Network Scaling Invariants Help to Elucidate Basic Topological Principles of Proteins

The structural architecture of proteins continues to be an area of active research. Despite the difference
in models dealing with the way proteins fold into their tertiary structures, it is recognized that small
regions of proteins tend to fold independently and are then stabilized by interactions between these
distinct subunits. However, there are a number of different definitions of what comprises an independent
subunit. In the belief that an unequivocal definition of a domain must be based on the most fundamental
property of protein 3D structure, namely, the adjacency matrix of inter-residues contact, we adopt a
network representation of the protein. In this work, we used a well-established, global method for
identifying modules in networks, without any specific reference to the kind of network being analyzed.
The algorithm converges toward the maximization of the modularity of the given protein network and,
in doing so, allows the representation of the residues of the protein in terms of their intramodule degree,
z, and participation coefficient, P. We demonstrate that the labeling of residues in terms of these
invariants allows for information-rich representations of the studied proteins as well as to sketch a
new way to link sequence, structure, and the dynamical properties of proteins. We discovered a strong
invariant character of protein molecules in terms of P/z characterization, pointing to a common
topological design of all protein structures. This invariant representation, applied to different protein
systems, enabled us to identify the possible functional role of high P/z residues during the folding
process. Additionally, we observe a hierarchical behavior of protein structural organization that provides
a sequence-secondary-tertiary structure link. The discovery of similar and repeatable scaling laws at
different level of definitions going from hydrophobicity patterning along the sequence up to the size
of an autonomous folding unit (AFU) and general contact distribution of the entire molecule suggest
a hierarchical-like behavior of protein architecture. This implies the possibility to select different
privileged scales of observation for deriving useful information on protein systems.
Keywords: proton topology • protein networks • autonomous folding units • protein folding • residue centrality •
modules • protein structure graphs