On the Analysis of Elastic Deformations in Hexagonal Phases

Abstract

The elastic parameters of lipid mesophases, such as bending modulus and intrinsic curvature, have commonly been obtained from the response of inverted hexagonal (HII) phase to osmotic stress. HII phase consists of a large number of hexagonally packed rods, one of which is shown in cross section in Fig. 1. In the original work on this approach (Rand et al., 1990), in which these rods were assumed to be circular cylinders, only two terms of the free energy of HII phase were accounted for, the elastic bending energy of the HII monolayer and the osmotic energy (normalized per lipid molecule): (1) where Kb is the bending modulus, Ap is the molecular area, and Rp and R0p are the actual local radius of curvature and the intrinsic radius of curvature at the pivotal plane, respectively. Π is the difference in osmotic pressure between the outside and inside of the HII cylinder, and Vw is the volume of water phase per lipid inside the HII cylinder. Since hexagonal phase is normally studied in distilled water in the absence of solutes, the osmolality inside does not change when the cylinder radius changes, i.e., Π is normally constant. The pivotal plane is defined as a surface inside the lipid phase such that both Ap and Vp are constant when the distance between rod axes varies. Vp is the volume per molecule between the Luzzati plane (a plane that divides the lattice into purely water and lipid phases) and the pivotal plane. More rigorous and complex definitions of these planes can be made (Leikin et al., 1996), but the practical definitions of Rand et al. (1990) suffice for this discussion. The minimum of the free energy as a function of only Rp at any given osmotic pressure Π can be resolved analytically and is defined by the simple expression: (2) A plot of versus 1/Rp yields a straight line, whose slope gives 2Kb and whose intercept defines 1/R0p. As a confirmation of this expression, experimental data from several studies are reasonably well described by straight lines. Open in a separate window FIGURE 1 Cross section of the HII phase. The cross-sectional structure of hexagonally packed cylinders (delimited by thick solid lines) in HII phase is illustrated. Labeled are the Luzzati (W) and pivotal (P) planes and the distances from the center to these two planes along the interaxial distance (dw and dp). The imaginary hydrophobic plane (I) is a constant distance, h, from the Luzzati plane, leading to the hydrophobic interstices, which are shaded gray. The locations of these planes in the conventionally assumed circular cylindrical geometry is illustrated by dashed boundaries, whereas the solid boundaries illustrate the more complex planes predicted by our calculations.