Numerical approximation of diffusive capture rates by planar and spherical surfaces with absorbing pores

TL;DR

This paper refines the mathematical modeling of diffusive capture rates of molecules by small pores on spherical and planar surfaces, improving upon the classical Berg-Purcell approximation by including pore interactions and curvature effects, verified through advanced numerical methods.

Contribution

It introduces asymptotic corrections to the Berg-Purcell formula accounting for pore interactions and curvature, and develops a spectral boundary element method for accurate numerical solutions.

Findings

Asymptotic corrections improve capture rate estimates.

Curvature enhances the capture efficiency.

Numerical method achieves high accuracy for realistic receptor densities.

Abstract

In 1977 Berg & Purcell published a landmark paper entitled "Physics of Chemoreception" which examined how a bacterium can sense a chemical attractant in the fluid surrounding it. At small scales the attractant molecules move by Brownian motion and diffusive processes dominate. This example is the archetype of diffusive signaling problems where an agent moves via a random walk until it either strikes or eludes a target. Berg & Purcell modeled the target as a sphere with a set of small circular targets (pores) that can capture a diffusing agent. They argued that, in the limit of small radii and wide spacing, each pore could be modeled independently as a circular pore on an infinite plane. Using a known exact solution, they showed the capture rate to be proportional to the combined perimeter of the pores. In this paper we study how to improve this approximation by including inter-pore competition effects and verify this result numerically for a finite collection of pores on a plane or sphere. Asymptotically we have found the corrections to the Berg-Purcell formula that account for enhancement of capture due to the curvature of the spherical target and inhibition of capture due to spatial interactions of the pores. Numerically we develop a spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. Our formulation reduces the problem to a linear integral equation, specifically a Neumann to Dirichlet map, which is supported only on the individual pores. The difficulty is that both the kernel and the flux are singular, a notorious obstacle in such problems. A judicious choice of singular boundary elements allows us to resolve the flux singularity at the edge of the pore. In biological systems there can be thousands of receptors whose radii are 0.1% the cell radius. Our numerics resolve this realistic limit with an accuracy of roughly one part in 10^8.