The Geometry of Biological Time

543 Pages·1980·21.698 MB·English

by  Arthur T. Winfree (auth.)

Biomathematics
Volume 8
Managing Editors
K. Krickeberg S.A. Levin
Editorial Board
H.J. Bremermann J. Cowan W.M. Hirsch S. Karlin
J. Keller R.C. Lewontin R.M. May
J. Neyman S.1. Rubinow M. Schreiber L.A. Segel
Arthur T. Winfree
The Geometry
of Biological Time
With 290 Illustrations
Springer Science+Business Media, LLC
Arthur T. Winfree
Department of Biological Sciences
Lilly Hall of Life Sciences
Purdue University
West Lafayette, Indiana 47907
USA
AMS Subject Classification (1980): 92A05
Library of Congress Cataloging in Publication Data
Winfree, Arthur T
The geometry of biological time.
(Biomathematics ; 8)
Bibliography: p.
Includes index.
1. Biological rhythms-Mathematics. I. Title.
QH527.W55 574.1 79-12375
This material is based upon work supported by the National Science Foundation under Grants No. GB
16513, GB 37947, BMS 73-06888A 01, CHE 77-24649, and by National Institutes of Health Research
Career Development Award 5 K04 GM 70660.
Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the
author and do not necessarily reflect the views of the National Science Foundation or the National Institutes
of Health.
All rights reserved.
No part of this book may be translated or reproduced in any form, without written permission
from Springer-Verlag.
© 1980 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York Inc. in 1980.
Softcover reprint ofthe hardcover 1st edition 1980
9 8 7 6 543 2 1
ISBN 978-3-540-52528-8 ISBN 978-3-662-22492-2 (eBook)
DOI 10.1007/978-3-662-22492-2
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.
Preface
As 1 review these pages, the last of them written in Summer 1978, some retrospec
tive thoughts come to mind which 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 1 call "gathering " was done somewhat in anticipation in the form of
cönjecture, query, and prediction based on analogy between developments in different
fields. The consequence throughout 1979 was that our long-suffering publisher re
peatedly had to replace such material by citation of unexpected flurries of papers giving
substantive demonstration. 1 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. A residue of loose ends is largely collected
in the index under "QUERIES."
It is c1ear to me already that the materials I began to gather several years ago
represented only the first flickering of what turns out to be a substantial conflagration.
According, I took a liberty with the reference list. You will notice that about 30% 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.
And what comes next? Well, 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 in purging it for a later volume. You will find
here almost no mention of rhythmic driving of biological dynamics. Plainly that must
contain the essence of any practical application, be it in hormonal gating of cell
division, in cardiac or gastric pacemaking, or in agricultural photoperiodism. Many
vii
viii Acknowledgments
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, mathemati
cians, and engineers, and will be riper for harvest a few years hence.
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 form 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 presenta
tion. I suspect that substantial improvements of content and clarity as weil s 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.
April 1980 Arthur Winfree
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 re mark that without
the impact of Ralph Abraham, Arthur BrilI, Robert DeHaan, Wolfgang Engelmann,
Brian Goodwin, Herman Gordon, Joseph Higgins, Stuart Kauffman, Richard Levins,
Robert MacArthur, Graeme Mitchison, Jay Mittenthai, George Oster, Theodosios
Pavlidis, Colin Pittendrigh, John Platt, Kendall Pye, John Rinzel, Frank Rosenblatt,
Otto Rössler, Rene Thom, John Tyson, and Trisha Winfree, my explorations into the
dynamics of evolved life would have lacked the special richness and color that I here
seek to share.
Perseverance in this line of enquiry 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, Henry Koffler, and Struther Arnott, for safe escort through the three grades of
professorship while I remained lost in the dreamworld here described.
Finally, I wish to acknowledge the frequent restoration of my sanity by the turquoise
waters, white sands, and blinding sunlight of the lsle uf Palms, South Carolina, where
most of these pages were first drafted in 1977.
Contents
Introduction 1
1. Circular Logic 4
A: Spaces 4
B: Mappings 6
C: Phase Singularities of Maps 25
D: Technical Details on Application to Biological Rhythms 30
2. Phase Singularities (Screwy Results of Circular Logic) 40
A: Examples 40
B: Counterexamples 70
C: The Word "Singularity" 71
3. The Rules of the Ring 74
A: Basic Principles, Paradigms, Language Conventions, Epistemology 74
B: Dynamics on the Ring 77
•
C: Derivation of Phase-Resetting Curves 82
D: Historical Appendix 91
4. Ring Populations 95
A: Collective Rhythmicity in a Population of Independent Simple Clocks 95
B: Communities of Clocks 112
C: Spatially Distributed Independent Simple Clocks 121
D: Ring Devices Interacting Locally 125
ix
x Contents
5. Getting OfT the Ring 131
A: Enumerating Dimensions 131
B: Deducing the Topology 132
C: The Simplest Models 134
D: Mathematical Redescription 136
E: Graphical Interpretation 140
F: Summary 144
6. Attracting Cycles and Isochrons 145
A: Unperturbed Dynamics 145
B: Perturbing an Attractor Cycle Oscillator 159
C: Unsmooth Kinetics 168
7. Measuring the Trajectories of a Circadian Clock 176
A: Introduction 176
B: The Time Machine Experiment 178
C: Unperturbed Dynamics 185
D: The Impact of Light 191
E: Deriving the Pinwheel Experiment 194
F: So What? 198
G: In Conclusion 204
8. Populations of Attractor Cycle Oscillators 205
A: Collective Rhythmicity in a Population of Independent Oscillators: How
Many Oscillators? 205
B: Collective Rhythmicity in a Community of Attractor Cycle Oscillators 207
C: Spatially Distributed Independent Oscillators 212
D: Attractor Cycle Oscillators Interacting Locally in Two-Dimensional
Space 225
9. Excitable Kinetics and Excitable Media 231
A: Excitability 231
B: Rotors 235
C: Three-Dimensional Rotors 250
10. The Varieties of Phaseless Experience: In Which the Geometrical
Orderliness of Rhythmic Organization Breaks Down in Diverse Ways 258
A: The Physical Nature of Diverse States of Ambiguous Phase 259
B: The Singularities of Unsmooth Cycles 273
C: Transition to Bestiary 275
Contents xi
11. The Firefly Machine 277
A: Mechanics 277
B: Results 280
C: Historical 283
12. Energy Metabolism in Cells 285
A: Oscillators 285
B: The Dynamics of Anaerobic Sugar Metabolism 286
C: The Pasteur Effect 288
D: Goldbeter's PFK Kinetics 289
E: Phase Control of the PFK! ADP Oscillator 291
F: More Phase-Resetting Experiments 291
G: Results: The Time Crystal 292
H· A Repeat Using Divalent Cations 296
I: A Repeat Using Acetaldehyde 296
J: Phase Compromise Experiments 298
13. The Malonic Acid Reagent ("Sodium Geometrate") 300
A: Mechanism of the Reaction 302
B: Wave Phenomena 304
C: Excitation in Non-oscillating Medium 307
D: Wave Pattern in Two- and Three-Dimensional Context 308
E: Pacemakers 312
14. Electrical Rhythmicity and Excitability in Cell Membranes 315
A: Rephasing Schedules of Pacemaker Neurons 317
B: Mutual Synchronization 323
C: Waves in One Dimension 325
D: Rotating Waves in Two Dimensions 329
15. The Aggregation of Slime Mold Amoebae 337
A: The Life Cycle of a Social Amoeba 337
B: Questions of Continuity 339
C: Chemistry in the Single Cell 340
D: Phase Resetting by a cAMP Pulse 342
E: Historical Note 343
16. Growth and Regeneration 345
A: The Clockface Model 345
B: An Alternative Description 349