Generalized Crick equations for modeling noncanonical coiled coils

Crick envisaged the alpha-helical coiled coil to result from systematic bending of an alpha-helix such that every seventh residue was structurally equivalent, and he derived equations for the coordinates of the backbone atoms. Crick's predictions were vindicated experimentally and coiled-coil sequences were shown to have hydrophobic residues alternately spaced 3 and 4 residues apart. Nonetheless, in some coiled coils such canonical heptad repeats are interrupted by inserts of 3 or 4 residues generating decad and hendecad motifs. The supercoiling of the coiled coils varies with the sequence pattern, being left- or right-handed in purely heptad-based or hendecad-based motifs, respectively. To model coiled coils with a mixture of motifs, we describe how Crick's equations can be modified for cases where the pitch is not constant. Using the analogy of the bending of a beam, we took the tilt angle to change linearly with distance along the major helix and the pitch of a motif to be affected by neighboring motifs depending on the rigidity of the alpha-helical strands. We tested our approach by fitting the two-, three-, and four-stranded noncanonical coiled coils of GrpE, hemagglutinin, and tetrabrachion. The backbone atoms of the model and crystal structures agreed with root mean square deviations of <1.1 A.