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Final Thesis Defense
Biophysical applications of Lie groups and moving frames
Date: Thursday, August 17th
Time: 2:00 pm
Place: 5310 Chamberlin
Speaker: Wilson B Lough, Department of Physics Graduate Student
Abstract: Living matter is often composed of microstructures which possess rotational degrees of freedom, in addition to the translational degrees of freedom which describe the point-like material particles of classical continuum mechanics. Dynamics of these materials must account for interactions between microstructures which are mediated by couple stresses. Lie theory and Cartan's method of moving frames provides a natural framework in which both translational and rotational aspects of microstructured materials can be treated in a unified manner. In this framework, the state of the material is described by a collection of fields which take values in the special Euclidean group and its Lie algebra. These fields are governed by Euler-Poincaré equations which enforce the local balance of linear and angular momentum. In this dissertation, we develop theoretical and computational tools for modeling these generalized continua. We then apply these methods to a number of biophysical systems.

Remodeling of biological membranes often involves interactions with helical ribbon-like protein filaments which polymerize on the membrane surface. Using a combination of moving frame and level-set methods to describe membrane geometry, we derive the constrained Euler-Poincaré equations governing a surface-bound flexible filament. We provide direct numerical evidence that the filament can undergo growth-induced buckling which results in highly localized forces and moments being applied to the membrane. This lends support to the conjecture that buckling of surface-bound polymers plays a role in overcoming energy barriers which resist topological transitions during cell division and vesicle formation. Our simulations also suggest that the chirality of curvature-sensing proteins plays a crucial role in their ability to navigate the membrane surface.

The twisting and writhing of a cell body and associated mechanical stresses is an underappreciated constraint on microbial self-propulsion. Multi-flagellated bacteria can even buckle and writhe under their own activity as they swim through a viscous fluid. New equilibrium configurations and steady-state dynamics then emerge which depend on the organism's mechanical properties and on the oriented distribution of flagella along its surface. Modeling the cell body as a semi-flexible Kirchhoff rod and coupling the mechanics to a dynamically evolving flagellar orientation field, we derive the Euler-Poincaré equations governing dynamics of the system and rationalize experimental observations of buckling and writhing of elongated swarmer cells of the bacterium Proteus mirabilis. We identify a sequence of bifurcations as the body is made more compliant, due to both buckling and torsional instabilities. Our analysis reveals a minimal stiffness required of a cell, below which its motility is severely hampered.
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