Theory of curvature-dependent kinetics of diffusion-limited reactions and its application to ligand binding to a sphere with multiple receptors.

We present a simple theory that explains how surface curvature affects the reaction kinetics of diffusion-limited reactions on spherically curved surfaces. In this theory, we derive a quadratic equation under the conditions that the rate constant satisfies the Hill and Smoluchowski rate constants at the lowest and highest curvatures, respectively, and that at a certain intermediate curvature, there should be a maximum value of the rate constant, which was recently found in our previous work. We find that the result obtained from our theory is in good agreement with the corresponding one obtained from numerical calculation. In addition, we show that our theory can be directly applied to the Šolc-Stockmayer model of axially symmetric reactants, which can be considered as a spherical reactant with a single reaction site. Furthermore, we discuss using our theory to improve the formula for the rate constant in the Berg-Purcell ligand-binding model of a cell membrane covered by multiple receptors. Our simple theory yields insight into the effect of curvature on diffusion-influenced reactions and provides a useful formula for easily and quantitatively evaluating the curvature effect.