How Receptor Surface Diffusion and Cell Rotation Increase Association Rates

Many biological processes are initiated when diffusing extracellular reactants reach receptors on a cell membrane. Calculating this arrival rate has therefore attracted theoretical interest for decades. However, previous work has largely ignored the fact that receptors diffuse on the two-dimensional cell membrane in a process called surface or lateral diffusion. In this work, we derive an analytical formula for this arrival rate that takes into account receptor surface diffusion and cell rotational diffusion. Our theory predicts that the impact of these diffusive processes can be quantitatively described in terms of the relative size of the cell and the reactant. As applications, our theory predicts that surface and rotational diffusion have a negligible impact on bacterial chemoreception and a moderate impact on bacteriophage adsorption. Mathematically, our model is a three-dimensional anisotropic diffusion equation coupled to boundary conditions that are described by stochastic differential equations. We first apply matched asymptotic analysis to this stochastic partial differential equation and then use probabilistic methods to show that the solution can be represented by a Brownian particle in a stochastic environment. This representation enables efficient numerical computation of solution statistics and validation of our analytical results.