## 3 Results and Discussion
### Incorporating Waters into the Computations
Figure 6a-c report the correlations ordered by increasing values of \(\rho_{W,FL}\), whereas Figure 6d displays the statistical distribution for all four Pearson coefficients, whose median values and standard deviations are shown in the key. From the results reported in Figure 6, it follows that applying forces on the protein network slightly (<10%) enhances the prediction of the B-factors with respect to the traditional mode-based fluctuations. As a matter of fact, the median value of \(\rho_{W,FL}\) was found to be 0.57 for the selected dataset, whereas the median values of \(\rho_{W,FR,ALL}\), \(\rho_{W,FR,EXT}\) and \(\rho_{W,FR,WAT}\) were 0.60, 0.66, and 0.59, respectively. It is evident that applying random forces on the surface of the network (which now considers also the layer of water molecules) yields a significant 10% gain in the correlation with the experimental fluctuations, compared to the mode-based fluctuations. Figure 5: Surface representation of infrared fluorescent protein (PDB: 5 AJG): (**a**) protein structure (light gray) + water molecules (blue spheres); (**b**) protein structure alone. Figure 6: Experimental B-factors vs. mode-based fluctuations and average displacements due to random perturbations (wpfANM—water molecules included): (**a–c**) correlation coefficients for the 919 protein structures (blue curve for \(\rho_{W,FL}\), orange curve for \(\rho_{W,FR,ALL}\), red curve for \(\rho_{W,FR,EXT}\) and green curve for \(\rho_{W,FR,WAT}\)); (**d**) statistical distribution of the correlation coefficients. The highest correlations are seen when perturbations are applied on the surface. Also in the case of the wpfANM, the surface of the network is not unique but depends on the adopted shrink factor. The water molecules contained in the network play a major role in the definition of the surface since they are mostly placed on the exterior of the structure (see Figure 5). As an example, Figure 7 shows the different surfaces generated for the infrared fluorescent protein (PDB: 5 AJG, with water molecules included) with shrink factors equal to 0, 0.2, 0.4, 0.6, 0.8 and 1. As can be seen by comparing Figure 7 to Figure 2, other than the selected value of the shrink factor, the shape of the external surface also depends on the presence of the water molecules within the network. Similarly to what was shown in the previous section, Figure 8 shows the optimal shrink factors (with the best correlation with B-factors) obtained for the 919 water-enriched protein structures. Again, the distribution of the optimal shrink factor is rather uniform, although in this case a slight bias towards the convex hulls surfaces appears, i.e., with a shrink factor equal to 0. This probably reflects a preference for smoother structures when water molecules are added (see Figure 5). Figure 8: Statistical distribution of the optimal shrink factor, resulting from the comparison between experimental B-factors and average displacements due to the application of random forces on the external protein surface, with PDB water molecules included. Figure 7: Dependence of the generated external boundary (external protein surface) on the value of the shrink factor. Infrared fluorescent protein (PDB: 5 AJG), with water molecules included from the PDB structure file, 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 + water molecules), while the external surface is represented by Delaunay triangles (in light blue), which connect the nodes in the external boundary. Figure 9 shows the correlation coefficients of the wpfANM of the infrared fluorescent protein (PDB: 5 AJG), and how it depends on the adopted shrink factor for the surface.