Theory of the diffusion-influenced substrate binding rate to a buried and gated active site

Abstract

The effects of stochastic gating on the diffusion-influenced substrate binding rate to a buried active site are studied. An approximation introduced by Samson and Deutch [J. Chem. Phys. 68, 285 (1978)] is shown to be equivalent to making the constant-flux approximation on the entrance to the active site. The constant-flux approximation is then extended to the case where the entrance to the active site is stochastically gated because of conformational fluctuations of the enzyme. The stochastically gated rate constant, ksg, is found to be given by the relation 1/ksg=1/k+wo/wc(wo+wc)ĥ(wo+wc), where k is the rate constant in the absence of gating, ĥ(s) is the Laplace transform of the total flux across the entrance after the substrate is started from an equilibrium distribution outside the entrance, and wo and wc are the transition rates between the open and closed gating states. This relation reduces to an approximate relation derived earlier for a more restrictive situation, where the reactivity within the active site is gated. The leading term in the expansion of sĥ(s) for large s is DA〈exp(−βU)〉(s/D)1/2/2, where D is the diffusion coefficient of the substrate, A is the total area of the entrance, and 〈exp(−βU)〉 is the average Boltzmann factor on the entrance. The time scale of conformational fluctuations, ∼ a few picoseconds, is much shorter than the time scale of diffusion, so this leading term is useful for estimating (wo+wc)ĥ(wo+wc). A further consequence of the disparity in time scales is that the value of (wo+wc)ĥ(wo+wc) is much larger than k. As a result the decrease of the rate constant due to gating is relatively small (unless the entrance to the active site is closed nearly all the time). This suggests that a buried and gated active site may play the important role of controlling enzyme specificity without sacrificing efficiency.