REACTION-SUBDIFFUSION EQUATIONS WITH SPECIES-DEPENDENT MOVEMENT

Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than normal diffusion, anomalous subdiffusion is observed in many systems and is especially prevalent in cell biology. What are the reaction-sub diffusion equations describing a system that involves first-order reactions and subdiffusive motion? In this paper, we derive fractional reaction-sub diffusion equations describing an arbitrary number of molecular species that react at first-order rates and move subdiffusively. Importantly, different species may have different diffusivities and drifts, which contrasts previous approaches to this question that assume that each species has the same movement dynamics. We derive the equations by combining results on time dependent fractional Fokker-Planck equations with methods of analyzing stochastically switching evolution equations. Furthermore, we construct the stochastic description of individual molecules whose deterministic concentrations follow these reaction-sub diffusion equations. This stochastic description involves subordinating a diffusion process whose dynamics are controlled by a subordinated jump process. We illustrate our results in several examples and show that solutions of the reactionsubdiffusion equations agree with stochastic simulations of individual molecules.

Automated summary

Reaction-diffusion equations are commonly used mathematical models in natural sciences, but anomalous subdiffusion is observed in many systems, especially in cell biology. This paper derives fractional reaction-sub diffusion equations to describe systems involving first-order reactions and subdiffusive motion. The study shows that different molecular species may have varying diffusivities and drifts, unlike previous assumptions of uniform movement dynamics for each species.