Large deviations of the average shapes of vesicles from equilibrium:Effects of thermal fluctuations in the presence of constraints

In the absence of external stresses, the surface area and the volume of a closed, flaccid lipid vesicle are practically constant. Thermal shape fluctuations of vesicles that are subject to these constraints recently have been shown to induce a shift of the average shapes away from the equilibrium (zero temperature) shapes. Since only the average shapes can be determined from observations by optical microscopy, it is important to establish the magnitude of their deviation from the well-studied equilibrium shapes. In this paper we develop a formalism to calculate this thermal shift, and we demonstrate that nonlinearities in the constraints may cause it to be unexpectedly large. Allowing for arbitrary shape deformations, we present numerical calculations revealing a logarithmic dependence of the thermal shift on the number of fluctuational degrees of freedom of the vesicle membrane or, equivalently, on the number of lipid molecules constituting the membrane. As a consequence, the surface area ('projected area') and, to a lesser extent, the volume ('projected volume') of the average shape are smaller than their true values. These numerical results are in general agreement with theoretical predictions that have been made so far only for pieces of flat membranes but not for closed lipid membranes subject to the constraints of both constant area and volume. Furthermore, we derive an expression for the correlation function of deviations from equilibrium including terms of the order of (${\mathrm{k}}_{\mathrm{B}}$ T${)}^2$ that involve the (quadratic) thermal shift. We demonstrate that these terms may actually exceed the commonly used leading term of the correlation function. This analysis suggests that the determination of the membrane bending modulus ${\mathrm{k}}_{\mathrm{c}}$ from observations of thermal vesicle shape fluctuations should be based on the variances rather than the correlation functions.