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From tiger stripes to leopard spots, the animal world is full of distinctive and intricate patterns.

In a new study, �鶹��Ѱ�����Boulder scientists refined their previous theory of how animal patterns form and successfully recreated imperfections in natural designs, like irregular spots on a leopard. The new mechanism, , could lead to materials that can respond to their environment, such as fabrics that change color on demand for camouflage.

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Top: A male ornate boxfish (Aracana ornata). Bottom left: A close-up picture of the fish’s natural hexagonal pattern. Bottom center: Fish pattern simulation based on Turing’s reaction-diffusion theory. Bottom right: Diffusiophoresis-enhanced reaction-diffusion simulation. (Credit: The Birch Aquarium/Scripps Institution of Oceanography and Benjamin Alessio/�鶹��Ѱ�����Boulder)

“Imperfections are everywhere in nature,” said Ankur Gupta, the study’s lead researcher in the Department of Chemical and Biological Engineering. “We proposed a simple idea that can explain how cells assemble to create these variations.”

For decades, scientists have been trying to crack the code of how different animal patterns emerge from a mass of developing cells. In 1952, mathematician Alan Turing hypothesized that as tissue develops, it produces chemical agents that diffuse in the system in a process similar to pouring milk into coffee. Some of these chemicals activate pigment-producing cells, forming spots. Other chemicals inhibit these cells, creating the blank spaces in between.

But just as milk clouds the coffee, computer simulations based on Turing’s theory produced spots that were blurrier than those found in nature.

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Updated simulations generated imperfect and textured hexagon and stripe patterns. (Credit: Siamak Mirfendereski and Ankur Gupta/�鶹��Ѱ�����Boulder)
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In 2023, Gupta and his collaborators improved upon Turing’s theory by adding another mechanism called diffusiopherosis, a process where diffusing particles pull other particles along with them. It’s the same principle that helps laundry get clean: As soap diffuses out of the laundry into water, it drags dirt out from the fabric.

When Gupta simulated the purple-and-black hexagon pattern seen on ornate boxfish, a flashy species found in the seas off Australia, he found that diffusiopherosis could generate patterns with sharper outlines than Turing’s original model.

But the team’s results were a little too perfect. All the hexagons were the same size and shape, and the spaces between them were identical.

In nature, no animal has flawless patterns. A zebra’s black stripes vary in thickness, and the hexagons on the boxfish are never perfectly uniform.��
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So Gupta and his team set off to improve the diffusiopherosis model.��
They found that by giving individual cells defined sizes and modeling how each one moved through tissue, their simulations began producing imperfect patterns and textures.

Imagine ping-pong balls of different sizes traveling through a tube. Larger balls would create thicker outlines than smaller ones. When bigger cells cluster, they form patterns that are broader. Sometimes the balls bump into one another and jam the tube, breaking up a continuous line. When cells experience that, they create breaks in the stripes.

“We are able to capture these imperfections and textures simply by giving these cells a size,” Gupta said. Their simulations showed breaks and grainy textures that look far more like what’s found in nature.

In the future, Gupta plans to incorporate more complex interactions among cells and with the background chemical agents to improve their simulations.

Humans have always drawn inspiration from nature. ��Bats’ ability to navigate using echoes led to sonar technology, which locates objects through sound. Gupta said understanding how pattern-making cells assemble could help engineers design synthetic materials that can change colors based on the environment, much like a chameleon’s skin. It could also help design effective approaches to deliver medicine to a specific part of the body.

“We are drawing inspiration from the imperfect beauty of natural systems and hope to harness these imperfections for new kinds of functionality in the future,” Gupta said.��
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