We report the results of x-ray-scattering studies of individual helical ribbons formed in multicomponent solutions of cholesterol solubilized by various surfactants. The solutions were chemically defined lipid concentrate (CDLC) and model bile. In these and many analogous multicomponent surfactant-cholesterol solutions, helical ribbons of two well defined pitch angles, namely 11°and 54°, are formed. We have suggested previously that this remarkable stability results from an underlying crystalline structure of the sterol ribbon strips. Using a synchrotron x-ray source, we have indeed observed Bragg reflections from individual ribbons having 11°pitch angle. We have been able to deduce the parameters of the unit cell. The crystal structure of these ribbons is similar to that of cholesterol monohydrate, with the important difference that the length of the unit cell perpendicular to the cholesterol layers is tripled. We discuss possible origins for this triplication as well as the connection between the crystalline structure and the geometrical form of the helical ribbons.crystal structure ͉ x-ray diffraction ͉ crystallization ͉ surfactants S elf assembly of helical ribbons in complex fluids is an interesting phenomenon, which poses fundamental questions about the molecular structure, elastic properties, and kinetic evolution of these objects. In particular, quaternary solutions, which contain cholesterol, nonionic surfactants, and lipids, spontaneously form helical ribbons with characteristic pitch angles of 11°and 54°. These helical ribbons are long rectangular strips, which curl along a cylindrical surface. These objects were discovered in human gallbladder bile, where they form spontaneously upon the dilution of bile. This dilution produces a solution supersaturated with respect to cholesterol (1). Formation of similar helical ribbons has been later reported in Ͼ20 different solutions with various sterols analogous to cholesterol, surfactants, and phospholipids or fatty acids (2). These helical ribbons form in a variety of axial lengths, widths, and radii. Remarkably, however, almost all have pitch angles of either 11°o r 54°. Several theoretical models have been proposed to explain the formation and properties of helical ribbons, which sometimes form in complex fluids containing chiral amphiphilic molecules (3-10). These theories were designed to describe the helical ribbons in solutions containing a single species of phospholipids, which can form bilayers. Therefore, the ribbons were modeled theoretically as fluid bilayers, where hydrophobic carbon chains are sandwiched between hydrophilic head groups. Calculations of the properties of such membranes are usually based on the curvature elasticity model (11), which is founded on general physical arguments about the dependence of the fluid membrane elastic free energy on its curvature. Based on this model, subsequent theories attempted to explain the geometrical and elastic properties of the helical ribbons. For example, the formation of helices is attributed to b...

Using quantitative phase microscopy, we have discovered a quadratic relationship between the radius R and the thickness t of helical ribbons that form spontaneously in multicomponent cholesterol-surfactant mixtures. These helical ribbons may serve as mesoscopic springs to measure or to exert forces on nanoscale biological objects. The spring constants of these helices depend on their submicroscopic thickness. The quadratic relationship (R ؔ t 2 ) between radius and thickness is a consequence of the crystal structure of the ribbons and enables a determination of the spring constant of any of our helices solely in terms of its observable geometrical dimensions.biological force spectroscopy ͉ elasticity of thin films ͉ phase-contrast microscopy in biophysics T he elastic properties of meso-and nanoscale thin elastic strips forming helical ribbons or tubules, have been the focus of active recent research in both biophysics and nanoscience communities (1-7). We have discovered that in a number of complex aqueous solutions containing a sterol (cholesterol in particular) and a mixture of surfactants, the sterol molecules may self-assemble into ribbons of helical shape (8). The geometry of the helical ribbons is characterized by the radius, width, thickness, contour length, and pitch angle, see figure 1a in ref. 9. Remarkably, the pitch angle is always either 11°or 54°, whereas axial length, width, and radius vary by two orders of magnitude in the range from 1 to Ϸ100 m. These helical ribbons are fascinating objects for fundamental studies (2,(8)(9)(10). Furthermore, because low-pitch helical ribbons have spring constants in the range of 0.5 to 500 pN/m (2), and the elongation of these springs from 1 m up to 100 m can easily be observed microscopically, it follows that they can be used as mesoscopic spring scales to measure forces between nanoscale biological objects in the range from 0.5 pN to 50 nN. For this and other applications, the ability to readily determine the spring constants of individual helixes is of crucial importance. In this article, we establish the relationship between the spring constant of the low-pitch cholesterol helical ribbons and its readily observable dimensions: width, radius, and length.Originally, it had been thought that cholesterol helical ribbons formed in surfactant mixtures had liquid crystalline structure and that their shape was governed by elastic properties of liquid crystalline layer (9,11,12). Recently, we have shown by X-ray diffraction that these helical ribbons are, in fact, single crystals with structure closely resembling that of cholesterol monohydrate (10). Having in mind the single-crystal nature of our ribbons, we have proposed that their helical shape is determined by a balance between two terms in the free energy of deformation of the cholesterol crystalline strip (2). The first term, the spontaneous bending energy, favors curling toward one of the two faces of the ribbon and is linear in curvature, ϪK s /R. The second term is the elastic energy of bending a strip. Thi...