Theory of helical structures of tilted chiral lipid bilayers

Analogous to the theory of cholesteric liquid crystal, a theory for tilted chiral lipid bilayers (TCLB) [briefly reported in Phys. Rev. Lett. 65, 1679 (1990)] is developed. The tilt equation and the surface-shape-equilibrium equation have been derived. Application of the theory shows that on a cylindrical surface, the field of tilt directions of TCLB molecules forms a right-handed helix with a 45° gradient angle for ${\mathit{k}}_2$>0 or a left-handed helix with -45° gradient angle for ${\mathit{k}}_2$${\mathit{k}}_2$ is the chiral curvature modulus of the cholesteric liquid crystal. In addition, there exists another type of helical structure that looks like twisted strips or helicoids. It is also shown that on a spherical surface, the tilt field has at least two singular points. Based on these results, the succession of transitions from a vesicular dispersion to a phase involving helical structures and then to tubes observed in several laboratories can be reasonably explained. The general formula that explains the aggregation of narrow and prolate ribbon structures is derived. By including more terms of elasticity free energy into the theory, the size of the helical structure is obtained and is shown to agree with experimental observations.