Landscape description of the dynamics of Turing patterns

Turing patterns are a well-studied model of reaction-diffusion equations for developmental patterning. Their applicability has often been limited by the difficulty in identifying candidate molecules that satisfy the requisite criteria for patterning. Here, we build on recent work on geometric models to describe Turing patterning as a potential flow. We show how the universal dynamics of Turing patterning is described by a landscape, largely independent of the underlying reaction-diffusion equations. We apply our framework to three-component systems and demonstrate that we can accurately capture the dynamics of any given component. We extend our framework to larger networks and to models of Turing patterns coupled with external morphogens that provide positional information. We provide a quantitative description of the dynamics of chosen markers and apply it to the dynamics of SOX9 expression during digit patterning.