Results and Discussion

Results are presented for RT, p38, and Cdk2, using the datasets, I, II, and III, comprised of 112 (RT), 74 (p38) and 106 (Cdk2) X-ray crystallographic structures, respectively. The complete list of PDB identifiers may be found in the supporting information (SI) Table S1, and the corresponding distributions of RMSDs may be seen in the Fig. S1. The RMSD distributions for the examined NMR ensembles are presented in the Fig. S2. Below is an outline of the approach adopted for each dataset. More details can be found in the Methods and the SI Text and Fig. S3.

The goal is to compare two sets of data: experimental structural data for the same protein in different functional forms, including various ligand-bound forms; and computational data predicted using a representative unliganded structure in the dataset.

The experimental structural data are analyzed as follows: (i) The ensemble of structures are superimposed using the Kabsch algorithm in an iterative procedure (see SI Text), and mean positions 〈R_i〉 = [〈x_i〉 〈y_i〉 〈z_i〉]^T are determined for α-carbons 1 ≤ i ≤ N (or those with known coordinates), (ii) Departures from their mean positions, ΔR_i^s = [Δx_i^s Δy_i^s Δz_i^s]^T (where Δx_i^s = x_i^s − 〈x_i〉) are organized in a 3N-dimensional deformation vector ΔR^s (where (ΔR^s)^T = [(ΔR₁^s)^T (ΔR₂^s)^T…. (ΔR_N^s)^T]), for all structures, s, in the dataset; and their cross-correlations, averaged over the entire set are combined in a 3N x 3N covariance matrix C, and (iii) C is diagonalized to determine the principal modes of structural variations, p⁽ⁱ⁾, observed in experiments. The principal modes (m of them, for an ensemble of m < 3N-6 structures) are rank-ordered: PCA mode 1 (PC1), p⁽¹⁾, refers to the direction of maximal variance, succeeded by PC2, etc. Of interest is to view the distribution of dataset structures in the subspace spanned by PC1 and PC2, which permit us to discriminate, or cluster, the conformations based on their most distinctive structural similarities and/or dissimilarities.

The computational data are generated using the ANM (see Methods). In this case, C is simply the inverse of the 3N × 3N Hessian matrix H (17, 18, 24) (see the SI Text). Eigenvalue decomposition of H gives the ANM modes u⁽¹⁾, u⁽²⁾, …, u^(3N-6), intrinsically favored by the native contact topology. Each eigenvector represents a normalized direction of motion away from the original energy minimum in the 3N-dimensional space. The easiest (or energetically least expensive) direction is, by definition, u⁽¹⁾. This is because the associated uphill curvature of the energy surface or effective force constant (eigenvalue λ₁) is the smallest.

Our objective is to compare the functional variations in structures observed experimentally, and those expected from a physical theory and method (ANM) based on native contact topology. We focus on the top-ranking PCA modes and top-ranking ANM modes, which reflect the dominant features in each group of results. In all three proteins, we will show how the ensembles of conformations observed in experiments (in the presence of different ligands) may be explained by the intrinsic dynamics of the protein (in the absence of ligands). These findings lend support to the view that the “functional changes in structure” predominantly obey ‘structure-encoded preferences’. Alternatively, one might argue that structures have evolved to preferentially sample functional changes in conformation. We now proceed to detailed analyses of the three cases.

HIV-1 RT.

HIV-1 RT is a multifunctional enzyme composed of two subunits, p66 and p51 (25). p66 contains the polymerase and RNase H domains. The polymerase domain is described as a right hand containing fingers, palm, thumb, and connection subdomains. The connection subdomain links the polymerase and RNase H domains. The p51 subunit bears only the polymerase domain, comprised of the same subdomains, arranged in a different tertiary structure. Non-nucleoside RT inhibitors (NNRTIs) are known to act allosterically by interfering with the functional motions of p66 fingers and thumb upon binding a pocket at a global hinge site (25–28).

PCA Results.

Fig. 1A shows the projection of RT structures onto the subspace spanned by the first two principal axes, PC1 and PC2, determined for the examined dataset (Table S1). The points therein represent 112 RT structures. These two PCA modes were found to account for 71% of the total variance in structure. Notably PC1 provides a clear separation of the structures into three clusters according to the types of ligands.

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Fig. 1.

Results for HIV-RT. (A) Projection of 6 unliganded (red), 97 NNRTI-bound (blue), 8 dsDNA/RNA-bound (green), and 1 ATP-bound (black) RT structures onto PC1 and PC2. (B) Structural variation along PC1, illustrated using selected structures labeled in A. (C) Structural variation along PC2. Inset shows a closer view of the thumb subdomain. Inhibitors are shown by the same color as the corresponding RT conformation. (D) Comparison of the weighted sum of square displacements along PC1 and PC2, with those predicted along ANM modes 2 and 3. (E) Projections of the 112 structures onto PC1 and ANM2 directions. (F) Projections onto PC2 and ANM3.

The structural variation represented by PC1 is shown in Fig. 1B. The most distinctive feature is the large movement of the thumb. More careful examination reveals the anti-correlated displacements of the fingers and thumb of ≈10Å and 20Å at their most exposed regions, respectively. The fluctuations along this mode contribute by 47% to the total variance, and span a cumulative displacement of 176Å along this axis. Note that this value refers to the cumulative displacement summed over all residues (see Methods and Fig. S3). The tendency of RT to sample conformations along this mode in the absence of ligands is evidenced by site-directed spin labeling experiments (29) and supported by ENMs (26, 27) and MD simulations (10).

PC2 describes the out-of-plane fluctuations of the thumb (Fig. 1C), which would not be obvious from comparisons of arbitrarily chosen structures. These fluctuations are approximately orthogonal to those described by PC1. Resulting structural differences are illustrated in Fig. 1C using two structures separated by 81 Å. Fig. 1C Inset shows a close up view of the thumb. The thumb and the polymerase primer grip (part of the palm), move together as a rigid body.

As a further test, we analyzed the NNRTI-bound subset. The PC1 in this case was found to be almost identical (correlation coefficient of 0.99) to the PC2 of the complete set, and contributed by 50% to the total variance (Table S2). It was also interesting to note that the PC1 from the complete ensemble did not have a counterpart in the NNRTI-bound subset. This supports the view that NNRTI binding depresses the anti-correlated fluctuations of the fingers and thumb, and stimulates thumb fluctuations in an orthogonal direction. This observation is in parallel with the previously proposed view that NNRTI inhibition is achieved by imparting a change in the direction of the thumb movements (26, 27).

How does a small molecule perturb the global dynamics of such a large structure? The answer lies in the location of the NNRTI binding pocket. As shown in the displacement profile in Fig. 1D, the binding pocket residues show minimal, if any, variations in their positions in these two dominant PCA modes. ANM calculations presented below also confirm that the NNRTI binding residues are severely constrained in the global modes of RT. Perturbation of such a severely constrained region in the global modes has a dramatic effect, hence the inhibition of functional movements by NNRTI binding at this critical site.

Comparison with ANM Results.

Fig. 1D displays the joint effect of ANM modes 2 and 3. These two ANM modes were found to yield the highest correlation (among all ANM modes) with PC1 and PC2, respectively (see Table 1). ANM mode 1, however, refers to the anticorrelated fluctuations of fingers subdomain and RNase H domain. Comparison with PCA modes showed that this mode shows a weak correlation (0.52) with the PC5. The directions of first three ANM modes are shown in Fig. S4A. As a direct comparison of the structural changes along PC1 and motions induced in ANM mode 2 (ANM2), we examined the level of correlation between the projections of the structures onto these two collective displacement directions. Fig. 1E displays the results. Strikingly, the structures perfectly align along these two axes (correlation coefficient of 0.99), demonstrating the equivalence of the predicted (ANM2) and experimentally observed (PC1) global modes. Similarly, by projecting the structures onto ANM3, and its PCA counterpart, PC2, we find a correlation coefficient of 0.94, again supporting the view that the most distinctive structural changes assumed by the NNRTI-bound RTs simply originate from intrinsically favorable ANM modes. When put together, these results suggest that RT samples conformations predisposed to NNRTI binding, and NNRTI binding shifts RT dynamics from one mode to another, both being intrinsically favored.

p38 MAP Kinase.

The p38 MAP kinases are serine/threonine kinases activated in response to external stress, regulate the production of proinflammatory cytokines, and hence serve as targets in the treatment of inflammatory diseases (30). The structure and interactions of p38s with different inhibitor classes are well characterized. Their dynamics, however, is not as well understood and poses challenges in inhibitor docking (31).

PCA Results.

The results from the PCA are presented in Fig. 2. Fig. 2A displays the distribution of 74 structures of p38 isoform α. The p38 structure has a canonical kinase fold composed of a β-sheets rich N-terminal lobe (N-lobe) and an α-helix rich C-terminal lobe (C-lobe) (32). The catalytic site, the binding site for competitive inhibitors, is the cleft between these two lobes. Example structures from different subgroups are displayed in Fig. 2 B and C to illustrate the major structural changes along PC1 and PC2, respectively. PC1 refers to anticorrelated movements of the two lobes, as shown by the superimposition of an unliganded (red) and an inhibitor-bound (blue) structures in panel B. These movements map to a separation of >25 Å between the two conformers along PC1 axis (Fig. 2A). PC2, however, involves twisting motion of N-lobe with respect to the C-lobe illustrated by a glucoside- and an inhibitor-bound structure (Fig. 2C). The size of the motions along PC2 is comparable with that along PC1 (Fig. 2A). The global backbone changes observed in the ensemble are well described by the first two PCs. Notably, although these modes represent only 2 degrees of freedom of a total of 963, they account for 45% of the variance in the backbone structure observed in the dataset (Table 1), and reconfigurations along these two directions allow for reducing the average RMSD with respect to the mean structure from 1.0 ± 0.3 Å to 0.53 ± 0.17 Å.

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Fig. 2.

Results for p38 MAP kinase. (A) Projection of 4 unliganded (red dots), 56 inhibitor-bound (blue), 10 glucoside-bound (yellow), and 4 peptide-bound (violet) p38 structures onto PC1 and PC2. Distant structures along PC1 and PC2 are selected to illustrate in the respective B and C the structural variations represented by these PCs. (D) Square displacements of residues along PC1 and PC2, compared with those driven by ANM modes 1 and 3. (E) Projections of the 74 structures onto PC1 and ANM3. (F) Projections onto PC2 and ANM1.

Fluctuations along PC1 determine the exposure of the binding cleft. In the unliganded state, the cleft is wide open, presumably to facilitate ATP/inhibitor recognition and optimal interaction. Upon ligand binding, the cleft closes down. The closure of the cleft increases the packing interactions with the bound molecule. Movements along PC1 are therefore functionally relevant. In one of the peptide-bound structures, p38 is complexed with its primary substrate MAPK-activated protein kinase 2 (MK2), which is located 20 Å away from the unliganded p38 along PC1 (labeled as 1OZA in Fig. 2A). Other peptide-bound structures contain short stretches, corresponding to docking sites of p38 regulators/substrates, with comparably much less impact on p38. The conformational change observed when p38 is crystallized with MK2 potentially affects its function (33). Fluctuations along PC2, however, change the relative positioning of the atoms lining the cleft between the lobes, affecting in particular the β-strands 1–3 and the connecting hairpins.

Fig. 2D displays the square displacements of residues resulting from the weighted contributions of movements along PC1 and PC2. The binding pocket residues are marked by the blue dots. Some of them, corresponding to hinge sites in these global modes, are extremely constrained, and others show moderate displacement. The structural diversity of inhibitors may be a primary reason for the conformational heterogeneity along these modes. The average pairwise similarity among the 55 compounds in the dataset is 0.40 ± 0.15 based on standard cheminformatics metrics. Apparently, the movements of the lobes allow for optimizing their interactions with the small-molecules, and hence the poor results in in silico cross-docking experiments, when the target backbone is assumed to be rigid (31).

Comparison with ANM Results.

The correlations between the lowest three ANM modes and the top-ranking three PC directions are listed in Table 1. The counterparts of PC1 and PC2 are found to be ANM3 and ANM1, with respective correlation coefficients of 0.71 and 0.79. ANM2, a twisting motion of the two lobes, was found to correspond to PC3 with a coefficient 0.57 (Fig. S4B illustrates these three ANM modes). The displacements of residues induced by ANM modes 1 and 3 are compared in Fig. 2D with those driven by PC1 and PC2. The two profiles agree with a correlation coefficient of 0.79. Also important is the directionality of ANM predicted fluctuations. Hence, we projected the ensemble onto these ANM modes and distributions along corresponding PCs. Distributions of the structures along the PC1-ANM3 (Fig. 2E) and PC2-ANM1 (Fig. 2F) yielded correlation coefficients of 0.95 and 0.82, respectively. This excellent correspondence underscores the robustness of the low-frequency modes, and shows their functional importance in accommodating the binding of structurally diverse inhibitors and the substrate MK2 which presents a much larger interface, at the P38 active site cleft.

Cdk2.

Cdks are serine/threonine kinases involved in the regulation and progression of the cell cycle. Cdk activity is regulated by activator (cyclin family) or inhibitor (INK and Cip families) protein binding and phosphorylation (34). In cyclin- or Ink-bound structures, the N-lobe adopts a host of distinct conformations intrinsically favored by Cdk (6). This conformational flexibility impacts the ATP/small-molecule pocket, and hence needs be considered in designing drugs (31).

Results from PCA.

We have assembled an ensemble of 106 Cdk2 structures. This set was more narrowly distributed compared with the previous two cases (average RMSD of 0.50 ± 0.14 Å with respect to the reference structure; Fig. S1) and therefore the structural variations may not be large and precise enough to classify them into distinctive PC modes. Yet, the first two PCA modes were able to account for 39% of the variance in the dataset (Table 1). The projection of Cdk2 structures onto PC1 and PC2 showed a diffuse distribution, not clearly distinguishing the different bound forms, although the two unliganded structures clustered together at a distinctive end of the subspace (Fig. 3A). We selected the Cdk2 structures that fall in the extremes of this distribution to visualize the conformational differences along these two PCA directions. PC1 describes the twisting motion of N and C lobes, which is comparable with p38 movements along PC2. This is illustrated by two structures that are 13.2 Å apart along this axis (Fig. 3B). In Fig. 3A, the unliganded structures fall close to the center on PC1 coordinate, which indicates that twisting motion occurs in either direction. The structural variation described by the PC2, however, was localized at chain termini and flexible loops. Fig. 3C illustrates such a local movement for the so-called glycine loop that lines the binding cleft. The two structures in Fig. 3C are 10.8 Å apart when projected onto PC2. The heterogeneity of the ensemble along these modes presumably originates from the physicochemical diversity of bound inhibitors. The ensemble contained 100 different compounds with an average pairwise similarity metric of 0.43 ± 0.14.

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Fig. 3.

Results for Cdk2. (A) Projection of 2 unliganded (red), 3 ATP-bound (green), and 101 inhibitor-bound (blue) Cdk2 structures onto PC1 and PC2. (B) Structural variation along PC1. (C) Structural variation along PC2. (D) Comparison of the square displacements of residues along PC1 and ANM2. (E) Projection of 106 Cdk2 structures onto PC1 and ANM2.

Comparison with ANM Predictions.

The top-ranking ANM modes predicted for the unliganded structure (1HCL) are displayed in Fig. S4C. Among them, ANM2 yields a correlation of 0.73 with PC1 (Table 1). The square displacements of residues along these modes are compared in Fig. 3D. Some of the binding pocket residues are located at the minima of these profiles, while the rest are located in highly mobile regions of N-lobe. Those at the minima include Val-64, Phe-80, Asn-132, and Ala-144-Asp-145, which are buried deep into the cleft between the two lobes, while those exhibiting moderate-to-high flexibility are at the periphery of the cleft. This indicates that fluctuations along this mode allow for the positioning of recognition site residues, while other functional residues, such as Asn-132 and Asp-145 that coordinate the metal atom for the catalytic reaction, and the catalytic residues Asp-127 and Lys-129 are almost fixed. The projections of the ensemble of structures onto ANM2 and PC1 yield a correlation of 0.97 (Fig. 3E). These suggest that despite the minimal changes in structure, the observed structural heterogeneity is not random, but geared toward the intrinsically favored mode of motion. The changes along the experimentally observed PC2, however, seem to be rather local and would rather be viewed as induced by inhibitor binding.

NMR Ensembles.

The ensembles of NMR models deposited for ubiquitin (9) and for CaM (8) have been pointed out to populate conformations comparable to those observed in their substrate-bound forms. The question is: how do principal modes of structural change extracted from the PCA of these ensembles correlate with those predicted by the ANM?

Results are presented in Fig. 4. The projection of the NMR models onto PC1 and PC2 are shown in both cases to exhibit a close agreement with one or two ANM global modes. We note that CaM samples a very broad conformational space in its unliganded form (Fig. S2) as the N- and C-terminal domains (NTD and CTD) are connected by a helix that is readily flexed. We examined the principal modes accessible to the NTD and CTD in relation to their ANM counterparts, as a further comparison. Correlations in the range 87% to 94% were observed in both cases (Fig. S5A–D), demonstrating that the ANM provides a good representation of the variability of the structures even within domains, provided that they are sufficiently decoupled.

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Fig. 4.

Comparison of the principal modes from NMR ensembles with ANM-predicted global modes. The projections of NMR models for ubiquitin (panels A and B), and CaM (C and D) onto PC1 and PC2 are compared with the projections onto ANM global modes. ANM calculations are performed for the model that has the lowest RMSD with respect to all others in each ensemble. See Fig. S2 and Fig. S6 for the respective RMSD distributions and ANM modes.

Finally, we also analyzed the complex CaM-MLCK with a myosin light chain kinase (MLCK) peptide, which further confirmed that principal modes derived from NMR models are in accord with ANM predictions (correlations of 0.88 and 0.77 in Fig. S5 E and F). The results for all examined NMR ensembles are compiled in Table S3 and Table S4, and illustrated in Fig. S6. The cumulative overlaps between subsets of 3, 6, and 20 ANM modes with the top three PCs (Table S3) further support the consistent correlation between the subspace of conformations seen in the experimental structures and those predicted by computations.