## 3 Results and Discussion
### Fluctuations and Force Application on the pfANM
Figure 1: Correlations of experimental B-factors with ENM-based fluctuations and the average displacements due to random perturbations (pfANM): (**a,b**) correlation coefficients for the analyzed 919 protein structures (blue for \(\rho_{FL}\), orange for \(\rho_{FKL}\), and red for \(\rho_{FLEX}\)); (**c**) statistical distribution of the obtained correlation coefficients. The results are all very similar, showing relatively little differences among them. Figure 2: Dependence of the generated external boundary (external protein surface) on the value of the shrink factor. Infrared fluorescent protein (PDB: 5 AJG) is reported as an example, with shrink factors equal to 0, 0.2, 0.4, 0.6, 0.8 and 1. Red points represent the nodes of the network (C\({}^{\alpha}\) atoms), while the external surface is represented by Delaunay triangles (in light blue), which connect the nodes in the external boundary. For each of these generated surfaces, the random forces were applied only to the nodes lying on the external boundary. The resulting network displacements were then evaluated from Equations (4)-(7). It follows that the correlation of the average displacements (from 10,000 samples) due to the application of the external forces, i.e., \(\rho_{FR,EXT}\), also depends on the adopted shrink factor. The shrink factor that leads to the maximum value of \(\rho_{FR,EXT}\) for each protein is then selected as the optimal one. Figure 3 reports the statistical distribution of the optimal shrink factors obtained for the 919 single-chain proteins. As can be observed, the optimal shrink factor assumes almost all values, meaning that it is strongly protein-specific. Nevertheless, a slight preference towards shrink factors equal to 1.0 is observed. In Figure 4 we show results for the example of the infrared fluorescent protein (PDB: 5 AJG), where the correlation coefficients \(\rho_{FL}\), \(\rho_{FR,ALL}\) and \(\rho_{FR,EXT}\), are shown depending on the shrink factor for the external surface representation. From these calculations, we obtain a correlation between the B-factors and the traditional mode-based fluctuations \(\rho_{FL}\) equal to 0.56, a correlation with the displacements resulting from the application of forces to the entire structure \(\rho_{FR,ALL}\) equal to 0.62, and a correlation derived from perturbations only on the external surface \(\rho_{FR,EXT}\) which varies with the shrink factor and reaches a maximum value of 0.62 for an optimal shrink factor of 1.0. As can be seen from the results, in this case, applying random forces on the protein network enhances the correlation with the experimental B-factors of 6% (0.62 vs. 0.56) compared with the usual mode-based fluctuations. This result points out the cohesive nature of the protein structure, showing that the point of application of forces does not matter much, with the result of applying forces in all possible directions on the surface yielding nearly the same result as applying them in all directions throughout the structure when the surface representation is detailed enough. Figures S1-S5 in the Supplementary Material report similar results, obtained by adopting different values of \(p\) for the decay exponent of the ENM spring constants (\(p=1\), \(2\), \(4\), \(6\) and \(12\)). As can be seen there, similar conclusions can be drawn for these cases. Note that, for this protein, higher values of \(p\), e.g., \(p=4\) and \(6\), lead to a greater enhancement in the correlation with experimental B-factors when the ENM forces are applied, rather than just looking at the intrinsic dynamical fluctuations. Figure 3: Statistical distribution of the optimal shrink factor, based on a comparison between experimental B-factors and average displacements due to the application of random forces on the exterior protein surface.