A general expression for bimolecular association rates with orientational constraints

We present a general expression for the association rate for partially diffusion-controlled reactions between spherical molecules with an asymmetric reactive patch on each surface. Reaction can occur only if the two patches are in contact and properly aligned to within specified angular tolerances. This extends and generalizes previous approaches that considered only axially symmetric patches; the earlier solutions are shown to be limiting cases of our general expression. Previous numerical results on the rate of protein–protein association with high steric specificity are in very good agreement with the value computed from our analytic expression. Using the new expression, we investigate the influence of orientational constraints on the rate constant. We find that for angular constraints of ∼ 5–15, a typical range for example in the case of protein–protein interactions, the reaction rate is about 2 to 3 orders of magnitude higher than expected from a simple geometric model.