Today mathematician Alan Turing is world-famous because he helped the Allies achieve victory against the Axis powers by deciphering an encryption that was considered unbreakable. That story inspired the 2014film The Imitation Game. Turing’s cryptographic work remained under wraps until the 1970s, however, so his incredible achievements only became known after his death.
During his lifetime, Turing was known among certain experts. He developed the mathematical model of a computer and explained which mathematical quantities it could calculate—and which tasks would exceed even the most sophisticated algorithms. He is also well known for a test that he developed, later named after him, that assesses how “human” artificial intelligence appears to be. For instance, if people cannot tell whether they are chatting to a real person or an AI, then the machine has passed the Turing test.
The list of Turing’s scientific contributions is long. But one area of his research is rarely mentioned: his work on mathematical biology that dealt with the formation of patterns. He was interested in the question of how animals develop their impressive stripes and spots, and he was convinced that there must be a mechanism by which pigments in skin cells arrange themselves into these patterns.
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How Does the Tiger Get Its Stripes?
When I first heard about this, I was puzzled. One of my physics professors mentioned a link between abstract mathematical operators and a tiger’s stripes in a first-semester lecture, a connection that made me and my fellow students laugh rather than think. After all, what could the pattern of a tiger’s skin have to do with abstract mathematics? Until then, I had assumed that some complex biochemical processes led to the tiger’s impressive patterns of dots and stripes—not something that could be represented by a tensor (a kind of high-dimensional table).
I now realize that I lacked Turing’s imagination. According to his mother, even as a child, he was a dreamer who marveled at the natural world around him. He wanted to understand his surroundings. Mathematics lent itself as a language to reduce even the most complex relationships to the essentials. And so Turing found a very simple mechanism that could explain nature’s patterns.
To understand Turing’s ideas, you first need a little biological background. A tiger’s coat pattern is already determined before it is born. In the embryo, pigment-producing cells emerge at the point where the spinal column will later develop. From there, they migrate through the entire body. Although research into these cells was lacking in Turing’s time, he recognized that there was a developmental process that formed skin patterns, and he wanted to find out how this occurred.
It was impossible to model all the interacting molecules of an animal embryo. Moreover, Turing was not an expert in biochemistry. Therefore, as is usual for mathematicians, he started with a very simple model. He investigated how two different pigment-producing molecules, which he generally called morphogens, spread from cell to cell.
A Story of Two Morphogens
Let’s assume that one morphogen is responsible for the color black and another for orange. The more black or orange morphogens there are, the more of these molecules are generally produced. In addition, these two substances influence each other: the orange morphogens can inhibit the production of the black ones.
Such an interplay is often found in ecology. For example, the black morphogens can be thought of as hares, which reproduce rapidly and thus attract foxes (the orange morphogens). The foxes, however, eat the hares and thus limit their population.
This complex interaction can lead to a variety of situations. Sometimes small colonies of hares are kept in check by different foxes in the area. Translating that example to morphogens, one can imagine how a dotlike pattern similar to a cheetah’s fur could reflect that one morphogen has limited the spread of another.
Turing did not use the fox-and-hare visualization when he described how morphogens could move through an embryo’s cells, but he did take into account the phenomenon of diffusion. If a cell harbors many black morphogens, for example, and a neighboring cell harbors few of them, then the molecules strive to move such that they are distributed as evenly as possible.
All of these processes can be described by so-called differential equations. These equations contain one or more derivatives and can indicate how the number of morphogens per cell changes. Turing used the equations to investigate how two morphogens spread in the cells and what distribution occurs in the end. In doing so, he was able to adjust several parameters. To put this in terms of our animal analogy: How many foxes and hares are there at the beginning? How quickly do the hares reproduce, and how many foxes do they attract? How quickly do they spread? How are the cells arranged through which the molecules migrate? All these factors influence the pattern that emerges at the end.
When Turing investigated this problem, he did not have a powerful computer at his disposal and had to carry out many of the calculations by hand. He solved the differential equations, and he recorded how the two morphogens were ultimately distributed in the cells and observed how patterns emerged. In some cases, there were stripes; in others, he found dots or sometimes spots similar to those on cows. (If you would also like to experiment with the Turing mechanism but don’t feel like doing lengthy calculations, here is a simulator.)
As Turing discovered, the type of pattern depends on the arrangement of cells. Stripes tend to form in smaller, elongated structures, while dots form on large surfaces. Many years later British mathematical biologist James Murray applied Turing’s idea to big cats. A grown tiger is not a small animal. But, per Turing’s theory, the cat’s patterning indicates that the distribution of morphogens takes place at a time when the tiger embryo is still very small. The situation is different in leopards. And both effects can be seen in cheetahs: their body is spotted, but their tail is striped.
Nice Theory, but What Happens in Practice?
Unfortunately, this work by Turing attracted little attention during his lifetime. Shortly after he published his research in 1952, the discovery of the DNA’s double-helix structure overshadowed everything else. It took about 20 years for biologists to rediscover Turing’s work, inspiring a new generation to test whether the Turing mechanism really does occur in nature. But the technologies required have only been available since the 2000s.
To prove Turing’s hypothesis, the corresponding morphogens must be identified in animals. Although this is not easy, a few cases are now known. For example, scientists identified two proteins in mice that produce the striped structure of their palate, and the arrangement of the animals’ hair follicles appears to be influenced by two other proteins. In the same way, the coloration of zebra fish is probably caused by the Turing mechanism, but it involves a complex interaction of three rather than two morphogens. These examples show that Turing’s ideas are not limited to color patterns but also apply to other structures.
And what about the tiger and its stripes? To date, morphogens have been most clearly detected in mice, which are much easier to study in the lab. But hopefully a future scientific strategy can be found to prove Turing’s mechanism beyond a doubt in big cats as well.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.