Jürgen Jost - Mathematical Methods in Biology and Neurobiology

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Jrgen Jost Universitext Mathematical Methods in Biology and Neurobiology 2014 10.1007/978-1-4471-6353-4_1

Springer-Verlag London 2014

1. Introduction

Jrgen Jost 1

(1)

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Jrgen Jost

Email:

Abstract

Questions:

• What can mathematics contribute to biology, and which mathematical theories are useful for that purpose?

Biology does not have the clear structure of mathematics. Nevertheless, it possesses some fundamental concepts. The gene is the unit of coding, function, and inheritance. It contains the information for a phenotypic trait that is realized in interaction with contributions from the environment and transmitted to offspring. The cell is the basic unit within which metabolic processes can take place. The species is the dynamic pool for genetic recombination. An organism is a carrier of genes, an organized ensemble of cells and a member of a population or species. Mathematical methods to study biological phenomena can be taken from algebra, analysis, stochastics, or geometry, but should always be developed with a clear vision of the biological problems to be addressed.

1.1 Theses About Biology

Thesis 1.

Biological structures are aggregate structures. Therefore, biological laws are not basic ones that do not admit exceptions, but rather emerging from some lower scale.

Thesis 2.

Biological entities are discrete, but biological structures are situated in continuous space and biological processes take place in continuous time.

Thesis 3.

Biological processes intertwine stochastic effects and deterministic dynamics. Randomness can support order while deterministic processes can be unpredictable, chaotic. The question then is at which level regularities emerge.

Thesis 4.

Large populations of discrete units can be described by continuous models and, conversely, invariant discrete quantities can emerge from an underlying continuous substrate.

Thesis 5.

Fundamental biological concepts, like fitness or information, are relative and not absolute ones.

Thesis 6.

Fundamental biological quantities do not satisfy conservation laws. Those rather appear as external constraints.

Thesis 7.

Biological systems interact with their environments and are thermodynamically open. Biological structures sustain the processes that reproduce them and are therefore operationally closed.

Thesis 8.

Biological structures are results of historical processes. It is the task of biological theory to distinguish the regularities from the contingencies.

Thesis 9.

The abstract question posed to mathematics by biology is structure formation. This needs to be understood as a process because living structures are not at thermodynamic equilibrium.

Thesis 10.

Gathering biological data without guiding concepts and theories is useless.

1.2 Fundamental Biological Concepts

The gene is the unit of coding, function, and inheritance. As such, it links molecular biology and evolutionary biology. The Neodarwinian Synthesis combined Mendel and Darwin. Modern molecular biology seems to offer a more basic perspective.

The cell is the unit of metabolism. It constitutes the basic operationally closed, autopoietic system in biology. Modern biology struggles to understand cells on the basis of their molecular constituents, DNA, RNA, and polypeptides (proteins). Multicellular organisms emerge through a partial suppression of the autonomy of the constituting cells.

The species represents the balance between the diverging effects of genetic mutations and selection at the organismic or other levels and the converging mechanism of sexual recombination. It is the arena of population biology, a child of the Neodarwinian Synthesis and the first success of mathematical models in biology. It is also important in ecology.

The organism , in fact, is the carrier of genes, the organization of cells, and the member of a species. It thus links the three fundamental biological concepts. It is also a, but not the exclusive, unit of selection.

It seems that neurobiology has not yet identified such a fundamental concept, but perhaps the spike can be considered as the basic event of information transmission, and the synapse as the basic structure supporting this.

1.3 A Classification of Mathematical Methods

The following is a somewhat incomplete list, arranged partly with relevance for biology in mind.

Discrete structures

(a)

Static structures

i.

Algebraic concepts: Combination and composition of objects

ii.

Graphs and networks, including phylogenetic trees

iii.

Information

iv.

Discrete invariants of continuous structures and dynamical processes

(b)

Discrete processes (Cellular automata, Boolean networks, finite state machines,...)

(c)

Game theory as the formalization of competition

Spatial relations

(a)

Geometry of (three-dimensional) physical space

(b)

Abstract notions of space for expressing relationships (discrete ones like graphs and continuous ones like Hilbert spaces; state spaces of dynamical systems)

(c)

Symmetries and invariances

Continuous methods

(a)

Deterministic dynamical processes

i.

Continuous states enable phase transitions and bifurcations, that is, qualitative structural changes resulting from small underlying variations

ii.

Continuous states and time: Ordinary differential equations and other dynamical systems

iii.

Continuous spatial structures: Partial differential equations (example: Reaction-diffusion equations)

(b)

Stochastic analysis

i.

Stochastic processes (while stochastic processes may also operate on discrete quantities, the concept of probability is a continuous one)

ii.

Population processes: averaging over stochastic fluctuations in lower level dynamics

iii.

Optimization schemes with stochastic ingredients: Genetic and other evolutionary algorithms, swarm algorithms for distributed search, certain neural networks,...

iv.

Statistical methods for the analysis of biological data

Hybrid models

(a)

Difference equations (continuous states, but discrete time)

(b)

Dynamical networks (dynamical systems coupled by a graph), in particular neural networks

System theory as a global unifying perspective?

According to the preceding list, not all mathematical subjects seem to be relevant for biology. Classical algebraic structures occur in a cursory manner at best, and one of the deepest branches, number theory and arithmetics, is entirely absent. Three-dimensional physical space constitutes an important constraint for biological organization. Organisms and their constitutive biological structures like cells are living and interacting in space, and are defining and shaping their own spaces like architectural structures which is constitutive for morphology. Symmetries and invariances, the merging ground of algebra and geometry, are important issues for the neurobiology underlying cognition, as well as for many classification purposes. In any case, the branches of algebra, geometry and analysis are often interwoven.

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