The application of graph theory in structural biology offers an alternative means of studying 3D models of large macromolecules such as proteins. 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 (MND) 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. 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 match along 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 analyze the distribution of the central and peripheral amino acids/nodes of the macromolecular structure.

中文翻译：

---

3D 蛋白质结构图的缩放规律

图论在结构生物学中的应用为研究大分子（如蛋白质）的 3D 模型提供了另一种方法。回转半径，随指数缩放～0.4，提供有关蛋白质结构紧凑性的定量信息。在这项研究中，我们结合了两种经过验证的方法，即图论和基本比例定律，来研究 3D 蛋白质模型。这项研究表明，蛋白质图的平均节点度 (MND) 以指数 0.038 为尺度，是尺度不变的。此外，大小不同的蛋白质具有高度相似的节点度分布。线性回归分析表明，图形参数（半径、直径和平均偏心率）可以解释高达 90% 的总回转半径方差。因此，半径、直径和平均偏心率比例的图形参数与回转半径的指数相同。