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Stochastic reaction-diffusion models have become an important tool instudying how both noise in the chemical reaction process and the spatialmovement of molecules influences the behavior of biological systems. There aretwo primary spatially-continuous models that have been used in recent studies:the diffusion limited reaction model of Smoluchowski, and a second approachpopularized by Doi. Both models treat molecules as points undergoing Brownianmotion. The former represents chemical reactions between two reactants throughthe use of reactive boundary conditions, with two molecules reacting instantlyupon reaching a fixed separation (called the reaction-radius). The Doi modeluses reaction potentials, whereby two molecules react with a fixed probabilityper unit time, $\lambda$, when separated by less than the reaction radius. Inthis work we study the rigorous relationship between the two models. For thespecial case of a protein diffusing to a fixed DNA binding site, we prove thatthe solution to the Doi model converges to the solution of the Smoluchowskimodel as $\lambda \to \infty$, with a rigorous $O(\lambda^{-1/2 + \epsilon})$error bound (for any fixed $\epsilon > 0$). We investigate by numericalsimulation, for biologically relevant parameter values, the difference betweenthe solutions and associated reaction time statistics of the two models. As thereaction-radius is decreased, for sufficiently large but fixed values of$\lambda$, these differences are found to increase like the inverse of thebinding radius.

A Comparison of Bimolecular Reaction Models for Stochastic Reaction Diffusion Systems