A microscopic approach to nonlinear Reaction-Diffusion: the case of morphogen gradient formation

TL;DR

This paper introduces a microscopic theory for nonlinear reaction-diffusion processes, deriving generalized equations that explain diverse behaviors in morphogen gradient formation, including long-range power laws and finite support solutions, validated by experimental data.

Contribution

It develops a microscopic framework for nonlinear reaction-diffusion equations, extending classical models and analyzing their solutions in morphogen gradient contexts.

Findings

Long-range power law solutions for certain reaction orders

Finite support solutions indicating rapid extinction

Agreement with experimental morphogen gradient data

Abstract

We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the $n$-th order annihilation reaction $A+A+A+...+A\rightarrow 0$, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior (for $n>\alpha$) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for $n<\alpha<n+1$) with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.