# Spider Silk and Melodies Have a Common Mathematical Thread

One of the reasons I love my job is the fantastic correspondence that arrives in my e-mail inbox every day. The information I get allows me to live on the leading edge of technology as it comes through the door, so to speak. This morning was no different. The subject line in this particular email read: MIT: Researchers link patterns seen in spider silk, melodies. When I opened this missive, a subhead read: Analogy could help engineers develop materials that make use of repeating patterns.

According to the release, this work involves a new mathematical methodology called ontology logs or “ologs.” This concept was introduced about a year ago by David Spivak, a post doctoral candidate in the Department of Mathematics. He specializes in a branch of mathematics called category theory. Ologs provide an abstract means for categorizing the general properties of a system—be it a material, mathematical concept or phenomenon—and showing inherent relationships between function and structure.

At first glance, an olog can look deceptively simple, much like a corporate organizational chart that shows reporting relationships using directional arrows. But ologs demand scientific rigor to break a system down into its most basic structural building blocks, define the functional properties of the building blocks with respect to one another, show how function emerges through the building blocks’ interactions, and do this in a self-consistent manner. With this structure, two or more systems can be formally compared.

To build the ologs, Spivak along with Associate Professor Markus Buehler of the Department of Civil and Environmental Engineering (CEE), and CEE graduate student Tristan Giesa used information from Buehler’s previous studies of the nanostructure of spider silk and other biological materials.

The researchers showed the similarity between the physical structure of spider silk and the sonic structure of a melody, proving that the structure of each relates to its function in an equivalent way. The step-by-step comparison began with the primary building blocks of each item—an amino acid and a sound wave—and moved up to the level of a beta sheet nanocomposite (the secondary structure of a protein consisting of repeated hierarchical patterns) and a musical riff (a repeated pattern of notes or chords). The study explains that structural patterns are directly related to the functional properties of lightweight strength in the spider silk and, in the riff, sonic tension that creates an emotional response in the listener.

You can think what you may about likening spider silk to musical composition, but the methodology behind the research apparently represents a new approach to comparing research findings from disparate scientific fields. Such analogies could help engineers develop materials that make use of the repeating patterns of simple building blocks found in many biological materials that, like spider silk, are lightweight yet extremely failure-resistant. The work also suggests that engineers may be able to gain new insights into biological systems through the study of the structure-function relationships found in music and other art forms.

Here’s what Geisa said about the research: “The fact that a spider’s thread is robust enough to avoid catastrophic failure even when a defect is present can be explained by the very distinct material makeup of spider-silk fibers. It’s exciting to see that music theoreticians observed the same phenomenon in their field, probably without any knowledge of the concept of damage tolerance in materials. Deleting single chords from a harmonic sequence often has only a minor effect on the harmonic quality of the whole sequence.”

According to Spivak, the seemingly incredible gap between spider silk and music is no wider than the gap between the two disparate mathematical fields of geometry—think of triangles and spheres—and algebra, which uses variables and equations. Yet, Spivak said, category theory’s first success, in the 1940s, was to express a rigorous mathematical analogy between these two domains and use it to prove new theorems about complex geometric shapes by importing existing theorems from algebra.

“It remains to be seen whether our olog will yield such striking results; however, the foundation for such an inquiry is now in place,” Spivak said.

This project was funded by the U.S. Air Force Office of Scientific Research, a Department of Defense Presidential Early Career Award for Scientists and Engineers, the U.S. Office of Naval Research, and the German National Academic Foundation. The researchers published their findings in the December issue of BioNanoScience.