  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-subdiffusion equations describing a system that
involves first-order reactions and subdiffusive motion? In this paper, we
answer this question. We derive fractional reaction-subdiffusion equations
describing an arbitrary number of molecular species which react at first-order
rates and move subdiffusively with general space-dependent diffusivities and
drifts. Importantly, different species may have different diffusivities and
drifts, which contrasts previous approaches to this question which 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-subdiffusion equations. This
stochastic description involves subordinating a diffusion process whose
dynamics are controlled by a subordinated Markov jump process. We illustrate
our results in several examples and show that solutions of the
reaction-subdiffusion equations agree with stochastic simulations of individual
molecules.
