Here is an equation to describe lizard patterns

The subject of the study reaches 70 cm, from head to tail. It’s a lizard, but it’s not just a lizard. Dressed in a costume with green and black maze patterns, the so-called eyed lizard is the largest lizard in Europe. It’s a media monster in more ways than one. Classified as a threatened species in France, this conservative inhabitant of the Atlantic coast, though climate guard. And for good reason, they are not very mobile and their living environment, sand dunes, is very sensitive to climatic hazards. His second title of champion, the most famous and universal, comes from his camouflaged clothing that allowed him to improve his survival and finally continue today. Focus on the winning style.

>> Video: Nature does the math

Lizard skin: a “reason” to survive

Evolution selects the most suitable species. So each individual improves their own survival and organization to the best of their ability. “Save from the skin” is one option. Camouflage goal: blend into the background so you can’t be seen. Therefore, to deceive the eye of a predator or prey distinguishing features, Mother Nature offers several styles of invisibility cloaks. The strategy is to camouflage the figure with lines, spots, or even maze patterns like a lizard. Species suggest a “coloring pattern,” even if the details of the pattern are customizable at the individual level. And if the model is, in most cases, “one size fits all”, it happens that in some species of animals, the little ones do not dress up as their parents.

for four years, Team UNIGE The coloration of many lizards follows, from hatching to adulthood. Thanks to the high-precision robotic system developed in the laboratory, the researchers reconstructed the geometry and color of the scale grid.
balance sheet? The maze of brown and white scales of the young gradually turns green and black as they grow. But strangely enough, in adults, some scales “turn” from green to black and vice versa, maintaining the same general pattern.

R2OBBIE-3D is an integrated system that combines a robotic arm, a high-resolution digital color camera and high-intensity LED illumination to create accurate 3D models of biological objects between 1 and 100 cm.

Scale, “unit” changes the entire equation

Alan Turing, suggested in 1952 that physical and chemical processes involving activators and inhibitors are the origin of repetitive structures in animals. It will form by self-assembly, spontaneously, and without a genotype! Its equation calledreaction-diffusionLeopard spots style like zebra stripes, but oddly enough, they don’t work on the eyed lizard.

To unravel this mystery, Swiss researchers next looked at its anatomical features. Unlike many other lizards and snakes, the color is switched over the entire scale. Each of them changes color depending on its neighbors. Modelcellular humaninvented it John von Neumann , then it seems more convenient to them. And computer simulations prove this to be true.

Narrated by Michael C. Milinkovic, director of the laboratory, presented this mathematical research: “I was convinced that there was a link between the Turing equation and the mathematics of human cellularity. Otherwise, how to reconcile the interactions at the level of chromosomes and the macroscopic pattern? “. The researcher notes that the skin is thicker under the crusts, but very thin between them. He then puts forward the hypothesis that interactions are more limited at the edge of the scales. By reducing the diffusion coefficients around them in the Turing equation, Michel Milinkovitch and his team obtained the same results as the cellular human.

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© UNIGE

Left, modeling of lizard skin with the hybrid Turing-Cellular Automaton equation. On the right, modeling using the Ising equation.

Equation: Towards the shock of simplification

Turing models and cellular automata are complex and vary by at least 14 parameters. Michael Milinkovic, Zabolex Zacani and Stanislav Smirnov (Fields Medal2010) search for a new, simpler mathematical model. They set their sights on a model from the 1920s, a modelI sing. Describes the behavior of magnetic particles oscillating between two states [spin up et spin down], in the same way that the unit of measure moves between the two colors. The probability of a color change is assigned to each scale. The latter depends on the energy of its interactions with its neighbors and the “weight” of color [cf. modèle d’Ising]. Then the system chooses the state in which the power is the lowest. Modeldesigned for a square grid, Swiss researchers have successfully resized it to the hexagonal grid measurements of lizards’ scales.

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“The beauty of this simple two-parameter equation is that it can generate complexity and diversity. Species of contact lens lizards are all different, but their general pattern gives them the same advantage over natural selection. The work has shown that evolution also plays into these simple self-assembly systems. The evolution of patterns It is not a genetic process, and it does not depend on the environment. This is completely new for biologists. We will now see if the model can be generalized to more species,” concludes Michel Milinkovitch.

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© UNIGE

The Ising model makes it possible to create 7 qualitatively different color patterns using only two parameters. The first parameter is the neighbor energy (βJ). It will determine the probability of converting one scale from one color to another based on its interaction energies with each of the six adjacent scales. The hexagonal grid generates conflicts that make the system dynamic. The second is the color modulus (βB). Determines if one color is preferred over the other. In the ocular orchid, a little black is preferred. The parameters of the orchid place it in the area marked with a red star.