Interdisciplinary Applied Mathematics

Springer Science+Business Media, LLC

Interdisciplinary Applied Mathematics

Volume 12

Editors J.E. Marsden L. Sirovich S. Wiggins

Mathematical Biology L.Glass, J.D. Murray

Mechanics and Materials S.S. Antman, R. V. Kohn

Systems and Control S.S. Sastry, P.S. Krishnaprasad

Geophysics and Planetary Science

Problems in engineering, eomputational scienee, and the physical and biological seienees are using inereasingly sophistieated mathematieal teehniques. Thus, the bridge between the math­ematieal seienees and other disciplines is heavily traveled. The eorrespondingly increased dialog between the diseiplines has led to the establishment of the series: Interdisciplinary Applied Mathematics.

The purpose of this series is to meet the eurrent and future needs for the interaction between varlous seienee and technology areas on the one hand and mathematies on the other. This is done, ftrstly, by ~neouraging the ways that mathematics may be applied in traditional areas, as weIl as point towards new and innovative areas of applieations; and, seeondly, by eneouraging other scientifie disciplines to engage in a dialog with mathematicians outlining their problems to both aeeess new methods and suggest innovative developments within mathematics itself.

The series will consist of monographs and high-level texts from researehers working on the interplay between mathematics and other fields of seienee and teehnology.

Arthur T. Winfree

The Geometry of Biological Time Second Edition

With 336 lllustrations

, Springer

Arthur T. Winfree University of Arizona Department of Ecology and Evolutionary Biology 326 Biological Sciences West Tucson, AZ 85721 USA art@cochise.biosci.arizona.edu

Editors J.E. Marsden Control and Dynmnical Systems Mai! Code 107-81 California Institute of Technology Pasadena, CA 91125 USA

S. Wiggins Control and DynaIDical Systems Mai! Code 107-81 California Institute of Technology Pasadena, CA 91125 USA

L.Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA

Mathematics Subject Classification (2000): 92-02, 34C35, 58Fxx, 92Bxx

Library of Congress CataIoging-in-Publication Data Winfree, Arthur T.

The geometry of biological time I A.T. Winfree. - 2nd ed. p. cm. - (Interdisciplinary applied mathematics ; v. 12)

IncIudes bibliographical references (p. ).

1. Biological rhythms-Mathematics. I. TItle. 11. Series. QH527.W55 2000 57 1.7 '7--dc21 00-026155

Printed on acid-free paper.

ISBN 978-1-4419-3196-2 ISBN 978-1-4757-3484-3 (eBook) DOI10.1007/978-1-4757-3484-3

e 2001, 1980 Springer Science+Business Media New York OriginaIly published by Springer-Verlag New Vork, Inc. in 2001. Softcover reprint ofthe hardcover 2nd edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrievaI, electronic adaptation, computer software, or by similar or dissirnilar methodology now known or here­after developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Lesley Poliner; manufacturing supervised by Jerome Basma. Photocomposed copy prepared from the author's MS Word files by SPI Technologies Inc., Ashland, VA.

987654321

SPIN 10750851

• I dedicate this book to my parents,

Dorothy and Van, who first gave me tools.

And I dedicate this book to those readers who, expecting wonders to follow so grand a title as it flaunts, may feel cheated by its actual content. I will be delighted if you take this beginning as a serious challenge .

•

After any interval of fru itfu I innovation catches the spotlight there seems to be a refractory period during which few care to further investigate whatever recently attracted attention. The first edition of this book captured its biological topics in such a spotlight, and the spotlight moved on. But it has recently shifted back to some of the main topics of The Geometl}' of Biological Time, e.g., to phase singularities in the heart, to circadian docks, and to metabolie oscillations, in some cases correcting misimpres­sions frozen into the 1978 writing, and in others ca ses fulfilling rash inferences largely based on the analogy of one or another topological concept. What? T opological concepts in real biology?! Yes, and illuminating a remarkably broad range of biological rhythms, from the central neural mechanisms of breathing to the collective organization of f1ashing among fireflies. Three concepts parvade this diversity. You are invited here to intimacy with the three notions that:

• "Continuous behavior can be thought of in terms of states serially connected through a vector field, thus biological dyn .. mics in time can be seen as geometry on an unchanging timeless manifold; and

• "If cell-cell coupling enforces spatial conti~uity of states, then the state manifold of an excitable tissue during wave circulation becomes equivalent to that of a spontaneous oscillator; and

• "No manifold can be smoothly retracted onto its whole boundary while leaving that boundary unmoved."

We are not talking about a mere reconceptualization of known biology, or worse, just redescribing it in fancy language. Rather, these concepts exposed errors of fact and pointed to unforeseen facts that no one believed on occasions when they were encountered by accident, facts that no one would have ever considered looking for. They present a way of thinking geometrically about biological dynamics, and about the laboratory experiments implemented to check its counter-intuitive implications. This way led to intriguing discoveries in several fields, and to questions and puzzles that still await your attention. I have no doubt that the adventures illustrated here represent only the crudest beginning.

Tinkering with a book written 21 years aga is a peculiar project, more akin to history of science than research in biological dynamics. Actually, I am as fascinated by the finding out - especially the processes of asking a different question, of recognizing hallucinations for what they are, of reformulating reality - as by the final factual answers. Johannes Kepler stated in the Preface to Astronomia Nova (1609) that 'What matters to me is not merely to impart to the reader what I

vii

After any interval of fru itfu I innovation catches the spotlight there seems to be a refractory period during which few care to further investigate whatever recently attracted attention. The first edition of this book captured its biological topics in such a spotlight, and the spotlight moved on. But it has recently shifted back to some of the main topics of The Geometry of Biological Time, e.g., to phase singularities in the heart, to circadian docks, and to metabolie oscillations, in some cases correcting misimpres­sions frozen into the 1978 writing, and in others ca ses fulfilling rash inferences largely based on the analogy of one or another topological concept. What? T opological concepts in real biology?! Yes, and illuminating a remarkably broad range of biological rhythms, from the central neural mechanisms of breathing to the collective organization of flashing among fireflies. Three concepts pervade this diversity. You are invited here to intimacy with the three notions that:

• "Continuous behavior can be thought of in terms of states serially connected through a vector field, thus biological dyn .. mics in time can be seen as geometry on an unchanging timeless manifold; and

• "If cell-cell coupling enforces spatial conti~uity of states, then the state manifold of an excitable tissue during wave circulation becomes equivalent to that of a spontaneous oscillator; and

• "No manifold can be smoothly retracted onto its whole boundary while leaving that boundary unmoved."

We are not talking about a mere reconceptualization of known biology, or worse, just redescribing it in fancy language. Rather, these concepts exposed errors of fact and pointed to unforeseen facts that no one believed on occasions when they were encountered by accident, facts that no one would have ever considered looking for. They present a way of thinking geometrically about biological dynamies, and about the laboratory experiments implemented to check its counter-intuitive implications. This way led to intriguing discoveries in several fields, and to questions and puzzles that still await your attention. I have no doubt that the adventures illustrated here represent only the crudest beginning.

Tinkering with a book written 21 years ago is a peculiar project, more akin to history of science than research in biological dynamics. Actually, I am as fascinated by the finding out - especially the processes of asking a different question, of recognizing hallucinations for what they are, of reformulating reality - as by the final factual answers. Johannes Kepler stated in the Preface to Astronomia Nova (1609) that 'What matters to me is not merely to impart to the reader what I

vii

..• Preface to the 2000 Edition

viii PREFACE TO THE 2000 EDITION

have to say, but above all to convey to him the reasons, subterfuges, .and lucky hazards which led '" to discoveries. When Columbus, Magelhaen, and the Portuguese relate how they went astrayon their journeys, we not only forgive them, but would regret to miss their narration because without it the whole, grand entertainment would be lost." In late December of 1999 at Tucson, sitting in rebuilt remains of the black swivel chair I bought in 1977 for writing on the Isle of Palms, I have just spent a few months flipping through my tattered copy of the 1980 printing. Its accumulated marginal notations have now been transferred and expanded as inserts to the retyped original text. My purpose is to follow its original topics to maturation or demise during the two intervening decades, telling some of the unexpected twists and tums taken in the dance of experiment and theory.

During the 21-year lifespan of today's college senior, two to three Ph.D. dissertations per year specifically aimed at questions formulated in the old book, and dose to 5000 new technical papers annotated in my files now inform continuing interest in those questions. That is too much new material, even for this mostly verbal history skipping lightly though fascinating adventures to get to the end of the century and answer, "What is my impression today of answers to queries raised here in 1978, mostly about the singularities of biological timing, and what are the outstanding questions inviting answer today?" I did this update in the same way as the original text, by first writing what I think I understand, then checking in the library for published instances of the same material, and for counterexamples or unforeseen discoveries. After finding and studying 300 new research reports during this sabbatical semester and summer, I replaced by citations more draft text than I could last time: there are about twice as many per new page, as befits matured research areas. Two new chapters (16 and 17) constitute almost half the update. To make room, former Chapters 16 and 17 were compressed into Examples 11 and 12 of Chapter 10, and most of their references and figures were deleted. I also deleted some papers that appeared merely in topical lists or otherwise entered into no specific argument. So about nine hundred works cited in the 1980 printing are still cited, without replacement by more comprehensive reviews or corrections. The topics of 1978 were largely those I knew from first-hand involvement in formulation of theory and execution of experiments. Because that involvement intensified through 1999, almost 6% of the new references are inevitably 'Winfree, 19[80 .. 00]." I apologize for the appearance of braggadocio (ameliorated here by stressing the known errors of those publications), or worse, for echoing the stereotypical "Did I ever tell you about my adventures in The Great War ... " of a doddering old fossil. I retain the references anyway to direct you to data that do not appear elsewhere, without which some current interpretations might seem unsupportable. They also direct you to seven hundred excellent papers of intermediate vintage that are not cited again here despite their direct relevance to these topics. This update, with as many new references entering through new text as remain in the original (see the vintage histogram atop the bibliography) presents Nature mostly through the achievements of multitudes of gifted experimentalists and insightful theorists, many of whom I do not know personally, and some of whom may even be among this update's readers. I hope you sense my admiration of the amazing fruits of your patient ingenuity.

PREFACE TO THE 2000 EDITION ix

5witching perspectives back and forth across so many years every day while writing highlighted for me a feature of the sociology of science that I had not previously appreciated: that it seems rare for any experiment to be explicitly repeated in print either for confirmation or refutation. Repeats are expensive and time-consuming and seldom make a welcome publication: confirmations would seem redundant, while refutations irritate and might merely point to a need for explicitness in a Materials and Methods section. This scarcity of checkups would be okay if the fiction could be maintained that there is no need, every refereed publication being a permanent stone in the Cathedral of 5cience. But in the discovery business outside of pure mathematics both facts and the theoretical frameworks in wh ich we provisionally mount them evolve as though in a house of mirrors, so that a decade after their publication many things are no longer "true" in all their originally intended meanings and connotations. There is seldom found a clear statement that something has been checked and generalized, or that something else was amistake, or that some seemingly obvious implication in fact should not be drawn. Many unspoken implications contradict one another, and anomalous reports tend to be simply omitted from review papers. How can students simultaneously believe that, once published, not much needs rechecking, and that literature predating their high-school years has little interest or validity? Reality is so different! The student naively entering the research library to learn new things (my role during both writings) finds every proposition and its converse confirmed by one or another computer model and quite a lot of ostensible contradiction and other inconsistency in laboratory data, if not always in the words surrounding them. This introduces a disconcerting uncertainty, particularly when we work in the mode of mathematical biology, stressing logical consistency over statistical probabilities, and emphasizing the decimation of models by a decisive test of their unique predictions. It becomes hard to know what to make of the abundant ostensible counterexamples to almost every assertion. General Douglas MacArthur is said to have remarked, "Expect only 5% of an intelligence report to be correct. The trick of a good commander is to isolate the 5%." With "5" considerably upgraded in application to scientific research papers crystallized without military haste, one way to adapt to this reality it is to maximize awareness not only of currently fashionable ways of thinking and choices of emphasis, but also of how they got that way, and so to appreciate their vulnerability to reappraisal. The improbable challenge to update across two decades presents a unique opportunity to try doing just that, coloring the factual content with an interest in the evolution of understanding. 50 I approached the task of modemizing this old monograph for graduate students almost as a literary experiment, hopefully with some interest outside its several narrowly technical specialties. I deliberately refrained from hiding errors of concept and fact. Instead, lightly shaded inserts show how those now recognized were corrected. And I draw explicit attention to confirmations or dlsconfirmations discovered in the literature, wherever they bear directly on the central themes of the book. The emphasis is more on what we don't understand than on what we do. I won't know whether the experiment worked until you read it and inform me at art@cochise.biosci.arizona.edu or the newer winfree@u.arizona.edu.

All new text is in sans-serif font and flagged by shading, except for a few one-liners flagged only by some such phrase as "In 1999 ... " and for new chapters 16 and 17, which appear in sans-serif but with shading restricted to page headers. Old phrases like "at this writing," "not yet," "recent," "remains to be seen" are now flagged with an "in

X PREFACE TO THE 2000 EDITION

1978" or a "still so in 1999" without shading. I corrected spelling, unclear phrases, etc. wherever my high-school English teacher would have done so, without flagging these mutations as such; only changes of scientific content are shaded. A few figures are improved or discarded; this forced a renumbering only in Chapter 13. These updates are limited to developments that I perceive as gr6wth from seeds already germinating in 1978, i.e., limited to topics in the geometry of timing, principally as it involves phase singularities. The old text was largely about circadian and glycolytic oscillators, topics both substantially updated here, but the more extensive change is an expansion of Chapter 9 (Excitable Media) into updates of Chapter 13 (Chemical) and new Chapters 16 (Three-Dimensional) and 17 (Cardiac). These might also be viewed as an update of Winfree (1987a, The Three-dimensional Dynamics of Chemical Waves and Cardiac Arrhythmias).

Though I don't believe any of them have yet percolated into textbooks, I am pleased to see that some of the notions proffered in the 1978 writing are now more widely accepted. Kenneth J.W. Craik, in his 1943 book The Nature of Explanation, succinctly captured my present feeling as to how this happened: "If lever conceive any original idea, it will be because I have been abnormally prone to confuse ideas ... and have thus found remote analogies and relations which others have not considered! Others rarely make these confusions, and proceed by precise analysis." Many questions requiring precise analysis are still outstanding, and more have accumulated. Explicit questions that seem ripe for research today are flagged in the page margins. While I was busy with the first version of this book, Frank Hoppensteadt also invited "24 Hard Problems about the Mathematics of 24-hour Rhythms," a chapter of puzzles to which I added a reference in the page proofs of 1979 without remembering to cite it anywhere. Section D of Winfree (1998a) has more. Most of those questions remain still unanswered, so I advertise them again here.

Narrow expertise is a kind of provincialism, inducing arrogance, which, expressed as deafness, entrenches provincialism. This is a feedback loop familiar in most human clubs, whether composed of countryside villagers, physicists, doctors, molecular biologists, or prelates. In attempting this book I have to draw eclectically from such mutually isolated sources, abandoning aspiration to expertise ... but not abandoning the will to correct my errors of fact and of perspective. I hope the reader will forgive many blunders by instructing me to correct them. Your feedback will also be helpful in maintaining the list of known errors and updates to this book at http://cochise.biosci. arizona.edu/-artlY2KGBT.html.

Who knows? There might even be a third edition.

Acknowledgments

I confess that of all men I am the least able to traverse so vast a field .... I pray that men of ability and leaming will examine my work with good will, and when they find faults will indulgently correct them. That which I offer to the public will have little value in the eyes of scholars ... but one should always be able to count on the courtesy of his colleagues.

-Abd-er-Rahman ibn-Khaldun of Tunis, Muquaddama-al-Alamat (lntroduction to the Universe), 1377

PREFACE TO THE 2000 EDITION xi

The last exam I had the fun of working through in high school was intended to inform students of their psychological bias toward one or another career. In my case, the retum pointed strongly and almost exclusively to two careers: adventurer/explorer, or else research scientist. I made the choice at the end of the Indian Ocean Expedition in 1964 by not joining a further oceanographic cruise to Antarctica. But a flavor of "The Excellent Adventures of A TW" from college to full professorship pervades the first edition of this book, and this update perseveres in the same style with "what happened next." Though as narrow as any other specialty, this un-named borehole though the world of science slants across the deep mineshafts of a wide variety of named specialties, traversing a fjeld too vast for my solo scholarship. I thank my many correspondents and colleagues since 1978 for graciously responding to a chronic barrage of naive questions, and especially those, e.g., Dwight Barkley, Ding Da-fu, Stephen Dillon, Ce es Diks, Igor Efimov, Paul Fife, Leon Glass, Jaganathan Gomatam, Michael Guevara, Morris Hirsch, Raymond Ideker, Jose Jalife, Alain Karma, James Keener, Richard Krasnow, Milos Marek, Ehud Meron, Alexandre Panfilov, David Paydarfar, Arkady Pertsov, Eric Peterson, Theo Plesser, Timothy Poston, Brad Roth, Jack Rogers, Steven Scott, John Tyson, Wilbert van Meerwijk, Brian Welsh, and the late Frank Witkowski, who undertook to rectify my confusions and find out real answers by analysis, computation, or laboratory experiments. I am especially grateful for the opportunity to kibbitz medical experiments (at Syracuse and at Duke until about 1990, at Edmonton until 2000), suggest new ventures, and see the results: this humbling experience has usefully eliminated a lot of nonsense. In my own lab, results since 1978 would have been much slower to arrive without the dedicated technical assistance of Harold Aberman, Scott Caudle, C. C. Chen, Gang Chen, Marc Courtemanche, Mark Gallagher, William Guilford, Chris Henze, Wolfgang Jahnke, Patrick McGuire, Erszebet Lugosi, Pramod Nandapurkar, Victor Reiner, Dean Schulze, William Skaggs, David Stone, Steven Strogatz, Zoltan Szilagyi, George Twaddle, Erik Winfree, and Michael Wolfson, half of whom took the initiative to venture into my lab as advanced undergraduates. You all went on to greater things, followed by my gratitude for your willingness to help me out during the earliest stages of your own adventures.

Histories of science give me the impression that the actual sources of innovative ideas are commonly forgotten in later publications. I was taken aback to discover a striking instance in the original version of this book: no allusion to the works of Nicholas Rashevsky. In fact, I still find no specific place to use any particular one of his results. But the Dover paperback of his two-volume Mathematical Biophysics seems in long retrospect the clear source of many of my biases since that 1961 encounter, while a sophomore in the Engineering Physics Department at Comell. In particular, Rashevsky pointed me to reaction-diffusion systems of both chemical and electro­physiological sorts, and to the variety of their solutions in two- and three-dimensional geometries. I owe him at least this belated acknowledgment.

The Institute for Natural Philosophy has been the mainstay of my efforts since 1980. I thank the US National Science Foundation for persistently funding my lab through these two additional decades, and more importantly for funding the many younger scientists elsewhere whose efforts made this update possible and interesting. I thank David Paydarfar and others who sent me corrections to the original text. Leon Glass, John Guckenheimer, Dwight Barkley, and my patient editor, Ach i Dosanjh, collectively read most of the new insertions; without their patient critique, much that

xii PREFACE TO THE 2000 EDITION

embarrassed me would still be visible. Needless to say, my sparkling wife, Ji-Yun, put up with a lot of neglect while science magnetized her husband's attention, and without her love it would have all come to ruin. She also read the manuscript for comments that I need not retain, saw to it that I delete them, and made the 900 new references presentable.

AT. Winfree Tucson, Arizona Winter Solstice 1999

Preface to the 1980 Edition

As I review these pages, the last of them written in summer 1978, some thoughts come to mind that put the whole business into better perspective for me and might aid the prospective reader in choosing how to approach this volume.

The most conspicuous thought in my mind at present is the diversity of wholly independent explorations that came upon phase singularities, in one guise or another, during the past decade. My efforts to gather the published literature during the last phases of actually writing a whole book about them were almost equally divided between libraries of Biology, Chemistry, Engineering, Mathematics, Medicine, and Physics.

A lot of what I call "gathering" was done somewhat in anticipation in the form of conjecture, query, and prediction based on analogy between develop­ments in "different fields." The consequence throughout 1979 was that our long­suffering publisher repeatedly had to replace such material by citation of unexpected flurries of papers giving substantive demonstration. I trust that the authors of these many excellent reports, and especially of those I only found too late, will forgive the brevity of allusion I feit compelled to observe in these substitutions.

It is clear to me already that the materials I began to gather several years aga represented only the first flickering of what turns out to be a substantial conflagration. Accordingly, I took a liberty with the 1978 reference list. You will notice that about 20% of its entries are not to be found in the page index of publications cited. That is because they are not explicitly cited. Readers who like to browse will easily find these extra papers: they lie among papers on similar topics by much-cited authors. They lead in the directions of significant expansion .

.. In eed they did, so t ey are less interesting today: I ave accordingly deleted them. The new and shertened eid reference lists, each new with about 900 iterns, are al habetized tegether. ...

And what comes next? WeIl, one never knows; that is half the fun of doing science. But one inevitable development is especially conspicuous by its absence here. In fact, the original 30 chapters came down to 23 by purging 7 to ripen further into a later volume 1. You will find here almost no mention of rhythmic drlving of

L This became The Timing ofBiological Clocks Freeman. 1986 and When Time Breaks DOWn: The Three­Dimensional namics ofElectrochemical Wayes and Cardiac Arrh>: hmi{lS, Princeto,n Univ. Press, 1987.

xiii

xiv PREFACE TO THE 1980 EDITION

biological dynamics. Plainly that must contain the essence of any practical application, be it in honnonal gating of cell division, in cardiac or gastric pacemaking, or in agricultural photoperiodism. Many surprises await discovery in connection with alternative modes of entrainment, the consequences of synchro­nization, and evolution in periodic environments. This topic is the natural successor to the present volume on autonomous periodicity. It is now undergoing rapid development, mainly at the hands of neurobiologists, mathematicians, and engineers, and will be riper for harvest a few years hence.2

A residue of loose ends (questions for research) is largely collected in the index under "Queries."

.. Surv1vors and replacernents are instead flasged in the margins for-the Y2K Edition ....

It has been my good fortune to visit lively investigators in many laboratories. I have been stimulated by early exposure to their discoveries (which fill out so much of the following chapters), and their critical attention to my own seminars has refined into presentable fonn most of what is presented here. But I have never found an opportunity to teach on these subjects, as you can see by the lack of problem sets in this presentation. I suspect that substantial improve­ments of content and clarity as weIl as significant new directions would inevitably emerge through contact with students who are eager and ready to study living systems in a mathematical spirit. That is a hard clientele to locate; I could use some help.

I wish you good reading and wish you to send me marginal notations to collect on my copy. Who knows? There might even be a second edition.3

Acknowledgments

I wrote this book but its authors live all over the world. In a broad sense the list of authors is the bibliography. But in a more precise sense this gathering of facts and ideas was shaped by about twenty-five individuals whose conversation and correspondence molded every topic represented here. Many others also will recognize in these pages the distorted reflection of their own imagination and skepticism. Rather than belabor the apologies and disclaimers usual in books of this sort, let me just remark that without the impact of Ralph Abraham, Arthur Brill, Rohert DeHaan, Wolfgang Engelmann, Brian Goodwin, Hennan Gordon, Joseph Higgins, Walter Kaufman-Bühler, Stuart Kauffman, Richard Levins, Robert MacArthur, Graeme Mitchison, Jay Mittenthal, George Oster, Theodosios Pavlidis, Colin Pittendrigh, John Platt, Kendall Pye, John Rinzel, Frank Rosenblatt, Otto Rössler, Rene Thom, John Tyson, and Trisha Woollcott, my explorations into the dynamics of evolved life would have lacked the special richness and color that I here seek to share.

See Glass, L. and Mackey, M.C. From Clocks & Chaos: TheRhyrhms vf Li{e Princeton Uni\'. Press. 1988.

After a second printing in 1990 that fixed many typos, Ihis is it.

PREFACE TO THE 1980 EDITION XV

Perseverance in this line of inquiry was made possible by the generous financial support of the National Science Foundation since 1965 and of the National Institutes of Health during 1973-1978. I am especially indebted to my department chairmen, Jack Cowan and Henry Koffler for safe escort through the three grades of professorship while I remained lost in the dream world here described.

Finally, I wish to acknowledge the frequent restoration of my sanity by the turquoise waters, white sands, and blinding sunlight of the Isle of Palms, South Carolina, where most of these pages were first drafted in 1977.

A. T. Winfree West Lafayette, Indiana April 1980

xvi PREFACE TO THE 1980 EDITION

• Imagine that we are living on an intricately patterned carpet. It may or may not extend to infinity in all directions. Some parts of the pattern appear to be random, like an abstract expressionist painting; other parts are rigidly geometrical. A portion of the carpet may seem totally irregular, but when the same portion is viewed in a larger context, it becomes part of a subtle symmetry.

The task of describing the pattern is made difficult by the fact that the carpet is protected by a thick plastic sheet with a translucence that varies {rom place to place. In certain places we can see through the sheet and perceive the pattern; in others the sheet is opaque. The plastic sheet also varies in hardness. Here and there we can scrape· it down so that the pattern is more clearly visible. In other places the sheet resists all efforts to make it less opaque. Light passing through the sheet is often refracted in bizarre ways, so that as more of the sheet is removed the pattern is radically transformed. Everywhere there is a mysterious mixing of order and disorder. Faint lattices with beautiful symmetries appear to cover the entire rug, but how far they extend is anyone's guess. No one knows how thick the plastic sheet is. At no place has anyone scraped deep enough to reach the carpet's surface, if there is one.

•

-Martin Gardner Scientific American March 1976, p. 119

Contents

Preface to the 2000 Edition vii Preface to the 1980 Edition xiü mmOOu~oo Dill

1. Circular Logic 1

A: Spaces 1 B: Mappings 4 C: Phase Singularities of Maps 24 D: Technical Details on Application to Biological Rhythms 30

2. Phase Singularities (Screwy Results of Circular Logic) 41

A: Examples 42 B: Counterexamples 75 C: The Word "Singularity" 77

3. The Rules of the Ring 80

A: Basic Principles, Paradigms, Language Conventions, and Epistemology 80 B: Dynamics on the Ring 83 C: Derivation of Phase-Resetting Curves 88 D: Historical Appendix 97

4. Ring Populations 101

A: Collective Rhythmicity in a Population of Independent Simple Clocks 101

B: Communities of Clocks 119 C: Spatially Distributed Independent Simple Clocks 134 D: Ring Devices Interacting Locally 138

xvii

xviii Contents

5. Getting off the Ring 146

A: Enumerating Dimensions 146 B: Deducing the Topology 147 C: The Simplest Models 149 D: Mathematical Redescription 151 E: Graphical Interpretation 155 F: Summary 159

6. Attracting Cycles and Isochrons 161

A: Unperturbed Dynamics 161 B: Perturbing an Attracting-Cycle Oscillator 177 C: Unsmooth Kinetics 188

7. Measuring the 'frajectories of a Circadian Clock 198

A: Introduction 198 B: The Tune Machine Experiment 200 C: Unperturbed Dynamics 206 D: The Impact of Light 213 E: Deriving the Pinwheel Experiment 216 F: So What? 220 G: In Conclusion 228

8. Populations of Attracting-Cycle Oscillators 229

A: Collective Rhythmicity in a Population of Independent Oscillators: How Many Oscillators? 230

B: Collective Rhythmicity in a Community of Attracting-Cycle Oscillators 231

C: Spatially Distributed Independent Oscillators 236 D: Attracting-Cycle Oscillators Interacting Locally in

Two-Dimensional Space 250

9. Excitable Kinetics and Excitable Media 258

A: Excitability 258 B: Rotors 264 C: Three-Dimensional Rotors 291

10. The Varieties of Phaseless Experience: In which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways 303

A: The Physical Nature of Diverse States of Ambiguous Phase 304 B: The Singularities of Unsmooth Cycles 333 C: Transition to Bestiary 336

11. The Firefly Machine

A: Mechanics B: Results C: Historical

Contents

12. Energy Metabolism in Cells

A: Oscillators B: The Dynamics of Anaerobic Sugar Metabolism C: The Pasteur Effect D: Goldbeter's PFK. Kinetics E: Phase Control by ADP F: More Phase-Resetting Experiments G: Results: The TIme Crystal H: A Repeat Using Divalent Cations I: A Repeat Using Acetaldehyde J: Phase Compromise Experiments

13. The Malonic Acid Reagent ("Sodium Geometrate")

A: Mechanism of the Reaction B: Wave Phenomena C: Exdtation in Nonoscillating Medium D: Wave Patterns in Two- and Three-Dimensional Context E: Pacemakers

14. Electrlcal Rhythmicity and Excitability in Cell Membranes

A: Rephasing Schedules of Pacemaker Neurons B: Mutual Synchronization C: Waves in One Dimension D: Rotating Waves in Two Dimensions

15. The Aggregation of Slime Mold Amoebae

A: The Life Cycle of a Sodal Amoeba B: Questions of Continuity C: Ceil Chemistry and Cell-Ceil Coupling D: Phase Resetting by a cAMP Pulse E: Historical Note

16. Numerical Organizing Centers

A: Seroil Filaments in 1Wo Dimensions B: Seroll Rings Shrink and Drift C: Linked and Knotted Ring Anatomy

xix

338

338 341 344

347

347 348 350 351 353 354 355 359 360 366

368

371 377 382 383 406

411

413 422 428 432

443

443 446 449 451 453

455

456 467 469

xx Contents

D: Linked and Knotted Ring Dynamies 472 E: Efforts to Analytically Derive and Numerically Confirm the

Laws of Filament Dynamies 474 F: The Discovery of Persistent Ring Configurations 483 G: For the Future 490 H: Transition to Chapter 17 About Heart Muscle 492

17. Electrical Singular Filaments in the Heart Wall 495

A: Rotors in a Field of Coupled Oscillators 498 B: The Pinwheel Experiment 505 C: Pinwheel Experiment Revisited (PER) 518 D: Two- vs. Three-dimensional Instabilities of the Singular Filament

in Heart Muscle 525 E: A Quick Summary of the April 1997 Experiments 532 F: What's Next? 533

18. Pattern Formation in the Fungi 537

A: Breadmold with a Circadian Clock 538 B: Breadmolds in 1Wo-Dimensional Growth 539 C: Pattern Polymorphism in Bourret's Nectria 540 D: Integration of Pattern 543

19. Circadian Rhythms in General 545

A: Some Characteristics of Circadian Rhythms 561 B: ClockEvolution 576 C: The Multioscillator View of Circadian Rhythms 585

20. The Circadian Clocks of Insect Eclosion 592

A: Basics of Insect Eclosion Clocks 592 B: Phase and Amplitude Resetting in Drosophila Pseudoobscura 606 C: Other Diptera 620

21. The Flower of Kalanchoe 624

A: Type 0 Resetting 624 B: Resetting Data at Many Stimulus Magnitudes 627 C: A Phase Singularity 630 D: Arrhythmicity Not an Artifact of Populations 630 E: Amplitude Resetting 631

22. The Cell Mitotic Cycle 632

A: Three Basic Concepts and Some Models 634 B: Regulation of Mitosis by the Circadian Clock 640

Contents xxi

C: Further Developrnents in the Area of Circadian Rhythrns, Applied Back to the Cell Cycle 642

D: Physarum Polycephalum 644

23. The Female Cycle

A: Wornen, Hormones, and Eggs B: Statistics ("Am I Overdue?!") C: Rephasing Schedules D: The Question of Srnoothness E: Circadian Control of Ovulation

Index of Author Citations and Publications Index of Subjects

650

650 653 655 657 658

661 761

Introduction

Ubi materia, ibi geometria. -Johannes Kepler, 1571-1629

This is a story about dynamics: about change, flow, and rhythm, mostly in things that are alive. My basic outlook is drawn from physical chemistry, with its state variables and rate laws. But in living things, physical and chemical mechanisms are mostly quite complex and confusing, if known at all. So I'm not going to deal much in mechanisms, nor even in cause and effect. Instead I will adopt the attitude of a naturalist-anatomist, describing morphology. The subject matter being dynamics, we are embarked upon a study of temporalmorphology, of shapes not in space so much as in time. But by introducing molecular diffusion as a principle of spatial ordering, we do come upon some consequences of temporal morphology for the more plainly visible shapes of things in space.

This is a story about dynamics, but not about all kinds of dynamics. It is mosdy about processes that repeat themselves regularly. In living systems, as in much of mankind's energy-handling machinery, rhythmic return through a cyele of change is a ubiquitous principle of organization. So this book of temporal morphology is mostly about circles, in one guise after another. The wordpwe is used (over 1,200 times in 1978) to signify position on a cirele, on a cycle of states. Phase provides us with a banner around which to rally a weiter of diverse rhythmic (temporal) or periodic (spatial) patterns that lie elose at hand all around us in the natural world. I will draw your attention in particular to "phase singularities": peculiar states or places where phase is ambiguous but plays some kind of a seminal, organizing role. For example in a chemical solution aphase singularity may become the source of waves that organize reactions in space and time.

This book is intended primarily for research students. Readers who come to it seeking crystallized Truth will go away irritated. I suspect that the most satisfied readers will be those who come with revisionist intentions, seeking the frayed ends of new puzzles and seeking outright errors that might lead to novel perspectives. I am confident that you will find plenty of both, since this project has ramified into more specialty areas than I can keep abreast of, ranging from topology through biochemistry. I've done all I can to eliminate nonsense from earlier drafts, with indispensable help from my very critical friends, especially Herman Gordon and

xxiii

xxiv INTRODUCTION

Richard Krasnow. But the material is kaleidoscopic. As long as I work at it, the pieces keep rearranging themselves into tantalizing new patterns. Most of these dissolve under continued scrutiny but more remain than I can pursue. I have chosen to lay them out in the one-dimensional way inevitable to written communication as follows.

~ At the time of this Y2K update I am more impressed y human ability to narrate long about visions seen from one direction only, that are often mirages. We begin to feel that a new thing has reality only when we see it from several directions or stumble across it along multiple paths crossing through that part of a multi-dimensional space. So this update strives to embed each new item in several pertinent contexts. The Index is useful for finding them all. ...

This volume has two roughly equal parts. The first half mainly develops a few themes in an order natural to the fundamental concepts involved. The second half is organized more like a "dramatis personae." I call it the Bestiary. It teIls about the organisms or other experimental systems from which the conceptual themes arose. In more detail:

Experiments with docks and maps constitute the principal theme of this book. Phase singularities figure prominently in these experiments. Secondary themes will be played through again and again in different contexts: the progression from dynamics in a single unit (a cell, an organism, a volume element) to collective dynamics in populations of independent units, then to populations of promiscu­ously interacting units, and to populations arranged in space with interaction restricted to immediate neighbors (as by molecular diffusion). New phenomena emerge at each level.

Apart from those themes, the material gathered here might at first seem to have few unifying features. I have chosen examples from a diversity of living organisms and nonliving experimental models. Each recurs in several places, illustrating different points. Dur trail through this jungle of exotic flowers intersects itself in several places as these themes surlace again and again in new combinations, in new experimental contexts.

The material is handled in three ways:

1. A single thread of text proceeds through the first half of the book under 10 chapter headings. By the following two devices this part is kept as lean as possible to enable the reader to seoot through for perspective before choosing where to invest more critical thought.

2. Along the way, frequent allusion is made to endosed "boxes" of finer print, each elaborating a particular point, raising questions for exercise or research, or offering an anecdote. These stand aside from the main thread of text like scenic overlooks along the toll road. I think they contribute interest and perspective, but you may want to pass over them until you locate the chapters dosest to your particular interests.

3. The second half of the book is a Bestiary of 13 chapters about particular experimental systems. These provide background facts about the organisms or phenomena most frequently alluded to in the first half of the book. These might be the most interesting chapters for readers with little use for theories

INTRODUCTION xxv

and models and for readers unfamiliar with the experimentallaboratory. I had to put one or the other perspective first and naturally some people think I should have chosen the other way around. I think that only your taste should determine whether you read the Bestiary first, or the preceding abstract notions that inter-relate its contents.

The whole is sandwiched between a table of contents and an extensive bibliography.

In your first glance through these pages you will notice a mathematical flavor about some topics. Though my aim is to avoid mathematical "models" wherever possible, some reasoning with symbols seems inevitable to this subject matter. Mathematics enters in four ways:

1. Simply using numbers to quantify experimental data for presentation as graphs and for comparison with quantified ideas about their meaning ("models").

2. Using digital computer languages to implement data handling and to extract implications from numerical statements of an hypothesis. For example, we might assume that in some useful approximation ceIls divide when some constantly synthesized substance accumulates to a critical concentration, and then digitally seek the implications for a compact tissue in which this substance leaks between adjacent ceIls.

3. Using standard undergraduate mathematics to extract such implications when conjectures can be formulated in terms tractable to geometry, elementary differential equations, and so on. I've made a big effort to avoid mathematical equations. This is only in part because I seldom manage to get equations right. My deeper motivation is a feeling that numerical exactitude is alien to the diversity of organic evolution, and pretense of exactitude often obscures the qualitative essentials that I find more meaningful. My aim is to get those right first. And that purpose seems better served by words and pictures supplemented by occasional numerical simulations and lots of experimental measurements.

4. Employing a little of the language of topologists in an effort to extract from models or from observations what seems to be the essence of their behavior, independent of quantitative variations. Such efforts are fraught both with technical difficulties and with the ever-present danger of extracting mathema­tically trivial tautologies while discarding as "mere quantitative variations" all that is of factual interest about a particular phenomenon. Nonetheless, I believe such efforts constitute an indispensable prelude to explanation: "What is the phenomenon to be explained, as distinct from incidentals to be glossed over? What aspects require explanation in terms of empirical cause and effect, and what aspects are merely mathematical consequences of fads already known?"

It is my belief that the life sciences in particular have much to gain from, and perhaps something to contribute to, mathematical developments in the general area of topology. I wish the reader to consider this. Thus I will dweIl on such topological notions as I have found useful in designing experiments and in interpreting their results. From heuristic beginnings, my own efforts seldom go far toward logical rigor, yet I have found much satisfaction in the fruitful dialog between theory and experiment that this approach has fostered.

xxvi INTRODUCTION

,.. A caution: the word "topological" is (mis}use in recent biomedical literature as a synonym for "qualitative" in geometrical contexts, e.g./ about the shapes of proteins or of cortical maps. This book stiJI uses it as in most 'prior biological literature, in physics, in mathematics, and in chemistry, to connote certain precise features (having three holes rather than one, being connected like-a sphere not Iike a torus, having two dimensions rather than one) that are independent of continuous quantitative variations of shape. <c

We turn now to the simplest abstractions about "rhythms," "cycles," and "docks," with a few examples. Examples are merely mentioned here, pending their fuller description in later chapters, where the context is riper.