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Bimolecular binding rate constants are often used to describe the associationof large molecules, such as proteins. In this paper, we analyze a model forsuch binding rates that includes the fact that pairs of molecules can bind onlyin certain orientations. The model considers two spherical molecules, each withan arbitrary number of small binding sites on their surface, and the twomolecules bind if and only if their binding sites come into contact (suchmolecules are often called "patchy particles" in the biochemistry literature).The molecules undergo translational and rotational diffusion, and the bindingsites are allowed to diffuse on their surfaces. Mathematically, the model takesthe form of a high-dimensional, anisotropic diffusion equation with mixedboundary conditions. We apply matched asymptotic analysis to derive thebimolecular binding rate in the limit of small, well-separated binding sites.The resulting binding rate formula involves a factor that depends on theelectrostatic capacitance of a certain four-dimensional region embedded in fivedimensions. We compute this factor numerically by modifying a recent kineticMonte Carlo algorithm. We then apply a quasi chemical formalism to obtain asimple analytical approximation for this factor and find a binding rate formulathat includes the effects of binding site competition/saturation. We verify ourresults by numerical simulation.

Bimolecular binding rates for pairs of spherical molecules with small binding sites