New study shows how to learn the equations of cell migration

When you cut yourself, mass migration begins in your body: skin cells flock by the thousands to the wound site, where they soon shed new layers of protective tissue.

In a new study, researchers at the University of Colorado Boulder have taken an important step in unraveling the drivers behind this collective behavior. The team has developed an equation-learning technique that could one day help scientists understand how the body repairs skin and potentially inspire new therapies to speed up wound healing.

“Learning the rules of how individual cells respond to the proximity and relative motion of other cells is critical to understanding why cells migrate into a wound,” said David Bortz, 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 natural world, said study lead author Dan Messenger.

“Although this article is about cells, the math is also applicable to a variety of areas, including how flocks of birds avoid both predators and each other,” said Messenger, a postdoctoral researcher in Bortz’s lab.

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

The research draws on a range of tools from the field of ‘data-driven modelling’, an emerging field at the intersection of applied mathematics, statistics and data science. Using this approach, the group designed computer simulations of hundreds of cells moving towards an artificial wound, and then developed a method to learn the 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 infection, we want our wounds to close as soon as possible,” Liu said. “We plan to use these learned models to test pharmaceuticals and drug therapies that have the potential to stimulate wound healing.”

trial and error

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

For example, in 2020 Bortz joined a team of scientists using models to try to predict the spread of COVID-19 in Colorado. But, he noted, it can take a lot of trial and error and even supercomputers to validate these equations.

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

In this new study, he and his colleagues extended their recently developed WSINDy method to use data directly to learn models from individuals.

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

cells to particles

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

Liu and his colleagues have observed in the laboratory how skin cells grow together as a group. They found that migrating skin cells follow certain rules: like a herd of wild buffalo, skin cells align their direction with the cells in front of them, but also try not to bump into 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 tandem. The team used their WSINDy approach to create precise equations that describe the movement of each of these cells.

“With WSINDy, you can learn 1,000 different models with 1,000 cells,” Bortz said.

They then resorted to even more mathematics to group these models. Bortz noted that WSINDy is particularly good at finding patterns hidden in data. For example, if the researchers mixed two or more types of cells that moved in different ways, their tools could accurately identify the cells and sort them into groups.

“Not only do we learn models for each cell, but these models can be sorted, revealing the dominant categories of cell behavior that play a role in wound healing,” Messenger said.

In the future, the collaborators hope to be able to use their approach to study the behavior of real cells in the laboratory. Liu noted that the 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 usually don’t have a quantitative way to describe this cell migration,” Liu said. “But now we do it.”