Prof

by Dermot Turing

  I am afraid I am very far from the stage where I feel inclined to start asking any anatomical questions. According to my notions of how to set about it that will not occur until quite a late stage when I have a fairly definite theory about how things are done.

  At present I am not working on the problem at all, but on my mathematical theory of embryology, which I think I described to you at one time. This is yielding to treatment, and it will so far as I can see, give satisfactory explanations of –

  i) Gastrulation.1

  ii) Polygonally symmetrical structures, e.g., starfish, flowers.

  iii) Leaf arrangement, in particular the way the Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13, …) comes to be involved.

  iv) Colour patterns on animals, e.g., stripes, spots and dappling.

  v) Patterns on nearly spherical structures such as some Radiolaria, but this is more difficult and doubtful.

  I am really doing this now because it is yielding more easily to treatment. I think it is not altogether unconnected with the other problem. The brain structure has to be one which can be achieved by the genetical embryological mechanism, and I hope that this theory that I am now working on may make clearer what restrictions this really implies. What you tell me about growth of neurons under stimulation is very interesting in this connection. It suggests means by which the neurons might be made to grow so as to form a particular circuit, rather than to reach a particular place.

  Yours sincerely,

  A.M. Turing

  Alan Turing was trying something nobody had done before: to explain, in mathematical terms, the reasons for patterns seen in organisms. In February 1951 Alan wrote to his former colleague Mike Woodger at the NPL:

  Dear Woodger,

  Our new machine is to start arriving on Monday. I am hoping as one of the first jobs to do something about ‘chemical embryology’. In particular I think one can account for the appearance of Fibonacci numbers in connection with fir-cones.

  Yours,

  A.M. Turing

  Another professor may have had a hand in the changing direction of Alan’s work. M.H.A. Newman had also moved back from the development of computing machinery as an end in itself. He was turning his attention back to topology, which was the subject of a textbook he had written in 1939. Topology is about shapes, spaces and surfaces, knots and folding. The maths involved is hard, but the subject is very visual, and it’s about multi-dimensional spatial problems.

  Together with Bertrand Russell, Newman was also pushing for Alan’s election to a fellowship of the Royal Society. Alan was elected in March 1951 on the basis of his Computable Numbers paper. Congratulations came in by every post: from Mr Darlington, his headmaster from Hazelhurst, from Sir Charles Darwin and from Sir Geoffrey Jefferson. Mother sent a Greetings Telegram: ‘– GREETINGS ALAN TURING FRS HOLLYMEADE ADLINGTON ROAD WILMSLOW = LOVING CONGRATULATIONS WELL DESERVED HONOUR = MOTHER +++’ Alan also wrote to Philip Hall, his former tutor at King’s in Cambridge, thanking him for his congratulations.

  It is very gratifying to be about to join the Olympians.

  The ‘waves on cows’ is just an example in my mathematical theory of embryology which I am busy on now. ‘Waves on leopards’ are rather more elementary. A leopard skin, before the spots arrive is supposed an infinite thin sheet containing two chemical substances with concentrations U, V which react and diffuse. [Alan then sets out some equations.] What particular solution you get will depend on random disturbances just before instability started. Roughly speaking you get a random solution. By assuming there is black where Z > Z0, yellow for Z < Z0 you get very reasonable leopard skins. Certain slight variations of assumptions give you giraffes, zebras, cows. Cows are dappled. […]

  I hope I am not described as ‘distinguished for work on unsolvable problems’.

  A horse is not spherically symmetrical

  Living things tend to start life as formless, spherical objects, like fertilised egg-cells or globs of embryonic cells. But as they grow and develop, shape and differentiation appear – the process of morphogenesis. How can that be? How would the embryo know to make itself symmetrical for some things (left and right) and to limit the number of things like arms and fingers, and yet allow blotchy patterns on animal skins? To answer these questions needed a theory. Rudyard Kipling’s Just So Stories, with their nonsense explanations for how the elephant got its trunk and how the leopard got its spots, had been nursery reading at Baston Lodge. Alan Turing knew about the problem from the very beginning.

  An embryo in its spherical blastula stage has spherical symmetry, or if there are any deviations from perfect symmetry, they cannot be regarded as of any particular importance, for the deviations vary greatly from embryo to embryo within a species, though the organisms developed from them are barely distinguishable. One may take it therefore that there is perfect spherical symmetry. But a system which has spherical symmetry, and whose state is changing because of chemical reactions and diffusion, will remain spherically symmetrical for ever. It certainly cannot result in an organism such as a horse, which is not spherically symmetrical.

  Now, Alan thought he had the solution. Tiny fluctuations in the concentration of biochemicals could make a difference. The trick is to introduce some instability, which, after a period, restabilises. This could be done by imagining two different biochemicals – morphogens – which were produced, diffused, and destroyed, at different rates. Now Alan could bring on the equations. There was only one problem. Equations would frighten the biologists. It would not be wise to assume that the biologists would all be as receptive as the Ratio Club members or J.Z. Young to being kept in order by mathematicians like A.M. Turing. So, in his paper, Alan would have to present both the key – the maths to demonstrate rigour – and also the decrypt, to convince the biologists.

  Certain readers may have preferred to omit the detailed mathematical treatment of §§ 6 to 10. For their benefit the assumptions and results will be briefly summarized, with some change of emphasis. The system considered was either a ring of cells each in contact with its neighbours, or a continuous ring of tissue. The system was supposed to be initially in a stable homogeneous condition, but disturbed slightly from this state by some influences unspecified, such as Brownian movement or the effects of neighbouring structures or slight irregularities of form. It was supposed also that slow changes are taking place in the reaction rates (or, possibly, the diffusibilities) of the two or three morphogens under consideration. These might, for instance, be due to changes of concentration of other morphogens acting in the role of catalyst or of fuel supply, or to a concurrent growth of the cells, or a change of temperature. […] The conclusions reached were as follows. After the lapse of a certain period of time from the beginning of instability, a pattern of morphogen concentrations appears which can best be described in terms of ‘waves’. There are six types of possibility which may arise.

  (a) […] This is the least interesting of the cases. It is possible, however, that it might account for ‘dappled’ colour patterns, and an example of a pattern in two dimensions produced by this type of process is shown in figure 2 for comparison with ‘dappling’. […]

  (d) There is a stationary wave pattern on the ring, with no time variation, apart from a slow increase in amplitude, i.e. the pattern is slowly becoming more marked. In the case of a ring of continuous tissue the pattern is sinusoidal, i.e. the concentration of one of the morphogens plotted against position on the ring is a sine curve. The peaks of the waves will be uniformly spaced round the ring. […] Biological examples of this case are discussed at some length below.

  Alan’s examples caused him some trouble, because ‘isolated rings of tissue are very rare’; but he pointed to the tentacles of the freshwater polyp Hydra and the leaves of the woodruff plant as capable of being explained by his theory. What Alan’s paper discussed next was ‘chemical waves on spheres’. This was the problem of gastrulation – how a ball of embryonic cells folds in on itself – the beginning of tur
ning a sphere into a horse.

  The treatment of homogeneity breakdown on the surface of a sphere is not much more difficult than in the case of the ring. The theory of spherical harmonics, on which it is based, is not, however, known to many that are not mathematical specialists. Although the essential properties of spherical harmonics that are used are stated below, many readers will prefer to proceed directly to the last paragraph of this section.

  You bet they would.

  The operator 2 will be used here to mean the superficial part of the Laplacian, i.e. 2V will be an abbreviation of

  where θ and ϕ are spherical polar co-ordinates on the surface of the sphere and ρ is its radius.

  The computing of morphogenesis

  With his new status as a fellow of the Royal Society, Alan’s ground-breaking paper on The Chemical Basis of Morphogenesis was published in the Philosophical Transactions of the Royal Society in August 1952. Within days of publication, he received this from C.H. Waddington, Professor of Genetics at the University of Edinburgh:

  I was extremely interested to read your recent paper on the chemical basis of morphogenesis. It is very encouraging that some really competent mathematician has at last taken up this subject. Although parts of your discussion were rather above my head, I found the general arguments extremely interesting and suggestive.

  Illustrations to Alan Turing’s 1952 Royal Society paper on The Chemical Basis of Morphogenesis, showing how a reaction-diffusion mechanism can build up localised concentrations of morphogens in a wave pattern, and explain dappling.

  I rather doubt, however, whether the kind of processes with which you were concerned play a very important role in the fundamental morphogenesis which occurs in early stages of development. Even in a case like the regeneration of tentacles in Hydra the final result seems to me more regular than one would expect from your type of mechanism.

  The most clear-cut cause of your type of mechanism seems to me to 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 skin of tigers, leopards, etc.

  Although, as predicted, the maths was going to defeat the biologists, they had understood the conclusion clearly enough. Reaction-diffusion and maths could explain some of the basic observations about development of living organisms. And, what was more, it might be possible to go beyond the mere theory by using the Manchester computer to try out some numerical examples. That was going to be the next stage in Alan Turing’s work.

  The computing potential of the Manchester Mark 1 machine deserved its own conference, which took place in July 1951. Maurice Wilkes and Tommy Flowers attended, and a variety of papers were presented; consistent with the ambitions of Newman and Turing, these weren’t entirely focused on technicalities of hardware design, logic or programming. Newman himself put forward ideas, which seem to have been rooted in the Bletchley experience, that automatic computers made new approaches to mathematical problems possible. For example, that a random walk ‘Monte Carlo’ method could be used to solve partial differential equations, and that ‘probability methods might throw at any rate a feeble light on problems so large that rigorous methods leave them in complete darkness’.

  The final paper was one of great significance, presented by J.M. Bennett and J.C. Kendrew. As with Alan’s developing ideas on morphogenesis, Bennett and Kendrew were exploring the zone where biology and computing might cross over. Bennett was Wilkes’s first research student, who had been programming the EDSAC since its first days; Kendrew went on to win a Nobel Prize for unravelling the atomic structure of myoglobin using X-ray crystallography. Their paper at the Manchester computing conference described how you could get to the structure from the pattern of spots on an X-ray film, via a program run on the EDSAC. What they were describing was how to use the computer to create a ‘contour map’. The contour map was exactly what Alan wanted the Manchester computer to produce to help him with the next stage in his work on the mathematics of organism development. Having set out the basics in his theory of morphogens, he was now going to tackle the remaining parts of the agenda he had set himself in that letter to J.Z. Young.1 This was a much tougher assignment: to set out the mathematical theory underlying the shapes of plants.

  How did this happen? Alan’s exasperation at a problematic routine. But the computer was not the only machinery going wrong for Alan Turing at the beginning of 1952.

  The first few months of 1952 buzzed with activity. In January Alan’s conversation with Sir Geoffrey Jefferson, M.H.A. Newman and Professor Braithwaite of King’s was broadcast. The first draft of The Chemical Basis of Morphogenesis came back from the reviewers with comments, and had (as is usual with scientific papers) to be revised and resubmitted. On 8 February Alan was to go to London to present his (as yet unpublished) work on reaction-diffusion models of morphogenesis to the Ratio Club, explaining how it could be developed through computer modelling. But, despite their apparent importance, none of these things was foremost in Alan Turing’s life at that time. Alan Turing was an established logician, mathematician, and computer scientist. He had just created an entirely new science applying mathematics to developmental biology. He had been given an OBE for his secret war work and elected to a fellowship of the Royal Society. Yet, on the day before he was to give the talk at the Ratio Club, Alan Turing’s entire world was turned upside down, because Alan Turing had met a young man in a pub.

  Notes

  1 Cathode Ray Tube – see Chapter 8

  1 Manchester University Computer

  1 Another codebreaker from Bletchley Park days

  1 The assumption of a cup shape by an embryo

  1 See here

  10

  MACHINERY OF JUSTICE

  JOHN TURING wrote that ‘if the episode of “the burglar” had not proved, ultimately, so fatal to Alan, I suppose this might have been regarded as farcical’. Indeed, it was tragedy, not farce. The facts were set out in the police statement which was read out in the Wilmslow Magistrates’ Court on 27 February 1952.

  Detective Constable Robert Wills said he went to Turing’s home with Detective Sergeant Rimmer on February 7. He said to Turing: ‘On February 3 you visited Wilmslow Police Station and gave information about two men, who you alleged, had broken into your house. We have made inquiries, and now have some information. Would you please give us his description?’ Turing replied: ‘He’s about 25 years of age, 5 ft. 10 inches, with black hair’. Constable Wills said: ‘We have reason to believe your description is false. Why are you lying?’

  The episode of the burglar

  Alan’s house had indeed been burgled. Alan had lost ‘2 medals, 3 clocks, 2 shavers, 2 pairs of shoes, 1 compass, 1 watch, 1 suitcase, 1 part bottle of sherry, 1 pair of trousers, 1 shirt, 1 pullover, and 1 case of fish knives and forks together of the value of £50.10.0.’ The worst of these was the watch. Julius Turing, who had died in 1947, hadn’t left riches to his sons, but he had bequeathed ‘To my younger son the said Alan Mathison Turing the Gold Watch which I inherited from my father’. This wasn’t some wristwatch (Alan’s wristwatch wasn’t stolen, presumably because he was wearing it; in fact, it is now in the museum at Bletchley Park) but a half-hunter, and its heirloom status meant that Alan couldn’t ignore the break-in. Accordingly, on 20 February 1952, one Harold Arthur Thacker had been committed to stand trial for the burglary at the forthcoming Knutsford Quarter Sessions. The following week, on 27 February, the court was not concerned with the burglary. It was concerned with Alan Turing.

  An Affair. Turing, it was alleged, replied: ‘I tried to mislead you about my informant. I have been an accessory to an offence in this house. I have had an affair with him and I have regarded his conduct as a form of blackmail and have consulted my solicitor about him. His name is Arnold Murray. I picked him up in Oxford Street, Manchester.’ Constable Wills read out a statement alleged to have been made by Turing in which he said he had committed an offence at his home with Murray.
In an alleged statement Murray said he met Turing in Oxford Street and ‘knew what he was by the way he talked’.

  If Thacker was the burglar, who was Murray, and what on earth was going on? Murray was the young man Alan had met in a pub in the Oxford Road and brought home to Wilmslow. Murray had presumably tipped off Thacker about the riches strewn around in Alan’s house. In talking to the police about the burglary, Alan had told the constable how he thought he knew the identity of the thief – one of Murray’s unpleasant friends – and had tried to protect Murray from liability as an accessory with the false description. But now the police were far more interested in why Alan should be harbouring low-lifers like Murray. They were bound to ask themselves what possible reason there could be for Murray to be in Alan’s house in the first place. And the possible reason made Alan an accomplice to whatever Murray had done together with Alan.

  In February Alan wrote to Norman Routledge in reply to his jokey letter about Christmas cards and Norman’s career:

  My dear Norman,

  I don’t think I really do know much about jobs […]. However I am not at present in a state in which I am able to concentrate well, for reasons explained in next paragraph.

  Norman Routledge while at King’s in the early 1950s. One of Alan’s more flamboyant friends, he later went on to become a revered maths teacher at Eton.