Patterns in Time and Space from a Single Morphogen via Nonlinear Layering

TL;DR

This paper demonstrates that a single morphogen in layered media with nonlinear coupling can generate stable, complex patterns in space and time, broadening understanding of pattern formation in biological systems.

Contribution

It introduces a novel layered reaction-diffusion model with nonlinear coupling, showing it can produce diverse patterns with only one morphogen, analyzed via a thin-domain limit.

Findings

Layered media with nonlinear coupling can produce stable patterns from a single morphogen.

The reduced model exhibits Turing, Hopf, and Turing-wave instabilities.

Numerical simulations confirm pattern persistence beyond the thin-domain approximation.

Abstract

Spatial and temporal pattern formation in reaction-diffusion systems is typically studied with two or more equations, as scalar reaction-diffusion equations confined to convex domains do not admit stable inhomogeneous states in time or space on long timescales. Here, we show that a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space. This $N$-layer model is analysed via a thin-domain limit, which reduces to an $N$-component reaction-diffusion system on a homogeneous one-dimensional domain. This reduced model can be analysed via linear stability techniques, showing that non-diffusive, or reactive, coupling between regions is necessary for pattern-forming instabilities, at least in the reduced model. This reduced system can exhibit Turing, Hopf, and Turing-wave instabilities, with emergent structures that are numerically shown to persist even away from the thin-domain regime of the full 2D single-morphogen system. These results suggest that heterogeneous stratification and nonlinear coupling can broaden the class of systems which exhibit complex spatiotemporal behaviours, which may be relevant in scenarios where only a single morphogen is known to act.