Discrete-continuous reaction-diffusion model with mobile point-like sources and sinks

In many applications in soft and biological physics, there are multiple time and length scales involved but often with a distinct separation between them. For instance, in enzyme kinetics, enzymes are relatively large, move slowly and their copy numbers are typically small, while the metabolites (being transformed by these enzymes) are often present in abundance, are small in size and diffuse fast. It seems thus natural to apply different techniques to different time and length levels and couple them. Here we explore this possibility by constructing a stochastic-deterministic discrete-continuous reaction-diffusion model with mobile sources and sinks. Such an approach allows in particular to separate different sources of stochasticity. We demonstrate its application by modelling enzyme-catalysed reactions with freely diffusing enzymes and a heterogeneous source of metabolites. Our calculations suggest that using a higher amount of less active enzymes, as compared to fewer more active enzymes, reduces the metabolite pool size and correspondingly the lag time, giving rise to a faster response to external stimuli. The methodology presented can be extended to more complex systems and offers exciting possibilities for studying problems where spatial heterogeneities, stochasticity or discreteness play a role