Extending the mathematical palette for developmental pattern formation: Piebaldism

📝 Abstract

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.

💡 Analysis

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.

📄 Content

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Extending the mathematical palette for developmental pattern formation: Piebaldism

Michaël Dougoud1, Christian Mazza1, Beat Schwaller2, Laszlo Pecze2

1Department of Mathematics, University of Fribourg, Chemin du Musée 23, CH-1700 Fribourg, Switzerland
2Anatomy, Department of Medicine, University of Fribourg, Route Albert-Gockel 1, CH-1700 Fribourg, Switzerland

To whom correspondence should be addressed: Laszlo Pecze, Anatomy, Department of Medicine, University of Fribourg, Route Albert-Gockel 1, CH-1700 Fribourg, Switzerland. Tel. ++41 26 300 85 11 Fax: ++41 26 300 97 33, E-mail: laszlo.pecze@unifr.ch

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Abstract 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.

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Introduction

The various patterns on the surface of animals (fur, skin, feathers) have always been intriguing because of their diversity and presumed functions. The coloration of animal skin is due to melanin pigments that are produced by melanocytes. Melanocytes are located in the stratum basale layer of the skin’s epidermis and in hair follicles; melanocytes secrete mature melanosomes to surrounding keratinocytes (Lin and Fisher, 2007). The localized changes in the homogeneous distribution of melanocytes or in the pigment synthesis pathway result in different patterns (Mills and Patterson, 2009). The biological function of these patterns is likely as camouflage or decoy devices rather than for communication or other physiological functions (Allen et al., 2011). Zebra stripes most probably serve as a dazzle pattern. Unlike other forms of camouflage, the intention of a dazzle pattern is not to conceal, but to make it difficult to estimate a target’s range, speed, and the direction of motion (How and Zanker, 2014). Zebra stripes not only confuse big predators like lions, but also make zebra fur less attractive for flies (Egri et al., 2012). The pattern formation mechanisms have been largely debated since the pioneering work of Alan Turing, who proposed a reaction-diffusion model to explain how very distinctive patterns may arise autonomously (Turing, 1952). Almost all natural occurring patterns can be recapitulated by this model; the seminal idea has served many different purposes (Murray, 1989; Salsa et al., 2013). However, different kinds of patterns exist that can’t be recapitulated by Turing’s original model, an example is the fur patterns of Dalmatian dogs. These so-called piebald patterns show randomly distributed dots of different sizes. Because of their natural irregularity, piebald patterns are not considered to be the result of cell-cell interactions based on reaction-diffusion models. The pigmented spots are the result of the stochastic migration of primordial pigment cells (melanoblasts) from the neural crest to their final locations in the skin of the early embryo (Li et al., 2011; Mort et al., 2016). However the mathematical model considering only the random movement of melanoblast do not produce sharp edges between the colored and the melanocyte-free regions (Mort et al., 2016). The spots on Dalmatian dogs appear after their birth with increasing size and contrast, indicating that other mechanisms are also involved in their formation. The original model of Turing considers two diffusing factors, an inhibitor and an activator, which require short-range activating and long-range inhibitory effects (Turing, 1952). However, cells must translate the diffusing factor-encoded information into a biological signal. The importance of the transcription f

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