The ‘Elegant’ Math Model That Could Help Rescue Coral Reefs

“Every computational biologist wants to do something like this … this kind of Occam’s razor idea that with the simplest model you try to explain as much as possible,” said Jaap Kaandorp, a computational biologist with the University of Amsterdam who has modeled corals but was not part of Llabrés’ project. “The emergence of growth and form is one of the fundamental questions in biology.”

Coral modeling has immediate applications. Marine heat waves, sea level rise and ocean acidification — consequences of climate change — threaten coral animals, their calcium carbonate structures and the ecosystems they anchor. If scientists can understand the rules of how these organisms grow, they can better predict how to keep them alive and thriving through the changes to come.

Polyp Predictions

Llabrés’ foray into quantitative ecology began with a different marine species that shapes shallow-water ecosystems. Since the 1970s, computational biologists have modeled the theoretical growth of plants, such as grasses and trees. Llabrés joined the Institute for Cross-Disciplinary Physics and Complex Systems in Mallorca to help with a similar effort with seagrass, led by the institute’s Tomàs Sintes and the marine biologist Carlos Duarte from King Abdullah University of Science and Technology in Saudi Arabia. During the research, one collaborator noticed that seagrasses grow complex colony patterns from budding clones — just like corals. “So then we said: Let’s try to apply what we know to corals,” Llabrés said.

The team wanted to home in on the mathematical rules that conjure the most common coral structures. A logic that explains the difference between growing into a tall and narrow column and a domelike “massive” coral must be buried deep in a polyp’s biological programming, they figured. It can be seen as an optimization problem: What’s the minimum number of variables needed to simulate the maximum number of shapes?

They started with the marine biologists’ expertise. When scientists say that coral “grows,” they’re referring to two processes: expansion and cloning. In expansion, individual polyps deposit calcium carbonate beneath their bodies in a cuplike shape, which enlarges as the polyp grows. Then, when the distance between polyps gets large enough, and there is empty space nearby, new polyps will bud off asexually — cloning — to expand the structure in a new direction.

This told them that all coral structures take their shape from individual polyps’ microscopic inclinations. One polyp could grow and then clone up, down or sideways, but collectively they appear organized — fanning out into sheets or protruding like tendrils. “Massive” colonies grow outward horizontally and vertically at comparable rates, like inflated balloons; polyps of column colonies secrete their skeletal ingredients more or less vertically. Examples like these cued Llabrés into a biological logic that she could translate into mathematical language.

First, she reimagined a basic coral structure: Instead of being built from polyps, it is made of hexagonal, pyramidlike objects — pointy like a cone with a six-edged base — which she called “hexacones.” Each vertex (corner) represents a polyp, and the lines connecting them form a patchwork of triangles. Llabrés wrote rules to govern what happens to hexacones as the digital coral expands.

One rule describes cloning: Polyps grow apart until the space between them reaches a critical size, at which point a new polyp generation appears in that space. Another rule governs the expansion of the hexacone based on how and where polyps deposit calcium. And a third rule encodes how a subset of polyps can construct branches that jut out laterally from the rest of the coral.

The principles of cloning, expansion and branching guided Llabrés toward the most important variables for the model. The calcium carbonate deposit rate could partially describe expansion, and the distance between polyps was crucial to simulating cloning. For branching, both the angle at which branches protrude and the distance between branches mattered. This gave her four variables, each of which played a unique role.

Llabrés suspected that she was missing another variable that could skew a structure’s overall growth vertically or horizontally — a factor that distinguishes tables, massives and columns. She worried that this was asking too much of one variable with a value between zero and 1.

After hours and hours of clacking at code on her keyboard, it came together. A “growth mode” variable she devised was powerful. It allowed polyps in Llabrés’ model to grow differently based on their position in the colony. Adjusting its value, “very fast I got a massive and then a column,” she recalled. Then cauliflowers and tables and branches. “I was like, wow, I think I might have something here.”

Study co-author Eleonora Re, a doctoral candidate on Duarte’s marine biology team, recently conducted experiments in the Red Sea to validate the team’s five-variable model with real coral data. So far, the model’s predictions of coral shape match real coral, according to preliminary results she expects to publish this year.

The set of five variables reproduces more coral forms than any model before it, including those made by Kaandorp. However, it reproduces only five of the many known shapes. “To re-create the whole diversity in growth forms you see in nature is an immense challenge,” Kaandorp said. “They cannot simulate everything, but it’s still an impressive range of growth forms.”

Vinton found the work exciting despite its limits. “It’s a model to represent the real world, so it’s not going to capture all of the complexity that we see in coral,” she said. “But it does [capture] a fantastic amount given how simple their mathematical framework is.”

It’s an encouraging illustration of theoretical ecology, she added. “People call it the ‘headlights’ of ecology and evolution,” she said. “It can guide your hypotheses for what you might see in the real world.”

From Model to Real World

Coral reefs have been around for millions of years, and many of today’s living reefs are thousands of years old. Clearly, corals are survivors. That’s because a coral is biologically programmed to adapt to new conditions — an ability called plasticity — by adjusting its physiology and growth to cope with change within certain bounds.

Plasticity differs from evolution because it happens within an individual’s lifetime. Understanding a polyp’s adjustments can therefore help biologists grasp the limits of adaptation in an era of unprecedented change. How quickly does coral grow? How densely can polyps pack together? What shapes do colonies assume to adapt to different environments?

Vinton wonders whether certain shapes are inherently more adaptable than others. “Their shape can determine their fitness in different environments,” she said. “Their survival, but also their reproduction.” When a chunk of coral breaks off in a strong wave, it can grow on its own into a new colony — a form of asexual propagation that lets species colonize new areas. Shape and density matter; a coral with fragile branches is more likely to reproduce this way than one in a massive boulder form. “Are they breaking off more, or are they not?” she asked. That difference between two structures could determine a reef’s future.

However, polyps’ internal growth programming isn’t everything. While Llabrés’ model represents an imagined genetic predisposition for certain shapes, in real life the environment is just as important to coral growth, if not more so. For example, if you grow one species of coral in sunlit shallow water, Kaandorp said, it will grow very differently than in deeper, darker water.

“There’s a direct connection between the growth process and environmental influence,” he said. “This issue is very important.”

Llabrés’ next step is to include environmental factors such as water flow or light intensity. “These are the two main things known to influence coral,” said Llabrés, who is now a postdoc at the Hawai‘i Institute of Marine Biology. “When it’s working, then the model can be a tool to predict what’s going to happen in changing conditions.”

Such tools can guide biologists to rebuild reefs with shapes optimally equipped for the long-term, large-scale ecological restoration that’s so far been elusive. “This kind of understanding is crucial for predicting how coral ecosystems and marine ecosystems might respond to climate change,” Vinton said, “and which species might need more attention and restoration.”

Llabrés has witnessed decades of impacts from climate change in the waters she grew up snorkeling. “I’ve seen the change — the system degrading,” she said. “There’s some species that I don’t see there anymore.” But her experience in the water has also evolved, thanks to a physics-tinted lens on marine life.

“I find myself asking more questions whenever I’m snorkeling,” she said. “I see even more clearly how resilient nature is; it often finds ways to adapt and thrive, even in ways we might not expect.”