Transformation of the ϕ-ψ plot for proteins to a new representation with local helicity and peptide torsional angles as variables

A new representation of protein structure is obtained by the angular coordinate transformations η_i = (?_i+1?ψ_i)/2 and ξ_i = ?_i+1+ψ_i with careful mathematical attention to the cyclical boundary conditions of all of the variables involved. From published ?-ψ data it is possible to obtain a new η-ξ plot. As the angle ξ_i is varied from – 180° through 0° to + 180° in this plot, the local helicity of the polypeptide chain changes continuously and contiguously without sudden reversals in handedness. The variable, η_i, gives the torsional position of the ith peptide group. Some peptide groups in proteins, such as the second peptide residue in a type II β-turn, are nonhydrogen-bonded and can undergo considerable torsional oscillation. In such cases the η angle should be represented by a line whose length reflects the allowed dynamical variations in the peptide torsional position. Certain peptide residues in proteins may be able to undergo a complete torsional rotation of 360°. Such residues would be represented on the η-ξ plot as a straight line across the plot parallel to the abscissa. Other examples of the possible usefulness of this plot are also given.