[CCoE Notice] Dissertation Defense: Variational formulation of charged curves on a sphere

[Grayson, Audrey A]

PhD DEFENSE STUDENT: Vikash Chaurasia

DATE: Tuesday, July 12, 2018

TIME: 10am – 1pm

PLACE:  Mechanical Engineering Large Conference Room

COMMITTEE: Prof. Yi Chao-Chen, Prof. Shailendra Joshi., Prof. Kalyana Babu Nakshatrala, Prof. Gemunu Gunaratne, Prof. Ralph Metcalfe, Prof. Eliot Fried



ADVISORS: Prof. Yi Chao-Chen and Prof. Eliot Fried



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TITLE:

Variational formulation of charged curves on a sphere





In biological structures, charged proteins that are constrained to a surface are ubiquitously encountered. These proteins interact among each other to maintain their shapes and drive the functionality of the overall structure. Among the various such situations, we particularly focus on mature HDL (high density lipoprotein) molecules which are key components of Reverse Cholesterol transport that regulate the cholesterol level in our body. Mature HDL molecules are spheroidal shells with cholesterol encapsulated in their core, the outer shell being constituted of head groups of lipid molecules and charged proteins that are present on the outer surface of it. How these charged proteins arrange themselves on the surface of the sphere could be of key importance in the functionality of HDL molecules.



In our study, we model charged curves that are restricted to the surface of a rigid sphere. Our model consists of two interacting, inextensible elastic loops (closed curves) that are constrained to lie on a sphere. Each loop is endowed with bending energy, a distance-dependent self-interaction energy, and a distance-dependent energy that ac- counts for interactions with the other loop. The first and second variation conditions are obtained in coordinate-free form. A trivial equilibrium solution exists for such a problem in form of a pair of parallel circular loops situated in opposite hemispheres. We discuss how the interplay between the electrostatic interactions of the loops and their bending rigidity governs the stability of the trivial solution. Motivated by the understandings developed with this analysis, we solve the equilibrium equations numerically and obtain the post-buckled equilibrium configurations.



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