Revising Berg-Purcell for finite receptor kinetics

TL;DR

This paper develops a new mathematical framework that couples bulk diffusion with finite receptor kinetics, revealing that the classical Berg-Purcell formula often overestimates cellular uptake rates in realistic biophysical conditions.

Contribution

It introduces a boundary homogenization approach to model receptor kinetics coupled with diffusion, providing an explicit formula for cellular uptake rate that improves upon classical models.

Findings

The classical Berg-Purcell formula overestimates uptake in many scenarios.

The new model accurately predicts receptor-mediated uptake rates.

Numerical simulations confirm the analytical results.

Abstract

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.