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Enzyme-catalysed reactions involve two distinct timescales. There is a short timescale on which enzymes bind to substrate molecules to produce bound complexes, and a comparatively long timescale on which the complex is transformed into a product. The rate at which the substrate is converted into product is characteristically non-linear and is traditionally derived by applying singular perturbation theory to the system's governing equations. Central to this analysis is the assumption that complex formation is effectively instantaneous on the timescale over which significant substrate degradation occurs. This prevents accurate modelling of enzyme kinetics by many particle-based simulations of reaction-diffusion systems as they rely on proximity-based reaction conditions that do not correctly model the fast reactions associated with the complex on the long timescale. In this paper we derive a new proximity-based reaction condition that correctly incorporates the reactions that occur on the short timescale for a specific enzymatic system. We present proof of concept particle-based simulations and demonstrate that non-linear reaction rates typical of enzyme kinetics can be reproduced without needing to explicitly simulate reactions on the short timescale., Comment: 18 pages, 6 figures

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 last century, Smoluchowski reaction theory has been applied widely in the physical, chemical, environmental and, more recently, the biological sciences. Despite the widespread utility of the Smoluchowski theory, it only describes the rates of second order reactions and is inadequate for the description of higher order reactions for which there is no equivalent method for theoretical investigation. In this paper, we derive a generalised Smoluchowski framework in which we define what should be meant by proximity in this 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 using 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.

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, due to recent advances in computational power, the simulation, and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist. The stochastic models with which we shall be concerned in this manuscript are referred to as `compartment-based'. These models are characterised by a discretisation of the computational domain into a grid/lattice of `compartments'. Within each compartment particles are assumed to be well-mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. We develop two hybrid algorithms in which a PDE is coupled to a compartment-based model. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: `how are individual particles transported between the vastly different model descriptions?' First, we present an algorithm derived by carefully re-defining the continuous PDE concentration as a probability distribution. Whilst this first algorithm shows strong convergence to analytic solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations to analytic solutions of PDEs for mean concentrations., Comment: MAIN - 24 pages, 10 figures, 1 supplementary file - 3 pages, 2 figures

The two-regime method (TRM) has been recently developed for optimizing stochastic reaction-diffusion simulations. 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 modelling 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 modelling regimes contain either flat parts or right-angled 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.

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 to accurately compute variances using the PBD simulation requires the overlap region. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented., Comment: submitted to SIAM Journal on Applied Mathematics