Protein backbone motions viewed by intraresidue and sequential H^N-H^α residual dipolar couplings

Summary

Triple resonance E.COSY-based techniques were used to measure intra-residue and sequential H^N-H^α residual dipolar couplings (RDCs) for the third IgG-binding domain of protein G (GB3), aligned in Pf1 medium. Measurements closely correlate with values predicted on the basis of an NMR structure, previously determined on the basis of a large number of one-bond backbone RDCs measured in five alignment media. However, in particular the sequential H^N-H^α RDCs are smaller than predicted for a static structure, suggesting a degree of motion for these internuclear vectors that exceeds that of the backbone amide N-H vectors. Of all experimentally determined GB3 structures available, the best correlation between experimental ¹H-¹H couplings is observed for a GB3 ensemble, previously derived to generate a realistic picture of the conformational space sampled by GB3. However, for both NMR and X-ray-derived structures the ¹H-¹H couplings are found to be systematically smaller by ca 10% than expected on the basis of alignment tensors derived from ¹⁵N-¹H amide RDCs, assuming librationally corrected N-H bond lengths of 1.041 Ǻ.

Introduction

¹⁵N relaxation experiments, routinely used to study protein backbone dynamics, capture motions on time scales shorter than the rotational correlation time of the system in question, which is typically in the nanosecond range (Lipari and Szabo, 1982). A variety of more recent, rotating frame experiments add motions slower than ca 10⁴ s^-1 to this accessible range, although in most cases the amplitude of such motions remains difficult to evaluate (Ishima and Torchia, 2000; Mulder et al., 2001; Palmer et al., 2001; Eisenmesser et al., 2005). Residual dipolar couplings (RDCs), usually measured in an anisotropic environment, most commonly a liquid crystalline suspension or a compressed hydrogel (Tjandra and Bax, 1997; Clore et al., 1998b; Hansen et al., 1998a; Sass et al., 2000; Tycko et al., 2000), report on time-averaged internuclear vector orientations but also carry information on their motions, integrated over the entire range of time scales faster than ca 100 s^-1 (Meiler et al., 2001; Tolman et al., 2001; Peti et al., 2002; Tolman and Ruan, 2006; Zhang et al., 2006). Provided RDCs can be measured under five linearly independent alignment orientations, their values provide information not only on the amplitude of motions, but also on their asymmetry and the direction of this asymmetry (Tolman, 2002; Tolman and Ruan, 2006).

The presence of ¹H-¹H RDCs in weakly aligned proteins was first demonstrated by the presence of COSY cross peaks between protons far apart in the ubiquitin sequence (Tjandra and Bax, 1997), and similarly for DNA by TOCSY cross peaks between protons attached to different bases (Hansen et al., 1998b). The potential of ¹H-¹H RDCs for structural studies of oligosaccharides (Bolon and Prestegard, 1998; Tian et al., 1999), DNA (Wu et al., 2003), and proteins (Cai et al., 1999; Pellecchia et al., 2000; Tian et al., 2000; Tjandra et al., 2000) also has been well documented. Peti and Griesinger assessed relative magnitudes of order parameters of H-H and H-N bonds by comparing RDCs (Peti and Griesinger, 2000). The ¹H-¹H RDCs, which they obtained from a 3D NOESY-based exclusive-correlation type experiment (Griesinger et al., 1986; Peti and Griesinger, 2000), suggest that they are subject to significantly stronger rescaling as a result of internal motion than H-N RDCs.

Here, we use E.COSY type experiments (Griesinger et al., 1985; Montelione and Wagner, 1989; Wang and Bax, 1996) for measurements of intraresidue and sequential D_HNHα RDCs in GB3. The precisely measured RDCs, corrected for the effect of passive spin flips together with an NMR structure calculated on the basis of a very extensive set of RDCs, measured in five alignment media (Ulmer et al., 2003), allows re-evaluation of the impact of motions on backbone D_HNHα RDCs.

Ensembles of structures that are constrained to yield a simultaneous fit to both the structural (NOE, J coupling, and RDCs) and dynamical (¹⁵N relaxation rates) data have been shown to yield improved cross validation statistics when compared with a single static structure (Lindorff-Larsen et al., 2005). A variation on this approach has previously been used by Clore and Schwieters (2006) to derive an ensemble representation of GB3, and we therefore also evaluate whether this ensemble yields a better prediction of the D_HNHα RDCs than any single static representation.

Theoretical Section

Residual dipolar couplings in presence of motions

Assuming the absence of correlation between internal motion and molecular alignment, a residual dipolar coupling between nuclei i and j is described by (Prestegard et al., 2000):

$$D_{\mathrm{ij}}=-\left(\frac{{\mu }_0}{4\pi }\right)\frac{{\gamma }_{\mathrm{i}}{\gamma }_{\mathrm{j}}\mathrm{h}}{2{\pi }^2}\sum _{q=-2}^2\langle \frac{{\mathrm{Y}}_{2q}^{\ast }({\theta }_{\mathrm{ij}}^{\prime }\left(t\right),{\varphi }_{\mathrm{ij}}^{\prime }\left(t\right))}{r_{\mathrm{ij}}^3\left(t\right)}\rangle \langle {\mathrm{Y}}_{2q}(\zeta \left(t\right),\xi \left(t\right))\rangle$$ (1)

The angular brackets denote time averages, Y_2q are the second rank spherical harmonics, γ_i is the gyromagnetic ratio of nucleus i, r_ij is the distance between nuclei i and j, μ₀ is the permeability of free space, and h denotes Planck’s constant. The term Y_2q(ζ(t), ξ(t)) describes the time dependence of the magnetic field orientation in molecular frame, and can be recast as an alignment tensor (Bax et al., 2001). The first term in angular brackets describes the angular and radial fluctuations of the bond vector ij within the molecular frame. A transformation into the principal axis system of the bond vector can be carried out using the Wigner rotation matrix elements ${\mathrm{D}}_{q{q}^{\prime }}^{\left(2\right)}$ and yields (Prestegard et al., 2000):

$$D_{\mathrm{ij}}=-\left(\frac{{\mu }_0}{4\pi }\right)\frac{{\gamma }_{\mathrm{i}}{\gamma }_{\mathrm{j}}\mathrm{h}}{2{\pi }^2}\frac{4\pi }{5}\sum _{q{q}^{\prime }=-2}^2{\mathrm{D}}_{q{q}^{\prime }}^{\left(2\right)}(-\psi ,-{\theta }_{\mathrm{ij},\mathrm{av}},-{\varphi }_{\mathrm{ij},\mathrm{av}})\langle \frac{{\mathrm{Y}}_{2q^{\prime }}^{\ast }({\theta }_{\mathrm{ij}}\left(t\right),{\varphi }_{\mathrm{ij}}\left(t\right))}{r_{\mathrm{ij}}^3\left(t\right)}\rangle \langle {\mathrm{Y}}_{2q}(\zeta \left(t\right),\xi \left(t\right))\rangle$$ (2)

where (θ_ij,av, ϕ_ij,av) define the averaged internuclear orientation, and ψ is the principal direction of the internal motion anisotropy. The Σ_q<Y_2q(ζ(t), ξ(t))> term can be transformed into a diagonal matrix, with order parameters as their elements, where

$$\sqrt{\frac{4\pi }{5}}\langle {\mathrm{Y}}_{20}\left(\zeta \left(t\right)\right)\rangle \equiv S_{\mathrm{zz},\mathrm{mol}}$$ (3a)

$$\sqrt{\frac{24\pi }{5}}(\langle {\mathrm{Y}}_{22}(\zeta \left(t\right),\xi \left(t\right))\rangle +\langle {\mathrm{Y}}_{2-2}(\zeta \left(t\right),\xi \left(t\right))\rangle )\equiv S_{\mathrm{xx},\mathrm{mol}}-S_{\mathrm{yy},\mathrm{mol}}$$ (3b)

Eq 3a is commonly referred to as the axially symmetric (axial) component of the alignment tensor, and the ratio of the right hand sides of eq 3b and 3a as the rhombicity (Clore et al., 1998a). Motion of the internuclear vector relative to the molecular frame involves both angular and radial fluctuations, and the term describing this motion in eq 2 can be rewritten as

$${\left(r_{\mathrm{ij}}^{\mathrm{eff}}\right)}^3\sqrt{\frac{4\pi }{5}}\langle \frac{{\mathrm{Y}}_{20}\left({\theta }_{\mathrm{ij}}\left(t\right)\right)}{{\left(r_{\mathrm{ij}}\left(t\right)\right)}^3}\rangle \equiv S_{\mathrm{zz},\mathrm{int}\left(\mathrm{ij}\right)}^{3\mathrm{d}}$$ (3c)

$${\left(r_{\mathrm{ij}}^{\mathrm{eff}}\right)}^3\sqrt{\frac{24\pi }{5}}\left(\langle \frac{{\mathrm{Y}}_{22}({\theta }_{\mathrm{ij}}\left(t\right),{\varphi }_{\mathrm{ij}}\left(t\right))}{{\left(r_{\mathrm{ij}}\left(t\right)\right)}^3}\rangle +\langle \frac{{\mathrm{Y}}_{2-2}({\theta }_{\mathrm{ij}}\left(t\right),{\varphi }_{\mathrm{ij}}\left(t\right))}{{\left(r_{ij}\left(t\right)\right)}^3}\rangle \right)\equiv S_{\mathrm{xx},\mathrm{int}\left(\mathrm{ij}\right)}^{3\mathrm{d}}-S_{\mathrm{yy},\mathrm{int}\left(\mathrm{ij}\right)}^{3\mathrm{d}}$$ (3d)

The symbol r_ij^eff absorbs the radial motion and eqs 3c and 3d reduce to the standard order parameters only if the time dependence of r_ij is not correlated with that of Y₂₀. Additionally assuming axially symmetric internal motion of the bond vector about the mean orientation, eq 2 can be written in the commonly used form (Tjandra and Bax, 1997):

$$D_{\mathrm{ij}}=-\left(\frac{{\mu }_0}{4\pi }\right)\frac{{\gamma }_{\mathrm{i}}{\gamma }_{\mathrm{j}}\mathrm{h}}{2{\pi }^2{\left(r_{\mathrm{ij}}^{\mathrm{eff}}\right)}^3}{S}_{\mathrm{zz},\mathrm{int}\left(\mathrm{ij}\right)}^{3\mathrm{d}}\left[S_{\mathrm{zz},\mathrm{mol}}\frac{3{\cos }^2{\theta }_{\mathrm{ij},\mathrm{av}}-1}{2}+\frac{S_{\mathrm{xx},\mathrm{mol}}-S_{\mathrm{yy},\mathrm{mol}}}{2}{\sin }^2{\theta }_{\mathrm{ij},\mathrm{av}}\cos 2{\varphi }_{\mathrm{ij},\mathrm{av}}\right]$$ (4)

Tensor fits assuming a rigid structure

The molecular alignment tensor is commonly derived by fitting a set of RDCs to an a priori known set of $({\theta }_{\mathrm{ij},\mathrm{av}}^{\prime },{\varphi }_{\mathrm{ij},\mathrm{av}}^{\prime })$. If the fit assumes the structure to be rigid, motion actually present is absorbed into an “effective” alignment tensor and eq 1 can be rewritten as

$$D_{\mathrm{ij}}^{\mathrm{eff}}\equiv -D_{\mathrm{ij}}\left(\frac{4\pi }{{\mu }_0}\right)\frac{2{\pi }^2{\left(r_{\mathrm{ij}}^{\mathrm{eff}}\right)}^3}{{\gamma }_{\mathrm{i}}{\gamma }_{\mathrm{j}}\mathrm{h}}=\sum _{q=-2}^2{\mathrm{Y}}_{2q}^{\ast }({\theta }_{\mathrm{ij},\mathrm{av}}^{\prime },{\varphi }_{\mathrm{ij},\mathrm{av}}^{\prime }){\langle {\mathrm{Y}}_{2q}(\zeta \left(t\right),\xi \left(t\right))\rangle }^{\mathrm{eff}}$$ (5)

As can be seen from the complexity of eq 2, physical interpretation of eq 5 is difficult because each element of the effective alignment tensor is impacted to a different extent by the orientation and amplitude of the fluctuation of each internuclear vector. However, if internal motion is assumed to be axially symmetric, eq 4 shows that all elements describing the effective alignment tensor are scaled in a uniform manner by real coefficients, and therefore the tensor elements obtained from fits to a data set are scaled by a factor that reflects an “average” over all vectors

$$S_{\mathrm{aa},\mathrm{mol}}^{\mathrm{eff}}={\left\{S_{\mathrm{zz},\mathrm{int}\left(\mathrm{ij}\right)}^{3\mathrm{d}}\right\}}_{\mathrm{SVD}}^{\mathrm{ij}}{S}_{\mathrm{aa},\mathrm{mol}}$$ (6)

where aa = xx, yy, zz, and the curled brackets describe a single value obtained from a singular value decomposition (SVD) fit of an RDC set to their corresponding ij vectors.

Predicting an RDC set B using a tensor obtained from set A

For a protein of known structure, a molecular alignment tensor can be obtained by fitting eq 5 to a set of RDCs, A, consisting of couplings D^A_ij, usually carried out by SVD (Losonczi et al., 1999). If only one alignment medium is used, it is not straightforward to rigorously extract dynamics as well as structural information from RDCs, and a static structure is commonly assumed. As described above, motional effects of set A then are absorbed into the effective alignment tensor ${\langle {\mathrm{Y}}_{2q}(\zeta \left(t\right),\xi \left(t\right))\rangle }^{\mathrm{eff}\left(\mathrm{A}\right)}$. A second set of RDCs, D^B_kl, now can be predicted using eq 5 and the effective tensor obtained from set A. However, because set B may be subject to different types of motion, in particular when sets A and B comprise different types of interactions, the prediction may under- or overestimate the true values. This effect can be quantified by calculating a linear regression through pairs of experimental and predicted kl internuclear RDCs, (D^Bexp_kl, D^Bpred_kl), yielding a slope referred to as $\kappa ={\left\{\frac{{{\mathrm{D}}^{\mathrm{Bexp}}}_{\mathrm{k}l}}{{{\mathrm{D}}^{\mathrm{Bpred}}}_{\mathrm{k}l}}\right\}}_{\mathrm{LinReg}}^{\mathrm{k}l}$.

Assuming axially symmetric internal motion, the expression for the slope simplifies considerably:

$${\left\{\frac{{{\mathrm{D}}^{\mathrm{Bexp}}}_{\mathrm{kl}}}{{{\mathrm{D}}^{\mathrm{Bpred}}}_{\mathrm{kl}}}\right\}}_{\mathrm{LinReg}}^{\mathrm{kl}}=\frac{{\left\{S_{\mathrm{zz},\mathrm{int}\left(\mathrm{kl}\right)}^{3\mathrm{d}}\right\}}_{\mathrm{LinReg}}^{\mathrm{kl}}}{{\left\{S_{\mathrm{zz},\mathrm{int}\left(\mathrm{ij}\right)}^{3\mathrm{d}}\right\}}_{\mathrm{SVD}}^{\mathrm{ij}}}=\frac{{\left\{{\left(r_{\mathrm{kl}}^{\mathrm{eff}}\right)}^3\langle \frac{{\mathrm{Y}}_{20}\left({\theta }_{\mathrm{kl}}\left(t\right)\right)}{{\left(r_{\mathrm{kl}}\left(t\right)\right)}^3}\rangle \right\}}_{\mathrm{LinReg}}^{\mathrm{kl}}}{{\left\{{\left(r_{\mathrm{ij}}^{\mathrm{eff}}\right)}^3\langle \frac{{\mathrm{Y}}_{20}\left({\theta }_{\mathrm{ij}}\left(t\right)\right)}{{\left(r_{\mathrm{ij}}\left(t\right)\right)}^3}\rangle \right\}}_{\mathrm{SVD}}^{\mathrm{ij}}}$$ (7a)

This can be further simplified if the bond lengths are not correlated with the orientation:

$$\kappa =\frac{{\left\{\langle 3\cos {\left({\theta }_{\mathrm{kl}}\left(t\right)\right)}^2-1\rangle \right\}}_{\mathrm{LinReg}}^{\mathrm{kl}}}{{\left\{\langle 3\cos {\left({\theta }_{\mathrm{ij}}\left(t\right)\right)}^2-1\rangle \right\}}_{\mathrm{SVD}}^{\mathrm{ij}}}=\frac{{\left\{S_{\mathrm{zz},\mathrm{int}\left(\mathrm{kl}\right)}\right\}}_{\mathrm{LinReg}}^{\mathrm{kl}}}{{\left\{S_{\mathrm{zz},\mathrm{int}\left(\mathrm{ij}\right)}\right\}}_{\mathrm{SVD}}^{\mathrm{ij}}}$$ (7b)

and this result can therefore be conveniently used to evaluate relative amplitudes of internal motions of different types of internuclear vectors, while ensuring that the alignment tensor used for the two sets of dipolar couplings is the same.

Longitudinal flips of passive spins

Spin flips of passive spins, coupled to a spin of interest, result in different relaxation rates of the in-phase and anti-phase coherences. The time dependence of the spin states |α> and |β> of spin I, coupled to spin S, impacts the evolution of the transverse magnetization of I-spin doublet components:

$$\frac{d}{dt}\left(\begin{array}{c}\hfill I_{\mathit{trans}}{S}^{\alpha }\hfill \\ \hfill I_{\mathit{trans}}{S}^{\beta }\hfill \end{array}\right)=\left(\begin{array}{cc}-R_{\mathit{trans}I}\hfill & R_{1\mathrm{S}}∕2\hfill \\ R_{1\mathrm{S}}∕2\hfill & -R_{\mathit{trans}I}\hfill \end{array}\right)\left(\begin{array}{c}\hfill I_{\mathit{trans}}{S}^{\alpha }\hfill \\ \hfill I_{\mathit{trans}}{S}^{\beta }\hfill \end{array}\right)$$ (8)

where R_transI is the transverse autorelaxation rate of spin I, and R_1S is the longitudinal relaxation rate of spin S as observed in a selective inversion recovery experiment (Hu et al., 2006). In a spin state-selective experiment, one doublet component can be selected while the other is being suppressed. However, the two components cross relax when R_1S is non-zero and integration of eq 8 then shows an increase in the intensity of the unselected component and an attenuation of the selected component, with a time dependence given by

$$I_{\mathit{trans}}{S}^{\text{selected}}\left(t\right)∕I_{\mathit{trans}}{S}^{\text{selected}}\left(0\right)=1-\sinh (R_{1\mathrm{S}}t∕2)$$ (9.1)

$$I_{\mathit{trans}}{S}^{\text{unselected}}\left(t\right)∕I_{\mathit{trans}}{S}^{\text{selected}}\left(0\right)=\sinh (R_{1\mathrm{s}}t∕2)$$ (9.2)

where selected = α, unselected = β, or vice versa. As a consequence, the two components that are ideally suppressed in an E.COSY pattern (Griesinger et al., 1985) become observable. If these weaker components overlap the selected components, the apparent positions of the selected components and thereby the measured couplings are affected. When the passive spin S is ¹H, its relaxation is dominated by homonuclear dipolar couplings and their J(0) spectral density terms (Wang and Bax, 1996). If S is¹⁵N, R_1S will be considerably smaller, but long transfer delays (e.g. NOE mixing) increase the impact of this term (Peti and Griesinger, 2000), also resulting in an underestimation of the true (D + J) coupling value if the effect is not accounted for. Although experiments have been proposed to mitigate this effect (Rexroth et al., 1995; Vogeli et al., 2007), due to extensive homonuclear ¹H-¹H dephasing these schemes are less suitable for measuring couplings in the aligned state. On the other hand, if R_1S is known, the impact of the spurious component on the measured E.COSY splitting is easily simulated (Wang and Bax, 1996), and the result can be used to correct the experimental data. This approach is used in the present study.

As a second consequence of the finite lifetime of the passive spin, S, the apparent (D + J)_IS^app coupling between spins I and S measured from the splitting on spin I is given by (Harbison, 1993):

$${{(D+J)}_{\mathrm{IS}}}^{\mathrm{app}}=\sqrt{{{(D+J)}_{\mathrm{IS}}}^2-{(R_{1\mathrm{S}}∕2\pi )}^2}$$ (10)

Eq 10 can be used to calculate the impact of the passive spin relaxation during data acquisition, and thereby to apply a second correction to the measured apparent (D+J) coupling.

Experimental Section

Sample expression and purification

Escherichia coli BL21 (DE3*) cells transformed with a pET-11 vector containing the GB3 gene were grown in M9 minimal media, containing ¹⁵NH₄Cl and uniformly ¹³C-enriched glucose. Protein expression was induced by 0.2 mM IPTG. This culture was centrifuged and resuspended in 20 ml PBS. The cells were lysed at 80 °C and centrifuged again. The supernatant was loaded to a Superdex 75 HiLoad 26/60 (Amersham Biosciences) column, equilibrated with NMR buffer (50 mM sodium phosphate buffer, 50 mM NaCl, and 0.05% w/v NaN₃ at pH 7.0). The ¹³C,¹⁵N-enriched NMR sample contained 400 μl of a 2.3 mM protein solution (95%/5% H₂O/D₂O). For measurement of RDCs, the sample was titrated with Pf1 phage (http://www.asla-biotech.com) until the solvent quadrupolar ²H splitting reached 9.5 Hz (Hansen et al., 1998a).

NMR spectroscopy

All NMR experiments were carried out at 296 K on a Bruker DMX600 spectrometer, equipped with a three-axes gradient triple resonance probe, or on a Bruker DRX600 spectrometer, equipped with a z-axis gradient cryogenic probe.

The 3D HNCA[HA]-E.COSY experiments for measurement of ³(J+D)_HNHα (Wang and Bax, 1996) were recorded with 32(t₁) × 38(t₂) × 400(t₃) complex points, t_1max = 19.2 ms, t_2max = 12.9 ms, t_3max = 51.4 ms, an interscan delay of 1.0 s and 8 scans per FID. The time domain data were apodized with a squared, 90°-shifted sine bell function in the direct dimension and with regular 90°-shifted sine bell functions in both of the indirect dimensions, and zero-filled to 128 × 256 × 2048 complex points. The ¹H^N selective sinc-shaped pulses used by (Wang and Bax, 1996) were replaced by selective 90° EBURP and 180° ¹H^N ReBURP pulses (Geen and Freeman, 1991).

The 3D HN(CO)CA[HA]-E.COSY experiment for measurement of ⁴(J+D)_HNHα (Figure 1) is very similar to the HNCA[HA]-E.COSY experiments for measurement of ³(J+D)_HNHα (Wang and Bax, 1996) but includes the two additional INEPT steps transferring magnetization from ¹⁵N(i) to ¹³C^α(i-1) via ¹³C’(i-1) and back (Vuister and Bax, 1994). As in the HNCA[HA]-E.COSY experiment, the ¹H^N selective pulses in the final reverse INEPT of the HNCA[HA]-E.COSY experiment are of the 90° EBURP and 180° ReBURP type (Geen and Freeman, 1991), thereby avoiding ¹H^α excitation which is advantageous in terms of preserving the spin state of H^α. More details are given in the figure caption of Figure 1. The experiments were recorded with 32(t₁) × 38(t₂) × 400(t₃) complex points, t_1max = 19.2 ms, t_2max = 12.9 ms, t_3max = 51.4 ms, an interscan delay of 1.0 s and 8 scans per FID. The time domain data were apodized with a squared, 90°-shifted sine bell function in the direct dimension and with regular 90°-shifted sine bell functions in both the indirect dimensions, and zero-filled to 128 × 256 × 2048 complex points.

Figure 1.

Pulse sequence of the 3D HN(CO)CA[HA]-E.COSY experiment for measurement of sequential D_HαHN couplings. The radio-frequency pulses on ¹H, ¹⁵N, ¹³C^α and ¹³C’ are applied at 4.7, 120, 56 and 174 ppm, respectively. Narrow and wide bars indicate non-selective 90° and 180° pulses. The triangular shape represents a ¹³C’-selective 180°-Gaussian pulse of duration p₁ = 150 μs; the shaped ¹H^N-selective EBURP and ReBURP pulses (Geen and Freeman, 1991) have durations p₂ = 1.0 ms and p₃ = 1.5 ms and are applied at 10.2 ppm and 9.9 ppm, respectively, on a 600 MHz instrument. The duration of the shaped pulses needs to be adjusted inversely with the strength of the static magnetic field. ¹⁵N-decoupling is achieved with WALTZ16 (Shaka et al., 1983) at a RF field strength γB₁ = 1.1 kHz, and optional ¹³C’-decoupling is also achieved with WALTZ16 (γB₁ = 600 Hz). Unless indicated otherwise, all radio-frequency pulses are applied with phase x. The phase cycle is: ϕ₁ = x; ϕ₂ = {x, y,-x,-y}; ϕ₃ = x; ϕ₄ = {x, x,-x,-x}; ϕ₅ = {x, x, x, x,-x,-x,-x,-x}; ϕ_rec = {x,-x,-x, x,-x, x, x,-x}. The delays have the following values: τ₁ = 1/(4J_HN) = 2.6 ms, τ₂ = 1/(4J_NC’) = 13.5 ms, τ₃ = 6.5 ms = 0.7 × 1/(2J_CαC’), τ₄ = 11 ms < 1/(4J_NC) and η is the length of the ¹³C^α 180° pulse. Pulsed field gradients (PFG) are applied along the z-axis with duration/strength of: G₁, 0.5 ms / 12 G/cm; G₂, 2 ms / 18 G/cm; G₃, 1.5 ms / -10 G/cm; G₄, 0.7 ms / -18 G/cm; G₅, 1 ms / 15 G/cm; G₆, 0.2 ms / 18 G/cm. Quadrature detection in the ¹⁵N (t₁) and ¹³C(t₂) dimension is achieved by the States-TPPI method, applied to the phases ϕ₁ and ϕ₃, respectively.

A 2D HACA(CO)NH experiment was used for measurement of longitudinal relaxation rates of C^α_z and 2C^α_zH^α_z. The decay of 2C^α_zH^α_z, observed after the H^α→C^α INEPT, and the decay of C^α_z after a refocused H^α-C^α INEPT transfer, was sampled at τ = 0, 100, 200, and 300 ms (Figure S1 in Supporting Information).

All spectra were processed and analyzed using the software package NMRPipe (Delaglio et al., 1995). Peak positions were determined by parabolic interpolation.

Results and discussion

Under weak alignment, resolution in the ¹H-dimension of multidimensional NMR spectra is adversely affected by the presence of large numbers of unresolvable ¹H-¹H residual dipolar couplings, superimposed on the regular ¹H-¹H J-multiplet patterns. Quantitative measurement of ¹H-¹H RDCs tends to be more challenging than for one-bond ¹H-¹³C or ¹H-¹⁵N couplings, not only because of this decreased resolution, but also because of their typically smaller size associated with the larger internuclear distance, which is only partially offset by both spins having a high magnetogyric ratio. Of the various approaches available, E.COSY experiments (Griesinger et al., 1985; Griesinger et al., 1986) appear best suited for measurement of RDCs because they do not require the coupling of interest to be resolvable, and when carried out in a heteronuclear fashion (Montelione and Wagner, 1989; Biamonti et al., 1994) they also yield the sign of the ¹H-¹H coupling relative to that of a (larger) one-bond ¹J_NH or ¹J_CH interaction. We recorded two separate E.COSY experiments: HNCA[HA] for intraresidue ¹H^N-¹H^α couplings and HN(CO)CA[HA] for measurement of sequential ¹H^α-¹H^N interactions. Accurate ³J_HNHα couplings for the isotropic sample were available from a recent set of multiple quantum experiments, and fit the Karplus equation with a root-mean-square difference (rmsd) of less than 0.4 Hz. Four-bond isotropic ⁴J_HαHN couplings are known to be vanishingly small (Vuister and Bax, 1994) and therefore were assumed to be zero. Hence, E.COSY experiments were carried out only for the aligned sample.

Figure 2 shows small regions of cross sections through the 3D HNCA[HA] and HN(CO)CA[HA] E.COSY spectra, exhibiting the ¹J_CαHα + ¹D_CαHα displacement in the ¹³C dimension, and the J_HαHN + D_HαHN splitting in the H^N dimension. The acquisition time in the ¹³C dimension, and hence the resolution, was adjusted to be sufficient for resolving the two ¹³C^α-{¹H^α}doublet components, but no effort was made to optimize the measurement for ¹J_CαHα + ¹D_CαHα couplings as these values are known at high accuracy from prior experiments (Ulmer et al., 2003). Due to the presence of multiple unresolved ¹H-¹H couplings, line widths in the ¹H^N dimension are relatively large (ca 25-45 Hz; Fig.2) but the pairwise rmsd of measured J_HαHN + D_HαHN splittings in a duplicate set of experiments was only 1.4 Hz for both intraresidue and sequential couplings, indicating random errors of 0.7 Hz in the averaged values.

Figure 2.

Small regions of (F₂,F₃) cross sections through the HNCA[HA] (top, intraresidue) and HN(CO)CA[HA] (bottom, sequential) E.COSY spectra of GB3 taken at the ¹⁵N frequencies of Asn8 (left) and Trp43 (right). Multiplets exhibit the ¹J_CαHα + ¹D_CαHα splitting in the ¹³C dimension, and the J_HαHN + D_HNHα displacement in the H^N dimension.

Correction for the effect of ¹H^α spin flips

As pointed out in the Theoretical Section, a change in ¹H^α spin state between ¹³C^α evolution and ¹H^N detection gives rise to weak additional but frequently unresolved components in the ¹H^N-¹³C^α multiplet structure (Wang and Bax, 1996). The impact of these ¹H^α spin flips on the measured J_HαHN + D_HαHN splitting is a function of the ¹H^α spin-flip rate, the ¹H^N line width, and the ¹H^α-¹H^N coupling, and is most easily evaluated by simulating the E.COSY multiplets (Wang and Bax, 1996). To accomplish this, the ¹H^α R₁^Hα were measured from the difference in the relaxation rates of C^α_z and 2C^α_zH^α_z (Peng and Wagner, 1992) (Supporting Information Table S1). These measured rates were found to be in fair agreement with rates simulated on the basis of the NMR structure (PDB entry 2OED), assuming isotropic rotational diffusion with a rotational correlation time of 3.4 ns (Hall and Fushman, 2003), and including all protons within a 5-Ǻ radius from ¹H^α (Supporting Information Table S1). The Pearson’s correlation coefficient between the experimental and calculated sets of R₁(¹H^α) values equals 0.82 (Figure S2 in Supporting Information), and for all final evaluations the experimental values were used to correct the observed splittings because they resulted in slightly better fits between RDCs and structure than did the calculated R₁(¹H^α) values.

The effect of ¹H^α spin flips during the period between the end of ¹³C^α evolution and the start of ¹H^N acquisition on the E.COSY cross peak multiplet was simulated using the program MATLAB and eq 9, assuming a uniform ¹H^N line width of 35 Hz. As seen in Figure 3, the effect of ¹H^α spin flips on the splittings can be considerably larger than the random measurement error, and always attenuates the total splitting measured. For intraresidue interactions, this can lead to an increase or decrease of the magnitude of the resulting RDC, depending on whether the RDC is smaller in magnitude and of opposite sign to ³J_HNHα or not. For sequential H^α-H^N RDCs, the effect of ¹H^α spin flips always reduces the magnitude of the RDC, by an amount roughly proportional to its size. In the following analysis, Lorentzian peak shapes were assumed, although internal RDCs may render the shapes partially Gaussian. As can be seen from Figure 3, corrections derived for Lorentzian shapes are slightly smaller than for Gaussian shapes and therefore may be considered a lower limit.

Figure 3.

Correction in Hz to be applied to the (D + J) value as measured from the E.COSY spectra, as a function of (D + J) and R₁^Hα. Opposite sign corrections of the same magnitude are required for interactions where (D + J) < 0. Corrections were derived using peak picking on multiplets simulated using eq 9, assuming H^N Lorentzian (left) or Gaussian (right) line shapes with line width at half height of 35 Hz, and H^α spin flips occurring between the end of the C^α evolution period and the start of direct detection (28.8 ms).

The frequencies of the ¹H^N-{¹H^α} doublet components in F₃ are affected by ¹H^α spin flips during the detection period, t₃ (Harbison, 1993). The impact of ¹H^α spin flips during t₃ is readily accounted for by eq 10 and tends to be somewhat smaller than the effect of ¹H^α spin flips taking place between the end of ¹³C^α evolution and the start of t₃. Supporting Information Table S1 reports both the measured “raw” splittings, the isotropic ³J_HNHα value, and the corrected RDC values. Note that the isotropic ³J_HNHα couplings were measured in the indirect dimension of a multiple quantum experiment, designed to minimize the effect of the finite life time of the passive spins (Rexroth et al., 1995; Vogeli et al., 2007), and therefore did not require any such corrections. Unfortunately, for weakly aligned proteins, such multiple quantum schemes are adversely affected by the extensive dephasing resulting from the multitude of homonuclear ¹H-¹H couplings. Therefore, such experiments were not used to measure RDCs.

Comparison of ¹H-¹H RDCs with values expected for static structure

The ¹³C’-¹³C^α and ¹³C’-¹⁵N one-bond RDCs for ubiquitin and GB3 were found to be in excellent agreement with those calculated based upon their respective X-ray structures (Ottiger and Bax, 1998; Ulmer et al., 2003). Considering that the impact of zero-point librations on these two types of RDCs is negligible, and that the corresponding interatomic distances are quite uniform and known at high accuracy (Engh and Huber, 1991) these couplings have previously been used to define the molecular alignment tensor. When subsequently evaluating the ¹D_HN and ¹D_CαHα couplings, these were found to be ca 6% smaller than anticipated for the standard bond lengths, r_NH = 1.02 Ǻ and r_CH = 1.09 Ǻ (Ottiger and Bax, 1998), in perfect agreement with predictions for the effect of zero point librations (Case, 1999). Therefore, librationally-corrected bond lengths of r_NH = 1.041 Ǻ and r_CH = 1.117 Ǻ are commonly used when deriving the molecular alignment tensor from ¹D_HN and ¹D_CH RDCs. This procedure is followed here too, and using ¹D_HN couplings the alignment tensor was determined by fitting these data either to the original X-ray structure (PDB entry 1IGD (Derrick and Wigley, 1994), with protons added by MOLMOL (Koradi et al., 1996)) or to the NMR-refined structure (PDB entry 2OED)(Ulmer et al., 2003), yielding better fits for the NMR structure (Table 1). In the following analysis, coordinates from the NMR structure are used. Using these alignment tensors and structures, values for the ¹H-¹H RDCs are readily calculated using eq 4. When deriving the molecular alignment tensor from an SVD fit to N-H or C^α-H^α RDCs, the tensor magnitude scales with the inverse cubed of the bond length, r_NH^-3 or r_CH^-3. However, whereas for one-bond couplings a change in the effective bond lengths simply scales the predicted RDC, for D_HH a change in N-H and/or C^α-H^α bond length also alters the orientation of the interproton vector. Rather than using the librationally corrected effective bond lengths, which artificially lengthen the internuclear N-H and C^α-H^α distances, we therefore use the standard r_NH = 1.02 Ǻ and r_CH = 1.09 Ǻ bond lengths obtained from neutron scattering for defining the ¹H positions when deriving the interproton vectors, even while using r_NH = 1.041 Ǻ for deriving the alignment tensor. Subsequently, we will also evaluate the impact of the NH bond lengths on the correlation between observed and predicted D_HH RDCs.

Table 1.

Comparison of measured H^N-H^a RDCs and values predicted from an alignment tensor obtained from ¹H-¹⁵N RDCs

Coupling type	# res	structure	rmsdd [Hz]	rmsd /range	r	slopee
intra DHNHα	41a	2OED	1.32	0.057	0.981	1.01±0.03
		1IGD	1.39	0.060	0.978	0.99±0.04
		DIDCb	1.22	0.053	0.984	0.99±0.03
		Ensemblec	1.45	0.062	0.980	1.06±0.04
sequential DHNHα	46a	2OED	2.04	0.047	0.978	1.10±0.03
		1IGD	3.11	0.072	0.943	1.02±0.05
		DIDCb	1.86	0.043	0.982	1.13±0.03
		Ensemblec	1.28	0.030	0.991	1.11±0.02

^aDynamic residues 12, 40, 41 are excluded.

^bFor the 2OED structure with the N-H and Cα-Hα vectors replaced by those derived from iterative DIDC analysis of six GB3 mutants, all aligned in Pf1 medium (Yao et al., 2008).

^c160-member ensemble from(Clore and Schwieters, 2006).

^drmsd relative to the regression line.

^eThe slope of the correlation between predicted (y axis) and observed RDCs.

Although both the intraresidue and sequential ¹H^N-¹H^α RDCs correlate well with values expected for a static structure (Figure 4), in both cases the pairwise rmsd exceeds the lower limit of the experimental uncertainty (estimated from reproducibility of the measurement, see above) by more than two-fold (Table 1). In particular, a best fit results in the scaling of observed D_HH values by factors of ca 0.99 and 0.91 for the intraresidue and sequential H^N-H^α couplings, respectively, indicating that these couplings are scaled down relative to backbone ¹³C’-¹³C^α and ¹³C’-¹⁵N one-bond interactions, presumably as a result of internal motion. For the intraresidue D_HNHα RDCs, this scaling is less pronounced than the 6% previously noted for the ¹D_HN and ¹D_CH RDCs (Ottiger and Bax, 1998). On the other hand, a slightly more pronounced scaling is observed for sequential D_HαHN interactions, suggesting more dynamic averaging for this interaction than for the one-bond N-H and C^α-H^α RDCs. Residues for which the prediction overestimates the magnitude of the observed RDC value are mostly located in strand β4 (residues 50-56). This observation is consistent with slightly elevated motions in edge strand β4 relative to core of the protein (Yao et al., 2008). On the other hand, our ¹H-¹H couplings do not systematically differ from predictions for the other edge strand, β2, which has also been assigned above average amplitude backbone motions (Hall and Fushman, 2003; Bouvignies et al., 2005; Yao et al., 2008).

Figure 4.

Correlation plots showing predicted versus experimental RDCs for (a) ³D_HNHα and (b)⁴D_HαHN for 2OED. The effective alignment tensor is obtained from the ¹D_HN RDCs and used for the prediction of all RDCs (r_NH^eff is 1.041 Ǻ). Table 1 presents statistics regarding the observed correlations. Highly flexible residues 12, 40 and 41 are excluded. Symbols embossed with red asterisks correspond to residues that fall more than one standard deviation from a linear regression best fit. Residue labeling corresponds to the residue on which H^N resides.

Comparison of ¹H-¹H RDCs with values expected for a dynamic ensemble

The use of eqs 4, 6 and 7 is based on the assumption of axially symmetric motion of the internuclear vectors, leading to a simple scaling of the internuclear dipolar coupling. True internal motions in many cases will be far more complex, and may include significant asymmetry. Only if RDCs can be measured in five linearly independent alignment orientations does it become possible to extract the amplitude and asymmetry of internal motion in a model free manner (Tolman, 2002; Tolman and Ruan, 2006; Yao et al., 2008). However, it is interesting to note that NMR structures also can be generated in an ensemble representation that reflects the dynamic behavior of the protein (Lindorff-Larsen et al., 2005; Clore and Schwieters, 2006). A problem in calculating an ensemble of conformers that simultaneously satisfy the experimental NMR restraints has been that the number of NMR observables is insufficient to uniquely define the coordinates when the number of members of the ensemble is greater than one. Incorporation of generalized order parameter restraints derived from relaxation data (Lindorff-Larsen et al., 2005; Clore and Schwieters, 2006), possibly in combination with crystallographically derived temperature factors (Clore and Schwieters, 2006), can mitigate this problem but at the same time limits the amplitude of motions that may be taking place on a time scale much slower than the molecular tumbling. Nevertheless, ensemble structures previously derived by Clore and Schwieters for GB3 yielded not only modest improvements in RDC cross validation statistics (Clore and Schwieters, 2006), they also provided an even better fit to ³J_HNHα couplings recently measured for this protein (rmsd = 0.36 Hz) than did the static NMR structure (rmsd = 0.42 Hz) (Vogeli et al., 2007). Below, we therefore also evaluate how well our newly measured D_HH couplings agree with this ensemble presentation.

Clore and Schwieters used RDCs from five media, ¹⁵N-relaxation derived order parameters, and crystallographic B factors to generate twenty 8-member ensembles of structures. In order to obtain a statistically better sampling of the conformational space accessible to GB3, all 160 members of these 20 ensembles are considered simultaneously in the analysis presented below. Eq 1 provides a straightforward avenue to predict RDCs for an ensemble of structures. First, spherical harmonics for the various internuclear interactions are calculated for the ensemble by averaging the values from all 160 conformers. Ensemble-averaged values for the bond vector rhombicities vary considerably from residue to residue, but mostly are quite small: 0.028 (¹D_HN), 0.007 (³D_HNHα), 0.015 (⁴D_HNHα) and 0.010 (¹D_CαHα). Note that these small values for the rhombicities do not necessarily exclude considerable asymmetry of the internal motions because, following the definition of eqs 3c and 3d, the accessible range of rhombicity falls between zero and 1-S^3d_zz,int(i,j). Eq 1 is then used to fit the alignment tensor simultaneously to all ¹⁵N-¹H vectors, using the averages of the corresponding spherical harmonics from 160 conformers. As expected, with dynamic averaging of the bond vector orientations, slightly larger alignment is observed than when fitting with a single, static structure (Supporting Information Tables S2 and S4). Figure 5 shows the agreement between the observed ¹H-¹H RDCs and values predicted for the 160-member ensemble. In particular for the sequential H^α_i-H^N_i+1 RDCs, a significantly better correlation between observed and predicted couplings is observed for this ensemble (Figure 5b) than for either the static X-ray (PDB entry 1IGD) or NMR structure (PDB entry 2OED), where the rmsd between observed and predicted couplings drops from 2.04 to 1.28 Hz. Remarkably, the difference in the slopes of the two regression lines that relate the predicted to the measured couplings for intraresidue and sequential H^N-H^α interactions drops from 0.09 to 0.05, and then falls within the experimental uncertainties of these slopes. Note, however, that for both intraresidue and sequential RDCs, predicted values remain larger than observed values. Therefore, even though the ¹H-¹H RDC data provide strong evidence that the ensemble representation provides a better description of the protein in solution, the ¹H-¹H RDC data either point to the presence of some additional motion, beyond what is seen in the ensemble, or to a remaining small systematic error that attenuates the measured ¹H-¹H RDCs.

Figure 5.

Correlation plots showing predicted versus experimental RDCs for (a) ³D_HNHα and (b) ⁴D_HαHN, for a 160 conformer ensemble (Clore and Schwieters, 2006). The effective alignment tensor is obtained from the ¹D_HN RDCs and used for the prediction of all other RDCs (r_NH^eff is 1.041 Ǻ). Table 1 presents statistics regarding the observed correlations. Symbols embossed with red asterisks correspond to RDCs that differ by more than 1.5 standard deviations from a linear regression best fit. Residue labeling for ⁴D_HαHN corresponds to the residue on which H^N resides.

N-H and C^α-H^α bond lengths from D_HH couplings

The orientation of ¹H^N-¹H^α vectors is directly impacted by the N-H and C^α-H^α bond lengths. D_HNHα couplings therefore provide an opportunity to evaluate which bond lengths are in best agreement with the observed RDCs, i.e., which bond lengths yield H-H vectors that correlate best with the RDCs. When systematically stepping the N-H and C^α-H^α bond lengths through a reasonable range, plus or minus a few percent of the commonly used equilibrium bond lengths, changes in ¹H-¹H vector orientations are very small, at most a few degrees. Therefore, in order to improve the statistics, we assume that for both N-H and C^α-H^α equilibrium bond lengths are uniform throughout the protein. As discussed above, variations in dynamics along the backbone also impact the measured ¹H-¹H couplings in a non-uniform manner, adversely affecting the correlation between structure and RDCs. However, considering the above results, the 160-member ensemble at least partially accounts for the effect of internal dynamics, and we therefore use this ensemble to evaluate which N-H and C^α-H^α bond lengths agree best with the RDCs.

Figure 6 shows two-dimensional contour plots, as a function of r_NH and r_CαHα, of the rmsd between observed ¹H-¹H RDCs and those predicted by the ensemble. If only sequential couplings in β-sheet are considered, contours run mostly parallel to the diagonal (Fig. 6). The reason for this correlation between r_NH and r_CαHα is that the C^α-H^α and N-H vectors in sequential β-sheet positions are approximately parallel, and an equal lengthening of both bonds has little impact on the ¹H^N-¹H^α vector. Also, because these sequential interactions correspond to short interproton distances, the impact of varying r_NH and r_CαHα on D_HNHα tends to be larger than for residues in helices, for example. When considering the 46 sequential D_HNHα RDCs, a minimum is observed that includes the canonical r_NH = 1.02 Ǻ and r_CαHα = 1.09 Ǻ values, but as can be seen this minimum is quite shallow, and the observed χ² is statistically compatible with a rather broad range of values: r_NH = 1.00-1.11 Ǻ and r_CαHα = 1.04-1.14 Ǻ.

Figure 6.

RMSD between experimental and predicted sequential D_HNHα couplings in GB3 versus r_HN and r_CαHα bond lengths for the 160-conformer ensemble. Equidistant lines are drawn in steps of 0.03 Hz above the minimal rmsd of 1.16 Hz (at r_HN = 1.05 Ǻ and r_CαHα = 1.09 Ǻ). The blue ellipse covers bond length pairs with an rmsd which is not statistically significantly larger than the minimal rmsd (<1.19 Hz).

Concluding Remarks

The E.COSY experiments used in our study yielded quantitative measurement of both intraresidue and sequential D_HNHα couplings. Corrections were made for the finite lifetime of the passive spin (¹H^α) in these experiments, and such corrections were found to be larger than the random measurement error. In principle, such corrections are also required in the analogous experiments by Peti and Griesinger (Peti and Griesinger, 2000), but there it is the lifetime of the ¹⁵N spin which gives rise to the analogous spurious multiplet components. Considering that ¹⁵N longitudinal relaxation rates tend to be considerably longer than selective ¹H^α relaxation rates, their experiment is intrinsically less sensitive to such passive spin flips. However, this favorable ¹⁵N relaxation dependence may be offset by the longer (NOE mixing) time that separates evolution in the indirect and direct dimensions. It therefore remains ambiguous why in their measurements a larger reduction (22% for H^N-H^α) for backbone ¹H-¹H couplings was observed than in our study. For GB3, the D_HNHα couplings are 1±4% (intraresidual) and 9±3% (sequential) smaller than what is expected on the basis of a single static structure with librationally averaged 1.041-Ǻ N-H bond lengths. Using standard r_NH = 1.02 Ǻ and the same alignment tensor, the observed ¹D_NH RDCs therefore are ∼6% smaller than those predicted for such a structure, whereas backbone ¹D_CαC’ and D_C’N couplings would correlate to predicted RDCs with a slope of one. Therefore, the intraresidue D_HNHα RDCs, which are attenuated by about 1±4% relative to what is predicted for the static structure, point to motions that are indistinguishable in amplitude from those of the C^α-C’ and C’-N bonds.

The sequential D_HNHα RDCs are 9±3% smaller than expected for a single static structure, suggesting larger fluctuations in the orientation of these vectors relative to those of the C-C and C-N bonds in the peptide backbone. In this respect it is interesting to note that these couplings agree considerably better with the ensemble structure of Clore and Schwieters (2006), where the rmsd between observed and predicted D_HNHα RDCs drops from 2.04 Hz for a static structure to 1.28 Hz for the ensemble. The ensemble representation also accounts for one half of the stronger rescaling observed for sequential RDCs compared to intraresidue interactions. On the other hand, a molecular dynamics trajectory does not provide any indication of systematically lower order parameters for sequential H^α-H^N interactions compared to intraresidue H^N-H^α vectors (Supplementary Material). Conceivably, the motions that cause a slight lowering in the sequential H^α-H^N interactions take place on a time scale longer than the 5-ns duration of the dynamics trajectory. Alternatively, it cannot be completely excluded that small systematic errors in either type of D_HNHα measurement are at the origin of the difference in the average order parameters.

In principle, our measurement of D_HNHα couplings presents a unique opportunity to evaluate N-H and C-H bond lengths without being significantly affected by high frequency librations about their average orientations. Although such librations may scale the magnitude of the ¹H-¹H RDC, albeit by a smaller factor than for the shorter one-bond N-H and C-H RDCs, they do not result in a systematic change in orientation of the ¹H-¹H vectors relative to the alignment frame. Librations therefore cause small perturbations to the correlations shown in Figures 4 and 5, equivalent to adding structural noise (Zweckstetter and Bax, 2002). Unfortunately, the scatter in the correlations of Figures 4 and 5 is so large that changing the N-H and C^α-H^α bond lengths does not lead to a statistically significant increase in χ² until these bond lengths fall well outside the canonical range.