Theory of diffusion-influenced reactions in complex geometries

TL;DR

This paper develops a comprehensive analytical theory for diffusion-influenced reactions in complex environments with obstacles and reactive regions, applicable to biochemical and nanotechnological systems.

Contribution

It introduces a general method using addition theorems for special functions to model DI reactions in arbitrary reactive geometries, providing explicit analytical formulas.

Findings

Able to describe reactions in environments with multiple spherical boundaries

Derives explicit formulas for reaction rates in complex geometries

Applicable to biochemical and nanotech systems

Abstract

Chemical reactions involving diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced" (DI). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DI reactions, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DI reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical formulas can be derived easily in most cases.