Protein secondary structure motifs: A kinematic construction

The kinematic geometry of protein backbone structures, constrained by either single or multiple hydrogen bonds (H-bonds), possibly in a periodic array, is discussed. These structures include regular secondary structure elements α-helices and β-sheets but also include other short H-bond stabilized irregular structural elements like β-turns. The work here shows that the variations observed in such structures have simple geometrical correlations consistent with constrained motion kinematics. A new classification of the ideal helices is given, in terms of the parameter α, the angle at a C_α atom to its two neighboring C_α 's along the helix, and shown how it can be generalized to include nonideal helices. Specifically, we derive an analytical expression of the backbone dihedrals, (ϕ, ψ), in terms of the parameter α subject to the constraint that the peptide planes are parallel to the helical axis. Helices constructed in this way exhibit near-vertical alignment of the C = O and N - H units and are the canonical objects of this study. These expressions are easily modifiable to include perturbations of parameters relevant to nonplanar peptide units and noncanonical angles. The addition of a second parameter, ε₀ , inclination of successive peptide planes along a helix with respect to the helical axis leads to a generalization of the previous expression and provides an efficient parametrization of such structures in terms of coordinates consistent with H-bond parameters. An analogs parametrization of β-turns, using inverse kinematic methods, is also given. Besides offering a unifying viewpoint, our results may find useful applications to protein and peptide design.