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...