Tweaking the pattern equations

Scientists offer update on Turing’s theory explaining the origin of stripes and spots

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DOING THE MATH  Spots like those on a cheetah’s coat may be governed by rules laid out by computer scientist Alan Turing. A developmental cell biologist now offers a slightly different take on Turing’s equations to explain how cell interactions may create natural patterns.

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SAN DIEGO — A mathematical tale of how tigers got their stripes and leopards acquired spots has undergone a slight revision.

In 1952, computer scientist and polymath Alan Turing devised a theory about how regular, repeating patterns — from the pigmentation on an animal’s coat to leaf arrangements in ferns — form in nature. His idea was that two chemicals, which he called morphogens, interact as they spread across a surface to create patterns.

Biologists have found scant evidence that patterns in nature are created as Turing described. For one thing, chemicals don’t diffuse freely in bodies.

But Shigeru Kondo of Osaka University in Japan thought there was something to Turing’s idea. “I saw that this theory was too good to discard,” Kondo said December 15 at the annual meeting of the American Society for Cell Biology.

Turing’s equations require that one of the morphogens be an “activator” that stimulates a chemical reaction and the other be an “inhibitor” that blunts the activator. The activator should stimulate its own activity over short ranges, while the inhibitor should work against the activator in a long-distance feedback loop, Turing proposed. The idea captured scientists’ interests because its simple formula so faithfully reproduced natural patterns.

In 2014, Kondo and colleague Hiroaki Yamanaka showed that yellow and black pigment cells play “tag” in the skin of zebrafish (SN: 2/22/14, p. 9). The cells’ run-and-chase behavior is analogous to Turing’s diffusing morphogens.

That experiment was some of the only work to show that Turing reactions may actually play a role in making biological patterns, said H. Frederik Nijhout, a developmental physiologist at Duke University.

Now, Kondo proposes that diffusion isn’t necessary to create Turing patterns. To make Turing’s idea more compatible with actual biological systems, Kondo suggests replacing diffusion with varying profiles of cell-to-cell interactions. Communication between cells serves the activating and inhibiting functions of the Turing equations. When the cells touch, sometimes making full contact and sometimes through long-range projections, they set off biochemical chain reactions that influence pattern formation. Varying the type of interaction and the strength of cells’ responses could produce patterns similar to those that Turing’s diffusion model creates, Kondo discovered.

His update, which he calls Kondo’s Activation Profile Simulator, or KAPS, preserves the underlying math of the Turing equations. His simulations can produce a wide variety of skin patterns, including mottled shading reminiscent of an eel’s skin.

Giving diffusion the heave-ho doesn’t negate Turing’s idea, said Thomas Gregor, a physicist at Princeton University. “You’re just changing the mechanism by which information spreads.”

Nijhout notes that Turing’s idea doesn’t depend explicitly on diffusion. It only needs a way to transfer information across a distance, and that the inhibitor acts over greater distances than the activator. “So a bucket-brigade mechanism by which cells pass molecules or signals along, or by which cells stimulate adjoining cells to produce activators and inhibitors could work just as well,” he said in an e-mail.

Kondo said his updated equations generate stable patterns like those seen on adult animals, but can’t simulate dynamic patterns, such as stripes that change as fish grow. And his idea only works in two dimensions. Neither his nor Turing’s equations seem to explain the development of 3-D patterns, such as tree limbs or fingers and toes.

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