Thickness-radius relationship and spring constants of cholesterol helical ribbons.

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 proportional to 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.

Structure of cholesterol helical ribbons and self-assembling biological springs.

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 degrees and 54 degrees, 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 degrees 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.