Forging patterns and making waves from biology to geology: a commentary on Turing (1952) 'The chemical basis of morphogenesis'

The qualitative essentials of his model did not emerge until two decades later. In 1972, developmental biologists Hans Meinhardt and Alfred Gierer at the Max Planck Institute for Virus Research in Tübingen, Germany, devised a theory of biological pattern formation caused by diffusing reagents that paralleled Turing's [13]. They did not knowing about Turing's work until a referee of their paper pointed it out (Meinhardt 2012, personal communication). In Meinhardt and Gierer's model, stationary chemical patterns can result from two interacting ingredients—equivalent to Turing's morphogens—if they have specific characteristics. One is an ‘activator’, which is autocatalytic and so introduces positive feedback. The other is an ‘inhibitor’, which suppresses the autocatalysis of the activator. Crucially, they must have different rates of diffusion, the inhibitor being faster. In effect, this means that the activator's self-amplification is corralled into local patches, whereas the inhibitor prevents another such patch from growing too close by. Meinhardt and Gierer found that Turing's equations describe just this situation [14]. By a curious coincidence, a theoretical model similar to this activator–inhibitor scheme was also proposed in 1972 for a predator–prey system [15], although it tends to be unjustly overlooked today.

The availability of computers made it easier to deduce what are the generic patterns produced by activator–inhibitor systems: they generate quasi-ordered spots and stripes, with pattern features all of roughly the same size and separation (figure 4). These outcomes—a chemical leopard and chemical zebra—made it all the more plausible that Turing patterns might explain animal markings. In the 1980s, Meinhardt and mathematical biologist James Murray at the University of Washington in Seattle worked independently to show that Turing's theory offered a possible explanation for a wide range of animal pigment patterns, from zebras to giraffes to seashells [16–18]. The idea here is that the morphogens turn on or off genetic pathways that stimulate the production of pigments—in mammal skins this is the pigment melanin, which generates colours from tawny to black.

More recently, Turing models have been shown capable of reproducing some of the specific fine details of animal markings. For example, two coupled activator–inhibitor processes can produce the broken ring markings characteristic of jaguars (figure 5) [19], and a Turing scheme implemented on the curved shells of ladybirds can produce patterns looking very such like those seen in nature [20].

The stripes of the angelfish offer a particularly suggestive example. These are unusual in that they continue to grow and develop as the fish grow, rather than just being laid down during embryogenesis and then getting blown up like markings on a balloon. The detailed changes in these patterns, such as a characteristic ‘unzipping’ of two merging stripes, is perfectly mimicked by a Turing model [21]. However, later work with the similar patterning system of the zebrafish has shown that although this pattern does seem to result from autoactivation and long-range inhibition between the two pigmented cell types (black melanophores and yellow xanthophores)—making it a genuine Turing pattern—these interactions do not depend on the diffusion of morphogens, but result from the properties of the network of direct cell–cell interaction [22].

Turing's travelling waves might also produce pigmentation patterns. Meinhardt has shown that an activator–inhibitor scheme with a third morphogen that creates short-ranged but long-lasting inhibition can reproduce the kinds of complex patterns seen on some mollusc shells, which are in effect frozen traces of two-dimensional travelling waves on the rim of the growing shell (figure 6) [23].

It is worth pointing out that there is still no real consensus on what many of these animal markings are for. The default assumption has tended to be that they conceal the animal as camouflage, but this is by no means obviously true in many cases. Some molluscs, for example, spend much of their life cycles covered in mud, with the pigmentation invisible. Meinhardt speculates that here pigmentation might be a side-effect of mechanisms for removing wastes from the mollusc itself.

Even for the zebra, the classic example of a striking quasi-regular pigmentation pattern, it is not clear that the markings are for concealment. Several other functions of the stripes have been proposed, ranging from heat regulation to deterrence of biting insects [24,25]. For wildcats there does seem to be a correlation between those with spots or other markings and a habitat with a variegated appearance—as Rudyard Kipling put it in the Just so stories (1902), ‘full of trees and bushes and stripy, speckly, patchy–blatchy shadows'. But there are also outliers, such as the cheetah, which are spotted despite a preference for open habitats. It also seems that the available ‘pattern space’ provided by a Turing-type model accounts fairly well for the range of markings seen in 32 species of wildcats [26]. But it remains unclear how well the precise details of the skin patterns match the visual appearance of the environment in which the animals dwell—in other words, how ‘carefully’ evolution is selecting from the available palette, if indeed that is what is going on.

Even in purely chemical systems, Turing patterns proved elusive for long after they were proposed. It was not until 1990 that they were first reported in a real chemical system: an oscillating reaction somewhat similar to the BZ reaction, known as the chlorite–iodide–malonic acid (CIMA) reaction. Patrick De Kepper, Jacques Boissonade and their collaborators at the University of Bordeaux saw a band of stationary spots develop in a strip of gel into which the CIMA reagents diffused from opposite sides [27,28]. The following year, extended Turing structures were generated in a two-dimensional layer of the CIMA mixture, and switching between spots and stripes was achieved by altering the concentrations of the reagents (figure 7) [29].

In biology, it turns out that Turing's mechanism is in fact not generally necessary to break the symmetry of a fertilized egg—in many organisms this is disrupted from the outset by maternal proteins diffusing from one side of the embryo. There is evidently a great deal more symmetry-breaking that must happen for a zygote to acquire its full body plan—but it remains far from clear that Turing patterns have much to do with this. For example, the stripe patterns of protein expression that appear in the fruitfly embryo look superficially like Turing's stripes, but they are not generated by the self-organization inherent in his model. Instead there is a hierarchical cascade of patterning steps involving the straightforward diffusion of morphogenetic proteins. Here, a concentration threshold for activating particular genes converts a smooth gradient into an abrupt interface, which becomes a more complex pattern by sequential elaboration of the same simple mechanism [30,31].

For such reasons, until recently Turing's work had rather little impact on developmental biology outside the special case of animal pigmentation. Over the past decade or so, however, good evidence has emerged that Turing patterns and related reaction–diffusion mechanisms do feature in other patterning processes during developmental morphogenesis [32]. For example, the proteins Nodal and Lefty appear to operate as an activator–inhibitor pair during the induction of the mesoderm of metazoans, as demonstrated in experiments on zebrafish and mice [33,34]. And there is some evidence that the hair follicles of mice are positioned in their quasi-regular array on the skin by a process of activation and inhibition involving proteins called Wnt (the activator of follicle formation) and Dkk2 and Dkk4 (inhibitors of Wnt) [35]. For example, genetic mutant mice that produce Dkk proteins in abnormally high amounts develop follicle patterns that match those predicted theoretically from Turing-style activator–inhibitor models of the diffusion and interaction of Wnt and Dkk. However, the details of this patterning process seem likely to be complex, involving various networks of protein and gene interactions rather than a simple activator–inhibitor pair. Something analogous to the patterning of hair follicles may also be at work in the regular arrangement of feathers in birds and the scales of lizards and butterfly wings.

The diverse family of Wnt-type developmental proteins seem likely to produce a range of different patterning mechanisms. Meinhardt [36] has argued that one such protein morphogen organizes the formation of tentacles around the cylindrical gastric column of the hydra—essentially the ring symmetry explored by Turing. The periodic positioning of bird feather barbs can be explained if the protein product of a gene called Sonic hedgehog (Shh)—a common patterning gene in many species—behaves as an activator while the bone morphogenic protein 2 is an inhibitor [37]. Through the interaction of these components, the uniform epithelium of the developing feather bud becomes divided into a series of stripe-like ridges that prefigure its break-up into distinct barbs. Meanwhile, the regularly spaced ridges of the mammalian mouth palette seem to be arranged by a Turing-type reaction–diffusion mechanism involving the proteins fibroblast growth factor and Shh as the activator and inhibitor, respectively—albeit with the possible involvement also of other proteins, including those of the Wnt family [38].

Perhaps the most striking challenge to the predominant view that biological development is largely dictated by smooth, long-range biochemical gradients comes from recent evidence that digit formation in the embryo can be regarded as arising from striped Turing patterns [39]. Here too, Wnt proteins play a role. Digit formation is ultimately under the control of a gene called SOX9, which triggers differentiation of soft tissue towards the formation of bone and cartilage. WNT gene products and bone morphogenetic proteins (products of BMP genes) influence the activity of SOX9 in a manner described by an activator–inhibitor scheme, so that drug-induced suppression of WNT or BMP leads to predictable changes in the number or spacing of digits.

The important unanswered question for many of these systems seems to be that, while the central elements of activation and inhibition, and reaction and diffusion, do appear to provide a useful basis for framing the problem theoretically, to what extent can the details be reduced and simplified to the two-component scheme proposed by Turing?

Turing's discussion of the whorled leaf arrangements of the woodruff plant in his 1952 paper shows that he suspected his theory might have something to say about a biological patterning process quite different from morphogenesis of the embryo, namely phyllotaxis: the arrangement of leaves or related features on plant stems. He promised that ‘the morphogen theory of phyllotaxis' would be ‘described in a later paper’, and he had already drafted that paper by the time of his death—which, however, prevented its publication at that time [40].1 Turing's basic idea was that an activator–inhibitor system of hormones acting at the growing tip of a plant lays down the spots that grow into buds on the cylindrical stem. This seems plausible in the light of current knowledge of plant biology. It has been known since the 1930s that the plant hormone auxin can function as an activator to initiate the growth of new leaf buds (primordia). In 2003 it was shown that phyllotaxis is regulated by proteins that ferry auxin through the outer ‘skin’ of the stem up towards its apex [41]. Existing leaf buds soak up auxin and thus act as sinks, inhibiting the formation of any new buds nearby.

Phyllotaxis is one of the oldest and most compelling problems of biological pattern formation, not least because it has a mathematical character that seems at first deeply mysterious. The leaves or florets are typically arranged around the stem in a spiral pattern, and when this is projected onto a horizontal plane—as it is in the plant itself for the arrangement of florets in the head of a sunflower or daisy (figure 8)—one finds that there are in fact two groups of counter-rotating spirals. In each of the two groups, the numbers of spirals are always successive numbers in the Fibonacci series, generated from the pair {0,1} by adding together the two preceding numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …

Why should the spirals obey this rule? A common view is that the Fibonacci arrangement enables the most efficient packing of the florets. This may well be true, but it does not explain the mechanics of how the growing plant ‘finds' this solution, any more than explaining the leopard's spots in terms of camouflage accounts for how the pigmented spots actually form on the leopard foetus. It seems conceivable, at least, that Turing's scheme might provide the biochemical mechanism: a Turing process involving two sets of activators and inhibitors operating on a cylindrical stem may produce primordia in a (2,3) spiral phyllotaxis pattern [42]. Whether the model can generate higher-order Fibonacci spirals is not clear, however, and there is as yet no direct evidence that such a double activator–inhibitor process really operates in plants.

Turing's stripe patterns resemble not only the skin of a zebra, tiger or angelfish, but also some patterns in inanimate nature, such as the ripples in wind-blown sand (figure 9). This may be no coincidence. Meinhardt [23] suggests that, at root, the formation of these sand patterns is akin to an activator–inhibitor system. The mounds and ridges of sand are formed by deposition of wind-blown grains. As a ridge gets bigger, it enhances its own growth by capturing more sand from the moving air. But in doing so it acts as a sink, removing sand from the wind and suppressing the formation of other ripples nearby. The balance between these two processes establishes a roughly constant mean distance between ripples.

The feedbacks involved in replication, competition and predation might set up Turing-type patterns in animal and plant communities. These might account for the patchiness of zooplankton in the sea and the phytoplankton on which they graze [43]. And there is rather compelling evidence that a Turing-type mechanism accounts for structures formed by a species of ant [44]. Mediterranean Messor sancta ants collect the bodies of expired colony members and arrange them in piles. The ants constantly pick up and redistribute the corpses, producing a kind of ‘diffusion’ of bodies. Nonetheless, after a certain time the locations of the piles stay fixed. Because ants are more likely to drop a body on a pile as the pile gets larger, there is a positive feedback (activation) controlling their growth, analogous to that by which sand ripples form. There is also long-range inhibition, because the region surrounding a big pile gets swept clear of bodies, making it less likely for a new one to be started in the vicinity. The result is a series of stationary clusters of corpses, and if the ants are confined in a Petri dish then these clusters are created around the perimeter, with precisely the ring geometry that Turing first explored in 1952 (figure 2). Mechanisms like this might underlie many other aspects of habitat formation and grouping, such as nest construction, in higher organisms.

Even human communities, orchestrated by social feedbacks on behaviour and movement, might organize themselves into Turing patterns. A reaction–diffusion model has been proposed to explain the phenomenon of crime hotspots: districts in which the crime rate is anomalously high [45]. Here criminal offenders are modelled as predators who seek ‘prey’ (victims), while both agents move around (diffuse) in the available space. The ‘reaction’—predation of criminals on victims—can be potentially suppressed by an inhibiting agency such as a security measure or a police force.

This model produces two types of hotspots through the competing influences of activation and inhibition. The first are aggregates of individual crimes with overlapping spheres of influence. The second type of hotspot is caused more directly by positive feedback: by the known phenomenon in which crime induces more crime (figure 10). The first sort of hotspot can be eradicated completely by a sufficiently strong inhibiting influence: that is, by locally concentrated policing. But the second kind, corresponding to Turing-type patterns, is harder to eliminate. Focused inhibition may merely cause these hotspots to move or mutate, breaking up into smaller spots or rings in the close vicinity. If this picture is an accurate reflection of the world, it suggests that not all hotspots will yield to the same style of policing, but that different strategies might be needed in different situations.

Alan Turing's 1952 paper, proposed by an author with no real professional background in the subject he was addressing, put forward an astonishingly rich idea. The formation of regular structures by the competition between an autocatalytic activating process and an inhibiting influence, both of which may diffuse through space, now appears to have possible relevance not just for developmental biology but for pure and applied chemistry, geomorphology, plant biology, ecology, sociology and perhaps even astrophysics (a reaction–diffusion mechanism has, for example, been suggested as the origin of spiral galaxies). Turing seems to have identified one of nature's general mechanisms for generating order from macroscopic uniformity and microscopic disorder. Several of the putative Turing structures in nature remain speculative, and indeed, it is notoriously difficult to distinguish between different candidate processes for generating a particular pattern. But, there seems little question that nature, including the living world, does use Turing's mechanism as one way of producing its rich and often beautiful panoply of forms.

From a biological perspective, the broader question is how a spontaneous process such as that deduced by Turing, which gives rise to a particular palette of shapes and patterns, interacts with natural selection. To what extent can evolution adapt and modify Turing structures? Are all such structures necessarily adaptive at all? Or are we too readily tempted, when we descry order and regularity in nature, to attribute a ‘purpose’ to it? Might some of it, at least, represent nothing more than a kind of intrinsic creative potential in the natural world?

The author gratefully acknowledges helpful conversations with Jeremy Green and Hans Meinhardt.

Philip Ball is a freelance writer. He worked as an editor at Nature for many years, and now writes on all areas of science, and in particular on interactions between science, art and culture. His books include The self-made tapestry (Oxford University Press, 1998), a comprehensive survey of natural pattern formation, which was reissued in 2009 as a trilogy called Nature's patterns, comprising Shapes, Flow and Branches (Oxford University Press). His most recent book is Invisible: the dangerous allure of the unseen (2014).