A Lizard's Color Shifting Is Based in Mathematical Formulas (von Neumann and Turing)

Remember how your teachers always said that you needed an basic understanding of mathematics regardless of what profession you wanted to pursue as an adult? It's sound advice; from basic addition and subtraction to complex formulas and equations, you're going to be exposed to some form of math at some point (even if you don't completely realize it). Working with snakes and other reptiles, I'm expected to do my fair share of math; from using weight logs to map out animal nutrition to calculating the concentrations of medications to be administered, I am constantly checking and rechecking numbers to be sure our animals are kept in top condition. But a new study reveals that mathematics may play a greater role in herpetology; it seems as though some species of reptiles follow mathematical rules.

The colorful fellow pictured above is an ocellated lizard, a medium sized lizard native to Europe. Now these guys aren't born with their remarkable patterns and colors, they begin life a dull shade of brown speckled with white polka dots. It's as they mature and grow that this unimpressive pattern shifts towards a brilliant green and black labyrinthine work of art. Researchers studying this transformation are still not entirely sure why the lizard makes this colorful transition, but they may have stumbled upon how the change takes place: the scales appear to change based on a mathematical model. More bizarrely, this model is different than the one that scientists long believe dictated what patterns animals had.

One major theory suggesting how biological patterns form was brought to us by an unexpected source: codebreaker Alan Turing. That's right, Turing, the guy that invented computers, came up with a biological theory that has persisted for over half a century. 65 years ago, it was Turing who suggested that the huge variety of biological patterns, such as stripes and spots (and even appendages like tentacles and fingers) were the result of chemical reactions between two theoretical substances: an activator and an inhibitor. He hypothesized that these two substances spread out across an animals skin in different patches, and the varying interactions between them resulted in the many patterns seen in the animal kingdom. The video below briefly illustrates the process Turing hypothesized.

So we know that, while some patterns seem to deviate from the Turing mechanism, others seem to fall right in line with it. Today biologists, mathematicians and programmers are working together to figure out exactly what types of patterns abide by Turing's mechanism and uncover the identity of these so-called "activators and inhibitors".

Now, back to our ocellated lizard friend. An evolutionary biologist by the name of Michel Milinkovitch set out to explore whether or not Turing's model explained how the lizards could change the colors of their scales to their ornate pattern. What interested the biologist most was that these color patterns were formed at the level of scales (the scales are either entirely black or entirely green) and not at the level of biological cells, which would be more in line with the Turing mechanism (Source). Milinkovitch and his colleagues took periodic 3D scans of three ocellated lizards from the time of their birth up until about 3 years of age. What they discovered was that the scales didn't simply change and settle on a color, rather they would continuously flip from green to black and back to green; their patterns seemed to be in a constant state of flux. However, at any given moment, the color of each scale seemed dependent of the color of neighboring scales. Green scales were typically surrounded by four black and two green scales, and black scales were bordered by three black and three green scales (blue spots and ventral scales were excluded).

The patterning seemed to resemble a mathematical model developed by John von Neumann in the 1940's: the cellular automaton. To put it in simple terms, the cellular automaton is basically "a grid of connected units, and the state of each unit is governed by the state of its neighbors" (source). (To fully understand the concept, you can play with a famous automaton simulation here). So this observation begged an important question: 

 “What if each scale is changing color as a function of the state of its neighbor?” (Source)

After looking over their findings and crunching the numbers, Milinkovitch and his colleagues confirmed that the scales were behaving as von Neumann's automaton dictated.

 “Each scale senses the color of its neighbors and then takes a decision on the basis of its neighborhood.” -Milinkovitch Source

Now the team has a new task at hand: it is still not known what biological signals or processes are dictating the autonomy of the cells. 

So, was Turing's mechanism wrong when it comes to the ocellated lizard?

Actually, when the lab team adjusted their model to incorporate the thickness of each scale (and the thinness of skin connecting them), the Turing model suddenly seemed to work! Under the right conditions, the team's data suggests that Turing's mechanism could create a von Neumann cellular automaton...on the skin of an animal.

 This study reveals that "a cellular automaton is not just an abstract concept, but corresponds to a process generated by biological evolution.”-Leah Edelstein -Keshet, mathematician at the University of British Columbia Source

When asked why the findings matter, Milinkovitch has an elegant response.

 We just want to understand better the universe. And in this case, our universe is this beautiful lizard.” -Milinkovitch Source

Image Links: 1, 2, 3, 4, 5, 6

Video Link: 1

Articles: https://www.sciencedaily.com/releases/2017/04/170412132330.htm

                https://www.theverge.com/2017/4/16/15306962/ocella...