  From nutrient uptake, to chemoreception, to synaptic transmission, many
systems in cell biology depend on molecules diffusing and binding to membrane
receptors. Mathematical analysis of such systems often neglects the fact that
receptors process molecules at finite kinetic rates. A key example is the
celebrated formula of Berg and Purcell for the rate that cell surface receptors
capture extracellular molecules. Indeed, this influential result is only valid
if receptors transport molecules through the cell wall at a rate much faster
than molecules arrive at receptors. From a mathematical perspective, ignoring
receptor kinetics is convenient because it makes the diffusing molecules
independent. In contrast, including receptor kinetics introduces correlations
between the diffusing molecules since, for example, bound receptors may be
temporarily blocked from binding additional molecules. In this work, we present
a modeling framework for coupling bulk diffusion to surface receptors with
finite kinetic rates. The framework uses boundary homogenization to couple the
diffusion equation to nonlinear ordinary differential equations on the
boundary. We use this framework to derive an explicit formula for the cellular
uptake rate and show that the analysis of Berg and Purcell significantly
overestimates uptake in some typical biophysical scenarios. We confirm our
analysis by numerical simulations of a many particle stochastic system.
