Geometrical criteria for formation of coiled-coil structures of polypeptide chains

Crick's general formulas describing a coiled coil are expressed in a different form to combine the parameters of a coiled coil with the backbone dihedral angles of a polypeptide chain, assuming that the bond lengths and bond angles of the chain are fixed. While the existence of a low-energy coiled-coil conformation depends on energetic considerations, these formulas, which pertain to single-stranded structures and, by application of symmetry operations, to multistranded structures, provide the geometrical criteria for the existence of coiled coils. The concept of "the averaged structure of the minor helix", introduced here, makes it possible to relate the shape of the major helix to that of the minor helix. It is shown, in the analysis of a simple model of a single-stranded coiled-coil beta structure, that strong geometrical restrictions exist for the formation of coiled-coil structures from a given minor helix conformation of a polypeptide chain; these restrictions are expressed in a general form that is applicable to any coiled-coil of any number of residues in a repeat unit. As an application, the possible existence of a two-stranded coiled-coil antiparallel beta structure is considered, both geometrically and energetically, and discussed in relation to the observed twisted beta structures in globular proteins. The proposed coiled-coil models of alpha-helical proteins are also examined briefly.