## 3 Results and Discussion
### Incorporating Waters into the Computations  
In this section, we show results obtained by also including the localized waters given in the crystal structure as part of the structure for defining the elastic network. There is some ongoing debate about whether or not these bound waters should be considered as an actual part of the structure. Each high-quality protein structure available in the PDB contains a substantial number of waters that were present in the crystal formed at low temperatures. The open question is whether these remain bound at higher temperatures. These molecules often typically form a network of hydrogen bonds with the side chains of polar amino acids on the protein surface and thus can appear to be quite stable. It follows that these waters may possibly cause some changes to the overall flexibility and dynamics of any given protein. Moreover, since we are interested in looking at the responses of each protein structure due to external perturbations, the inclusion of these external waters would be expected to affect the motions to some extent. Figure 5 shows a surface representation of the infrared fluorescent protein (PDB: 5A)G, with and without the addition of water molecules in Figure 5a,b, respectively. The protein structure is shown in light-gray, with the surface depiction highlighting the external surface and cavities. Water molecules available in the PDB are shown in Figure 5a as blue spheres. As can be seen from the comparison between the two figures, most of the crystallized water molecules are bound in concave parts of the protein surface, and thus act to smooth the structure [79]. This smoothing might restrict the flexibility of certain parts of the proteins that might cause problems for the mechanisms otherwise; this can be looked at as flowing liquid that represents protein motion, if a certain direction of the flow is restricted, it may change the overall flow of the water, therefore, to have an optimal flow path (specific functional protein motion), the restrictions (waters) are as important as the structure itself. As mentioned in the previous section, the wpfANM is built as the usual pfANM with \(p\) equal to 3 (see Equation (2)), except that both the coordinates of the C\({}^{\alpha}\) atoms and the water molecules are now considered as a part of the whole structure. Based on the resulting wpfANM, the mode-based fluctuations are evaluated from Equation (3), whereas the  
Figure 4: Experimental B-factors vs. mode-based fluctuations and average displacements due to random forces for the infrared fluorescent protein (PDB: 5A)G). The solid blue line refers to \(\rho_{FL}\), the dashed orange line to \(\rho_{FR,ALL}\) and the dotted red line to \(\rho_{FR,EXT}\). The correlation arising from the application of forces only on the external surface, i.e., \(\rho_{FR,EXT}\), depends on the selected shrink factor, which is 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 301 nodes of the network. This shows that the best result is the structure representation with the greatest level of detail, and interestingly the most detailed structure with forces applied to the surface only leads to nearly the same result as the application of forces throughout the structure. average displacements resulting from the random perturbations are computed according to Equations (4)-(7). In the latter cases, as explained above, three types of force application are considered, as shown in Table 1. From the calculations, we then obtain four correlation coefficients to compare with the experimental B-factors, namely \(\rho_{W,FL}\), \(\rho_{W,FR,ALL}\), \(\rho_{W,FR,EXT}\) and \(\rho_{W,FR,WAT}\), with these being described in Table 1.  
Figure 6 shows the correlation coefficients for the 919 proteins investigated.