrose (1990)]). The reason
is that determinism and in-principle predictability are not the same things.
There are deterministic theories in which systems are unpredictable even
in principle because there are in-principle limitations on how much any
physical observer can find out about the initial conditions of the system.
Moreover, these theories are not mere philosopher’s toys. One of the more
popular interpretations of quantum mechanics, known as Bohmian mechan-
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
397
ics [Bohm (1952); Bohm and Hiley (1993); Holland (1993); Cushing et al.
(1996)], is just such a theory.8
In the Bohmian interpretation of quantum mechanics, quantum particles have definite positions that move guided by the universal wave-function
Ψ. The latter evolves according to Schrödinger’s equation; it never collapses. All that “collapses” is an observer’s knowledge of the positions of
particles, and this “collapse” is nothing but Bayesian updating based on
correlations between the particles in the system of interest and the particles
from which the observer is constituted (and on which the observer’s consciousness supervenes). Because of the way the particles’ motion is guided
by Ψ, it can be shown that the observer’s knowledge of the position x of a
particle for some system is limited by quantum uncertainty in exactly the
same way as in orthodox quantum mechanics. But, since Bohmian mechanics is a deterministic theory, probability enters only through observer’s lack
of knowledge about the position of particles, due in part to their chaotic
motion [Valentini (1991)].
In the biological context, this interpretation says that the behaviour
of humans is determined, by the initial positions of the particles in the
person’s brain, and its environment. The latter is naturally regarded as
a random influence, while the former is more naturally regarded as the
source of an individual’s will. It is impossible for an outside observer, no
matter how skilled, to find out precisely the positions of the particles in
an individual’s brain, without making a precise quantum measurement of
the positions. Such a measurement would instantly destroy the brain by
creating states with unbounded energy. Thus, in the Bohmian interpretation, the actions of an individual derive from the physical configuration
of their brain, but quantum mechanics makes this configuration unknowable in-principle to anyone else. For compatibilists,9 the picture offered
by Bohmian mechanics—a deterministic yet unpredictable quantum free
will—may be an appealing one.
Acknowledgements
HMW acknowledges discussion with Eric Calvalcanti regarding free will
in quantum mechanics. HMW was supported by the Australian Research
8 Please
note that the following discussion only reflects the opinions of one of us (HMW).
is, those who hold that determinism is compatible with—or even a precondition
of—free will [Dennett (1984)].
9 That
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Quantum Aspects of Life
Council Federation Fellowship scheme and Centre for Quantum Computer
Technology, and the State of Queensland. JE was supported by the DFG,
Microsoft Research, the EPSRC, and the EURYI Award Scheme.
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Biology 6, pp. R191–8.
Nielsen M. A., and Chuang, I. L. (2000). Quantum Computation and Quantum
Information (Cambridge University Press).
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Patel, A. (2001). Why genetic information processing could have a quantum basis,
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of Consciousness Studies 2, pp. 98–111.
Penrose, R. (1990). The Emperor’s New Mind (Vintage).
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Schadé, J. P., and Ford, D. H. (1973). Basic Neurology, 2nd ed. (Elsevier).
Shor, P. W. (1995). Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, p. 2493.
Steane, A. (1996). Error-correcting codes in quantum theory, Phys. Rev. Lett. 77,
p. 793.
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Qβ replicase, Proceedings of the National Academy of Sciences 72, pp. 162–
166.
Susskind, L.(2005). The Cosmic Landscape: String Theory and the Illusion of
Intelligent Design (Little Brown and Company).
Tegmark, M. (2000). Importance of quantum decoherence in brain processes,
Phys. Rev. E 61, pp. 4194–4206.
Valentini, A. (1991). Signal-locality, uncertainty, and the subquantum H-theorem
(part I), Phys. Lett. A 156, pp. 5–11; ibid. (part II) 158, pp. 1–8.
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About the authors
Howard Wiseman is a theoretical quantum physicist. His principle interests are quantum measurements, quantum feedback control, quantum information, fundamental questions in quantum mechanics, and open quantum
systems. He completed his PhD under Gerard J. Milburn at the University
of Queensland in 1994, and then undertook a a postdoc under Dan Walls
at the University of Auckland. Since 1996 he has held Australian Research
Council research fellowships. He is currently Professor and Federation Fellow at Griffith University, where he is the Director of the Centre Quantum
Dynamics. He is also a Program Manager in the ARC Centre for Quantum Computer Technology. He has over 120 refereed journal papers, and
his awards include the Bragg Medal of the Australian Institute of Physics,
the Pawsey Medal of the Australian Academy of Science and the Malcolm
Macintosh Medal of the Federal Science Ministry.
Jens Eisert is a lecturer and holder of the European Young Investigator Award at Imperial College London in the UK (Diploma, University of
Freiburg, Germany; MSc, University of Connecticut, USA; PhD, University
Nontrivial Quantum Effects in Biology: A Skeptical Physicists’ View
401
of Potsdam, Germany). His research interests are in quantum information
science and related fields: This includes formal aspects of entanglement theory and computational models, as well as quantum optical implementations
and the study of complex quantum systems.
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Chapter 18
That’s Life!—The Geometry of
π Electron Clouds
Stuart Hameroff
18.1.
What is Life?
Historically, two broad approaches have attempted to characterize the essential nature of living systems: 1) functional emergence, and 2) vitalism.
Functionalism characterizes life by its organizational arrangements leading to purposeful behaviours,1 implying that non-biological systems which
exhibit at least some of these behaviours may be considered living. For
example certain types of self-organizing computer programs have lifelike functions, and “artificial life” proponents view such systems as alive
[Langton (1991)].
But living systems seem to have an essential uniqueness, often ascribed to an emergent property of biochemical and physiological processes.
Emergence implies a hierarchical organization in which a novel property
(e.g. life) arises from complex interactions of relatively simple components
at lower, reductionist levels. As weather patterns and candle flames emerge
from complex interactions among simple gas and dust particles, life is
said to emerge from complex interactions among biomolecules, ions and
atoms [Scott (1995)]. However biomolecules are not simple, and emergent
Received May 1, 2007
self-organization, 2) homeostasis (maintaining a steady-state internal environment),
3) metabolism (energy utilization), 4) growth, 5) adaptive behaviour, 6) response to
stimuli, 7) replication/reproduction and 8) evolution. Adapted from [Margulis and Sagan
(1995)].
1 1)
403
404
Quantum Aspects of Life
phenomena like candle flames and weather patterns manifest selforganization, homeostasis and adaptive behaviours. Are they also alive?
Functionalist and emergent approaches based on simple reductionism
dominate molecular biology. An historical, opposing viewpoint is that
such descriptions fail to capture a fundamental (i.e. non-emergent) essential uniqueness, or “unitary one-ness” specific to living systems. Many
nineteenth-century scientists ascribed this quality to a “life force,” “elan
vital,” or energy field: “vitalism” (or “animism”). But as molecular biology
revealed biochemical and physical processes involved in cellular activities,
vitalism fell from favour. The notion of a life force or field has become almost taboo in modern science—although fields apparently organize mitosis,
[Karsenti and Vernos (2001)]. Nineteenth-century vitalism was based either
on electromagnetism or on forces outside the realm of science. Quantum
mechanics was as yet undiscovered.
In his famous book “What is Life? ” quantum pioneer Erwin Schrödinger
(1944) suggested a quantized basis for living systems, concluding that life’s
essential framework was in “aperiodic lattices”.2 Schrödinger’s description
applies to DNA and RNA, and also to cytoskeletal protein assemblies
such as microtubules and actin gels that extend throughout cell volumes.
Schrödinger suggested further that globally unified behaviour inherent
in living systems might involve non-local quantum correlations among
biomolecules (see the Foreword by Penrose).
The conventional wisdom in modern science is that biological systems
are too “warm, wet, and noisy” for seemingly delicate quantum processes,
i.e. that “decoherence” precludes supra-molecular or systemic roles. However recent evidence suggests otherwise, that biomolecules can harness heat
and energy to promote functional quantum states, rather than cause decoherence [Engel et al. (2007); Ouyang and Awschalom (2003)].
Since Schrödinger’s time scientists, including Szent-Györgyi (1960),
Pullman and Pullman (1963), Fröhlich (1968, 1970, 1975), Conrad (1994)
and others, have in various ways proposed that life is related to organized
quantum processes in π electron resonance clouds within biomolecules. For
example Fröhlich suggested that living systems utilize heat and biochemical energy to pump quantum coherent dipole states in geometrical arrays
of non-polar π electron resonance clouds in protein lattices. Russian biochemists Alberte and Phillip Pullman described cooperative π electron resonance as the essential feature of living systems.
2 In geometry, “aperiodic tiling” of a plane. A shifted copy of such a tiling matches only
locally with its original. The best known examples are Penrose tilings, described by Sir
Roger Penrose.
That’s Life!—The Geometry of π Electron Clouds
405
In today’s terms it is proposed that life involves cooperative quantum
processes in geometric lattices of non-polar π electron resonance clouds.
Such π clouds are isolated from cell water and ions, buried within nonpolar subspaces of components of biomolecular assemblies, e.g. membranes,
microtubules, nucleic acids and organelles. In each π cloud, electron dances
known as quantum London forces (a type of van der Waals force) govern
local nanoscale biomolecular conformational states.
In repetitive structures like the cytoskeleton and DNA, π clouds arrayed in specific periodic and aperiodic lattice geometries are separated
by less than two nanometres and thus conducive to electron tunnelling,
exciton hopping, long-range classical, and non-local quantum processes,
e.g. entanglement, superposition, and quantum computation [Hameroff
and Tuszynski (2003); Hameroff (2004)]. In particular, cytoskeletal-based
quantum processes can couple to biomolecular mechanical resonances, and
extend throughout cell volumes and possibly between cells and throughout
organisms via tunnelling through gap junctions [Hameroff (1997)]. Thus
geometric distributions of non-polar π electron clouds can enable a collective, cooperative quantum process—a unitary wave function—mediating
perception and governing purposeful behaviour of living organisms.
That’s life.
18.2.
Protoplasm: Water, Gels and Solid Non-polar
Regions
Living cells are composed of protoplasm, which in turn consists of
cytoplasm—the internal, aqueous cell milieu—as well as structures in and
around cytoplasm, e.g. membranes, organelles, nucleic acids and protein
assemblies.
Comprising sixty percent water, a cytoplasm alternates between two
or more aqueous polar forms. One is a liquid electrolyte solution (“sol”)
which may rapidly change to a quasi-solid “gel” upon polymerization of
actin proteins, as water molecules become aligned and ordered on actin
surfaces.
More solid structures including membranes, protein assemblies and nucleic acids are embedded in and around sol or gel cytoplasm. Buried and
isolated within these structures are specific non-polar regions, e.g. waterexcluding confluences of oil-like aromatic ring structures consisting largely
of π electron resonance clouds. Within these isolated regions, quantum
processes occur.
406
Quantum Aspects of Life
Here we consider the three types of media that comprise protoplasm:
Liquid cytoplasm: Aqueous solutions are characterized by H2 O molecular dipoles with different electrical charges on opposite ends, or
“poles”.3 Positive poles of each water molecule happily interact with
negative poles of others in a stable liquid phase which is chaotic at
the molecular level, water molecules adopting disordered configurations which maximize entropy.4 Charged molecules or ions such as
salt/sodium chloride easily dissolve in water via “hydrophilic” (water
loving) polar interactions.5 In such a polar, active aqueous medium,
significant supra-molecular quantum states are highly unlikely.
Gel cytoplasm: Cytoplasm may also adopt “gel” phases upon polymerization of the ubiquitous and versatile cytoskeletal protein actin
which self-polymerizes to form dense and complex filamentous networks. Negative charges are periodically arrayed on actin filament
surfaces, attracting positive poles of water molecules and arranging
them in ordered layers. This converts cell interior cytoplasm from a
liquid solution (“sol”) to a quasi-solid state gelatin (“gel”), such that
living cytoplasm transiently exists as an ordered dipole field [Pollack
(2001)]. Rapid sol-gel transitions drive “amoeboid” and other intracellular movements. Water molecules also become ordered, or aligned
on charged surfaces of membranes, organelles or microtubules. In some
situations, charged surfaces attract counter-ions, resulting in plasmalike Debye layers, e.g. on membrane and microtubule surfaces [Hameroff
and Tuszynski (2003); Hameroff (2004)].
Solid structures—Hildebrand solubility λ: In and around liquid and
water-ordered gel cytoplasm are solid structures including membranes,
protein assemblies, organelles and nucleic acids. At smaller scales
within these structures are “non-polar” regions comprised of molecules
(or portions of molecules) which do not easily dissolve in water, e.g.
oily compounds which form oil slicks on top of water puddles. These
oily hydrophobic (water repelling) non-polar groups are electrically neutral and stabilized from within. They include cyclic carbon rings with
delocalized electrons (“π electron resonance clouds”) exemplified by
benzene, also known as phenyl rings.
3 Two positively charged hydrogen atoms protrude from one “pole”, and one doubly
negative charged oxygen protrudes from the other.
4 Transient meta-stable H O complexes of varying geometry also occur.
2
5 Types 1 and 2 van der Waals forces and hydrogen bonds.
That’s Life!—The Geometry of π Electron Clouds
407
The degree of non-polarity for a particular solvent is given by the Hildebrand solubility coefficient λ, a measure of the amount of energy needed
for a dissolving molecule to break the van der Waals forces in a solvent.
18.3.
Van der Waals Forces
Van der Waals forces are attractions or repulsions between atoms or
molecules, differing from biochemical covalent forces (based on sharing of
electrons), or ionic and electrostatic attractions (between opposite charges).
Instead, van der Waals interactions occur between neutral electron clouds
of non-polar atoms and molecules. They were discovered in the behaviour
of gases.
In the 17th century British scientist Robert Boyle studied the relation
between pressure, volume and temperature of a gas. He arrived at Boyle’s
law P V = RT in which P is pressure, V is volume, R is a constant and T
is temperature.6 But measurement of actual gases showed smaller volumes
than predicted. Numerous attempts failed to account for the discrepancy
until an explanation by Dutch physicist Johannes Diderik van der Waals
several hundred years after Boyle.
Van der Waals reasoned that the smaller volume was the result of an attraction among the gas molecules, “pulling” them together. Gas molecules
and atoms are neutral with non-polar electron clouds. But such clouds
are “polarizable”—dipoles may be induced within them. Van der Waals
attractions and repulsions are based on dipole couplings of nearby electron
clouds.
There are three types of van der Waals forces. The first occurs between
permanent dipoles in polar molecules, like two tiny bar magnets attracting or repelling each other’s opposite poles. The second type is between
such a permanent dipole and a neutral atom or molecule with a non-polar
(but polarizable) electron cloud. The permanent dipole induces a temporary dipole in (“polarizes”) the non-polar electron cloud; the permanent
and temporary dipoles then attract or repel each other depending on their
relative positions.
The third type of van der Waals interaction is the London force that
occurs between non-polar electron clouds of two or more neutral atoms,
molecules or molecular groups (Fig. 18.1). London forces are instantaneous
6 For one mole of a gas, P is in atmospheres, V is in litres, R = 0.08207, and T is
temperature in degrees Kelvin.
408
Quantum Aspects of Life
Fig. 18.1. The van der Waals (Type 3) London force. (a) Filled electron clouds of
two neutral, non-polar atoms or molecular groups induce dipoles in each other; the
mutually-induced dipoles then attract each other. (b) The coupled dipoles oscillate
between different orientations. (c) Being quantum mechanical, the coupled dipoles can
exist in quantum superposition of both orientations.
dipole couplings—electrons in one cloud repel those in the other, forming
temporary dipoles. The dipoles then attract each other, leading to coupled
dynamics.7 The clouds are resonant, or probabilistic electron distributions,
so London forces are inherently quantum mechanical, occurring in regions
described by a low Hildebrand solubility coefficient λ.
Water, the most polar solvent, utilizes Types 1 and 2 van der Waals
forces (and hydrogen bonds) and has a very high λ coefficient of 48 SI units.8
In water, non-polar oily molecules such as benzene disrupt the entropymaximized state and self-associate, being pushed together by water—the
“hydrophobic effect”—and attracting each other by London forces. The
7 London
force attractions are exquisitely sensitive to distance, varying inversely with
the 6th power of the distance between electron clouds. Thus the forces are small (though
numerous) until clouds are almost adjacent. However if the clouds become too close and
begin to overlap, even stronger repulsive forces occur which vary with the 12th power of
the radius.
−3/2
Solubility Parameters: δ/MPa1/2 = 2.0455 × δ/cal1/2 cm
Standard
Hildebrand values from Hansen, Journal of Paint Technology Vol. 39, No. 505, Feb
1967 SI Hildebrand values from Barton, Handbook of Solubility Parameters, CRC Press,
1983 and Crowley, et al. Journal of Paint Technology Vol. 38, No. 496, May 1966.
http://sul-server-2.stanford.edu/byauth/burke/solpar/solpar2.html
8 Hildebrand
That’s Life!—The Geometry of π Electron Clouds
409
non-polar molecules aggregate into stable regions (λ equal to 18.7 SI units
for benzene) shielded from polar interactions with water. Within such
environments, Type 3 van der Waals-London forces play important roles in
protein folding and conformational dynamics, protein-protein interactions,
membranes and nucleic acid structure.
We can conclude that living protoplasm includes three types of phases:
1) liquid, 2) gel (with or without ordered water), and 3) solid structural
phase (membranes, protein assemblies, nucleic acids, organelles). Within
the latter, i.e. buried in interiors of solid structural phase components, are
non-polar, hydrophobic oil-like regions of low δ, e.g. like that of benzene
18.7 SI units. Such regions occur as planes in membranes, as continuous
cores or stacks in nucleic acids and as discrete pockets or islands within
proteins. Non-polar regions are typified by benzene-like “oily” aromatic
rings with π electron resonance clouds.
Oil slicks or bulk benzene bear no resemblance to living systems. But
when discrete regions are arrayed in periodic (or aperiodic) geometric lattices (and particularly when cytoplasm adopts an ordered-water gel phase
to minimize decoherence and extend non-polar lattice sites throughout cell
interiors), π electron clouds have an opportunity for cooperative resonances,
non-local interactions and quantum computation.
18.4.
Kekule’s Dream and π Electron Resonance
Life is based on carbon chemistry, including organic carbon ring molecules
with electron resonance clouds in which London forces play significant roles.
The flagship biomolecular organic structure is the phenyl ring, also known
as benzene.
The hexagonal ring structure of benzene (and the field of organic chemistry) was discovered in a dream by the 19th century German chemist
Friedrich August von Kekule. At that time, carbon atoms were known
to have valence 4 (four electrons in the outer shell) and thus able to share
electrons with four other atoms—form four covalent single bonds, e.g. with
hydrogen or other carbons in hydrocarbon chains. Carbons can also use
two valence electrons and form double bonds in which neighbouring carbons
share two electrons, the extra, mobile electrons being known as π electrons.
Hydrocarbons with only single bonds are called alkanes (methane, alkane
etc.) and have the generic formula of Cn H2n+2 . Hydrocarbons with a
carbon-carbon double bond and one π electron are known as alkenes and
follow the formula Cn H2n (Fig. 18.2).
410
Quantum Aspects of Life
Fig. 18.2. (a) Alkanes with carbon-carbon single bonds and generic formula Cn H2n+2 .
(b) Alkenes with one carbon-carbon double bond and π electron.
Kekule and his colleagues knew that benzene had the structure of C6 H6 ,
generically Cn Hn , and thus didn’t fit in either alkanes nor alkenes. Plus
benzene was far more hydrophobic (“oily”) and water insoluble than alkenes
and alkenes. While the C6 H6 chemical formula of benzene was known, the
structure was not. Finally, Kekule reported that he had a dream in which
snakes of various lengths represented the different hydrocarbon chains and
one snake swallowed its tail, forming a ring.9 This, Kekule concluded, was
benzene in which three carbon-carbon double bonds occur among the six
carbons in the ring.
There are two alternative configurations of the three double bonds and π
electrons among the six possible locations. Where are the extra π electrons?
Two types of explanations approach the question. According to valence
bond theory, the double bonds and π electrons shift locations, resonating
between two equally stable configurations (Fig. 18.3a). As double bonds are
shorter than single bonds, the carbon atoms are pulled and pushed slightly,
vibrating with electron states. In valence bond theory, benzene is a linear
superposition of wave functions of the two states.
In molecular orbital theory, π electrons are considered delocalized over
the entire carbon ring, as toroidal π electron cloud orbitals above and below
the plane of the hexagonal carbon ring, with carbon locations also slightly
delocalized. Thus in both valence bond and molecular orbital approaches,
the structure of benzene/phenyl rings is best described quantum mechanically and the electron locations represented as a ring (Fig. 18.3b).
9 In
The Act of Creation, Arthur Koestler called this “probably the most important
dream in history since Joseph’s seven fat and seven lean cows”. The snake eating its tail
is also represented in the mythical “Ourabouris”.
That’s Life!—The Geometry of π Electron Clouds
411
Fig. 18.3. (a) Benzene/phenyl structure with two possible electron configurations. The
structure is said to resonate between the two states. (b) Quantum resonance representation of the two states. (c) and (d) Resonance states of the indole ring of tryptophan.
Benzene/phenyl rings, along with more complex indole rings (Figs. 18.3c
and d), are often called aromatic rings10 and comprise several amino acid
side groups (e.g. phenyl rings of phenylalanine and tyrosine, and the indole
ring of tryptophan). These are the aromatic amino acids.
Aromatic ring π electrons can also be excited into higher energy orbitals
(“excited states”) by absorption of photons of specific wavelengths. As the
excited states return to lower energy ground states, photons are emitted
as fluorescence. Becker et al. (1975) showed fluorescence resonance energy
transfer between subunits in a microtubule lattice (i.e. photon transfer from
tryptophan on one subunit to tyrosine on another).
Many biomolecules are amphiphilic—having both polar/hydrophilic and
non-polar/hydrophobic regions.11 For example components of lipid membranes, proteins and nucleic acids may have an oil-like, non-polar aromatic
ring region at one end of a linear molecule, with the opposite end having
polar charges.
Consider a toy amphiphilic biomolecule (Fig. 18.4) that can represent
aromatic amino acids in proteins, lipids in membranes or nucleic acids
in DNA and RNA. The biomolecule has a non-polar end consisting of a
phenyl ring, and a polar end consisting of separated positive and negative
charges.12
10 Original preparations were fragrant, however the smell was found to come from contaminants.
11 Hydrophobic is equivalent to lipophilic—highly soluble in a non-polar lipid medium.
12 As in a carboxyl COOH− .
412
Quantum Aspects of Life
Fig. 18.4. (a) An amphiphilic toy biomolecule consistent with lipids in membranes, nonpolar amino acids in proteins and nucleic acids in DNA and RNA. (b) Two amphiphilic
biomolecules coalesce, driven by the hydrophobic effect and van der Waals London forces.
(c) (Left) Polar ends of the amphiphilic biomolecule interact with water.
In aqueous solution, non-polar hydrophobic ends avoid the polar water
environment, coalesce by the hydrophobic effect and, when close enough,
attract each other by van der Waals London forces. Opposite end hydrophilic groups face outward into the polar, aqueous environment, stabilizing the hydrophobic core. This is rudimentary self-organization, precisely
how membranes and proteins form.
Membranes are double layers of amphiphilic biomolecules (Fig. 18.5a).
The planar internal region is non-polar and hydrophobic, and exteriors are
polar and hydrophilic, interacting with water. Membranes allow compartmentalization of cells and internal regions, and with ion channels, pumps
and other proteins regulate the cell milieu. But lipid membranes are rather
fluid, not lattice-like, and may lack abilities to represent and control discrete
information.
To be functionally significant, quantum states of π electron resonance
clouds must couple to mechanical or chemical states of their host molecules,
e.g. protein conformation. It appears that certain proteins may be optimally
designed as quantum levers, amplifying quantum processes to exert causal
efficacy in the classical world.
That’s Life!—The Geometry of π Electron Clouds
413
Fig. 18.5. (a) Amphiphilic biomolecules can form double layers, as in cell membranes,
with an internal non-polar hydrophobic planar region and external polar, hydrophilic
groups which interact with water. (b) Non-polar groups can coalesce to form pockets,
as in proteins.
18.5.
Proteins—The Engines of Life
Proteins are the molecular machines of living systems, exerting force and
causing movement by changing shape, e.g. opening and closing of ion channels, bending and sliding of filaments in muscle, movement of motor proteins along microtubules, assembly of actin gel, grasping of molecules by
enzymes and receptors, and flexing of tubulin subunits within microtubules.
These coordinated and purposeful molecular movements are the currency
of real-time living processes. Their organization and regulation are poorly
understood, and lie close to the essential feature of living systems.
Proteins are produced as linear strings of amino acid molecules linked
by chemical peptide bonds to form peptide chains. Twenty different amino
acids comprise our proteins, each distinguished by a particular chemical
“side group” (or “residue”) attached to the peptide chain like charms on
a bracelet. The specific sequence in which various amino acids are strung
together is coded in our genes, and the number of possible sequences, and
thus the number of possible proteins is enormous.
414
Quantum Aspects of Life
Fig. 18.6. (a) A linear chain of amino acids, each with a specific residue (R1-R8). (b)
The chain folds as non-polar hydrophobic residues coalesce, interacting cooperatively by
London forces.
But proteins do not remain as linear peptide chains. Because of van
der Waals attractions and repulsions among various side groups, proteins
“fold” into three dimensional conformational structures that minimize the
protein’s energy. The number of possible attractions and repulsions among
side groups is huge, and predicting three dimensional conformation and energy minimum from amino acid sequence is a computational feat of colossal
proportions. But proteins fold quickly using the hydrophobic effect and van
der Waals forces (Fig. 18.5b, Fig. 18.6). During folding, non-local interactions among aromatic rings suggest quantum mechanical sampling of all
possible folding pathways [Klein-Seetharaman et al. (2002)].
Once formed, protein 3-dimensional structure is stabilized and dynamically regulated in the aqueous phase by outwardly-facing polar side groups,
and from within by non-polar regions. Coalescence of two or more non-polar
amino acid side groups, e.g. a stack of two aromatic rings13 form extended
electron cloud regions called hydrophobic pockets with distinctively low λ
(e.g. benzene-like) solubility values. The largest hydrophobic pockets are
relatively small, (∼0.3 cubic nanometers, roughly 1/100 to 1/30 the volume
of single proteins) yet enable quantum London forces to regulate protein
dynamical functions.
13 Along
with non-aromatic, non-polar amino acids including glycine, alanine and valine.
That’s Life!—The Geometry of π Electron Clouds
415
Fig. 18.7. Seven tubulin proteins in the skewed hexagonal lattice in microtubules
(Figs. 8 and 9). Dark circles within each tubulin are non-polar regions, e.g. π electron
clouds. Large circles are non-polar binding sites of the drug paclitel (taxol); small circles
are sites of indole rings of tryptophan (from Hameroff, 2002), whose 3-d locations are
here projected onto 2-d. Each tubulin is 8 nanometres by 4 nanometres by 4 nanometres,
so non-polar π electron resonance regions are separated roughly by 2 nanometres.
One particular protein is tubulin, a 110 kiloDalton peanut-shaped
dimer, which self-assembles into skewed hexagonal cylindrical lattices in
microtubules (Figs. 18.7 and 18.8). Tubulin has one large non-polar region
below the narrow “hinge,”14 which binds anesthetic gas molecules, as well
as the anti-cancer drug paclitaxel (Taxol), used to paralyze microtubules
in out-of-control mitosis.15 Tubulin has other smaller non-polar regions,
for example 8 tryptophans per tubulin, with π electron-rich indole rings
distributed throughout tubulin with separations of roughly 2 nanometres
(Fig. 18.7).
Periodic placement of tryptophans and other non-polar π cloud regions
within 2 nanometres of each other in microtubule subunits can enable
14 Each tubulin is a “dimer” composed of alpha and beta monomers. The paclitaxel site
is in the beta monomer, below the narrow hinge.
15 Tubulins also bind non-polar anesthetic gas molecules, presumably in the same site at
which paclitaxel binds.
416
Quantum Aspects of Life
Fig. 18.8. A microtubule composed of peanut-shaped tubulin proteins (the microtubule
is surrounded by a Debye layer formed by negatively charged tubulin C-termini and positively charged ions). Right, top: A single tubulin switches between two conformational
states coupled to London force dipole states in a non-polar hydrophobic pocket. For simplicity, one large π cloud (4 rings) represents the nine or more shown in Fig. 7. Right,
bottom: Quantum superposition of alternate conformations: tubulin as protein qubit.
electron tunnelling, “through-bond” exciton hopping or quantum coherence and entanglement. Conventional wisdom suggests that electron tunnelling or exciton hopping in proteins is only possible over distances under
1 nanometre. This is the “Förster distance” (maximum length of an excitation to travel). However the Förster distance pertains to free hopping via an
inert medium like an ionic solution. Within proteins, electron movements
may be facilitated by “through bond hopping” over distances of 2 nanometres or more, e.g. from π cloud to an adjacent π cloud in a non-polar phase.
In some enzymes, electron hopping between amino acid residues may span
3.5 nanometres or more [Wagenknecht et al. (2000)].
In repetitive structures like the cytoskeleton, π clouds separated by less
than two nanometres are in lattice geometries which extend throughout cell
volumes. Thus electron tunnelling, exciton hopping, long-range classical
and non-local quantum processes can lead to entanglement, superposition
and quantum computation extending throughout cell volumes, and possibly
between cells and throughout organisms via tunnelling through gap junctions [Hameroff (1997); Hameroff and Tuszynski (2003); Hameroff (2004)].
Moreover cytoskeletal-based quantum processes can couple to biomolecular mechanical resonances, as evidenced by simulated coherent phonon
That’s Life!—The Geometry of π Electron Clouds
417
resonances which match functional sites in microtubule lattice structure
[Samsonovich et al. (1992)]. Thus geometric distributions of non-polar π
electron clouds can enable a collective, cooperative quantum process—a
unitary wave function—mediating perception and purposeful behaviour of
living organisms through the governing of conformational states of individual proteins.
Proteins and their components move between different conformations/energy minima at multiple size and time scales. Amino acid side
chains wiggle in femtoseconds (10−15 seconds), and longer time scale transitions last many seconds. Spontaneous global and functional protein transitions occur in the range from 10−6 to 10−11 seconds (microseconds to 10
picoseconds), with nanoseconds (10−9 secs) being a representative approximation.
Proteins have large energies with thousands of kiloJoules per mole available from amino acid side group interactions, but are only marginally stable
against abrupt unfolding (i.e. exploding) by approximately 40 kiloJoules
per mole. Consequently protein conformation is a “delicate balance among
powerful countervailing forces” [Voet and Voet (1995)]. At least in some
proteins, higher energy, longer time scale chemical and ionic bonds cancel
out and London forces acting in hydrophobic pocket π electron clouds tip
the balance; they are the switch or lever amplifying quantum electron states
to choose conformations and energy minima.
Electron cloud states couple to protein nuclear motions (and thus conformation) via the Mossbauer recoil effect [Sataric et al. (1998); Brizhik et
al. (2001)]. But because of the extremely small electron mass relative to
nuclear protons and neutrons, the conformational movement due to recoil
is slight: a one nanometre shift of a single electron moves a carbon atom by
only 10−8 nanometres, the diameter of its nucleus.16 However the electrical
charge on each electron is equivalent in magnitude to that on each nuclear
proton. Collectively acting London dipole forces are thus able to influence
nuclear motion and protein conformation by charge movements and, to a
lesser extent, recoil [Conrad (1994)].
Biophysicist Herbert Fröhlich (1968, 1970, 1975) proposed that fluctuating electron dipoles—London forces—in “non-polar regions” of proteins
in geometrical lattices constrained in a voltage gradient (e.g. membrane or
cytoskeletal proteins) would oscillate collectively, forming a laser-like quantum coherent state (essentially a pumped Bose-Einstein condensate). Some
16 The Fermi length—also the separation distance for tubulin superposition in the Orch
OR model.
418
Quantum Aspects of Life
evidence supports biological “Fröhlich coherence”, for example in geometrically arrayed protein scaffoldings in photosynthesis [Engel et al. (2007)].
Additional support for the essential importance of London forces in nonpolar hydrophobic protein pockets of π electron resonance clouds is the
mechanism of anesthesia.
18.6.
Anesthesia and Consciousness
When inhaled into the lungs, and then dissolving in blood and then brain
at a specific concentration, anesthetic gas molecules have the remarkable
property of selectively erasing consciousness while having very few effects
on other brain and bodily functions. It turns out that anesthetic gases act
solely via quantum London forces in hydrophobic pockets in a subset of
brain proteins.
At the turn of the 20th century, Meyer and Overton showed that anesthetic potency of a wide variety of gas molecules correlated with their solubility in a non-polar, lipid-like medium of low Hildebrand solubility λ
15.2 to 19.3 SI Units, resembling olive oil and benzene.17 Consciousness
emanates from this low λ, non-polar medium.
After Meyer and Overton, the medium for anesthetic action (and thus
consciousness) was assumed to be lipid membranes. But Franks and Lieb
(1984) demonstrated that anesthetic gases act in protein hydrophobic pockets of low λ, binding by London forces. Other work demonstrated a cut-off
effect; molecules expected to be anesthetic have no effect if they are larger
than a critical sub-nanometre size threshold (∼0.3 cubic nanometre). Anesthetic gases must be small enough to fit into π electron cloud pockets,
blocking endogenous London forces that are, in some way, responsible for
consciousness.
Other non-polar regions may be too small for anesthetic gas molecules,
e.g. single aromatic rings, π electrons in alkenes and electron clouds in
methyl groups. As consciousness is erased from large hydrophobic pockets,
nonconscious quantum processes can continue among these smaller regions.
Life goes on.
Mapping of intra-cellular regions according to Hildebrand solubility parameter λ would identify a non-polar, low λ phase in which London forces
operate in π electron clouds. Included would be internal planar regions
17 Lambda values of e.g. diethyl ether 15.4, chloroform 18.7, trichcloroethylene 18.7 SI
Units.
That’s Life!—The Geometry of π Electron Clouds
419
of lipid membranes, π electron stacks in DNA and RNA, and discrete hydrophobic pockets and smaller non-polar regions in proteins, including cytoskeleton. Because of their geometric distribution throughout cell volumes,
we will consider primarily electron clouds in protein hydrophobic pockets
in cytoskeletal protein assemblies.
18.7.
Cytoskeletal Geometry: Microtubules, Cilia and
Flagella
Cellular movements and activities including mitosis, growth and adaptive
behaviours are organized by microtubules—self-assembling cylindrical protein polymers, which are the main girders in the cell’s three dimensional
cytoskeleton. But in addition to bone-like support, microtubules also appear to function as the cellular nervous system and on-board computer.
In the 1950s neuroscientist Charles Sherrington (and in the 1970s biologist Jelle Atema) suggested microtubules and other cytoskeletal structures
might process information.
Microtubules are hollow cylindrical polymers of individual peanutshaped protein subunits called tubulin. The cylinders have internal hollow
cores of 15 nanometre diameter, with an external diameter of 25 nanometres; lengths may vary from hundreds of nanometres to meters in the case
of peripheral nerve axons. Thirteen filamentous tubulin chains (“protofilaments”) align so that cylinder walls are slightly skewed hexagonal lattices
(Fig. 18.7). Microtubule lattice geometry is suitable for computation if
states of each tubulin correspond with information, i.e. bit states (Figs. 18.7
and 18.8). Neighbour interactions allow simulated microtubules to act as
cellular automata and process information, with capacity for long term
programming (memory) encoded in post-translational modifications of individual tubulins [Hameroff and Watt (1982); Hameroff (1987); Rasmussen
et al. (1990)].
As computational lattices, microtubles are unique. Not only are they
cylindrical with hexagonal lattices, but their skewed geometry results in
winding pathways whose intersection on any protofilament reflects the Fibonacci series. Intersections of winding patterns coincide with attachment
sites of microtubule-associated proteins (“MAPs”) which interconnect microtubules to form 3-dimensional scaffoldings which determine particular
cellular architecture, function and behaviour [Lee et al. (1986)]. Such
MAP attachment patterns correspond with simulated coherent phonon resonances in microtubule lattice structure [Samsonovich et al. (1992)].
420
Quantum Aspects of Life
Fig. 18.9. (a) Classical cellular automata information processing model in microtubules
based on neighbour dipole interactions (e.g. Rasmussen et al. 1990), (b) Winding
patterns along adjacent tubulin dimers or monomers. Intersections of the patterns (not
shown) match attachment sites of MAP proteins.
Microtubules in neuronal brain dendrites have a distinct and unique arrangement and play essential roles in brain functions. Penrose and Hameroff
(1995); Hameroff and Penrose (1996a,b); Hameroff (1998a,b); Woolf and
Hameroff (2001); Hameroff (2007) have proposed a model of consciousness
based on quantum computation in microtubules mediated by Penrose objective reduction, and timed to gamma synchrony EEG. A summary of the
model is included in Appendix 2.
Applying quantum field theory, Del Giudice et al. (1982, 1983) concluded that electromagnetic energy penetrating into cytoplasm would selffocus inside filaments whose diameters depended on symmetry breaking
(“Bose condensation”) of ordered dipoles (e.g. those occurring in actin gels).
They calculated a self focusing diameter of about 15 nanometres, precisely
the inner diameter of microtubules. Along similar lines, Jibu et al. (1994);
Jibu and Yasue (1995) calculated that Fröhlich dynamics of ordered water
on microtubule surfaces, particularly the internal hollow core of microtubules, would result in quantum optical modes termed super-radiance and
self-induced transparency.
That’s Life!—The Geometry of π Electron Clouds
421
Applying zero point energy of the quantum vacuum, Hall (1996) calculated the Casimir force on a microtubule. As the force depends on d−4
(where d is the gap excluding certain photons), and setting d equal to the
microtubule inner core of 15 nanometres, Hall calculated a Casimir pressure of 0.5 to 20 atmospheres on a microtubule depending on its length,
c.f. [Hameroff (1998b)]. London forces are closely related to Casimir forces
and the quantum vacuum.18
Microtubules occur not only as individual cylinders, but also fuse longitudinally into doublets and triplets. In turn, nine microtubule doublets
or triplets align longitudinally to form indispensable barrel-like structures
known as cilia, centrioles and flagella found widely in living cells. In some
cases a microtubule doublet occurs in the middle of the nine doublets/or
triplets barrel—the well known “9&2” structure.
Hundreds of membrane-covered cilia may project outward from cell surfaces, acting as sensors to transmit information about the outside environment (sensory cilia), and like motile oars to efficiently move cells (motor
cilia). Cilia are anchored to cytoskeletal structures within cytoplasm. Some
living cells use long, single flagella for whip-like movements, e.g. flagellates and spirochetes.19 Centrioles (part of the cell centrosome) contain
two cilia-like barrels in perpendicular tandem which act as the focal point
of the cytoskeleton, located adjacent to the cell nucleus. During cell division/mitosis, centriole barrels separate and each form new barrels—the
new centriole tandems move to become the focal points of daughter cells,
pulling chromosomes apart in a precisely choreographed dance.20 In many
cells (including brain pyramidal cell neurons) one centriole barrel elongates
and, covered by cell membrane, pushes outward above the cell surface to
form the “primary cilium” thought to function as a chemical antenna.
Motor cilia and flagella bend by contractions of (“dynein”) protein struts
connecting the doublets. The movement requires dynein consumption of
ATP, but the coordination and timing is unknown. Atema (1974) suggested
propagating conformational changes along microtubules provide timing to
organize ciliary contraction and cell movements. Centrioles move differently, rotating through cell interiors like an Archimedian screw.
18 A string theory approach to microtubules has also been developed (e.g. [Nanopoulos
(1995)]).
19 The proto-organisms suggested by Margulis (1975) as contributing to formation of the
animal cell by endosymbiosis.
20 The precision is poorly understood, prompting suggestion of quantum entanglement
between daughter centrioles [Hameroff (2004)].
422
Quantum Aspects of Life
Fig. 18.10. Structure of cilia, flagella and centriole barrel (i.e. half centriole). Nine
triplets (or doublets) of longitudinally fused microtubules align and are connected by
linking proteins to form a mega-cylinder. In some cases (e.g. motor cilia), a central
pair of fused microtubules, or other structures, occupy the inner core. The dimensions
of the inner core of the mega-cylinder are roughly 750 nanometres in length, and 150
nanometres diameter, suitable for optical waveguide and detection functions depending
on the inner core permittivity. These structures are found in primitive photoreception
and vision, and within all retinal rod and cone cells, and may detect quantum properties
of photons.
In addition to sensing pressure and chemicals, sensory cilia also detect
light, being the main components of primitive visual systems. In our retinas, light passes through cilia in rods and cones to reach photodetectors.21
In euglena and other organisms, the flagellar base detects light, and centrioles also detect photons and orient cell movement accordingly. AlbrechtBuehler (1992) has shown that centrioles respond to infra-red photons generated by other cells.
Cilia, centrioles and flagella are all capable of photodetection and share
the same general structure—a barrel or cylinder whose dimensions are precisely related to wavelengths of light from infra-red through visible and
ultra-violet. Waveguide properties have been suggested for cilia, centrioles, and flagella that could include the ability to detect quantum optical
properties of photons including polarization, angular and orbital momenta.
Overall, the cytoskeleton (microtubules, actin, intermediate filaments,
linking proteins, centrioles, cilia etc.) organizes real-time intracellular activities. In polymerized gel form, actin (which also binds anesthetics and
contains hydrophobic pockets) fills the cell interior and orders cell water,
21 The possibility has been raised that retinal cilia extract quantum information from
incoming photons [Hameroff (2004)].
That’s Life!—The Geometry of π Electron Clouds
423
providing a quasi-solid state environment. Microtubules and MAPs are
embedded in the actin gel, their non-polar, hydrophobic pocket electron
clouds especially isolated from electrostatic water interactions.
Thus π electron resonance clouds are geometrically arrayed as lattices
of isolated pockets in cytoskeletal polymers as well as membrane interiors
and nucleic acids. Quantum states in the interior of one particular cell
may become entangled with those in adjacent cells and throughout the
organism by tunnelling through gap junctions, window-like openings which
also electrically couple membranes. Such cooperative quantum processes
suggest a promising explanation for an underlying unifying feature in living
systems—“quantum vitalism”. However to most scientists, such quantum
processes seem unlikely in biological conditions due to decoherence.
18.8.
Decoherence
The enticing possibility of quantum interactions unifying and regulating living systems faces the seemingly daunting issue of decoherence. Quantum
computing requires superposition of information states (quantum bits, or
“qubits”), which interact/compute by nonlocal entanglement. When measured, quantum superpositions reduce/collapse to classical states as the
solution.
Technological quantum computing (e.g. ion trap quantum computing)
is plagued by decoherence—disruption of seemingly delicate quantum processes by thermal and other environmental interactions. So decoherence
must be avoided long enough for quantum computations to proceed. For
technological quantum computing, this necessitates extreme cold and isolation to avoid decoherence and loss of qubit superposition before the computation is completed (although quantum entanglement occurs in ambient
temperatures in atmosphere). Thus biological systems are assumed too
“warm, wet and noisy” for useful supra-molecular quantum processes.22
22 Physicist Max Tegmark (2000) calculated that microtubule quantum states decohere
far too quickly (10−13 seconds) at brain temperature to exert useful neurophysiological
effects. However Tegmark’s calculations ignored stipulations of the Penrose-Hameroff
model to avoid decoherence. These include 1) transiently encasing bundles of dendritic
microtubules in actin gel—an isolated, shielded, and water-ordered non-liquid environment for quantum processes, 2) quantum states extending among dendritic gel environments via quantum tunnelling and/or entanglement through window-like gap junctions
of dendritic webs, 3) microtubule quantum error correction topology [Hameroff et al.
(2002)] and 4) biomolecular quantum states pumped by, rather than disrupted by, heat
energy. Hagan, Hameroff & Tuszynski used Tegmark’s decoherence formula with Orch
OR stipulations and calculated microtubule decoherence times in hundreds of milliseconds or longer—sufficient for neurophysiological effects [Hagan et al. (2002)].
424
Quantum Aspects of Life
Fig. 18.11. (Top) Quantum spin transfer occurs between quantum dots (QD1 and QD2 )
via π electron rings. (Bottom) Quantum spin transfer increases with temperature from
77 K to brain temperature 300 K. (Modified after Ouyang and Awschalom (2003)).
But π electron resonance clouds avoid water and are hence not “wet”.
Moreover evidence suggests that biological systems can utilize heat energy (“warm, noisy”) to promote quantum processes. Indeed, Engel et al.
(2007) demonstrated quantum coherence at significantly warm temperatures in photosynthesis. Specifically, photon energy is transported through
all possible pathways of the protein scaffolding surrounding the photosynthetic chlorophyll, demonstrating quantum coherent “beating”. Ouyang
and Awschalom (2003) showed that quantum spin transfer through organic
benzene π electron resonance clouds is enhanced as temperature is increased
(Fig. 18.11).
Microtubules and microtubule assemblies appear to have specific attributes to avoid decoherence and functionally utilize quantum processes:
(1) At physiological pH, microtubules are covered by a Debye plasma layer
due to protruding tubulin C-termini tails which attract counter ions
(Fig. 18.7), thus screening microtubule quantum processes from interactions with surrounding cytoplasm.
(2) When embedded in actin gel with ordered water, microtubules may
be strongly coupled to this environment, limiting degrees of freedom
That’s Life!—The Geometry of π Electron Clouds
(3)
(4)
(5)
(6)
425
and engendering a “decoherence-free subspace” via the quantum Zeno
effect.
Microtubules appear to have quantum error correction topology based
on Fibonacci series winding pathways which match biomechanical resonances. Qubits based on winding pathways (Fig. 18.9b) would thus
be very robust.
Microtubules appear to be suited to quantum field effects via symmetry
breaking and Casimir force effects.
Microtubules have non-polar π electron clouds spatially arrayed within
2 nanometres of each other in tubulin subunits, and thus throughout
microtubules.
Assemblies of microtubules (cilia, centrioles, flagella) may utilize quantum optical waveguide effects.
18.9.
Conclusion
As reductionist biology reveals ever-greater detail, the essential nature of
living systems—what life actually is—becomes more and more elusive.23
It is time to reconsider pioneering suggestions by Erwin Schrödinger, Albert Szent-Györgyi, Alberte and Phillip Pullman, Herbert Fröhlich, Michael
Conrad and many others that life’s essential feature involves quantum effects in and among geometrically arrayed π electron resonance clouds in
non-polar regions of carbon-based chemistry.
Interiors of living cells are composed of cytoplasm, the aqueous phase of
cell water and ions. Cytoplasm alternates between a true liquid medium and
a quasi-solid gel, as cytoskeletal actin polymerizes to form dense meshworks
of protein filaments on which water molecules become ordered. Within and
surrounding both liquid and gel cytoplasmic states are solid structures:
membranes, protein assemblies, nucleic acids and organelles (in cytoplasmic
gels, actin meshworks are also solid structures). Buried within the solid
structures are non-polar, hydrophobic oil-like sub-nanometre regions of low
Hildebrand solubility coefficient λ. These non-polar islands are typified by
benzene-like aromatic rings with π electron resonance clouds. Isolated from
the aqueous, polar environment (particularly when cytoplasm is in orderedwater actin gel states), quantum London forces within these clouds operate
to govern biomolecular function.
23 The
Myth of Sisyphus.
426
Quantum Aspects of Life
Particular geometric distributions of π cloud London forces can extend
throughout cells and couple to biomolecular mechanical resonances. Quantum states in cytoplasm may extend to neighbouring cells, and throughout organisms by tunnelling through window-like gap junctions [Hameroff
(1997)], enabling long-range and non-local collective quantum processes,
e.g. superpositions, entanglement and computation. Unitary quantum wave
functions can govern purposeful activities of living organisms.
Conventional wisdom (decoherence theory) predicts heat and environmental interactions prevent supramolecular quantum states in biology.
However decoherence theory cannot account for warm quantum states in superconductors and semiconductors [Lau et al. (2006); Stern et al. (2006)],
c.f. [Amin et al. (2006)]. More importantly, the only experiments that
have tested decoherence in biomolecules have shown 1) quantum coherence
in photosynthesis scaffolding proteins [Engel et al. (2007)] and 2) quantum
spin transfer through π electron clouds [Ouyang and Awschalom (2003)] are
both enhanced by increased temperatures. The evidence—still meager—is
on the side of biomolecular quantum processes.
Organized quantum processes would offer huge advantages to living systems. Quantum computations could enable biomolecules, cells and organisms to sample all possible energy and information transfers, aspects of perceptions and possible responses before choosing particular actions.24 Entanglement allows immediate communication or correlations. Thus, if feasible, organized quantum processes would have emerged during the course
of evolution to enhance survival. However it is also possible that life began
in simple oil-water interfaces, that living systems are processes on the edge
between the quantum and classical worlds, and that biology evolved to best
utilize cooperative quantum processes.
That’s life.
Acknowledgements
I thank Dave Cantrell for artwork, the YeTaDeL Foundation for support,
Professor Jack Tuszynski and Sir Roger Penrose for collaboration and inspiration.
24 And the potential for consciousness, assuming it depends on the Orch OR mechanism
or something similar.
That’s Life!—The Geometry of π Electron Clouds
427
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About the author
Stuart Hameroff is an anesthesiologist and Professor of Anesthesiology
and Psychology at the University of Arizona in Tucson, Arizona. He received his MD from Hahnemann College, Philadelphia, Pennsylvenia, in
1973. He has teamed with Sir Roger Penrose to develop the “Orch OR”
(orchestrated objective reduction) model of consciousness based on quantum computation in brain microtubules, and has also researched the action
of anesthetic gases. As Director of the University of Arizona’s Center for
Consciousness Studies, Hameroff organizes the biennial “Tucson conferences” Toward a Science of Consciousness, among other Centre activities.
His website is www.quantumconsciousness.org.
Appendix 1 Quantum computing in DNA π electron stacks
DNA may utilize quantum information and quantum computation. DNA
base pairs all have π electron rings with inducible dipoles, forming a π
electron stack in the central core of the double helix. Superpositions of
base pair dipole states consisting of purine and pyrimidine ring structures can play the role of qubits, and quantum communication (coherence,
entanglement, non-locality) can occur in the “π stack” region of the DNA
molecule.
That’s Life!—The Geometry of π Electron Clouds
431
The “π electron stack” is the internal core of the DNA molecule comprised of the purine and pyrimidine ring structures of the base pairs which
are always either Adenine (purine) and Thymine (pyrimidine, “A-T”), or
Guanine (purine) and Cytosine (pyrimidine, “G-C”). Purines have a double ring structure, with a 6 member ring fused to a 5 member ring, whereas
pyrimidines have a single 6 member ring. (The complementary base pairs
are held together by hydrogen bonds—2 between A and T, and 3 between
G and C.) Thus each base pair always consists of one 6/5 purine ring and
one 6 pyrimidine ring.
Each A-T and G-C base pair also has a dipole—a type of van der Waals
London force due to mutually induced polarizations between electron clouds
of the purine and pyrimidine rings. At any particular time an electron
negative charge may be shifted either toward the purine ring, or toward the
pyrimidine ring (with corresponding conformational shifts).
For the A-T base pair we can have negative charge more localized
toward the adenine purine ring, e.g. A→T, or more toward the thymine
pyrimidine ring A←T. For the base pair G-C we can similarly have G→C,
or G←C.
But as these dipole couplings are quantum mechanical they can exist
in superposition of both possibilities. So quantum mechanically we can
have: Both A→T and A←T which eventually reduce to either A→T or
A←T
As well as:
Both G→C and G←C which eventually collapse to either G→C or
G←C. Using quantum nomenclature we can refer to the quantum superpositions of both possible states | A→T + A←T .
And similarly: | G→C + G←C .
Such superpositions may act as “qubits”, bit states which can exist in
quantum superposition of, e.g. both 1 AND 0. DNA could function as a
quantum computer with superpositions of base pair dipoles acting as qubits.
Entanglement among the qubits, necessary in quantum computation is accounted for through quantum coherence in the π stack where the quantum
information is shared. Consider a string of three base pairs: A-T G-C G-C
A-T can be either A→T or A←T, or quantum superposition of both
| A→T + A←T
G-C can be either G→C or G←C, or quantum superposition of both
| G→C + G←C .
As each pair may be in two possible dipole states mediated by quantum mechanical interactions, the 3 base pairs may be seen as a quantum
432
Quantum Aspects of Life
superposition of 8 possible dipole states:
A→T A→T A→T A→T
G→C G←C G→C G←C
G→C G→C G←C G←C
A←T A←T A←T A←T
G→C G←C G→C G←C
G→C G→C G←C G←C.
Each dipole differs slightly due to structural differences, so for example A←T and G←C have slightly different dipoles though pointing in the
same general direction whereas A←T and G→C have more or less opposite
dipoles. The slight differences will introduce irregularities in the π stack
quantum dynamics, and couple to mechanical/conformational movements
of the DNA strand. Net and complex dipoles within the π stack may show
emergent phenomena. Particular dipoles corresponding to loops, hairpins,
dyads etc. may have specific properties. Superconductive DNA loops, for
example, could function in a way analogous to SQUIDs (superconductive
quantum interference devices). Squids have a superconductive ring with
one segment of lower conductance; current through the ring is highly sensitive to dipoles. DNA loops may serve as quantum antenna, with nonlocal
communication with other DNA, and perhaps cell machinery. We can then
consider DNA as a chain of qubits (with helical twist). Output of quantum
computation would be manifest as the net electron interference pattern in
the quantum state of the π stack, regulating gene expression and other
functions locally and nonlocally by radiation or entanglement.
Appendix 2 Penrose-Hameroff Orch OR model
The Penrose-Hameroff Orch OR model proposes that microtubule (MT)
quantum computations in neurons are orchestrated by synaptic inputs and
MT-associated proteins (MAPs), and terminate (e.g. after 25 msec, 40 Hz)
by the Penrose objective reduction (“OR”) mechanism. Hence the model
is known as orchestrated objective reduction, “Orch OR”. Complete details may be found in [Penrose and Hameroff (1995); Hameroff and Penrose
(1996a,b)] and [Hameroff (1998a)]. The key points are:
(1) Conformational states of tubulin protein subunits within dendritic MTs
interact with neighbor tubulin states by dipole coupling such that MTs
process information in a manner analogous to cellular automata, regulating neuronal activities (trigger axonal spikes, modify synaptic plasticity and hardwire memory by MT-MAP architecture etc.).
That’s Life!—The Geometry of π Electron Clouds
433
(2) Tubulin conformational states and dipoles are governed by quantum
mechanical London forces within tubulin interiors (non-polar hydrophobic pockets of low Hildebrand solubility) so that tubulins may exist as
quantum superpositions of differing conformational states, thus acting
as quantum levers and qubits.
(3) While in superposition, tubulin qubits communicate/compute by entanglement with other tubulin qubits in the same MT, other MTs in
the same dendrite, and MTs in other gap junction-connected dendrites
(i.e. within a dendritic web or hyper-neuron). Thus quantum computation occurs among MTs throughout macroscopic regions of brain via
tunnelling through gap junctions of adjacent neurons or other mechanisms.
(4) Dendritic interiors alternate between two states determined by polymerization of actin protein: a) In the liquid (solution: sol) state, actin
is depolymerized and MTs communicate/process information classically
(tubulin bits) with the external world. During this phase synaptic activities provide inputs via MAPs which orchestrate MT processing and
(after reduction) MT (output) states regulate axonal firing and synaptic plasticity. b) As actin polymerizes (e.g. triggered by glutamate
binding to receptors on dendritic spines), dendritic cytoplasm enters a
quasi-solid gelatinous (gel) state in which cell water is ordered in cytoskeletal surfaces. MTs become isolated from environment and enter
quantum superposition mode in which tubulins function as quantum
bits or qubits. The two states alternate e.g. at 40 Hz.
(5) Quantum states of tubulin/MTs in gel phase are isolated/protected
from environmental decoherence in shielded non-polar regions of low
Hildebrand solubility, encasement by actin gelation and ordered water, Debye screening, coherent pumping and topological quantum error
correction.
(6) During quantum gel phase, MT tubulin qubits represent pre-conscious
information as quantum information—superpositions of multiple possibilities, of which dream content is exemplary.
(7) Pre-conscious tubulin superpositions reach threshold for Penrose OR
(e.g. after 25 msec) according to E = /t in which E is the gravitational
self-energy of the superpositioned mass (e.g. the number of tubulins in
superposition), is Planck’s constant over 2π, and t is the time until
OR. Larger superpositions (more intense experience) reach threshold
faster. For t = 25 msec (i.e. 40 Hz) E is roughly 1011 tubulins, requiring
a hyper-neuron of minimally 104 to 105 neurons per conscious event.
434
Quantum Aspects of Life
The makeup of the hyper-neuron (and content of consciousness) evolves
with subsequent events.
(8) Each 25 msec OR event chooses ∼1011 tubulin bit states which proceed by MT automata to govern neurophysiological events, e.g. trigger axonal spikes, specify MAP binding sites/restructure dendritic architecture, regulate synapses and membrane functions. The quantum
computation is algorithmic but at the instant of OR a non-computable
influence (i.e. from Platonic values in fundamental spacetime geometry)
occurs.
(9) Each OR event ties the process to fundamental spacetime geometry, enabling a Whiteheadian pan-protopsychist approach to the “hard problem” of subjective experience. A sequence of such events gives rise to
our familiar stream of consciousness.
Index
A and J gates, 328
Abbott, Derek, 249, 290, 314, 327,
338, 339, 349, 351, 378, 381
ADP, 129, 136
Aharanov-Bohm effect, 354
Al-Khalili, Jim, 48
allometric rule, 130
Alzheimer’s disease, 116
amino acid, 14, 196
chiral L-type, 197
classes, 198, 205
doubling, 210
R-groups, 196
aminoacyl-tRNA synthetases, 198
ammonia, 43
amyloid plaques, 115
anthropic principle, 384
anticodon, 149, 175, 179, 206
Archaean rocks, 33
Arima, Akito, 290
Arizona State University, 18, 376
artificial intelligence, 396
ATP, 10, 15, 129, 136, 214
Australian National University, 93
automata, 233
one-dimensional, 234
partitioned, 234, 235
quantum, 237
one-dimensional, 238
reversible, 234, 238
Awschalom, David, 354
Baker, Marshall, 345
Bartel, David, 38
base
Adenosine, 149
classification, 152
complementary, 150
conformational states, 175
Cytosine, 149
Guanine, 149
nucleotide, 149
Thymine, 149
Uracil, 149
Bashford, Jim, 185
Battle of Sexes (BoS), 256, 258
Baumert, Mathias, xvii
Bayesianism, 397
Bell inequality, 363
Bell, John, 362
Bennett, Charles H., 229, 355
benzene, 46
ring, 354, 411
Berber, Robert Benny, 345
Berry phase, 4
Berryman, Matthew John, 148, 375
Bezrukov, Sergey, 313, 347, 349, 379
Biham, Ofer, 345
binding problem, viii, 112
Bio-Info-Nano-Systems, xiv
biochemical assembly, 201, 211
biogenesis, 3, 12
biomembranes, 129
435
436
Quantum Aspects of Life
Bioy Casares, Adolfo, 291
Birrell, Nicholas, 18, 377
Bohmian mechanics, 397
Bohr, Niels, 3, 45, 369
Boltzmann, Ludwig, 23, 132
bombardier beetle, 355
Bose-Einstein condensate, 318
Bothma, Jacques, 93
box, 161, 172
family, 150, 166, 179
mixed, 150, 166, 179
Boyle, Robert, 407
branching rule, 159, 162, 169
Brandt, Howard E., 313, 345
Braunstein, Samuel L., 231
bubble nucleation, 12
Buck, Ken, 49
Caldeira-Leggett Hamiltonian, 79
Cartan subgroup, 169
Casimir force, 421
Casimir operators, 169
Cavalcanti, Eric, 397
Caves, Carlton M., 231, 313, 344
cellular automaton, 13, 233
quantum, 13
centriole, 422
Chaitin, Gregory, 26
chaotic dynamics, 395, 397
Cheon, Taksu, 290
Chou-Fasman parameters, 171
chromophore, 80
Chuang, Isaac L., 229
Church-Turing thesis, 223, 325
cilia, 422
classical error correction, 387, 389
Clinton, William Jefferson, 335
codon, 206
coherent excitation, 382
coherent photosynthetic unit
(CPSU), 54
compatabilism, 397
computability, 392, 396
computation
analog, 395
quantum, 395
consciousness, 392, 397
convex linear combination, 259
Conway, John Horton, 233
cosmic imperative, 352
Coulomb interaction, 134
Crick, Francis, vii, 113
cyanobacteria, 36
cytoskeleton, 115, 357, 405
Darwin, Charles, 34
Darwinian competition, 13
Darwinian evolution, 229
Darwinian fitness, 252
Darwinian selection, 189, 193, 210
Davies Limit, 332
Davies, Paul C. W., xiii, 17, 100, 349,
376
Davis, Bruce Raymond, 249, 378
Dawkins, Richard, 291, 383
de Broglie wavelength, 104
de Broglie, Louis, ix, 71
de Duve, Christian, 352
Debye layer screening, 354
Debye model, 82
Debye, Peter, 133
decoherence, 9, 10, 22, 42, 203, 213,
362, 386, 423
time, 44, 386, 394
decoherence-free subspace, 10, 46,
336, 356, 388
Delbourgo, Robert, 186
Demetrius, Lloyd, 146
dense coding, 298
density matrix, 362
Descartes, René, 292
determinism, 396
Deutsch multiversum, 303
Deutsch, David, 293
diamond code, 147
digital
computation, 395
computer, 22
information, 20
life, 39
Digital Equipment Corp, 323
Dirac, Paul, ix
437
Index
directionality theory, 142
DNA, vii, x, 3, 8, 38, 46, 190, 405
lambda-phage, 15
polymer chain, 104
relaxation times, 104
replication, 13
Doering, Charles R., 313, 344
Dombey, Norman, 124
double potential well, 42
doublet code, 207
Dougal, Arwin A., 346
Drabkin, Giliary Moiseevich, 347, 379
Drosophila, 114
dynamic combinatorial library, 41
Dynkin label, 158
Earth, 18, 33, 377
Efficiency, 58
Ehrenreich, Henry, 70
Einstein, Albert, ix
Einstein-Podolsky-Rosen, viii, 119
Eisert, Jens, 349, 363, 379, 400
Elitzur-Vaidman
bomb tester, 300
circuit-breaker, 303
emergence, 403
ENIAC, 317
entanglement, 4, 54, 258
sharing, 60
entropy, 23
evolutionary, 268
relative negentropy, 268
environmental post-selection, 4, 11
enzyme, 214
polymerase, 8, 15
equilibrium
correlated, 252
perfect, 252
sequential, 252
Eshraghian, Kamran, 249, 378
ESS, 255, 263
evolution, 190, 216
cooperation, 193
direction, 192
mutation, 191
selection, 191
evolutionary game theory, 252
evolutionary stability, 253, 261
evolutionary stable strategy (ESS),
251
excitation lifetime, 58
transfer time, 58
Förster distance, 416
Förster interaction, 54
factoring, 389
Faraday, Michael, viii
Fassioli Olsen, Francesca, 69
fault tolerance, 388
Fenna-Matthews-Olson (FMO)
bacteriochlorophyll complex, 52
Fermi length, 361
Fermi’s golden rule, 213
Ferry, David Keane, 313, 346
Fibonacci series, 371
firefly, 355
Fisher, Ronald Aylmer, 253
flagella, 355, 422
Flitney, Adrian Paul, 249
Fox, Geoffrey, 219
Fröhlich coherence, 418
Fröhlich, Herbert, 134, 417
Frauenfelder, Hans, 349, 380
free will, 392, 393, 395, 396
frozen accident, 191
fullerene, 46, 71, 391
G-protein, 111
Gödel’s theorem, 369
Game of Life, 13, 233, 234
birth, 239, 243
classical structures, 236, 237
death, 239, 243
quantum, 238
semi-quantum, 242
interference, 244
structures, 245
survival, 239, 243
Gamow, George, 147
Gardner, Martin, 294
Gea-Banacloche, Julio, 313, 347, 376
general relativity, 393
438
Quantum Aspects of Life
genetic code, 147, 204, 390
co-evolution theory, 153
doublet predecessor, 151
duplication, 204
evolution, 209
hypotheses for evolution of, 153
mitochondrial, 152, 179, 206
physico-chemical hypothesis, 154
predecessor, 207
universal, 205
universal (eukaryotic), 151
variations, 166
genetic information, 190
genetic languages, 193
genotype, 192
Gibbs, Willard, 23
Gilmore, Joel, 93
god of the gaps, 110
Goel, Anita, 16, 107
gravitational objective state
reduction, 363
Green, Frederick, 331
Griffiths, Robert B., 320
Grover’s algorithm, 155, 174, 201,
203, 211, 212, 317, 390
Grover, Luv, 317
Hänggi, Peter, 333
Haldane, John Burdon Sanderson,
vii, 34
Hameroff, Stuart, 109, 349, 377, 430
Hamilton, Alex, 313, 346
Harvard University, 70
Hawking, Stephen, xi
Heisenberg, Werner, 3
Herschbach, Dudley, 107
Heszler, Peter, 369
Hilbert space, 10, 276, 382
Hildebrand solubility, 406
Hoffman, Michael, 372
holographic principle, 332
Hoyle, Fred, 17, 37, 376
human language, 28
hydration correlation function, 83
hydrogen tunnelling, 356
hydrophobic pocket, 85, 353
Hyperion, 396
Imperial College, 49
information
biochemical, 188
information theory, 188, 215
internal combustion engine, 364
inverse-Zeno effect, 11
ion-channels, 336
Iqbal, Azhar, 289
Jacobson, Boris, 359
Jarvis, Peter, 186
Johnson, Neil F., 70
Johnson, Ronald C., 48
K-meson, 43
Kane quantum computer, 338
Kaufman, Stuart, 12
Kelly, Michael, 346
Kish, Laszlo Bela, 314, 347, 375
Kitaev, Alexei, 354
knowledge, 189, 193
Koch, Christoff, 113
Kolmogorov, Andrey, 26
Landauer, Rolf, 316
language, 189
efficiency, 189
Laplace, Pierre Simon, 128
Last Universal Common Ancestor
(LUCA), 151
Laughlin, Robert B., 331
Lavoisier, Antoine, 128
Lebedev, Pyotr Nikolaevich, 365
Lee, Chiu Fan, 69
Lee, Kotik, 330
Lewinsky, Monica Samille, 335
Lidar, Daniel, 313, 345
Lie superalgebra, 148, 157
life, 188, 403
origin, 382
principle, 382
lipid-protein complexes, 136
Lloyd, Seth, 30, 229
439
Index
Macquarie University, 17, 376
Mars, 18, 377
Mathematica, 244
Maxwell, James Clerk, viii, 23
Mayer-Kress, Gottfried, 335, 343
Maynard Smith, John, 253, 355
McFadden, Johnjoe, 49
McKay, Chris, xiii
McKenzie, Ross, 93
meme, 291
memory, 114
Mershin, Andreas, 124
metabolic
efficiency, 141
rate, 127, 140
metabolism, 383
Meyer, David A., 256
Microsoft Office, 321
microtubule, x, 45, 109, 112, 357, 371,
387, 393
Milburn, Gerard J., 16, 379, 400
Miller, Stanley, 35
Miller-Urey experiment, 3, 35
minimal language, 194
DNA, 201
proteins, 195
mRNA, 149
multi-level scale hypothesis, 140
mutant strategy, 264
mycoplasma, 37
capricolum, 166
myoglobin, 361
Nanoarchaeum equitans, 37
nanomotor, 98
Nanopoulos, Dimitri V., 124
NASA, xiii
Nash equilibrium, 251, 252
Nash, John, 256
National Institutes of Health, 359
Building 5, 359, 373
natural selection, 356, 383
neuron, 45, 387, 395
neurotransmitter, 366
dopamine, 366
serotonin, 366
neutrino oscillations, 46
Newcomb’s problem, 396
Newcomb, William, 294
no-cloning theorem, 7, 223
Nostoc punctiforme, 37
NP incomplete algorithms, 325
Ockham’s razor, 26, 291
Ockham, William of, 26
Olaya-Castro, Alexandra, 69
Oparin, Alexander, 34
Oparin-Haldane theory, 34
operational RNA code, 205, 208
operator
Casimir, 156
Hamiltonian, 167, 168, 179
optimal solution, 191
oracle, 390
Orch OR model, 109, 417, 432
orchestrated objective reduction
(Orch-OR), 393
organic chemistry, 3
Ouyang, Min, 354
Pareto optimum, 258
Parrondo, Juan M. R., 369
Patel, Apoorva D., 155, 175, 219
Pati, Arun K., 230
Pauli group, 300
Pauli’s Exclusion Principle, 39
Pawlikowski, Andrzej, 309
payoff
matrix, 272, 277, 280
operator, 260, 276, 282
Penn State University, x
Penrose, Roger, x, 109, 369, 377, 396,
404, 430
Pepper, Michael, 346
philosophy, 382, 395, 396
phosphorylation, 130, 136
photosynthesis, 51
Piotrowski, Edward W., 309
Planck, Max, 131
Plato, 292
polypeptide chains, 196
Popescu, Sandu, 16
440
Quantum Aspects of Life
Poppelbaum, Wolfgang, J., 329
postmodernism, 395
PQ penny flip, 256
predictability, 395
Price, George R., 253
primordial soup, 34
Princeton University, 93
Prisoners’ Dilemma, 256
probabilistic computing, 329
Prochlorococcus marinus, 37
Prodan, 85
protein, 151, 157
amino-acyl tRNA synthetase,
(aaRS), 151, 154, 164, 173,
175
assembly, 155
elongation factor (EF-Tu), 149, 175
porphyrin, 373
structure, 171
tubulin, 370, 373
protein folding, 198, 200
proto-cell, 34
proton
tunnelling, 41, 43, 46
protoplasm, 405
Pullman, Alberte, 404
Pullman, Phillip, 404
purple bacteria, 58
Q-life, 6–8, 15
Q-replicase, 8, 15
qualia, 112
quantization scheme
Eisert-Wilkens-Lewenstein (EWL),
256
EWL, 261
Marinatto-Weber (MW), 256
quantum
uncertainty, 396
algorithm, 389
anti-Zeno effect, 303
coherence, 60, 387
computer, 42, 315
consciousness, 382
consciousness idea, 109
cryptography, 334
decoherence, 40
dots, 324
entanglement, 5, 40, 46, 263
error correction, 317, 371, 387
topological, 361
field theory, 12
fluctuations, 13, 25
gates, 388
gravity, 393, 394
information, 385
jump approach, 56
mechanics
orthodox, 397
mind, 109
mutants, 264
number, 369
replicator, 6
strategy, 263
superposition, 5, 40, 42, 361
teleportation, 317
trajectory, 56
no-jump trajectory, 56
tunnelling, 14, 40, 41, 382
uncertainty, 397
Zeno effect, 10, 44, 46, 75, 302
quantum computing, 385
brain, 382
general purpose, 325
genetics, 382
quantum Darwinism, 292
Quantum Game Model of Mind
(QGMM), 305
quantum games, 261
quantum life
principle, 382, 385
quantum mechanics
interpretation, 397
qumeme, 293
quvirus, 305
radio, 355
Ramachandran map, 196
rapid single flux quantum electronics,
328
Rauch, Helmut, 377
religion, 360
441
Index
Renninger, Mauritius, 301
reproduction, 5
rhodopsin, 111
ribozyme, 38, 204, 209
RNA, 38, 190, 384
RNA world, 204, 216
RNA world hypothesis, 37, 38
Rock-Scissors-Paper (RSP), 280
RSA encryption, 334
Sauls, Jim, 93
Scherrer, Paul, 380
Schrödinger’s cat, ix, 374
Schrödinger, Erwin, vii, xiii, 3, 97,
188, 404
Scott, Tim, 290
Scully, Marlan Orvil, 347, 376
search algorithm, 390
Seaton, Michael, 17, 376
self-replication, 44
semi-quantum life, 238
Sherrington, Charles, 419
Shor’s algorithm, 317
Siemion’s rings, 171
Solmonoff, Ray, 26
Solvay Conference, 320
spectral density, 79
power law, 80
Spiegelman’s Monster, 383
spin-boson model, 79
Stokes shift, 82, 85
Strategy, 254
string theory, 384
structural language, 198
superalgebra, 148
superconductors, viii
superposition, 155, 175, 177, 203, 212,
385, 386
supersymmetric quantum mechanics,
167
supersymmetry
algebras, 157
supersymmetry model, 148
surface plasmons, 120
symmetry breaking, 156, 167, 179
partial, 166, 168
synaptic transduction, 322
Sladkowski, Jan, 309
tautomerization, 41
Tegmark, Max, 110, 356, 387
teleology, 11, 42, 392
television, 364
tetrahedral geometry, 196
Thermus aquaticus, 101
Thomas, Anthony W., 185
Thorne, Kip, 344
Tollaksen, Jeff, 16
Toor, Abdul Hameel, 289
topological quantum memories, 388
Torok, Miklos, 348
transition states, 214
tRNA, 149, 198, 204
acceptor stem, 208, 209
tubulin, 45, 413, 415
tunnelling, 4
Turing machine, 324
quantum, 224
Tuszynski, Jack, 351
universal constructor, 223
University College London, 17, 376
University of Adelaide, 17, 376
University of Cambridge, x, 17, 376
University of Newcastle-upon-Tyne,
17, 376
University of Oxford, x, 70
University of Queensland, 93
University of Surrey, 48
Unruh, William G., 316
Urey, Harold, 35
Uridine, 166, 177
modified, 177
van der Waals force, 46, 365, 382, 407
van der Waals, Johannes Diderik, 407
van der Waals-London force, 353
Vazirani, Umesh, 330
vitalism, 370, 404
Vize, Laszlo, 348
von Kekule, Friedrich August, 409
von Neumann, John, xiv, 224, 251
442
Quantum Aspects of Life
watchdog effect, 4
Watson, James, vii
Watson-Crick pair, 179
wave mechanics, 202
wave-function
collapse, 393, 397
weight label, 160, 162, 169
Wheeler, John Archibald, 300
Wigner inequality, 15, 98
first, 105
second, 102, 105
Wigner’s clock, 14
Wigner, Eugene (Jenő) Paul (Pál), 3,
14, 100, 353
Wilkins, Maurice, vii
Wiseman, Howard, 329, 343, 349,
379, 400
wobble hypothesis
modified, 177
wobble pairing, 150, 152, 164, 177,
206
wobble position, 208, 209
Wolfram, Stephen, 234
Woolf, Nancy, 113
xenon, 361
Yang, Chen Ning Franklin, 359
Zeilinger, Anton, 71, 237, 349, 353,
377, 386
Zienau, Sigurd, 17, 376
Zimm model, 104
Zralek, Marek, 309
Zurek, Wojciech H., 104