Scaling laws of graphs of 3D protein structures

Abstract

AbstractThe application of graph theory in structural biology offers an alternative means of studying 3D models of large macromolecules, such as proteins. However, basic structural parameters still play an important role in the description of macromolecules. For example, the radius of gyration, which scales with exponent ~0.4, provides quantitative information about the compactness of the protein structure. In this study, we combine two proven methods, the graph-theoretical and the fundamental scaling laws, to study 3D protein models.This study shows that the mean node degree of the protein graphs, which scales with exponent 0.038, is scale-invariant. In addition, proteins that differ in size have a highly similar node degree distribution, which peaks at node degree 7, and additionally conforms to the same statistical properties at any scale. Linear regression analysis showed that the graph parameters (radius, diameter and mean eccentricity) can explain up to 90% of the total radius of gyration variance. Thus, the graph parameters of radius, diameter and mean eccentricity scale with the same exponent as the radius of gyration. The main advantage of graph eccentricity compared to the radius of gyration is that it can be used to analyse the distribution of the central and peripheral amino acids/nodes of the macromolecular structure. The central nodes are hydrophobic amino acids (Val, Leu, Ile, Phe), which tend to be buried, while the peripheral nodes are more hydrophilic residues (Asp, Glu, Lys). Furthermore, it has been shown that the number of central and peripheral nodes is more related to the fold of the protein than to the protein length.