Fig 1. Simulation of a bimolecular reaction using a particle-based method and the spatial Gillespie approach.

In our particle-based simulations (A) molecules undergo a random walk in continuous space and discrete time intervals Δt. If a pair of reacting molecules are within a distance ρ, they can react with probability rate λ. In the spatial Gillespie approach (B), the domain is discretized using a grid, here with square elements of size h. Molecule jumps to adjacent grid elements and reactions take place within grid elements at random times. The propensity of a reaction within a grid element is proportional to the number of molecules (n_A, n_B) and a mesoscopic rate constant k_meso. (C) The scale-dependent mesoscopic rate k_h ensures that the mean association time of two molecules in the reaction-diffusion master equation (τ_c) matches an analogous microscopic representation (τ_R). (D) A concentration-dependent mesoscopic rate constant is defined as k_c = A_c/τ_Rc where A_c is the mean free area between molecules of the most abundant reactant in the grid element, estimated as A_c = h²/max(n_A, n_B), and τ_Rc is the mean association time calculated from a microscopic model of two molecules that react in a circular domain with area πR_c² = A_c. See Methods section for further details.