By Lisa Ann Jackson, '97

A BYU math professor is using complex equations to help solve the problem of scarring.

A BYU math professor envisions a world with no scars. Burns, scrapes, gashes, and slices would heal without mark or lasting tissue damage. Blemishes would be prevented with the rub of an ointment.

Admittedly, John C. Dallon, assistant professor of mathematics, is years away from realizing his dream. Maybe even a lifetime. But he is laying the theoretical foundation to help lab researchers zero in on the cause and cure of scarring.

In the early 1980s there was a resurgence of interest in wound healing and scarring. After surgeons performed the first in utero operations, an unexpected discovery was that the infants treated as fetuses had no scars from the surgery.

Working in the relatively new field of math biology, Dallon has developed equations that model the cell-level interactions taking place as a wound heals. His research suggests that biologists may have the right chemical but the wrong interaction.

Results of his ongoing wound-healing studies have recently been published in the journal Wound Repair and Regeneration and are turning the heads of mathematicians and biologists alike.

Dallon began his research by studying what biologists had already discovered about the process of scarring. When it was discovered that fetuses didn't scar, biologists naturally started looking for differences in the ways adult wounds and infant wounds heal. They narrowed it down to a chemical called Transforming Growth Factor-beta (TGF-b), a substance present in adult wounds but not in fetal wounds.

The next step for biologists was to remove TGF-b to see if it reduced scarring, and in experiments it has. "As clinicians, this is exactly what they want to know. They want to know what to do to reduce scarring," says Dallon. "But as a theoretician, I want to know what's going on."

So Dallon decided to focus on the differences between normal tissue and scar tissue. Normal tissue fiber is randomly aligned, creating a basket weave–like affect. Scar tissue, however, aligns in a more parallel fashion, making it stand out from the normal tissue around it.

Using what biologists have already discovered about the molecular process that causes scar-tissue alignment and combining it with existing equations, Dallon was able to create mathematical models of the processes taking place in a wound.

His simulations, however, are not consistent with the hypothesis that removing TGF-b from a wound will produce less alignment of the tissue and therefore less scarring. In fact, his models suggest that reducing TGF-b may actually produce more alignment and, therefore, more scarring. In addition, further simulations reveal that recently discovered effects of one type of TGF-b not found in skin cells may be integral to the random alignment of normal tissue fibers.

"The known effects biologists are looking at do not explain this phenomenon—that by removing TGF-b you get less scarring," Dallon concludes.

Burgeoning Field

Convincing biologists of his conclusions isn't easy. Although math has been used for decades to study biology, the field of math biology is still gaining recognition.

"It's something that has been done in physics for a long time," Dallon says of combining math and science. "And I think biologists are going to have to resort to using much more math than they have in the past."

Some biology disciplines already rely on mathematical modeling. Ecological studies of fish populations, for instance, are typically simulated mathematically, and cancer researchers are also turning to mathematical models.

"There's an increasing awareness among medical scientists about math biology," says Jonathan Sherratt, professor of math at Heriot-Watt University in Edinburgh, Scotland, and Dallon's coresearcher. But Sherratt says that if you asked the average biologist on the street about math biology, you might still get a blank stare.

Dallon's work has been well received at conferences and in journals, but there is still a common thread among biologists' responses: "They say it is way too simplistic," Dallon says.

"The biological process is extremely complicated," Dallon concedes. "Biologists are right when they say, 'You are simplifying it so much. How can we believe anything you say?' To which I say, 'You can believe some of what I am saying.'"

Not that Dallon's equations are only half right; rather, they are intentionally stripped down. His goal as a mathematician is to reduce a system to its simplest form and then add complicating variables back into the equations as he learns from the simpler ones.

For example, an initial equation may involve just collagen (a protein found in skin) and fibroblasts (cells that produce collagen). As he manipulates the equation, he is able to gain insight into how these two elements interact. Then he can add other elements to the mix, complicating the equation and interaction.

"Then I can put all these effects together, which I hope will model the real situation," Dallon says.

Zeroing In

Simplifying interactions allows researchers to produce situations biologists cannot create themselves.

"I can change how much collagen is produced and do just that one thing," says Dallon. "It is much easier to manipulate a mathematical model than it is to manipulate the real system. It takes biologists years to do these sorts of experiments. And in the real system, if you want the wound to heal, you can't eliminate everything."

The promise of mathematical modeling is drawing more attention to math biology. Speaking specifically about Dallon's work with TGF-b, William J. Linbald, editor-in-chief of Wound Repair and Regeneration, notes that these studies are difficult to perform in organic settings. "Unfortunately, it is very difficult to do these types of studies in vivo. A mathematical model that would correctly predict the effect of different TGF-beta profiles on scar formation would be extremely informative."

The models provide an element of navigation not commonly available to biologists. They are able to hone their experiments to the areas pinpointed by the models.

As Dallon and colleagues blend known biological processes with known equations, at points along the way the analogy breaks down and they have to adjust the model. "We're refining the model and making it more realistic," Sherratt says. "There is obviously a lot more to be done. It's ongoing."

The Real World

So the question remains, when someone scrapes a knee or undergoes surgery, will Dallon's equations help him avoid the resultant scar? That's the ultimate goal. But Dallon is, admittedly, a few steps removed from a real-world application. Pharmaceutical companies can't bottle Dallon's equations, but Dallon hopes to point biologists in the right direction so they can get to the treatment-development stage faster.

"Eventually this will lead to developing anti-scarring therapy," says Sherratt. "That's the big picture of what we're doing."