The fur and scales of the animal kingdom are far from boring and full of wild colors and patterns. These mathematically inspired designs like leopard spots and tiger stripes are as interesting as they are intricate. But how did animals get their spots, stripes, and everything in between? It’s a question that has puzzled scientists and mathematicians for decades, but one group might be closer to an answer.
A puzzle even a famous code-breaker couldn’t crack
In 1952, British mathematician Alan Turing hypothesized that as tissue develops, it generates chemical agents that move about, similar to how white milk spreads when it is poured into black coffee. In Turing’s theory, some of these chemicals then activate pigment-producing cells, which creates spots. Other chemicals will stop these cells, creating the blank spaces in between them. However, computer simulations based on Turing’s idea created spots that were blurrier than those found in nature.
In 2023, University of Colorado at Boulder chemical engineer Ankur Gupta and his collaborators improved upon Turing’s theory by adding another mechanism called diffusiopherosis. This is a process where diffusing particles pull other particles along with them. It’s similar to how dirty clothes are cleaned in the laundry. As the soap dispenses out of the clothing and into the water, it drags dirt and grime away from the fabric.
Gupta turned to the purple-and-black hexagon pattern seen on ornate boxfish, a bright species found off the coast of Australia, as a test. He found that diffusiopherosis could generate patterns with sharper outlines than Turing’s original model, but these results were just a little too perfect. All of the hexagons were still the same size and shape and had identical spaces between them. In nature, no pattern is perfect. For example, a zebra’s black stripes vary in thickness, while hexagons on the boxfish are never perfectly uniform. So Gupta and the team sought out to refine their diffusiopherosis theory.
“Imperfections are everywhere in nature,” Gupta said in a statement. “We proposed a simple idea that can explain how cells assemble to create these variations.”Â
Like balls in a tube
In a study published today in the journal Matter, Gupta and the team detail how they were able to mimic the imperfect patterns and texture. After giving individual cells defined sizes and then modelling how each one moved through tissue, the simulations began to make less uniform patterns.
It’s similar to how balls of different sizes would move through a tube. The larger ones like a basketball or bowling ball would create thicker outlines than golf balls or ping-pong balls. It’s the same with cells–when bigger cells cluster, they make broader patterns. If the same balls traveling in a tube bump into one another and clog it, it will break up a continuous line. When cells experience that same traffic jam, the result is the breaks in the stripes.
A mixture of two types of pigment-producing cells undergoes diffusiophoretic transport to self-assemble into a hexagonal pattern. CREDIT: Siamak Mirfendereski and Ankur Gupta/CU Boulder
“We are able to capture these imperfections and textures simply by giving these cells a size,” Gupta said.Â
Their new simulations showed breaks and grainy textures that are more similar to those found in nature.Â
Why it matters
In the future, the team plans to use more complex interactions among cells and background chemical agents to improve the accuracy of their simulations.Â
Understanding how pattern-making cells assemble could help engineers develop materials that can change colors based on their environment the way that a chameleon’s skin does. It may also help create more effective approaches for delivering medicine to a specific part of the body.
“We are drawing inspiration from the imperfect beauty of [a] natural system and hope to harness these imperfections for new kinds of functionality in the future,” Gupta said.
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