A mathematical mystery of lizard camouflage – ScienceDaily


The shape-shifting clouds of starlings, the organization of neural networks or the structure of an anthill: nature is full of complex systems whose behavior can be modeled with mathematical tools. The same is true for the maze-like patterns formed by the eyed lizard’s green or black scales. A multidisciplinary team from the University of Geneva (UNIGE) explains the complexity of the system that generates these patterns, thanks to a very simple mathematical equation. 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 always results in an optimal pattern for animal survival. These results are published in the journal Physical Verification Letters.

A complex system consists 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 produce unexpected behavior of the whole. The group of Michel Milinkovitch, Professor at the Institute of Genetics and Evolution, and Stanislav Smirnov, Professor at the Department of Mathematics, Faculty of Science at UNIGE, were interested in the complexity of the distribution of colored scales on the skin of ocellid lizards.

labyrinths of scales

The individual scales of the eyed lizard (Timon Lepidus) change color (from green to black, and and vice versa) over the course of the animal’s life and gradually forms a complex labyrinthine pattern as it reaches adulthood. The UNIGE researchers have previously shown that the labyrinths arise on the skin’s surface because the scale network represents a so-called “cellular automaton”. “This is a computer system invented by the mathematician John von Neumann in 1948, in which each element changes its state according to the states of the neighboring elements,” explains Stanislav Smirnov.

In the eyed lizard, the scales change color – green or black – depending on the color of their neighbors, according to a precise mathematical rule. Milinkovitch had shown that this cellular automaton mechanism arises from the superposition of the geometry of the skin (thick within the scales and much thinner between the scales) on the one hand and the interactions between the skin’s pigment cells on the other.

The path to simplicity

Szabolcs Zakany, a theoretical physicist in Michel Milinkovitch’s lab, teamed up with the two professors to see if this change in the color of the scales might 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 with a single equation with only two parameters: the energy of the aligned or misaligned neighbors and the energy of an external magnetic field that tends to push all particles towards + press 1 or -1 state,” explains Szabolcs Zakany.

A maximum mess for a better survival

The three UNIGE scientists found that this model can accurately describe the scale color change phenomenon in the eyed lizard. More specifically, they fitted the Lenz-Ising model, which is usually organized on a square lattice, to the hexagonal lattice of skin scales. For a given mean energy, the Lenz-Ising model favors the formation of all magnetic particle state configurations that correspond to that same energy. In the eyed lizard, the process of color change favors the formation of all distributions of green and black scales, each time resulting in a labyrinthine pattern (rather than lines, dots, circles or solid areas…).

“These maze-like patterns, which give pearl lizards optimal camouflage, were chosen during evolution. These patterns are generated by a complex system, but one that can be simplified to a single equation, where what matters is not the exact location of the green and black scales, but the general appearance of the final pattern,” enthuses Michel Milinkovitch. Every animal has a different exact location of its green and black scales, but all of these alternative patterns share a similar appearance (i.e., very similar “energy” in the Lenz-Ising model), giving these different animals equal chances of survival.

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