Normal mode analysis calculation for a full-atom model with a smaller number of degrees of freedom for huge protein molecules

Abstract

The number of degrees of freedom (DOF), N, in normal mode analysis (NMA) calculations of proteins is a crucial problem in huge systems because the eigenvalue problem of an N-by-N matrix must be solved. If it were possible to perform the analysis with a smaller number of DOF for the same system with minimal deterioration in accuracy, this would make a significant impact on the computational study of protein dynamics. We examined two models in which the number of DOF was reduced. Both of them adopted a full-atom model with dihedral angles as independent variables. In one model, side-chain dihedral angles, χ’s, and a main-chain dihedral angle, ω, were fixed and only the main-chain dihedral angles, ϕ and ψ, were variable. In another model, the dihedral angles around virtual bonds that connect neighboring Cα atoms were tested. The number of DOFs for the two models was two and one per residue, respectively. The residue-by-residue fluctuation profiles for atoms and dihedral angles were well reproduced in both models. The motion of atoms in the individual lowest-frequency normal modes of the two models was also very similar to those of the original model in which all rotatable dihedral angles were variable. Consequently, these models could predict large-amplitude concerted motion. These results also imply that proteins in a full-atom model can undergo only limited large-scale conformational changes around the native conformation, and consequently, NMA results do not strongly depend on the independent variables adopted.

Significance.

The number of degrees of freedom in normal mode analysis calculations of protein is a crucial problem in a huge system. We examined two models in which the number of variables was reduced. They both adopted a full-atom model with dihedral angles as independent variables. The numbers of variables of the two models were two and one per residue, respectively. The fluctuation profiles and the motions of the atoms in the individual lowest-frequency normal modes were well reproduced, implying that they can predict large-amplitude concerted motions of the huge protein systems.

Normal mode analysis (NMA) of protein molecules is a well-established and computationally efficient method to quickly survey their dynamic structures in biophysics and biochemistry. In particular, the development of the elastic network model-based NMA (ENM-NMA) had an impact because it was not expected that such a crude model could reproduce the protein motion associated with their function to a significant extent [1–6].

The computational processes of the NMA are composed principally of three steps. Firstly, the protein structure must be represented with appropriate variables. It is natural and easy to use the Cartesian coordinates of constituent atoms as independent variables (CC system). However, it is also possible to use dihedral angles around rotatable chemical bonds as independent variables for a molecule with a fixed geometry, i.e., with fixed bond lengths and bond angles (DA system). In this system, however, it is necessary to design an appropriate format for the input data to describe the protein structure, whereas the Cartesian coordinates of constituent atoms are sufficient in the CC system. In this study, we used a computer program developed to investigate the ENM-NMA in the DA system [7] and demonstrated its advantage.

In the second step of NMA, a Hessian matrix of conformational energy functions was calculated. Finally, the generalized eigenvalue problem involving the Hessian matrix was solved. In this computational process, the number of degrees of freedom (DOF), N, in the NMA calculations of a protein is a crucial problem as a system gets larger because the calculation of eigenvalues and eigenvectors of the N-by-N Hessian matrix in the final step is the most time-consuming process with a complexity of O(N³). If it were possible to perform NMA with a smaller N for the same system without reducing the degree of accuracy, this would make a significant impact on computational studies of protein dynamics.

The use of a coarse-grained molecular model is one possible strategy. The most common model is one in which each residue is represented by one atom, usually a Cα atom (Cα model). However, the interactions between atoms other than Cα are completely ignored and the movements of atoms apart from the Cα cannot be calculated in the analysis except in some research [8]. Another possible strategy is to fix some of the variables in changing protein conformations. In this strategy, it is still possible to retain a full-atom model. If selecting the variables that affect only local conformations were possible for fixing them, it is expected that global motion could be well reproduced. Given that the Cα model is very popular and many studies have been undertaken using this model [4–6,8–10], we examined the latter in this study.

The variables employed to describe the conformations of a protein molecule are also of interest. The Cartesian coordinate system (i.e., CC system) is one option. Three variables per atom are required, and thus 3n variables are necessary for an n-atom system. The dihedral angles around rotatable chemical bonds in a molecule with fixed geometry (i.e., DA system) are another option. Approximately one-fifth of the number of variables in the CC system is sufficient for the DA system for a full-atom system if hydrogen atoms are not considered. In this study, two models will be presented in the DA system. The number of variables is two per residue in one model and one per residue in another, even though both adopt the full-atom model. The numbers of variables of the two models are less than three per residue of the Cα model in the CC system.

In addition, the DA system has an advantage over the CC system in reducing the number of variables. It is easier to classify variables into two categories: global and local. The main-chain dihedral angles, ϕ and ψ, are global variables that affect the global conformations of a protein. Another main-chain dihedral angle, ω, is also a global variable, but can be fixed because its rotation is strongly restricted by a peptide plane. However, side-chain dihedral angles, χ’s, are local variables that affect only side-chain conformations. Given that the protein conformation can be essentially represented by rigid-body parts and the rotatable bonds that connect them in the DA system (see Method for details), the treatment of rigid bodies that result from fixing some of the variables is much easier in the DA system than the CC system.

Two kinds of models are introduced and examined to reduce the number of variables, compared to the original model in which all dihedral angles are variable. It should be emphasized again that they all adopt a full-atom model. The difference between the models is an independent variable.

Method

ENM-NMA in a DA system

In this study, ENM-NMA in the DA system was performed by our in-house program PDBETA [7]. The bond lengths and bond angles were fixed, and the dihedral angles around the rotatable chemical bonds were used as independent variables to describe protein conformations. The potential energy functions between atoms have a harmonic form and their parameters do not depend on the atomic type. PDBETA regards a protein molecule system including a monomer and oligomer as a tree in a graph-theory sense; i.e., a system comprises a rigid-body portion that contains one or more atoms (node) and a rotatable covalent bond that connects the rigid-body portions (edge). In our previous works, the rigid-body portion and rotatable bond are simply referred to as a unit and bond, respectively [11,12]. This unit-bond representation of the molecular system is an essential aspect of the models discussed in this report.

The calculated atomic fluctuations were calibrated such that the mean fluctuation was matched with the mean fluctuation estimated from temperature factors in the PDB data because temperature is not applicable to the ENM-NMA.

Models examined

Three models, referred to as FULL, PP, and VB models were considered. Any model is a full-atom model (more precisely, all atoms in the PDB data are considered in the computation but no hydrogen atoms are involved in the model) and has a fixed geometry with fixed bond lengths and bond angles. They are defined as follows:

1. FULL model: All rotatable dihedral angles are variable.

2. PP model: Only the main-chain dihedral angles, ϕ and ψ, are variable; the other dihedral angles, i.e., main-chain dihedral angles, ω, and side-chain dihedral angles, χ’s, are fixed. Any rotatable dihedral angles in a ligand, if any, are considered to be variable.

3. VB model: The dihedral angles defined in a virtual-bond system are variable, but the virtual bond angles are fixed. The regular dihedral angles, ϕ, ψ, and χ’s, are fixed. The peptide bonds are hypothetically broken. The rotatable dihedral angles in a ligand are considered as variables in the same way as the PP model.

The virtual-bond system is defined as follows: a virtual bond connects C^α atoms of neighboring residues, i and i+1, i.e., C^α_i−C^α_i+1. The four consecutive C^α atoms, C^α_i−1−C^α_i−C^α_i+1−C^α_i+2 , define a dihedral angle around a bond, C^α_i−C^α_i+1. The virtual bond angle is an angle between two bonds, C^α_i−1−C^α_i and C^α_i−C^α_i+1.

In the DA system, a protein structure is represented with units and bonds. The virtual bond was regarded as a bond in the VB model. The unit is defined as a set of atoms in which the mutual distances between atoms are fixed. If this condition is satisfied, any set of atoms can be a unit. Figure 1 illustrates units and bonds for the three models. In the VB model, we defined a set of atoms as indicated in Figure 1, i.e., a set of atoms in a residue, as a unit. The virtual bond angles are fixed because the computer program used could not treat bond angles as independent variables. However, this did not appear to significantly affect the results.

Figure 1.

A unit-and-bond representation of a protein structure. (A) FULL, (B) PP, and (C) VB models. A rotatable bond is represented by a red line. A unit, i.e., a set of atoms with fixed mutual distances, is enclosed by a dashed line. A red line in (C) indicates a virtual bond.

Proteins examined

We chose 42 proteins from the PDB entries [13] that were relatively large and whose structures were determined with high resolution (their resolutions are less than 1.0 Å and the numbers of residues are from 400 to 2030; see Supplementary Table S1 for the proteins chosen). There were 12 monomers, 14 dimers, 4 trimers, 9 tetramers, and 3 hexamers. If the data have some ligands, they were also considered in the computations. If the PDB entry contained data for only one subunit, even though its biological unit was suggested to be a multimer rather than a monomer, the atomic coordinates for biological assembly were obtained from the biounit archived in PDBj.

Results

Fluctuation of atoms

Firstly, residue-by-residue atomic fluctuations were examined. The fluctuation profile is referred to as Prof(atom, model) in this report, where “model” can be FULL, PP, VB, or Exp. The argument, model=Exp, means that this is a fluctuation profile that was estimated from temperature factors given in the PDB data. In the first argument, “atom” is specified because we consider dihedral angle fluctuation in which “ϕ” or “ψ” is specified in place of “atom.” In Figure 2, the root-mean-square (RMS) deviations of Prof(atom, PP) and Prof(atom, VB) from Prof(atom, Exp) are plotted against the RMS deviation of Prof(atom, FULL) from Prof(atom, Exp) for 42 proteins. With regard to residue-by-residue fluctuations, the results indicate that any model can reproduce them with the same level of precision. In Supplementary Figure S1, the fluctuation profiles of the three models, FULL, PP, and VB, are shown together with those estimated from the temperature factors given in the PDB data for Mn catalase (PDB ID: 2v8t) [14] and D-xylose isomerase (2glk) [15]. The good correspondence between these profiles can be seen in the superimposed profiles.

Figure 2.

RMS deviations of Prof(atom, PP) and Prof(atom, VB) from Prof(atom, Exp) plotted against the RMS deviation of Prof(atom, FULL) from Prof(atom, Exp) (panels, (A) and (B), respectively) for the 42 proteins examined. The red line indicates that both RMS deviations along the horizontal and vertical axes are identical.

There are three exceptional cases in the VB model in which the RMS deviations are greater than 0.2 Å (the right panel of Fig. 2); they are EDGP (3vla) [16], endosialidase NF (3ju4) [17], and Cu nitrite reductase (5akr) [18]. The large deviations arise from the first two residues at their N-terminals that show extraordinary fluctuations in any proteins because they can move rather freely in the FULL and VB models. If these residues are omitted in the RMS deviation calculations, their values are reduced to nearly 0.1 Å. Therefore, the difference between the FULL and VB models do not have significant meaning for large RMS deviations.

Mode-to-mode correspondence

The reproduction of the atomic movements of individual lowest-frequency normal modes is an important point in the assessment of the presented models. The ten lowest-frequency normal modes from the two models were compared for any possible pairs. The cosine similarity of the two modes, the ith mode of model X and jth mode of model Y, are defined as the mean cosine of the two displacement vectors:

$${\mathrm{\gamma }}_{ij}(\text{X},\text{Y})=\frac{1}{M}\sum _{k=1}^M\frac{\mid u_k^i(\text{X})·v_k^j(\text{Y})\mid }{\mid u_k^i(\text{X})\mid \mid v_k^j(\text{Y})\mid }$$

where $u_k^i(\text{X})$ and $v_k^j(\text{Y})$ are the displacement vectors of atom k of the ith mode of model X and the jth mode of model Y, respectively (i, j=1, 2, ..., 10; X, Y=FULL, PP, or VB); M is the number of constituent atoms (only Cα atoms were considered in the computation process). The cosine similarity is a similarity measure between two modes with respect to the directional correspondence of the displacement vectors of atoms. A value of 1 indicates perfect agreement between the motions of the two modes, and a value of 0 implies that there is not similarity between the motions.

The mode-to-mode correspondence between two models is thus defined as follows; i.e., if γ_ij(X, Y) for j=j_m is the largest for a specified i, the corresponding mode of mode i of model X is mode j_m of model Y. The j_mth mode of model Y is referred to as a counterpart mode of the ith mode of model X and the cosine similarity is denoted as Γ_i(X, Y)=γ_ijm(X, Y). A mode number gap between the ith mode of model X and the counterpart mode of model Y, g_i(X, Y), is j_m–i.

The tables of γ_ij(FULL, PP) and γ_ij(FULL, VB) (i, j=1, 2, ..., 10) for Mn superoxide dismutase (1ix9) (Anderson, B. F., et al., unpublished work) are shown in Figure 3, and those for another two proteins, liver alcohol dehydrogenase (2jhf) [19] and GH47 α-mannosidase (4ayo) [20], are presented in Supplementary Figure S2. In Mn superoxide dismutase and liver alcohol dehydrogenase, most of the modes in one model correspond to the modes with the same mode numbers in the other. GH47 α-mannosidase is a significantly worse example because the mode-to-mode correspondence is rather irregular in the VB model; the second, third, and fourth modes of the FULL model correspond to the third mode of the VB model. The tables for the other proteins are not shown here, but the statistical results of the 42 proteins are presented herein instead.

Figure 3.

The cosine similarity between the ith mode of model X indicated in the first column and the jth mode of model Y indicated in the first row, γ_ij(X, Y) for Mn superoxide dismutase (1ix9). The maximum value of γ_ij(X, Y) for a given mode number i, i.e., the maximum value of each row, is colored; red, yellow, green, and blue to indicate the ranges of the values, >0.9, 0.8–0.9, 0.7–0.8, and <0.7, respectively. This is referred to as Γ_i(X, Y) in the text.

The relative frequency distributions of the cosine similarities, Γ_i(FULL, PP) and Γ_i(FULL, VB) (i=1, 2, ..., 10) are shown in Figure 4. In the comparison between the FULL and PP models (Fig. 4A), most of the first to fifth lower-frequency modes (solid lines in Fig. 4) have cosine similarities with their counterpart modes greater than 0.9. However, the 6th to 10th modes (dashed lines in Fig. 4) also have high similarities, but they are slightly smaller than those of the top five lowest-frequency modes. There are few modes with exceptionally small cosine similarities. They are mostly the 10th modes; this is because their true counterpart modes would be higher-frequency modes beyond the 10th mode, which were not considered in this study.

Figure 4.

Mode-to-mode correspondence of the PP model (A and C) and VB model (B and D) to the FULL model. The panels A and B show the relative frequencies of the cosine similarities between the modes with a bin width of 0.05 for the 42 proteins examined. The panels C and D show the relative frequencies of the mode number gaps. See text for detail.

The relative frequency distributions of Γ_i(FULL, VB) (Fig. 4B) are similar to those of Γ_i(FULL, PP). Although the frequencies of the cosine similarities from 0.5 to 0.8 in Γ_i(FULL, VB) are slightly larger than the values for Γ_i(FULL, PP), they are not sufficiently significant such that the VB model can be evaluated as a properly approximated method.

Figures 4C and 4D show the relative frequency distributions of the mode number gaps, g_i(FULL, PP) and g_i(FULL, VB) (i=1, 2, ..., 10), respectively. There are large peaks at g=0 in both distributions, indicating that most of the modes in the PP and VB models correspond to the modes with the same mode numbers in the FULL model. Some modes have g=±1; i.e., the ith mode in model X corresponds to the (i±1)th mode in model Y, respectively. The modes with larger differences, i.e., |g|≥2, are few. Most of the first modes from the 42 proteins have g=0, and the numbers of modes of g=0 decrease as the mode numbers increase. The correspondence of the PP model to the FULL model is better than the VB model.

Fluctuation of dihedral angles

The fluctuation profiles of main-chain dihedral angles, ϕ and ψ, in the FULL and PP models, Prof(ϕ, FULL) and Prof(ψ, FULL), and Prof(ϕ, PP) and Prof(ψ, PP), respectively, are interesting to compare. To what extent does fixing the side-chain dihedral angles in the PP model affect fluctuations of ϕ and ψ? Figure 5 shows examples of the dihedral angle profile for histidine ammonia lyase (1gkm). Figure 6 shows the correlations between Prof(ϕ, FULL) and Prof(ϕ, PP) and those between Prof(ψ, FULL) and Prof(ψ, PP), γ^ϕ and γ^ψ, respectively, for 42 proteins.

Figure 5.

Dihedral angle profiles for histidine ammonialyase (1gkm). (A) Prof(ϕ, FULL) and Prof(ϕ, PP) and (B) Prof(ψ, FULL) and Prof(ψ, PP). Blue and orange lines represent the FULL and PP models, respectively. Given that this protein is too large to show all the data (2,028 amino acid residues), results are shown only for the residues 400–509 of chain A and 1–100 of chain B. The remaining regions, that are not shown, also exhibited good correspondence between the two models.

Figure 6.

Correlation coefficients between fluctuations of ϕ and ψ for the FULL and PP models for the 42 proteins examined.

Figure 5 shows that Prof(ϕ, FULL) and Prof(ψ, FULL) are well reproduced by Prof(ϕ, PP) and Prof(ψ, PP), respectively. Statistically, γ^ϕ and γ^ψ in Figure 6 are also significantly high, indicating that the effect of fixing the side-chain conformations is small. The correlation of ϕ (γ^ϕ=0.7–0.8) is slightly weaker than ψ (γ^ψ=0.8–0.9). In the Ramachandran plot, the allowed region of ϕ is smaller than ψ, mainly owing to the interactions between the main-chain and side-chain atoms. The fixation of the side chains may affect ϕ more than ψ.

The correlations were worse in some proteins; For example, γ^ψ is less than 0.7 for Ketol-acid reductoisomerase (4ypo) [21], nitrogenase MoFe protein (3u7q) [22], lysl endoproteinase (4nsv) [23], and transketolase (4kxu) [24]. In these proteins, ψ angles fluctuate extraordinarily in the C-terminal residues that have large temperature factors because PDB structures of these proteins have many missing residues in their C-terminal ends. We did not find any correlation of either γ^ϕ or γ^ψ with the composition of amino acids with long side chains (Glu, Gln, Lys, Arg, and Met). Consequently, we can conclude that the correlation between fluctuations of ϕ and ψ in the FULL and PP models is good, and thus the fluctuations of ϕ and ψ are well reproduced in the PP model.

Discussion

In this study, we examined the PP and VB models in comparison with the FULL model. In the PP model, the side-chain conformations and peptide planes in the main-chain are fixed to the native conformations. In the VB model, the conformation of individual residues is fixed to the native one and only the mutual positions between residues are allowed to vary. In spite of the fixation of the local conformation, the residue-by-residue fluctuation profiles for the atoms were well reproduced except for some terminal residues of a polypeptide chain with an exceptionally large fluctuation. The fluctuation profile for dihedral angles in the PP model was also reproduced to a significant extent.

The displacement vectors of atoms in the lowest-frequency normal modes in the PP and VB models were also very similar to those of the FULL model with respect to the directional correspondence of motion. As such, it was shown that the PP and VB models can be used to predict the direction of large-amplitude concerted motion. In the mode-to-mode correspondence between different models, most of the mode pairs from different models have the same mode number. Given that the motions of the individual lowest-frequency normal modes are frequently interpreted in relation to the function of a protein, the good correspondence between the models is significant.

The reason why some modes in one model correspond to the modes with different mode numbers in another model in the mode-to-mode correspondence should be noted. Most of these cases involve swapping of neighboring modes; i.e., the modes, i and i+1, of one model corresponds to the modes, i+1 and i, of another, respectively; i.e., they are the mode pairs with the mode number gap=±1 in Figures 4C and 4D. Sometimes, the cosine similarity between two modes is slightly lower. These cases mainly arise from degeneracy or the very close frequencies of the two modes. The degeneracy of modes is frequently observed in homo-oligomers owing to their symmetric properties. The motions of such modes obtained in the NMA are principally the same, but are given at different phases. This is the reason why the neighboring modes are swapped and the cosine similarity between them becomes smaller. Therefore, the difference in the mode number correspondence does not necessarily indicate a defect in the models that fix the local conformation.

In fact, the mode-to-mode correspondence between the FULL and VB models of GH47 α-mannosidase shown in Supplementary Figure S2(B) as the worst case is a typical example in which the close frequencies of neighboring modes cause swapping of the correspondent modes. In the FULL model, the frequencies of the first to fourth modes are very close to each other, including the eighth and ninth modes. Supplementary Figure S2(B) shows irregular correspondence between these modes. This protein is a monomer, but forms a ring consisting of seven units of inner and outer helices referred to as an (α/α)₇ barrel. Presumably, the structure with a rotational quasi-symmetry is responsible for the close frequencies of these modes.

There are several possible reasons why the PP and VB models are good approximation methods for reducing the number of variables in the NMA computations.

The most important point is that we assessed only the reproduction of large-scale conformational changes around the native conformation (i.e., the minimum-energy conformation). The motion of constituent atoms is constrained because the atoms are composed of a chain molecule and are packed densely in the native conformation. In addition, NMA can reveal only motion in the neighborhood of the native conformation. Consequently, the possible large-scale conformational changes are so limited that the NMA results do not strongly depend on the independent variables chosen.

In this sense, we believe that a full-atom model may play an essential role. The computational load of the conformational energy calculations even for the full-atom model is much less than that of the eigenvalue problem in the NMA. In addition, the number of variables is not affected by the number of constituent atoms in the models considered here. Consequently, there is no reason to use a coarse-grained model in which many constituent atoms are discarded.

Na, et al. [25] compared three NMA programs developed independently, sbNMA [26], ATMAN [27], and ProMode [7,28], whose atom sets include all heavy atoms. In addition, sbNMA includes all hydrogen atoms, and ATMAN contains main-chain amide hydrogen atoms. The independent variables are the Cartesian coordinates in sbNMA and the dihedral angles in ATMAN and ProMode. It was shown that the lowest-frequency normal mode motions are nearly identical, not merely similar. These results also show that the full-atom model can provide consistent features of the lowest-frequency normal modes.

The variables used in the NMA calculation, i.e., dihedral angles as independent variables, are another important point. Proteins are chain polymers. Therefore, it is typical that the main-chain dihedral angles would be considered principally responsible for the concerted motion of proteins that are dominant in the low-frequency normal modes. The same applies to the virtual bonds, although they are artificial. As such, given that the dihedral angles can be definitely classified into global and local groups, it is less ambiguous to decide which dihedral angles should be fixed.

The algorithm for rapid calculation of the Hessian matrix with dihedral angles as independent variables was developed by Professor Gō’s group [29–30]. The coding of a computer program for NMA for the DA system is complicated compared to that of the CC system. However, it is possible to code the program independent from the molecular structure. Therefore, once the computer program for NMA calculation is written, the remaining task is to create structural data of a given molecular system for input to that program. The point in constructing the structural data is to represent the molecular system, including not only a monomer, but also an oligomer as in a tree in a graph theory sense, as previously described [11–12]. A tree is one of the most familiar data structures in computer programming. In fact, it is essentially automatable for the creation of molecular structure data from PDB data in our program, PDBETA [7], even though the data contain non-protein molecules such as DNA, RNA, and any ligand molecules. Furthermore, because the program can work as long as a molecular system is represented as a unit-bond system, it was easy to apply the program to the VB system examined in this case.

The assessment of the models proposed here strongly depends on what is expected in the normal mode analysis. It is obvious that we cannot expect to obtain information about conformational fluctuations that deviate significantly from the native conformation. We must be careful not to exceed the limits of the model when interpreting the results. Nevertheless, the NMA results are still useful for quickly surveying protein dynamics. We believe that the models with a smaller number of variables, and consequent speed-up of computing time, can provide reliable views of the concert motion of huge protein systems.