Stochastic Simulation of Coupled Reaction-Diffusion Processes

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.