Abstract
The zebrafish is a model organism for pattern formation in vertebrates. Understanding what drives the formation of its coloured skin motifs could reveal pivotal to comprehend the mechanisms behind morphogenesis. The motifs look and behave like reaction–diffusion Turing patterns, but the nature of the underlying physico-chemical processes is very different, and the origin of the patterns is still unclear. Here we propose a minimal model for such pattern formation based on a regulatory mechanism deduced from experimental observations. This model is able to produce patterns with intrinsic wavelength, closely resembling the experimental ones. We mathematically prove that their origin is a Turing bifurcation occurring despite the absence of cell motion, through an effect that we call differential growth. This mechanism is qualitatively different from the reaction–diffusion originally proposed by Turing, although they both generate the short-range activation and the long-range inhibition required to form Turing patterns.
Highlights
The zebrafish is a model organism for pattern formation in vertebrates
A Turing bifurcation can occur in the system of partial differential equations (PDEs) ruling the spatiotemporal evolution of a set of dynamical variables x 1⁄4 {xi(r, t)}
To test whether this hypothesis is correct, we derive and analyse in this work a mathematical model for immobile cells based on the observed regulatory network. We show that this model can undergo a Turing bifurcation and reproduce the experimental patterns. We provide in this way a proof of principle for the idea that Turing patterns can appear in systems made of immobile agents
Summary
The zebrafish is a model organism for pattern formation in vertebrates. Understanding what drives the formation of its coloured skin motifs could reveal pivotal to comprehend the mechanisms behind morphogenesis. We mathematically prove that their origin is a Turing bifurcation occurring despite the absence of cell motion, through an effect that we call differential growth This mechanism is qualitatively different from the reaction–diffusion originally proposed by Turing, they both generate the short-range activation and the long-range inhibition required to form Turing patterns. Turing shows with a theoretical model that a chemical reaction coupled with Fickian diffusion can give rise to a spontaneous symmetry-breaking phenomenon, in which an initial state having a uniform distribution of chemicals is converted into a regular pattern of concentrations He formulated his ideas in the framework of the reaction–diffusion (RD) formalism, with a mechanism that can be shortly explained as follows. A Turing bifurcation can occur in the system of partial differential equations (PDEs) ruling the spatiotemporal evolution of a set of dynamical variables x 1⁄4 {xi(r, t)}
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