by Nadia Pisanti

What do a leopard skin and a sunflower head have in common? Very little, should they not both exhibit patterns whose generation is the result of the interaction of components, the morphogens, according to mathematical laws described sixty years ago by Alan Turing.
Patterns have always been observed in nature, and Turing certainly was not the first to detect mathematical models and geometrical schemes in, for example, plants’ phyllotaxis. In the days of Turing, half a millennium had passed since the geometric studies of Leonardo da Vinci on plants [1], and D'Arcy Thompson Wentworth seemed to have almost exhausted, with his book On Growth and Form [2] in 1917, whatever could be mathematically told about morphogenesis, that is, the biological process of the creation of shapes in living organisms.

When, in 1952, Alan Turing wrote The Chemical Basis of Morphogenesis [3], the morphogenesis was that of D’Arcy Thompson Wentworth, whose contribution was mainly that of highlighting the importance of mechanisms and physical laws in determining certain shapes in living beings. In his book [1], D'Arcy Thompson lists an enormous number of correlations among observed shapes in living organisms, mechanical phenomena, and their corresponding mathematical models. The contribution of Alan Turing to morphogenesis is less extensive than that of Thompson, but no less important, as it exhibits several innovative ideas. First of all, as the title of his paper (The Chemical Basis of Morphogenesis) suggests, it moves the attention from the mechanics of morphogenesis to the chemistry of its components. Such components, the morphogens, are the actual agents of a system in which they co-operate for the mechanisms of morphogenesis.

Morphogens are substances that diffuse, and thus propagate, depending on their concentration, signals that control cell differentiation. Turing gives precise mathematical models for the fluctuations of these chemical quantities and applies the corresponding laws. The model he suggests was a novelty for the theory of morphogenesis (Turing introduces it as “mathematically convenient, though biologically unusual”): a continuous system of differential equations representing non-linear dynamics. With it, Alan Turing captures his brilliant intuition that patterns observable in nature, with their repetitiveness and variability, are the result of the interaction between chemical substances whose concentrations have iterative reciprocal effects. Such effects are, at each iteration, qualitatively similar but quantitatively slightly different, thus driving the pattern creation.

Turing describes his model as a reaction-diffusion system: starting from a conformation determined by the quantities of the chemical components in an initial state, it evolves by changing these quantities according to mathematical laws. Within these fluctuations, the system can reach various kinds of stable equilibria, and stability is due to a substantial symmetry. Breaking this symmetry leads to a (continuous, not discrete) transition from one stable state to another. Turing observes that the variety of irregularities that can break the symmetry is large and hard to investigate, but that fortunately the variety of equilibrium states that can be reached is limited to six. In his paper Turing analyses them all, outlining hypotheses on the kind of morphogenesis they can possibly give rise to. In particular, Turing’s non-trivial observation is that in a system with several components, the diffusion of a morphogen in space can itself be a cause of instability of the global system, whereas, in the local system of each single component there would otherwise have been stability.

In the conclusion of [3], Turing modestly talks about the “relatively elementary mathematics used in this paper”. This is unfair: among the remarkable aspects of his model, there is the pioneering notion of exponential drift that he uses for the onset of instability. Turing observes that “a drift away from the equilibrium occurs with almost any small displacement from the equilibrium condition”: a very similar concept to what later (after Ruelle's work in the seventies [4]) will be named sensitivity to initial conditions.

Earlier, in [5], Alan Turing had explained discrete-state machines as “machines which move by sudden jumps or clicks from one quite definite state to another”. By adding a few lines later that “strictly speaking there are no such machines. Everything really moves continuously”, he somehow anticipated the motivations of his work on morphogenesis. The inventor of the discrete-state machines had caught, besides their potential, their limits for modelling physical processes. With the reaction-diffusion system he describes in “The Chemical Basis of Morphogenesis”, Alan Turing somehow fills this gap.

This extraordinary scientist, whose polyhedral abilities are reminiscent of those of Leonardo da Vinci, has made fundamental contributions in very distinct scientific disciplines, with discoveries based on very different concepts, such as discrete and continuous. Maybe the key of his genius lies precisely in the depth of the observations that can emerge by investigating and comparing such opposite concepts.

[1] Leonardo da Vinci, manuscript “Studi di geometria, ritratto di pianta in fiore, e studi di saldatura”
[2] D.W. Thompson. “On growth and Form”, Cambridge University Press, 1917.
[3] A.M. Turing. “The Chemical Basis of Morphogenesis”, Philosophical Transactions of the Royal Society of London 237(641), 37-72, 1952.
[4] D. Ruelle, F. Takens, “On the nature of turbulence”. Communications in Mathematical Physics 20(3), 167-192, 1971.
[5] A.M. Turing, “Computing Machinery and Intelligence” Mind 49, 433-460, 1950.

Please contact:
Nadia Pisanti,
University of Pisa, Italy and Leiden University, The Netherlands
E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

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Next issue: January 2023
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