## 2 Methods
### Anisotropic Network Model (ANM) and the Calculation of Normal Mode-Based Fluctuations  
ANM relies on the assumption that proteins can be modeled as simple elastic networks, made up of point nodes connected by linear elastic springs, allowing insights regarding fluctuations and global mechanisms [32; 34; 74]. For a system of \(N\) points, e.g., \(N\) residues in the one-bead-per-residue coarse-grained representation, the \(3N\times 3N\) Hessian matrix of the system takes the following form:  
\[\mathbf{H}=\left[\begin{array}{cccccccc}\mathbf{H_{1,1}}&\ldots&\mathbf{H_{1,i}}&\ldots &\mathbf{H_{1,j}}&\ldots&\mathbf{H_{1,N}}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \mathbf{H_{i,1}}&\ldots&\mathbf{H_{i,i}}&\ldots&\mathbf{H_{i,j}}&\ldots&\mathbf{H_{i,N}}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \mathbf{H_{j,1}}&\ldots&\mathbf{H_{j,i}}&\ldots&\mathbf{H_{j,j}}&\ldots&\mathbf{H_{j,N}}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \mathbf{H_{N,1}}&\ldots&\mathbf{H_{N,i}}&\ldots&\mathbf{H_{N,j}}&\ldots&\mathbf{H_{N,N}}\\ \end{array}\right], \tag{1}\]  
where each \(3\times 3\) submatrix \(\mathbf{H_{i,j}}\) contains the stiffness coefficients of the springs connecting nodes \(i\) and \(j\). The off-diagonal submatrix \(\mathbf{H_{i,j}}\) is computed based on the harmonic potential of the elastic spring with force constant \(\gamma_{ij}\). The diagonal submatrices \(\mathbf{H_{i,i}}\) are calculated as the summation involving all the nodes linked to \(i\)th node with a negative sign [32]. The model, and consequently the Hessian matrix, depends on some numerical parameters: the usual model uses a cut-off limit in distance to define the network topology. The original ANM was developed by considering equal spring constants for all connections, i.e., \(\gamma_{ij}=\gamma\), and a geometrical cut-off \(r_{c}\) was applied in order to consider springs placed only between close nodes. Typical values of \(r_{c}\) in the ANM are around 15 A. Later on, distance-dependent force constants were introduced [33; 34], as:  
\[\gamma_{ij}\propto\frac{1}{\left(r_{i,j}\right.^{0})^{p}}, \tag{2}\]  
where \(p\) represents an inverse number for the decay parameter that allows connecting all points in a structure, with springs with variable strength--lower spring constants for longer inter-node distances \(r_{ij}^{~{}~{}0}\). This distance-dependent spring network was shown to provide an improved agreement between the results and experimental data [33; 34].  
Once the Hessian matrix is computed based on the protein coordinates from the PDB file [73] and the spring connectivity, the \(3N\) eigenvalues \(\lambda_{n}\) and \(3N\) eigenvectors \(\mathbf{U}_{n}\) are obtained by solving the eigenvalue-eigenvectors decomposition. Due to the fact that the protein structure is usually not externally constrained, the first six eigenvalues are found to be zero, with corresponding mode shapes associated with the six rigid-body motions of the whole molecule. Therefore, these six motions are factored out and singular value decomposition is used to obtain the normal modes. Hence, the fluctuations based on the normal modes can be easily calculated as [22; 75; 76]:  
\[B_{i}=\frac{8}{3}\pi^{2}k_{B}T\sum_{n=7}^{3N}\frac{U_{i,n}}{\lambda_{n}}^{2}, \tag{3}\]  
where \(B_{i}\) represents the computed B-factor for residue \(i\), \(k_{B}\) is the Boltzmann constant, \(T\) is the absolute temperature, \(U_{i,n}\) stands for the displacement of node \(i\) in the \(n\)th mode, and \(\lambda_{n}\) is the eigenvalue for the \(n\)th mode.