The humble ray-finned fish for many years never ceases to amaze both scientists and ocean lovers.
Despite their angular shape, they swim with surprising agility, leaving researchers baffled as to how they do it. What's more, they're very attractive: with plump lips and cube-shaped bodies, painted in bright, eye-catching patterns – polka dots, stripes and more.
Researchers from University of Colorado Boulder I'm especially interested in the spots, stripes and hexagons on the painted body of the Australian Aracana ornata. They discovered that all of these patterns on his skin could be described and reproduced mathematically using the theory developed by Alan Turing, considered the father of modern computing, in the middle of the last century.
A mathematical model that accurately reproduces even the smallest details of the paint body's pattern, including natural grain and other “imperfections,” is presented in the magazine Problem. It brings scientists closer to understanding exactly how such complex patterns are formed in nature on the skin of fish and other living organisms, explains chemical engineer Ankur Gupta, one of the authors.
“This helps bridge the gap between rigorous mathematical models and the chaotic, vibrant beauty of the real world,” he said.
This knowledge could help create camouflage fabrics that mimic natural colors or promote the development of soft robots that use flexible materials instead of rigid structures.
Developed ideas 70 years ago
This research develops a theoretical model that Turing published in 1952. It is based on the interaction of two processes: diffusion, when particles try to evenly fill all available space, and chemical reactions between them.
Usually, diffusion leads to uniformity – for example, if you drop paint in water, it will gradually form a solution of one color. But Turing demonstrated that, under certain conditions, the combination of diffusion and chemical reaction, on the contrary, can spontaneously create ordered structures: stripes, spots and other patterns. These later became known as Turing models.
The mathematics behind these patterns help explain how nature draws leopard spots, curves on mollusk shells, and many other biological patterns. The same model describes the formation of human fingerprints, the pattern of sand ripples at the bottom, and the distribution of matter in galaxies.
Computer programs that simulate diffusion and reaction processes can reproduce some biological models. But according to Gupta, their results often look too sterile and perfect, without the inherent roughness of nature – fault lines, uneven thickness or graininess. Even their own model, which simulates the behavior of pigment cells in body skin, initially produced blurry, indistinct patterns.
“Diffusion systems inherently tend to be blurry,” says Gupta. “So how does nature create clear patterns?”
In 2023, one of Gupta's students found a solution by adding a different type of cell movement to the model. According to the researcher, cells in the liquid can clump together and move together, swept away by the flow of other diffusing particles. This process, called diffusion, is the same process that causes soap to draw dirt out of fabrics when washed.
Thanks to that, the simulated patterns on the skin of the car body have become much clearer and more expressive. And to add to their natural imperfections, the model was complicated by taking into account random collisions between individual cells.
And along with the patterns, similar “errors” also appear: the stripes become unevenly thick, with breaks in places; the edges of the hexagon do not meet in places and look twisted or lumpy; Points inside the hexagon extend or merge together. The best part is that the degree of these defects can be adjusted.
A simplified version of reality
The authors admit that their model is a simplified version of reality. It does not take into account the more complex interactions between cells and, like Turing's original model, does not reveal the specific biological mechanisms of pigment production.
However, Turing's model laid the foundation, allowing scientists to manipulate model formation for many practical purposes. Researchers have used it to create patterns in growing E. coli colonies, change the stripes of zebrafish, develop more efficient seawater filters, and even understand human settlement patterns.
“We study how nature does this so we can replicate it later,” says Gupta, however, making no secret that we are motivated primarily by simple scientific curiosity. He is eager to understand how nature creates “unusual but unique patterns that have fascinated biologists for decades.”
