Self-similar dynamics of morphogen gradients

TL;DR

This paper identifies and analyzes self-similar solutions in nonlinear models of morphogen gradient formation, providing analytical and numerical characterizations of the dynamics in biological tissue development.

Contribution

It introduces a new class of self-similar solutions for nonlinear morphogen models and proves their uniqueness using a variational approach.

Findings

Derived a nonlinear boundary value problem for the solutions.

Proved the existence and uniqueness of solutions.

Provided analytical approximations for the solutions.

Abstract

We discovered a class of self-similar solutions in nonlinear models describing the formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differention in developing tissues. These models account for diffusion and self-induced degration of locally produced chemical signals. When production starts, the signal concentration is equal to zero throughout the system. We found that in the limit of infinitely large signal production strength the solution of this problem is given by the product of the steady state concentration profile and a function of the diffusion similarity variable. We derived a nonlinear boundary value problem satisfied by this function and used a variational approach to prove that this problem has a unique solution in a natural setting. Using the asymptotic behavior of the solutions established by the analysis, we constructed these solutions numerically by the shooting method. Finally, we demonstrated that the obtained solutions may be easily approximated by simple analytical expressions, thus providing an accurate global characterization of the dynamics in an important class of non-linear models of morphogen gradient formation. Our results illustrate the power of analytical approaches to studying nonlinear models of biophysical processes.