# [088] Microscopic approach to nonlinear reaction-diffusion: The case of morphogen gradient formation

Jean Pierre Boon, James F. Lutsko, and Christopher Lutsko, "Microscopic approach to nonlinear reaction-diffusion: The case of morphogen gradient formation", *Phys. Rev. E*, **85**, 21126 (2012) __http://jimlutsko.github.io/files/Boon_Lutsko__Lutsko_PRE_2012.pdf__

## 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->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>α) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for n<α<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.