  Folded proteins have a modular assembly. They are constructed from regular
secondary structures like alpha-helices and beta-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 only
describea 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-alpha 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 non-glycyl amino acids. This
region corresponds to an isolated and highly localized residue independent
sector in the direction of the C-beta carbons on the two-sphere. We show that
the residue independent localization extends to C-gamma and C-delta 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 beta-turns display a regular
and largely amino acid independent structure that can extend to seven
consecutive residues. This collective pattern is duu 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.
