Mapping all-atom models onto one-bead coarse-grained models: General properties and applications to a minimal polypeptide model

In the one- and two-bead coarse-grained (CG) models for proteins, the two conformational dihedrals phi and psi that describe the backbone geometry are no longer present as explicit internal coordinates; thus, the information contained in the Ramachandran plot cannot be used directly. We derive an analytical mapping between these dihedrals and the internal variable describing the backbone conformation in the one- (two-) bead CG models, namely, the pseudo-bond angle and pseudo-dihedral between subsequent C alpha's. This is used to derive a new density plot that contains the same information as the Ramachandran plot and can be used with the one- ( two-) bead CG models. The use of this mapping is then illustrated with a new one- bead polypeptide model that accounts for transitions between alpha helices and beta sheets.