The Turing instability has been a popular theme in mathematical biology. There is no doubt that it is nice mathematics but how much does it really explain in biology? Recently a detailed proposal was made by researchers from the Max Planck Institute of Immunobiology and the University of Freiburg to explain the development of hairs in mice on the basis of a Turing mechanism. (S. Sick, S. Reinker, J. Timmer and T.Schlake. WNT and DKK determine hair follicle spacing through a reaction diffusion mechanism. Science 314, 1447.)
Here I want to take this as a stimulus to think again about the status of these ideas. On my bookshelf at home I have the biography of Turing by Andrew Hodges (‘Alan Turing: the enigma’). I now reread the parts concerning biology, which are not very extensive. On the basis of thistext it seems that one of the things in biology that fascinated Turing most was the occurrence of Fibonacci numbers in plants. This seems to have little to do with the contribution to biology for which he became famous. He himself seems to have hoped that there would be a connection. I looked at the original paper of Turing (‘The chemical basis of morphogenesis’) but I did not learn anything new compared to modern accounts of the same subject I had seen before. The basic mathematical input is a system of reaction-diffusion equations, as described briefly in a previous post. A homogeneous steady state is considered which is stable within the class of homogeneous solutions. Then a growing mode is sought which describes the beginning of pattern formation. This is similar to what is done for the Keller-Segel system. There is a mouse in Turing’s paper but it has nothing to do with the mouse in the title of this post. Its role is to climb on a pendulum and thus illustrate ideas about instability.
Another book I have at home is ‘Endless forms most beautiful‘ by Sean Carroll. In this book, which appeared in 2005, the author explains recent ideas about embryonic development and their connections to the evolution of organisms on geological timescales. Turing is mentioned once, on p. 123, but only to dismiss the relevance of his ideas to embryology, the central theme of his paper. Carroll writes, ‘While the math and models are beautiful, none of this theory has been borne out by the discoveries of the last twenty years’. As a remaining glimmer of hope for the Turing mechanism, the diagrams on pp. 104-105 of the book might fit a Turing-type picture but concern small-scale structures. The large-scale architecture of living bodies is claimed to arise in a quite different way. The picture of the development of individual organisms presented by Carroll has a character which strikes me as digital. I do not find it attractive. I should emphasize that this is an aesthetic judgement and not a scientific one. I suppose I am just in love with the continuum.
Now I return to the article of Sick et. al. A key new aspect of what they have done in comparison to previous attempts in this direction is that they are able to identify definite candidates for the substances taking part in the reaction-diffusion scenario and obtain experimental evidence supporting their suggestion. These belong to the classes Wnt and Dkk (Dickkopf). An accompanying article by Maini et. al. (Science 314, 1397) is broadly positive but does also add a cautionary note. It is pointed out that similar predictions can be produced by different mathematical models. A model of Turing type may produce something that looks a lot like what is provided by a model involving chemotaxis. This is a generic danger in mathematical biology. In a given application it is important to be on the lookout for experimental data which can help to resolve this type of ambiguity.
The reaction-diffusion model used for modelling and numerical simulation is related to a classical model of Gierer and Meinhardt. The original paper from 1972 and a great deal of information on related topics are available from this web page.