  The reaction-diffusion master equation (RDME) is a lattice stochastic
reaction-diffusion model that has been used to study spatially distributed
cellular processes. The RDME is often interpreted as an approximation to
spatially-continuous models in which molecules move by Brownian motion and
react by one of several mechanisms when sufficiently close. In the limit that
the lattice spacing approaches zero, in two or more dimensions, the RDME has
been shown to lose bimolecular reactions. The RDME is therefore not a
convergent approximation to any spatially-continuous model that incorporates
bimolecular reactions. In this work we derive a new convergent RDME (CRDME) by
finite volume discretization of a spatially-continuous stochastic
reaction-diffusion model popularized by Doi. We demonstrate the numerical
convergence of reaction time statistics associated with the CRDME. For
sufficiently large lattice spacings or slow bimolecular reaction rates, we also
show the reaction time statistics of the CRDME may be approximated by those
from the RDME. The original RDME may therefore be interpreted as an
approximation to the CRDME in several asymptotic limits.
