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For details see paper https://doi.org/10.1142/S021972002050050X File: DataSet.csv Column description PDBid ... Protein Data Bank identification CAatoms ... number of CA atoms (residues) Rg-rgyr ... bio3d (R) Radius of gyration MND ... Mean Node Degree Radius ... min eccentricity - Radius MEC ... mean eccentricity Diameter ... max eccentricity - Diameter central ... number of central nodes in graph peripheral ... number of peripheral nodes in graph tot.solv.A. ... total solvent-accessible area of protein File: DataSetHistogram.csv column ... histogram breaks row ... PDBid Abstract The 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.

Averaged kick maps are the sum of a series of individual kick maps, where each map is calculated from atomic coordinates modified by random shifts. These maps offer the possibility of an improved and less model-biased map interpretation., Use of reliable density maps is crucial for rapid and successful crystal structure determination. Here, the averaged kick (AK) map approach is investigated, its application is generalized and it is compared with other map-calculation methods. AK maps are the sum of a series of kick maps, where each kick map is calculated from atomic coordinates modified by random shifts. As such, they are a numerical analogue of maximum-likelihood maps. AK maps can be unweighted or maximum-likelihood (σA) weighted. Analysis shows that they are comparable and correspond better to the final model than σA and simulated-annealing maps. The AK maps were challenged by a difficult structure-validation case, in which they were able to clarify the problematic region in the density without the need for model rebuilding. The conclusion is that AK maps can be useful throughout the entire progress of crystal structure determination, offering the possibility of improved map interpretation.