M11: Stochastic modelling of reaction-diffusion processes

There are two fundamental approaches to the mathematical modelling of chemical reactions and diffusion: deterministic models which are based on partial differential equations (PDEs), and stochastic simulation algorithms (SSAs). Stochastic models provide a more detailed understanding of reaction-diffusion processes, and are often necessary for the modelling of biological systems where small numbers of molecules of some chemical species make deterministic models inaccurate or even inapplicable. SSAs are also necessary when biologically observed phenomena depend on stochastic fluctuations [5].

Several reaction-diffusion models have been proposed in the literature but they each have their limitations. The first objective of the project is to investigate under which conditions different stochastic reaction-diffusion models are equivalent and under which conditions they differ [1,2]. We investigate scenarios where lattice-based models provide the correct description of physical reality, and where they fail (and more computationally intensive off-lattice models must be used instead) [2,3]. The second major objective is to design a reliable, correct and efficient method for the stochastic simulation of reaction-diffusion processes in biology. It is often the case that one is interested in a detailed description of only a small subset of the reaction-diffusion system (for example, one wants to understand the intracellular processes close to a cellular membrane [1]). Consequently, there is a possibility of decreasing the computational intensity of SSAs by using the most detailed modelling approach only in the small subdomain which one wants to study, and by using a less computationally intensive (less detailed, coarser) model in other parts of the computational domain [4]. To achieve this goal, we investigate how combining stochastic reaction-diffusion models with different levels of detail (or even combining SSAs with deterministic PDEs) can be done accurately and efficiently. This will ultimately lead to reliable software which will be of use to scientists outside mathematics, for example, to computational biologists and computational chemists.

[1] Radek Erban and Jonathan Chapman, "Reactive boundary conditions for stochastic simulations of reaction-diffusion processes", Physical Biology, Volume 4, Number 1, pp. 16-28 (2007)

[2] Radek Erban and Jonathan Chapman, "Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions", Physical Biology, Volume 6, Number 4, 046001 (2009)

[3] Jana Lipkova, Konstantinos Zygalakis, Jonathan Chapman and Radek Erban, "Analysis of Brownian Dynamics Simulations of Reversible Bimolecular Reactions", submitted to SIAM Journal on Applied Mathematics (2010)

[4] Radek Erban and Jonathan Chapman, "Time scale of random sequential adsorption", Physical Review E, Volume 75, Number 4, 041116 (2007)

[5] Radek Erban, Jonathan Chapman and Philip Maini, "A practical guide to stochastic simulations of reaction-diffusion processes", Lecture Notes, (2007)

This project is funded by the European Research Council Starting Independent Researcher Grant awarded to Dr Erban.