A mathematical secret of lizard camouflage

Date:
January 27, 2022
Source:
Universite' de Gene`ve
Summary:
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 explains, thanks to 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 always leads to an optimal
pattern for the animal survival.



FULL STORY ==========================================================================
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, thanks to 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 always leads to an optimal pattern for the animal survival. These results are published in the journal Physical Review
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 an unexpected behavior of the whole. The group of Michel Milinkovitch, Professor at the Department of Genetics and Evolution,
and Stanislav Smirnov, Professor at the Section of Mathematics of the
Faculty of Science of the UNIGE, have been interested in 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
course of the animal's life, gradually forming a complex labyrinthine
pattern as it reaches adulthood. The UNIGE researchers have previously
shown that the labyrinths emerge on the skin surface because the network
of scales constitutes a so-called 'cellular automaton'. "This is a
computing system invented in 1948 by the mathematician John von Neumann
in which each element changes its state according to the states of the neighboring elements," explains Stanislav Smirnov.

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

The road to simplicity Szabolcs Zakany, a theoretical physicist in Michel Milinkovitch's laboratory, teamed up with the two professors to determine whether this change in the color of the scales could obey an even simpler mathematical law. The researchers thus turned to the Lenz-Ising model
developed in the 1920's to describe the behavior of magnetic particles
that possess spontaneous magnetization. The particles can be in two
different states (+1 or -1) and interact only 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 the
aligned or misaligned neighbors, and the energy of an external magnetic
field that tends to push all particles toward the +1 or -1 state,"
explains Szabolcs Zakany.

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

"These labyrinthine patterns, which provides ocellated lizards with an
optimal camouflage, have been selected in the course of evolution. These patterns are generated by a complex system, that yet can be simplified as
a single equation, where what matters is not the precise location of the
green and black scales, but the general appearance of the final patterns," enthuses Michel Milinkovitch. Each animal will have a different precise location of its green and black scales, but all of these alternative
patterns will have a similar appearance (i.e., a very similar 'energy'
in the Lenz-Ising model) giving these different animals equivalent
chances of survival.

========================================================================== Story Source: Materials provided by Universite'_de_Gene`ve. Note:
Content may be edited for style and length.


========================================================================== Related Multimedia:
* Ocellated_lizard ========================================================================== Journal Reference:
1. 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 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2022/01/220127114348.htm

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