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

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 $A+A+A+...+A\to 0$, 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.

Article, 10 pages, 5 figures

References in corpus (3)

• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Development of anomalous diffusion among crowding proteins
• Generalized Diffusion

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