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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]

In this paper, three multiscale methods for coupling of mesoscopic (compartment-based) and microscopic (molecular-based) stochastic reaction–diffusion simulations are investigated. Two of the three methods that will be discussed in detail have been previously reported in the literature; the two-regime method (TRM) and the compartment–placement method (CPM). The third method that is introduced and analysed in this paper is called the ghost cell method (GCM), since it works by constructing a “ghost cell” in which molecules can disappear and jump into the compartment-based simulation. Presented is a comparison of sources of error. The convergent properties of this error are studied as the time step Δ t (for updating the molecular-based part of the model) approaches zero. It is found that the error behaviour depends on another fundamental computational parameter h , the compartment size in the mesoscopic part of the model. Two important limiting cases, which appear in applications, are considered: (i) Δ t → 0 and h is fixed; (ii) Δ t → 0 and h → 0 such that Δ t / h is fixed. The error for previously developed approaches (the TRM and CPM) converges to zero only in the limiting case (ii), but not in case (i). It is shown that the error of the GCM converges in the limiting case (i). Thus the GCM is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model. [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]