# B5.1 Stochastic Modelling of Biological Processes - Material for the year 2020-2021

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Part A Probability and Part A Integral Transforms

16 lectures

### Assessment type:

- Written Examination

"Stochastic Modelling of Biological Processes" provides an introduction to stochastic methods for modelling biological systems, covering a number of applications, ranging in size from molecular dynamics simulations of small biomolecules to stochastic modelling of groups of animals. The focus is on the underlying mathematics, i.e. it is not assumed that students took any advanced courses in biology or chemistry.

The course discusses the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields (including mathematical biology, non-equilibrium statistical physics, computational chemistry, soft condensed matter, physical chemistry or biophysics). New mathematical approaches and their analysis are explained using simple examples of biological models.

The course starts with stochastic (non-spatial) modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods which can be used for analysis of stochastic models. Different stochastic spatio-temporal models are then studied, including models of diffusion and stochastic reaction-diffusion modelling. The methods covered include molecular dynamics, Brownian dynamics, velocity jump processes and compartment-based (lattice-based) models.

The student will learn: (i) mathematical techniques for the analysis of stochastic models; (ii) how stochastic models can be efficiently simulated using a computer; (iii) connections and differences between different stochastic methods, and between stochastic and deterministic modelling.

Stochastic simulation of chemical reactions in well-stirred systems: Gillespie algorithm, chemical master equation, analysis of simple systems, deterministic vs. stochastic modelling.

Stochastic differential equations: numerical methods, Fokker-Planck equation, first exit time, backward Kolmogorov equation, chemical Fokker-Planck equation.

Stochastic reaction-diffusion modelling: compartment-based (lattice-based) models, reaction-diffusion master equation, Brownian dynamics, diffusion-limited reactions.

Molecular dynamics: molecular mechanics, generalised Langevin equation.

Stochastic models of dispersal in biological systems: velocity-jump processes, bacterial chemotaxis, collective animal behaviour.

R. Erban and S. J. Chapman, *Stochastic Modelling of Reaction-Diffusion Processes* (Cambridge University Press, 2020). Available at https://www.cambridge.org/core/books/stochastic-modelling-of-reactiondif...

R. Erban, S. J. Chapman and P. K. Maini, *A practical guide to stochastic simulation of reaction-diffusion processes* (2007). Available at http://arxiv.org/abs/0704.1908

Students are by no means expected to read all these sources. There are suggestions intended to be helpful to students interested exploring the subjects covered in further detail.

- H. Berg,
*Random Walks in Biology*(Princeton University Press, 1993). - D. T. Gillespie,
*Markov Processes, an Introduction for Physical Scientists*(Gulf Professional Publishing, 1992). - P. Attard,
*Non-Equilibrium Thermodynamics and Statistical Mechanics*(Oxford University Press, 2012). - A. Nitzan,
*Chemical Dynamics in Condensed Phases*(Oxford University Press, 2006). - P. Krapivsky, S. Redner and E. Ben-Naim,
*A Kinetic View of Statistical Physics*(Cambridge University Press, 2010). - D. Anderson and T. Kurtz,
*Stochastic Analysis of Biochemical Systems*(Springer, 2015). - B. Leimkuhler and C. Matthews,
*Molecular Dynamics: with Deterministic and Stochastic Numerical Methods*(Springer, 2015).