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Reveal a mathematical secret of lizard camouflage

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The patterns of the ocellated lizard are predictable by a mathematical model. Credit: UNIGE / Michel Milinkovitch

The shape-shifting clouds of starling birds, the organization of neural networks or the structure of an anthill: nature is full of complex systems whose behaviors can be modeled using mathematical tools. The same is true for the labyrinthine patterns formed by the green or black scales of the ocellated lizard. A multidisciplinary team from the University of Geneva (UNIGE) explains, using a very simple mathematical equation, the complexity of the system that generates these patterns. This discovery contributes to a better understanding of the evolution of skin color patterns: the process allows for many different locations of green and black scales but still leads to an optimal pattern for the animal’s survival. These results are published in the journal Physical examination letters.

A complex system is composed of several elements (sometimes only two) whose local interactions lead to global properties that are difficult to predict. The result of a complex system will not be the sum of these elements taken separately since the interactions between them will generate unexpected behavior of the whole. The group of Michel Milinkovitch, professor in the Department of Genetics and Evolution, and Stanislav Smirnov, professor in the Mathematics Section of the UNIGE Faculty of Sciences, studied the complexity of the distribution of colored scales on the skin of ocellated lizards.

Labyrinths of scales

The individual scales of the ocellated lizard (Timon lepidus) change color (from green to black, and vice versa) over the animal’s life, gradually forming an intricate labyrinthine pattern as it reaches adulthood. UNIGE researchers have previously shown that the labyrinths emerge on the surface of the skin because the network of scales constitutes what is called a “cellular automaton”. “It is a computer system invented in 1948 by mathematician John von Neumann in which each element changes state depending on the states of neighboring elements”, explains Stanislav Smirnov.

In the case of the ocellated lizard, the scales change state – green or black – according to the colors of their neighbors according to a precise mathematical rule. Milinkovitch had demonstrated that this cellular automaton mechanism emerges from the superposition, on the one hand, of the geometry of the skin (thick inside the scales and much thinner between the scales) and, on the other hand, of the interactions between the pigment cells of the skin.

The path to simplicity

Szabolcs Zakany, a theoretical physicist in the laboratory of Michel Milinkovitch, joined forces with the two professors to determine whether this change in the color of the scales could obey an even simpler mathematical law. Researchers therefore turned to the Lenz-Ising model developed in the 1920s to describe the behavior of magnetic particles that possess spontaneous magnetization. The particles can be in two different states (+1 or -1) and only interact with their first neighbors.

“The elegance of the Lenz-Ising model is that it describes these dynamics using a single equation with only two parameters: the energy of aligned or misaligned neighbors, and the energy of an external magnetic field which tends to push all particles towards the +1 or -1 state,” says Szabolcs Zakany.

Maximum chaos for better survival

The three UNIGE scientists determined that this model could accurately describe the phenomenon of scale color change in the ocellated lizard. More precisely, they adapted the Lenz-Ising model, generally organized on a square lattice, to the hexagonal lattice of the skin scales. At a given average energy, the Lenz-Ising model favors the formation of all the state configurations of the magnetic particles corresponding to this same energy. In the case of the ocellated lizard, the process of color change promotes the formation of all distributions of green and black scales which each time result in a labyrinthine pattern (and not lines, spots, circles or unicolor areas… ).

“These labyrinthine patterns, which provide ocellated lizards with optimal camouflage, have been selected during evolution. These patterns are generated by a complex system, which can however be simplified into a single equation, where what matters is not not the precise location of the green and black scales, but the overall look of the end patterns,” enthuses Michel Milinkovitch. Each animal will have a different precise location of its green and black scales, but all of these alternate patterns will look different. similar (i.e. very similar “energy” in the Lenz-Ising model) giving these different animals equal chances of survival.


How to color a lizard: from biology to mathematics


More information:
Szabolcs Zakany et al, Lizard skin models and Ising model, Physical examination letters (2022). DOI: 10.1103/PhysRevLett.128.048102

Provided by the University of Geneva


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