Theory and simulation of stochastically-gated diffusion-influenced reactions

The kinetics of the irreversible diffusion-influenced reaction between a protein (P) and a ligand (L) is studied when [L] much greater than [P] and the reactivity is stochastically gated due to conformational fluctuations of one of the species. If gating is due to the ligand, we show that the Smoluchowski rate equation, d[P(t)]/dt = -k(t)[L][P(t)], can be generalized by simply using a stochastically-gated time-dependent rate coefficient, k(sg)(t). However, if gating is due to the protein, this is no longer true, except when the gating dynamics is sufficiently fast or the Ligand concentration is very low. The dynamics of ail the ligands around a protein become correlated even when they diffuse independently. An approximate theory for the kinetics of protein-gated reactions that is exact in both the fast and slow gating limits is developed. In order to test this theory, a Brownian dynamics simulation algorithm based on a path-integral formulation is introduced to calculate both k(sg)(t) and the time dependence of the protein concentration. Illustrative simulations using a simple model are carried out for a variety of gating rates. The results are in good agreement with the approximate theory.