Biology and Mathematics: History and Challenges

Description

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. Biology and Mathematics: History and Challenges Contents Foreword Introduction 1 On the Status of Biology: On the Definition of Life 1.1. Causality in biology 1.1.1. Vitalism 1.1.2. Teleology 1.2. Variability in biology 1.2.1. Time-dependence of biological processes 1.2.2. Environment-dependence of biological processes 2 On the Nature of the Contribution Made by Mathematics to Biology 2.1. On the affinity of mathematics with biology 2.2. Mathematics, an instrument of work and thought on biology 3 Some Historical Reference Points: Biology Fashioned by Mathematics 3.1. The first remarkable steps in biomathematics 3.1.1. On the continuous in biology 3.1.2. On the discrete in biology 3.1.3. The notion of laws in biology 3.1.4. The beginning of classical science: Descartes and Pascal 3.1.5. Buffon and hesitations relating to the utility of mathematics in biology 3.1.5.1. On Buffon’s epistemological position 3.2. Some pertinent contributions from mathematics in the modern era 3.2.1. The laws of growth 3.2.2. Formal genetics 3.2.2.1. Mendel’s laws 3.2.2.2. The genetics of populations 3.2.2.3. Quantitative genetics 3.2.2.4. The probabilistic point of view 3.2.2.5. The algebraic approach 3.3. Introduction of the notion of a probabilistic model in biology 3.4. The physiology of C. Bernard (1813–1878): the call to mathematics 3.5. The principle of optimality in biology 3.6. Introduction of the formalism of dynamic systems in biology 3.7. Morphogenesis: the need for mathematics in the study of biological forms 3.7.1. General principles from D’Arcy Thompson 3.7.2. Turing’s reaction–diffusion systems (1952): morphogenesis, a “break of symmetry” 3.8. The theory of automatons and cybernetics in biology 3.8.1. The theory of automatons 3.8.2. The contribution of cybernetics 3.8.3. The case of L-systems 3.8.4. Petri’s networks 3.9. Molecular biology 3.9.1. On genetic information 3.9.2. The linguistic model in biology 3.10. Information and communication, important notions in biology 3.11. The property of self-organization in biology 3.11.1. Structural self-organization 3.11.2. Self-reproductive hypercycle 3.12. Systemic biology 3.12.1. On the notion of system 3.12.2. Essay in relational biology 3.12.3. Emergence and complexity 3.12.4. Networks 3.12.4.1. Random networks 3.12.4.2. Small-world network 3.12.4.3. Scale-free networks 3.12.5. Order, innovation and complex networks 3.13. Game theory in biology 3.14. Artificial life 3.14.1. Biomimetic automatons 3.14.2. Psychophysiology and mathematics: controls on learning 3.15. Bioinformatics 4 Laws and Models in Biology 4.1. Biological laws in literary language 4.1.1. The law of Cuvier’s organic correlations (1825) 4.1.2. The fundamental biogenetic law 4.2. Biological laws in mathematical language 4.2.1. Statistical laws 4.2.1.1. The law of allometry 4.2.1.2. Physiological or metabolic allometry 4.2.1.3. Physiological allometry and growth: Bertalanffy’s theory 4.2.1.4. Laws of action of a factor: effect/concentration relationships 4.3. Theoretical laws 4.3.1. Formal genetics 4.3.2. Growth laws 4.3.3. Population dynamics 5 Mathematical Tools and Concepts in Biology 5.1. An old biomathematical subject: describing and/or explaining phyllotaxis 5.2. The notion of invariant and its substrate: time and space 5.2.1. Physical time/biological time 5.2.1.1. The physiological time of Lecomte du Noüy (1936) 5.2.1.2. Backman’s biological time (1942) 5.2.1.3. Plastochronic time in higher plants (according to Erickson 1957) 5.2.1.4. The change of state of the system 5.2.2. Metric space/non-metric space 5.2.2.1. Metric space 5.2.2.2. Non-metric space 5.2.3. Multi-scale processes 5.3. Continuous formalism 5.3.1. Dynamics of a univariate process 5.3.2. Structured models 5.3.2.1. The univariate function of Gompertz 5.3.2.2. Multi-compartment models 5.3.3. Oscillatory dynamics 5.3.4. On the stability of dynamic systems 5.3.4.1. Structural stability, an implicit assumption of dynamic systems 5.3.4.2. The paradox of enrichment: the Rosenzweig–MacArthur model 5.3.4.3. Stability and optimal process control 5.3.5. Multivariate structured models 5.3.5.1. Droop’s model 5.3.5.2. Models of enzymatic kinetics 5.3.6. Dynamics of spatio-temporal process 5.3.6.1. Growth–diffusion–advection models 5.3.6.2. Reaction–diffusion models 5.3.6.3. Field models 5.3.6.4. The notion of hypercycle 5.3.7. Multi-scale models 5.4. Discreet formalism 5.5. Spatialized models 5.5.1. Multi-agent models: dynamics of a biological association of the individual-centered type 5.5.2. Electrophysiological models: transmission of electrical signals 5.6. Random processes in biology 5.6.1. Poisson process 5.6.2. Birth–death processes 5.7. Logic kinetics of regulation Conclusion Mathematics and the perception of the singularities of the living Discrete versus continuous Microscopic versus macroscopic levels The new trends in biology Glossary References Index