Curvature Induced Patterns: A Geometric, Analytical Approach to Understanding a Mechanochemical Model

Abstract

The exact mechanisms behind many morphogenic processes are still a mystery. Mechanical cues, such as curvature, play an important role when tissue or cell shape is formed. In this work, we derive and analyze a mechanochemical model. This particular spatially one-dimensional model describes the deformation of a tissue- or cell surface over time, which is driven by a morphogen that locally induces curvature. The model consists of two PDEs with periodic boundary conditions; one reaction-diffusion equation for the morphogen and one PDE that describes the dynamics of the curve, derived by taking the $L^2$-gradient flow of the Helfrich energy. We analyze the possible steady states of this model using geometric singular perturbation theory. It turns out that the strength of interaction between the morphogen and the curvature plays a key role in the type of possible steady state solutions. In the case of weak interaction, the geometry of the slow manifolds allows only for (in space) slowly changing periodic orbits that lay completely on one slow manifold. In the case of strong interaction, there exist multiple front solutions: periodic orbits that jump between different slow manifolds. The singular skeletons of the steady state solutions do not meet the required consistency conditions for the curvature, a priori indicating that the solutions might not be observable. The observability and stability are investigated further using numerical simulation.

Keywords: mechanochemical model, pattern formation, morphogenesis, curvature, geometric singular perturbation theory