Quaternion-based definition of protein secondary structure straightness and its relationship to Ramachandran angles

We describe here definitions of "local helical axis" and "straightness" that are developed using a simple quaternion-based analysis of protein structure without resort to least-squares fitting. As part of this analysis, it is shown how quaternion differences can be visualized to depict accurately the local helical axis relating any two adjacent amino acid residues in standard, nonidealized proteins. Three different options for the definition of amino acid residue orientation in terms of quaternion frames are described. Two of these, the "C(α) frame" and the "P frame," are shown to be correlated strongly with a simple approximate measure derived solely from Ramachandran angles. The relationship between quaternion-based straightness and recognized DSSP-derived secondary structure motifs is discussed.