The biological Turing

There has recently been a lot of interest in Alan Turing, but you may not know this . . .

A planet like the Earth is bathed in the flow of energy from a star, which makes the whole surface of the planet an open, dissipative system. All life on the surface of the Earth makes use of this energy to maintain itself far from equilibrium, on the edge of chaos.[1] Plants get their energy directly from sunlight through photosynthesis; grazers get their energy from plants; carnivores get their energy from other animals. But it is all originally from the Sun, and it is all, originally, thanks to gravity. But the way in which systems organise themselves, using this flow of energy, into what seem to be complex forms is really quite simple. We can see this clearly by following the lead of the brilliant mathematician Alan Turing (1912-1954), who had the chutzpah to attempt to explain, more than half a century ago, what happens in the most complicated process we know of, the development of an embryo from a single living cell. Turing was way ahead of his time, and the importance of his work in this field was not recognised until long after his death.

Turing, who was born at Paddington, in London, on 23 June 1912, is best known as a cryptographer, the leading member of the team at Bletchley Park, in Buckinghamshire, which cracked the German codes (including the famous Enigma code) during World War Two.

After his war work, Turing spent the academic year 1947-1948 working in Cambridge, on secondment from the NPL, and wrote a paper, never published in his lifetime, on what we would now call neural nets (more of these in Chapter Five), an attempt to demonstrate that any sufficiently complex mechanical system could learn from experience, without actually being programmed by an outside intelligence. By 1950, settled in Manchester, he was ready to begin to apply the knowledge he had gained about mechanical systems and electronic computers to biological systems and the human brain. The jump from there to his work on how embryos develop wasn’t as great as it might seem, since Turing wasn’t only interested in how brains grow and form connections; his interest in the way the variety of living things develop from simple beginnings had been stimulated in his youth by reading D’Arcy Thompson’s classic book On Growth and Form. So at the time Turing was elected as a Fellow of the Royal Society in 1951, for his contributions to computer science, he was already working on what would probably, had he lived, have been an even greater contribution to science.

Even Turing couldn’t leap straight from the understanding of biology that existed at the beginning of the 1950s to a model of how the brain itself develops its network of connections – after all, the double helix structure of DNA, the life molecule, was not determined until 1953, by Francis Crick and James Watson, working in Cambridge. Instead, he decided to tackle the fundamental problem of how structure emerges in the developing embryo from what is an almost spherical, almost featureless initial bob of cells, the blastocyst formed from the fertilised egg. In mathematical terms, the problem was one of broken symmetry, a phenomenon already familiar to physicists in other contexts (not least, Bénard convection). A good example of symmetry breaking occurs when certain kinds of magnetic substances are heated and then cooled down. Magnetic materials such as iron can be regarded as made up of a collection of tiny dipoles, like little bar magnets. Above a critical temperature, known as the Curie point (after Pierre Curie, who discovered the effect in 1895), there is enough heat energy to break any magnetic links between these dipoles, so that they can spin around and are jumbled up in a random fashion, pointing in all directions, so that there is no overall magnetic field. In magnetic terms the material can be said to be spherically symmetric, because there is no preferred magnetic direction. As the temperature drops below the Curie point (760 oC for iron), the magnetic forces between adjacent dipoles overcome the tendency of their declining heat energy to jumble them up, and the dipoles line up to produce an overall magnetic field, with a north pole at one end and a south pole at the other end. The original symmetry has been broken. Such a change is called a phase transition, and is similar to the way water freezes into ice in a phase transition at 0 oC. The concept of a phase transition also has important applications in particle physics, which we need not go into here; the relevant point is that although such ideas had not been widely applied in biology before 1950, at that time it was natural for a mathematician moving into the theory of biological development to think in terms of symmetry breaking, and to have the mathematical tools describing the general nature of such transitions available.

In 1952, Turing published a paper which described in principle how the symmetry of an initially uniform mixture of chemicals could be spontaneously broken by the diffusion of different chemicals through the mixture. The anticipated relevance of this to biology was clear from the title of the paper, “The chemical basis of morphogenesis,”[3] and Turing’s proposal was that something like the process that he described mathematically might actually take place in the developing embryo, to produce patterns where none existed originally.

At first sight, Turing’s proposal seems utterly counter-intuitive. We expect diffusion to mix things up, and to destroy patterns, not to create patterns where none exist originally. The obvious example is the way a drop of ink placed in a glass of water spreads out to fill the glass with a uniform mixture of water and ink; it seems almost as if Turing is suggesting a reversal of the thermodynamic processes that operate on this scale, with time running backwards and a uniform mixture of water and ink separating out into a glass of clear water surrounding a single droplet of ink. But that is not the case, and the key to Turing’s insight is that the pattern-forming process he described involves at least two chemicals interacting with one another.

It all depends on the process known as catalysis, whereby the presence of a particular chemical substance (the catalyst) encourages a particular chemical reaction to take place. In some cases, the presence of a chemical compound (which we can represent by the letter A) in a mixture of chemicals encourages reactions which make more of itself. The reaction is said to be autocatalytic, and since the more A there is, the more A is produced, we can see that this is another example of positive feedback at work in a nonlinear process. On the other hand, there are chemicals which act in the opposite way, to inhibit certain chemical reactions. Logically enough, they are called inhibitors. And there is nothing to say that a single substance cannot encourage more than one chemical reaction at the same time. Turing calculated that patterns could arise in a mixture of chemicals if the catalyst A not only encouraged the production of more A, but also encouraged the formation of another compound, B, which was an inhibitor that acted to slow down the rate at which more A is produced. His crucial suggestion was that once A and B formed they would diffuse through the mixture of chemicals at different rates, so that there would be more A than B in some parts of the mixture, and more B than A in other places. In order to calculate just how much A and B there would be in different places, Turing had to use the simplest equations he could, since electronic computers were still highly limited in their abilities, and in very short supply, so he was working everything out on paper. This meant working with linear approximations to the real nonlinear equations describing the situation, and these equations turn out to be very unstable, in the sense that a small error in one part of the calculation leads to a big error later on. As a result, Turing could only calculate what was going on for the simplest systems, but that was enough to hint at the possibilities. Turing himself acknowledged that a full investigation of his ideas would have to await the development of more powerful digital computers, but in developing his ideas as best he could beyond those sketched out in his 1952 paper he showed how the competition between A and B was the key to pattern formation, and that it was essential that B must diffuse through the mixture more quickly than A, so that while the runaway production of A by the autocatalytic feedback process is always a local phenomenon, the inhibition of A by B is a widespread phenomenon. It also means that the rapid diffusion of B away from where it is being made means that it doesn’t entirely prevent the manufacture of A at its source.

To picture what is going on, imagine the mixture of chemicals sitting quietly in a glass jar. Because of random fluctuations, there will be some spots in the liquid where there is a slightly greater concentration of A, and this will encourage the formation of both A and B at those spots. Most of the B will diffuse away from these spots, and prevent any A forming in the spaces between the spots, while the autocatalytic process ensures that more A (and B) continues to be produced at the spots. (There will also be places in the original mixture where random fluctuations produce an excess of B to start with, but, of course, nothing interesting will happen there.) Now suppose that chemical A is coloured red and chemical B is coloured green. The result will be that an initially uniform , featureless jar of liquid transforms itself spontaneously into a sea of green dotted with red spots that maintain their positions in the liquid (as long as the liquid is not stirred up or sloshed around). The pattern is stable, but in this particular case it is a dynamic process, with new A and B being produced as long as there is a source of the chemicals from which they are being manufactured, and as long as there is a “sink” through which the end products can be removed. In the terminology that ought to be becoming familiar by now, the pattern is stable and persistent provided that we are dealing with an open, dissipative system which is being maintained in a non-equilibrium state. Turing also described mathematically systems in which there is a changing pattern of colour rippling through the liquid, where it would be more obvious to any observer (if such systems could be replicated in real experiments) that a dynamic process is going on. Today, an autocatalytic compound such as A is called an actuator, while B is indeed known as an inhibitor; Turing himself, though, didn’t use these terms, and referred to B as a “poison,” which now has chilling echoes of his own death.[4] Although it may seem far removed from the development of an embryo (let alone a brain), the essential point about Turing’s discovery was that it provided a natural chemical way in which symmetry could be broken to spontaneously create patterns in an initially uniform system – if there were real chemical systems that behaved in this way.

Intriguing though Turing’s ideas were, although his paper is seen as being of seminal importance to theoretical biology today, in the 1950s and through most of the 1960s it attracted little interest among chemists and biologists, precisely because nobody knew of any real chemical system which behaved in the way that this mathematical model described. Nobody, that is, except one person, the Russian biochemist Boris Belousov, and he, not being a reader of the Philosophical Transactions of the Royal Society didn’t know about Turing’s work, just as Turing never learned about Belousov’s work before his own untimely death. At the beginning of the 1950s, Belousov was working at the Soviet Ministry of Health, and interested in the way glucose is broken down in the body to release energy. He was already in his early fifties, an unusually advanced age for any scientist to make a major new discovery, and had a background of work in a military laboratory, about which little is recorded, but where he reached the rank of Combrig, roughly equivalent to a Colonel in the army and an unusually high distinction for a chemist, before retiring from this work after World War Two. Like many other metabolic processes, the breakdown of glucose that Belousov was interested in is facilitated by the action of enzymes, different kinds of protein molecules which act as catalysts for different steps in the appropriate suite of chemical reactions. Belousov concocted a mixture of chemicals which he thought would mimic at least some features of this process, and was utterly astonished when the solution in front of him kept changing from being clear and colourless to yellow and back again, with a regular, repeating rhythm. It was as if he had sat down with a glass of red wine, only to see the colour disappear from the wine, then reappear, not once but many times, as if by magic. This seemed to fly in the face of the second law of thermodynamics, as it was understood at the time. It would be entirely reasonable for the liquid to change from clear to yellow, if the yellow represented a more stable state with higher entropy. And it would be entirely reasonable for the liquid to change from yellow to clear, if clear represented a more stable state with higher entropy. But both states couldn’t be at higher entropy than the other! It was as if, using the original nineteenth century ideas about the relationship between thermodynamics and time, the arrow of time itself kept reversing, flipping backwards and forwards within the fluid.

[1] Energy also comes from within the Earth, chiefly as a result of the decay of radioactive elements in the Earth’s core. This radioactive material was produced in previous generations of stars, and spread through space when those stars exploded, becoming part of the interstellar cloud from which the Solar System formed. So this energy source, too, ultimately owes its origin to gravity. Life forms that feed off this energy, which escapes through hot vents in the ocean floor, may do so entirely independently of the energy from sunlight, but they are as much a product of gravity as we are.

[3] Philosophical Transactions of the Royal Society, volume B237, page 37; this is now regarded as one of the most influential papers in the whole field of theoretical biology.

[4] Turing seems to have had an obsession with poison. His biographer Andrew Hodges describes how Turing went to see the movie Snow White and the Seven Dwarfs in Cambridge in 1938, and was very taken “with the scene where the Wicked Witch dangled an apple on a string into a boiling brew of poison, muttering: ‘Dip the apple in the brew. Let the Sleeping Death seep through.’” Apparently, Turing was fond of chanting the couplet ”over and over again”.

Adapted from my book Deep Simplicity (Penguin).

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