Stochastic Simulation of Coupled Reaction–Diffusion Processes

• Published in: Journal of computational physics 1996-08, Vol.127 (1), p.196-207
• Main Authors: Stundzia, Audrius B., Lumsden, Charles J.
• Format: Article
• Language: English
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1006/jcph.1996.0168

The stochastic time evolution method has been used previously to study non-linear chemical reaction processes in well-stirred homogeneous systems. We present the first treatment of diffusion, in the stochastic method, for non-linear reaction–diffusion processes. The derivation introduces mesoscopic rates of diffusion that are formally analogous to reaction rates. We map, using Green's function, the bulk diffusion coefficient D in Fick's differential law to the corresponding transition rate probability for diffusion of a particle between finite volume elements. This generalized stochastic algorithm enables us to numerically calculate the time evolution of a spatially inhomogeneous mixture of reaction–diffusion species in a finite volume. The algorithm is equivalent to solving the time evolution of the spatially inhomogeneous master equation. A unique feature of our method is that the time step is stochastic and is generated by a probability distribution determined by the intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method, we consider the biologically important nonlinear reaction–diffusion process of calcium wave propagation within living cells.