# Stochastic processes with applications to biological systems

This course takes place in the summer term 2008 every Tuesday from 14:15 - 15:45 in the large seminar room of the BIOQUANT-building (INF 267) and is given jointly with Christian Korn. On April 8 we will have a short meeting discussing content and literature. The first lecture will take place on April 15 and the last one on July 15. The course is given in English and divided into two parts.

The first part offers a detailed introduction into the theory of stochastic processes, similar to the first chapters of the textbook by Honerkamp, but supplemented by more recent developments and the special requirements for applications to biophysics. The following subjects are covered:

• Fundamental concepts: random variables, probability distribution, moments and cumulants, central limit theorem, conditional probability, stochastic (Markov) processes, white and colored noise, Chapman-Kolmogorov equation
• Examples for probability distributions: binomial, Gauss, Poisson
• Equations for stochastic processes: Fokker-Planck, master, Langevin
• Additive versus multiplicative noise, Ito versus Stratonovich interpretation, equivalence of Fokker-Planck and Langevin equations
• Examples for stochastic processes: random walks, radioactive decay, chemical reactions, birth and death processes
• Advanced subjects: first passage time problems, Kramers theory, bistable systems, noise-induced transitions, fluctuation-dissipation theorem, detailed balance, Kramers-Moyal expansion, fluctuation theorems and Jarzynski equation

The second part deals with modelling of biological systems. Here we follow mainly the recent literature. The following subjects are covered:

• Biomolecular bonds under force: binding in biological systems is always transient and stochastic; we discuss mean first passage time in one-dimensional energy landscape, escape over a transition state barrier, Kramers theory, coupling to an external force, adiabatic approximation and Bell equation, master equation for cooperative processes, Jarzynski equation, bond heterogeneity, models for catch bonds, analogy to protein folding
• Ion channels: these proteins allow ions to pass through biomembranes and are the basis for neuronal excitability; opening and closing is stochastic and can be modelled with mean first passage time methods. We also explain the relation to the famous Hodgkins-Huxley model for action potentials.
• Molecular motors: these proteins are responsible for force production and transport in cells, eg myosin II in muscle and kinesin for axonal transport; they move stochastically and different kinds of models have been developed to describe their motion, including ratchet models and the asymmetric exclusion process (ASEP); we also discuss recent models for cooperative transport by many motors, tug-of-war in bidirectional transport and mean first passage time problems for the motor-based transport of viruses to the nucleus
• Noise in gene expression: transcription and translation are stochastic events and recently a large body of experimental data has been measured in bacterial systems. For example, it has been shown that increasing cell (=system) size in E. Coli leads to a decreased noise level. We will review these experiments and simple models for this context.

# Literature

• J. Honerkamp, Stochastische Dynamische Systeme, VCH 1990
• N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier 1992
• C.W. Gardiner, Handbook of stochastic methods, Springer 2004
• W. Horsthemke und R. Lefever, Noise-induced transitions. Theory and Applications in Physics, Chemistry, and Biology, Springer 1984
• H. Risken, The Fokker-Planck Equation, Springer 1996
• H. C. Berg, Random Walks in Biology, Princeton University Press 1993
• P. Nelson, Biological Physics, Freeman 2003
• R. Phillips, J. Kondev and J. Theriot, Physical Biology of the Cell, to appear fall 2008