Work

The Effective Reactivity for Capturing Brownian Motion by Partially Reactive Patches on a Spherical Surface

Year: 2026

Type: article

Abstract: . We analyze the trapping of diffusing ligands, modeled as Brownian particles, by a sphere that has [Formula: see text] partially reactive boundary patches, each of small area and arbitrary shape, on an otherwise reflecting boundary. For such a structured target, the partial reactivity of each boundary patch is characterized by a Robin boundary condition, with a local boundary reactivity [Formula: see text] for [Formula: see text]. For any spatial arrangement of well-separated patches on the surface of the sphere, the method of matched asymptotic expansions is used to derive explicit results for the capacitance [Formula: see text] of the structured target, which is valid for any [Formula: see text]. This target capacitance [Formula: see text] is defined in terms of a Green’s matrix, which depends on the spatial configuration of patches, the local reactive capacitance [Formula: see text] of each patch and another coefficient that depends on the local geometry near a patch. The analytical dependence of [Formula: see text] on [Formula: see text] is uncovered via a spectral expansion over Steklov eigenfunctions. For circular patches, the latter are readily computed numerically and provide an accurate fully explicit sigmoidal approximation for [Formula: see text]. In the homogenization limit of [Formula: see text] identical uniformly-spaced patches with [Formula: see text], we derive an explicit scaling law for the effective capacitance and the effective reactivity of the structured target that is valid in the limit of small patch area fraction. From a comparison with numerical simulations, we show that this scaling law provides a highly accurate approximation over the full range [Formula: see text], even when there is only a moderately large number of reactive patches.

Source: Multiscale Modeling and Simulation

DOI: 10.1137/25m180562x

Authors: Denis S. Grebenkov, Michael J. Ward

Institutions: Centre National de la Recherche Scientifique, École Polytechnique, Institut Polytechnique de Paris, Laboratoire de Physique de la Matière Condensée, University of British Columbia

Language: English

Topic: Diffusion and Search Dynamics

Subfield: Molecular Biology

Field: Biochemistry, Genetics and Molecular Biology

Domain: Life Sciences

Open Access status: green

Funders: Simons Foundation, Alexander von Humboldt-Stiftung, Natural Sciences and Engineering Research Council of Canada

OpenAlex ID: W7161992652

FWCI: 1.477

Cites: 52

Cited by: 1

Related to: 10