An unstructured mesh convergent reaction–diffusion master equation for reversible reactions

The convergent reaction–diffusion master equation (CRDME) was recently developed to provide a lattice particle-based stochastic reaction–diffusion model that is a convergent approximation in the lattice spacing to an underlying spatially-continuous particle dynamics model. The CRDME was designed to be identical to the popular lattice reaction–diffusion master equation (RDME) model for systems with only linear reactions, while overcoming the RDME's loss of bimolecular reaction effects as the lattice spacing is taken to zero. In our original work we developed the CRDME to handle bimolecular association reactions on Cartesian grids. In this work we develop several extensions to the CRDME to facilitate the modeling of cellular processes within realistic biological domains. Foremost, we extend the CRDME to handle reversible bimolecular reactions on unstructured grids. Here we develop a generalized CRDME through discretization of the spatially continuous volume reactivity model, extending the CRDME to encompass a larger variety of particle–particle interactions. Finally, we conclude by examining several numerical examples to demonstrate the convergence and accuracy of the CRDME in approximating the volume reactivity model.