  A variety of simulation methodologies have been used for modeling
reaction-diffusion dynamics -- including approaches based on Differential
Equations (DE), the Stochastic Simulation Algorithm (SSA), Brownian Dynamics
(BD), Green's Function Reaction Dynamics (GFRD), and variations thereon -- each
offering trade-offs with respect to the ranges of phenomena they can model,
their computational tractability, and the difficulty of fitting them to
experimental measurements. Here, we develop a multiscale approach combining
efficient SSA-like sampling suitable for well-mixed systems with aspects of the
slower but space-aware GFRD model, assuming as with GFRD that reactions occur
in a spatially heterogeneous environment that must be explicitly modeled. Our
method extends the SSA approach in two major ways. First, we sample bimolecular
association reactions following diffusive motion with a time-dependent reaction
propensity. Second, reaction locations are sampled from within overlapping
diffusion spheres describing the spatial probability densities of individual
reactants. We show the approach to provide efficient simulation of spatially
heterogeneous biochemistry in comparison to alternative methods via application
to a Michaelis-Menten model.
