## 3 Results and Discussion  
### Fluctuations and Force Application on the pfANM  
In this section, the flexibility of the protein structure will be investigated by using traditional pfANM mode-based fluctuations as well as from the outcome of the random force applications to the protein. Figure 1 shows the correlation coefficients obtained from a comparison of the experimental B-factors with the mode-based fluctuations as well as the average displacements due to the applied random forces. Figure 1a,b report the Pearson coefficients obtained for the 919 single-chain protein structures, with the values ordered by ascending values of \(\rho_{FL}\). Figure 1c shows the statistical distributions of the correlation coefficients, whose median values and standard deviations are reported in the keys. Figure 1a,b shows the distributions of correlation coefficients \(\rho_{FR,ALL}\) and \(\rho_{FR,EXT}\), that are due to the random force applications, and these are observed to oscillate near the population of \(\rho_{FL}\). This means that the average displacements of the protein elastic network due to the force application are indeed well correlated with the experimental B-factors, with a similar agreement as for the traditionally used mode-based fluctuations. The same conclusions can also be drawn by looking at Figure 1c, where the three statistical distributions exhibit the same pattern and similar median values, i.e., 0.63, 0.63 and 0.64, for \(\rho_{FL}\), \(\rho_{FR,ALL}\) and \(\rho_{FR,EXT}\), respectively. Therefore, it cannot be concluded that applying random forces on the protein structures always enhances the correlation with the experimental B-factors, while it can be concluded that perturbing the protein structure by applying random forces leads to a good estimate of the experimentally determined fluctuations, at least as good as those found with the normal modes. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & & **pfANM** & & & **wpfANM** & \\ \hline Nodes in the network & & C\({}^{\alpha}\) atoms & & & C\({}^{\alpha}\) atoms + water molecules \({}^{\star}\) & \\ Analysis & Mode-based fluctuations & Force application & Mode-based fluctuations & & Force application & \\ Nodes perturbed & - & All nodes & External nodes & - & All nodes & External nodes ** & *** \\ Correlation coefficient & \(\rho_{FL}\) & \(\rho_{FR,ALL}\) & \(\rho_{FR,EXT}\) & \(\rho_{W,FL}\) & \(\rho_{W,FR,ALL}\) & \(\rho_{W,FR,EXT}\) & \(\rho_{W,FR,WAT}\) \\ \hline \hline \end{tabular}  
* All waters in the pdb are included, without checking whether they are exterior. ** All nodes lying on the external surface (boundary geometry) of the protein-water network are perturbed by random forces; *** Only water molecules in the protein-water network are perturbed by random forces.  
\end{table} 
Table 1: The Designators for the Pearson Collection Coefficients for the Various Elastic Network Models. As mentioned in the previous section, the application of random forces on the external protein surface requires the selection of the nodes that lie on the exterior. For this purpose, various external boundaries were generated by changing the shrink factor of the surface, varying from 0 to 1 with steps of 0.1. Figure 2 shows an example of different external surfaces generated with shrink factors of 0, 0.2, 0.4, 0.6, 0.8 and 1 for the infrared fluorescent protein (PDB: 5 AJG). As can be observed, increasing the shrink factor leads to considering a higher number of nodes lying on the surface, which in turn has a more detailed structure. Since the primary determinant of a structure's dynamics is its shape, clearly the most detailed structure would be expected to be the best [78].