Mathematics Getting More Important in Science

UCSB’s Paul Atzberger uses mathematics to advance problems in the sciences

In popular culture, mathematics is often deemed inaccessible or esoteric. Yet in the modern world, it plays an ever more important role in our daily lives and a decisive role in the discovery and development of new ideas — often behind the scenes.

UC Santa Barbara’s Paul Atzberger, a professor in the Department of Mathematics and in mechanical engineering, often works in areas where science and math intersect. Some of his recent research published in the Proceedings of the National Academy of Science (PNAS) and featured on the cover of the journal Soft Matter focuses on problems specific to lipid bilayer membranes. These microscopic structures can form a sheet that envelopes the outside of a biological cell in much the same way that human skin serves as the body’s barrier to the outside environment.

In the PNAS paper, Atzberger and his graduate student Jon Karl Sigurdsson worked in collaboration with the experimental laboratory of Patricia Bassereau and David Lacoste at the Institut Curie in Paris, to develop new mathematical approaches to gain insights into how proteins move around within lipid bilayer membranes.

“Proteins are not just passive voyagers within the bilayer, but rather their very presence can change the local properties of the lipid bilayer membrane in interesting ways,” Atzberger said. “This includes bending the bilayer with a local preferred curvature or changing the nature of a viscous flow. This dual coupling of responding to the local geometry while also affecting it makes it very difficult to formulate concise models and to make predictions.”

To address these issues, Atzberger developed a statistical mechanics description of the membrane sheet and the proteins based on his past work on immersed-boundary approximations. The idea is to treat the heterogeneous membrane-protein material uniformly but use a moving marker to demarcate the parts associated with the proteins. This approach allows for a simple and reliable description, which captures many of the essential features of membrane-protein dynamics. This not only facilitates performing analytic calculations but also carrying out efficient computational simulations.

“It used to be just theory and experiment,” Atzberger added. “Now computation serves an ever more important third branch of science. With simulations, one can take underlying assumptions into account in detail and explore their consequences in novel ways. Of course, theory and abstraction are still very important to gain understanding. What computation provides is the ability to grapple with a level of detail and complexity that is often simply beyond the reach of pure theoretical methods.”

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