Of Turing and zebras

Abstract

The Turing diffusion model emerges as an explanation for pattern formation in many species and across biological scales.

Alan Turing has attained celebrity status for his pioneering work in computer science and cryptographic code breaking during World War 2 and for his eponymous test applied to Open AI's ChatbotGPT chatbot to distinguish a computer from a human. What is less publicly known is his influential work on chemical diffusion gradients. Recently though there has been growing interest in developmental biology and reference to Turing's two‐protein chemical diffusion model as an explanation for a variety of pattern formations across biological scales, such as human fingerprints, leopard spots, zebra stripes or pigmentation spots on plant flowers.

… there has been growing interest in developmental biology and reference to Turing's two‐protein chemical diffusion model as an explanation for a variety of pattern formations…

Popular media interest in the diffusion model was eventually elicited by a 2023 paper identifying the genes responsible for the formation of fingerprints at the early prenatal stage and linking these to analogous processes in some animals, such as colour patterns in cats' fur. Spearheaded by developmental biologists Denis Headon and James Glover at Edinburgh University in the UK, the paper on fingerprint patterns is the culmination of years of work that is also associated with a broader effort to develop a more general theory of pattern formation in biology (Glover et al, ²⁰²³). The aim was partly to confirm or eliminate various hypotheses for the formation of the unique fingerprint patterns that have rendered this a valuable tool in forensic science.

Fingerprints

Such hypotheses include the idea that fingerprints are just a consequence of natural skin wrinkling processes or that they arise from interactions between epidermal cells in the basal layer around hair follicles, so‐called Merkel cells, and larger‐scale mechanical stresses as skin is formed (Düring et al, ²⁰¹⁹). This research yielded insights into mechanistic processes, but the Edinburgh study sought to link underlying genes with fingerprint formation through the Turing diffusion model. The authors found that fingerprints arise from the same roots as hair follicles. These roots comprise discs of cells on the epithelium that activate the genes for various proteins that determine subsequent development. Two such proteins are Ectodysplasin A receptor (EDAR) and WNT, which play key interacting roles in early follicle development by enabling the cells to migrate, differentiate and mature (Zhang et al, ²⁰⁰⁹).

Small divergences in gene expression then determine whether the incipient hair follicles will go on to form a deep tube from which hair will eventually grow, or whether they become a sweat gland. As a third option—and if the gene expression is right—this development is blocked and instead the discs evolve into the precursors of fingerprints, according to the Edinburgh study.

Headon explained that the group did not set out to identify a Turing‐type system originally, but to explore how human fingerprint ridges develop, what guides their formation, and how they create the familiar patterns. But the results strongly indicated the underlying actions of Turing diffusion. “The findings of how the pattern behaves in model organisms, producing spots or stripes, or thick or thin stripes with variable spacing, were indicative of the action of a Turing type system in their formation,” Headon explained. “Since our characterizations did not identify a role for the motile mesenchymal cells in producing the ridge patterns, this strongly supported the action of a Turing or reaction‐diffusion system in their formation, which occurs entirely in the much less motile epithelial cells.”

… fingerprints emerge during early development as a result of Turing diffusion in the epithelial layer without involvement of motile cells from the underlying mesenchymal layers.

The bottom line is that fingerprints emerge during early development as a result of Turing diffusion in the epithelial layer without involvement of motile cells from the underlying mesenchymal layers. The other key point is that the Turing pattern appears to emerge from the same differences in gene expression that determine whether the precursor cell discs differentiate into hair follicles. On that basis, the genes associated with fingerprint formation can be said to have been identified.

Zebra stripes

While there is obvious interest in fingerprint formation, the natural question is whether the same underlying principles apply to other surface patterns in animals or plants. In that regard, the Edinburgh study follows earlier work on the role of Turing diffusion in pattern formation, as noted by Cheng‐Ming Chuong from the University of Southern California, whose laboratory focusses on tissue development and regeneration. “Several papers have been published using Turing principles from fish pigment stripes, to feather or hair patterns, and most recently Headon's Cell paper on fingerprints,” he said. “These papers gave new understanding on how Turing principles can be used to generate patterns. Furthermore, they show Turing can work with other global gradients to generate more complex or more organized patterns.”

Given that Turing patterning appears to be widespread in nature and occurs in many different tissues, there can be no common underlying molecular or genetic elements, Headon noted. “But many elements and themes do seem consistent,” he added, referring to other pattern formations including zebra stripes (Fig 1). “There has been next to no direct study of zebra pattern formation, to my knowledge, so the evidence that their stripes are produced by a Turing type system is limited to observations of how the pattern changes. Two points are particularly relevant – the first is that the stripes are transverse to the long axis of the body and also transverse to the long axis of the legs, so they are running perpendicular to one another from trunk to leg. At the shoulder, where these two pattern types meet, you have the formation of chevrons. Turing type systems will readily produce such a chevron where two orthogonally aligned stripe forming systems meet at a junction, so this is a sign that a Turing system can explain a signature feature of zebra striping.”

Figure 1. Beautiful example of the Turing diffusion model.

Equus zebra zebra in the Mountain Zebra National Park, South Africa. George Brits/Wikimedia Commons.

Given that Turing patterning appears to be widespread in nature and occurs in many different tissues, there can be no common underlying molecular or genetic elements…

The second line of evidence follows from the fact that zebra species that have evolved more recently have different numbers of stripes. “A Turing system can readily change number of stripes or spots depending on when in development it becomes ‘fixed’ or finalised, and shifting this window of fixation to a later timepoint can easily explain the greater number of stripes on some species compared to others,” Headon explained. He cited domestic cats (Felis catus) as an animal species in which coat pattern formation has been more thoroughly studied. “Genetic studies on domestic cats have identified the basis of some variants in colour pattern, including changes to stripe thickness, and spot to stripe transitions. These are again hallmarks of the operation of a Turing type system, and single‐cell sequencing has implicated WNT signaling as key to the distinction between dark and light‐coloured regions of cat skin, originating long before the pigments are actually produced,” Headon said. “Again, in this system the epithelium appears to be defining its own pattern without local instructions from the mesenchyme, simplifying the system compared to hair or feather follicle patterning, where both tissues contribute.”

In many of these cases, the unique individual pattern is not itself subject to natural selection, since it confers no advantage. However, the ability to form patterns at all can itself be advantageous and subject to strong selection. In humans, fingerprints as 3D structures allow better tactile control than would be possible with smooth skin; fingerprints are not universal across human epithelial surfaces but occur only on the fingers. Zebra stripes provide camouflage to evade predators, while cats' fur pattern may help them to get closer to prey.

The clearest example of unique patterns without obvious evolutionary advantage occurs in sea snails that spend most of their lives underground with a shell to afford protection. “While these beautiful seashell patterns are fine for collectors, many likely serve no purpose for the organism, with several species with beautiful shell patterns apparently spending their lives buried in sand or mud,” agreed Headon (Box 1).

Box 1. Turing diffusion explained.

Alan Turing's reaction–diffusion model has been widely accepted as an explanation for the underlying basis of early biological developmental processes. It comprises two proteins, an activator and an inhibitor, the interactions of which breaks the initial symmetry and allows heterogeneous tissues to emerge that later coalesce into distinct organs and other structures. The model also relies on a third external component to generate 2D structures such as stripes from initial one‐dimensional spots.

The process starts with the expression of the activator, which is required not just for its own production but also of the inhibitor protein. The inhibitor, when expressed, turns down the activator, so in the absence of any other factors, there is a see‐sawing between both proteins being off and on with a transient state in between where only the activator is expressed, followed by just the inhibitor.

However, when, as in a real system, diffusion enters the equation, the dynamics become more subtle and interesting. The activator protein then diffuses away from its point source, activating itself in nearby cells. This triggers the expression of the inhibitor, which also diffuses away from the point source. Depending on the precise timing of activation and diffusion, this can generate an expanding “wave” of activator and inhibitor expression.

However, as Turing realized, two more layers of complexity are required to translate these expanding rings into stable spots. The first is that the activator's expression must fluctuate randomly to provide the basis for variable spot formation. Second, the activator must diffuse more slowly than the inhibitor, or else both the activator and inhibitor spots would continue to grow (see diagram). Each of the small spots activates the expression of activator, which diffuses away more slowly, and of the inhibitor, which diffuses away too quickly to eliminate activator expression from the initial point source.

This gradient of inhibitor diffusion from each spot prevents nearby cells from making activator and can result in the development of regular patterns, the exact form of which depends on several variables. These include the rates of activator and inhibitor diffusion, as well as other elements nearby that influence their distribution across cells. Although Turing focussed on molecular diffusion, the same underlying mathematical principle involving interaction between a short‐range and a long‐range force applies to many other systems both inside and outside biology.

Feathers and flight

In the case of birds, Turing patterning plays a role both in colour and in arrangement of the feathers. Colour patterning is relevant for display as males often have more elaborate or eye‐catching plumage for mating, but feather arrangement is directly related to the aerodynamic properties of wings. For this reason, there is likely strong selection to minimize air resistance and maximize control. Indeed, flightless birds such as ostriches and emus have irregular feather arrays, presumably because of lower selective pressure for regularly arranged contour feathers (Inaba et al, ²⁰¹⁹).

Philip Maini, a mathematical biologist at the University of Oxford, UK, noted that a range of models has been proposed, some based on chemical patterning, some on cell movement and some on mechanics to explain particular pattern mechanisms. “Crucially these models have the same short‐range activation and long‐range inhibition mechanism, but also the patterns form behind a ‘wave of competence’, he said. “That is, the hexagonal 2‐dimensional pattern forms as rows of 1‐dimensional spots. The mathematical explanation for this is that these mathematical models, if solved on a 2‐D domain, will give irregular patterns and this could be an issue in the case of feathers for flight. However, if the pattern forms behind a propagating wave of competence, then it will robustly be of a certain type basically because the patterning competent domain is then a long thin strip, upon which spots are robust as patterns.” In the case of flying birds, there would be strong selection for feather arrangements that form regular patterns behind this wave of competence, but this falls away in species that have lost the power of flight, similar to many past adaptations in nature that have lost their utility.

Turing himself acknowledged that his diffusion model is a simplified representation of the systems involved and often only part of the story. “The Turing theory has been very influential, but we have to see it as a ‘toy model’, and a vast simplification of any actual biological system,” commented Leah Edelstein‐Keshet from the University of British Columbia, Vancouver, Canada. She referred to the Kaelin et al (2021) paper on cats as an example of where understanding additional complexity is important. “The paper shows that it is in practice a lot more complicated, though there are likely to be some common principles,” she explained. “A core principle of pattern forming mechanisms is something like ‘local activation, and long‐range inhibition’. In the Turing theory, this is coming directly from the interaction and diffusion ranges of two chemicals. More broadly, it could be any physical system where some components locally activate and inhibit or get depleted on a broader spatial range.” Examples of such systems outside biology include sand dunes, clouds and heated fluids, all of which according to Edelstein‐Keshet have similar mechanisms of local reinforcement.

Turing himself acknowledged that his diffusion model is a simplified representation of the systems involved and often only part of the story.

A general theory and applications

In many of these cases, there are as yet no practical applications, and the first may follow from the work on dermatology and fingerprint formation. As Headon noted, humans have eccrine‐type glands, the major sweat glands, all over the body, rather than just on the soles of paws as in dogs, for example. For therapeutic applications, the key question is how Turing mechanisms help sweat glands distinguish from hair follicles. “This is relevant to generating such glands, either trying to restore their development in conditions such as hypohidrotic ectodermal dysplasia where they fail to form, or to try to promote their formation in healing skin,” he said. “Our report that TGF (Transforming Growth Factor) alpha is important for sweat gland formation and distinct from the early hair follicle could help design approaches here.”

Further progress in understanding pattern formation in nature will require extending beyond breaking the original symmetry towards transformations that occur in systems that are already disordered. Again, Turing himself understood this as the next step, having written in that seminal paper that breaking symmetry merely leads to the first appearance of distinctive patterns in an organism. “Most of an organism, most of the time, is developing from one pattern into another, rather than from homogeneity into a pattern,” Turing noted. “One would like to be able to follow this more general process mathematically also.”

That is the focus of work by, among others, Andrew Krause at Durham University, UK, who tries to generalize Turing's mechanism to heterogeneous systems where homogeneous patterns are locally confined (Krause & Woolley, 2020). The idea is to consider the Turing diffusion model as a local special case, which, the authors argue, extends the concept to a wider class of systems closer to those in nature. Such systems involve sequential evolution of heterogenous states that might have originated with symmetry breaking. “This helps separate out which kinds of structures are due to Turing‐type instabilities, and which come from prior asymmetries,” Krause explained. “Of course, it is difficult to do this precisely given the biological complexities involved, but some progress has been made on extending Turing's ideas beyond the realm of pure symmetry breaking.”

Krause thinks that this could have wide‐ranging implications, or applications, a few of which have already emerged: “For example, there have been recent advances in manufacturing desalination filters and other materials from essentially these kinds of processes, as well as much more theoretical ideas such as viewing cosmological formations from a reaction‐diffusion theory perspective.” On the biology front, Krause reckoned there is potential for such approaches in tissue engineering and the creation of organoids. “These are all very ‘pie in the sky’ ideas of course, and it is very likely that this specific work is of most relevance in dermatology directly,” he said.

EMBO reports (2023) 24: e57405