How the zebra got its stripes: Curvature-dependent diffusion orients Turing patterns on 3D surfaces

TL;DR

This paper introduces a curvature-dependent diffusion mechanism in reaction-diffusion models to explain the orientation of animal patterns like zebra stripes, linking local surface geometry to global pattern formation.

Contribution

It proposes a novel coupling between surface curvature and diffusion rates to control pattern orientation in reaction-diffusion systems, eliminating the need for morphogen gradients.

Findings

Curvature influences diffusion rates, aligning patterns with surface geometry.

The model reproduces zebra stripe orientation on 3D animal models.

Local geometry can control global pattern orientation without gradient cues.

Abstract

Many animals have patterned fur, feathers, or scales, such as the stripes of a zebra. Turing models, or reaction-diffusion systems, are a class of mathematical models of interacting species that have been successfully used to generate animal-like patterns for many species. When diffusion of the inhibitor is high enough relative to the activator, a diffusion-driven instability can spontaneously form patterns. However, it is not just the type of pattern but also the orientation that matters, and it remains unclear how this is done in practice. Here, we propose a mechanism by which the curvature of the surface influence the rate of diffusion, and can recapture the correct orientation of stripes on models of a zebra and of a cat in numerical simulations. Previous work has shown how anisotropic diffusion can give stripe forming reaction-diffusion systems a bias in orientation. From the observation that zebra stripes run around the direction of highest curvature, that is around the torso and legs, we apply this result by modifying the diffusion rate in a direction based on the local curvature. These results show how local geometry can influence the reaction dynamics to give robust, global-scale patterns. Overall, this model proposes a coupling between the system geometry and reaction-diffusion dynamics that can give global control over the patterning by using only local curvature information. Such a model can give shape and positioning information in animal development without the need for spatially dependent morphogen gradients.