Microscopic approach to nonlinear reaction-diffusion: the case of morphogen gradient formation.

We develop a microscopic theory for reaction-diffusion (RD) processes based on a generalization of Einstein's master equation [Ann. Phys. 17, 549 (1905)] with a reactive term and show how the mean-field formulation leads to a generalized RD equation with nonclassical solutions. For the nth-order annihilation reaction A+A+A+···+A→0, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and nonscaling formulations. We find steady states with solutions either exhibiting long-range power-law behavior showing the relative dominance of subdiffusion over reaction effects in constrained systems or, conversely, solutions that go to zero a finite distance from the source, i.e., having finite support of the concentration distribution, describing situations in which diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.