Gaussian fluctuations of lipid bilayer vesicles: a numerical study

Fluid-phase phospholipid vesicles are abundant in biological cells and play an essential role in cellular function. Artificial phospholipid “liposomes” can be synthesized in the lab. Observed vesicle shapes are governed by the elastic energy of the enclosing membrane. Contributions to this energy come from bilayer bending and from the so-called area-difference-elasticity (ADE) energy associated with the relative stretching/compression of membrane monolayers generated by the requirement of vesicle closure. We construct a shape-energy functional representing these elements and identify observable vesicle shapes as local energy minima at fixed values of membrane area and vesicle volume. As control parameters vary, low-energy observable shape classes may be organized into a “phase diagram” in analogy with thermodynamics. At any temperature $T>0,$ thermal excitations generate shape fluctuations. Such fluctuations are generally small at room temperature but become enhanced close to the instability thresholds. In this thesis, we calculate numerically vesicle shapes and Gaussian-level shape fluctuations. Our focus is on axisymmetric shapes of spherical topology with volumes significantly less than the maximum allowed by given membrane area. For each minimum-energy shape, we show how to calculate the full spectrum of thermal shape fluctuations available at strictly fixed membrane area and volume. Euclidean modes, not associated with shape change, are systematically removed. The hard geometric constraints lead to thermally-induced shifts in the average shape, which would not occur in a standard Gaussian analysis without constraints. We show how to calculate these shifts but conclude that they are generally negligible at laboratory temperatures. Results are illustrated by the case study of a stable prolate vesicle, for which energy eigenvalues, shape eigenmodes, and static correlations are all given. A final chapter develops specific applications: (a) Constructing the phase diagram by analyzing energy levels; (b) Finding instability boundaries by tracking soft modes; (c) Illustration of characteristic spectra and eigenmodes for representative shape classes; (d) Calculation of mean-square shape fluctuations and two-point correlation functions. In particular, we confirm the expected ``stiffness" of narrow-necked shapes.