## 3 Results and Discussion
### Comparison between pfANM and wpfANM Results
However, the distribution with the highest median value and the lowest standard deviation was found for the analysis involving the application of random perturbation on the external surface of the wpfANM, i.e., \(\rho_{W,FR,EXT}\). A direct comparison between the statistical distribution of \(\rho_{W,FR,EXT}\) and the one related to the traditional mode-based internal protein fluctuations of the no-water pfANM, i.e., \(\rho_{W,FL}\), is reported in Figure 11b. From the direct comparison of the two  
Figure 10: Infrared fluorescent protein (PDB: 5 AJG): correlation coefficients obtained from the seven types of analyses, as reported in Table 1 and in the previous sections. distributions, it is clear that applying random forces on the surface of the wpfANM leads to an overall, yet slight, improvement of the correlation with the experimental data. Based on the correlation coefficients obtained for our entire dataset of protein structures, the analysis among the seven performed ones (Table 1) that gave the highest correlations with the experimental data was noted. Figure 12a shows the relative number of such occurrences for each type of analysis. It was found that, out of the 919 investigated protein structures, the mode-based fluctuations of the pfANM provided the highest correlation coefficient in 107 cases (11.6%), the application of random forces on the entire pfANM in 74 cases (8.1%), the application of forces on the external surface of the pfANM for 172 cases (18.7%), the mode-based fluctuations of the wpfANM in 60 cases (6.5%), the application of forces on the entire wpfANM in 70 cases (7.62%), the application of forces only on the water molecules of the wpfANM in 93 cases (10.1%), while the application of forces on the external surface of the wpfANM were provided in 343 cases (37.3%). As can be seen and has already been discussed concerning Figure 11, the application of random forces on the surface of the water-enriched protein network is statistically the best performing, although not the only one, with regards to better predicting protein fluctuations and flexibility in terms of experimental B-factors. Moreover, looking cumulatively at the analyses _FR,EXT_ and _W,FR,EXT_, in 515 cases (56.0%), the application of random forces on the external surface of the protein network yields the highest correlation coefficients, thus confirming that perturbing the external protein surface can induce a response in good agreement with the experimental B-factors. Figure 11: (**a**) Correlation coefficients obtained from the seven types of analyses (median values and standard deviations of the statistical distributions) for the 919 single-chain protein structures; (**b**) direct comparison between the statistical distributions of \(\rho_{FL}\) and \(\rho_{W,FR,EXT}\). Figure 12: (**a**) Relative frequency of the highest correlation coefficient for each type of analysis for the dataset of 919 single-chain protein structures; (**b**) relative frequency of the highest correlation coefficient for the pfANM or the wpfANM. Figure 12b shows the occurrence of the highest correlation coefficients for the two models, i.e., the pfANM vs. wpfANM. From the outcomes, it was obtained that in 353 cases (38.41%) the highest correlation with the B-factors was obtained with the pfANM, whereas in the remaining 566 cases (61.59%) the wpfANM was allowed to reach the highest correlation with the experimental data. This confirms that considering the PDB water molecules might actually enhance the prediction of the protein fluctuations and therefore the numerical evaluation of the experimental B-factors.