Multiscale stochastic reaction-diffusion algorithms combining Markov chain models with SPDEs

Abstract

When large chemical systems are investigated and if molecules are not well-stirred, spatial distribution of the molecules of chemical species can be modeled as reaction-diffusion systems. To describe stochastic fluctuations, we split the spatial domain into compartments and assume each species is well mixed in each compartment. Then, we model the system state as a Markov chain and a sample path can be obtained using Gillespie’s Stochastic Simulation Algorithm [1]. The algorithm provides one realization of the exact trajectory of the system state, but it can be computationally expensive due to high dimensionality. Large chemical systems have multiscale nature so it is possible that one region has more abundance in the number of molecules of chemical substances than that in the other region. In this talk, I will introduce a multiscale algorithm for stochastic simulation of reaction-diffusion processes. The algorithm is applicable to systems which include regions with a few molecules and regions with a large number of molecules. A domain of interest is divided into two subsets and two regions are connected by interface. In our algorithm, one set of regions with a small number of molecules is modeled as a continuous-time Markov chain and simulated by Gillespie’s algorithm. On the other hand, the other set of regions with a large number of molecules is described using stochastic partial differential equations (SPDEs) as derived in [2] and a solution of SPDEs is numerically computed. Matching interface between two regions was challenging and mean and variance of the number of molecules are matched. In this talk, we suggest two multiscale schemes for reaction diffusion processes, and several examples will be shown.