The role of phospholipid asymmetry in the transition from the lamellar (L(alpha)) to the inverted hexagonal (H(II)) phase upon the temperature increase was considered. The equilibrium configuration of the system was determined by the minimum of the free energy including the contribution of the isotropic and deviatoric bending and the interstitial energy of phospholipid monolayers. The shape and local interactions of a single lipid molecule were taken into account. The minimization with respect to the configuration of the lipid layers was performed by a numerical solution of the system of the Euler-Lagrange differential equations and by the Monte Carlo simulated annealing method. At high enough temperature, the lipid molecules attain a shape exhibiting higher intrinsic mean and deviatoric curvatures, which fits better into the H(II) phase than into the L(alpha) phase. Furthermore, the orientational ordering of lipid molecules in the curvature field expressed as the deviatoric bending provides a considerable negative contribution to the free energy, which stabilizes the nonlamellar H(II) phase. The nucleation configuration for the L(alpha)-H(II) phase transition is tuned by the isotropic and deviatoric bending energies and the interstitial energy.

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In this study we present experimental and theoretical results which concern the deviations from circularity of the pivotal plane in the inverse hexagonal phases (H(II)) of phospholipid self-assemblies. Due to packing constraints, the cross-section of the polar/apolar interface deviates from a circle, which we studied in minute detail by analysing small-angle X-ray diffraction data of dioleoyl-phosphatidylethanolamine (DOPE) and stearoyl-oleoyl-phosphatidylethanolamine (SOPE), respectively. On this structural basis, Monte Carlo (MC) simulated annealing variations of the free energy were carried out, both on the formation of the H(II)-phase and on the particular shape of the cross-section in the H(II)-phase. The equilibrium of the H(II)-phase pivotal plane contour and the corresponding values of the mean intrinsic curvature, H(m), and the hydrocarbon chain stiffness, τ, were determined from MC calculations. The results of these calculations were tested by solving the corresponding system of non-linear differential equations derived using variational calculus. Here our main aim is to predict the range of possible values of H(m) and τ. Comparing the measured structural data with predictions from MC calculations including lipid anisotropy, and accounting for the elastic deformations of the pivotal plane allowed us to determine a relationship between the bending deformation and stretching of hydrocarbon chains.

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In this paper we describe the general forms of surjective isometries of the space of all probability distribution functions on R with respect to the Kolmogorov-Smirnov metric.