Diffusion-controlled reaction rates for two active sites on a sphere

The diffusion-limited reaction rate of a uniform spherical reactant is generalized to anisotropic reactivity. Previous work has shown that the protein model of a uniform sphere is unsatisfactory in many cases. Competition of ligands binding to two active sites, on a spherical enzyme or cell is studied analytically.

The reaction rate constant is given for two sites at opposite ends of the species of interest. This is compared with twice the reaction rate for a single site. It is found that the competition between sites lowers the reaction rate over what is expected for two sites individually. Competition between sites does not show up, until the site half angle is greater than 30 degrees.

Competition between sites is negligible until the site size becomes large. The competitive effect grows as theta becomes large. The maximum effect is given for theta = pi/2.