## 3 Results and Discussion
### Comparison between pfANM and wpfANM Results  
In the previous sections, it has been shown that perturbing the protein structure with random forces, either on the entire structure or on the surface, generally leads to a fairly accurate prediction of the protein fluctuations and flexibility. In a large number of cases, it has also been found that the agreement with the experimental B-factors was improved  
Figure 9: Experimental B-factors vs. mode-based fluctuations and average displacements due to random forces for the infrared fluorescent protein (PDB: 5 AJG), with PDB water molecules included. The continuous blue line refers to \(\rho_{W,FL}\), the dashed orange line to \(\rho_{W,F,F,ALL}\), the dotted red line to \(\rho_{W,F,F,EXT}\), and the dashed-dotted green line to \(\rho_{W,F,WAT}\). The correlations from the application of forces only on the protein’s external surface, i.e., \(\rho_{W,F,EXT}\), depends on the selected shrink factor, in the range 0–1. For each shrink factor, the values reported close to the marker represent the fraction of external nodes out of the total of 533 nodes (301 C\({}^{\alpha}\) atoms + 252 water molecules) of the network. with respect to considering the traditional fluctuations of the usual elastic network. As an example, the results shown in the previous sections for the infrared fluorescent protein (PDB: 5 AJG) are reported together in Figure 10 in terms of correlation coefficients with the experimental B-factors. Several observations can be made regarding this figure. First, it is evident that the traditionally employed internal protein fluctuations provide the lowest correlation with the experimental data, with a correlation of about 55% (\(\rho_{FL}\)). Conversely, applying random perturbations on the same elastic network leads to a 5% gain in the correlations with the B-factors (\(\rho_{FR,ALL}\) and \(\rho_{FR,EXT}\)). Furthermore, adding the PDB water molecules to the elastic network further improves the correlation with the experimental data. In this case, considering the internal protein fluctuations of the water-enriched elastic network or applying random forces to it yields correlation coefficients of about 85% (\(\rho_{W,FL}\), \(\rho_{W,FR,ALL}\) and \(\rho_{W,FR,EXT}\)). Also, applying random forces only on these water molecules, i.e., not perturbing the protein molecule directly but only the water molecules attached to the network (see Figure 5a), leads to a correlation of about 60% (\(\rho_{W,FR,WAT}\)), which is still higher than the correlation obtained with the traditional mode-based internal protein fluctuations of the ENM (\(\rho_{FL}=55\%\)). Figures S11-S15 in the Supplementary Material show similar outcomes, but are obtained by changing the exponent \(p\) of the ENM. As can be seen, similar conclusions are drawn. In all cases, it is found that perturbing the protein with random forces provides an enhancement of the correlation with experimental B-factors, rather than considering the traditional ENM with only intra-protein interactions. Moreover, the inclusion of waters in the ENM leads to further improvements in the correlation, with some gains in the Pearson coefficient being as high as 30-35%. The example shown in Figure 10 obviously refers to one single case, but these results were found for a decent amount of protein structures. For other protein structures, the addition of water molecules to the protein network led to results which were quite similar to the ones obtained in the classical way, i.e., calculating the mode-based fluctuations of the traditional no-water elastic network. Figure 11a reports the median values and standard deviations of the seven correlation coefficients obtained for the dataset of 919 single-chain protein structures, as reported in the keys of Figures 1c and 6d. As can be observed, the median values lie in the range 0.60-0.65, and present a similar standard deviation (15-20%).