Reaction Diffusion Master Equation (RDME) framework, characterized by the discretization of the spatial domain, is one of the most widely used methods in the stochastic simulation of reaction-diffusion systems. Discretization sizes for RDME have to be appropriately chosen such that each discrete compartment is "well-stirred" and the computational cost is not too expensive.

An efficient discretization size based on the reaction-diffusion dynamics of each species is derived in this dissertation. Usually, the species with larger diffusion rate yields a larger discretization size. Partitioning with an efficient discretization size for each species, a multiple grid discretization (MGD) method is proposed. MGD avoids unnecessary molecular jumping and achieves great simulation efficiency improvement.

Moreover, reaction-diffusion systems with reaction dynamics modeled by highly nonlinear functions, show large simulation error when discretization sizes are too small in RDME systems. The switch-like Hill function reduces into a simple bimolecular mass reaction when the discretization size is smaller than a critical value in RDME framework. Convergent Hill function dynamics in RDME framework that maintains the switch behavior of Hill functions with fine discretization is proposed.

Furthermore, the application of stochastic modeling and simulation techniques to the spatiotemporal regulatory network in Caulobacter crescentus is included. A stochastic model based on Turing pattern is exploited to demonstrate the bipolarization of a scaffold protein, PopZ, during Caulobacter cell cycle. In addition, the stochastic simulation of the spatiotemporal histidine kinase switch model captures the increased variability of cycle time in cells depleted of the divJ genes.