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On a March day in 2023, Ben Alessio, then a research assistant at ���Ҵ�ý�ƽ������, was wandering around the Birch Aquarium in La Jolla, California, when he clocked a surprising sight: a male ornate boxfish undulating in the water, tessellating with violet and tangerine hexagons. It was dazzling — but more importantly, it was vindicating.��

The rare fish’s markings were a real-life example of something that he and��Ankur Gupta, ���Ҵ�ý�ƽ������ assistant professor of chemical and biological engineering, had previously only modelled mathematically. It confirmed that they were onto a scientific breakthrough.��

Turing Patterns��

To understand Gupta’s research, one must first understand morphogenesis. Morphogenesis is the process by which cells, tissues and organisms develop their shapes.��

Nearly 75 years ago, the famed British mathematician Alan Turing published a paper titled�� Since then, Turing’s work has been key to our understanding of how many — but not all — patterns form in nature.��

“That’s sort of��the foundational thinking, mathematically speaking, for this area of work,” said Gupta.��

Turing had an uncommonly innovative mind — his ability to think beyond the status quo and make connections between various fields of thought led to breakthroughs in electronic computing, artificial intelligence, code breaking in WWII and, in this case, mathematical biology.��

When it came to morphogenesis, Turing was interested in how heterogeneity, or diversity, arises out of homogeneity, which is when something is composed of all one type of thing. In other words, why does a zebra have both black and white stripes instead of a coat with hairs that are all one solid color, like gray?��

The reason is diffusion — which is central to Turing’s theory. Diffusion is the movement of molecules from areas of higher concentration to areas of lower concentration; molecules tend to spread out until there’s an even distribution (much like people in an elevator). In chemistry, diffusion often dominates systems, especially when particles are tiny.��

“Diffusion essentially promotes homogeneity,” said Gupta.��

What he means is that if you drop blue dye into clear, still water, for example, it will slowly diffuse, in a gradient, until the whole container is equally blue. Similarly, when mixing red and blue dye in a diffusion-dominated system, one expects the colors to blend, ultimately yielding a homogenous purple hue. However, when a chemical reaction also occurs, something different may happen. In certain conditions, even a diffusion-dominated system can promote heterogeneity from homogeneity.��

“Essentially, what [Turing] argued was that under the right conditions, if there is diffusion as well as [a chemical] reaction between different components — if I have five or six dyes, or three or four dyes, and they’re reacting with each other — then essentially it’s just a delicate dance between these two processes.”��

These days, the term “Turing pattern” is generally applied to any reaction-diffusion pattern. This natural pattern forms when chemicals react with one another and spread out, often resulting in wavy lines or spots. A notable example of a Turing pattern in nature is the sparkling blue zebra fish, a slender creature that’s gilded with horizontal, blurry-edged golden stripes.��

However, some wild animals feature very crisp markings.��

“Why would a diffusion model describe something that is so striking and sharp?” Gupta wondered.

An Accidental Discovery

Gupta didn’t initially set out to answer that question. His focus was on diffusiophoresis, which is the combination of diffusion, described earlier, and phoresis, which describes how ultra-small dissolved particles — around a tenth or even a hundredth the width of a single human hair — can sometimes drag other things along with them in a solution. So, if diffusion is the way that blue dye spreads through clear water, phoresis is the movement of particles that happens because they’re temporarily dragged by that dye.��

Alessio, who was doing computational research at the time, had been running mathematical simulations of reaction-diffusion systems that also had a diffusiophoretic element. The resulting visuals were notably defined, unlike the fuzzier ones that emerge from reaction-diffusion models (as seen in the zebrafish). And it was the striking violet and tangerine hexagon boxfish pattern that caught his attention in the Southern California aquarium.��

“I was just literally simulating something like this on my computer,” thought Alessio when he saw it. He snapped a slew of photos and messaged Gupta excitedly. “I have something exciting to show you.”��

Until that point, Gupta and Alessio had the models, but they didn’t have an example of them in nature.��

“I didn’t have any sort of idea about this fish or anything like that,” said Gupta. “He showed me this, and then we sort of reverse-engineered the missing link.”��

They dove into existing research and realized that chromatophores — cells that create pigment in the bodies of fish, reptiles and some other animals — can be carried by dissolved chemicals. In other words, they can move diffusiophoretically (like the particles temporarily dragged by dye).��

In late 2023, Gupta and Alessio published a paper titled�� in the peer-reviewed journal��Science Advances. Their research advances Turing’s theory by describing how more precise patterns — like the one seen on the ornate boxfish — come to exist.��

While Alessio is now working on a PhD in mechanical engineering at Stanford, Gupta intends to continue researching how diffusiophoresis factors into Turing patterns.��

“On the pattern-formation side, it would be useful to see if we can replicate some of this synthetically,” he said.��

It’s a tall order, but more investigation can potentially help us understand how to control things synthetically.��

Gupta is now investigating this phenomenon at an individual-cell level, which he likens to studying a single human versus a population of people.��

“If I’m thinking about a population, then one option is to track individual people, and one is to say, ‘What is the population density?’” he said. “It was the population density approach that we were taking in our first paper. But now, we’re examining individual cells, and that has been interesting, because now what we start to see is imperfect Turing patterns.”��

While mathematical models tend to be perfect, in reality, you often see imperfections: deformed hexagons or hexagons sliced in half. Taking an individual-cell-level approach to diffusiophoretic Turing patterns could provide more insight into why patterns sometimes don’t appear as expected based on mathematical models.��

“We think it’s exciting, because real systems actually are not perfect,” Gupta said.��

A Pilgrimage

In October 2023, before the “Diffusiophoresis-enhanced Turing patterns” paper was published, Gupta’s wife was traveling to a conference in San Diego. With anticipation, he packed his bags, hoping to set eyes on the ornate boxfish that inspired their discovery.��

Inside the Birch Aquarium, he did a lap around the right side, where most of the fish seemed to be. But he wasn’t sure exactly where to look, and he struggled to spot its telltale scales.��

“I couldn’t see it,” he said. “It was hidden.”��

Ten minutes passed, then 20. Increasingly worried, he considered enlisting a staff member to help him track it down. Finally, in a last-ditch effort, he ventured off in the direction of the children’s area, toward the other side of the building. There, at long last, he caught his glimpse of the elusive fish.

Eureka.

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Illustration by Petra Péterffy