Stochastic Reaction–Diffusion Models

Abstract

This chapter discusses stochastic approaches for modelling chemical reactions (introduced in Chapter 1) and molecular diffusion at the same time. The presented stochastic reaction–diffusion processes add chemical reactions to the two position-jump models of molecular diffusion that are introduced in Chapter 4: the compartment-based approach (described by the reaction–diffusion master equation) and the SDE-based approach, which gives the Brownian dynamics. Basic principles of each approach are explained using an example that includes only zeroth- and first-order chemical reactions. This is followed by discussion of more complicated systems when some chemical species are subject to higher-order chemical reactions. The reaction radius, reaction probability and the choice of the compartment size are studied in detail. The chapter concludes with the discussion of applications to pattern formation in biology, including stochastic French flag model and stochastic Turing patterns.