A lizard’s scales switch colors according to the same rules as spin flips
Timon lepidus, also known as the ocellated lizard, has a patterned skin of black and green scales. During the lizard’s adolescence, each scale can flip from green to black and vice versa. As it ages, its youthful spots change one scale at a time to form labyrinthine patterns, which become fixed in adulthood.
Five years ago Michel Milinkovitch of the University of Geneva and his colleagues found that they could model the formation of T. lepidus skin patterns with a cellular automaton. (See “The ocellated lizard is a computer game come to life,” Physics Today online, 24 April 2017 .) Perhaps best known from John Horton Conway’s Game of Life, the model consists of a grid of cells that each evolve according to a set of rules. Now the researchers have found a simpler description for the pattern evolution: the Ising model, a statistical mechanical model of ferromagnetism.
S. Zakany, S. Smirnov, M. C. Milinkovitch, Phys. Rev. Lett. 128, 048102 (2022)
The odds that any given scale changes its color depend on its nearest neighbors; scales seem to avoid sharing the same color with too many adjacent ones. The cellular automaton model of the lizard’s skin requires many parameters in the form of the probabilities of the scales changing from black to green or from green to black as a function of the number of black or green neighbors. But the behavior is also reminiscent of a lattice of antiferromagnetic spins: Each spin antialigns with as many neighbors as possible.
Milinkovitch and his colleagues found that an Ising model for an antiferromagnet at finite temperatures in an external magnetic field matched the observed lizard-scale behavior and the results from cellular automaton models, as shown to the right. (The external magnetic field serves as a surrogate for the scales’ slight preference for being black over green, and the finite temperature reflects the role of randomness in the process.) The model correctly predicts the pattern evolution with time, the final patterns, and the overall balance of green and black scales. And it does so with a single equation and two parameters.
The Ising model doesn’t require any microscopic information about how cells interact, but future studies could explore how cell-level parameters influence the fitting parameters of the Ising model. The model may also apply to the Gila monster and other species with color-flipping scales. (S. Zakany, S. Smirnov, M. C. Milinkovitch, Phys. Rev. Lett. 128, 048102, 2022 .)
