Extending the mathematical palette for developmental pattern formation: Piebaldism

Piebaldism usually manifests as white areas of fur, hair or skin due to the absence of pigment-producing cells in those regions. The distribution of the white and colored zones does not follow the classical Turing patterns. Here we present a modeling framework for pattern formation that enables to easily modify the relationship between three factors with different feedback mechanisms. These factors consist of two diffusing factors and a cell-autonomous immobile transcription factor. Globally the model allowed to distinguishing four different situations. Two situations result in the production of classical Turing patterns; regularly spaced spots and labyrinth patterns. Moreover, an initial slope in the activation of the transcription factor produces straight lines. The third situation does not lead to patterns, but results in different homogeneous color tones. Finally, the fourth one sheds new light on the possible mechanisms leading to the formation of piebald patterns exemplified by the random patterns on the fur of some cow strains and Dalmatian dogs. We demonstrate that these piebald patterns are of transient nature, develop from random initial conditions and rely on a system's bi-stability. The main novelty lies in our finding that the presence of a cell-autonomous factor not only expands the range of reaction diffusion parameters in which a pattern may arise, but also extends the pattern-forming abilities of the reaction-diffusion equations.