New study shows how to learn the equations of cell migration

When you cut yourself, a mass migration begins within your body: Skin cells flood by the thousands to the site of the wound, where they will soon lay down fresh layers of protective tissue.

In a new study, researchers from the University of Colorado Boulder have taken an important step toward uncovering the drivers behind this collective behavior. The team has developed an equation-learning technique that may one day help scientists understand how the body rebuilds skin and could potentially inspire new treatments to speed wound healing.

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

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

“While this work is about cells, the math can also be applied to a wide range of fields, including how flocks of birds avoid both predators and each other,” said Messenger, a postdoctoral researcher in the lab. by Bortz.

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

The research draws on a set of tools from the field of “data-driven modeling,” an emerging area at the intersection of applied mathematics, statistics, and data science. Using this approach, the team designed computer simulations of hundreds of cells moving toward an artificial wound and then built a method to learn the equations to describe and examine 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 physical phenomena such as wound healing.

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

Trial and error

Mathematical models come in many shapes and sizes, but most use a complex series of equations to try to capture a real-world phenomenon.

Bortz, for example, joined a team 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, even supercomputers, to validate those equations.

“Developing an accurate and reliable model can be a very long and arduous process,” Bortz said.

In this new study, he and his colleagues extended the recently developed WSINDy method to use direct data to learn models of individuals.

“It’s about putting the data first and letting the math follow,” Bortz said.

Particle cells

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

Liu and his colleagues observed how skin cells grow together as a group in the laboratory. Migrating skin cells, they discovered, tend to follow certain rules: Like a herd of buffalo, skin cells will align their direction with the cells in front of them, but also try not to hit the leaders from behind.

To see if WSINDy could shed light on this mass movement, Bortz and Messenger designed computer simulations showing hundreds of digital cells moving in parallel. The team developed 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 models,” Bortz said.

They then used even more math to start grouping these models together. Bortz noted that WSINDy is particularly suited to finding patterns hidden in data. When the researchers, for example, combined two or more types of cells that moved in different ways, their suite of tools could precisely identify and sort the cells into groups.

“Not only are we learning models for each cell, but these models can be classified, thereby revealing the dominant categories of cellular behaviors that play a role in wound healing,” Messenger said.

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

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