  Stochastic reaction-diffusion models have become an important tool in
studying how both noise in the chemical reaction process and the spatial
movement of molecules influences the behavior of biological systems. There are
two primary spatially-continuous models that have been used in recent studies:
the diffusion limited reaction model of Smoluchowski, and a second approach
popularized by Doi. Both models treat molecules as points undergoing Brownian
motion. The former represents chemical reactions between two reactants through
the use of reactive boundary conditions, with two molecules reacting instantly
upon reaching a fixed separation (called the reaction-radius). The Doi model
uses reaction potentials, whereby two molecules react with a fixed probability
per unit time, $\lambda$, when separated by less than the reaction radius. In
this work we study the rigorous relationship between the two models. For the
special case of a protein diffusing to a fixed DNA binding site, we prove that
the solution to the Doi model converges to the solution of the Smoluchowski
model as $\lambda \to \infty$, with a rigorous $O(\lambda^{-1/2 + \epsilon})$
error bound (for any fixed $\epsilon > 0$). We investigate by numerical
simulation, for biologically relevant parameter values, the difference between
the solutions and associated reaction time statistics of the two models. As the
reaction-radius is decreased, for sufficiently large but fixed values of
$\lambda$, these differences are found to increase like the inverse of the
binding radius.
