Residue Level Inverse Kinematics of Peptide Chains in the Presence of Observation Inaccuracies and Bond Length Changes

Dihedral angles as generalized coordinates define the geometric conformation of a peptide chain. Given the exact coordinates of the atoms, it is possible to rigorously calculate the dihedral angles. We will refer to this calculation process as the residue level inverse kinematics of peptide chains. However uncertainties and experimental observation inaccuracies in the atoms’ coordinates handicap this otherwise simple and straightforward process. In this paper, we present three new efficient methodologies to find all the dihedral angles of a peptide chain for a given conformation. Comparison of these results with the dihedral angle values reported in the PDB (Protein Data Bank) indicates significant improvements. While these improvements benefit most modeling methods in protein analysis, it is in particular, very significant in homology modeling where the dihedral angles are the structural variables. The first method presented here fits a best plane through five atoms of each peptide unit. The angle between the successive planes is defined as the dihedral angle. The second method is based on the Zero-Position analysis method. Successive links in this method rotate by the dihedral angles so as to minimize the structural error between respective atoms in the model conformation with given atoms’ coordinates. Dihedral angle final values correspond to the minimum structural error configuration. In this method, singular value decomposition (SVD) technique is used to best fit the atoms in the two conformations. The third method is a variant of the second method. In this instead of rotating all the links successively only three links are matched each time to extract the dihedral angle of the middle link. By doing so, the error accumulation on the successive links is reduced. This paper focuses on the Euclidean norm as the measure of merit (structural error) to compare different methods with the Protein Data Bank (PDB). This Euclidean norm is further, minimized by optimizing the geometrical features of the peptide plane.