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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]

In 1917, Marian von Smoluchowski presented a simple mathematical description of diffusion-controlled reactions on the scale of individual molecules. His model postulated that a reaction would occur when two reactants were sufficiently close and, more specifically, presented a succinct relationship between the relative proximity of two reactants at the moment of reaction and the macroscopic reaction rate. Over the past century, the Smoluchowski reaction theory has been applied widely in the physical, chemical, environmental, and, more recently, biological sciences. Despite the widespread utility of the Smoluchowski theory, it only describes the rates of second order reactions and is inadequate for describing higher order reactions for which there is no equivalent method for theoretical investigation. In this paper, we derive a generalized Smoluchowski framework in which we define proximity in the context when more than two reactants are involved. We derive the relationship between the macroscopic reaction rate and the critical proximity at which a reaction occurs for higher order reactions. Using this theoretical framework and through numerical experiments, we explore various peculiar properties of multimolecular diffusion-controlled reactions which, due to there being no other numerical method of this nature, have not been previous reported. [ABSTRACT FROM AUTHOR]

The two-regime method (TRM) has been recently developed for optimizing stochastic reaction-diffusion simulations [M. Flegg, J. Chapman, and R. Erban, J. Roy. Soc. Interface, 9 (2012), pp. 859--868]. It is a multiscale (hybrid) algorithm which uses stochastic reaction-diffusion models with different levels of detail in different parts of the computational domain. The coupling condition on the interface between different modeling regimes of the TRM was previously derived for one-dimensional models. In this paper, the TRM is generalized to higher dimensional reaction-diffusion systems. Coupling Brownian dynamics models with compartment-based models on regular (square) two-dimensional lattices is studied in detail. In this case, the interface between different modeling regimes contains either flat parts or right-angle corners. Both cases are studied in the paper. For flat interfaces, it is shown that the one-dimensional theory can be used along the line perpendicular to the TRM interface. In the direction tangential to the interface, two choices of the TRM parameters are presented. Their applicability depends on the compartment size and the time step used in the molecular-based regime. The two-dimensional generalization of the TRM is also discussed in the case of corners. [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]