The shapes of backbones of chain molecules: three-dimensional characterization by spherical shape maps

In this work we present a new method to characterize the shape of flexible chain molecules. The procedure associates a sphere and a spherical shape map with each given molecular backbone. Each point on the sphere is classified according to the crossing pattern obtained when the backbone is looked at along a direction defined by the center of the sphere and the chosen point. The approach is simple to implement. It consists of the following steps: (a) enclosing the backbone by a sphere, whose center is the geometrical center of the backbone; (b) projecting the backbone onto a plane tangent to the sphere at the given point; and (c) characterizing the resulting planar curve by a vector specifying the number and handedness type of the crossings. When the procedure is repeated for all points on the spherical surface, the latter is divided into regions that are equivalence classes of points, corresponding to directions from where the backbone has the same overcrossing pattern. The computation of these equivalence classes is performed automatically by the computer, by determining the boundary of the regions characterized by different crossing vectors. The characterization provided is thus direction independent since it takes into account all possible directions from where a backbone can he analyzed. The procedure is illustrated for a number of supersecondary protein structures and small proteins. We find that a characterization of substructures can be obtained in terms of the arrangement of the equivalence region for the viewing directions from where the backbone shows no crossings. For instance, an alpha-helix is represented by a spherical map with a small "band" region of no crossings perpendicular to the helical axis. Other supersecondary structural features are described in a similar fashion. A number of refinements of the method, based on the distances between crossings, are also given for the case of irregular backbones.