Chapter 03 · Reaction–diffusion

Spots and stripes

Pour two chemicals into a dish and let them spread. You would expect them to blend into an even grey. Under the right conditions, they paint spots instead.

In 1952 Alan Turing, better known for code-breaking and the foundations of computing, published a paper called The Chemical Basis of Morphogenesis. It asked how an embryo, which starts as a roughly uniform ball of cells, comes to have structure. Turing coined the word morphogen for a chemical that could carry that kind of information. Then he showed that a few morphogens that react and spread through tissue can break an even starting state into a regular pattern on their own.

The odd part is that the spreading helps. Spreading by diffusion, molecules drifting from where they are crowded to where they are scarce, is why a drop of ink fades into a glass of water. Turing saw that two chemicals diffusing at different speeds can do the opposite.

The rule

The model in Figure 1 comes from chemistry. Peter Gray and Stephen Scott studied it in the 1980s as a reaction in a stirred tank, and in 1993 John Pearson put it on a grid and catalogued the patterns it makes. It is not a model of any animal.

There are two chemicals, U and V, and three things happen at every point of the dish:

V feeds on U and makes more of itself. When one U meets two V, it becomes a third V: U + 2V → 3V. A reaction like this, whose product speeds up its own making, is called autocatalytic.

Fresh U is fed in at a steady rate, the feed F, topping each point back up toward full.

V is removed at a steady rate, the kill k, plus the feed rate, as if the dish were being drained as it is refilled. Left alone, V dies out.

Both chemicals also spread to their neighbours, and U spreads twice as fast as V. Each square looks only at itself and its eight neighbours. That is the whole model.

no V a little more most V V growing

feed kill

feed F—

kill k—

pattern—

steps0

grid—

Figure 1. A dish of chemical U with a few drops of chemical V, running the Gray–Scott model. Colour shows how much V there is. Click or drag to add more V. The preset buttons load settings for different patterns, and the feed and kill sliders move between them. The dish wraps around at its edges.

What the dish does

On the default setting, Mitosis, each drop swells into a spot. The spot stretches, pinches in the middle and splits in two, and the daughters drift apart and split again until the dish is full. Nothing in the simulation knows what a spot is. Press Maze and the same drops grow into winding stripes that fill the dish like the folds of a brain coral.

Other settings give other patterns, and most give nothing at all. Figure 2 is a map of which is which.

feed F = 0.08↑ more feed

F = 0, k = 0.040more kill →k = 0.070

under pointer—

sent to Figure 1—

map—

Figure 2. One big dish in which the settings change from place to place: the kill rate k grows from left to right and the feed rate F from bottom to top. Lettered dots mark the presets in Figure 1, and the ring marks Figure 1's current setting. Click anywhere to send that setting to Figure 1. Neighbouring regions leak into each other, so the borders are rough.

Most of the map is flat. On the left, where the kill rate is low, V survives everywhere as an even wash. On the right and along the bottom, V is removed faster than it can grow, and only U is left. The patterns live in a narrow band between the two, where V can only just hold on. Click along it and watch Figure 1 change. Near its lower end you'll find restless, travelling waves. Further up come mazes and coral. Along the band's right-hand edge, where V is closest to dying out, are spots that divide, lone spots and short worms.

Why spreading makes pattern

Turing's argument uses an activator and an inhibitor. The activator makes more of itself and also makes the inhibitor. The inhibitor suppresses the activator. Without any spreading, the two settle into a steady balance.

Now let them spread, the inhibitor faster. Suppose one spot gets a little extra activator by chance. The bump grows. It also makes extra inhibitor, but that diffuses away quickly and lands in a ring around the bump, where it suppresses the activator. The result is a peak with a moat around it, and beyond the moat another peak can form. The spacing between peaks is set by how far the inhibitor travels, not by the size of the dish.

Short-range activation, long-range inhibition.

Alfred Gierer and Hans Meinhardt worked this out in general form in 1972. If the inhibitor spreads as slowly as the activator, it stays on top of every bump and flattens it. The pattern exists only because the two speeds differ. Figure 3 checks this in one dimension, with an activator–inhibitor pair of their kind.

inhibitor speed

speed ratio—

peaks—

Figure 3. A row of 160 cells, starting almost flat. The solid line is the activator, the dashed line the inhibitor, and the faint line the balance they would sit at without diffusion. inhibitor speed sets how many times faster the inhibitor spreads. Below a ratio of about 12, any bumps die away. Above it, the row breaks into evenly spaced peaks. Stir starts over from noise.

Drag inhibitor speed down to 1 and the peaks melt back into a flat line. Push it past about 12 and they return, in about the same number whatever the random start. The chemistry has not changed, only how fast one of the chemicals moves.

In Gray–Scott, V is the activator, since it makes itself. U plays the inhibitor by its absence: wherever V grows it uses up the local U and starves itself, and because U spreads faster, the shortage is felt over a wider area. That is why the model needs U to diffuse faster than V.

Spots on animals

In the 1980s the mathematical biologist James Murray ran reaction–diffusion models on surfaces shaped like animal skins and found many familiar coats. Shape matters. A narrow surface only has room for stripes running across it, so his model predicted that a spotted animal could have a striped tail but not the reverse. That roughly matches what is seen in big cats.

A sharper test came from fish. In 1995 Shigeru Kondo and Rihito Asai watched the stripes of the marine angelfish Pomacanthus as the fish grew. Instead of stretching, the stripes split and new ones appeared between them, keeping the spacing roughly constant. A Turing mechanism predicts that, and a pattern drawn once and then enlarged would not.

The honest caveats

Looking like a Turing pattern is not proof of one. The stripes of gene activity that lay out a fruit-fly embryo's body segments come from a cascade of genes reading chemical gradients laid down by the mother. In zebrafish, Nakamasu and colleagues found in 2009 that the stripes come mostly from pigment cells signalling to one another by direct contact, though the logic of local activation and longer-range inhibition seems to survive there in a different form.

Turing died in 1954, before his mechanism was ever observed. The first clear laboratory demonstration came in 1990, when Vincent Castets, Étiennette Dulos, Jacques Boissonade and Patrick De Kepper ran a reaction known as CIMA (chlorite, iodide and malonic acid) in a thin gel and saw a stationary pattern of spots appear. It took nearly forty years partly because real inhibitors rarely diffuse much faster than their activators. The gap was opened by accident: starch, added to show colour changes, binds iodide, the activator, and slows its diffusion.

In living tissue the best evidence comes from changing the mechanism and predicting the result. In 2012 Rushikesh Sheth and colleagues worked with mice that lacked a gene called Gli3 and so had extra digits. Removing copies of certain Hox genes as well gave still more digits, each thinner and more closely packed. That is what you'd expect if the digits were laid out by a Turing-type mechanism whose wavelength those genes help set. Even there, the full picture involves several molecules and the growth of the limb.

So Turing found a real and general way for structure to come from uniformity. It is confirmed in chemistry, supported in some living systems, and one tool among several that development uses. Squares that mind only their own chemistry and their neighbours can lay down a design that no single square contains.

Try this

Choose Mitosis in Figure 1, press Clear, and write a letter on the blank dish with your finger or the mouse. Each stroke breaks into spots that multiply until they fill the dish. Then pick Coral, press Clear and write the letter again. This time it grows branches.

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