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In 1977 Berg & Purcell published a landmark paper entitled "Physics ofChemoreception" which examined how a bacterium can sense a chemical attractantin the fluid surrounding it. At small scales the attractant molecules move byBrownian motion and diffusive processes dominate. This example is the archetypeof diffusive signaling problems where an agent moves via a random walk until iteither strikes or eludes a target. Berg & Purcell modeled the target as asphere with a set of small circular targets (pores) that can capture adiffusing agent. They argued that, in the limit of small radii and widespacing, each pore could be modeled independently as a circular pore on aninfinite plane. Using a known exact solution, they showed the capture rate tobe proportional to the combined perimeter of the pores. In this paper we studyhow to improve this approximation by including inter-pore competition effectsand verify this result numerically for a finite collection of pores on a planeor sphere. Asymptotically we have found the corrections to the Berg-Purcellformula that account for enhancement of capture due to the curvature of thespherical target and inhibition of capture due to spatial interactions of thepores. Numerically we develop a spectral boundary element method for theexterior mixed Neumann-Dirichlet boundary value problem. Our formulationreduces the problem to a linear integral equation, specifically a Neumann toDirichlet map, which is supported only on the individual pores. The difficultyis that both the kernel and the flux are singular, a notorious obstacle in suchproblems. A judicious choice of singular boundary elements allows us to resolvethe flux singularity at the edge of the pore. In biological systems there canbe thousands of receptors whose radii are 0.1% the cell radius. Our numericsresolve this realistic limit with an accuracy of roughly one part in 10^8.

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