Twisted hyperboloid (Strophoid) as a model of beta-barrels in proteins

Using a least-squares fitting procedure, polypeptide backbones of one parallel and seven antiparallel beta-barrels were approximated with various curved surfaces. Although the hyperboloid gave better approximations to all the beta-barrel backbones than the ellipsoid, elliptical cylinder or catenoid, the best approximations were obtained with a novel surface, a twisted hyperboloid (strophoid). The root-mean-square errors between individual beta-barrels and the fitted strophoid surfaces ranged from 0.75 A to 1.64 A. The parameters which determine the strophoid surface allow groups of beta-barrel shapes to be defined according to their barrel twists (i.e. angles subtended by directions of the long axis of cross-section at the top and the bottom of the barrel), course of elliptical cross-sections (either monotonically increasing along the barrel axis, as in cones, or having a middle "waist", as in hyperboloids), and types of backbone curvatures (either convex or concave). The curvatures at individual points of strophoid surface are local, variable quantities related to the local helicity (coil) of the polypeptide backbone, in contrast to values of beta-sheet twist (i.e. dihedral angles subtended by adjacent beta-strands) known to be virtually identical in all the beta-sheets. The variability found in parameters such as barrel shapes and curvatures suggests that simple models (isotropically stressed surfaces, principle of minimal surface tension) proposed in the past to account for beta-barrel shapes are not sufficient. Rather, the complex nature of best-fit theoretical surfaces points to an important role played by a local variability of the forces involved.