deodands

Where do spots come from? Turing's answer

Turing's 1952 paper, The Chemical Basis of Morphogenesis has become celebrated in its own right for introducing what is now known as the Turing instability, and provides a framework for understanding Turing's later, unfinished work. This section provides a brief non-technical discussion of the Turing instability.

Turing provided a hypothesis to explain the generation of pattern when smooth sheet of cells develop pattern during development in a wide variety of settings including the formation of leaf buds, florets, skin markings, and limbs. According to this hypothesis, chemicals called morphogens generate organs when present in sufficient density, and the pattern is created through mechanisms of reaction and diffusion. The corresponding reaction-diffusion models are by now well known to mathematical biologists, and for the mathematically inclined the books by Meinhardt and Murray (1993) can give much more detail.

Reaction...

More.

One way of understanding the reaction-diffusion process is to borrow an analogy which Turing himself used at least once, although in a slightly different model: cannibals and missionaries. An island is supposed to be populated by a population of cannibals and missionaries. The missionaries are all celibate and thus depend on recruitment from the external world to maintain the population as its members gradually die. Cannibals also die, but can also reproduce, so that the population naturally increases. However when two missionaries meet a cannibal, the cannibal is converted to missionary status. (If this seems a rather imperialist island it might be worth pointing out that under a commoner interpretation the cannibals are the growth promoters and the missionaries are the poison). This tension between production and transformation means that a balance is reached when both populations are mixed together. If this balance is disturbed by a small amount of noise, the tension will act to restore the balance: the system is stable.

...and diffusion

Pattern emerges on a ring.

Now we imagine that the two populations, instead of mixing completely together, are spread out in a thin ring around the rather narrow beach of the island. Now individuals react (that is, reproduce or convert) only with their immediate neighbours, but they also move around at random in a diffusive way. Moreover the members of the two populations move at different speeds: the missionaries have bicycles and move faster. This is enough to destabilize the system. For if there is at any point a small excess of cannibals, say, then this will be followed by excess 'production' of more cannibals, and then of more missionaries (since they have more targets for conversion). Without the spatial dimension the extra production of missionaries would in turn reduce the cannibal excess and the system would return to balance. But because the missionary excess is transported away more quickly, a pattern develops in which there is a near excess of cannibals and a far excess of missionaries. Moreover the distance between these zones of relative excess is determined by the interaction between the reaction and the diffusion: a length scale, which is what is required for the emergence of pattern from non-pattern, has emerged from the dynamics.

Where did the idea come from?

The key to making this idea work is the missionaries' bicycles: more technically that the inhibitor morphogen has a higher coefficient of diffusivity. Once the reaction-diffusion system is set up, a simple linear analysis makes this an obvious requirement for heterogeneity, but that was an analysis that no one, to Turing's knowledge, had done at that time. We have no record of Turing's thought process in developing the model idea and whether the diffusivity constraint came before or after the reaction-diffusion model itself. His analysis in Turing (1952) and Turing (1992)in terms of Fourier modes would have been second nature to him: for example his pre-war project to compute the zeroes of the Riemann zeta function using an analogue computer used a similar basic analysis. The formal theory in The Morphogen Theory of Phyllotaxis shares some structure with the then new quantum mechanics he had learned as a student in Cambridge, but presumably many of the techniques were in the armoury of any applied mathematician at the time. As Allaerts (2003) points out, Jeans’ 1927 book on Electricity and Magnetism, which Turing cited, is a source for many of the techniques, particularly spherical harmonics. But the source of the key scientific innovation is harder to pin down. It's not even obvious that Turing himself appreciated it that it was key: it is hardly emphasised in either Turing paper.

Turing was not alone in arriving at these ideas. Jean states that similar ideas were also introduced by Kolmogorov, Petrovski and Piskunov (1937), and Rashevsky (1940). Nanjundiah (2003) discusses the (lack of) influences in more detail. And the splendid Ralph Abraham's website, in an article on the mathematical formalism of Rupert Sheldrake's morphic resonance fields, mentions (but gives no reference for) as relevant the 1924 work of Roland Fischer, about which I know nothing. (Can't find that refeence now, but it is mentioned here

The 1952 paper actually dealt with a number of important and more complex issues usually glossed over in the standard undergraduate accounts. Turing also discussed tricky issues of mode selection and the effect of noise. He also extended the model to two dimensions to produce an example of dappling. It's hard to understand from his description exactly how the example was drawn, but it's possible to understand enough to see why Turing thought this was this least interesting of the cases. This paragraph will make more sense one day.

By the time of the drafting of The Morphogen Theory of Phyllotaxis, the theory had been developed yet further, particularly by a representation in terms of spherical harmonics, and by an application to the particular case of a sphere, done as an MSc project by Bernard Richards (1998). This more general theory, which has been recently reviewed by Allaerts (2003), is, though relatively technical, conceptually a fairly straightforward development of the original idea.