Evaluating Elastic Network Models of Crystalline Biological Molecules with Temperature Factors, Correlated Motions, and Diffuse X-Ray Scattering

Abstract

In this study, the variance-covariance matrix of protein motions is used to compare several elastic network models within the theoretical framework of x-ray scattering from crystals. A set of 33 ultra-high resolution structures is used to characterize the average scaling behavior of the vibrational density of states and make comparisons between experimental and theoretical temperature factors. Detailed investigations of the vibrational density of states, correlations, and predicted diffuse x-ray scatter are carried out for crystalline Staphylococcal nuclease; correlations and diffuse x-ray scatter are also compared to predictions from the translation, libration, screw model and a liquid-like dynamics model. We show that elastic network models developed to best predict temperature factors without regard for the crystal environment have relatively strong long-range interactions that yield very short-ranged atom-atom correlations. Further, we find that the low-frequency modes dominate the variance-covariance matrix only for those models with a physically reasonable vibrational density of states, and the fraction of modes required to converge the correlations is higher than that typically used for elastic network model studies. The practical implications are explored using computed diffuse x-ray scatter, which can be measured experimentally.

Introduction

X-ray scattering from biological crystals provides a wealth of information about the time-averaged coordinates of atoms. Under the harmonic approximation, the matrix of coordinate variances and covariances is intimately related to the intensity of x-ray scattering through modulation of the atomic pair-distribution. Conventional experimental x-ray crystallography protocols utilize only the sharply localized Bragg scatter that describes the coordinate variances via the Debye-Waller factor (1). In this approximation, the background intensity surrounding each Bragg peak is subtracted out. In addition to removing artifacts (e.g., air scattering), this process removes contributions to the intensity associated with correlated motions, imposes symmetry, and reduces the accessible information about dynamics. The efficacy of this approach is apparent considering the vast number and value of biomolecular structures determined with x-ray crystallography, but additional dynamic information can be gleaned from crystallography experiments by studying both the Bragg peaks and the diffuse x-ray scattering that is associated with variations about the average coordinates caused by correlated displacements (1-6). The development of efficient and accurate computational models of the dynamics of biological molecules should enable such experimental advances while at the same time improving our understanding of molecular dynamics. In this study, the theoretical framework of x-ray crystallography is used to compare several elastic network models in detail.

The pioneering work of Tirion (7) showed with normal mode analysis (8-10) that the low-frequency vibrations of all-atom potentials could be well reproduced using simplified potentials that invoke elastic networks. Elastic network models with varying cutoffs, which define a maximum interaction distance, were compared to the all-atom L79 potential (11). The low-frequency region of the cumulative density of states

$$G\left({\omega }_f\right)={\sum }_i^{f}g\left({\omega }_i\right)$$

(where g(ω_i) is the density of states at frequency ω_i) was found to increase curvature with increasing cutoff distance, where relatively short cutoffs were closest to the all-atom result (G_ω ∝ ω²) (12) and were recommended for future applications. Subsequently, other researchers have developed several modifications that differ in the representation of the molecule and/or the spring force constant between two interacting atoms. Because low-frequency modes are usually collective in nature, they can be reasonably described with relatively coarse-grained models (13-15), where residues are represented by C_α atoms (16-18) or rigid blocks (19,20). Atomic interactions are modulated with either distance-based cutoffs or a functional dependence of the force constant on the distance between atoms (17,21-23). For validation, elastic network models were typically developed either with respect to all-atom potentials (HCA) (17) and simulations, REACH (Realistic Extension Algorithm via Covariance Hessian) (22) using empirical force fields or by comparisons between predicted and crystallographic temperature factors (16,24) along with characterizations and comparisons of low-frequency modes (18,19,21,23).

For practical applications, elastic network models have the distinct advantage of having the energy minimum defined with respect to a given structure. This allows direct application to molecules of various resolutions and is an attractive approach for structural refinement (25). On the other hand, elastic network models are not necessarily transferable from one system to another; the network varies from protein to protein and the force constants typically require different scaling constants for magnitudes to be comparable to each other or to experiment. In validation of elastic network models, consideration of the environment is also important. Comparisons of temperature factors predicted for isolated (ISL) biological molecules are not necessarily representative of those determined from x-ray experiments carried out for the crystalline state. The effect of treating crystalline environment on the dynamics has been studied in detail (26-29), and the explicit inclusion of crystal contacts significantly affects dynamics and improves temperature factor predictions (24,27,29,30). In fact, those models that were optimized (in terms of cutoffs and force constants), with respect to temperature factor comparisons without regard for the crystal environment, are most likely far too restrictive (29), wherein the crystal effects are parameterized into the isolated molecule.

In this study, the dynamics and calculated diffuse x-ray scattering, as described by the variance-covariance matrix (VCOV) (31,32) of crystal structures, is used to compare several popular elastic network models with representations ranging from all-atom (nonhydrogen) to those projected into blocks (BNM) and reduced to include only C_α atoms. The investigation is carried out at two levels: a bioinformatics-like approach where a set of 33 ultra-high resolution structures is used to determine average behavior of the density of states and theoretical temperature factors; more-detailed investigations of the density of states, correlations, and diffuse x-ray scatter are carried out for crystalline Staphylococcal (staph) nuclease (PDBID: 1STN (33)). Comparisons are made where possible to all-atom simulations carried out previously by Meinhold and Smith (34) and diffuse x-ray scattering experiments carried out by Wall et al. (35). When the crystal environment is included, periodic boundary conditions (PBC) are invoked about the primitive unit cell (P1PBC).

Methods

VCOV and power laws of G_ω and C_V

The VCOV is straightforward to compute (31) from the eigenvectors (L) and eigenvalues (Λ, diagonal matrix of square circular frequencies ω²) of the elastic network model force constant matrix,

$$\langle \mathbf{u}{\mathbf{u}}^{\mathit{T}}\rangle ={\mathit{k}}_{\text{B}}\mathit{T}\times \mathbf{L}{\Lambda }^{-1}{\mathbf{L}}^{\mathit{T}},$$ (1)

where k_B is the Boltzmann constant and T is the temperature. This expression highlights that the contributions of all vibrations are proportional to the inverse square of the circular frequencies, which leads to the dominance of low-frequency vibrations. The zero frequency modes are not included. Similar to our previous study (29), the scaling behavior of cumulative density of states (G_ω ∝ ω^b) and heat capacity (C_V ∝ T^b′) at low frequency and temperature, respectively, are compared for several elastic network models.

Force constants

The elastic network model force constant matrix (Hessian) is constructed from interatomic spring interactions. For simpler elastic network models, the spring constant, k(R), for interacting atoms may depend on the distance between them if they are within a preset cutoff distance, R_cut,

$$k\left(R\right)=\{\begin{array}{ll}{R}^{-c}\hfill & \text{if}R\le R_{cut}\hfill \\ 0\hfill & \text{if}R>R_{cut}\hfill \end{array}.$$ (2)

In the main text, all but one of the models stem from two distance-dependent elastic network models (ENMs) with c = 2 and c = 6. The first is referred to as anisotropic network model (ANM), ANM²₅₀, and ENM²₅₀ for C_α and all-atom (nonhydrogen) representations where the cutoff distance is 50 Å. The cutoff is set to be sufficiently long to satisfy, within reason for crystalline conditions, the recent suggestions of the distance dependence for the parameter free elastic network model (pfENM) and parameter free anisotropic network model (pfANM), where the cutoff distance is discarded (R_cut = ∞) (23); invoking cutoffs is convenient when including the crystal environment, but these cutoffs (even the long ones) necessarily change the interaction model recommended for pfANM (23). It is expected that the 50 Å cutoff is representative of how the pfANM would perform under crystalline conditions (see Fig S3 in the Supporting Material for additional discussions and calculations using longer cutoffs than those included here). Inspired by the C_α model of Hinsen et al. (17) referred to as HCA, we introduce a similar all-atom potential with a bonded region (within 3 Å) that has a constant value and a nonbonded region with an R^–6 dependence, ${\text{ENM}}_{3:10}^6$ and ${\text{ENM}}_{3:25}^6$ for 10 and 25 Å cutoffs, respectively; to characterize the effect of the bonded region, ${\text{ENM}}_{25}^6$ (very similar to pfENM with c = 6) is also carried out. All-atom calculations are carried out using the block normal mode (BNM) approach, where all-atoms of a given residue are projected into a rigid blocks with rotational and translational degrees of freedom; they are referred to as ${\text{BNM}}_{50}^2$ and ${\text{BNM}}_{3:10}^6$. Finally, REACH is used, which was parameterized by Moritsugu et al. (22) to capture the variance-covariance behavior of all-atom molecular dynamics simulations using C_α atoms. REACH has constant-bonded (pseudo bonds, angles, and dihedrals) and nonbonded interactions, which fall off exponentially. The parameters of the force constants for isolated and periodic boundary conditions were determined in Moritsugu et al. (22) for the solvated myoglobin monomer and crystalline tetramer, respectively. See the Supporting Material for more-detailed descriptions and additional models.

Boundary conditions

All of the elastic network models discussed above have been implemented for isolated (ISL) and crystalline state calculations, which has been discussed in detail previously (29) and will be briefly described here. In the main text, the crystal environment is included using a primitive unit cell with periodic boundary (P1PBC) conditions for all-atom elastic network models; for C_α elastic network models, Born-von Kármán (BVK) boundary conditions are also used. BVK expands on P1PBC to include lattice vibrations. The classical theory of lattice vibrations was formulated by Born and von Kármán in the early twentieth century and has been discussed in detail in several classical texts (1,36). Eighty-one wavevectors (permutations of 0.00, 0.25, and 0.50 along reciprocal lattice vectors) are distributed throughout one-half of the Brillouin Zone, which constitutes a finite sampling of the full region due to the frequency dispersion symmetry and relationships between eigenvectors and their complex conjugates. The VCOV is averaged over all sampled wavevectors. A larger number of wavevectors (425 and 3700) is used with REACH for staph nuclease to determine the convergence behavior of the atom-atom correlations. The crysFML library (37) is used for all necessary symmetry operations required for crystal construction.

Block normal mode

For this study, the all-atom elastic network models for all boundary conditions are also projected into rigid blocks of atoms, block normal mode (BNM) (19,20), where each residue constitutes a block. For staph nuclease, parallels are drawn between BNM and the translation, libration, screw model (TLS) (38) where the TLS tensor is projected into the covariance-matrix using similar projection matrices (see the Supporting Material). The TLS tensor for the entire molecule (denoted by TLS₁) is determined from the TLSMD webserver (39).

X-ray scattering

The total intensity at a point in reciprocal space (Q) can be separated into Bragg (I_B) and diffuse (I_D) contributions,

$${\mathit{I}}_{\mathit{T}}\left(\mathbf{Q}\right)={\mathit{I}}_{\mathit{B}}\left(\mathbf{Q}\right)+{\mathit{I}}_{\mathit{D}}\left(\mathbf{Q}\right),$$ (3)

which, under harmonic approximation, is directly related to the VCOV and has been described in detail previously (1,3). The Bragg intensities may be computed from the square modulus of the structure factor (S) that is nonzero (for the crystal) at nodes of the reciprocal lattice (2πH) and depends on the atom scattering factors (f_k) and locations (r_k), occupancy (O_k, ranges from 0 to 1), and the temperature factor (T_k). The experimental and theoretical temperature factors are compared with three metrics that have been used previously (29) (see the Supporting Material): linear correlations for isotropic temperature factors; anisotropies (ratio of small/large eigenvalues of the ADP) for atoms with full occupancy (O_k = 1.0); and modified overlap score, cc_mod (21,29,40) for the theoretical and experimental distributions that vary from 0.0 to 1.0 for perfect misalignment and alignment, respectively, for atoms with full occupancy and an experimental anisotropy ≤0.5. Each is averaged over the set of 33 ultra-high resolution crystal structures.

The diffuse scattering intensity (1,3) is computed for a single unit cell of staph nuclease at the nodes of the reciprocal lattice (2πH),

$$I_D\left(2\pi \mathbf{H}\right)={\sum }_k{\sum }_{k^{\prime }}{f}_k{T}_k{f}_{k^{\prime }}{T}_{k^{\prime }}{e}^{i2\pi \mathbf{H}·\left({\mathbf{r}}_k-{\mathbf{r}}_{k^{\prime }}\right)}\times \left(e^{4{\pi }^2{\mathbf{H}}^2{\Phi }_{k,k^{\prime }}}-1\right),$$ (4)

where the VCOV has been substituted by the isotropic atom-atom correlation matrix (Φ_{k, k′},) weighted by the experimental atomic fluctuations determined from the experimental B factors; the atomic scattering and temperature factors (f and T, respectively) depend on the reciprocal lattice vector, 2πH. The weighted atom-atom correlation matrix is constructed from the VCOV by replacing each 3×3 block with the average of its diagonal (〈δr_iδr_j〉),

$${\Phi }_{ij}=\sqrt{u_i^2{u}_j^2}\frac{\langle \delta r_i\delta r_j\rangle }{\sqrt{\langle \delta r_i^2\rangle \langle \delta r_j^2\rangle }},$$ (5)

where the entries are first normalized by the theoretical fluctuations and then weighted by the experimental atomic fluctuations (u²). Using this approach allows a more direct comparison between correlations determined with dynamic models. By setting different blocks of regions of Φ to zero, the diffuse intensity is decomposed into different contributions. For example, the diffuse intensity from interatomic correlations can be determined as

$${\mathit{I}}_{\mathit{D},\text{interatomic}}={\mathit{I}}_{\mathit{D}}-{\mathit{I}}_{\mathit{D},\text{self}},$$ (6)

by subtracting the contribution from atomic self-correlations (I_{D, self}) computed with

$${\Phi }_{i,j}={\delta }_{i,j}\sqrt{u_i^2{u}_j^2},$$

where δ_i,j is Kronecker's delta.

Correlations

The distance-dependent correlations are determined by averaging the normalized atom-atom correlation matrix binned by distance and fit to a modified liquid-like model (2),

$$C\left(r\right)=C_0\left(1-{\lambda }_0\right)e^{-\frac{r}{\lambda }}+{\lambda }_0,$$ (7)

introduced by Meinhold and Smith (34) to determine the correlation length (λ, which is greater than zero), correlation offset (λ₀), and intercept (C₀) for all-atom simulations of a staph nuclease unit cell where both intra- and intermolecular correlations were characterized. The liquid-like model has been applied to several diffuse x-ray scattering studies (2,5,35,41,42) and the model itself has been thoroughly explored in previous theoretical studies (3,34). The main role of the model in this work is to use the fitted parameters to compare elastic network models to each other and to the parameters found in all atom simulations (34) and diffuse x-ray scattering experiments (35).

Results and Discussion

We recently studied the effects of the crystalline environment on predicted temperature factors and the scaling behavior of the cumulative density of states and the heat capacity for a set of 83 high resolution proteins for the following C_α-based models (29): Gaussian network model (16), anisotropic network model (ANM) (18), and the C_α-based model of Hinsen et al. (17) referred to as HCA. In that study, a subset of 33 crystal structures (those with <500 residues in the primitive unit cell) was used to include the more expensive BVK boundary conditions and was found to be representative of the larger set. The distance-dependent HCA (interactions fall off with R^–6 for nonbonded interatomic distances) was recommended for future applications. In this study, additional distance-dependent elastic network models are compared: ${\text{ANM}}_{50}^2$ and ${\text{ENM}}_{50}^2$, which have interactions (50.0 Å cutoff) that fall off with R^–2 for C_α and all-atom (nonhydrogen), respectively; and REACH and ${\text{ENM}}_{3:10}^6$, which are C_α and all-atom models, respectively, that have bonded and nonbonded regions (e.g., ${\text{ENM}}_{3:10}^6$ has a strong distance-independent force constant below 3 Å, which then falls off as R^–6 before the 10 Å cutoff; see Methods). For average behaviors, the same set of 33 proteins is used, and the results are briefly described. More-detailed comparisons of the density of states and correlations are made for crystalline staph nuclease. Additional elastic network models are discussed with results tabulated in the Supporting Material.

Characterizing the density of states and the impact on the VCOV

Those elastic network models that have very strong long-range interactions relative to local interactions (e.g., ${\text{ANM}}_{50}^2$ and ${\text{ENM}}_{50}^2$) yield unphysically high scaling exponents for G_ω and C_V at low frequency and temperature, respectively (see Table 1 for G_ω). The results for heat capacity scaling at low temperature are parallel to that of G_ω. The expected range is 1.6–1.8 for crystalline proteins (43), and all-atom crystalline normal mode calculations on ribonuclease A (with axial sampling of the Brillouin Zone) by Meinhold et al. (44) found that C_V ∝ T^1.68 at low temperature. For the set of 33 proteins, REACH with BVK and ${\text{ENM}}_{3:10}^6$ with P1PBC have average values of 2.20 and 1.82, respectively. Increasing the cutoff of ${\text{ENM}}_{3:10}^6$ to 25 Å (${\text{ENM}}_{3:25}^6$) yields a slight increase (1.87), and when all interactions are treated the same (${\text{ENM}}_{25}^6$), the exponent increases further (2.36). This trend implies that increasing the relative strength of long-range interactions yields steeper scaling exponents; the ratio of force constants at 3 and 5 Å are 1626.3 and 21.4 for ${\text{ENM}}_{3:10}^6$ and ${\text{ENM}}_{25}^6$, respectively. Yielding a significantly higher exponent of 5.03 (P1PBC), this observation holds for the more extreme ${\text{ENM}}_{50}^2$ where the force constant at 3 Å is only 2.8 times stronger than that at 5 Å.

Table 1.

Temperature factor comparisons and the scaling exponent for the cumulative density of states (G_ω) averaged for the set of 33 ultra high-resolution structures

Model	〈Isocorr〉	〈Aniso〉	〈ccmod〉	Gω ∝ ω〈b〉
${\text{ENM}}_{3:10}^6$	0.55 (0.70)	0.42 (0.51)	0.66 (0.72)	1.52 (1.76)
${\text{BNM}}_{3:10}^6$	0.54 (0.66)	0.38 (0.48)	0.64 (0.71)	1.82 (2.23)
REACH	0.50 [0.60]	0.35 [0.51]	0.61 [0.69]	1.85 [2.12]
${\text{ENM}}_{50}^2$	0.60 (0.67)	0.58 (0.86)	0.63 (0.33)	4.73 (19.94)
${\text{BNM}}_{50}^2$	0.45 (0.20)	0.53 (0.62)	0.61 (0.42)	4.66 (23.76)
${\text{ANM}}_{50}^2$	0.59 [0.64]	0.48 [0.85]	0.68 [0.37]	3.87 [32.85]

The correlation coefficient (Isocorr), anisotropy (Aniso), distribution overlap score (cc_mod), and G_ω are shown for isolated and crystalline conditions (see Methods). P1PBC is shown in parentheses (for all-atom elastic network models) and BVK is shown in brackets (for C_α models).

The cumulative density of states and relative contributions to the VCOV are computed for staph nuclease with P1PBC in Fig. 1. The characteristics of the g_ω are reflected in the shape of G_ω: for example, the leveling in the midfrequency region of REACH is due to the g_ω having two distinct frequency regions. In addition to providing direct comparisons between elastic network models, Fig. 1 yields two significant observations that underlie the rest of this investigation.

Figure 1.

Cumulative density of states and the relative contribution to the VCOV as function of the normalized frequency for a unit cell of staph nuclease with P1PBC . For each elastic network model, the density of states (g_ω) is computed as a function of the frequency normalized by its maximum frequency; from this, each cumulative density of states (dashed lines) and the relative contribution of the frequency to the VCOV are computed (solid lines). The relative VCOV contribution is computed using the histogram values of the g_ω divided by the square of the frequency for each bin $\left(g\left(\omega \right)/{\omega }^2\right)$. ${\text{ANM}}_{50}^2$ and ${\text{ENM}}_{50}^2$ have a narrow region of growth for G_ω, in which nearly all frequencies significantly contribute to the VCOV. The opposite is true for the other models. These contrasting behaviors lead to many of the contrasting observations (e.g., anisotropy, atom-atom correlations, and diffuse scatter) discussed in the main text. (Color figure online.)

First, the elastic network models with high scaling exponent for G_ω (and C_V) do not yield a clear dominance of low frequency modes; for ${\text{ENM}}_{50}^2$, G_ω grows from zero to one in a very narrow region from 0.7 to 1.0, and the relative contribution to the VCOV is much larger than zero in that same region; this implies that all ${\text{ANM}}_{50}^2$ modes have very similar frequencies relative to other elastic network models, which have a clear dominance of low-frequency modes (see Fig. 1).

Second, for elastic network models with dominant low frequencies, the nonzero region of the modes with significant covariance contributions is larger than the number of modes (∼100) that is typically utilized in elastic network model studies. For example, approximately half of the REACH modes have significant contributions (G_ω ∼0.5 at 0.2 of maximum frequency, Fig. 1), corresponding to ∼816 modes.

Diagonal entries of the VCOV: temperature factors for the set of 33 high-resolution structures

The effect of boundary conditions is significant (see Table 1). The average linear correlations between experimental and theoretical isotropic temperature factors for C_α atoms improve 27% going from ISL to 0.70 for ${\text{ENM}}_{3:10}^6$ with P1PBC boundary conditions; REACH and ${\text{BNM}}_{3:10}^6$ had similar improvements of 20% and 22% with correlations of 0.60 (BVK) and 0.66 (P1PBC), respectively. For isolated boundary conditions, ${\text{ANM}}_{50}^2$ and ${\text{ENM}}_{50}^2$ had the highest linear correlations of ∼0.60, and including the crystal environment yields slight improvements of 8% and 12%, respectively. ${\text{BNM}}_{50}^2$ yields a significantly worse agreement with experiment (see Table 1) going from ISL to P1PBC. When comparing the experimental and theoretical ellipsoidal distributions (cc_mod), including the crystalline environment yields similar improvements (∼10%) for ${\text{ENM}}_{3:10}^6$, ${\text{BNM}}_{3:10}^6$, and REACH . In contrast, ${\text{ENM}}_{50}^2$, ${\text{BNM}}_{50}^2$, and ${\text{ANM}}_{50}^2$ all yield significant reductions in cc_mod when the crystal environment is included. This implies that these elastic network models are not as transferrable between different treatments of boundary conditions when compared to REACH, ${\text{ENM}}_{3:10}^6$, and ${\text{BNM}}_{3:10}^6$ .

The average theoretical anisotropy for C_α atoms (experimental value is 0.52, where an anisotropy value of 1.0 describes a purely isotropic spherical distribution) reflects observations discussed above with respect to the scaling behavior of G_ω (see Fig. 1). The average anisotropy is 0.34 and 0.28 for REACH and ${\text{ANM}}_{50}^2$ with P1PBC, respectively, when only the lowest 5% of the modes (average is ∼50 for the set of 33) are included; including all modes increases those values to 0.46 and 0.85. BVK increases the anisotropy slightly for REACH from 0.46 to 0.51 while that for ${\text{ANM}}_{50}^2$ remains the same (see Table 1). The significant increase for ${\text{ANM}}_{50}^2$ is due to the similarity of all mode frequencies when compared to REACH, which has a clear dominance of low-frequency modes (see Fig. 1). Next, we shift from the average behavior of elastic network models determined for the set of 33 proteins to more-detailed comparisons of the atom-atom correlations for crystalline staph nuclease.

Off-diagonal entries of the VCOV: staph nuclease atom-atom correlations and diffuse x-ray scatter

The experimental diffuse x-ray scattering was determined for staph nuclease crystals by Wall et al. (35) and modeled using the liquid-like model of Caspar et al. (2) (Eq. 7 with C₀ = 1.0 and λ₀ = 0.0; see Methods). A correlation length of 10 Å and root-mean-square fluctuation of 0.36 Ų (Φ_ij = 0.36 exp(– r_ij/10), see Methods) were found to best represent the scattering. Using all-atom molecular dynamics (MD) simulations of a single unit cell with periodic boundary conditions (P1PBC), Meinhold and Smith (34,45) obtained qualitative agreement with the experimental scattering profile and characterized protein and solvent contributions by setting the respective atomic scattering factors to zero (45). Characterizing the distance-averaged atom-atom correlations, they found similar exponential dependence to that determined by Wall et al. (35); fitting to the modified liquid-like model (see Eq. 7) yielded intra- and intermolecular correlations of 10.9 ± 0.3 with C₀ = 1.0, λ₀ = –0.11, and 14.0 ± 3.2 Å with C₀ = 0.43 ± 0.05, λ₀ = –0.12, respectively. The reduced intermolecular C₀-value reflects weaker short-ranged correlations for atoms belonging to different proteins; the negative values of λ₀ were designated as artifactual anticorrelations caused by the removal of the center of mass translations and rotations (34). In the next section, we carry out similar analyses for the distance-averaged atom-atom correlation function (separated into intramolecular and intermolecular correlations).

Atom-atom correlations

The distance-averaged atom-atom correlation is sensitive to the number of modes included, the boundary conditions (see the Supporting Material), and the different elastic network models in a way that is consistent with above discussions regarding the dependence of the VCOV on the density of states. Unless otherwise stated, P1PBC is used and all modes are included. For all-atom elastic network models, the atom-atom correlation analysis is carried out using all nonhydrogen atoms or only the subset of C_α atoms. In Fig. 2, the distance-averaged atom-atom correlation is plotted for TLS (with the tensor corresponding to rotations and translations of one rigid staph nuclease molecule), ${\text{ANM}}_{50}^2$, and REACH; for comparison, the atom-atom correlation for the liquid-like model (C(r) = e^–λ, Eq. 7 with C₀ = 1.0 and λ₀ = 0.0) is plotted for correlation lengths of 2.0 and 10.0 Å. Because TLS assumes that each rigid block of atoms moves independently (zero interblock correlations), the intramolecular distance-averaged correlation is plotted in Fig. 2, while the total correlation (intra- and intermolecular) is plotted for the elastic network models. The different models yield dramatic differences in the atom-atom correlations. Due to the long-range rigidity of the model, the TLS correlations do not fall-off exponentially; as one may expect this is similar to observed for elastic network models when only low-frequency, large-scale modes are included (see Fig. S1), or when large blocks are utilized in BNM. Considering the experimental and simulation results, TLS does not accurately represent the interatomic correlations for crystalline staph nuclease; this may be true for most biological molecules.

Figure 2.

Distance-dependent correlation function for ${\text{ANM}}_{50}^2$, REACH, and TLS. For ${\text{ANM}}_{50}^2$, only the P1PBC boundary condition is plotted. For REACH, BVK with different numbers of wavevectors sampled are plotted; the zero wavevector (solid circle) corresponds to P1PBC . The correlation assuming the liquid-like correlation (2) with a correlation length of 2 and 10 Å is plotted (dashed and solid lines, respectively); for each elastic network model, the fitted line (dotted) for Eq. 7 is also plotted. (Color figure online.)

More-detailed comparisons of the atom-atom correlations between elastic network models and to the all-atom simulations (34) can be attained by examining the parameters (λ, C₀, λ₀) for fitting the distance-averaged correlation with Eq. 7. Overall, the correlation lengths determined with REACH, ${\text{ENM}}_{3:10}^6$, and ${\text{BNM}}_{3:10}^6$ are in reasonable agreement with those determined from experiment, λ = 10 (35), and all-atom MD simulations (34) discussed above. The most striking result is for ${\text{ANM}}_{50}^2$, where very strong long-range interactions yield very short intramolecular correlation lengths of 1.22 Å, which is similar to that expected for an Einstein crystal where all-atoms vibrate independently (see Fig. 2). With ${\text{ENM}}_{50}^2$, using the correlations from all-atoms or only the subset of C_α atoms, yields similar results to ${\text{ANM}}_{50}^2$ (see the Supporting Material). The source of this observation again connects back to the density of states and Fig. 1, where, because there is no clear dominance of modes, all modes (low and high) contribute similarly such that anisotropy is removed from the system. When all-atoms are used for ENM⁶_3:10, the intramolecular correlation lengths determined with ${\text{ENM}}_{3:10}^6$ and ${\text{BNM}}_{3:10}^6$ are decreased from the C_α subset values of 10.30 and 11.18 to 7.43 and 8.70 Å, respectively. The correlations become more similar to TLS (see Fig. 2) and each other (e.g., REACH and ${\text{ANM}}_{50}^2$) as fewer modes are included. Carrying out the same fits of Eq. 7 when using 5% of the modes reduces the quality of the fit (see the Supporting Material); the correlation length and intercept for all elastic network models are significantly increased. For example, the correlation length increases to 10.71 and 17.41 and the intercepts increase to 1.16 and 1.12 for ${\text{ANM}}_{50}^2$ and REACH, respectively, which is due to correlations not decaying as quickly at short interatomic distances. The practical implications of these correlation discussions will be seen as we explore the diffuse x-ray scatter in the next section.

The intercepts (C₀) are close to 1.0 for intramolecular correlations (see Table 2) when the subset of C_α atoms (for all-atom elastic network models) and all modes are used (see Table 2). For the all-atom correlation fit, the intramolecular C₀ is reduced to 0.93, which implies that intraresidue interactions may be relatively weak for ${\text{ENM}}_{3:10}^6$; on the other hand, C₀ is slightly increased to 1.03 for ${\text{BNM}}_{3:10}^6$, which is due to the rigidity (by construction) of the residues. The short-range intermolecular correlations (as characterized by C₀) are higher than the all-atom MD simulations (34) for both REACH and ${\text{ENM}}_{3:10}^6$ (see Table 2). The all-atom intermolecular C₀ are 0.56 and 0.65 for ${\text{ENM}}_{3:10}^6$ and ${\text{BNM}}_{3:10}^6$, respectively, which imply stronger short-range correlations between molecules than those described by the MD simulation value of 0.43. This may result from the assumed harmonicity of the low frequency modes that were found to be anharmonic in the all-atom simulations (34). In contrast, intermolecular correlations are extremely weak for ${\text{ANM}}_{50}^2$ or ${\text{ENM}}_{50}^2$ with C₀ ∼0.04 (see Table 2).

Table 2.

Fitted parameters for Eq. 7 using the distance-averaged intra- and intermolecular correlations for P1PBC

Model	Intramolecular	Intermolecular
Cα atoms
${\text{ANM}}_{50}^2$	1.21 (0.00) [1.00]	5.74 (0.00) [0.04]
REACH	9.57 (−0.08) [0.98]	9.10 (−0.10) [0.83]
${\text{ENM}}_{3:10}^6$	10.30 (−0.10) [0.98]	11.07 (−0.12) [0.68]
${\text{BNM}}_{3:10}^6$	11.18 (−0.12) [1.00]	10.70 (−0.13) [0.76]
All-atoms
${\text{ENM}}_{3:10}^6$	7.43 (−0.04) [0.93]	9.64 (−0.08) [0.56]
${\text{BNM}}_{3:10}^6$	8.70 (−0.07) [1.03]	10.24 (−0.10) [0.65]
${\text{BNM}}_{50}^2$	2.08 (−0.01) [1.04]	5.26 (−0.00) [0.03]

The parameters are as follows: λ (λ₀) [C₀]. See the Supporting Material for additional results. As denoted, the fits are carried out using only C_α atoms or all nonhydrogen atoms.

The long-range correlations (λ₀) are also informative. For all-atom correlations computed with ${\text{BNM}}_{3:10}^6$, intra- and intermolecular atoms are anticorrelated at long distances with values of −0.07 and −0.10, respectively. In contrast to the all-atom simulations carried out for the primitive unit cell (P1PBC), because there are no corrections to center of mass translations and rotations for elastic network models, the negative long-range correlations are, in part, due to the combined rotations and translations of molecules within the unit cell. The center-of-mass corrections for the all-atom simulations may account for the slight increase in anticorrelation magnitudes (34). As expected, the introduction of acoustic modes (with BVK) converts the anticorrelation of the purely optical modes of P1PBC to a positive correlation as atoms within the unit cell move together over longer wavelength, lattice vibrations. In fact, λ₀ provides a reasonably sensitive measure of convergence with the number of wavevectors as compared to previous analyses of convergence using temperature factors (see Supporting Material of reference 29). For C_α correlations computed with REACH, the overall long-range correlation (both intra- and intermolecular combined) increases from −0.11 to 0.04, 0.09, and 0.12 for BVK with 81, 425, and 3700 wavevectors, respectively (Fig. 2). The corresponding correlation lengths slightly increase from 9.31 Å to 9.99 going from P1PBC to BVK with 81 wavevectors and remains nearly constant for 425 and 3700 wavevectors with values of 10.04 and 10.05, respectively. The value of the correlation length is well converged with 81 wavevectors, but the long-range correlation requires larger sampling of the Brillouin Zone.

Diffuse x-ray scatter

In this study, diffuse x-ray scatter is computed using the atom-atom correlation matrix renormalized by the experimental temperature factors (see Methods). Using a common diagonal allows more direct comparisons that stem from differences in the atom-atom correlations and sidesteps scaling of the force constant. The simplifications inherent in this approach have the disadvantage that the atom-atom correlation matrix does not include anisotropic correlations and the similarity between scattering patterns is maximized. The effects of correlations described in previous section on diffuse scatter is very clear. For the a^∗ = 0 plane in Fig. 3, ANM²₅₀ with all modes yields the spherical shell characteristic of an Einstein crystal due to minimal interatomic correlations. For REACH the pattern agrees well with that calculated with the liquid-like model (2) using a correlation length of 10 Å. Reducing the number of modes included in the calculation of the atom-atom correlation to eight (Fig. 3 f) yields very strong features that dominate the pattern and are very similar for REACH and ${\text{ANM}}_{50}^2$ and are qualitatively similar to that computed with TLS. REACH and ${\text{ANM}}_{50}^2$ diverge when the number of modes is increased to 100 where the features of the ${\text{ANM}}_{50}^2$ pattern are much less distinct; the features of the REACH pattern, while more intense, is similar to the final pattern with all modes included.

Figure 3.

Diffuse x-ray scatter in the a^∗ = 0 plane computed from C_α atoms and their correlations. (a) REACH with BVK (81 wavevectors); (b) liquid-like model (2) with correlation length of 10 Å; (c) TLS₁; (d) ${\text{ANM}}_{50}^2$ with BVK (81 wavevectors); (e) REACH and ${\text{ANM}}_{50}^2$ (BVK) with the 100 lowest modes on left and right of split vertical split; and (f) same as panel e, using a small number of modes (8). (Color figure online.)

As briefly discussed in Methods, the diffuse x-ray scatter can be decomposed into different contributions by computing the intensity with appropriate regions of the atom-atom correlation matrix elements set to zero; a schematic of a regionally highlighted atom-atom correlation matrix is included in Fig. S4. The simplest decomposition is to separate the diffuse x-ray intensity into interatomic and atomic-self contributions. For such decomposition, two intensity calculations are needed:

• 1.The total diffuse intensity computed from the full atom-atom correlation matrix, and
• 2.The diffuse intensity computed with all off-diagonal elements set to zero (this can be carried out very quickly, because most of the terms in Eq. 4 are zero).

Subtracting the atomic self-contribution (as discussed in Methods) from the total yields the interatomic contribution. In Fig. 4, a and b, the C_α and all-atom x-ray scattering, respectively, are decomposed into atomic self- and interatomic contributions. For C_α diffuse x-ray scattering, the atomic self-contribution dominates the magnitude of the scatter beyond 0.1 Å⁻¹, and all approaches yield similar regions of positive and negative interatomic contributions but with different magnitudes. The strong correlations of TLS₁ (see Fig. 2) yield the highest magnitudes with very similar scattering profiles to ${\text{ANM}}_{50}^2$ and REACH using the lowest eight modes. In contrast, ${\text{ANM}}_{50}^2$ has an insignificant interatomic contribution when all modes are included.

Figure 4.

Isotropically averaged diffuse x-ray scatter intensity decomposed into various contributions. The intensity is in units of 10³ electrons², and panels b and c are on the same scale. (a) The interatomic contribution to the C_α scattering: REACH and ${\text{ANM}}_{50}^2$ are computed with BVK. REACH and ${\text{ANM}}_{50}^2$ using only the eight lowest modes (dashes) both nearly overlay the TLS₁ profile. The atomic self-contribution, ${\sum }_k{f}_k{\left(2\pi \mathbf{H}\right)}^2\left(1-T_{k^2}\right)$ (see Eq. 4), is the same for all approaches (see Methods). (b) All-atom diffuse scatter decomposed into interatomic contributions. (Dashes) Total scatter. (Solid lines) Corresponding interatomic contribution to the diffuse scatter. The total scatter of ENM²₅₀ nearly overlays the Atom_self scatter, which results in a near-zero interatomic contribution. (c) All-atom diffuse scatter decomposed into interresidue contributions. (Dashes) Total scatter. (Solid lines) Interresidue contributions. (Solid dots) Intraresidue contribution for ${\text{BNM}}_{3:10}^6$. Note that it is not straightforward to compare the profile features of the all-atom and C_α diffuse x-ray scattering: the atomic pair distribution functions of the coarse-grained C_α and all-atom models are inherently different. (Color figure online.)

For the all-atom (nonhydrogen) diffuse x-ray scattering (Fig. 4, b and c), there are large interatomic contributions below 0.3 Å^–1 for all approaches except ${\text{ENM}}_{50}^2$, which has an insignificant interatomic contribution. For all-atom (nonhydrogen) diffuse x-ray scatter, the ${\text{ENM}}_{3:10}^6$ (Fig. 4 b) and ${\text{BNM}}_{3:10}^6$ (Fig. 4 c) scatter is in qualitative agreement with that of the all-atom MD simulations (34,35) with a strong peak and shoulder found at ∼0.21 and ∼0.45 Å⁻¹, respectively. The first strong peak at ∼0.1 Å^–1 is not present in the all-atom MD simulations, and this discrepancy can be due to the harmonic limitations of elastic network models or the lack of convergence of large-scale motions of the all-atom simulations (45). Unlike ${\text{ENM}}_{50}^2$, ${\text{BNM}}_{50}^2$ has significant interatomic contributions, which may be surprising; the difference is due to the rigid residue blocks defined with BNM. To show this, and to better understand the large interatomic contribution for the all-atom diffuse scatter, we further deconstruct the pattern for ${\text{BNM}}_{3:10}^6$ and ${\text{BNM}}_{50}^2$ into intra- and interresidue contributions (see Fig. 4 c). Similar to the calculation of the interatomic contributions to the diffuse scatter, the interresidue contribution is computed by subtracting the intraresidue contribution from the total diffuse intensity. See the Supporting Material for more details and additional analyses (Fig. S5) of how residues within the secondary structure contribute to the diffuse x-ray scattering as predicted by ${\text{BNM}}_{3:10}^6$ .

Conclusions

In the development of elastic network models or any other dynamic models, the identification of benchmarks that probe different aspects of the physical system is paramount. In this study, the VCOV is examined in detail for several elastic network models of proteins using the theoretical framework of x-ray scattering from protein crystals. The diagonal of the VCOV is explored by comparing the theoretical and experimental temperature factors for a set of 33 ultra-high resolution structures. The off-diagonal of the VCOV is explored using atom-atom correlations for crystalline staph nuclease, which had been studied previously using all-atom simulations (34) and diffuse x-ray scattering (35). Throughout, the impact of the density of states, which defines how modes contribute to the VCOV, is highlighted. It is clear that only using temperature factors and the characteristics of low-frequency displacement vectors is not sufficient (32) and can be misleading. Large rigid blocks (e.g., TLS) and elastic network models with very strong long-range interactions (relative to short-range interactions, e.g., ${\text{ENM}}_{50}^2$) provide extremes: the parameters of TLS can be adjusted to yield very good agreement with experimental temperature factors, but the correlations between atoms of these rigid blocks is much too strong; on the other hand, there are no long-range correlations for ${\text{ENM}}_{50}^2$, where atoms vibrate independent of one another. Further, limiting elastic network models to include low-frequency modes, which are less sensitive to the model of the force constant (21), makes steps toward the TLS extreme (see the Supporting Material). In agreement with models proposed by Hinsen et al. (17), using R⁻⁶ or an exponential dependence, as is the case for REACH (22), for nonbond interactions yields reasonable treatment of atomic interactions and characteristics for the density of states for isolated and crystalline molecules; also, the rapid fall-off of these force constants with distance naturally avoids cutoff parameters (23). A major shortcoming of elastic network models is that the effects of solvent and anharmonicity is lacking, and investigations of such effects will continue to provide valuable insights into the atomic details of biomolecular dynamics (34). Further, it is important to consider the resolution of the model and the relevance of that model to a given phenomenon. For example, diffuse x-ray scatter results from motions of all protein and solvent atoms where correlated motions may extend over multiple unit cells. Clearly, to predict such experimental phenomena, care is required in the application of multiscale elastic network models.

Elastic network models provide an efficient platform for the exploration of how interatomic interactions and surroundings impact the dynamics of proteins, and may be further improved by adding more atomic detail to the force constants without significant increase in cost. Accurate elastic network models are a promising avenue to understanding correlated motions of large biological systems. X-ray diffraction experiments for biological crystals and cocrystals will continue to provide the bedrock for protein structure and protein-protein interactions, from which dynamics models may be constructed and validated. Accurate models can help advance current experimental protocol by enabling us to step off the Bragg peaks and into regions of reciprocal space where the intensity is governed by disorder and dynamics.