  To model bio-chemical reaction systems with diffusion one can either use
stochastic, microscopic reaction-diffusion master equations or deterministic,
macroscopic reaction-diffusion system. The connection between these two models
is not only theoretically important but also plays an essential role in
applications. This paper considers the macroscopic limits of the chemical
reaction-diffusion master equation for first-order chemical reaction systems in
highly heterogeneous environments. More precisely, the diffusion coefficients
as well as the reaction rates are spatially inhomogeneous and the reaction
rates may also be discontinuous. By carefully discretizing these
heterogeneities within a reaction-diffusion master equation model, we show that
in the limit we recover the macroscopic reaction-diffusion system with
inhomogeneous diffusion and reaction rates.
