Intrinsic protein geometry with application to non-proline cis peptide planes

The shape of a protein can be modeled by the C ^α atoms of its backbone, the mathematical description employing the notion of extrinsic geometry of a discrete piecewise linear chain. We advance differential geometry of a natively framed discrete chain to argue the existence of two additional, independent and intrinsic geometric structures, provided by the peptide planes and side chains, respectively. We develop our general methodology within a case study: analysis of the intrinsic geometry of atoms that are located around a non-proline cis peptide plane. We show that the native peptide plane framing allows for revealing of atomic positions anomalies. That way, we identify a number of entries that display such anomalies around their non-proline cis peptide planes within the ultrahigh-resolution structures in PDB. We propose that our approach can be extended into a visual analysis and refinement tool that is applicable even when resolution is limited or data is incomplete, for example when there are atoms missing in an experimental construct.