In this work we will discuss on the contribution made by Alan Turing (1912-1954) towards a mathematical foundation of Developmental Biology. To do so, we will briefly review the approach he laid out in his only published work on the subject, and then describe the impact of his work on Mathematics on one hand, and on Biology on the other.

En este artículo debatiremos la contribución hecha por Alan Turing (1912-1954) a la fundamentación matemática de la Biología del Desarrollo. Para ello, repasaremos brevemente su punto de vista en el único trabajo que publicó sobre este tema, y describiremos su impacto tanto en las Matemáticas como en la Biología.

Alan Turing published a single paper on what we now call Mathematical Biology (Turing,

In his work (Turing,

It has been long suspected before Turing that chemical substances should be instrumental in morphogenesis. Such assumption is however made precise in the very first lines of (Turing,

And now a key technical point is stated:

In Turing´s approach simplicity is crucial. A few lines below, our author makes clear that he only intends to make use of a few basic physical principles to address his ambitious goal:

Simple his proposal may be, but mastering it requires some background, which is precisely described:

These few remarkable lines encode the gist of Turing´s ideas on morphogenesis. Chemical species, which mediate the shaping of unfolding living patterns (such as organs or limbs), are postulated to operate according to a number of physical mechanisms, which are reduced to a bare minimum. In fact, only two such principles are discussed in detail. The first of them is diffusion, which accounts for random, unbiased molecular motion at the microscopic scale. The second is represented by chemical reactions, whereby new molecular substances are generated from the interactions between existing ones. It was shown in (Turing,

As recalled before, biological patterns are proposed to be the consequence of instabilities arising from a homogeneous state, which was initially stable from the point of view of reaction kinetics (described mathematically in terms of ordinary differential equations, ODEs). Such instabilities are a result of (arbitrarily small) random disturbances which introduce heterogeneity in such medium. This in turn induces diffusion (represented in mathematical terms by partial differential equations, PDEs). Diffusion is a mass transport mechanism, which in an attempt to suppress heterogeneity, moves chemicals from regions where their concentrations are higher (with respect to the global spatial average) to others where they are lower. Oddly enough, and this was one of Turing’s significant mathematical contributions (Turing,

Turing was well aware that his suggestion that symmetry breaking could be induced by arbitrarily small random disturbances might (and actually did) look surprising at first glance. In (Turing,

Turing´s answer to his own carefully worded statement goes as follows (Turing,

We have already noticed that the seemingly counterintuitive fact that symmetry breaking could be induced by any small, random perturbation is supported in (Turing,

Turing´s description of the initial stages of Pattern Formation in an embryo as a diffusion-induced destabilization of an initially stable, homogeneous steady state instantly appeals to a mathematically-minded reader for various reasons. It is elegant, simple, looks quite general and can be illustrated by means of elementary arguments. However, it just looks too good to be true, and the limitations of his approach were apparent to Turing himself. As a matter of fact, in page 37 in (Turing,

Indeed, the Mathematics behind Turing´s arguments were known to their author to be insufficient to fully achieve his purpose. A key technical limitation in his work is that significant results could only be obtained under the assumption of linearity in the equations involved. This is plainly declared in a final Section 13 of (Turing,

The last goal is acknowledged to be out of reach by the author at that time (1952).

(Turing,

It may well be guessed from the previous sentences that Turing clearly foresaw the impending power of computers, a technology that he himself contributed to develop, as a tool to address biological problems, developmental and otherwise. However, state-of-the-art computers in 1952 where unable to yield any scientific breakthrough in a mathematically-based theory of Pattern Formation.

How was Turing´s work received at his time? It would be fair to say that his proposals remained largely forgotten during the fifties and the sixties of previous century. Several facts conspired to that effect. To begin with, his untimely death in 1954, only two years after (Turing,

Indeed, only one year after (Turing,

It could well be that Crystallography had been grounded on Mathematics and Physics since its very beginning, and that Neurosciences were ripe enough to benefit from mathematical tools while Developmental Biology was not. In a way, the question was related to that of selecting a problem which was at the same time important and within reach of already existing mathematical techniques. It is interesting in this respect to read what Francis Crick, a physicist turned biologist, revealed to Lewis Wolpert, a distinguished developmental biologist, during a scientific interview. Their dialogue went as follows:

(see Wolpert and Richards,

In the context of morphogenesis, one had to wait for twenty years until a further significant step with respect to (Turing,

A key point is now immediately made:

As a striking difference with the situation in 1952, the basic nonlinear activator-inhibitor proposed in Gierer and Meinhardt (

As a matter of fact, there are understandable reasons for this shift from theory to simulations. It is beyond question that biological processes are exceedingly complex, and therefore most models, if manageable, have to keep to a few assumptions and disregard the rest, including obviously those that remain unknown at the time. To compare various modelling alternatives, there seems to be no better way than to check what their predictions are, and computational methods help in this respect more than any other mathematical technique. In particular rigorous analysis, with its characteristically long time scale to achieve results, and the difficulties inherent to its own method, is no match for educated computer guesses when it comes to exploring a largely unknown field. This has contributed to widening the gap between theory and simulations that shows no signs as yet of being filled.

In the years since Gierer and Meinhardt (

While the interest in using mathematical models in Biology has been steadily increasing among mathematicians and physicists ever since (and arguably before) the publication of (Turing,

To begin with, many biologists consider living beings to be simply too complex to be amenable to anything close to mathematical modelling, whose realm should be accordingly confined to that of inorganic matter. This belief runs deep both in science and philosophy since the very origins of scientific thought. No less a master than Aristotle (ca 384 BC - 322 BC) pronounced Mathematics valid only to deal with immaterial things with these terse words:

(see Aristotle,

An illuminating view on the stance taken by many biologists with respect to Mathematics is given by Evelyn Fox Keller in (Keller,

Moreover, to summarize the exchanges between Rashevsky and Wilson, E. Ponder, director of CSH at that time wrote:

These words, written less than twenty years

A bit later in Gordon and Beloussov (

Despite a general unfriendly welcome, the use of Reaction-Diffusion models to address biological problems has slowly gained ground as time has elapsed. Even hardliners nowadays consider it appropriate to refer to it, although not without due reservations. For instance, in the book

However, a number of molecular agents have been identified as candidates to mediate such processes, and their number continues to increase (Meinhardt,

Despite the obstacles found, some of which have been briefly addressed here, there is currently a larger interaction between biomedical scientists, mathematicians and physicists that has ever been in the past. It is beyond the scope of this note to report on some of the exciting scientific frontiers where this collaboration is already bearing fruit. Of course, the sheer complexity of living beings represents a formidable barrier that has to be overcome case by case for this collaboration to succeed. That was certainly foreseen in the concluding sentence in (Turing,

There is a long way to go following the path laid out in Turing´s celebrated paper. This seems to be a task for many generations of scientists, and it appears to be currently well on its way.

This work originated from a lecture given at the symposium “The Alan Turing legacy”, held at Fundación Ramón Areces in Madrid, in October 2012. The author is grateful to the organizers for their invitation to participate in that event. Partial support from MINECO Grant MTM2011-22656 is also acknowledged.

The following argument, taken from pages 42 and 43 in Turing (

Consider two cells I and II, and two substances (morphogens) X, Y present on them (c.f. Figure A below).

Chemical reactions will be assumed among X and Y of a linear nature and given by

(A1)

(A2)

Note that if both morphogens have concentrations X = Y = 1 in both cells, there is equilibrium of a stable nature, as can be seen from the study of the eigenvalues associated to (A1), (A2). Suppose now that, due perhaps to some fluctuation, the initial values in I and II turn out to be:

(A3)

According to (A1), (A2), X and Y are produced by chemical action at rates 0.18 and 0.22 in the first cell, and destroyed at the same rate in the second. As a consequence of the heterogeneity introduced by (A3), a diffusion mechanism sets in. Suppose that flow due to diffusion from the first cell to the second occurs at a rate 0.5 for the first morphogen , and 4.5 for the second (Note that the ratio of the second diffusivity to the first one is 9, about one order of magnitude). The combined reaction-diffusion process may now be represented by the following equations:

(A4)

(A5)

Let ξ(t) be the perturbation induced in each morphogen starting from an initial value ξ(0)=0.02 in (A3). It then follows from (A4), (A5) that for any t>0,

(A6)

so that an exponential drift away from equilibrium is induced by the initial perturbation (A3). Note that negative values for morphogen concentrations should be discarded. Actually, when application of the previous formulae result in the concentration of a morphogen in a cell becoming negative, it should be understood that it is instead removed only at the rate at which it is reaching that cell by diffusion.