Patterns Drawn by Diffusion

Two Substances

In 1952 Turing published “The Chemical Basis of Morphogenesis”. He imagined a few substances in an embryo that react and diffuse, and called them morphogens.

In 1972 Gierer and Meinhardt wrote the idea as a pair of equations that is easy to read. An activator makes more of itself and more of an inhibitor; the inhibitor holds the activator down and spreads much faster.

Above the gel stand two profiles along one line. The activator’s peaks are narrow and tall, the inhibitor’s wide and low: each spot clears a ring around itself, and the next spot can only grow outside it.

Diffusion That Makes Pattern

Without diffusion, these equations have an even state that is stable: a little extra anywhere falls back. Diffusion alone only smooths. Put the two stable things together and the result can be unstable.

Write a tiny ripple as a sine wave of wavelength 2π/k, and how fast it grows can be computed exactly; the curve is called the dispersion relation. When the inhibitor spreads fast enough, the curve rises above zero over a band of k: ripples of those wavelengths grow, all others die away.

Now and then a front sweeps the gel and resets it to the even state plus 1% noise. A few seconds later the noise has grown into spots at the same spacing again.

The dispersion relation: growth rate for each wavenumber

The Spacing Is Computed

The wavenumber that grows fastest gives the pattern its spacing. With these parameters the prediction is 11.7 cells.

Each step of the white ruler on the gel is that predicted wavelength. The “measured” readout comes from a Fourier transform of the whole gel, taking its strongest wavelength, recomputed every two seconds.

The two are close but not equal. The linear analysis only describes ripples while they are small; once the spots are grown, nonlinear terms settle them at a slightly different spacing.

Spots or Stripes

The same equations can grow spots or stripes. Meinhardt pointed out that when the activator’s self-enhancement saturates, stripes are favoured.

The two halves of this gel differ in one parameter only: on the left the self-enhancement never saturates, on the right it saturates at high concentration. Spots on the left, winding stripes on the right, at almost the same spacing.

Diffusion and reaction speeds set the spacing; the nonlinearity sets the shape.

A Map of Parameters

In 1993 John Pearson swept the Gray–Scott model through two parameters on a Connection Machine: the feed rate F and the removal rate k. Each run was 256 × 256 cells, started from one small disturbance and ran 200,000 steps. He sorted what grew into 12 classes named by Greek letters: spots, mazes, spots that divide, travelling waves and more.

On the gel F grows from bottom to top and k from left to right, and the whole map runs at once. Inside the solid line on the figure, two more even states exist besides the trivial one with fuel only.

Every run starts from the trivial state (all fuel, no self-copier), and that state is stable against small ripples: the pattern is lit by a disturbance of finite size. Pearson noted himself that at the diffusion ratio he used there are no stable Turing patterns. The small dark patch on the figure is where the even state does turn Turing-unstable. Turing’s mechanism is not the only way to grow a pattern.

The Gray–Scott parameter plane

Evidence in a Gel

Thirty-eight years passed before anyone saw Turing’s mechanism in a chemical reaction. In 1990 Castets, Dulos, Boissonade and De Kepper ran the chlorite–iodide–malonic acid reaction in a polyacrylamide gel fed continuously from two sides, with starch to show the colour, and saw spots that did not change in time. They called it the first unambiguous experimental evidence of a Turing structure.

In 1991 Ouyang and Swinney saw hexagonal spots, stripes and mixtures of the two in a thin disc of gel in contact with a reservoir of the reagents.

The same year Lengyel and Epstein explained the key: starch binds iodide, the activator, into a complex that cannot move, which in effect slows the activator down. The gel’s pattern is computed with their model; without starch the reaction in this model oscillates as a whole and grows no standing pattern.

A Growing Fish

In 1995 Shigeru Kondo and Rihito Asai reported that the stripes of marine angelfish (Pomacanthus) are not fixed in the skin. Mammal patterns enlarge in proportion to the body; the angelfish keeps the spaces between its stripes by rearranging them continuously.

When the fish doubles in size its stripes do not double in width: new ones appear between the old. With a Turing-type model they predicted the patterns that came later.

The strip in the brass frame grows to twice its size, in length and width together, while diffusion and reactions stay the same. The readouts are the number of stripes and their mean spacing (in cells of the grown strip).

Cells, Not Molecules

The “activator” and “inhibitor” in a living body need not be two molecules. In 2009 Kondo’s group found that in zebrafish, black and yellow pigment cells push each other away at close range, while at a distance the black cells need the yellow ones to survive. Inhibition near, support far: the conditions of reaction–diffusion.

In 2012 and 2014 two studies found a Turing-type network behind the spacing of digits in the mouse limb bud: Bmp, Sox9 and Wnt, with the spacing tuned by gradients of Hox genes and growth factors. The ridges of the mouse palate were traced to the same kind of mechanism.

The strip in this chapter has stripes along the body, like a zebrafish. It is the same pair of equations from a different start.

The Leopard’s Tail

In 1988 James Murray wrote in Scientific American that a spotted animal can have a striped tail, but a striped animal cannot have a spotted tail.

The reason is geometry. A pattern that varies around the tail needs room for at least a whole wavelength. Where the tail’s circumference is less than 2π/k₊ (k₊ the top of the unstable band), every pattern that varies around it dies away and only variation along the tail is left: rings.

The glaze of the ceramic tail is computed on the tail’s own surface. The wide end has spots; at the red line the circumference equals 2π/k₊, and beyond it there are only rings. Murray also pointed out that a pattern appears only when the embryo is the right size: too small and none forms, and very large animals are plain again.