Diffusion-controlled bimolecular reaction rates. The effect of rotational diffusion and orientation constraints

A new approach to the calculation of bimolecular association constants for partially diffusion-limited reactions between asymmetric species (e.g. the ligand binding site of a macromolecule covers only a portion of its surface) is presented. The usual formulation, which is almost always analytically intractable, is based on the solution of a steady-state rotational-translational diffusion equation subject to the mixed boundary conditions that (A) the ligand concentration vanishes over the reactive part of the macromolecular surface and (B) the flux vanishes over the remainder. We show that if A is replaced by the requirement that the flux is a constant over the reactive part of the macromolecular surface and this constant is evaluated by requiring the concentration to vanish on the average over the sink region, a whole class of problems can be solved analytically. We consider both the translational and rotational diffusion of the reactants and treat partially diffusion-controlled reactions using the so-called radiation boundary condition. To establish the validity of our approach, we study a simple model using the usual mixed as well as our boundary conditions. As illustrations of our method, we analytically solve and analyze the properties of two models that have been previously studied using numerical methods.