Even the mildest form of a traumatic brain injury (TBI), better known as a concussion, can cause permanent, irreparable damage.

To better predict what kinds of impacts can lead to this lasting injury—and potentially develop strategies to protect against it—an interdisciplinary team of researchers from the School of Engineering and Applied Science and the Perelman School of Medicine turned to mathematical modeling. Armed with a new understanding of the elastic properties of the protein known as tau, they were able to boil down the mechanical response of its related brain structures to equations with only a handful of variables.

The team consists of Vivek Shenoy, a professor of in the Department of Materials Science and Engineering at Penn Engineering, Hossein Ahmadzadeh, a member of Shenoy’s lab, and Douglas Smith, a professor of neurosurgery at Penn Medicine and director of the Penn Center for Brain Injury and Repair. 

Their recent findings hinge on the mechanical properties of tau and its role in the elasticity of axons, the long, tendril-like part of brain cells.

Smith had previously studied the mechanical properties of axons as a whole and found that they can stretch to at least twice their length with no signs of damage, but only when force was applied slowly. However, fast applications of force, like what would accompany a fall or a car accident, caused the axons to bulge and swell.

These bulges are the hallmark of a potentially fatal TBI; they are the result of broken structures within the axon known as microtubules. Microtubules are like train tracks for transporting molecular cargo; when the tracks break, the cargo piles up into a bulge.

Microtubules are the stiffest parts of the axon, so the researchers still did not understand why they were the parts that were breaking under stress. To solve that puzzle, the researchers had to model the tracks' crossties: the protein tau.

“You need to know the elastic properties of tau,” Shenoy says, “because when you load the microtubules with stress, you load the tau as well. How these two parts distribute the stress between them is going to have major impact on the system as a whole.”

Shenoy and his colleagues had a sense of tau’s elastic properties but did not have hard numbers until a 2011 experiment from a Swiss and German research team physically stretched out lengths of tau by plucking it with the tip of an atomic force microscope.

“This experiment demonstrated that tau is viscoelastic,” Shenoy says. “Like Silly Putty, when you add stress to it slowly, it stretches a lot. But if you add stress to it rapidly, like in an impact, it breaks.”

This behavior is due to the fact that the strands of tau protein are coiled up and bonded to themselves in different places. Pulled slowly, those bonds can come undone, lengthening the strand. This allows the microtubules and entire axon to stretch. A rapid application force, however, causes all of the strain to transfer to the microtubules, breaking them.

With a comprehensive model of the tau-microtubule system in hand, the researchers were able to represent it in relatively simple equations. This allowed them to produce a phase diagram that shows the dividing line between strain rates that leave permanent damage versus safe and reversible loading and unloading of stress.

This mathematical understanding can now be used to make computer models of the brain more realistic and potentially can be applied toward tau-related diseases, such as Alzheimer’s.