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.