  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 degrees, 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.
