Stochastic reaction-diffusion problems in modeling biochemical systems

The dynamics of many biological processes rely on an interplay between spatial transport and chemical reactions. In particular, spatial dynamics can play a critical role in the successful functioning of cellular signaling processes, where as basic a prop- erty as cell shape can significantly influence the behavior of signaling pathways. The inside of cells is a complex spatial environment, filled with organelles, filaments and proteins. We investigate the question of how cell signaling pathways function robustly in the presence of such spatial heterogeneity for the most basic of chemical signals. Due to the noisy environment of a cell, particle-based stochastic reaction-diffusion models are a widely used approach for studying such cellular processes, explicitly modeling the diffusion of, and reactions between, individual molecules. However, the computational expense of such methods can greatly limit the size of chemical systems that can be studied. To overcome this challenge, we rigorously derive coarse-grained deterministic partial integro-differential equation models that provide a mean field ap- proximation to the particle-based stochastic reaction-diffusion model. Relationships between the mean field models and standard reaction-diffusion partial differential equation models are further investigated for general biochemical reaction systems. Comparisons between these models are illustrated through mathematical analysis and numerical examples