The Mathematics Secret Behind Lizard Camouflage

The shape-shifting clouds of starling birds, the order of neural networks, or the anthill structure: Nature has many complicated systems whose behaviors can be modeled using mathematical tools. The same holds for the labyrinthine patterns created by the environment-friendly or black scales of the ocellated lizard.

A multidisciplinary group from the University of Geneva (UNIGE) describes the system’s complexity that produces these patterns thanks to a straightforward mathematical equation. This discovery contributes to a greater understanding of the evolution of skin color patterns: the process enables many different spots of green and also black scales but constantly brings about an optimal pattern for animal survival. These outcomes are published in the journal Physical Review Letters.

A complex system comprises several elements (occasionally just two) whose local interactions result in global properties that are hard to predict. The outcome of an complex system will certainly not be the sum of these elements taken individually since the interactions between them will generate an unpredictable behavior of the whole. The team of Michel Milinkovitch, Professor at the Department of Genes and Evolution, and Stanislav Smirnov, Professor at the Section of Mathematics of the Faculty of Science of the UNIGE, have wanted the complexity of the circulation of tinted ranges on the skin of ocellated reptiles.

Mazes of scales

The individual scales of the ocellated lizard (Timon Lepidus) modification of color (green to black, and the other way around) throughout the animal’s life, progressively creating an intricate labyrinthine pattern as it reaches adulthood. The UNIGE scientists have previously revealed that the labyrinths arise on the skin surface area because the network of scales composes a ‘mobile automaton.’ “This is a computing system created in 1948 by the mathematician John von Neumann in which each component transforms its state according to the states of the nearby components,” clarifies Stanislav Smirnov.

According to a precise mathematical rule, ocellated lizard’s scales change states– green or black– depending on the colors of their neighbors. Milinkovitch had shown that this cellular automaton mechanism arises from the superposition of the geometry of the skin (thick within ranges and much thinner between ranges), and the communications amongst the pigmentary cells of the skin.

The roadway to simplicity

Szabolcs Zakany, a theoritical physicist in Michel Milinkovitch’s laboratory, joined both professors to determine whether this shift in the color of the scales can follow a simpler mathematical law. The scientists hence turned to the Lenz-Ising model established in the 1920s to describe the behavior of magnetic particles that have spontaneous magnetization. The particles can be in two different states (+1 or -1) and connect just with their very first neighbors.

” The elegance of the Lenz-Ising design is that it defines these dynamics utilizing a single equation with just two parameters: the energy of the aligned or misaligned neighbors, and also the energy of an outer magnetic field that often tends to push all of the particles towards the +1 or -1 state,” clarifies Szabolcs Zakany.

An optimal disorder for a far better survival

The three UNIGE scientists established that this model could accurately explain the phenomenon of scale color change in the ocellated lizard. They adapted the Lenz-Ising design, normally arranged on a square lattice, to the hexagonal lattice of skin scales. At a provided average energy, the Lenz-Ising model prefers the formation of all state configurations of magnetic particles matching to this same energy. In the case of the ocellated lizard, the procedure of color change favors the formation of all distributions of green and black scales that each time causes a labyrinthine pattern (and also not in lines, spots, circles, or circles single-colored zones…).

“These labyrinthine patterns, which give ocellated lizards with an optimum camouflage, have been selected in the course of evolution. These patterns are created by a complex system that can not be simplified as a single equation yet. In this equation, what matters is the general appearance of the final patterns and not the specific location of the green and black scales”, enthuses Michel Milinkovitch. Each animal will have different specific locations of its green and black scales, yet every one of these alternative patterns will undoubtedly have a similar appearance (i.e., a very comparable ‘energy’ in the Lenz-Ising version), offering these other animals equal chances of survival.

Read the original article on Science Daily.

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Reference: Szabolcs Zakany, Stanislav Smirnov, Michel C. Milinkovitch. Lizard Skin Patterns and the Ising Model. Physical Review Letters, 2022; 128 (4) DOI: 10.1103/PhysRevLett.128.048102