Many plants and animals display fascinating multicolor geometric patterns, spots, stripes, swirls, and so on. How do these patterns develop? How does one bit of skin know it is supposed to be one color, and the bit next door another color?
At the dawn of the computer age, this question was recognized as a form of “computation”, in which the molecules, cells, and other structures of the organism somehow “calculate” a geometric equation.
Sensei Alan Turing himself explored the mathematics of reaction-diffusion (RD) equations, which describe some chemical reactions. He proved that some RD systems can produce patterns such as spots or swirls that resemble patterns seen in nature.
Around the same time, the other founding Olympian of computing, Johnny Von Neuman, explored cellular automata. These discrete systems are closely related to digital technology, and have become familiar through John Conway’s Game of Life. These systems can also produce geometric patterns that resemble natural biological systems.
Intuitively, these two systems seem similar (at least to me), but the math is not in any way similar. Like many interesting computer science problems, we have a continuous and a discrete “solution”, with a conceptual abyss between. They may both be correct, but it isn’t easy to get from one to the other.
This month Liana Manukyan, Sophie A. Montandon and a multidisciplinary team from University of Geneva and other institutions published a very beautiful study in Nature . They examined the skin color of the ocellated lizard (Timon lepidus). The young lizard is brown with white spots, and as it matures, the skin becomes black with green spots. More interesting, the spots are a pattern of colored scales, not a continuous patterns of cells. I.e., “clumps” in a nearly hexagonal grid. This looks like a cellular automaton, at the level of these scales. How does this develop?
The team observed maturing lizards for several years, and discovered an additional wrinkle: the pattern changes continuously, with scales becoming green, and then switching to black over time, maintaining the spotted pattern. (I.e., when a scale turns gree, the surrounding scales turn black. Clearly, there is some sort of dynamic process that is calculating “rules” analogous to a cellular automaton.
“scale colour change in ocellated lizards follows a probabilistic CA process” (, p. 176)
Attacking this problem with computer simulations, the team found that the pattern could indeed be described as a cellular automaton (CA). But what short of physical or chemical process could produce these “rules”?
Calling in the heavy cavalry from the Math department (or maybe they are the Pros from Dover), the team discovered that a RD system could produce this CA when the continuous functions are constrained by “interactions (cell–cell contacts) are substantially reduced between scales compared to within scales” (p. 177).
In other words, the 3D nubbiness of the skin moderates the continuous physical diffusion processes to create a discrete “computation”. The scales are thicker in the middle, forming small islands with troughs between. This geography forms natural boundaries, and turns out to be a critical feature.
I love this study for many reasons.
The result is a deep and brilliant description of this biological process. As Phys.org put it,
“The highly multidisciplinary team of researchers had closed the loop in this amazing journey, from biology to physics to mathematics … and back to biology.” 
It is also a beautiful example of both computational thinking (conceiving the lizard’s spots as a biological computation) and computational methods (simulations of several types were essential).
Achieving this kind of beautiful result was only possible with a multidisciplinary collaboration, which is the great strength of major research universities. (And, by the way, this is much maligned “curiosity driven research”, with no commercial spin off in sight.)
Finally, the entire enterprise depended on a careful and long term observation of the natural system in question. All the theory and computation could not even begin without the solid empirical observations of the biologists.
“Starting from the new- born stage (about 2 weeks after hatching), animals were scanned for a period of 3–4 years and with a frequency of two weeks to four months” ()
Google might have the math and computing resources, but they are unlikely to observe live lizards for four years as a run up.
- Liana Manukyan, Sophie A. Montandon, Anamarija Fofonjka, Stanislav Smirnov, and Michel C. Milinkovitch, A living mesoscopic cellular automaton made of skin scales. Nature, 544 (7649):173-179, 04/13/print 2017. http://dx.doi.org/10.1038/nature22031
- Phys.org. How to color a lizard: From biology to mathematics. 2017, https://phys.org/news/2017-04-lizard-biology-mathematics.html.