While statisticians are driven by real-world problems, U of S mathematics professor Chris Soteros is motivated by the more esoteric behavior of long-chain molecules, such as polymers and DNA, and the mathematical problems they pose.
Her work involves analyzing the folding and “packaging” behavior of these molecules. Given that two meters of DNA is folded into each cell in our body, studying the behavior is daunting indeed.
To help unpack the problem, Soteros simplifies and simulates these molecules on a three-dimensional lattice, then uses mathematical tools such as random and self-avoiding walks to model their behavior.
The sporadic path of a random walk is often described as “a drunkard’s walk home,” and is used to model random movements in large data sets—from stock market fluctuations to particle physics. A self-avoiding walk is a random walk that cannot cross the same path or retrace steps. Since no two atoms can occupy the same space, in three dimensions it is an ideal tool to model polymer behavior.
To study polymer behavior, Soteros models a polymer solution by using a lattice walk to represent the polymer and the empty spaces surrounding it to represent the solvent molecules of the solution.
In an experimental solution at high temperatures, the polymer behaves like a self-avoiding walk. “At these temperatures, the polymer prefers to be close to the solvent molecules, but if you decrease the temperature, the polymer prefers to be closer to itself,” explains Soteros.
Surprisingly, at a specific lower temperature the polymer behaves like a random walk, and below that temperature, a “collapse” transition occurs, and the polymer folds in on itself.
“It wasn’t until the late ’70s that the collapse transition was observed in the lab, and you had to have a very large molecule in a very dilute solution to see the transition,” says Soteros. “This is an example of mathematics predicting a behavior before it was confirmed by experiments.”
Read more: A tango with tangled polymers
thumbnail courtesy of phys.org