1. Introduction
Proteins affect virtually every biological process occurring in the human body [
1]. Their correct functioning is pivotal for a variety of tasks, such as delivery of nutrients throughout and across cells, recognition and neutralization of pathogenic bacteria and viruses, providing of suitable strength and rigidity to tissues, activation of signaling pathways and catalytic reactions, etc. [
2]. All these activities are performed within the physiological environment and in a highly dynamic fashion. This explains why so much research has been carried out in the last decades regarding protein dynamics and its relationship with the biological functionality. One of the main computational approaches used to investigate protein dynamics is molecular dynamics (MD) [
3,
4]. MD is based on the numerical integration of Newton’s laws of motion of the molecular system under scrutiny, subjected to the forces arising from the gradients of the interatomic potentials [
5]. Despite the high potential of MD simulations, its applicability to large peptide chains and protein complexes, especially for the investigation of the large-scale slow dynamics, remains quite elusive and requires cautious analysis of the results.
While trying to overcome the limitations of MD simulations and come up with more simplified approaches which could be of value for a general understanding of protein functionality, it was found that elastic models based on single-parameter Hookean potentials are still able to describe the slow protein dynamics in good detail [
6]. These models treat the protein as a network of elastic springs, connecting the atoms whose positions in the reference structure are assumed to be at the equilibrium, around which the thermal fluctuations take place [
7,
8,
9]. Despite the simplicity of this model, the predicted fluctuations as well as the obtained vibrational frequencies were found in good agreement with those obtained by considering more complex semi-empirical potentials [
6]. This discovery paved the way for the development of the coarse-grained elastic network models (ENMs), such as the Gaussian network model (GNM) [
10,
11,
12,
13,
14,
15,
16] and the anisotropic network model (ANM) [
17]. The GNM assumes that the protein structure undergoes isotropic fluctuations around the equilibrium position, therefore it predicts the amplitude of these fluctuations and hence it can be identified as a unidimensional model. Conversely, the ANM takes also into consideration the directionality of the expected motion, the protein structure being modelled as an actual three-dimensional network.
The ANM was extensively used for the investigation of protein dynamics for three main reasons. Firstly, the computed fluctuations are found to exhibit a good agreement with the B-factors obtained from crystallographic experiments, thus providing good estimates of the protein flexibility [
17,
18,
19,
20,
21,
22,
23]. Secondly, the ANM low-frequency motions are found to describe fairly accurately the directionality of the protein conformational change [
20,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34]. These conformational changes usually occur when the protein switches its three-dimensional shape while performing its biological activity (e.g., during ligand-binding or phosphorylation phenomena) and, therefore, they are informative of the protein biological mechanism [
1]. Thirdly, this model allows to obtain useful insights on the low-frequency dynamics with small computational burden, especially if compared with the more time-consuming MD simulations.
From a structural mechanics viewpoint, we have recently shown that the ANM can be seen as a spatial truss elastic model, where the atoms of the protein network can be replaced by frictionless spherical hinges and the Hookean connections by linear elastic bars [
35,
36]. In the traditional formulation of the ANM, the Hessian matrix of the network is computed and diagonalized to obtain the eigenvalues and eigenvectors. The former are associated with the vibrational frequencies, while the latter identify the mode shapes of vibration [
17]. However, since the mass of the protein is not explicitly taken into account in the classical ANM, the eigenvalues are only qualitatively related to the vibrational frequencies. In our previous works, we have also added the explicit mass information, thus obtaining more quantitative information about the frequencies of vibration via a classical free-vibration modal analysis [
35,
36]. In particular, in the case of lysozyme, we observed that the lowest-frequency modes lie in the sub-THz frequency range, with frequency values of the order of few tens of GHz, in agreement with previous studies [
6,
37,
38,
39,
40].
Another powerful application of the ANM, which has been developed since the last decade, is based on application of perturbations on the protein elastic network, both to probe protein flexibility and conformational changes. Eyal and Bahar [
41] developed a methodology that made use of the ANM normal modes to assess the anisotropic mechanical resistance of proteins under external pulling forces. This analysis was able to explain the anisotropy of the mechanical resistance observed from single-molecule manipulation techniques, such as atomic force microscopy (AFM). More recently, we made use of two structural metrics, which are well-known in the field of structural mechanics (i.e., compliance and stiffness), to study the flexibility of protein structures under pairwise force application [
42]. These metrics enabled to predict the distribution of protein flexibility and rigidity throughout the protein chain and were verified against the experimental B-factors. Referring to protein conformational changes, Ikeguchi et al. [
43] observed that protein transitions can be numerically simulated by evaluating the linear response of the protein reference structure subjected to external forces applied at specific locations. Based on this finding, the Atilgan’s group developed the perturbation-response scanning (PRS) technique, where directed forces are applied to the protein structure at single residues and the protein response is calculated and compared with the conformational change observed experimentally [
44]. The method was shown to work well for the prediction of a variety of protein conformational changes [
45], as well as for the detection of allosteric sites [
46]. More recently, the PRS method was used by Liu et al. [
47], coupled with an energy-based Metropolis Monte Carlo (MMC) algorithm, in order to simulate the complete closed-to-open transition of the GroEL subunit, induced by directional forces applied at the ATP-binding site.
From what we have reported above, it is evident that, when starting from the knowledge of the reference structure only, protein conformational changes have been analyzed with the ANM by following two separate approaches: (1) Evaluating the normal modes of vibration of the reference elastic model, with subsequent comparison between each individual mode shape and the conformational change; (2) applying forces to the protein reference structure and evaluating the response of the network in terms of displacements, with subsequent comparison with the observed conformational change. Fundamentally, approach (1) considers the free-vibration dynamics of the protein, whereas approach (2) focuses on the static response of the protein structure under external forces. In this work, we put the two approaches together, thus applying forces to the protein structure in a dynamic fashion. In this way, we exploit the main ideas behind both approaches: (1) Conformational changes might be favored by the intrinsic protein dynamics along its low-frequency modes of vibration [
24,
29]; (2) conformational changes might be triggered by external perturbations [
43,
45].
In particular, we apply external harmonic perturbations, randomly distributed in the space-domain but with a well-defined frequency content in the time-domain, to the protein ANM. The equations of motions are numerically solved, by considering mass, viscous damping and elastic stiffness contributions, in order to assess the complete time-dependent dynamic response of the protein. The time-history of nodal displacements is then evaluated both in the coordinate space, as well as in the sub-space of principal components (PCs) via the application of principal component analysis (PCA). The obtained time-dependent displacements are then compared to the observed conformational change, in order to find in which conditions these external perturbations are able to drive the conformational change. Results are shown here for lysine-arginine-ornithine(LAO)-binding protein, considering different perturbation patterns, damping coefficients and frequencies of excitation. The results of the analysis reveal that, when the external perturbation is applied in the low-frequency range, the protein structure undergoes a displacement field closely aligned with the observed conformational change, with a remarkably high overlap score (up to 0.95).
3. Results and Discussion
In this section, the results are reported for the case of LAO-binding protein, a widely studied protein, known to exhibit two different conformations (i.e., an open form (PDB code: 2lao) and a closed form upon ligand-binding (PDB code: 1lst)) [
24]. In the
Supplementary Material, the results for other three proteins are reported (i.e., maltodextrin-binding protein (PDB codes: 1omp, 1anf), lactoferrin (PDB codes: 1lfh, 1lfg) and triglyceride lipase (PDB codes: 3tgl, 4tgl)). The coordinates of the open form are used to build the elastic network model, with a cutoff of 15 Å (
Figure 1a). Free-vibration modal analysis is run first, in order to obtain the theoretical B-factors from Equation (6) and the value of the spring constant
γ, which is found to be equal to 0.10 N/m (~0.15 kcal/molÅ
2). As a result, the frequency spectrum of the 3
N – 6 (
N = 238) non-rigid modes related to the coarse-grained elastic network is found to lie in the range 0.05–0.8 THz (
Figure 1b), the lowest frequency being equal to 50.9 GHz.
Figure 1c shows the displacement field involved in the open-to-closed conformational change (continuous line). By carrying out the traditional overlap comparison between the displacement field {
CC} and each individual normal mode {
δn}, it is found that the first non-rigid normal mode (i.e., {
δ7}) is the one exhibiting the highest overlap value (0.76), as shown in
Figure 1c,d. The second low-frequency mode {
δ8} agrees with the conformational change with an overlap of 0.55, while all the higher-frequency modes have lower overlap scores (
Figure 1d). From these results, it is clear that the low-frequency modes are strictly related to the observed conformational change, as already reported in the previous literature [
24,
29].
The results reported above are based on the traditional analysis aimed at evaluating the similarity between the individual mode shapes of the protein structure and its conformational change [
24]. What happens if we look at the complete time-dependent protein response upon harmonic random perturbations, as described in
Section 2.3?
Figure 2 shows the time-dependent response of LAO-binding protein, subjected to the random force pattern reported in
Figure 2a, with an exciting frequency of 0.05 THz and a damping coefficient
ξ of 0.01. The response is reported in
Figure 2b in terms of the global root-mean-squared-deviation (RMSD). The RMSD is a measure of the average displacements of the atoms from the initial position. It can be simply computed as:
where
ui(
t) is the absolute displacement of the
ith node at instant
t. As can be noticed from
Figure 2b, the response of the protein network exhibits a transitory response at the beginning, and then enters a steady-state oscillation approximately from 400 ps onwards. Note that, with the frequency of the external oscillation of 0.05 THz, its period is equal to 20 ps. Moreover, since this frequency value is very close to the natural frequency of the first mode (
f7 = 0.051 THz), high amplifications in the response occur, leading to RMSD values of about 20 Å (
Figure 2b). On the other hand, if we applied the same force pattern in a static way (i.e., by following the approach reported in
Section 2.2), we would obtain a total RMSD of about 2.3 Å. This leads to a dynamic amplification value, evaluated as the ratio between the dynamic RMSD and static RMSD, of about 8.4. Such dynamic amplification factors might also explain why protein vibrations and responses under external forces, which are supposed to be theoretically valid only in the small-amplitude regime, might actually trigger large-scale conformational changes.
Figure 2c shows the time-dependent overlap, obtained by comparing the calculated displacement field {
u(
t)} with the conformational change {
CC}, as described in
Section 2.4. The results are astonishing, since values as high as 0.94 are frequently met. This unequivocally suggests that, even if the applied force pattern is completely random in the space-domain (
Figure 2a), its dynamic application is able to drive the protein structure towards the known closed conformation, with remarkably high levels of agreement.
Additionally, it is interesting to observe how the overlap score is not maintained to these high values constantly, but it keeps oscillating between low and high values. This suggests that the direction of the protein motion {
u(
t)} from the open towards the closed conformation is not linear—see Equation (16). As a matter of fact, if this motion were linear, we would find a roughly constant value of the overlap throughout the entire simulation. The fact that this does not happen suggests that, while jiggling around its equilibrium position, the protein is sampling a variety of different conformations, among which lies the known closed form. These dynamic jumps between conformations happen in a continuous fashion and involve curvilinear pathways, as suggested here from our overlap calculations and already reported by previous authors [
28,
56,
57,
58,
59,
60].
The complete trajectory of LAO-binding protein upon the force pattern shown in
Figure 2a is represented in the
Supplementary Movie S1, which is available in the
Supplementary Materials. In the movie the blue structure refers to the protein conformation generated at each instant
t starting from the open form. Conversely, the red structure refers to the known closed form of the protein and it is kept fixed in all frames to help the visualization of the conformational change. As can be seen, after the motion enters in the steady-state regime, the perturbed protein structure keeps oscillating between open and closed conformations. Note that several times the known closed conformation (in red) is reached with high accuracy during the motion. The instants at which this occurs are the ones where high levels of overlap values have been obtained and reported in
Figure 2c. As an example,
Figure 3a shows the snapshot of the dynamic displacements evaluated at
t = 513.5 ps, compared with the displacements of the known open-to-closed conformational change. The overlap value and correlation coefficient between the two displacement fields are 0.935 and 0.904, respectively, showing high level of agreement. Higher than that found by following only the first normal mode of vibration (compare
Figure 3a with
Figure 1c).
In order to describe more quantitatively the ensemble of generated conformations, PCA has been applied to the set of structures obtained during the trajectory according to Equations (17) and (18).
Figure 3b reports the PC score plot of all conformations in the PC1-PC2 sub-space. Note that PC1 and PC2 account for 93.2% and 6.7% of the total variance, thus they account for 99.9% of the total variance. The black point refers to the open form (pdb: 2lao), the red point to the closed form (pdb: 1lst), while all the points associated with the generated conformations are in blue. A dynamical representation of
Figure 3b can be observed in the
Supplementary Movie S2, where the time-dependent evolution of the conformations in the PC1-PC2 plot is reported. From the movie and
Figure 3b we see that, after a transitory, the steady-state trajectory implies a harmonic motion of the protein around the open form, mostly along the first PC (green arrows in
Figure 3b). The information contained in
Supplementary Movies S1 and S2 also suggests that the direction of PC1 involves mostly an opening-closing mechanism of the protein. During this harmonic oscillation, the closed conformation (red point in the PC score plot) is closely approached several times throughout the motion.
It is also interesting to observe that, even though we are applying forces at a frequency very close to the first natural mode (
fF = 0.05 THz and
f7 = 0.051 THz), the time-dependent displacement field contains the information about the complete dynamics of the system, and not only that of the first natural mode. This can be immediately understood if one looks at the overlap values. By considering the trajectory which would be induced by the first natural mode alone, we would get a 0.74 overlap with the conformational change for the entire trajectory (
Figure 1d). On the other hand, applying forces dynamically excites all modes and eventually leads to a much higher agreement with the conformational change, with
Omax = 0.94 (
Figure 2c and
Figure 3a).
Here, it is also important to notice that the methodology developed in this work does not require any a priori knowledge of the target conformation. The closed conformation is only used to assess whether the conformations generated by perturbing dynamically the reference structure overlap properly with the target. Other methods have been developed in the existing literature based on the ANM, in order to find intermediate conformations given the two end structures [
47,
56,
57,
59,
61,
62]. Conversely, the methodology presented here relies only on the knowledge of the reference structure and aims at evaluating its dynamic response upon external harmonic perturbations. As a result, the generated conformations do not depend on the target form, but only on the intrinsic dynamics of the reference structure and how it responses to external perturbations. Nevertheless, the conformations generated by following this approach are able to reach the other form of the protein known experimentally with high levels of agreement (see
Figure 3a).
We have briefly mentioned above that the oscillating trend of the overlap values is a fingerprint of the non-linearity of the protein motion. This can also be assessed by a geometrical evaluation, as reported in
Figure 4a. For each residue
i, the coordinate difference between two subsequent conformations at time
t and
t + Δt provides the direction of the instantaneous motion at each time frame {
Δui(
t)} [
59]. By calculating the normalized cosine between vector {
Δui(
t)} and the direction of motion at time
t = 0 ps, i.e., {
Δui(0)}, as:
we can geometrically evaluate the non-linearity of the trajectory. If the motion were completely linear throughout the entire simulation, the cosine would only assume values +1 and −1, the former when the protein moves in the positive direction and the latter when it comes back (
Figure 4a). Conversely, non-linear motions imply cosine values different from unity, which are also supposed to change during the simulation (
Figure 4a). The more frequent the change, the stronger the non-linearity of the motion.
Figure 4b shows the values of the normalized cosine for all 238 residues of LAO-binding protein for the entire simulation. As can be seen, the cosine values assume all possible values in the range between −1 (dark blue) and +1 (bright yellow), suggesting that the motion is non-linear. Moreover, this variation appears to be cyclical, confirming what already observed visually from the
Supplementary Movie S1, namely that the motion presents a strong harmonic nature.
Figure 4c reports an enlargement of
Figure 4b for residues 17–27 (the highly flexible flap of LAO-binding protein in the first domain) in the time range between 480 and 540 ps. This figure shows more clearly that each residue experiences a wide range of cosine values between −1 and +1, thus the motion is non-linear.
In the analysis reported above, the protein was perturbed with a specific random force pattern (
Figure 2), pulsing at a selected frequency (
fF = 0.05 THz) and with a defined damping coefficient (
ξ = 0.01). What happens if these three variables are modified?
Figure 5 shows the obtained RMSD dynamic amplification, computed as the ratio between the maximum dynamic RMSD and the RMSD obtained under the application of the perturbation in a static fashion, for the whole investigated frequency range (0.001–0.5 THz) and for the three selected damping coefficients
ξ (0.001, 0.01 and 0.1). The different colored curves are associated with each of the 100 different random force patterns applied to the protein structure.
Clear peaks are recognizable in the low-frequency range, around 0.05–0.15 THz, where the low-frequency protein modes are found to occur (
Figure 1b). The intensity of the peaks is highly dependent on the adopted value of the damping coefficient. Very low values of
ξ, such as 0.001 and 0.01, lead to amplifications of the order of 20–30. Conversely, amplification coefficients lower than 5 are found for higher damping coefficients. It is also evident that the most intense peaks are associated with the first low-frequency modes, in the region 0.05–0.06 THz. Other pronounced peaks are also found for higher modes, especially in the region between 0.08 and 0.15 THz. In the higher region of the spectrum, the dynamic amplification gets lower, eventually leading to de-amplified responses (i.e., with an RMSD amplification factor lower than 1), especially in the presence of higher damping coefficients (
ξ = 0.1). Moreover, it can be seen that the specific force pattern has an influence on the overall system amplification. Nevertheless, highly amplified responses are always found in the low-frequency range (
Figure 5). As briefly mentioned above, this dynamic amplification might be one underlying reason which enables the protein to achieve the large-scale conformational changes when it gets triggered in the low-frequency range, despite the theory behind all these calculations being strictly valid in the small-amplitude regime.
Figure 6 shows the maximum overlap values obtained during ten cycles of dynamic perturbation by comparing the calculated displacement field {
u(
t)} with the observed conformational change {
CC}, as a function of the exciting frequency, damping coefficient and specific force pattern. As can be seen, remarkably high values up to 0.95 are found in the low-frequency range, especially between 0.02 and 0.08 THz. It is interesting to observe how in this low-frequency range the maximum overlap score is always very high, despite the specific force pattern. The upper panel of
Figure 7 shows the maximum overlap scores obtained for each of the 100 different force patterns, for each selected damping coefficient, while the lower panel reports the exciting frequency in correspondence of which the best overlap is met. As can be seen, despite the specific random force pattern, very high values of the overlap are always obtained (up to 0.95) with applied frequencies in the range 0.02–0.08 THz, which corresponds to the low-frequency end of the spectrum (
Figure 1b). From
Figure 6, it can also be seen that if the protein is excited at higher frequencies, say with frequencies higher than 0.1 THz, the obtained overlap scores become lower, suggesting that the closed conformation cannot be sampled by applying harmonic excitations in this frequency range. Additionally, it can be noticed that the overlaps are higher when the damping coefficients are relatively low (
ξ = 0.001 and 0.01).
Putting together the results obtained above, we can conclude that, by exciting the open structure with external dynamic perturbations in the low-frequency range, we can sample the closed conformation with remarkably high values of directionality correlations. In case of low damping coefficients (
ξ = 0.001 and 0.01), one also obtains high dynamic amplification factors in that frequency range, therefore potentially allowing to reach the closed conformation even with a small amount of force involved. Finally, it also seems that the specific force pattern, which is completely random in the space-domain (
Figure 2a), has not a huge influence on the results, almost always leading to high overlap scores, as long as the forces are applied with an exciting frequency in the lower part of the mode spectrum (
Figure 7).
The analysis reported above focused on LAO-binding protein. However, the same analysis was carried out with other proteins (i.e., maltodextrin-binding protein, lactoferrin and triglyceride lipase; see
Supplementary Material). For the maltodextrin-binding protein (PDB code of the open form: 1omp, PDB code of the closed form: 1anf,
N = 370), one obtains a maximum overlap of 0.81 when comparing the second normal mode with the open-to-closed conformational change (
Figure S1). However, overlap values as high as 0.95 can be found again when applying dynamic perturbations in the low-frequency range (
Figures S3 and S4). As in the case of the LAO-binding protein, the overlaps are higher for lower damping coefficients (
ξ = 0.001 and 0.01), which also lead to higher dynamic amplifications (
Figure S2). In the case of lactoferrin (PDB code of the open form: 1lfh, PDB code of the closed form: 1lfg,
N = 691), a maximum overlap of 0.46 is obtained between the third ANM mode and the conformational change (
Figure S5). However, if the structure is perturbed dynamically in the low-frequency range, overlap values up to 0.88 can be obtained (
Figures S7 and S8). From the comparison between lactoferrin and the two previous proteins, we understand that, when the individual modes have a higher agreement with the conformational change, the full dynamic response can sample the closed conformation better. Nevertheless, even when individual modes have lower similarities (
Omax = 0.46 for lactoferrin), the perturbation-based dynamic response allows to achieve a better agreement with the closed conformation (
Omax = 0.88). Finally, in the case of triglyceride lipase (PDB code of the open form: 3tgl, PDB code of the open form: 4tgl,
N = 265), one obtains a really low value of the overlap when comparing individual ANM modes to the conformational change (
Omax = 0.27 for the fourteenth mode,
Figure S9). As a result, the maximum overlap found by applying dynamic perturbation to the protein ANM is only 0.44 (
Figures S11 and S12), showing that, in this case, the closed conformation cannot be sampled with high accuracy by the proposed method. This shows that the method proposed here always leads to higher overlaps than those obtained through the classic individual mode comparison. However, the method works better when the low-frequency modes have already a relevant similarity with the conformational change. As Tama and Sanejouand showed in their seminal work [
24], this is a direct consequence of the degree of collectivity of the conformational change. Collective transitions are usually better captured by the low-frequency modes, whereas localized conformational changes usually are not. For the four proteins investigated here, the collectivity degree of their conformational transitions are 0.68 (LAO-binding protein), 0.67 (maltodextrin-binding protein), 0.48 (lactoferrin) and 0.07 (triglyceride lipase). As a result, the LAO-binding protein and the maltodextrin-binding protein reach very high values of the overlap from the full dynamic response (0.95), lactoferrin reaches a high value (0.88), while triglyceride lipase reaches a quite low value (0.44). This leads us to conclude that, with the proposed methodology, starting from the open conformation of the protein and without any a priori knowledge of the closed form, we are able to capture the closed conformations accurately as long as the conformational transition is quite collective in nature.
In all previous examples, we have investigated the conformational change from the open to the closed conformation.
Figures S13 and S14 show the results of the analysis for LAO-binding protein, this time considering the closed conformation (pdb: 1lst) as reference and the open conformation (pdb: 2lao) as target. In agreement with what said above for the open-to-closed conformational transitions, the time-dependent force application generally allows to reach higher overlap values than those obtained by comparison with individual modes. As a matter of fact, when only looking at individual modes, the third ANM mode is the one showing the highest overlap, with
Omax = 0.56 (
Figure S13). Conversely, applying dynamic forces can enhance this maximum overlap, reaching values of
Omax = 0.75 (
Figure S14). Again, this suggests that the target conformation can generally be captured better by considering the full dynamic response of the protein, rather than looking at the trajectory generated with individual modes. However, despite the improvement, this value is lower than the one obtained when looking at the open-to-closed conformational change (compare
Figure S14 with
Figure 6). This suggests that, as already noticed by Tama and Sanejouand [
24] and subsequent researchers, the closed-to-open transition is generally more difficult to generate than the open-to-closed one when working with ENMs.
From the lower panels of
Figure 7 and
Figures S4, S8 and S12, one can also observe that there is not a unique value of the exciting frequency leading to the maximum overlap score. In fact, there exists a range of low-frequency values, where each specific force pattern is able to sample the target conformation with the highest directionality correlation. This suggests that, although we are often in the range of the first fundamental frequency (i.e., the frequency associated to the first non-rigid mode shape), the optimal excitation frequency might be slightly different than the fundamental frequency, mostly due to the not negligible involvement of higher-order modes in the definition of the complete protein dynamic response. Moreover, the exact values of these frequencies must be treated carefully, as they are strongly dependent on the model parameters, such as the cutoff and the definition of spring network [
35,
36]. The absolute values of these frequencies strongly depend on the value of the adopted spring constant
γ, which in turn is defined upon comparison between the numerical and experimental B-factors (
Section 2.1). In doing such direct comparison, we are implicitly assuming that the experimental B-factors are dominated by the protein fluctuations, i.e., they only depend on the internal protein dynamics. Unfortunately, this is not always the case, since studies have shown that B-factors might also include other contributions coming from rigid motions, crystal disorder, refinement effects, etc. [
63,
64,
65,
66]. Therefore, for all the above mentioned reasons, when using ENMs for the understating of protein motions and their corresponding frequencies of vibration, we can only have some insights on the expected frequency range, and not on the individual frequency values.
Additional considerations need to be done regarding the damping coefficients adopted in the present analysis. In this work, values of
ξ = 0.001, 0.01 and 0.1 have been used, meaning that the problem is treated in the underdamped regime (
ξ < 1). Moreover, the value of
ξ has been kept constant for all the vibrational modes (i.e.,
ξn =
ξ). However, one should be careful about such choices. As a matter of fact, in the framework of NMA and ENMs, the dynamic response of proteins is studied with no damping at all (i.e.,
ξ = 0). Conversely, few studies using Langevin network models (LNMs) have shown that the dynamics of macromolecules and proteins might be strongly overdamped (i.e.,
ξ >> 1), at least for the lowest-frequency modes [
67,
68]. We might also reasonably expect that the damping characteristics change for the different modes of vibration, the lowest-frequency ones being the more damped, the highest-frequency ones the less damped [
68]. By using the LNM, Miller et al. [
68] showed that at normal water viscosities most of the protein modes should be overdamped. Nevertheless, recent studies based on optical Kerr-effect (OKE) spectroscopy revealed the existence of underdamped global protein vibrations in the THz frequency range [
69]. Therefore, it is evident that there is still little consensus nowadays about the damping nature of these functional protein vibrations. Previous numerical results showing that undamped vibrational modes correlate well with protein conformational changes [
24,
28,
29], as well as the outcomes of our calculations based on the underdamped assumption, seem to suggest that these functional conformational transitions can indeed be retrieved by working in the underdamped limit. Yet, extensive research efforts still need to be carried out in the future to address this open issue.
Remarks need to be given also regarding the physical meaning associated with the adopted perturbation scheme (i.e., harmonic excitations with a random direction in the space-domain but with a well-defined frequency content in the time-domain). In the previous literature, different force application patterns have been applied in a static fashion to probe protein flexibility [
41,
42] and protein conformational changes [
44,
45,
47]. The approach proposed here (i.e., applying random forces to the protein ANM in a dynamic fashion) is supposed to simulate the external perturbations to the protein structure mainly due to Brownian motions of the surrounding particles [
70]. These collisions can be numerically simulated as random forces both in the space- and time-domain [
68]. However, we know that every time-dependent variable can be decomposed according to their frequency component (e.g., via the Fourier transform (FT) or other signal transformation techniques). In this way, a random excitation in the time-domain can always be represented as a sum of harmonic excitations at specific frequencies, weighted by their FT amplitude. Based on these considerations, the application of harmonic forces with random directions can be seen as the attempt to investigate the response of the protein structure under the different frequency-based components of the complex time-dependent excitations due to the particle collisions. From the results of the present analysis, we obtained that the low-frequency components of these excitations are able to drive the protein conformational change. Conversely, high-frequency components are not particularly relevant for the conformational transition (see
Figure 6).