Mathematical Biology 1952

The Chemical Basis of Morphogenesis

Alan M. Turing

Two chemicals, reacting and spreading, can turn a uniform tissue into spots and stripes.

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In depth · the introduction

Before an embryo has any spots, stripes or limbs, it is a nearly featureless ball of identical cells. This paper explained how it could pattern itself — with no blueprint at all.

The idea, unpacked

Living things are full of regular patterns: the even spacing of a leopard's spots, a zebra's stripes, the bristles on a fly, the buds set around a stem. Where does the spacing come from, if the cells all start the same? Turing's answer was that a pattern can build itself out of chemistry.

Picture two chemicals drifting through a tissue and reacting as they go. One encourages itself and seeps only a short way; the other, which the first also creates, spreads much farther and shuts the first one down. The surprise is what they do together. Diffusion, which normally blurs differences into a uniform smear, here does the opposite — it sharpens tiny accidental bumps into a regular, repeating pattern. Turing proved this was possible and called it a diffusion-driven instability.

Where it came from

Alan Turing wrote this in Manchester in 1952, a few years after his wartime code-breaking and his founding work on the computer. Turning from machines to living form, he used one of the world's first electronic computers, the Ferranti Mark I, to calculate a dappled pattern — one of the earliest computer simulations in all of biology. It was among his last works. Prosecuted in 1952 for being homosexual and forced to undergo hormone treatment, he died in 1954, the project unfinished.

Why it mattered

It gave biology a way to get structure for free. You do not need a tiny architect placing each spot; one simple chemical rule, repeated everywhere at once, lays down the whole pattern by itself. This idea — that order can emerge from uniform beginnings, that a system can organise itself — became a foundation of how we understand development, and it reaches far beyond animal coats.

A way to picture it

Think of a rumour in a crowd. A juicy rumour makes more of itself and passes to the people right beside you — a short-range booster. But it also sets off a faster-travelling denial that races ahead and quiets the rumour further away — a long-range damper. Start from a calm, uniform crowd and you do not end up with one even murmur; you get pockets of believers, spaced regularly apart. Swap ‘rumour’ for ‘pigment,’ and you have the makings of a leopard.

Where it sits

Turing appears three times in this Library, each time at a beginning. He defined computation itself (1936) and asked whether machines can think (1950); here he asks how a body builds its shape. Where Darwin (1859) explained why forms change across generations and Mendel (1866) how traits are passed on, Turing asked how a single fertilised cell physically becomes a patterned organism. His reacting, spreading chemicals echo the predator–prey cycles of Lotka and Volterra (1926) — the same mathematics of things that make and consume one another.

The original document

Original source text

A. M. Turing · Philosophical Transactions of the Royal Society of London B, vol. 237, no. 641, pp. 37–72 · 14 August 1952

Abstract

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.

Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances.

A model of the embryo · morphogens

These substances will be called morphogens, the word being intended to convey the idea of a form producer.

This model will be a simplification and an idealization, and consequently a falsification.

Turing opens by stating his aim and his method: not to model any one organism in detail, but to ask whether the simplest possible chemistry — substances that react and diffuse — can in principle generate biological form. He is candid that the picture is idealised, and treats the cell as a vessel of reacting morphogens whose concentrations vary in space and time.

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The breakdown of symmetry and homogeneity

The heart of the paper is an argument that the spatially uniform state can be unstable. Turing linearises the reaction–diffusion equations about a homogeneous equilibrium and follows how a small disturbance grows. Crucially, diffusion — ordinarily a force that smooths differences away — is shown to be capable of doing the opposite when two morphogens diffuse at different rates: it can amplify disturbances of a particular size.

Reaction and diffusion on a ring of cells

To make the analysis concrete he studies a ring of discrete cells and, equivalently, a continuous medium, decomposing any disturbance into its spatial waves. Each wavelength grows or decays on its own; a band of them grows, and the fastest-growing wave fixes the spacing of the emerging pattern. Turing catalogues six qualitatively distinct outcomes, from a return to uniformity, through stationary waves of definite wavelength — the spots-and-stripes case — to oscillations both standing and travelling.

A computed dappled pattern

Going beyond the linear analysis, Turing integrates the full nonlinear equations numerically on the Ferranti Mark I computer at Manchester to produce a two-dimensional ‘dappled’ pattern — among the earliest computer simulations of a biological process. He also sketches how the same instability might govern gastrulation and the spiral, Fibonacci-numbered arrangement of leaves (phyllotaxis).

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The full thirty-six-page memoir — with its linearised equations, its catalogue of the six behaviours, the ring-of-cells calculation, and the computed dappled figure — is available in full at the source below.

Manchester · 1952