Control of reaction-diffusion equations on time-evolving manifolds

TL;DR

This paper models the coupled dynamics of morphogen diffusion and shape changes in organisms using evolving manifolds, introducing a Lie bracket to describe their interaction and providing numerical simulations.

Contribution

It introduces a mathematical framework for the coupling of diffusion and shape evolution on time-dependent manifolds, including a novel Lie bracket concept.

Findings

Transport and diffusion do not commute, as shown by the Lie bracket.

Numerical simulations demonstrate the non-commutativity phenomenon.

The model captures the interaction between morphogen diffusion and shape deformation.

Abstract

Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanwhile, the diffusion of the signal is itself affected by the changes in shape and size of the organism. In other words, there is a complete coupling between the diffusion of the signal and the change of the shapes. In this paper, we introduce a mathematical model to investigate such coupling. The shape is given by a manifold, that varies in time as the result of a deformation given by a transport equation. The signal is represented by a density, diffusing on the manifold via a diffusion equation. We show the non-commutativity of the transport and diffusion evolution by introducing a new concept of Lie bracket between the diffusion and the transport operator. We also provide numerical simulations showing this phenomenon.