We report measurements of the geometrical a In 1986, Helfrich (10) proposed a simple elastic model in which the optimal shape of a helical ribbon is controlled by a balance between bending of the ribbon and torsion of its edges, leading to a universal pitch angle of 450 for an elastically isotropic ribbon. Two years later Helfrich and Prost (11) generalized the bending free energy for anisotropic bilayers. To account for the lack of mirror reflection symmetry of bilayers of chiral molecules, they also added a term linear in the surface curvature to the elastic free energy (11).Despite these refinements, the analysis of the pitch angle was made only for an isotropic bilayer and the pitch angle remained 45°. In 1990, Ou-Yang and Liu (12) modeled the The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. helical ribbon as a cholesteric liquid crystal layer and also obtained the pitch angle of 45°.Herein, we describe experimentally observed helical ribbon structures with either of two distinctive pitch angles: 540 or 110. We also present a form for the elastic free energy of anisotropic chiral bilayers that provides analytical expressions for the geometrical features and growth of the helical ribbon structures observed in our biles.
MATERIALS AND METHODSAll model biles were composed of the common bile salt sodium taurocholate, lecithin, and cholesterol in a molar ratio of 97.5:0.8:1.7. Sodium taurocholate (Sigma) was recrystallized and found to be >98% pure by HPLC and TLC. Cholesterol (Nu Chek Prep, Elysian, MN) was >99% pure by TLC, GC, and HPLC. Grade I egg yolk lecithin (Lipid Products, Nutley, U.K.) was >99% pure by TLC. Synthetic sn-1-16:0-sn-2-18:1 lecithin (lecithin I) and sn-1-16:0-sn-2-16:0 lecithin (lecithin II) were obtained from Avanti Polar Lipids. Model biles A, B, C, and D contained lecithin I, a 1:1 mixture of lecithin I and lecithin II, egg yolk lecithin, and lecithin II, respectively. Native bile was a generous gift from M. Cahalane (Beth Israel Hospital, Boston) (2). Using TLC, no degradation of the components of bile was observed during our experiments.At a total lipid concentration of 70 mg/ml (0.15 M NaCl/3 mM NaN3), all four model biles formed micellar solutions with a mean hydrodynamic radius of 17 ± 5 A (2, 13).Meticulous precautions were taken to reduce the nucleating effect of dust. A 5-ml vial with a Teflon septum cap was acid-cleaned, rinsed, filled with Milli-Q water (Millipore), Abbreviation: ChM, cholesterol monohydrate crystal.§Present address: