Math and the lizard's coat
(appeared in April 2017)

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The spots on some lizards’ skin seem to speak with each other while they organise themselves into patterns, says S.Ananthanarayanan.

The legendary mathematician and pioneer of digital computing, Alan Turing, first applied the mathematical methods of his fields to examine how shapes or colours could appear in living things. The ideas of Turing, which are based on diffusion of possible form-giving agents, called morphogens, from cell to cell, have been carried forward and there are now ways to explain the appearance of colour patterns and even ridges or shapes in living things. Further work has suggested what these agents of form may be and even genetic factors that bring about specific patterns, like the stripes on rodents’ backs, have been identified.evident that the coloured stripes and spots on animal coats are adapted for camouflage and sometimes to help draw attention of prospective mates. The mechanism by which these patterns form, however has been a subject of conjecture.

Liana Manukyan, Sophie A. Montandon, Anamarija Fofonjka, Stanislav Smirnov and Michel C. Milinkovitch from the departments of genetics, bioinformatics and mathematics at the University of Geneva and the Skolkova Institute and the University of St Petersburg in Russia report in the journal, Nature, the working of a known computing model that shows up in the formation, and then variations, of the patterns of spots on the back of Timon Lepidus, a lizard, found in Europe, whose back is decorated with motifs that look like eyes. The lizard is also called the ocellated (with eyelike spots) lizard, eyed lizard or the jewelled lacerta.

Alan Turing considered that chemical agents may interact and diffuse from cell to cell, at different speeds, so that different patterns of growth appeared along lines or at spots where the agents reinforced. Mathematical treatment of the idea could explain shapes taken by small groups of cells, like the tentacles that grow on the animal, the Hydra and the patches that appear on the coats of a breed of cows, the Friesian cows. The principles have been extended and there is now explanation, based on the interplay of waves of different agents, for the stripes of the tiger or the zebra, the leopard’s spots and even the spots on the body, but the stripes on the tail of the cheetah.

The movement of the growth agents which lead to these patterns is prenatal, at the time cells are still being generated, in the embryo. The patterned coats are hence fully formed when the animal is born. In the case of the ocellated lizard, with the eyelike pattern on its back, however, the pattern is more than a distribution of stripes, spots or ridges, or even limbs and protrusions, but is a pattern of more complex shapes. And what is more, the pattern is not fixed at birth but changes as the animal grows!

The group of the researchers writing in Nature observe that lizards and snakes display a variety of colours and patterns. The lizard relies on different kinds of cells, those that create colours by structural effects and interference colours of light and on cells that have different pigments. Some lizards, like the chameleon, can muscularly adjust the position of units responsible for colours and change colours. The pattern on the skin, however, is generally a result of the distribution of cells, itself a result of the reaction-diffusion mechanism, as analysed by Turing. The same mechanism, however, does not seem to fit the case of the of the ocellated lizard, the Nature paper says. In this case, the patterns are created by the positioning of larger dimension scales of skin, rather than at the individual colour-giving cell level. The pattern even changes, from white eye-like dots on a brown background on the back of young lizards, to a ‘labyrinthine pattern’ in green and black when the lizard grows to adulthood, the paper says

The group used a mechanised observation arrangement to create a record of changes in the skin pattern of three male ocellated lizards, from the age of two weeks to three or four years. High resolution scans of the lizard skin were recorded, every two weeks at the start and once every four months towards the end, because the changes in the patterns are slower when the animal is older. The progression of the pattern can be made out in the third panel in the picture.

Analysis of how the pattern develops, the paper says, revealed that the spots are distributed according to the working of mathematical rule known as the ‘cellular automaton’. The cellular automaton, which finds application in other fields of study, like computability, complexity and physics, is a grid, or a framework of cells, each of which starts out in a state, like black, green, etc. The pattern so formed then evolves, with each cell changing its state at each instant by following some rule that depends on the states of the neighbouring cells. The same rule is applied to all the cells and they change state simultaneously. We can see that there is mutual dependency of clusters of cells and the distribution could tend towards a pattern that would have elements of regularity and still show a great degree of randomness, as seen in the ’labyrinths’ on the lizards’ backs.

With some 5,000 scales being detected at each scan, the team retrieved the information of the state of the neighbours of each scale. The pattern on the young lizard was of about 60 white ocelli (eye-like spots), each composed of 5-17 scales, surrounded by scales of almost uniform brown. Gradually the scales changed colour, the lighter ones turning green and the darker ones turning black. But the colours did not stay fixed, the individual scales continued to change colour, from green to black and from black to green. Some 1,500 were found to do this, throughout the animal’s life. “This causes a gradual qualitative change of the pattern: the outlines of the original ocelli become obliterated and the pattern turns into a labyrinthine assortment of contrasting black and green chains of scales,” the paper says.

The function of the adult pattern, the paper says, is unknown but may be for camouflage, as the pattern disrupts the outline of the animal itself. The fact that the evolution of the pattern corresponds to a mathematical procedure, however, is significant. It is another instance of nature, with the help of genetic mechanisms, using the most energy conserving method to attain configurations that match the needs for the survival of living things.

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