NMR Determination of Amide N-H Equilibrium Bond Length from Concerted Dipolar Coupling Measurements

Abstract

The N-H bond length in backbone peptide groups of the protein GB3 has been studied by liquid crystal NMR, using five structurally conserved mutants of this protein. In the absence of additional information, the impact of dynamic fluctuations of the N-H vector orientation on the ¹⁵N-¹H dipolar interaction cannot be separated from a change in N-H bond length. However, a change in N-H bond length directly impacts the orientation of C′-H vectors in the peptide group, and simultaneous analysis of ¹³C′-H^N and ¹⁵N-H^N residual dipolar couplings, measured under five different alignment orientations, permits modelfree determination of the average equilibrium N-H bond length in GB3, yielding r_NH^eq = 1.008 ± 0.006 Å. Anharmonicity of the bond stretching results in a slightly longer time-averaged bond length <r_NH> = 1.015 ± 0.006 Å, and an effective bond length r_eff = <r_NH^-3>^-1/3 = 1.023 ± 0.006 Å pertinent for NMR relaxation analysis, not including the impact of zero-point or other angular fluctuations in N-H orientation. Using a reference frame defined by the backbone C^α-C′ vectors of the protein, angular fluctuations for N-H vectors in elements of secondary structure are approximately 1.5 fold larger for out-of-plane fluctuations than motions within the peptide plane and not much larger than anticipated on the basis of quantum mechanical analysis of their zero-point librations.

Quantitative interpretation of ¹⁵N NMR relaxation rates in terms of motion hinges on the value of the ¹⁵N-¹H internuclear distance. Neutron diffraction of small model peptides suggested a value of 1.020-1.024 Å.¹ From an NMR relaxation perspective, spectral density terms resulting from ultrafast motions such as zero-point librations do not significantly contribute to relaxation and simply attenuate the effective internuclear ¹⁵N-¹H dipolar coupling. Their impact is therefore similar to a lengthening of the N-H bond, and an increased effective bond length can be used to remove the effect of the nearly uniform normal mode librations and vibrations from the modelfree order parameters, often extracted from such data.² Internal motions also scale the internuclear dipolar couplings observed in solid state and liquid crystal NMR, with both technologies potentially providing access to quantitative analysis of the amplitude and direction of motions integrated over the entire sub-ms time scale. Although the equilibrium N-H bond length is known to be sensitive to hydrogen bonding, DFT calculations indicate this dependence to be weak for normal, high barrier hydrogen bonds, with an approximate 0.01 Å decrease in r_NH^eq when r_H..O increases from 1.8 to 2.5 Å, in agreement with prior computational results (Supporting Information).^2,3

Quantum mechanics defines the lowest vibrational states of any molecular bond. A detailed description of the impact of normal mode vibrations (bond stretching) and librations (bending) on NMR-observable parameters has been presented by Case.² Different definitions of the bond length are required for different purposes (Fig. 1A): The equilibrium bond length, r_NH^eq, corresponds to the internuclear distance of lowest energy. The average <r_NH> distance includes averaging over the lowest vibrational bond stretching mode, which due to anharmonicity of the potential is calculated to be ca 0.015 Å longer than r_NH^eq. From an NMR relaxation perspective, where the strength of the dipolar coupling scales with r_NH^-3, the pertinent bond length equals r_eff = <r_NH^-3>^-1/3. As motions of the N-H bond vector caused by processes other than the universal and largely structure-independent zero-point normal mode librations and vibrations also scale the ¹⁵N-¹H dipolar interaction, it has remained difficult to establish experimentally precise baseline values for <r_NH> and r_eff. Here, we report concerted measurement and analysis of ¹³C′-¹³C^α, ¹³C′-¹H^N and ¹⁵N-¹H^N residual dipolar couplings (RDCs), measured at very high precision in perdeuterated, amide-protonated forms of five mutants of the small and previously well-studied protein GB3. These mutants, K19AD47K, K19ED40N, K19AT11K, and two which include a C- and N-terminal His-tag, K19EK4A-C-His₆ and K19EK4A-N-His₆, previously were shown to map the five-dimensional alignment space when dissolved in a liquid crystalline Pf1 suspension.⁴ Using an improved stability, iterative implementation⁵ of the DIDC method,⁶ alignment tensors are derived independent of a starting structure, i.e. with no adverse impact from coordinate uncertainty in the reference model. Alternate iterative approaches can also be used for this purpose.^7,8 Excluding the invisible N-terminus, four residues identified by iterative DIDC as highly dynamic, and five residues for which resonance overlap interfered with accurate measurement of ¹³C′-¹³C^α, ¹³C′-¹H^N or ¹⁵N-¹H^N RDCs in one or more of the five mutants, leaves 45 sets of three RDCs per mutant for analysis.

Figure 1.

Amide N-H bond length in peptide groups. (A) Bond energy versus r_NH bond length, and definitions of r_NH^eq, <r_NH>, and r_eff (<r_NH⁻³>^−1/3). (B) Total χ² fitting error of C^α-H^N RDCs to GB3 structure (see text) as a function of r_NH^eq, with ${\chi }^2={\sum }_i{\sum }_j{(RD{C}_{ij}^{\text{pred}}-RD{C}_{ij}^{\text{exp}})}^2$, where i is the alignment index and j is the residue index. Values for <r_NH> and r_eff follow directly from r_NH^eq (see Figure S2). The H^N cloud in the inset represents the proton distribution generated using eq S28.

Assuming uniform C′-C^α and N-H^N bond lengths, the alignment tensor coefficients derived from the iterative DIDC method can be written in matrix format, Amat, and their ratio (Fig. 2)

Figure 2.

Alignment matrix elements obtained using iterative DIDC⁵ from C^α-C′ RDCs (y axis) versus values obtained from N-H RDCs. Matrix elements result from five mutants, each measured twice with slightly different HNCO experiments and different GB3 and Pf1 concentrations.

$$\frac{\text{Amat}(C\prime C\alpha )}{\text{Amat}(NH)}=\frac{{\gamma }_C{\gamma }_C{S}_{C\prime C\alpha }\langle \frac{1}{r_{C\prime C\alpha }^3}\rangle }{{\gamma }_N{\gamma }_H{S}_{NH}\langle \frac{1}{r_{NH}^3}\rangle }=0.202$$ (1)

represents the ratio of the strengths of the ¹³C′-¹³C^α and ¹⁵N-¹H^N dipolar interaction constants, which falls very close to the value previously reported⁹ using ubiquitin RDCs and its X-ray structure. Besides the assumption of uniform C′-C^α and N-H^N bond lengths, two additional approximations are used in eq 1: First, all 45 included residues have uniform S_NH as well as S_C′Cα order parameters. Second, angular internal motion of C′-Cα and N-H^N vectors, which determines S_NH and S_C′Cα, is assumed to be axially symmetric. Using simulated data, we previously have shown⁵ that the impact of these approximations upon the derived Amat values, used to calculate the ratio in eq 1, is much smaller than the uncertainty caused by RDC measurement error. The very tight correlation observed in Fig. 2 also confirms the validity of these assumptions.

With values for the average order parameters S_C′Cα and S_NH being unknown, the RDC data do not uniquely define <r_NH⁻³> and <r_CαC′^-3>. The brackets, < >, denote vibrational averaging, where for C^α-C′ to a very good approximation,² <r_CαC′> = <r_CαC′⁻³>^−1/3 = 1.525 Å.¹⁰ By defining S′_NH=S_NH/S_C′Cα, a value of

$$S{\prime }_{\text{NH}}<r_{NH}^{-3}>=0.872\pm 0.006{\mathring{\text{A}}}^{-3}$$ (2)

is then obtained from the Amat ratios (Fig. 2). Note that S′_NH defines the order of the N-H vectors relative to the average order parameter of the C^α-C′ vectors, which is taken as a proxy for the motion of the entire backbone, including C′-N and N-C^α vectors. Fluctuations of the peptide plane about the C^α_i-1-C^α_i vector, so-called γ motions,^11-14 have a larger impact on the orientations of N_i-H_i vectors than on the backbone C^α_i-1-C′_i-1 , C′_i-1-N_i and N_i-C^α_i vectors. Together with zero-point librations of the N-H vector, these motions therefore dominate the impact on S′_NH. The problem of finding the average N-H bond length and angular motion is now reduced to solving the positional distribution of the H^N nuclei relative to the backbone frame.

Considering that N-H bond stretching motions occur at higher frequencies than librational motions, librational and stretching motions to first order can be considered uncorrelated, allowing factorization of the positional distribution of the proton associated with any given N-H vector:

$$\rho (r,u,v)\approx \rho (u,v){\rho }_1(r)$$ (3a)

where ρ₁(r) denotes bond stretching (Supporting Info. Eq S25) and

$$\rho (u,v)=\frac{1}{2\pi {\sigma }_u{\sigma }_v}\text{exp}(-u^2/2{\sigma }_u^2)\text{exp}(-v^2/2{\sigma }_v^2)$$ (3b)

separates the angular motion into in-plane (σ_v) and out-of-plane (σ_u) components, described by two independent Gaussian distributions. Bond stretching motions impact averaging of the inverse cubic dependence of ¹⁵N-¹H dipolar couplings on r_NH. Based on the narrow range of infrared N-H stretching frequencies, the amplitudes of such motions are quite uniform, and their impact can be calculated (Supporting Information) to yield:

$$r_{\text{eff}}\equiv <r_{NH}^{-3}{>}^{-1/3}\approx r_{\text{eq}}+0.007\mathring{\text{A}}=<r_{\text{NH}}>-\text{0}\text{.009}\mathring{\text{A}}$$ (4)

In our study, the time-averaged N-H orientations are taken from the iterative DIDC analysis of the very high precision RDC data, recorded on the perdeuterated mutants of GB3. High accuracy of these N-H orientations is exemplified by exceptionally good fits to the ³J_HNHα Karplus curve (rmsd 0.33 Hz; Supporting Information). Three independent parameters σ_u, σ_v and <r_NH> are then determined by fitting C′-H^N and ¹⁵N-¹H RDCs data to eq 3.

We employ a two-stage procedure when fitting the positional distribution of each amide proton to the ten RDCs (N_i-H_i and C′_i-1-H_i RDCs for each of the five GB3 mutants): First, the axially symmetric internal motion model is applied, assuming uniform σ_u = σ_v values for all 45 residues included in the analysis. Whereas in this model the internal motion would not be separable from a change in bond length if only N-H RDCs were considered, the two-bond C′-H^N RDCs lift this degeneracy because the N-H bond length also impacts the C′-H^N vector orientations. Second, after a global value for r_NH^eq has been determined, r_NH^eq is fixed at this value and the constraint σ_u = σ_v is removed. This then makes it possible to evaluate the N-H motional asymmetry by simultaneously fitting σ_u and σ_v to the five sets of C′-H^N and ¹⁵N-¹H RDCs. Below, we describe these two stages in more detail.

In the first stage, the amide protons of the previously determined GB3 structure (PDB entry 2OED)¹⁵ are replaced by axially symmetric Gaussian distributions centered around the N-H vector orientations obtained from the above iterative DIDC analysis of the very high precision RDC data. Since the orientation of the molecular frame obtained from iterative DIDC analysis is arbitrary, it first needs to be overlaid with that of the experimental 2OED structure. For this purpose, the ¹³C′-¹³C^α RDCs are best-fit to the 2OED coordinates. Then, using all 45 DIDC-accepted N-H vectors, the optimal value of <r_NH> follows from a one-dimensional search: For each value of <r_NH>, an axially symmetric Gaussian distribution of 33,635 N-H vectors is generated where the width is defined by S′_NH, obtained from S′_NH<r_NH^-3> = 0.872 Å^-3. The ρ₁(r) distribution in this ensemble (and thereby <r_NH>, as well as r_eff and r_NH^eq, cf eq 4) is described by standard quantum statistical mechanics (Supporting Information Eq S24), and a singular value decomposition (SVD) fit of the C′-H^N RDCs to this ensemble yields χ².

The <r_NH> obtained from this search (Fig. 1B), 1.023Å, is in excellent agreement with neutron scattering data (Table 1) and also agrees remarkably well with a prior NMR analysis of less precise RDCs, which neglected the effect of N-H bond stretching and libration but used a three-parameter model for motions of rigid peptide planes.¹³ The <S′_NH> = 0.910 value we obtain when using <r_NH> = 1.023Å is smaller than the 0.938 value expected for zero-point motion alone (Table 1). Backbone motions other than those affecting the reference C^α-C′ vectors are responsible for this difference. For example, γ motions around the C^α-C^α axis^11-14 have little impact on the C^α-C′ axis due to the small angle (∼20°) between these axes, but strongly affect the N-H vectors, oriented nearly perpendicular to C^α_i-1-C^α_i.

Table 1.

Average dynamics parameters and amide N-H bond lengths (Å).

	rNHeqa	reffb	<rNH> c	S′NH	σtrg	S′C′HNh
this work d	1.008 ± 0.006	1.015 ± 0.006	1.023 ± 0.006	0.910 ± 0.016	9.9° ± 0.9°	0.981 ± 0.004
Others			1.022e	0.938f	8.2°f	0.994f

^ar_eq is the equilibrium bond length;

^br_eff = <r_NH^-3>^-1/3, where <…> denotes the bond stretching average;

^c<r> is the stretching-averaged bond length with the difference between r_eq and <r> resulting from anharmonicity;

^dUncertainty in the averaged value based on jackknife analysis;

^e<r_NH> from neutron scattering¹;

^fValue for N-H zero point libration in N-methylacetamide, obtained from MP2 quantum calculation, divided by the corresponding S_CαC′ value of 0.990.

^gσ_tr = (σ_u² + σ_v²)^1/2.

^hS′_C′HN is the C′-H^N order parameter relative to the C^α-C′-N backbone.

In the second stage, we explore the motional asymmetry of the N-H bond vector. Using the ρ₁(r) function that corresponds to <r_NH> = 1.023Å, the N-H and C′-H^N RDCs are fit to ensembles of structures where the two parameters S′_NH and η (asymmetry of motion parameter)^5,6 define the ellipsoidal cone-shaped Gaussian distributions of the proton positions, corresponding to independent σ_u and σ_v values (eq 3b). Rather than fitting individual S′_NH and η values for each residue, which is adversely affected by error propagation, N-H vectors in each of the four β strands as well as the single α-helix are treated as having the same S′_NH and η, and a two-dimensional grid search for the S′_NH and η values yielding the best fit to C′-H^N and N-H RDCs is carried out. Results are summarized in Table 2. The principal axes of the ellipsoidal cones are defined such that positive η values correspond to out-of-plane motion being larger than in-plane motion, as is found for all five fragments. To provide a more intuitive picture of these motions, the corresponding rms angular fluctuations are also listed in Table 2. In agreement with earlier analyses,^5,12,16 we find the largest motional amplitude in solvent-exposed Igg-recognition strand β2.

Table 2.

Average dynamics parameters of individual secondary structure fragments. ^a

	β1	β2	β3	β4	α1
S′NH b	0.923±0.029	0.867±0.032	0.924±0.025	0.931±0.011	0.900±0.026
ηNH	0.016±0.026	0.048±0.026	0.024±0.020	0.017±0.011	0.025±0.045
In-planec	8 ± 3°	9 ± 1°	7 ± 3°	7 ± 1°	9 ± 5°
Out-of plane c	10 ± 1°	15 ± 3°	11 ± 2°	10 ± 1°	12 ± 3°

^aExtracted from 5 independent alignment N-H and C′-H^N RDCs fitting, comprising residues Y3-N8 (β1), K13-K19 (β2), E42-D46 (β3), T51-T55 (β4), and (A23-D36) α1, values include the effect of zero point motions;

^bThe error is the standard deviation within each element of secondary structure, obtained from standard jackknife analysis;

^cRoot-mean-square amplitude of motion.

The value <r_NH> = 1.023Å obtained from our study is equivalent to an effective bond length r_eff = 1.015Å. This is considerably smaller than the r_eff = 1.041 Å value obtained when a rigid model is used to relate ¹⁵N-¹H RDCs to ¹³C^α-¹³C′ and ¹³C′-¹⁵N RDCs, where both the zero-point librations and the impact of larger angular fluctuations about the C^α-C^α axis (γ motions) are absorbed into r_eff. For analysis of ¹⁵N NMR relaxation data, factoring out only the nearly uniform effect of zero-point N-H librations² increases r_eff by 2.2% to 1.037 Å. This latter value is appropriate when characterizing backbone dynamics from ¹⁵N relaxation data, but note that site-to-site deviations from this value can be as large as ±0.005 Å (Supporting Information Fig. S5). The effect of γ motions should not be absorbed into r_eff as such motions are relaxation active and can vary residue by residue.

The RDCs used in the present study were obtained for perdeuterated mutants of GB3, yielding narrower line widths and removal of very small systematic RDC errors resulting from cross-correlated relaxation.¹⁷ The new RDCs are therefore more accurate than values used in prior GB3 studies.^5,12 Analysis of these new RDC values in terms of N-H order parameters, in the exact same manner as previously carried out for the protonated GB3 mutants,⁵ yields values that lack the S > 1 outliers obtained previously. Excepting the three highly mobile residues K10-L12, RDC-derived S values agree remarkably well (pairwise rmsd 0.03) with those obtained from ¹⁵N relaxation¹⁶ (Supporting Information Fig. S4).