Predicted and Experimental NMR Chemical Shifts at Variable Temperatures: The Effect of Protein Conformational Dynamics

Abstract

NMR chemical shifts provide a sensitive probe of protein structure and dynamics but remain challenging to predict and interpret. We examine the effect of protein conformational distributions on ¹⁵N chemical shifts for dihydrofolate reductase (DHFR), comparing QM/MM predicted shifts with experimental shifts in solution as well as frozen distributions. Representative snapshots from MD trajectories exhibit variation in predicted ¹⁵N chemical shifts of up to 25 ppm. The average over the fluctuations is in significantly better agreement with room temperature solution experimental values than the prediction for any single optimal conformations. Meanwhile, solid-state NMR (SSNMR) measurements of frozen solutions at 105 K exhibit broad lines whose widths agree well with the widths of distributions of predicted shifts for samples from the trajectory. The backbone torsion angle ψ_i−1 varies over 60° on the picosecond time scale, compensated by φ_i. These fluctuations can explain much of the shift variation.

Graphical Abstract

1. INTRODUCTION

NMR chemical shifts provide a sensitive probe of protein structure and dynamics.^1-3 For example, prior studies indicate that ¹⁵N chemical shifts are strongly influenced by local protein structure,^1,4,5 including backbone torsion angles^6,7 as well as electrostatic or hydrogen bonding effects. Accurate ab initio prediction shifts would assist in interpretations of experimental data, but important challenges remain, including the frequently measured amidic ¹⁵N sites.

A number of studies suggest that conformational averaging contributes to the complexity of shift interpretation.^8-10 With hybrid quantum mechanics/molecular mechanics/molecular dynamics (MD-QM/MM), improved accuracy of calculations of ¹⁵N chemical shift tensors has been reported.¹¹ Strong support for the importance of conformational averaging comes from cryogenic solid-state NMR (SSNMR) studies of proteins for which significant NMR line broadening has been observed,^12-18 consistent with freezing out disparate members of the ensemble. Other studies support the hypothesis that broad line widths in low temperature NMR are due to static conformational heterogeneity^14-18 including direct measurements of distribution in the local torsion angles.¹⁹ This points to the possibility of characterizing the protein conformational ensemble using low temperature NMR line widths.

In this study, we demonstrate the impact of conformational ensembles on NMR shifts, focusing on the antibiotic target, dihydrofolate reductase (DHFR), which has been well studied in terms of NMR chemical shifts and conformational dynamics. We focus on a specific site for which abundant experimental data are available, and which is evidently stably folded in a β strand conformation, I60. As is shown in Figure S1, I60 is located in the middle of β-sheet C (residues 58–63) and tends to form hydrogen bond interaction with adjacent β-sheet B (residues 39–43) and β-sheet D (residues 73–75). E. coli DHFR protein (UniProt: P0ABQ4) was expressed in M9 minimal media supplemented with combinations of corresponding isotopic enriched amino acids. The purified protein was saturated with trimethoprim (TMP) which is a well-known inhibitor of DHFR. The protein sample was prepared as described in Materials and Methods section in Supporting Information before taking NMR experiments. Given that the observation of broad backbone amide ¹⁵N NMR line widths at low temperature is relatively general, we also make comparisons with other sites.

2. RESULTS

2.1. QM/MM Calculation Based on MD Simulation.

For protein systems, hybrid quantum mechanics/molecular mechanics (QM/MM) has been applied with success previously. This approach results in increasingly reasonable computational demands, showing excellent and improving accuracy.²⁰ Here the QM/MM approach was implemented with an automated fragmentation (AF) procedure, which was previously benchmarked with 20 proteins.²¹

To investigate the importance of conformational sampling in NMR shifts, we computed an MD trajectory and selected samples from it to prepare an appropriate ensemble of native and near native conformations. This group of conformations was then studied using QM/MM to predict chemical shifts. A 1000 ns MD simulation was performed based on a solved DHFR:TMP X-ray crystal structure.²² The backbone heavy atoms root mean square deviation (RMSD) referenced to the initial state is shown in Figure S2a, demonstrating the overall stability of the whole protein during the MD trajectory. QM/MM calculations were used to study chemical shifts for snapshots extracted from the MD simulation. To study the effects of solvent, the locations and orientations of water molecules from the MD simulation were imported to AF-QM/ MM. A DHFR:TMP X-ray crystal structure²² with a resolution of 2.5 Å and R-value of 0.23 was used for direct QM/MM calculation and MD simulation. Details of the MD simulation protocols and QM/MM calculation of the coordinates of each MD snapshot and X-ray crystal structure are presented in the Materials and Methods section in Supporting Information.

To represent solution-state NMR data, the predicted shifts were averaged over the ensemble. Figure S3 shows the comparison between the chemical shift prediction from the X-ray crystal structure²² and that averaged over the conformations sampled with MD. Both the X-ray crystal structure and the MD snapshots were energy minimized before QM/MM calculations. The correlation coefficient comparing experimental solution ¹⁵N shifts to single conformer (X-ray crystal structure) predicted shifts is 0.69 and for ¹³Cα is 0.78. We then extracted 20 snapshots from the 1000 ns MD simulation, regularly sampled with an interval between snapshots of 50 ns. QM/MM calculations for each of the 20 snapshots using AF-QM/MM resulted in predicted average chemical shift with a better correlation to experimental solution-state NMR shifts, 0.73 for ¹⁵N and 0.94 for ¹³Cα. By another metric, the root mean square error (RMSE) ($\text{RMSE}=\sqrt{\frac{{\sum }_{i=1}^N{\mid y(i)-\widehat{y}(i)\mid }^2}{N}}$, where $N$ is the number of data points, $y(i)$ is the $i$-th measurement, and $\widehat{y}(i)$ is its corresponding prediction) can be thought of as a prediction accuracy and was 4.8 ppm (¹⁵N) and 1.7 ppm (¹³Cα), respectively, when averaging was included, vs 7.9 and 3.6 ppm for the corresponding single conformation predictions. ¹³C′ QM/MM calculation results do not match the experimental chemical shift assignments as well as ¹³Cα (correlation coefficient r = 0.51, RMSE = 2.3 ppm). Possibly the large chemical shift anisotropy (CSA) of the carbonyl, the resonance effects, and the important effects of hydrogen bonds are not well captured in the MD sampling. The improvement in the ¹⁵N shift prediction accuracy due to averaging is significant, especially given the importance of ¹⁵N shift measurements in protein studies.

The shift prediction is relatively stable over the trajectory. Averaging snapshots from the different time ranges of the MD trajectory gives a similar chemical shift correlation with experimental results (Figure S4). The correlation is similar when extracting the snapshots from the first 200 ns, the last 200 ns, or the middle 200 ns in MD simulation. This result indicates that the whole MD calculation is stable, and the improvement due to hybrid QM/MM calculations and MD simulation is reproducible. Moreover, the key fluctuations that contribute to chemical shift averaging appear to operate on a relatively fast time scale consistent with the hypothesis that these fluctuations are averaged in the fast limit in solution NMR measurements.

In contrast with the narrow NMR spectra at room temperature, protein NMR peaks at low temperature are notably broad, putatively because various conformations freeze to create an inhomogeneous mixture. Since the QM/MM predicted shifts averaged over the MD simulation predict the solution-state NMR chemical shift at room temperature with good accuracy, we asked whether the ensemble of snapshots in the trajectory might be a good representation of the low temperature experimental ensemble. In order to address this question, we compared the amide ¹⁵N and ¹³C chemical shifts of each residue from MD snapshots with spectra collected at cryogenic temperatures. In Figure 1a, the distribution of calculated ¹⁵N and ¹³C isotropic shifts in the snapshots matches well with the experimental contour at 105 K in solid-state NMR (SSNMR). Both the ¹³C chemical shift range and ¹⁵N shift range agree well, comparing the peak pattern and the QM/MM scatter points distribution. For the prediction of the five sites (highlighted in Figure 1c) of IG-DHFR, 85% of the scatter points lay within the contours of DNP spectra (Figure 1b). The chemical shifts were calculated with restricted minimization before QM/MM; without minimization, the predictions are less similar to the low temperature SSNMR experiments, as shown in Figure S5 where the ¹⁵N, ¹³C′, and ¹³Cα shifts are uniformly distributed in the ranges 95–140, 165–185, and 53–68 ppm, respectively. The observation suggests that local minimization applied on MD snapshots before QM/MM allowing an adjustment with heavy atom RMSD < 0.3 ψ is a reasonable simple model for experimental annealing/cooling of the sample.

Figure 1.

Low temperature solid-state NMR (SSNMR) ¹⁵N─¹³C correlation spectra of (a) ¹³C,¹⁵N-Ile labeled E. coli DHFR:TMP (I-DHFR) and (b) ¹³C,¹⁵N-Ile, ¹⁵N-Gly labeled E. coli DHFR:TMP (IG-DHFR) measured using DNP enhanced SSNMR at 105 K. Ile60-Ile61(sheet) is the only residue pair expected in I-DHFR DNP spectra because of a sparse isotopic enrichment scheme (Ile only is enriched). In IG-DHFR, 4 more residue pairs are observed (Ile2-Ser3 (sheet), Ile14-Gly15 (coil), Ile50-Gly51 (helix), Ile94-Gly95 (coil)). The 5 pairs are highlighted with stick rendering of the side chains in cartoon of the structure of E. coli DHFR:TMP (c): I2-S3 (green), I14-G15 (cyan), I50-G51 (blue), I94-G95 (orange), and I60-I61 (red). The predicted chemical shifts for conformations sampled from MD and predicted with QM/MM are overlaid: 100 snapshots were minimized and calculated for Ile60-Ile61 in (a) and 20 snapshots were used for residue pairs in (b). Both C and N chemical shifts from QM/MM calculation were calibrated with the offsets from average chemical shifts of snapshots and solution-state NMR chemical shifts. The lowest contour levels in SSNMR spectra are set to be 15 times (a) and 3.5 times (b) the RMS noise level, and the multiplier is 1.3 for both (a) and (b). The peak at ¹³Cα 53–55 ppm in (b) results from a spinning sideband and was not predicted. The cross markers in (b) indicate the solution-state NMR chemical shifts for the 5 residue pairs.

2.2. Torsion Angle Variation in MD.

The MD sampling studies described above suggest that NMR shifts fluctuate on a fast time scale at room temperature and are frozen into a broad inhomogeneous static distribution at cryogenic temperatures. The question arises as to the origins of these fluctuations in terms of conformational degrees of freedom. Inspecting the snapshots extracted every 10 ns from the MD trajectory, the torsion angles ψ and φ exhibit wide variations for many residues, as is shown in Figure 2c, Figure S6a,b, and Figure S7a. We focus on Ile60-Ile61 which is located in the middle of a sheet structure in E. coli DHFR, Figure 2b. The Ile60ψ value is 118° based on the X-ray crystal structure of E. coli DHFR:TMP,²² and X-ray crystal structures of E. coli DHFR complexes²³ in its catalytic cycle show variation for I60ψ in a range from 118° to 130°, indicating it as a relatively rigid site with small structural variation among different functional states. Although the ψ angle varies widely in the MD trajectory, Ile60 and Ile61 preserve β-sheet structures throughout the trajectory, and the protein remains locally folded. The scatter points in Figure S8 represent the φ−ψ correlation of these two residues in the Ramachandran plot. Such large fluctuations in backbone torison angles are consistent with stably folded structures if the change in one torsion angle is compensated by another, to keep the flanking reisues in the same position. Although ψ and φ both vary by 50° in one amide, for example, Ile60-Ile61 (Figure 2c), the overall volume occupied by the protein chain is not much altered with these variations because of the correlation between ψ_i and φ_i+1, which together result in the amide plane “rocking in place” (Figure 2b). Comparing the RMSD of atoms in residue Ile60 Cα vs Ile60O (Figure S9b,c), Ile60Cα ($\overline{{\text{RMSD}}_{\mathrm{I}60\mathrm{C}\alpha }}=0.38$) is more stationary, while the O atom makes somewhat wider excursions ($\overline{{\text{RMSD}}_{\mathrm{I}60\mathrm{O}}}=0.48$). A strong correlation between ψ_i and φ_i+1 is observed also for several other solvent inaccessible residues, Figure S10 and Figure S7a, though some flexible solvent accessible residues showed more complex backbone dynamics. Similar observations in peptide^24-26 and protein^27,28 structures have been previously reported, where a dominant “crankshaft-like” motions in the plane of the amide results in a strong anticorrelated motion between φ_i+1 and ψ_i. It is noteworthy that other torsion angles such as the side chain torsion angle χ1_i also exhibit a variation up to 40° (Figure S6c) do not correlate strongly with backbone torsion angles. The ω_i angle variation in the MD simulation within one amide is ~7°, relatively small compared with other torsion angles, shown in Figure S6d.

Figure 2.

Protein structural variation during MD simulation at 300 K. (a) Dihedral angles and H bond angle/distance were analyzed. (b) Structure overlay of Ile60-Ile61 in snapshots at 200 ns in MD with Ile60ψ = 160° (green) and at 160 ns in MD with Ile60ψ = 113° (pink) indicates that the Cα position is unchanged while the N (blue) and carbonyl O (orange) move. (c) There is strong correlation between Ile61φ and Ile60ψ. Each scatter point represents one snapshot from MD simulation extracted every 10 ns. The correlation coefficient is −0.89. (d) The correlation plot of Ile60θ and Ile60ψ shows strong correlation between backbone torsion angle and H bond angle. The H bond angle θ is defined as the angle between the H bond (O---H) and C═O. Ile60 is located in a β-sheet structure, and its O forms a H bond with Met42 ^NH in MD simulation. The closest water molecule is ~7.5 Å away from the Ile60 O during the trajectory. The correlation between ψ_i+1, φ_i, and H bond angle θ_i is not unique for I60-I61, as shown in Figure S7.

2.3. Correlation between N Isotropic Chemical Shifts and Torsion Angles.

Numerous physical effects or structural features such as torsion angles are expected to influence ¹⁵N chemical shifts. We specifically focused on the implications of the fluctuation of ψ_i on the N_i+1 chemical shift for Ile60-Ile61 in E. coli DHFR (Figure 3a). This was partly motivated by the broad range of values for ψ observed in the MD simulations. Also, previous low temperature DNP-enhanced NMR experimental data show a correlation between this torsion angle and ¹⁵N shifts.²⁹ A strong correlation between the I61 ¹⁵N chemical shifts and I60ψ values is observed from 100 snapshots in the 1000 ns MD trajectory (Figure 3a). More importantly, both the isotropic chemical shifts and the torsion angles from MD/QM/MM agree with the experimental data. These data support the suggestion that variations in backbone torsion angles within the β basin can have strong effects on the ¹⁵N chemical shifts.

Figure 3.

(a) QM/MM calculated Ile61N isotropic shifts are plotted as a function of I60ψ from 100 minimized snapshots from the MD trajectory (red, circle). Three ψ values were measured experimentally at three ¹⁵N shift positions for Ile60 in low temperature DNP-enhanced NMR experiments¹⁹ and are shown here depicted with stars. The correlation between ψ angles and chemical shifts from both prior experimental and these computational results match well. The side chain torsion angle I60χ1 was plotted with ¹⁵N shifts for snapshots with similar backbone torsion angle ψ, 120° < I60ψ < 125° (b). Ile61N shifts vary up to 5 ppm with a 5° range of I60ψ, and I60χ1 is likely to contribute to this variation.

The observation of wide variation in both the torsion angle and the ¹⁵N shift is not unique for Ile60-Ile61; as shown in Figure S11, most residues also exhibit a broad torsion angle variation of 40° or greater. Most residues also show predicted ¹⁵N shift variation over 10 ppm, consistent with the broad lines observed for all residues in low temperature NMR. The correlation of ψ with ¹⁵N chemical shifts is apparent for many other residues, as shown in Figure S11d-f, although it is not evident for every residue (Figure S11a-c). It is likely that the correlation is apparent for Ile60-Ile61 because of the particularly broad range in ψ values sampled by Ile60. The effect of ψ on Ile61 ¹⁵N shift is most evident for conformers where the backbone makes excursions out of the most populated sheet basin around ψ = 120° to conformations with ψ > 130°. Typical sheet/strand residues with ψ of 100°−130° show ¹⁵N shifts of 127 ±6 ppm. By contrast, sheets that are closer to flat extended conformations with ψ > 150° exhibit shifts of 115 ± 5 ppm (Figure 3a). β strand residues with more restricted ranges of torsion angle ψ from 90° to 130°, such Y111, do not show a significant correlation between backbone torsion ψ and ¹⁵N shift and show a narrower span of both ψ and ¹⁵N shift (Figure S11a). A master plot QM/MM calculated ¹⁵N shifts of all the residues from 100 snapshots with their associated torsion angle ψ (Figure S12) shows that for β sheets the effect of ψ on ¹⁵N chemical shifts is most pronounced for ψ 130°−170°, while for sheets with 90°−130°, the dependence is shallow and not monotonic. Ile60-Ile61 is located close to the cofactor binding site, which is empty in the structure that we studied. Possibly the particularly high flexibility of Ile60 is because of this “cavity.” Analogously, the well populated basin for helices (ψ range −50° to 0°) has a mean ¹⁵N shift of 116 ± 5 ppm. Helical residues, including Phe31 and Lys109 (Figure S11b,c), have a relatively flat value for the ¹⁵N shift as a function of ψ with poorer correlation and a narrower span of both ψ and ¹⁵N shift.

Deviation in ¹⁵N chemical shifts within a defined range of ψ are as large as 10 ppm, indicating that other structural degrees of freedom besides ψ also contribute to variation in shift. The side chain torsion χ1 has been discussed in relation to influencing backbone chemical shifts and therefore was also examined for correlations with ¹⁵N and ¹³C.^1,30,31 We identified conformations from the MD trajectory with ψ (120° < I60ψ < 125° in Figure 3b, 150° < 160ψ < 155° in Figure S13b, 135° < I60ψ < 140° in Figure S13c). Within each group, ¹⁵N chemical shifts vary by more than 3 ppm and side chain torsion angles I60χ1 vary by up to 30° while staying within the typically preferred “gauche” basin. The correlation plot suggests that χ1 is another factor influencing ¹⁵N shifts. The situation is more complex for solvent exposed and flexible residues, such as His149 (Figure S14), which show very broad ranges in torsion angles (including ψ_i−1 and φ_i) that are not as well compensated by other backbone angles. Consequently, they exhibit broad ranges of ¹⁵N_i shifts that are not evidently correlated to any one torsion angle (with a span of 25 ppm).

We explored whether the effect of ψ on the ¹⁵N shifts might occur because of changes in hydrogen bonding. Variation of backbone torsion angles and rocking of the amide plane described above is expected to modulate the interactions with hydrogen bond partners. This variation is of interest and is expected to have implications for energetics particularly dielectric and electrostatic characteristics. The hydrogen bond between the backbone N─H and neighboring groups, as well as the hydrogen bond between the C─O and other neighboring groups is also expected to affect ¹⁵N and ¹³C′ chemical shifts; both ¹⁵N and ¹³C′ chemical shift tensors are sensitive to the hydrogen bond variations (Figure S15d-f), especially for ¹³C′ atoms, consistent with previous literature.^32-34 We investigated the hydrogen bond distance and angle over typical snapshots from the 1000 ns MD trajectory. For desolvated residues, such as Ile60-Ile61 in the middle of a sheet structure (Figure 2b), the identity of the protein residues that serve as hydrogen bond partners remains constant, and the pairwise trajectory was analyzed in terms of the distance and angle of the hydrogen bond. The shortest (or only) hydrogen bond partner for Ile60O is Met42 ^NH. This distance varies by 0.6 Å over the trajectory. The shortest (or only) H bond partner of Ile61^NH is the carbonyl O of Thr73. The correlation between the Ile61N chemical shifts and Ile60O---H distance, shown in Figure S15a,b, is not strong, though the hydrogen bond angle θ (relative to the C─O vector of I60) shows a strong correlation with the torsion angle ψ. Analysis of solvent-accessible residues is complex because more than one water molecule is normally located within 4 Å of the amide atoms. Overall, it is likely that the strong variation in hydrogen bonding environment is a major reason that ψ influences the ¹⁵N shift.

2.4. Time Scale Analysis of Backbone Torsion Angle Dynamics.

In order to assess the time scale of backbone fluctuations, we analyzed a 100 ps segment of the MD simulation with snapshots extracted every 100 fs. The variation of torsion angles for Ile60ψ is shown in Figure S16a. The angle ranges from 90° to 150°, which is similar to the range observed in the longer time MD simulation shown in Figure 2d. Two states were defined based on the torsion angle range, State 0 with Ile60ψ < 125° and State 1 with Ile60ψ > 125°. The structure of Ile60-Ile61 amide undergoes a dynamic and structural change at ~50 ps; before this time, State 0 (Figure S16b) was preferable with Ile60ψ smaller than 125°, and after that time, State 1 (Figure S16c) with larger ψ was much more favored. Thus, Ile60ψ is considered as switching between two states, shown in Figure 4a. The time scale of this exchange motion is then determined from the time when Ile60ψ stays in either state continuously. In the first 50 ps, State 0 occupied more time with a residence time of 0.89 ± 0.18 ps (Figure 4d), while the residence time of State 1 is 0.34 ± 0.05 ps (Figure 4b). In the second 50 ps, the residence time is 0.23 ± 0.02 ps (Figure 4e) for State 0. State 1 is more preferable, and its residence time is longer than what can be extracted from this MD simulation (Figure 4c). This dynamic time scale agrees with previous research about polypeptides,²⁴ in which the anticorrelated motion of ψ_i and φ_i+1 is reported to be on the time scale of 0.1 ps. This time scale was associated with protein backbone motions and intermolecular vibrations of the water hydrogen bond network,³⁵ and these fast time scale atomic fluctuations were observed to facilitate large-scale, slower motions during enzyme catalysis.³⁶

Figure 4.

(a) Two states were defined based on Ile60ψ variation (Figure S16a) in 100 ps MD simulation, State 0 (Figure S16b) with Ile60ψ < 125° and State 1 (Figure S16c) with Ile60ψ > 125°. State 0 is more dominant before (pink) 50 ps and State 1 after (green) 50 ps. The histograms of the residence times for Ile60ψ staying in State 0 (d, e) or State 1 (b, c) before (b, d, pink) and after (c, e green) 50 ps were fit with a single exponential function (black dash line). For the first 50 ps trajectory, the residence time is 0.89 ± 0.18 ps for State 0 and 0.34 ± 0.05 ps for State 1. After 50 ps, the residence time is 0.23 ± 0.02 ps for State 0. The residence time for State 1 is longer than what can be identified through this trajecotry.

The conformational variation of Ile60-Ile61 is likely to be influenced by its partner Thr73, whose O forms a hydrogen bond with Ile61^NH. Thr73 is located in another β-sheet structure and is partially solvated, flips conformations at 50 ps, and probably causes the change in its partner Ile60. Thr73 in turn might be affected by the solvent. It forms closest hydrogen bonds with protein residues (Thr73O---Ile61^NH, Thr73^NH---Asn59O), yet there are also typically water molecules within 3.5 Å of this residue (Figure S16b,c). Because the β-sheet is solvent-exposed, neighboring Trp74O and Val72O both form the shortest hydrogen bonds with water molecules. It is likely that the solvent interactions lead to increased structural variation in Thr73, which then causes it to influence nearby desolvated residues such as Ile60-Ile61 through hydrogen bonding interactions. Longer MD simulations (Figure S6) indicate that the two-state flipping happens regularly and reversibly throughout the trajectory.

3. DISCUSSION

In summary, for relatively buried and structured residues in globular proteins, picosecond time scale large angle fluctuations in backbone torsion angles leads to large dynamic variation in backbone shifts. The changes, despite being over 60° in amplitude, can be consistent with stably folded structures because they are compensated by other torsion angles, and the structure remains in the β basin. The fluctuations should be considered when predicting room temperature shifts. They also contribute significantly to the inhomogeneous low temperature line widths, because the conformations appear to be static (or interconvert very slowly) at 105 K.

The large variation of torsion angles we observed in the MD trajectory may be surprising. Based on our analysis of MD snapshots, it is likely that the compensations from other backbone torsion angles and hydrogen bond angles preserve the global protein secondary structures. Analogous large-amplitude backbone motions have been discussed previously in peptides,^24,37 polymers,³⁸ and proteins^39,40 with MD simulation in explicit or implicit solvent. In recent research of protein dynamics,⁴¹ several high resolution protein structures were chosen for analyzing the amplitudes of protein motions; the (φ, ψ) distribution of Lys83 captured during dynamics of mouse protein kinase A is up to 150°, and the drastic changes are enabled by minor adjustments to the backbone geometry including bond lengths and angles. Similarly, the conformational transitions observed in Ala₇ peptide are accomplished by a large change (64.4°) in the φ angle of the fourth residue; however, the entire chains are nearly perfectly superimposed before and after the torsion angle variation except for the atoms involved in ψ₄, φ₄, or ψ₃.²⁴ The observation of thermally elicited large-angle local torsion angle variations on the picosecond time scale is expected to have many experimental and computational implications. If the range of backbone torsion angles is as broad as is indicated here, it is likely that it would have implications for protein modeling, for solution NMR protein dynamics studies, and for fitting electron density in X-ray diffraction and other structure methods.⁴² Variation and interdependence of the backbone dihedral angles^25-28 was predicted to be associated with isotropic chemical shifts,^7,43 NMR cross-relaxation,^44,45 and scalar coupling constants.⁴⁶

We asked how fast local large-amplitude conformational fluctuations affect the chemical shifts of proteins, focusing on room temperature experimental solution shifts which reflect the average over the fluctuations. Although the empirical methods, like SHIFTX2⁴⁷ and SPARTA+,⁴⁸ can predict chemical shifts with high accuracy, leading to correlation coefficients of 0.98 (¹⁵N), 0.99 (¹³C α), and 0.96 (¹³C′) with RMSE of 1.12, 0.44, and 0.53 ppm compared to the experimental results in 61 proteins,⁴⁷ their usage can be limited to frequently observed and thermally averaged conformations and so might be ill suited to sampling thermal excursions during dynamics.⁴⁹ Fragment density functional theory calculation of NMR chemical shifts for protein, on the one hand, has become increasingly practical in terms of computational time²¹ and, on the other hand, can capture nuanced variations in the structure including rare events and provide accurate and physics-based predictions. The chemical prediction of GB3 using the AF-QM/MM resulted in mean unsigned errors (MUE) of 4.8 ppm with R² of 0.71 for ¹⁵N and 2.1 ppm with R² of 0.79 for ¹³C α.⁵⁰ Polenova’s group compared the QM/MM calculated chemical shifts of galection-3C with its MAS NMR chemical shifts.⁵¹ The correlation coefficient comparing predicted vs observed shifts for ¹³C α and ¹⁵N was 0.83 and 0.68 using an X-ray structure determined at 100 K; a structure determined at 293 K gives a similar correlation, which is 0.84 for ¹³C α and 0.65 for ¹⁵N. Moreover, the explicit solvent molecules in QM/MM allow the interaction between the solvent and the protein to be calculated.^52-54 An explicit–implicit solvation model with only one explicit water molecule per amine and amide proton is sufficient to yield accurate results for ¹H (MAD ~ 0.3 ppm) and for ¹³C (MAD ~ 3 ppm).⁵⁵ Here, we conclude that it is possible to simulate the distribution in chemical shifts at room temperature and at low temperatures using QM/MM calculations, assuming that the following technical details and procedures are followed: (1) Averaging the QM/MM calculated shifts over a long enough MD trajectory to obtain better prediction at room temperature. (2) Applying restricted minimization on room temperature MD snapshots before QM/MM to mimic the freezing out protein conformation distribution. Conformational averaging has been observed to be necessary for QM/MM prediction.^52,53,56 Previously, QM/MM prediction of a 32-amino-acid-long polypeptide averaging 500 snapshots from a 5 ns explicit solvent MD simulation gives MUE of 14.6 ppm with R² value of 0.81 for ¹⁵N and MUE of 5.2 ppm with R² value of 0.99 for ¹³C (not only ¹³C α). With our procedure, we obtained a correlation for ¹³C α (r = 0.94) and ¹⁵N (r = 0.73) atoms and RMSE of 4.8, 1.7, and 2.3 ppm for ¹⁵N, ¹³C α, and ¹³C′, respectively, which is significantly improved relative to the single conformation case where ¹³C α (r = 0.78) and ¹⁵N (r = 0.69) are comparable to previous reports. Thus, we conclude that, while more work remains to be done, conformational averaging improves predictions of solution shifts significantly.

We asked how the fluctuations in the chemical shift might affect low temperature NMR measurements. Combining the results of QM/MM calculated shifts with the snapshots from the MD simulation, we noticed that the ¹⁵N chemical shifts of protein exhibit a broad range. The variation can have a number of structural origins. The contribution of the backbone^7,57,58 and side chain^2,30,59 torsion angles to ¹⁵N isotropic chemical shifts has been previously discussed; thus, their effects were carefully considered for a more accurate chemical shift prediction. Previous work indicates that high solvent accessibility correlates with high line widths in low temperature DNP-enhanced NMR spectra. As indicated in the previous literature,⁶⁰ and from the QM/MM predictions reported here, the contribution of the hydrogen bonds to both ¹⁵N and ¹³C chemical shift tensors is large and the effect of isotropic shifts can be up to 8 ppm to ¹⁵N chemical shifts in model systems and smaller for ¹³C isotropic shifts (<2 ppm for ¹³C′ and <0.5 for ¹³C α). For ¹⁵N chemical shift tensors with the hydrogen bond, δ₁₁ appears to be sensitive to the presence of hydrogen bonding interactions due to its unique orientation in the molecular frame.³² For the ¹³C chemical shift, the tensor components depend strongly on the hydrogen-bond length and angle, particularly δ₂₂.⁶¹ These effects may explain the empirical trends regarding solvent accessibility, and including solvent interactions in chemical shift calculation, especially anisotropy prediction, is expected to be important.^30,34,62 Accordingly, we chose to retain water molecule locations from MD simulations and to apply restricted minimization of the MD snapshots, to simulate the effects of sample freezing and predict inhomogeneous conformational distributions and inhomogeneous chemical shift distributions in the frozen protein.

The research reported here supports the hypothesis that much of the variation in chemical shifts and broad NMR line widths for frozen samples is due to local conformational variation, where various backbone conformers are frozen into an inhomogeneous distribution at 105 K (with interconversions on a time scale slower than milliseconds).^14-18 By contrast, they are averaged into a homogeneous distribution at room temperature (with interconversions on the subnano-second time scale). This conclusion agrees well with low temperature experimental measurements of distributions in backbone ψ and correlations to the chemical shifts in various members for the ensemble. Backbone torsion angles, specifically correlated changes in ψ_i and φ_i+1 and a resulting motion of the amide plane, are associated with these changes in NMR chemical shifts. While other structural degrees of freedom are likely to be variable and important as well, we showed here that for the N isotropic shift, ψ is a significant cause of variation.

4. CONCLUSIONS

We explored the connection between dynamics predicted by MD and NMR observables and presented strong indications of elevated thermal motions of the backbone of proteins. For liquid samples, averaging the QM/MM shift prediction over the trajectory improves the chemical shift prediction accuracy compared to predictions based on single global minimum conformations. The distributions in predicted values also agree well with the breadth of low temperature NMR spectra. Variation in the backbone torsion angles, ψ specifically, appears to have marked effects on the ¹⁵N isotropic shifts.

Biographies

Xu Yi obtained a Bachelor of Science degree from Nanjing University in 2017, followed by a Ph.D. in Chemistry from Columbia University in 2022 under the supervision of Professor Ann E. McDermott. Currently, she is engaged in the field of laboratory virtualization for chemistry and biotechnology while concurrently pursuing a degree in Computer Science at the University of Illinois Urbana–Champaign.

Lichirui Zhang earned his B.S. in Chemistry from Nankai University in 2017 and a Ph.D. from Columbia University in 2023, mentored by Prof. Richard A. Friesner. He is currently a Senior Scientist at Schrödinger Inc., focusing on the development of deep learning and physics-based computational methods for modeling biomolecules.

Richard A. Friesner was born in New York, NY, in 1952. He received his BS degree in chemistry from the University of Chicago in 1973. He obtained his Ph.D. in 1979 at the University of California, Berkeley, working in the laboratory of Kenneth Sauer. He then spent three years as a postdoctoral fellow working with Robert Silbey at the Massachusetts Institute of Technology. He joined the Chemistry Department at the University of Texas at Austin in 1982 as an Assistant Professor. In 1990, he became Professor of Chemistry at Columbia University. His work is currently focused on computational modeling of complex systems biology and materials sciences. Specific interests include protein structure prediction, structure based drug design, modeling of enzyme reactions, and modeling of nanosystems such as silicon nanoparticles and carbon nanotubes with a particular focus on solar energy applications.

Ann E. McDermott obtained her Bachelor of Science in Chemistry from Harvey Mudd College in Claremont, CA, in 1981. In 1988, she obtained her doctoral degree at U.C. Berkeley in the Department of Chemistry with Kenneth Sauer and Melvin Klein. As a postdoctoral researcher she worked with Robert G Griffin at The Massachusetts Institute of Technology. She joined Columbia University as a Professor in Chemistry Department in 1991. Her research exploits Nuclear Magnetic Resonance to study the functions, structures, and dynamics of proteins including enzymes, viral proteins, membrane proteins and amyloid proteins.