Detailed Balance for Particle Models of Reversible Reactions in Bounded Domains

In particle-based stochastic reaction-diffusion models, reaction rate and placement kernels are used to decide the probability per time a reaction can occur between reactant particles, and to decide where product particles should be placed. When choosing kernels to use in reversible reactions, a key constraint is to ensure that detailed balance of spatial reaction-fluxes holds at all points at equilibrium. In this work we formulate a general partial-integral differential equation model that encompasses several of the commonly used contact reactivity (e.g. Smoluchowski-Collins-Kimball) and volume reactivity (e.g. Doi) particle models. From these equations we derive a detailed balance condition for the reversible $\textrm{A} + \textrm{B} \leftrightarrows \textrm{C}$ reaction. In bounded domains with no-flux boundary conditions, when choosing unbinding kernels consistent with several commonly used binding kernels, we show that preserving detailed balance of spatial reaction-fluxes at all points requires spatially varying unbinding rate functions near the domain boundary. Brownian Dynamics simulation algorithms can realize such varying rates through ignoring domain boundaries during unbinding and rejecting unbinding events that result in product particles being placed outside the domain.