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Bimolecular binding rate constants are often used to describe the association of large molecules, such as proteins. In this paper, we analyze a model for such binding rates that includes the fact that pairs of molecules can bind only in certain orientations. The model considers two spherical molecules, each with an arbitrary number of small binding sites on their surface, and the two molecules bind if and only if their binding sites come into contact (such molecules are often called "patchy particles" in the biochemistry literature). The molecules undergo translational and rotational diffusion, and the binding sites are allowed to diffuse on their surfaces. Mathematically, the model takes the form of a high-dimensional, anisotropic diffusion equation with mixed boundary conditions. We apply matched asymptotic analysis to derive the bimolecular binding rate in the limit of small, well-separated binding sites. The resulting binding rate formula involves a factor that depends on the electrostatic capacitance of a certain four-dimensional region embedded in five dimensions. We compute this factor numerically by modifying a recent kinetic Monte Carlo algorithm. We then apply a quasi chemical formalism to obtain a simple analytical approximation for this factor and find a binding rate formula that includes the effects of binding site competition/saturation. We verify our results by numerical simulation.