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Many animals have patterned fur, feathers, or scales, such as the stripes ofa zebra. Turing models, or reaction-diffusion systems, are a class ofmathematical models of interacting species that have been successfully used togenerate animal-like patterns for many species. When diffusion of the inhibitoris high enough relative to the activator, a diffusion-driven instability canspontaneously form patterns. However, it is not just the type of pattern butalso the orientation that matters, and it remains unclear how this is done inpractice. Here, we propose a mechanism by which the curvature of the surfaceinfluence the rate of diffusion, and can recapture the correct orientation ofstripes on models of a zebra and of a cat in numerical simulations. Previouswork has shown how anisotropic diffusion can give stripe formingreaction-diffusion systems a bias in orientation. From the observation thatzebra stripes run around the direction of highest curvature, that is around thetorso and legs, we apply this result by modifying the diffusion rate in adirection based on the local curvature. These results show how local geometrycan influence the reaction dynamics to give robust, global-scale patterns.Overall, this model proposes a coupling between the system geometry andreaction-diffusion dynamics that can give global control over the patterning byusing only local curvature information. Such a model can give shape andpositioning information in animal development without the need for spatiallydependent morphogen gradients.

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