Diffusion-controlled reaction rate to an active site in a spherical cavity: Extension of Berg's theory

TL;DR

This paper extends Berg's theory to calculate diffusion-controlled reaction rates for particles in spherical cavities with anisotropic reactivity, providing semi-analytical solutions and fast-converging numerical methods applicable to biological and physical systems.

Contribution

It generalizes Berg's theory to include anisotropic reactivity within spherical cavities and introduces a highly efficient semi-analytical dual series relations method.

Findings

Derived semi-analytical expressions for trapping probability and reaction rate.

Developed a fast-converging numerical method for anisotropic reactivity.

Established a connection between dual series relations and separation of variables.

Abstract

This study is due to various applications in physics, chemistry and especially in biology, where both bounded configuration domain and chemical anisotropy could play a great part. In fact we generalize the well-known Berg theory, which describes diffusion-controlled reactions occurring within a spherically symmetric absorber-cavity system. The trapping probability and the reaction rate at which a small diffusing particle is captured by an axially symmetric one reactive patch absorber inside a spherical cavity were found semi-analytically and numerically by means of the dual series relations method. This approach leads to such incredibly fast convergence, that it may rightly be referred to as exact one. The results obtained can be used to test numerical programmes that describe diffusion-controlled reactions in real physical systems for reactants with arbitrary anisotropic reactivity, which are located inside of various cavities as well as in the unbounded domains. Moreover, we managed to find a close connection between the dual series relations method and the generalized method of separation of variables.