Biology and Mathematics

To formalize the dynamics of living things is to search for invariants in a system that contains an irreducible aspect of "fuzziness", because biological processes are characterized by their large statistical variability, and strong dependence on temporal and environmental factors. What is essential is the identification of what remains stable in a "living being" that is highly fluctuating. The use of mathematics is not limited to the use of calculating tools to simulate and predict results. It also allows us to adopt a way of thinking that is founded on concepts and hypotheses, leading to their discussion and validation. Instruments of mathematical intelligibility and coherence have gradually "fashioned" the view we now have of biological systems. Teaching and research, fundamental or applied, are now dependent on this new order known as Integrative Biology or Systems Biology.

Auteur

Roger Buis, Emeritus Professor at the Université de Toulouse, France, has taught Statistical Biometry and Biomathematics at various universities and engineering schools (agronomy). His research initially focused on plant growth and morphogenesis has been developed in the context of differential dynamical systems.

Contenu

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 Some Historical Reference Points: Biology Fashioned by Mathematics 35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

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 (18131878): 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 reactiondiffusion 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

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. Mult...