# How the leopard gets its spots

## Alan Turing and the math behind biological development

Every computer scientist is familiar with some of British mathematician Alan Turing’s contributions to the field, from creating arguably the first computer to greatly advancing computability theory. However, outside the world of algorithms and electronics, one of Turing’s most important scientific legacies is in a seemingly disjoint field entirely: developmental biology.

For some background, one of the key questions in developmental biology is how a single cell, the zygote, can generate the complex pattern that is an organism. The consensus today relies on chemicals known as morphogens, whose distribution dictates the morphology, or overall form, of the organism as it develops. For example, the first discovered morphogen is a gene in fruit flies known as *Bicoid*, whose product concentrates in the front of an embryo and canonically helps establish what is known as the A-P axis — it demarcates which side is the front and which side is the back. Since the discovery of *Bicoid*, dozens of other morphogens have been reported, and how these compounds interact during the course of development remains an area of intense research.

But how did the concept of morphogens arise? As it turns out, the term itself was coined by Turing in his 1952 paper, “The Chemical Basis of Morphogenesis” (Turing, Alan Mathison. 1952 The chemical basis of morphogenesis. *Phil. Trans. R. Soc. Lond. B *237: 37–72). Within this paper, Turing outlines that in order for a set of chemicals to shape a complex organism, there must be a mechanism to break the symmetry of its embryo. The developing embryo is initially roughly spherical and homogeneous, meaning that no part of the cell is more likely to grow into a head or arm than any other. It’s seemingly impossible that random fluctuations within such a uniform system could lead to a patterned distribution of morphogens. However, Turing showed that with a combination of chemical kinetics (reaction speeds) and diffusion, structures now eponymously known as Turing patterns emerge.

Turing patterns rely on the synthesis of morphogens at a specific location, as well as the flow of morphogens between locations. The simplest such system to analyze relies on two morphogens within many different cells. The rate of synthesis of each morphogen is dependent on one anothers’ concentrations. However, if only chemical kinetics are considered, no spatial pattern would be produced, since no information is traveling between cells. This is where diffusion plays a role: each cell is able to exchange chemicals with its immediate neighbors based on the concentration differences of each of the two morphogens. With these two factors, this type of setup is known as a reaction-diffusion system.

Physics aficionados and those familiar with coupled differential equations will recognize this as similar to the setup for coupled oscillators and standing waves on a string. Without delving too deep into the math, these types of systems gradually shift towards a stationary pattern. As it turns out, reaction-diffusion systems also establish a semi-regular wave pattern of concentrations, which are the Turing patterns.

This type of reaction-diffusion system appears in many places throughout nature. In developmental biology, Turing patterns are commonly found on the skin. For example, the pattern found in a leopard’s spots have been rationalized as Turing patterns. Other animals can be found to have Turing patterns as well, such as the stripes on a zebra or the maze on a puffer fish. Even humans have examples of Turing patterns: your fingerprints can be modeled as Turing patterns, as can the distribution of sweat glands, etc.

Turing patterns have also been found to be far more pervasive than just in the realm of biology. Indeed, while the original “reaction-diffusion” system was proposed for a biological model of cells, many other phenomena exist because of a movement of material coupled with formation of structures. For example, the shape of sand ripples in a desert as well as star formations within galaxies can be traced back to Turing-like patterns. Even modern-day materials science has manifestations of Turing patterns. One instance is in liquid crystals, a phase of matter with applications including optical displays and biological membranes, among others. Recent studies have shown that even these materials exhibit Turing patterns, with isomerization reactions occurring in standing wave-like formations.

As powerful as Turing patterns are in explaining symmetry breaking processes, they do not fully explain how these strange designs give rise to something like an animal. One of the main challenges to moving from an elementary treatment of Turing patterns to explaining developmental patterns is exactly this: Turing patterns break from spherical symmetry to the strange symmetry possessed by the patterns themselves. Explaining how to break from this, in turn, requires focusing on the nature of the morphogens. In biology, most molecules are chiral, meaning that they are different from their mirror image. For example, while a basketball is achiral, since it’s the same as its reflection, your hands are not — your left and right hands are distinct. In the same way, biological morphogens are “left” and “right” handed, which biases them to react in certain ways. Although the details are complicated, it’s not difficult to see that morphogens with an intrinsic handedness can give Turing patterns polarity, which eventually leads to the form of an organism.

Turing patterns represent a fascinating understanding of symmetry and asymmetry in nature. And while scientists currently implement more complex modifications of the basic Turing model, due to the volatility of Turing patterns, this type of analysis still is quite powerful. The power of Turing’s application of mathematics in biology helped establish the canon to this day. Although his life was limited by his untimely death two years after he published this paper, Alan Turing’s contributions and the beauty of his patterns certainly are not.