Protein loops, solitons, and side-chain visualization with applications to the left-handed helix region

Folded proteins have a modular assembly. They are constructed from regular secondary structures like α helices and β strands that are joined together by loops. Here we develop a visualization technique that is adapted to describe this modular structure. In complement to the widely employed Ramachandran plot that is based on toroidal geometry, our approach utilizes the geometry of a two sphere. Unlike the more conventional approaches that describe only a given peptide unit, ours is capable of describing the entire backbone environment including the neighboring peptide units. It maps the positions of each atom to the surface of the two-sphere exactly how these atoms are seen by an observer who is located at the position of the central C_{α} atom. At each level of side-chain atoms we observe a strong correlation between the positioning of the atom and the underlying local secondary structure with very little if any variation between the different amino acids. As a concrete example we analyze the left-handed helix region of nonglycyl amino acids. This region corresponds to an isolated and highly localized residue independent sector in the direction of the C_{β} carbons on the two-sphere. We show that the residue independent localization extends to C_{γ} and C_{δ} carbons and to side-chain oxygen and nitrogen atoms in the case of asparagine and aspartic acid. When we extend the analysis to the side-chain atoms of the neighboring residues, we observe that left-handed β turns display a regular and largely amino acid independent structure that can extend to seven consecutive residues. This collective pattern is due to the presence of a backbone soliton. We show how one can use our visualization techniques to analyze and classify the different solitons in terms of selection rules that we describe in detail.