Stochastic Simulation of Coupled Reaction–Diffusion Processes

actants is used to compute the time evolution of reactant concentrations. The stochastic algorithm is rigorous in the The stochastic time evolution method has been used previously to study non-linear chemical reaction processes in well-stirred hosense that it provides an exact solution to the correspondmogeneous systems. We present the first treatment of diffusion, in ing master equation for chemical reaction in a homogethe stochastic method, for non-linear reaction–diffusion processes. neous, well-stirred reaction volume [6]. Because the GillesThe derivation introduces mesoscopic rates of diffusion that are pie method follows unit-by-unit changes in the total formally analogous to reaction rates. We map, using Green’s funcnumbers of each reactant species, it is especially well suited tion, the bulk diffusion coefficient D in Fick’s differential law to the to the study of systems in which reactant densities are corresponding transition rate probability for diffusion of a particle between finite volume elements. This generalized stochastic algolow and the application of methods based on continuum rithm enables us to numerically calculate the time evolution of a approximations, such as the traditional ordinary differenspatially inhomogeneous mixture of reaction–diffusion species in tial equations of chemical kinetics, is questionable. This a finite volume. The algorithm is equivalent to solving the time capability is especially relevant to biophysics and cell biolevolution of the spatially inhomogeneous master equation. A ogy. Within the intact living cell, number densities of key unique feature of our method is that the time step is stochastic and is generated by a probability distribution determined by the intrinsic proteins, polynucleotides, and intracellular signaling molereaction kinetics and diffusion dynamics. To demonstrate the cules are typically low [1–102 em23]. Stochastic methods method, we consider the biologically important nonlinear reaction– such as that developed by Gillespie are well suited to the diffusion process of calcium wave propagation within living computational study of such systems. cells. Q 1996 Academic Press, Inc. However, in contrast to a well-stirred reaction system, the intact living cell is based on a highly complex spatial organization of its constituents, rather than on homoge