How spots and stripes can draw themselves

A zebrafish embryo does not carry a drawing of its stripes. Somehow a sheet of cells that starts out much the same everywhere ends up striped, and the stripes come back after they are damaged. In 1952 Alan Turing proposed a way this can happen with no blueprint at all: two substances that react and spread at different speeds.

What Turing proposed

Turing's paper opens with the claim 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, he wrote, “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.” The surprise is the word diffusing. Spreading normally smooths things out. Turing found conditions under which it does the opposite.

Why it matters

If a pattern can arise from the chemistry itself, an embryo does not need a pre-existing map to say where each stripe or bud goes. Kondo and Miura, reviewing the idea in 2010, describe the experimental cases behind it. After a laser removes pigment cells from a zebrafish's stripes, the stripes regrow and shift in the way a simulation predicts. In mouse skin, experiments on the spacing of hair follicles suggest that Wnt acts as a short-range activator and Dkk as a long-range inhibitor. Feather buds in the chick re-form their regular spacing even from skin rebuilt out of separated cells.

The rule: help your neighbours, hold back the distance

Take two substances. The activator makes more of itself, and it also makes the inhibitor. The inhibitor shuts the activator down. Now let the inhibitor spread faster than the activator. A small bump of activator grows, because it feeds itself locally. The inhibitor it produces leaks outward, farther than the activator can, and suppresses activator in a ring around the bump. Just beyond that ring the inhibitor has thinned out, so another bump can grow there. The result is a row of peaks spaced a set distance apart.

Kondo and Miura credit Gierer and Meinhardt with showing that a system needs only “a short-range positive feedback with a long-range negative feedback” to form a Turing pattern, and this “is now accepted as the basic requirement for Turing pattern formation.” The pieces need not be molecules. In zebrafish the players are two kinds of pigment cell, black melanophores and yellow xanthophores, and their interactions have the same short-range, long-range shape.

Breaking the rule

The key is unequal spreading, not merely having two substances. In the sheet above, set the inhibitor's diffusion equal to the activator's and no ripple can grow: the page's stability calculation says so, and the sheet flattens. For our model the smooth sheet becomes unstable only once the inhibitor spreads about 7.29 times as fast for the stripe recipe, or 11.24 times for the spot recipe (our arithmetic). Above that line, ripples of one preferred length grow fastest, about 12 cells long at a ratio of 20, and they roughly set the spacing of the stripes. Turing called this a “chemical wave-length”: it comes from the reaction rates and diffusion, not from the size of the ring.

Stripes and spots are not different mechanisms. Kondo and Miura show spots, stripes and mazes “made by an identical equation with slightly different parameter values,” and the buttons above change one number. Real fish show the same switch: in zebrafish, losing the gap-junction gene connexin41.8, which is named leopard, “reduces the spatial periodicity and changes the pattern from stripes to spots.”

What a look-alike does not prove

A pattern that looks right is not evidence on its own. Kondo and Miura grant the skeptics’ point: “just because there is water, it doesn't mean there are waves.” Their own figure of candidate systems carries the note that “the involvement of the RD mechanism in some of the phenomena above has not been fully accepted by experimental researchers.” The spots on a leopard are not among the cases they give as demonstrated. Turing himself worked mostly on a ring of cells, which he called “a mathematically convenient, though biologically unusual system,” and his one two-dimensional example is a pattern “reminiscent of dappling.” He added that dappled patterns made this way would have to be laid down “when the foetus is only a few inches long,” since later “the distances would be greater than the morphogens could travel by diffusion.”

Not every reaction-diffusion pattern is a Turing pattern, either. Pearson's 1993 simulations of the Gray-Scott model grew spots that divide and stripes that lengthen, with one substance spreading twice as fast as the other. At that ratio, he writes, “there are no stable Turing patterns”: his patterns grew from a sizeable starting disturbance (a “finite-amplitude” one) rather than from tiny noise, and the smallest ratio that gives stable Turing patterns in that model is about 2.8. The sheet above can show a milder version of the same thing: grow spots at a ratio of 20, then drag the slider to 10. The smooth sheet is stable there, yet the spots already made can hold themselves up. Restart at 10 and nothing forms.

In short

Local self-help plus distant suppression turns a nearly uniform sheet into evenly spaced peaks. The inhibitor must spread farther than the activator; make them spread alike and the pattern cannot start. Zebrafish stripes and mouse hair follicles are the well-tested examples. The leopard's spots are not among the cases these sources show to be Turing patterns.