The auxiliary region method: a hybrid method for coupling PDE- and Brownian-based dynamics for reaction–diffusion systems

Reaction–diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE- and Brownian-based representations of reaction–diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based ‘auxiliary regions’. We demonstrate that the hybrid method is able to simulate reaction–diffusion dynamics for a number of different test problems with high accuracy. Furthermore, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.