Reaction rates for a generalized reaction-diffusion master equation
For researchers using spatial stochastic simulations of reaction-diffusion systems, this work removes a fundamental accuracy limitation, enabling simulations with finer meshes and broader applicability.
The authors extend the reaction-diffusion master equation to allow reactions between molecules in neighboring voxels, deriving rates that match the Smoluchowski model. This enables accurate simulations for a wider range of mesh sizes, including those below the standard method's lower bound, making previously intractable systems accessible.
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm, which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.