Synopsis for Stochastic Modelling of Biological Processes

This course provides an overview of stochastic methods which are used for modelling biological systems. The student will learn:

1. about biological systems which are often described in terms of stochastic models;
2. mathematical techniques which are used for the analysis of stochastic models;
3. how the models can be efficiently simulated on a computer;
4. connections and differences between different stochastic methods, and between stochastic and deterministic modelling.

The following topics will be covered:

• Introduction: deterministic vs. stochastic modelling, systems with multiple favourable states, self-induced stochastic resonance, stochastic focusing [1 lecture]
• Stochastic simulation of chemical reactions: well-stirred systems, Gillespie algorithm, chemical master equation, analysis of simple systems, moment generating function [2 lectures]
• Stochastic differential equations: numerical methods, Fokker-Planck equation, first exit time, chemical Fokker-Planck equation [2 lectures]
• Diffusion: Brownian motion, on-lattice and off-lattice models, compartment-based approach, velocity jump processes, diffusion to adsorbing surfaces, reactive boundary conditions, Einstein-Smoluchowski relation [2 lectures]
• Efficient stochastic modelling of chemical reactions: multiscale modelling, multiscale SSA with partial equilibrium assumption, first-reaction SSA, Gibson-Bruck algorithm, tau-leaping algorithm [1 lecture]
• Stochastic reaction-diffusion models: compartment-based reaction-diffusion algorithm, reaction-diffusion master equation, pattern formation [1 lecture]
• Brownian dynamics: stochastic reaction-diffusion modelling, reaction radius, modelling reversible reactions; Brownian dynamics with constraints (simulation of molecules): algorithm SHAKE. Monte Carlo: Metropolis algorithm, Markov chain Monte Carlo simulation [1 lecture]
• Bacterial chemotaxis: reaction-diffusion-advection processes, velocity jump process with internal dynamics [1 lecture]
• Ions and ion channels: Brownian dynamics vs. molecular dynamics, multiscale modelling; Macromolecular crowding [1 lecture]

1. R. Erban, S. J. Chapman, P. K. Maini, A practical guide to stochastic simulations of reaction-diffusion processes (Draft textbook available from http://people.maths.ox.ac.uk/erban/Education/ or http://arxiv.org/abs/0704.1908).