New research shows how to learn the equations of cell migration

When you cut yourself, your body begins a mass migration: thousands of skin cells rush to the wound site, where they soon lay down a fresh layer of protective tissue.

In a new study, researchers at the University of Colorado at Boulder have taken an important step toward elucidating the factors behind this collective behavior. The team developed a method of studying the equations that could one day help scientists understand how the body repairs skin and could potentially inspire new treatments to speed up wound healing.

“Learning the rules of how individual cells respond to the proximity and relative movement of others cells is critical to understanding why cells migrate into a wound,” said David Bortz, a professor of applied mathematics at CU Boulder and senior author of the new study.

The research is the latest in a decade-long collaboration between Bortz and Xuedong Liu, a professor of biochemistry at CU Boulder. The group’s method, called Weak form Sparse Identification of Nonlinear Dynamics (WSINDy), can be applied to a wide range of phenomena in the world of naturesaid lead study author Dan Messenger.

“Although this paper focuses on cells, the math is also applicable to a wide range of areas, including how flocks of birds avoid both predators and each other,” said Messenger, a postdoctoral fellow in Bortz’s lab.

He and his colleagues published their results on October 12 Journal of The Royal Society Interface.

The research draws on a set of tools from the field of “data-driven modeling,” an emerging field at the intersection of applied mathematics, statistics, and data science. Using this approach, the group developed computer simulations of hundreds of cells moving toward an artificial wound, and then created a method of learning equations to describe and study the movement of each individual cell. The team’s tools are potentially much faster and more accurate than traditional modeling approaches — a boon for understanding complex natural phenomena like wound healing.

“To prevent infections, we want our wounds to close as quickly as possible,” Liu said. “We plan to use these learned models to test pharmaceuticals and drug regimens that could stimulate wound healing.”

By trial and error

Mathematical models come in all shapes and sizes, but most of them use complex series of equations to try to represent some phenomenon in the real world.

Bortz, for example, joined a group of scientists in 2020 who relied on models to try to predict the spread of COVID-19 in Colorado. But, he noted, it may take a lot of trial and error and even supercomputers to confirm these equations.

“The development is accurate and reliable model can be a very long and time-consuming process,” Bortz said.

In this new study, he and his colleagues expanded on them the newly developed WSINDy method directly use the data to learn patterns of individuals.

“It’s about putting the data first, followed by the math,” Bortz said.

Cells to particles

In the current study, he and his colleagues, including biochemistry graduate student Graysen Wheeler, decided to turn this data-driven lens to the problem of cell migration.

Liu and his colleagues observed skin cells grouping together in the laboratory. They discovered that migrating skin cells follow certain rules: like a herd of buffalo, skin cells will match their direction with the cells in front of them, but also try not to bump into the leaders behind them.

To see if WSINDy could shed light on this grassroots movement, Bortz and Messenger designed computer simulation shows hundreds of digital cells moving in tandem. The team used the WSINDy approach to create precise equations that describe the movement of each of these cells.

“With WSINDy, if you have 1,000 cells, you can learn 1,000 different patterns,” Bortz said.

They then brought in even more math to begin clustering these patterns. Bortz noted that WSINDy is particularly good at finding patterns hidden in data. When the researchers, for example, mixed together two or more types of cells that moved in different ways, their toolkit could accurately detect and sort the cells into groups.

“Not only are we learning patterns for each cell, but these patterns can be sorted, thus revealing dominant categories of cell behavior that play a role in wound healing,” Messenger said.

Moving forward, the team hopes to use their approach to begin digging into the behavior of real cells in the lab. Liu noted that this technique could be particularly useful for studying cancer. Cancer cells, he said, undergo similar mass migrations as they spread from one organ to another.

“As biochemists, we don’t usually have a quantitative way to describe this cell migration,” Liu said. “But now we do.”