Why are patterns so common in nature

Pattern maker of nature

More than 50 years ago, the Briton Alan Turing developed a model for the formation of biological patterns, such as the one that helps zebras and leopards to obtain their characteristic fur color

The wave patterns on the sandy seabed look like modeled. But when sand ripples arise, only physics is at work. And according to their principles, dunes are also created in the desert.

Text: Christina Beck

The initially flat surface of the sand is physically unstable, both in the open air and under water. This means that small irregularities increase under certain conditions. So sand is deposited in the slipstream behind a stone or a shell. The dune or the ripple begins to grow and with it the slipstream at the same time. As a result, even more sand is deposited. However, once the sand is bound, it can no longer contribute to the formation of dunes or ripples elsewhere. The opposite reaction is the decrease in sand particles that can be carried away by the wind or the waves. With this decrease, the likelihood that new dunes will form in the area or that existing sand ripples will continue to grow on the sea floor decreases. The formation of a pattern is based on a local, self-reinforcing reaction. An opposing reaction is coupled to this, which limits the spread of the self-reinforcing reaction.

Every pattern in nature is a puzzle. Mathematics is a great way to help us solve these puzzles. It helps in a more or less systematic way to bring to light the rules and structures that are hidden behind the observed patterns. Then these rules and structures can be used to explain what is going on. Mathematics helps scientists to calculate natural processes.

Ripple-like patterns, such as those we encounter on a walk on the beach, are common in nature: the fur of a zebra also has such characteristic line patterns. But what pattern formation process may such a coat drawing be based on?

From the jungle book ...

In his amusing essay “How the leopard got its spots”, the jungle book author Rudyard Kipling tried to find an explanation. After that, the zebra and giraffe stood long enough in the partial shade under the trees: “... and after a long time, because they were half in the shade and half in the sun, and because of the flickering shadows of the trees that fell on them, became the giraffe spotty, and the zebra streaked. "

On the other hand, a person helped the leopard to get the spots with his fingerprints: “... then the Ethiopian pressed his five fingers tightly together (there was still a lot of black left on his new skin) and pressed them all over the leopard, and wherever When his five fingers got there, they made five little black prints, all very close together. You can see them on any leopard skin [...] if you look closely at any leopard you will see that there are always five spots - from five fat black fingertips. "

... to Alan Turing

This attempt, which is not entirely serious, to interpret the origin of the pattern, however, also shows that we do not necessarily arrive at an explanation intuitively in pattern formation processes. In 1952, the British Alan Turing designed a mathematical model in his contribution "The Chemical Basis of Morphogenesis" to explain how the leopard gets its spots and the zebra its stripes.

Turing was a gifted mathematician. As early as 1936 he had developed the theoretical concept of a machine that is capable of “solving every conceivable mathematical problem, provided that it can also be solved by an algorithm - and thus virtually anticipated the computer. Since then, he has been regarded by computer scientists as the founding father of their discipline.

He was best known, however, because he managed to crack the Enigma, the machine that the German military used to encrypt its messages during World War II. His considerations on the formation of patterns in biological systems have contributed significantly to the understanding of the processes, especially in developmental biology.

Player and opponent

In his model, Turing considers two substances that react with one another. A substance that works over short distances, the activator, promotes its own production - what chemists call autocatalysis - as well as that of its rapidly expanding counterpart, the inhibitor. The concentrations of both substances can be in a state of equilibrium.

However, this state is locally unstable. Any local increase in the activator will increase further due to the autocatalysis, but the amount of inhibitor also increases with it. Since the inhibitor spreads faster and thus quickly moves away from the source of the activator, it cannot stop the further local rise of the activator, but slows down its autocatalysis in the area: a halo forms around the areas where the activator is formed . In this way, for example, a point pattern is created.

If the various parameters such as production rate, degradation rate or diffusion speed are changed, one or the other substance can easily gain the upper hand in different areas of the surface. Almost any pattern can be created in this way. The parameters mentioned are included in a so-called partial differential equation. It is an indispensable tool when creating mathematical models.

We owe it to Newton and Leibniz, who independently developed differential calculus in the second half of the 17th century with the aim of calculating rates of change. Pattern formation is not about anything else: With the help of partial differential equations, it is possible to relate the changes in concentration of both substances, i.e. the activator and the inhibitor, over a short period of time as a function of their current concentration. If the changes in concentration are added to the given initial concentrations, the concentration is obtained at a somewhat later point in time. By repeating such a calculation step many times, the concentrations of the substances involved are obtained over time.