Navigational aids provide geometrical explanation for polymer spindles

From the intricate patterns of pollen grains to the logarithmic spirals of nautilus shells, biology is full of complex patterns, shapes and geometries. Many of these intricate structures play important roles in biological function, but they can be difficult to create in the lab without state-of-the-art equipment or expensive and energy-consuming processes and materials.

A new study now describes how spheres can be transformed into twisted spindles thanks to insights from 16th century navigational tools. Researchers show how polymers can contract into spiral structures, known as loxodromes, that have complex patterning 10 times smaller than the width of a human hair.

Reported in a paper in Physical Review Letters, the research was conducted by University of Pennsylvania graduate student Helen Ansell, postdoc Daeseok Kim, and professors Randall Kamien and Eleni Katifori in the School of Arts and Sciences, together with Teresa Lopez-Leon of the École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI) in France.

Kim, who worked on this project at ESPCI before coming to the University of Pennsylvania, was inspired by previous studies showing that a mixture of polymer and liquid crystal took on a new spindle shape when placed in a different solvent. This change was reversible and reproducible, with little-to-no energy required to cause it to happen.

To understand the interesting conformational changes that Kim had seen in the lab, he sought out theorists who could help make sense of how the polymer's geometry caused it to twist and contract. After seeing the microscopic images and data that Kim had collected and analyzed, Ansell had an initial idea of what the spindle structure might be: a loxodrome.

More commonly referred to as rhumb lines, a loxodrome is an arc that follows a constant angle as it cuts across a sphere. Sailors from the 16^th to 19^th centuries used these lines to navigate, as they allowed the sailors to set their compasses to a constant bearing so that their ship did not have to change its direction.

"We tried to figure out if this was the case," Ansell says, referring to her hypothesis. "We think we found these loxodromes, so we had to go about comparing what does it look like versus the data."

Ansell then developed a mathematical model that describes how the spheres become elongated and twisted using the geometry of the loxodrome as a starting point. By comparing the results of her theory to the data generated by Kim, she was able to show that changing the solvent prompted the polymer to shrink, which caused its shape to twist as the polymer chains along the sphere's lines of longitude became shorter.

At the top of the spindles are 1µm spirals, nearly 100 times smaller than the width of a human hair. Creating manmade patterns that small usually requires costly methods and equipment, but this method of making self-assembled small-scale structures from larger-scale starting materials is much simpler.

The polymer loxodrome is the latest finding that delves into the Kamien group's interests in the crossover between chemistry and geometry. Kamien says that many interactions in biology, like protein folding, immune responses and even smell, are usually depicted in terms of chemical bonds, but he emphasizes that geometry also drives much of what happens in biology.

"Think about proteins," says Kamien. "You have these different amino acids, and they attract in different ways, but when you're all done, you have this giant glob, and there's this little pocket that grabs the residues, so you think of it geometrically. Helen's explanation is completely geometrical: It doesn't involve anything specific about how the binding works."

For Kim, this research is an exciting first step for studying unique structures in other biological systems. By designing new types of polymer particles and testing them out in different conditions, he hopes to learn more about how shape drives function, especially in systems that twist and contract. "We could study some biological matter in nature by mimicking a similar topological model," he says, "And we may solve or study some complex problem in nature."

Now, entirely coincidentally, Ansell's efforts have laid the groundwork for another unrelated project she had been stuck on for some time, which also appears to have a loxodrome solution. "They just appear," she says about the twisted spindle shapes.

"As Pasteur said, luck favors the prepared mind," adds Kamien. "Now, we're primed to look for them."

This story is adapted from material from the University of Pennsylvania, with editorial changes made by Materials Today. The views expressed in this article do not necessarily represent those of Elsevier. Link to original source.