Biology and Mathematics: History and Challenges

247 Pages·2019·2.403 MB·English

by  Roger Buis

Biology and Mathematics

Series Editor
Marie-Christine Maurel
Biology and Mathematics

History and Challenges

Roger Buis
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Control Number: 2019943329

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ISBN 978-1-78630-483-4
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1. On the Status of Biology: On the Definition of Life . . . . . . . . 1
1.1. Causality in biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1. Vitalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.2. Teleology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2. Variability in biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1. Time-dependence of biological processes . . . . . . . . . . . . . . . . . . 15
1.2.2. Environment-dependence of biological processes . . . . . . . . . . . . . . 17
Chapter 2. On the Nature of the Contribution Made by
Mathematics to Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1. On the affinity of mathematics with biology . . . . . . . . . . . . . . . . . . . 20
2.2. Mathematics, an instrument of work and thought on biology . . . . . . . . . . 25
Chapter 3. Some Historical Reference Points: Biology Fashioned by
Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1. The first remarkable steps in biomathematics . . . . . . . . . . . . . . . . . . . 37
3.1.1. On the continuous in biology . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.2. On the discrete in biology . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.3. The notion of laws in biology . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.4. The beginning of classical science: Descartes and Pascal . . . . . . . . . . 44
3.1.5. Buffon and hesitations relating to the utility
of mathematics in biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
vi Biology and Mathematics
3.2. Some pertinent contributions from mathematics in the modern era . . . . . . . 48
3.2.1. The laws of growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.2. Formal genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3. Introduction of the notion of a probabilistic model in biology . . . . . . . . . . 56
3.4. The physiology of C. Bernard (1813–1878): the call
to mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5. The principle of optimality in biology . . . . . . . . . . . . . . . . . . . . . . . 60
3.6. Introduction of the formalism of dynamic systems in biology . . . . . . . . . . 61
3.7. Morphogenesis: the need for mathematics in the study
of biological forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7.1. General principles from D’Arcy Thompson . . . . . . . . . . . . . . . . . 64
3.7.2. Turing’s reaction–diffusion systems (1952): morphogenesis,
a “break of symmetry” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.8. The theory of automatons and cybernetics in biology . . . . . . . . . . . . . . 70
3.8.1. The theory of automatons . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8.2. The contribution of cybernetics . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8.3. The case of L-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.8.4. Petri’s networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.9. Molecular biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.9.1. On genetic information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.9.2. The linguistic model in biology . . . . . . . . . . . . . . . . . . . . . . . . 83
3.10. Information and communication, important notions in biology . . . . . . . . 84
3.11. The property of self-organization in biology . . . . . . . . . . . . . . . . . . . 86
3.11.1. Structural self-organization . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.11.2. Self-reproductive hypercycle . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.12. Systemic biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.12.1. On the notion of system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.12.2. Essay in relational biology . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.12.3. Emergence and complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.12.4. Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.12.5. Order, innovation and complex networks . . . . . . . . . . . . . . . . . . 104
3.13. Game theory in biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.14. Artificial life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.14.1. Biomimetic automatons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.14.2. Psychophysiology and mathematics: controls on learning . . . . . . . . . 111
3.15. Bioinformatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter 4. Laws and Models in Biology . . . . . . . . . . . . . . . . . . . . . . 115
4.1. Biological laws in literary language . . . . . . . . . . . . . . . . . . . . . . . . 118
4.1.1. The law of Cuvier’s organic correlations (1825) . . . . . . . . . . . . . . 118
4.1.2. The fundamental biogenetic law . . . . . . . . . . . . . . . . . . . . . . . . 118
Contents vii
4.2. Biological laws in mathematical language . . . . . . . . . . . . . . . . . . . . . 119
4.2.1. Statistical laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3. Theoretical laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3.1. Formal genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3.2. Growth laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.3. Population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Chapter 5. Mathematical Tools and Concepts in Biology . . . . . . . . . . . . 135
5.1. An old biomathematical subject: describing and/or
explaining phyllotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2. The notion of invariant and its substrate: time and space . . . . . . . . . . . . 140
5.2.1. Physical time/biological time . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.2. Metric space/non-metric space . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2.3. Multi-scale processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3. Continuous formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.1. Dynamics of a univariate process . . . . . . . . . . . . . . . . . . . . . . . 148
5.3.2. Structured models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.3. Oscillatory dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.3.4. On the stability of dynamic systems . . . . . . . . . . . . . . . . . . . . . . 154
5.3.5. Multivariate structured models . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3.6. Dynamics of spatio-temporal process . . . . . . . . . . . . . . . . . . . . . 163
5.3.7. Multi-scale models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.4. Discreet formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.5. Spatialized models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.5.1. Multi-agent models: dynamics of a biological association
of the individual-centered type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.5.2. Electrophysiological models: transmission of electrical signals . . . . . . 176
5.6. Random processes in biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.6.1. Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.6.2. Birth–death processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.7. Logic kinetics of regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Foreword
Roger Buis, professor emeritus at the University of Toulouse (INP – Institut
National Polytechnique) – already the author of a considerable amount of research
into the biomathematics of growth1 – delivers here a true panorama of the
relationships between biology and mathematics over time, and in particular over the
course of the last century, accompanied by a series of profound epistemological
thoughts, thereby creating a book of great rarity and value.
As we know, mathematics is ancient, just like the interest taken in living things.
But the word “biology” only appeared at the beginning of the 19th Century and,
whilst E. Kant has already confirmed – and as Roger Buis rightly reminds us – a
piece of knowledge is scientific insofar as mathematics has been integrated into
it, the explicit idea of applying mathematics to biology is found only with
C. Bernard2 – one of the great references in the book.
In contrast with physics, biology resists mathematization, for understandable
reasons: the variability of living things, their dependence on time and on the
environment, diversity and the complexity of biological processes, the diffuse aspect
of causality (sometimes circular) and the difficulty of mastering the operational
conditions of experiments have made obtaining consistencies problematic. Hence,

1 Buis, R. (2016). Biomathématiques de la croissance, le cas des végétaux. EDP-Sciences,
Les Ulis.
2 Let us note that, previously, G.-L. Buffon regretted that qualitative mathematics remained in
limbo and could not be applied in natural sciences. “Everything that has an immediate
relationship with a position is totally missing from mathematical sciences. The art that
Leibniz coined Analysis situs has not yet come to light and yet this art that would allow us to
discover the relationships of position between things would also be useful and perhaps more
necessary for natural sciences than the art whose only objective is the size of things; because
we more often need to know the form than the matter.” Refer to Buffon, G.-L. (1774). Œuvres
complètes, vol. IV, chap. IX. Imprimerie Royale, Paris, 73.