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In recent years, modeling of biochemical systems has attracted much attention. Traditionally, these systems have been modeled with differential equations. However, specific biological processes tend to produce a lot of intrinsic noise, which has led to the development of a number of stochastic models for such processes. One such framework is that of Smoluchowski, which is used to model bimolecular reactions. In this model, particles diffuse with Brownian motion until they come within a critical separation distance and undergo a chemical reaction. This theory has undergone a number of developments, notably the extension of the theory to deal with reversible reactions, which has made available various software packages for stochastic reaction-diffusion simulations. Recent work has extended Smoluchowski theory to irreversible reactions of order greater than two. In this paper, we build on that theory and develop a model that includes reversible reactions of any order. This type of chemical kinetics is critical to many catalytic systems in biology. We use our new mod- eling framework to simulate the response of the core destruction regulation cycle to sharp changes in Wnt signaling, generating behavior not seen in classical ODE models. [ABSTRACT FROM AUTHOR]

Two algorithms that combine Brownian dynamics (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface, which partitions the domain, and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that the overlap region is required to accurately compute variances using PBD simulations. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented. [ABSTRACT FROM AUTHOR]