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

Alan Turing was neither a biologist nor a chemist, and yet the paper he published in 1952, ‘The chemical basis of morphogenesis’, on the spontaneous formation of patterns in systems undergoing reaction and diffusion of their ingredients has had a substantial impact on both fields, as well as in other areas as disparate as geomorphology and criminology. Motivated by the question of how a spherical embryo becomes a decidedly non-spherical organism such as a human being, Turing devised a mathematical model that explained how random fluctuations can drive the emergence of pattern and structure from initial uniformity. The spontaneous appearance of pattern and form in a system far away from its equilibrium state occurs in many types of natural process, and in some artificial ones too. It is often driven by very general mechanisms, of which Turing's model supplies one of the most versatile. For that reason, these patterns show striking similarities in systems that seem superficially to share nothing in common, such as the stripes of sand ripples and of pigmentation on a zebra skin. New examples of ‘Turing patterns' in biology and beyond are still being discovered today. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

Keywords: pattern formation, morphogenesis, developmental biology, chemical kinetics, reaction–diffusion, animal markings

Let me begin my exploration of Alan Turing's paper [1] in what might seem an unlikely and unpromising place: the list of references. There are just six of these, and five are to books on rather broad and disparate topics: biological development, the permeability of membranes, the theory of elasticity and magnetism. Aside from standard textbooks, the only paper cited was canonical and almost 40 years old. It is true that many papers in the 1950s have a concise way with citations, but even so, this is a remarkably sparse list.

There are usually two reasons for such a state of affairs: either the authors are writing about a topic that no one cares (nor need care) about, or they have something of startling originality to say.

Turing's paper would hardly be in this collection if it did not fall into the latter category. But even by that measure, his contribution is extraordinary. While no one could reasonably assert that this paper has had as profound an impact on biology as that published a year later by Watson and Crick on the structure of DNA (which paper has?), it surpasses their achievement in this one regard: before Turing, no one had even really thought to ask the question that he poses.

At first it does not sound that way. ‘The purpose of this paper’, Turing writes, ‘is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism’ [1, p. 37]. Little new there, surely? The patterning and elaboration of an embryo, and how it is dictated by genes, are after all the matters discussed in the books by American zoologist Charles Manning Child and the British biologist Conrad Waddington that constitute two of the references. That body of work was, however, essentially experimental. Studies in the 1930s by Waddington and others had shown that some tissues can act as ‘organization centres' that, if implanted at a different location on the organism—or even sometimes on that of a different species—could induce the host tissue to undertake a new developmental path. Waddington spoke rather vaguely of a substance called an evocator that represents the ‘active principle’ of the organization centres; he seemed to regard it as a chemical compound of some kind. But how can chemicals, which simply diffuse through the fluid medium that contains them, give rise to order and structure?

That was the central question that Turing addressed. He presents a theoretical model in which chemicals that are diffusing and reacting may produce neither bland uniformity nor disorderly chaos but something in between: a pattern. This notion is, as we shall see, not entirely without precedent, but no one previously had thought to relate such an abstruse phenomenon to the question of biological growth and form—in short, to suggest how chemistry alone might initiate the process that leads from a ball of cells to a starfish, a horse or to us.

A decade ago, the English mathematician Alan Turing, despite being an almost legendary figure to many scientists and the subject of an excellent biography [2], was largely unknown to the general public. If recent television dramatizations and stories about his wartime code-breaking work at Bletchley Park had not already changed that, then the 2014 biographical movie The imitation game has surely done so. The centenary of his birth in 1912 was marked with a number of commemorative events and eulogies, which highlighted in particular the shameful treatment he received after the war, precipitating what is generally thought to have been his suicide.

Turing (figure 1) studied mathematics at Cambridge, where he published perhaps his most important work at the age of just 24. In that paper, he showed that there exist some numbers that are not computable, meaning that they cannot be calculated as decimal numbers within a finite time. To make this argument, Turing needed to invoke the concept of an automatic ‘computing machine’, which is now regarded as a blueprint for the digital computer—a universal computing device that may store and execute programs.

When war broke out in 1939, Turing was enlisted at Bletchley Park, where he helped to crack the Enigma code used by the Germany Navy. His contribution was vital. ‘I won't say that what Turing did made us win the war’, one of his Bletchley colleagues said later, ‘but I daresay we might have lost it without him’ [3]. When the war ended, Turing moved to the National Physical Laboratory in London to assist with the construction of an electronic digital computer along the lines he had outlined. As it happened, the first such device was in fact built in Manchester in 1948, where Turing became the head of the Computing Machine Laboratory. During this period, he outlined the basics of what would later become known as artificial intelligence. Perhaps his best known contribution to this field is the so-called Turing test for determining whether machines can think. It involves a human interrogator who poses questions to the machine and to a human foil, and seeks to identify which is which. If there is no discernible difference in the responses, we have no logical reason to deny that the machine is thinking. This idea (adapted by the science fiction writer Philip K. Dick as the ‘Voigt-Kampff test’) was famously used in the opening sequence of Ridley Scott's 1982 movie Blade runner to identify non-human ‘replicants'.

Turing was actively homosexual at a time when this was illegal in Britain. In 1952 he was prosecuted, and his sexual orientation was deemed to pose a security risk in view of his wartime work (some of which remained classified for the rest of the century). He was sentenced to take a course of ‘corrective’ hormone therapy, and although Turing is said to have borne this sentence with ‘amused fortitude’, the shame and the physical effects of the hormone seem to have driven him to take his life in 1954 by biting on an apple laced with cyanide (some have asserted that his death was accidental). In 2009 the British prime minister Gordon Brown issued an apology for the way he was treated, saying that ‘on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better’ [4]. During the 2012 centenary, a bill was introduced to the House of Lords asking for an official pardon for Turing's conviction for ‘gross indecency’, and in late December of 2013 the Queen signed a pardon, invoking the ‘royal prerogative of mercy’. There was much debate at the time about whether any kind of pardon can be appropriate for activities that are no longer, and never should have been, regarded as a crime. At any rate, at the time of his death, Turing was still highly productive, and the loss to British science was immense.

If you could have placed a bet, before the Second World War, on who might have asked the question that Turing posed, you would not have expected it to come from this mathematician. You should have been more likely to look to the Theoretical Biology Club of Cambridge University, a loose collection of innovative thinkers that included Waddington, J. Desmond Bernal, Joseph Needham and J. B. S. Haldane. All were sympathetic to the idea that biology might benefit from a reductive ‘physicist's' approach, rather than the purely statistical methods of classical genetics. They saw that the fundamental questions of biological development were about the generation of shape and form—of what was called morphogenesis—and Needham even approached the Rockefeller Foundation to fund a proposed Institute for Morphology. They recognized that the basic task was to understand how genetics, which they hoped might be ultimately understood at the atomic level, could give rise to aspects of symmetry and geometry in the whole organism. This notion of biological morphology was strongly felt in the first half of the twentieth century, not least because of the impact of the 1917 book by Scottish zoologist D'Arcy Wentworth Thompson, On growth and form, which is another of Turing's scant references. Thompson called for a more mathematical approach to biology that took account of the physical forces and constraints on an organism's development.

Turing left Cambridge in the year that the Theoretical Biology Club was formed, but he seems to have had no contact with Waddington's crowd while he was there, and it is not clear that he would have done so had he remained. All the same, the interests of this august group might have been expected to prepare the ground for a warm reception of a paper claiming to offer an explanation of the ‘chemical basis of morphogenesis’. That, on the contrary, Turing's paper was largely ignored for several decades is partly an ironic consequence of the fact that another Cambridge scientist with a physicist's pedigree—Francis Crick—reoriented interest in the ‘chemical basis' of developmental biology in another direction. For Crick and Watson's discovery of how genetic information may be encoded in DNA seemed at first to provide a more fruitful avenue for exploring the roots of growth and form. The task of decoding the molecular message of the genes, and the molecular mechanisms by which they interact and cooperate to orchestrate the cell, was to preoccupy biology for most of the second half of the century. It is only in recent years that this gene-centred view of biology has begun to reconnect with the mathematical and physico-chemical picture provided by Turing.

Quite aside from the biologist's notorious aversion to mathematics (there is plenty of it in Turing's paper, albeit not of a terribly advanced form, as he tried to reassure the reader), one can see why scientists familiar with the messy contingencies of cell cultures and assays might have been deterred from Turing's paper. Even he admits from the outset that ‘this model will be a simplification and an idealization, and consequently a falsification’ [1, p. 37]. Mentioning that at one point ‘the cells are idealized into geometrical points' [1, p. 37], Turing then begins to talk about Newton's laws of motion, elasticities and imaginary numbers in the kind of way that still today is apt to leave biologists despairing at the physical scientist's tendency to make biology abstract beyond all reach of reality.

Yet, the paper does not do that at all. Rather, it follows the physical scientist's instinct to express the basic problem in the simplest possible terms, and then to explore what the minimal requirements are of a model that captures the essential phenomena. Turing points out that the question of how an embryo develops and acquires shape and form—bilateral symmetry, say, and the budding of limbs—is one of symmetry-breaking. ‘An embryo in its blastula stage has spherical symmetry’, he writes [1, p. 41]. But how, if it consists simply of molecules randomly diffusing and reacting, can it ever escape that perfection? One might naively think that, as Turing writes, ‘It certainly cannot result in an organism such as a horse’—which he points out, in what looks like a sly meta-joke, ‘is not spherically symmetrical’ [1, p. 41]. (Whether or not this is an expression of Turing's playful nature, one of the paper's referees, J. B. S. Haldane, gently took him to task for occasionally belabouring the biologically obvious.)

Turing posits that among the molecular ingredients of this bundle of cells are components called morphogens (‘shape-formers'), which are somehow responsible for triggering development of a cell or tissue along a certain pathway. Waddington's vague chemical ‘evocators' of anatomical development were exactly what he had in mind here. Genes too, he said (and no one knew at that point quite what these inheritable entities were), are a ‘special class' of morphogen, albeit ones that are locked in the chromosomes and so are not free to diffuse through the cells. Hormones and skin pigments might also act as morphogens. All one really needs to know about them is that they diffuse and they react (triggering some developmental feature). This is why Turing's model is now known as an example of a reaction–diffusion system. It was not, however, the first such, as we shall see later.

Given the presence of these morphogens, one can see that there are deviations from spherical symmetry in the blastula. That is because random diffusion does not produce perfect uniformity: there are small, chance fluctuations in the concentrations of the substances, just as there are random local variations in national birth statistics or in the local temperature of a glass of water. Over time, these should average out and vanish—unless some process exists that will amplify such irregularities, so that they break the symmetry spontaneously and irrevocably.

Turing was not a physicist, and it is not clear that he realized that such symmetry-breaking by fluctuations was already well known in physical theory. It occurs, for example, in the behaviour of a magnet cooled through its so-called Curie point, where it switches from being non-magnetic to magnetic. Above the Curie point, the poles are randomly oriented; below it, they are aligned, creating a net magnetic field. But which way do they all point, given that any direction is equivalent to any other? The balance is tipped by chance fluctuations at the Curie point. In this way, symmetry (uniformity) is broken even though there is no driving force that specifies the direction in which it breaks. By the same token, a needle balanced on its tip (symmetrical along the vertical axis) breaks symmetry by falling over, even though no measurable force ‘pushes' it in any particular direction. Turing is, however, aware of this general kind of ‘unstable equilibrium’, to which he alludes with reference to a marble placed on top of a dome.

The question is then: into which kind of arrangements does a system of diffusing morphogens tip? What sort of patterns can we expect to see as a result? Much of Turing's paper is taken up with finding a mathematical answer to that question. To do so, he needed to assume a particular geometric arrangement of cells in which the morphogens move and interact. For the sake of simplicity, he chose to study a circular ring of cells. He wrote down equations describing the reaction and diffusion of morphogens, and showed that two situations might result. In both of them, the concentrations rise and fall around the ring in the form of waves. In one case, the waves are stationary, like acoustic standing waves: the peaks and troughs of concentration stay in the same place, creating a series of bands. In the other case, the waves are oscillatory, meaning that the peaks and troughs move around the ring: they are ‘travelling waves'.

In fact, there may be several such waves of different wavelengths, and their interference can result in a series of ‘blobs' that differ from one another in size—a pattern, but not a perfectly regular one. Because he had to solve his equations by hand—not difficult, but laborious—Turing could offer only a rather sketchy indication of what the patterns would look like. One of them, in a 20-cell ring, is shown schematically in figure 2a. Turing showed this rather less strikingly as a histogram of concentrations; but a two-dimensional pattern that he calculated for a particular case of his reaction–diffusion system was much more evocative. It is shown in figure 2b, and one can see at once what it evokes: a dappled animal marking (see, for example, figure 2c), as Turing himself hinted.

Given his initial intention to explain embryonic morphogenesis, Turing focused the discussion of biological implications on the matter of how the scheme could account for the shapes of organisms that approximated his ideal geometries: for example, the sprouting of shoots as radial whorls in plants such as woodruff, or of tentacles on the cylindrical stems of the primitive aquatic animals called hydra. But, the glory of dappled things has excited more interest, and indeed Waddington wrote to Turing in 1952 saying that the best application of his theory would be ‘in the arising of spots, streaks and flecks of various kinds in apparently uniform areas such as the wings of butterflies, the shells of molluscs, the skins of tigers, leopards, etc’ [5]. Much of the focus of interest in chemical ‘Turing structures', as his stationary-wave patterns are now known, has subsequently turned to these pigmentation patterns as the ideal test-bed for whether Turing's theory stands up. I will turn to that shortly. First, it is apt to look more closely at why the theory, for all its originality, had precedents [6].

The key attributes of the chemical system that Turing described, in which travelling or stationary waves in concentration can arise from a combination of reaction and diffusion, had already been discussed more than 40 years earlier. In 1910, the Austrian–American ecologist and mathematician Alfred Lotka described a chemical reaction that could undergo oscillations of this kind [7]. Lotka was not particularly interested in chemistry per se. For him, the reaction was an analogy for the dynamics of animal populations, and the collisions and reactions that may take place between molecules were just proxies for the interactions between predatory beasts and their prey. Chemical ingredients can be ‘consumed’ by reactions just as prey can be consumed by their predators. And both predators and prey can multiply their own kind by reproduction.

This last aspect is the crucial one: an ingredient of the process can produce more of itself. This is possible for a chemical reagent if it is autocatalytic, which means that it acts as a catalyst that speeds up the rate of its own formation. Autocatalysis is a positive feedback process: the more that is made, the faster it appears. That kind of blow-up allows small, random fluctuations in concentration to be amplified into large non-uniformities—into the ‘patches' that Turing's process can generate. Unchecked, positive feedback will blossom out of control, at least until the ingredients that feed it are exhausted. For chemical reactions, fresh reagents are brought into play at any location when they diffuse there, and so the outcome depends on a delicate balance between the rates of reaction and the rates of diffusion.

Lotka's paper described only a chemical system in which oscillations are transient—eventually they die out, like the damped oscillations of a ringing bell. But 10 years later he outlined conditions under which they could persist [8]. And in 1921, chemical oscillations were reported by William Bray of the University of California at Berkeley in a real reaction, that between hydrogen peroxide and an iodate salt to produce iodine. Although Bray was a distinguished chemist, no one knew what to make of his observations and they were largely ignored.

Travelling wavefronts resulting from a reaction–diffusion process were also identified in the 1930s by the geneticist Ronald Fisher, who devised a mathematical model [9]—extended by the Soviet mathematical physicist Andrei Kolmogorov [10] and now called the Fisher–Kolmogorov equation—describing how an advantageous allele spreads in a population. This model is now regarded as a very general prescription for a certain type of reaction–diffusion process, but at the time, with theoretical biology in its infancy (Fisher was one of the pioneers), there was little appreciation that an approach devised in one area of biology (population genetics) should have anything to say about another (morphogenesis).

In this respect, Lotka's perception that population dynamics could be rephrased as a problem of chemical kinetics was deeply insightful, and foreshadowed Turing's own recasting of a biological problem as a chemical one. It was also prescient, for at the same time that Turing was working out a theory that could account for pattern-forming reaction–diffusion processes, the first chemical system of this kind to receive serious attention was being discovered. In the 1950s, the Soviet biochemist Boris Belousov devised a cocktail of reagents as a simplified model of the process in which sugars are broken down in the body, and he found that the mixture oscillated back and forth between two states. His claims were met with scepticism, because they seemed at face value to violate the second law of thermodynamics, which can be interpreted as stipulating that any chemical process can only proceed in one direction. Specifically, the second law states that all change happens in the direction that increases the net entropy of the universe—this, you might say, is the ‘downhill’ direction. Belousov's results appeared to be tantamount to saying that his reaction was ‘downhill’ in both directions, which made no sense. Sadly for Belousov, no one had noted what Lotka had already made plain decades earlier, which was that thermodynamics can tell us only about the stable, unchanging equilibrium state of a system. If, Lotka said, a system is provided with an unceasing flux of energy and ingredients, it can be driven away indefinitely from an equilibrium state: it becomes a non-equilibrium system. Then, classical thermodynamics is silent about the outcome. The oscillating chemical reactions of both Bray and Belousov will, if left alone, eventually exhaust themselves and settle into an equilibrium state.

But this can take a long time, and there is no thermodynamic prohibition to their changing direction on the way. Moreover, if these reactions are constantly provided with fresh ingredients, and if the reaction products are carried away, the oscillations continue indefinitely, persisting in what is then a permanently out-of-equilibrium system.

In the absence of this understanding, Belousov's results were dismissed and he was barely able to publish them. But in the 1960s they were explored by Anatoly Zhabotinsky, a graduate student at Moscow State University, who discovered a variation of Belousov's mixture that would switch back and forth between red and blue. Such dramatic contrasts were impossible to deny, and Zhabotinsky gradually won acceptance within the Soviet Union that oscillating chemical reactions are real. When he discussed these findings at an international conference in Prague in 1967, chemists in the West were intrigued and began to figure out what was going on. During the 1970s, the mechanism of the so-called Belousov–Zhabotinsky (BZ) reaction was decoded, and it came to be seen that autocatalysis of some of the intermediate molecular reagents was the key [11].

Moreover, it became recognized that the BZ reaction could produce patterns not just in time—the red–blue oscillations—but in space too. In a well-mixed solution, the colour changes all at once. But, if the reaction mixture is placed in a shallow dish, and particularly if the diffusion of molecules is slowed down by infusing the chemical mixture in a gel, the colour change may be initiated at a particular location, thanks to some random fluctuation in concentration or another disturbing influence, from which it radiates as a ‘reaction front’—a travelling wave of a different colour from its surroundings. And because the reaction oscillates, subsequent waves follow at regular intervals, producing a series of concentric rings like ripples in a pond. Where these wavefronts touch, they annihilate one another. Other triggers can elicit spiral waves instead, making the BZ reaction a rich source of chemical patterns (figure 3) [12].

All the same, the BZ reaction is not quite the kind of phenomenon sketched by Turing that generates travelling waves. They are all variants of pattern-forming reaction–diffusion processes, but the states they produce depend on the details: on the relative rates of reaction and diffusion, for example, and the nature of the reactions themselves. Given the highly mathematical and abstract nature of Turing's analysis, none of this was clear from his paper; indeed, it is almost devoid of any intuitive, physical picture of how the patterns arise, and one can only speculate about whether Turing had such a picture himself, or deemed it necessary. His reaction–diffusion equations are not wholly consistent with a real molecular picture, for instance—they can lead to negative concentrations, although Turing prohibited this by fiat.