## 1 Introduction  
The B-factors of a protein, the Debye-Waller factors or temperature factors, are measures of the atomic displacements about their equilibrium position, i.e., atomic fluctuations [1, 2, 3], but also the effects of multiple conformations as well as errors in the structures. They are generally accepted to be mostly the result of internal protein dynamics and any static disorder [4]. They have also been shown to be associated with protein flexibility and to correspond closely to protein mechanisms [5, 6, 7, 8, 9]. B-factors have been associated with protein flexibility, which is strictly related to protein action and function [10, 11, 12]. The experimental B-factors obtained from X-ray crystallography have been reproduced fairly accurately by various computational models. One of the most widely used computational methods for investigating protein dynamics and fluctuations has been molecular dynamics (MD). MD simulations have proven their usefulness for investigations of protein folding, enzyme catalysis, and protein mechanisms in general [13, 14, 15]. Also, it has been shown that the MD-derived atomic fluctuations due to the internal protein motions show some degree of agreement with the experimental B-factors [16; 17]. However, due to the high computational burden of MD simulations, these can sometimes be expensive for investigating the large molecular complexes, especially regarding the slowest protein motions, accessible only at long simulation times. These slow motions are in fact usually the ones most closely related to the functional mechanisms of the protein and can take place on longer time scales than may be accessible in standard MD simulations. The harmonicity assumption has been exploited for the extraction of the low-frequency protein dynamics [18; 19; 20; 21]. Normal mode analysis (NMA) came into play as a simplified yet powerful tool to investigate the slower protein motions and for evaluating protein fluctuations and mechanisms [22; 23], even in torsional space [24]. The seminal work of [PERSON] [25] showed that even a single-parameter harmonic potential, only based on the elastic properties of a network of Hookean springs connecting the protein atoms, was sufficient to reproduce the slow dynamics in good detail. All of the elastic network models are essentially entropic models since there is not usually any distinction of atom or amino acid types, i.e., all springs are taken to be similar in character. A further step towards simplification came with the coarse-graining development for these elastic network models (ENMs). Among the ENMs, the gaussian network model (GNM) was developed to obtain insights into protein dynamics and fluctuations simply by diagonalizing the Kirchhoff matrix, built by using the network contacts between close neighboring C\({}^{\alpha}\) atoms [26; 27; 28; 29; 30; 31]. Despite the remarkable correlations obtained between the GNM-based fluctuations and the experimental B-factors, the GNM lacks the information about the directions of motions, since it assumes that the motions are fundamentally isotropic in all directions [28]. The anisotropic network model (ANM) was then developed to include the three-dimensional directionality in the calculation of protein motions [32]. The ANM was then improved by various research groups to achieve higher correlations between the computed fluctuations and the experimental B-factors [33; 34; 35; 36; 37]. These elastic models were subsequently used to study the conformational changes of proteins arising from sets of low-frequency modes [38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48] as well as to generate feasible pathways between two known conformations [48; 49; 50; 51; 52; 53; 54]. Structural elastic models, particularly the ANM, were applied widely for the investigation of protein dynamics, fluctuations, and mechanism. However, they are also well-suited for the analysis of the protein structural responses from the application of external perturbations. Based on the work from [PERSON] et al.