A non-memoryless stochastic simulation algorithm for modeling diffusion-reactions on biological membranes

Diffusion of biological molecules on 2D membranes can play an important role in the behavior of stochastic biochemical reaction systems. Yet, we still lack a fundamental understanding of circumstances where explicit accounting of the diffusion and spatial coordinates of molecules is necessary. In this work, we illustrate how time-dependent, non-exponential reaction probabilities naturally arise when explicitly accounting for the diffusion of molecules. We use the analytical expression of these probabilities to derive a novel algorithm which, while ignoring the exact position of the molecules, can still accurately capture diffusion effects. We investigate the regions of validity of this algorithm and show that for most parameter regimes, it constitutes an accurate framework for studying these systems. Our results constitute an important step in deriving a fundamental understanding of the role of diffusion in the operation of biochemical networks. Simultaneously, they provide a general computational methodology for simulating a broad class of systems whose behavior is influenced by diffusion on 2D membranes.