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Über dieses Buch

Mathematical models can be used to meet many of the challenges and opportunities offered by modern biology. The description of biological phenomena requires a range of mathematical theories. This is the case particularly for the emerging field of systems biology. Mathematical Methods in Biology and Neurobiology introduces and develops these mathematical structures and methods in a systematic manner. It studies:

• discrete structures and graph theory
• stochastic processes
• dynamical systems and partial differential equations
• optimization and the calculus of variations.

The biological applications range from molecular to evolutionary and ecological levels, for example:

• cellular reaction kinetics and gene regulation
• biological pattern formation and chemotaxis
• the biophysics and dynamics of neurons
• the coding of information in neuronal systems
• phylogenetic tree reconstruction
• branching processes and population genetics
• optimal resource allocation
• sexual recombination
• the interaction of species.

Written by one of the most experienced and successful authors of advanced mathematical textbooks, this book stands apart for the wide range of mathematical tools that are featured. It will be useful for graduate students and researchers in mathematics and physics that want a comprehensive overview and a working knowledge of the mathematical tools that can be applied in biology. It will also be useful for biologists with some mathematical background that want to learn more about the mathematical methods available to deal with biological structures and data.



Chapter 1. Introduction

  • 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.
Jürgen Jost

Chapter 2. Discrete Structures

  • How can the cells of an organism which all share the same genes can fulfill so many different functions?
  • Are there good mathematical tools to identify the important features in all those networks that modern biological data collection produces?
  • How long ago did the last common ancestor of two species or two individuals live?
A model of combinatorial gene regulation shows the power of combinatorics. Graphs are useful tools for network analysis, and their spectral theory is developed. Phylogenetic relationships between species are modeled by particular types of graphs, the trees. Descendence relations between individuals involve two parents and lead to genealogies. Coalescents treat the question of common ancestors. Such structures also naturally lead to the stochastic processes treated in the Chap. 3.
Jürgen Jost

Chapter 3. Stochastic Processes

  • How can the seemingly random firing pattern of a neuron encode any information about the inputs received?
  • What will eventually happen to a population when the number of offspring of each individual randomly fluctuates?
We introduce the theory of stochastic processes. The coding and decoding of input information in systems of neurons is then modeled in terms of Poisson processes. Whereas in the last chapter we have treated descendence relations backward in time, to trace the ancestors, here we use branching processes to predict the future of populations.
Jürgen Jost

Chapter 4. Pattern Formation

  • How will substances diffuse over time?
  • How does the biophysics of a neuron work?
  • How can we model reaction kinetics in a cell?
  • What will happen when two or more species interact, like predators and their prey?
  • How can oscillatory patterns emerge?
  • How can external stimuli trigger collective behavior within a population of independent individuals?
Understanding pattern formation requires tools from analysis. We introduce dynamical systems to model changes in time and partial differential equations to model distributions in physical or feature spaces. The combination of the two in reaction-diffusion systems leads to mathematical models like the Turing mechanism that can generate surprisingly rich patterns. Another example we treat is chemotaxis where organisms can be induced to collective behavior by following gradients of chemical substances.
Jürgen Jost

Chapter 5. Optimization

  • How do we best distribute our finite resources among different tasks?
  • Isn’t sexual reproduction wasteful? Why are there males?
  • What are the criteria for the optimality of a functional relationship?
Biological evolution is about getting an advantage by becoming better than others. That is, optimization problems arise and should be solved. A key problem is the best allocation of finite resources to different tasks. We develop this in a simple setting. As an application, we can explain why sexual reproduction is prevalent inspite of its apparent shortcomings. We also introduce the calculus of variations as the mathematical theory for the optimization of functional relationships.
Jürgen Jost

Chapter 6. Population Genetics

  • How does the distribution of alleles (genetic variants) change over time in a population when those alleles are randomly passed on to offspring?
This last chapter draws upon all the different methods discussed in the preceding, discrete structures, stochastics, analysis, and geometry. It introduces mathematical population genetics, the theory of the time course of the distribution of alleles in a population in the presence of mutation, selection, and recombination. The basic Wright-Fisher model is a discrete stochastic processes. In order to understand it better, it is advantageous to pass to its diffusion approximation which leads to a partial differential equation. For understanding this differential equation in turn a geometric approach is insightful.
Jürgen Jost


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