On the Measurement of ¹⁵N-{¹H} Nuclear Overhauser Effects

Abstract

Accurate quantification of the ¹⁵N-{¹H} steady-state NOE is central to current methods for the elucidation of protein backbone dynamics on the fast, sub-nanosecond timescale. This experiment is highly susceptible to systematic errors arising from multiple sources. The nature of these errors and their effects on the determined NOE ratio is evaluated by a detailed analysis of the spin dynamics during the pair of experiments used to measure this ratio and possible improvements suggested. The experiment that includes ¹H irradiation, is analyzed in the framework of Average Liouvillian Theory and a modified saturation scheme that generates a stable steady state and eliminates the need to completely saturate ¹H nuclei, is presented. The largest source of error, however, in ¹H-dilute systems at ultra-high fields is found to be an overestimation of the steady-state NOE value as a consequence of the incomplete equilibration of the magnetization in the so-called “reference experiment”. The use of very long relaxation delays is usually an effective, but time consuming, solution. Here, we introduce an alternative reference experiment, designed for larger, deuterated systems, that uses the fastest relaxing component of the longitudinal magnetization a closer approximation to the equilibrium state for shorter relaxation delays. The utility of the modified approach is illustrated through simulations on realistic spin-systems over a wide range of timescales and experimentally verified using a perdeuterated sample of human ubiquitin.

1. Introduction

The ability to characterize protein dynamics at atomic resolution is key to understand biological processes, such as enzymatic reactions, protein folding, the driving forces of interactions and allostery. Nuclear Magnetic Resonance (NMR) is a powerful tool to study protein dynamics over a wide range of timescales [1,2]. Hydrodynamic properties of proteins in solution and dynamics occurring faster than a few nanoseconds are usually measured by the quantitative analysis of ¹⁵N relaxation rates [2–5]. These rates depend both on the overall rotational diffusion in solution as well as fast backbone motions on a sub-nanosecond timescale through their effects on the spectral density functions [6–9]. Longitudinal and transverse relaxation rates largely sample the spectral density at low (¹⁵N Larmor frequency) and zero-frequency [6,7]. The accurate determination of the spectral density function at high frequencies (around the ¹H Larmor frequency) relies on the accurate measurement of ¹⁵N-{¹H} nuclear Overhauser effects (NOE)[10].

Several studies have underlined the effect on the measured value of the steady-state ¹⁵N-{¹H} NOE of incomplete relaxation to the steady state, or equilibrium, ¹⁵N polarization in the pair of experiments with and without (reference) ¹H saturation [11–13]. Ways to accelerate the evolution towards the equilibrium polarization have been found [11] and correction factors introduced [14]. However, Gong and Ishima [12] have recently shown that, on high-field spectrometers and particularly using highly deuterated proteins, cross-relaxation pathways due to the cross-correlation of chemical shift anisotropy (CSA) and dipole-dipole (DD) interactions could significantly slow down the recovery of equilibrium polarization. A simple way to overcome this problem is to introduce a longer relaxation delay in the reference experiment. This solution is simple to implement and is recommended for small, deuterated proteins. However, it is extremely time-consuming since durations above 10 s should be employed.

In this article, we introduce two modifications to commonly employed pulse sequences to reduce potential artifacts. These modifications are based on our theoretical treatment of the experiments. We first present a theoretical framework for the detailed analysis of spin-dynamics during the course of the steady-state ¹⁵N-{¹H} NOE experiment using a realistic spin-system. Then, we analyze the saturated experiment in the framework of Average Liouvillian Theory [15,16]. We evaluate the effects of the sequence of pulses and delays in the saturation block and provide suggestions to reduce sources of errors in the measured steady-state NOE value. We examine the non-saturated experiment next and introduce a simple and effective modification to commonly employed sequences which accelerate recovery towards equilibrium. The use of this sequence is recommended for larger, deuterated proteins. The improvements in accuracy resulting from the new sequences set (saturated and reference) are evaluated on a sample of perdeuterated human ubiquitin at high fields (16.5 T and 21.1 T).

2. Theory

2.1 Definitions

For cases where the polarization does not evolve towards an effective zero steady-state [17], the homogeneous Master equation (HME) [15,16,18] provides the most convenient theoretical framework to describe longitudinal relaxation. For instance, Di Bari and Levitt [15] have shown that it was possible to create several kinds of steady-state terms under multiple-pulse experiments. These results are described elegantly and efficiently in the framework of Average Liouvillian Theory [15,16]. Using the notation of Ghose [16], one can write the HME:

$$\frac{\text{d}\sigma (t)}{\text{d}t}=-\widehat{L}(t)\sigma (t),$$ (1)

where σ is the density operator and L̂(t) the Liouvillian superoperator, defined as:

$$\widehat{L}(t)=i\widehat{H}(t)+\widehat{\mathrm{\Gamma }}(t)+\widehat{\mathrm{\Theta }}(t).$$ (2)

Ĥ(t) is the Hamiltonian superoperator that contains all coherent terms, Γ̂(t) is the relaxation superoperator that accounts for the non-coherent evolution towards the steady state, and Θ̂(t) is the thermal correction superoperator. The steady state represents the state of thermal equilibrium in the absence of any significant perturbations to the Zeeman interactions.

In the present context, we treat coherent evolutions under radiofrequency (RF) fields as perfect instantaneous rotations, allowing the Liouvillian superoperators to be simplified to include only relaxation and thermal correction terms. All the relevant spin dynamics can then be analyzed using a simple 4-dimensional Liouville space for a two-spin system in the basis {E/2, N_z, H_z, 2N_zH_z}, where E is the identity operator. The Liouvillian superoperator, for an amide ¹H, ¹⁵N spin pair can be written as:

$$\widehat{L}=\left(\begin{array}{cccc}0& 0& 0& 0\\ {\theta }_N& {\rho }_N& {\sigma }_{NH}& {\delta }_N\\ {\theta }_H& {\sigma }_{NH}& {\rho }_H& {\delta }_H\\ {\theta }_{NH}& {\delta }_N& {\delta }_H& {\rho }_{NH}\end{array}\right)$$ (3)

where ρ_N is the longitudinal relaxation rate of the ¹⁵N nucleus, ρ_H is the longitudinal relaxation rate of ¹H, ρ_NH is the auto-relaxation rate of the two-spin order, σ_NH is the auto-correlated dipolar cross-relaxation rate, δ_N (or δ_H) is the longitudinal cross-relaxation rate due to the cross-correlation of the ¹⁵N (or ¹H) chemical shift anisotropy (CSA) and the ¹⁵N-¹H dipole-dipole (DD) interaction. If one normalizes [19] the equilibrium ¹⁵N polarization to be 1, thermal correction terms can be written as:

$$\begin{array}{l}{\theta }_N={\rho }_N+\frac{{\gamma }_H}{{\gamma }_N}{\sigma }_{NH}\\ {\theta }_H={\sigma }_{NH}+\frac{{\gamma }_H}{{\gamma }_N}{\rho }_H\\ {\theta }_{NH}={\delta }_N+\frac{{\gamma }_H}{{\gamma }_N}{\delta }_H\end{array}$$ (4)

2.2 Symmetry of the saturation block

The experimental scheme often employed for ¹H saturation while observing the steady state on ¹⁵N is depicted in Figure 1a. The pulse applied on ¹H is often a 2π/3 rotation, however, we will consider π rotations in this discussion for simplicity and this does leads to no loss of generality. An alternative symmetric saturation scheme is presented in Figure 1c. The scheme of Figure 1b is a formally identical expansion of the scheme of Figure 1a, and the schemes of Figure 1d and 1c are similarly identical. The average Liouvillian that describes the evolution of the density operator during the bracketed element of Figures 1b and 1d is given by:

$${\widehat{L}}_{av}={\widehat{L}}_{av}^{(0)}+{\widehat{L}}_{av}^{(2)}+\dots$$ (5)

with:

$${\widehat{L}}_{av}^{(0)}=\frac{1}{2}\left(\widehat{L}+\widehat{R}\widehat{L}{\widehat{R}}^{-1}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ {\theta }_N& {\rho }_N& 0& 0\\ 0& 0& {\rho }_H& {\delta }_H\\ 0& 0& {\delta }_H& {\rho }_{NH}\end{array}\right)$$ (6)

where R̂ represents the inversion due to a π ¹H pulse:

$$\widehat{R}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$$ (7)

Figure 1.

Schematic representation of ¹H saturation elements to measure the steady-state polarization on the ¹⁵N nucleus. a–b: Two forms of a typical saturation sequence when π pulses are used on the ¹H. c–d: Two forms of a symmetric saturation element.

Note that all odd-order terms in the average Liouvillian vanish, because the bracketed sequences (Figures 1b and 1d) are symmetric. The odd terms subspace, consisting of operators inverted by π ¹H pulses (i.e. H_z and 2N_zH_z) is decoupled from the even subspace consisting of operators invariant under ¹H pulses (i.e. E/2 and N_z). ${\widehat{L}}_{av}^{(0)}$ leads to a monoexponential evolution of ¹⁵N longitudinal polarization with a rate ρ_N towards the well-known steady-state polarization $\langle N_z^{ss}\rangle$:

$$\frac{\langle N_z^{ss}\rangle }{\langle N_z^{eq}\rangle }=1+\frac{{\gamma }_H}{{\gamma }_N}\frac{{\sigma }_{NH}}{{\rho }_N}$$ (8)

where $\langle N_z^{eq}\rangle$ is the equilibrium polarization of the ¹⁵N nucleus.

All higher order terms in the average Liouvillian may be neglected when the inter-pulse delay 2τ is short. The steady state described by Equation 8 will be reached after an integer number of saturation elements. One can notice that a delay τ after the last multi-pulse element, in Figure 1b, corresponding to one half of the inter-pulse delay is therefore not required and leads to an evolution of ¹⁵N polarization away from the steady-state towards equilibrium. This effect is often negligible when the steady-state and equilibrium polarizations are comparable and typical values of τ that range from 2.5 ms to 10 ms are used. However, systematic errors up to a few percent may be found for mobile residues with negative NOE ratios (Eq. 8 i.e when $\mid \frac{{\gamma }_H}{{\gamma }_N}\frac{{\sigma }_{NH}}{{\rho }_N}\mid >1$) and typical longitudinal relaxation rates of the order of 1 s⁻¹. It is therefore preferable that a symmetric saturation element like Fig 1.c be used, since no additional saturation period is required (in practical terms, an initial recovery delay is always employed to lock the magnetic field allowing the first delay τ to be deleted) but may suppress small systematic errors.

2.3. Is saturation of ¹H polarization necessary?

It is quite remarkable that the degree of saturation of the ¹H polarization does not appear in Equations 5–8. In fact, in an extreme case with no auto-relaxation of ¹H nuclei and perfect π pulses (i.e. no saturation), the system evolves exactly towards the steady-state polarization. Indeed, during every element of Figures 1c–d, a ¹⁵N nucleus sees its neighboring ¹H nucleus an equal amount of time as spin-up and spin-down. Thus, on average, the ¹H polarization appears saturated to the ¹⁵N nucleus.

If the rate of pulsing on ¹H is fast, the evolution of the ¹H polarization during the inter-pulse (2τ) delay is very small. Therefore, during each saturation element, the ¹⁵N nucleus sees also, on average, a saturated ¹H nucleus. There is thus no requirement for the ¹H polarization to be saturated. In all simulations displayed in Fig. 2, the residual polarization of ¹H is still larger than 5% of the equilibrium polarization after 5 seconds. Nevertheless, the ¹⁵N polarization is identically equal to the expected steady-state polarization when the inter-pulse delay (2τ) is short.

Figure 2.

Evolution of the ¹⁵N longitudinal polarization towards the steady-state under the application of the saturation sequence of Figure 1c with different values for the delay τ. For a rigid system (a), the Lipari-Szabo [20] form of the spectral density function was used with an order parameter S² equal to 0.85 and the correlation time for local motions is 50 ps. For a mobile residue (b), the order parameter is 0.5 and the local correlation time is 200 ps. The inserts show the details of the last 500 ms of this evolution. As in the experimental situations, the system is allowed to relax from a fully saturated state for two seconds before the beginning of the saturation scheme. A small spherical deuterated protein was taken as an example: a single global rotation correlation time of 4.5 ns was used and the ¹⁵N-¹H pair was supplemented with two ¹H nuclei located at 290 pm of the ¹⁵N bound ¹H. The relaxation superoperator used for this calculation is displayed in the Appendix.

When the ¹H polarization is allowed to relax significantly towards equilibrium during the inter-pulse delay, the situation is more complex. On average, the polarization of ¹H is different from zero and the ¹⁵N polarization does not evolve towards the “true” steady-state. Figure 2 illustrates this property. For both a mobile NH pair and a rigid one, a delay τ as long as 50 ms is only marginally detrimental to the accuracy of the steady-state shown in Eq. 8. If one uses a symmetric saturation scheme, as depicted in the inserts in Figure 2, the result is perfectly accurate. In the case of very long inter-pulse delays, the polarization of the steady-state is different from the one expected from Eq. 8. In the latter case, the ¹H polarizations are closer to equilibrium before each pulse on ¹H, so that, after the π pulse, the nitrogen polarization is driven towards −〈N_z〉^eq. After half an interpulse delay, the system is still biased towards this state so that if the NOE ratio (Eq. 8) is larger than −1, the nitrogen longitudinal polarization expectation value is smaller than the steady state (Fig. 2). If the NOE ratio is smaller than −1, the expectation value is larger than the steady-state polarization (data not shown). One should notice that, under the conditions of the simulations, the system is significantly closer to the expected steady state after half an inter-pulse delay, validating the use of a symmetric saturation scheme.

Our simulations clearly show that the saturation of ¹H polarization is not necessary if one chooses the proper symmetry properties for the saturation element. However, when one uses an asymmetric saturation scheme, the saturation of the ¹H polarization may become necessary. Indeed, if the saturation is complete, the expectation value of operators inverted by the pulses on ¹H is null: the longitudinal polarization of ¹H and the two-spin order polarization are zero. Therefore:

$${\widehat{R}}^{-1}{\sigma }^{ss}={\sigma }^{ss},$$ (9)

where σ^ss is the density operator at the steady-state. From Equations 6 and 9, it follows that:

$${\overline{N}}_z^{T}L{\sigma }^{ss}={\overline{N}}_z^T{L}_{av}^{(0)}{\sigma }^{ss}=0.$$ (10)

where ${\overline{N}}_z^T$ is the conjugate transpose of N_z. From Eqs. (1) and (10), one obtains:

$${\frac{\text{d}\langle N_z\rangle (t)}{\text{d}t}\mid }_{t=0}=0.$$ (11)

The initial slope for the evolution of the longitudinal polarization of the ¹⁵N nucleus is zero.

In the case where the ¹H polarization is fully saturated, i.e. the ¹H polarization is zero, schemes presented in Figures 1a and 1c will provide identical expectation values for the longitudinal polarization of the ¹⁵N nucleus unless the delay τ is much longer than values typically employed in experiments (5 to 10 ms). The outcome of these two schemes will only be different in fast-relaxing systems (i.e. when τ is comparable to the relaxation rates) or when the ¹H and two-spin order polarizations are significantly different from zero.

Nevertheless, complete saturation of the polarization of ¹H may be difficult to achieve [13], particularly in cases where cross-relaxation or exchange with the slow-relaxing ¹H nuclei from the solvent is efficient. By reducing the need for saturation of the ¹H nuclei, the use of a symmetric saturation scheme, which comes at no cost, reduces one source of systematic errors. Even though the benefits may be limited is most cases, its use is recommended.

2.4. Relaxation in the reference experiment

As discussed in section 2.2, the effective relaxation of the ¹⁵N longitudinal polarization during the saturation scheme is a mono-exponential evolution towards the steady state. Relaxation during the reference experiment is much more complex.

As shown in the Redfield theory of relaxation [8], the secular approximation suppresses many of the possible cross-relaxation pathways. All longitudinal operators commute with the Zeeman and scalar coupling Hamiltonian operators. Therefore, in the absence of radio-frequency fields, all possible cross-relaxation pathways are active in the relevant Liouville subspace. The consequence is that relaxation towards equilibrium is potentially highly multi-exponential. As shown by Gong and Ishima [12], when small, highly deuterated proteins are under study at high fields, dipole-dipole/CSA cross-correlated cross-relaxation [21] is sufficient to lead to a very slow evolution towards equilibrium for the ¹⁵N longitudinal polarization. In larger non-deuterated proteins, the lifetime of the two-spin order 2H_zN_z is short, which significantly reduces this effect.

Nevertheless, deuterated proteins are excellent tools for relaxation studies, e.g. when basic ¹⁵N relaxation experiments are supplemented by experiments where the ¹H polarization participates in the coherence being followed. For example, the characterization of chemical exchange parameters is significantly improved by the implementation of CPMG (Carr Purcell Meiboom Gill) schemes on amide ¹H [22,23], and the measurement of cross-relaxation under the cross-correlation of dipolar and CSA interactions to complement typical ¹⁵N relaxation experiments allows an accurate determination of relaxation under chemical exchange [17,24]. The accuracy of the quantitative analyses of these experiments benefit strongly from deuteration. Thus in addition to large proteins, where the benefits of deuteration have been discussed at length elsewhere [25] and its application to relaxation studies demonstrated [26–28], deuteration of small proteins is also greatly beneficial especially when detailed, quantitative analyses are required to interpret data from relaxation experiments.

An accurate method to measure ¹⁵N-{¹H} nuclear Overhauser effects adapted for deuterated proteins at high fields is, to the best of our knowledge, not available in the literature. A difficulty here is that manipulations of the spin system designed to alter cross-relaxation pathways would make the effective final state of the system different from the equilibrium state. In the absence of any manipulation of the spin system, it is still possible to accelerate the evolution towards equilibrium by an appropriate choice of the operator being followed. Indeed, the closer this operator is to the fastest relaxing relaxation modes of the density operator, the more efficient is its evolution towards equilibrium.

The development of spin-state-selective experiments [29–32] has made the manipulation of single-transition operators very convenient. Figure 3 compares the projections of three longitudinal operators that can be readily converted to observables: the longitudinal polarization of the ¹⁵N nucleus and the two single-transition longitudinal operators. The spin system employed for this calculation comprises additional protons (see Appendix), so that there are eight eigenvalues and not four, as expected from Eq. 3. It is quite remarkable that, for a rigid NH pair in a small protein (overall correlation time 5 ns) with a magnetic field of 21.1 T, the two single-transition operators are very close to either the slowest relaxing mode or the fastest relaxing mode of the spin-system, as can be seen in Fig. 5a. This is because the dominant contributions to relaxation rates are the ¹⁵N-¹H dipole-dipole and ¹⁵N CSA, while the static field is close to its optimal value for relaxation interference [33]. This makes of the fast-relaxing single transition operator 2N_zH^β the ideal operator to be followed in an experiment designed to reduce the recovery delay. In this study, we will refer to 2N_zH^β as the fast-relaxing single-transition. This is consistent with the definitions of relaxation rates provided in the Appendix but may be in contradiction with part of the literature on the subject.

Figure 3.

Projection of the accessible observables 2H^βN_z 2H^αN_z and N_z on the eigenvectors of the Liouvillian superoperator that governs the evolution towards equilibrium. Eigenvalues are reported on the x-axis while the projections of the observables on the corresponding eigenvectors are reported on the z-axis. (a,b) For a rigid group with the same parameters as those used in Figure 2. (c,d) For a rigid group in a small mobile domain with the same parameters for local motions and a domain motion characterized by a correlation time equal to 10% of the overall tumbling time and an order parameter for slow motions of 0.3. The overall tumbling correlation time employed is (a,c) 5 ns and (b,d) 30 ns.

Figure 5.

Relaxation rates calculated for the simulations displayed in Figure 4. See the appendix for detailed expressions of the rates.

In a rigid NH pair in a large protein (Fig. 3b), however, the longitudinal polarization of the ¹⁵N nucleus is almost equal to the eigenvector with the lowest non-zero eigenvalue. The projections on fast-relaxing modes are only slightly larger for the 2N_zH^β operator. In this case, ¹H-¹H cross-relaxation is the dominant mechanism for relaxation. One consequence is the effective decoupling of CSA/DD cross-relaxation pathways. If semi-local motions, such as small domain motions, occur on an intermediate time-scale (this assumes that the fast sub-nanosecond motions occur on two different timescales that differ by at least one order of magnitude), the projection of 2N_zH^β on fast-relaxing modes increases significantly (Fig. 3d), so that this operator is, again, the most relevant “observable”. Overall, Figure 3 shows that this operator should relax faster than the other accessible operators.

To confirm this trend, we have simulated the evolution towards equilibrium of all three operators under three classes of local dynamics and over a wide range of overall tumbling times. Results are displayed in Figure 4 while the calculated rates are plotted in Figure 5. As illustrated by the bold line in all simulations, the same single-transition is the closest to equilibrium, under all simulated circumstances and all times. In small proteins, the advantage is dramatic. In larger systems, the difference between the relaxation properties of the three operators for rigid NH pairs (as illustrated by Fig 3b) and very mobile NH pairs becomes smaller, as expected from Fig. 3. Figure 5 illustrates the dominance of ¹H-¹H dipolar cross-relaxation in this regime. However, in the presence of small-domain motions the quantification of the operator 2N_zH^β, provides a much closer estimate of the equilibrium polarization because it relaxes at a significantly faster rate than the other two operators over the entire range of overall tumbling times explored.

Figure 4.

Evolution of ¹⁵N longitudinal polarization operators towards equilibrium from a fully saturated state. The bold contour lines correspond to the expectation value <2H^βN_z>. The thin line that carries the labels corresponds to the expectation value <N_z> while the second thin line represents the expectation value <2H^αN_z>. The parameters for local dynamics are displayed on each figure. They correspond to (a) a rigid NH pair, (b) a mobile NH pair and (c) a rigid NH pair in a mobile domain. Contour lines are plotted for a polarization equal to 70% (blue); 90% (cyan); and 99% (green) of its equilibrium value.

For example, in a large deuterated protein with an overall tumbling time of 20 ns, for which the use of this pulse sequence is recommended, we predict that, in a rigid system (S² = 0.85 and τ_loc = 50 ps) studied on a 900 MHz spectrometer, 98 % of the equilibrium polarization is recovered after 14 seconds for the fast-relaxing single transition. For the same duration, 97.1 % of the equilibrium longitudinal ¹⁵N polarization is recovered, while 98 % would be obtained after 15.25 seconds. The difference for this system is not spectacular, but the reduction of 8 % of the relaxation delay should be welcome.

Figure 4 shows that, even if the advantage for the selection of one single-transition operator may be limited for specific cases, selection of this operator always leads to the closest estimate of equilibrium. Therefore, it should always be advisable to measure the polarization of the fast-relaxing longitudinal single-transition of the ¹⁵N nucleus as a standard for equilibrium in the experiment without ¹H saturation when measuring the steady-state ¹⁵N-{¹H} NOE with TROSY-based sequences.

3. Material and Methods

All simulations were performed using MATLAB (www.mathworks.com). The programs can be provided upon request and only a few major features are discussed here. The spin system considered comprised one ¹⁵N nucleus and three ¹H nuclei. The distance between the ¹⁵N nucleus and its bound ¹H nucleus is 102 pm. The distances between this ¹H nucleus and its two neighbors are both 290 pm [30]). The ¹H-¹H cross-relaxation rate was added to the auto-relaxation rate of one of the two neighboring ¹H nuclei to take into account the effect of cross-relaxation or exchange with ¹H from the solvent. Values for the CSA of ¹H and ¹⁵N nuclei are the average values provided by Loth et al. in their study of human ubiquitin [35]. The model-free form of the spectral density function was used to describe rigid and mobile residues [20]. The extended model-free form was employed to take into account small domain motions [36]. The spectral density function was adapted to include cross-correlated relaxation as in Loth et al. [35].

We have designed a new reference experiment, shown in Figure 6, where the element that suppresses all polarizations involving the ¹H nucleus [26,37] (the element between points A^sat and B^sat in the saturated experiment) is substituted by a conversion between the fast- and slow-relaxing components of the polarization. At point A^ref, after a relaxation delay with a total duration T_lock + T, the expectation value <2N_zH^β> is larger than <2N_zH^β>. A composite π pulse on the on the ¹H channel interconverts these two operators, so that, at point B^ref in the sequence, the expectation value for 2N_zH^β is closer to equilibrium. The latter operator is subsequently detected in a TROSY detection: a spin-state-selective transfer follows the ¹⁵N frequency labeling for the detection of a TROSY-like spectrum [31]. The conversion between the polarizations is achieved by a composite π pulse on the ¹H channel to reduce effects of RF-field inhomogeneities flanked by two π/2 very selective pulses to ensure an efficient restoration of the water magnetization to the +z axis.

Figure 6.

Pulse sequences used for recording steady-state ¹⁵N-{¹H} nuclear Overhauser effects. In all experiments, a GARP [34] decoupling scheme was initially applied on the ¹⁵N channel for 60 ms followed by a 2 second recovery delay (T_lock) used to lock the magnetic field. Saturated and reference experiments were run in an interleaved manner with the boxed schemes (a) and (b) respectively. (a) A train of 2π/3 pulses was applied on the ¹H channel for 4s with a delay τ_s equal to 5 ms. (b) A recovery delay T is followed by a composite pulse flanked by two water flip back pulses to invert all ¹H polarizations but water ¹H. The total recovery delay is T_lock+T. For comparison with the initial scheme for the reference experiment, bracketed saturation elements were replaced by the delay T. A classic TROSY detection follows for all experiments. Except in the saturation element, all narrow (filled) and (wide) open rectangles represent π/2 and π pulses respectively. Pulse phases are along the x-axis of the rotating frame unless otherwise mentioned. Gaussian shaped pulses are 2 ms and have a Gaussian shape truncated at 1% of the maximal amplitude. A longer duration of 3 ms is recommended if greater selectivity is required. Bell-shaped pulses are sinebell shaped pulses, of 1 ms. The delay τ is 1.792 ms. The phase cycles are φ₁ = {y;−y; −x;x} used with φ₂ = {y} and φ₃ = {x}; and φ₁ = {−y;y; −x;x} used with φ₂ = {−y} and φ₃ = {−x} alternated for every other free induction decay. The phase cycle for acquisition is φ_acq = {x; −x; −y;y}. The duration of acquisition is 120 ms. For the indirect dimension, 75 complex points were recorded with a spectral width of 34 ppm.

All experiments were performed on Bruker Avance NMR spectrometers operating at ¹H Larmor frequencies of 500, 700 and 900 MHz equipped with TXI cryoprobes with z-axis gradients. A sample of perdeuterated, uniformly ¹⁵N-labelled ubiquitin (0.5 mM, in 50 mM ammonium acetate, 300 mM NaCl, pH 4.8) was used for all experiments. R₁, R₂ and ¹⁵N-{¹H} NOE values (using the experiment presented here with a T_lock+T value of 8 seconds, 16 scans per transient) were obtained at 500 MHz. The following relaxation delays were used for the R₁ and R₂ experiments: R₁: 5 ms (x2), 20 ms, 100 ms (x2), 300 ms and 600 ms; R₂: 1 ms (x2), 50 ms, 100 ms (x2), 150 ms, 250 ms. A relaxation delay of 2 seconds was used between scans in the R₁ and R₂ experiments and 8 scans were collected per transient. All experiments were performed at 25° C. Analysis of the relaxation rates was performed in the usual way using the program DIFFTENS v2.0 [38] using an isotropic diffusion model and yielded a rotational correlation time of 4.48 ± 0.05 ns.

R₁, R₂ and ¹⁵N-{¹H} NOE ratio were simulated using standard expressions [36] for a ¹⁵N-¹H two-spin system using standard parameters for several motional models described in the legend to Figures 9 and 10. The “experimental” precisions of the R₁, R₂ and ¹⁵N-{¹H} NOE ratio were assumed to be 2.5 %, 2.5 % and 5 % respectively. The microdynamic parameters were obtained assuming an isotropic overall motion and fitting to the 8 different motional models described in detail elsewhere using the DYNAMICS package[39].

Figure 9.

Influence of overestimation of the ¹⁵N-{¹H} NOE ratio on derived Lipari-Szabo micro-dynamic parameters. Parameters were obtained from theoretical R₁, R₂ and ¹⁵N-{¹H} NOE ratio at 700 MHz assuming an overall correlation time of 4.5 ns, S²=0.85 and τ_loc=50 ps (a,c,e) and S²=0.50 and τ_loc=200 ps (b,d,f). “Experimental” errors of 2.5 % for the R₁ and R₂ rates and 5 % for the ¹⁵N-{¹H} NOE were assumed. Filled circles represent cases where the correct (input) model (S², τ_loc) was selected based on the relaxation rates using a statistical F-test while open circles represent cases where the incorrect model (S² only) was chosen. The dashed lines (panels a–d) represent the correct motional parameters.

Figure 10.

Influence of overestimation of the ¹⁵N-{¹H} NOE ratio on Lipari-Szabo micro-dynamic parameters for a small rigid domain in a large protein. Parameters were obtained from theoretical R₁, R₂ and ¹⁵N-{¹H} NOE ratio at 700 MHz and 900 MHz (analyzed simultaneously) assuming an overall correlation time of 30 ns, $S_f^2=0.85,S_s^2=0.3,S^2=S_f^2{S}_s^2=0.255$, τs =3 and τ_f = 50 ps. “Experimental” errors of 2.5 % for the R₁ and R₂ rates and 5 % for the ¹⁵N-{¹H} NOE were assumed. Black circles represent the case where the NOE ratio at 700 MHz was overestimated and the correct value was used at 900 MHz, the reverse case is represented by gray circles. Filled circles represent cases where the correct model (S², $S_f^2$, τ_s, τ_f) was selected based on the relaxation rates using a statistical F-test while open circles represent cases where the incorrect model (S², $S_f^2$, τ_s only) was chosen. The dashed lines represent the correct motional parameters. The black concentric circles in (e) represent cases for which none of the possible dynamic models passed the goodness-of-fit test based on the corresponding χ² distribution. The χ² values for the most physical model in these cases have been displayed for illustration but the microdynamic parameters (which are erroneous) have not.

4. Results and Discussion

4.1. Quantification of the equilibrium polarization in the reference experiment

The scheme of Zhu et al. [37] as modified by Chill et al. [26] was compared to our present scheme, with the new reference experiment. Data obtained on a spectrometer with a 700 MHz Larmor frequency for ¹H are displayed in Fig. 7. As can be seen from Fig. 7a, after 6 seconds of total recovery time neither the initial scheme nor the new one provide with an exact measurement of the polarization at equilibrium. However, the new scheme gives a closer estimate of the equilibrium polarization. For most NH pairs, the polarization is larger than 95% of the equilibrium polarization, assuming that the polarization is at equilibrium after a period of 12 seconds. For the initial scheme, many residues are still below 90% of the equilibrium polarization. Fig. 7b shows the NOE ratios derived from the two experimental approaches. As expected, ours provides smaller absolute values for the NOE ratio. This translates into larger values for the cross-relaxation rate σ_NH when the NOE ratio is positive and smaller values of σ_NH when the NOE ratio is negative (Figs. 7c,d).

Figure 7.

Comparison of the initial and new schemes for the reference experiment at 16.4 T. (a) Ratio of intensities after a total of 6 s and 12 s of recovery. The delay is T_lock + T. T_lock is 2 s in all experiments. (b–d) The durations of the delay T and of the saturation schemes are both 4 s. (b) NOE ratio. (c) ratio of the dipolar cross-relaxation rates and longitudinal auto-relaxation rate of the ¹⁵N nuclei derived from the NOE ratio of (b) as from Eq. 8. (d) Same as (c) with an expansion along the y-axis.

Since for most residues the NOE ratio is close to one, according to Eq. 8, a small error in this ratio may translate into large relative errors in the value of σ_NH: for several residues, this cross-relaxation rate can be underestimated by a factor of 2 (Fig. 7d). For very mobile residues that display a negative NOE ratio, the error is marginally attenuated, which explains why the dramatic error in the estimate of the polarization at equilibrium for the NH pair of residue 76 translates in a comparable error for σ_NH. The NOE ratios displayed in Fig. 7b for residue 36 deserve comment. These large values are due to the underestimation of the intensity in all reference experiments. The ¹H signal of residue 36 is the closest to the water resonance, so that the ¹H polarization is affected by 2 ms Gaussian water flip-back pulses. Therefore, in the original scheme, the polarization of the two-spin order is not properly suppressed, so that the slowly relaxing component has a major contribution to the detected signal. In our present scheme, the inversion of the polarization of the two-spin order is also not perfect, so that both single-transition polarizations are partially mixed. With the use of the same selective water flip-back pulses; the scheme introduced in this article is significantly more robust to this artifact than the original scheme.

Figure 8 presents experimental results obtained on a 900 MHz (¹H Larmor frequency) spectrometer. This high magnetic field provides a close to maximum interference effect between the ¹⁵N CSA and ¹⁵N-¹H dipolar interactions [33]. On average, our scheme provides a significantly larger intensity (i.e. closer to equilibrium) after a recovery delay as long as 8 seconds. The results depend significantly on the NH pair observed. Gly76 shows the most spectacular improvement, which can be linked to a high degree of flexibility. However, the preceding residues in the flexible C-terminal tail show very similar intensities in the two experiments, in spite of a large mobility.

Figure 8.

Comparison of intensities of the experiment of Zhu et. al.[37] as modified by Chill et al. [26] and the new scheme. Error bars are smaller than the symbols. Experiments were recorded on a 900 MHz spectrometer with a total relaxation delay of 8 s. Results of a linear regression are shown by the dashed line.

At both fields, the scheme introduced in this paper provides a better estimate of the equilibrium polarization for almost all residues. The number of residues for which the initial scheme provides a higher intensity is small and the extent of the difference is generally negligible. According to our simulations (Fig. 4) this tendency is verified under all dynamic regimes and all molecular masses explored. It is therefore recommended to employ the new TROSY-based scheme presented in Fig. 6, with more selective water flip-back pulses (for instance 3 ms Gaussian-shaped pulses truncated at one percent of the maximum amplitude at high fields).

Depending on the acquisition mode, the implementation of TROSY may lead to a loss of signal of a factor of 2 to $2\sqrt{2}$ if relaxation is negligible. ¹⁵N-{¹H} NOE experiments are often signal-limited so that the use of TROSY for small proteins may not be the most appropriate method. It would indeed be more useful to employ twice longer relaxation recovery delays in the reference experiment with twice less scans and a conventional detection. That would lead to a good level of accuracy and more precision. On the other hand, if TROSY-based methods are to be used, which is the case for studies of large proteins or protein assemblies, the scheme discussed here should be favored.

4.2. Effect of the overestimation of the NOE on local dynamics parameters

The effect of the overestimation of ¹⁵N-{¹H} NOE can lead to significant errors in the microdynamic parameters determined from an analysis of R₁, R₂ and ¹⁵N-{¹H} NOE using the Lipari-Szabo or the extended Lipari-Szabo models. As seen in Figure 9, for simple models, the overestimation has a profound effect on a rigid system with errors greater than 60 % for the τ_e value even when the NOE value is overestimated by a mere 9 % (Figure 9(c)). When the overestimation is greater than 9 %, the timescale of the local motion can no longer be obtained and the incorrect motional model is selected based on a statistical F-test with very large increases in the χ² value (Figure 9(f)). The effect is somewhat smaller in more dynamic systems. The transition point between different motional models being selected would necessarily depend on the software used to perform the Lipari-Szabo analysis and the particular algorithms utilized therein, however, the same general trends should be expected. The estimated S² values are less affected by the NOE overestimation with the errors being approximately 2 % in both the rigid and dynamic cases. Additionally, when the NOE is overestimated, and the degree of overestimation is much larger than the experimental precision (i. e. the random error in the experimental data), the complete set of relaxation data ceases to be self-consistent and large errors are obtained in the fitted microdynamic parameters for the same level of experimental precision. The magnitude of these errors become larger when more complex motional models are considered (Figure 10) even though the average values of the fitted microdynamic parameters do not deviate significantly from the correct values. In extreme situations, for example, Figure 10(a–d), when the error in the NOE ratio exceeds 15 % at 700 MHz (the correct value is used at 900 MHz) for an extended Lipari-Szabo model, the goodness-of-fit criterion is no longer satisfied and no motional model is found to be appropriate for the set of relaxation data and no microdynamic parameters can be obtained.

5. Conclusion

We have introduced a new scheme for the measurement of ¹⁵N-{¹H} NOE’s. We have shown that this experiment provides a better estimate of the equilibrium polarization than previous approaches even for long recovery delays (8 seconds at 900 MHz). We show that the use of this new scheme when TROSY detection is likely to avoid systematic errors. For small proteins, the use of long recovery delays for the reference experiment should provide better precision. Systematic errors arising from the underestimation of the ¹⁵N equilibrium polarization were shown to have variable impact on the parameters for local dynamics derived from a model-free approach. An improvement of the saturation schemes employed in the complementary experiment was introduced and evaluated through simulations. We are currently investigating in greater detail the nature of the saturation schemes utilized for ¹H saturation in the saturated experiment.

Appendix

In this section, we present the expression of the Liouvillian superoperators used here for the simulation of the recovery of the spin-system towards equilibrium. The spin system consists of one ¹⁵N-¹H pair supplemented by two neighboring ¹H nuclei. One of these two ¹H nuclei shows enhanced auto-relaxation rates to take into account cross-relaxation or exchange with solvent protons during the recovery.

We have neglected all DD/DD cross-correlation mechanisms, as well as all long-range DD/CSA cross-correlations. We have not taken into account any cross-relaxation mechanism between neighboring ¹H nuclei and their ¹⁵N partner. Under these conditions, we can restrict the basis of the subspace of the Liouville space to: {E/2, N_z, H_z,2N_zH_z, $H_z^I,2N_z{H}_z^I,H_z^{II},2N_z{H}_z^{II}$}. The Liouvillian superoperator is:

$$L_8=\left(\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ {\theta }_N& {\rho }_N& {\sigma }_{NH}& {\delta }_N& 0& 0& 0& 0\\ {\theta }_H& {\sigma }_{NH}& {\rho }_H& {\delta }_H& {\sigma }_{HH}& 0& {\sigma }_{HH}& 0\\ {\theta }_{NH}& {\delta }_N& {\delta }_H& {\rho }_{NH}& 0& {\sigma }_{HH}& 0& {\sigma }_{HH}\\ {\theta }_H^I& 0& {\sigma }_{HH}& 0& {\rho }_H^I& {\delta }_N& 0& 0\\ {\theta }_{NH}^I& 0& 0& {\sigma }_{HH}& {\delta }_N& {\rho }_{NH}^I& 0& 0\\ {\theta }_H^{II}& 0& {\sigma }_{HH}& 0& 0& 0& {\rho }_H^{II}& {\delta }_N\\ {\theta }_{NH}^{II}& 0& 0& {\sigma }_{HH}& 0& 0& {\delta }_N& {\rho }_{NH}^{II}\end{array}\right)$$ (A1)

where the expressions of the relaxation rates are:

$${\rho }_N=\frac{1}{10}{d}_{NH}^2(3J({\omega }_N)+J({\omega }_N-{\omega }_H)+6J({\omega }_N+{\omega }_H))+\frac{2}{15}{d}_N^{2}J({\omega }_N)$$ (A2a)

$$\begin{array}{l}{\rho }_H=\frac{1}{10}{d}_{NH}^2(J({\omega }_N-{\omega }_H)+3J({\omega }_H)+6J({\omega }_N+{\omega }_H))+\frac{2}{15}{(d_{Hx}+d_{Hy})}^{2}J({\omega }_H)\\ +\frac{2}{10}{d}_{HH}^2(J(0)+3J({\omega }_H)+6J(2{\omega }_H))\end{array}$$ (A2b)

$$\begin{array}{l}{\rho }_{NH}=\frac{3}{10}{d}_{NH}^2(J({\omega }_N)+J({\omega }_H))+\frac{2}{15}{d}_N^{2}J({\omega }_N)+\frac{2}{15}{(d_{Hx}+d_{Hy})}^{2}J({\omega }_H)\\ +\frac{2}{10}{d}_{HH}^2(J(0)+3J({\omega }_H)+6J(2{\omega }_H))\end{array}$$ (A2c)

$$\begin{array}{l}{\rho }_H^I=\frac{1}{10}{d}_{NH}^2(J({\omega }_N-{\omega }_H)+3J({\omega }_H)+6J({\omega }_N+{\omega }_H))+\frac{2}{15}{(d_{Hx}+d_{Hy})}^{2}J({\omega }_H)\\ +\frac{1}{10}{d}_{HH}^2(J(0)+3J({\omega }_H)+6J(2{\omega }_H))\end{array}$$ (A2d)

$$\begin{array}{l}{\rho }_{NH}^I=\frac{3}{10}{d}_{NH}^2(J({\omega }_N)+J({\omega }_H))+\frac{2}{15}{d}_N^{2}J({\omega }_N)+\frac{2}{15}{(d_{Hx}+d_{Hy})}^{2}J({\omega }_H)\\ +\frac{1}{10}{d}_{HH}^2(J(0)+3J({\omega }_H)+6J(2{\omega }_H))\end{array}$$ (A2e)

$${\rho }_H^{II}={\rho }_H$$ (A2f)

$${\rho }_{NH}^{II}={\rho }_{NH}$$ (A2g)

$${\sigma }_{NH}=\frac{1}{10}{d}_{HH}^2(6J({\omega }_H+{\omega }_N)-J({\omega }_H-{\omega }_N))$$ (A2h)

$${\sigma }_{HH}=\frac{1}{10}{d}_{HH}^2(6J(2{\omega }_H)-J(0))$$ (A2i)

$${\delta }_N=\frac{2}{5}{d}_{NH}{d}_N{J}_N({\omega }_N)$$ (A2j)

$${\delta }_H=\frac{1}{5}{d}_{NH}(d_{Hx}{J}_{Hx}({\omega }_H)+d_{Hy}{J}_{Hy}({\omega }_H))$$ (A2k)

with: $d_{NH}=({\mu }_0/4\pi )\hslash {\gamma }_N{\gamma }_H{r}_{NH}^{-3};d_{HH}=({\mu }_0/4\pi )\hslash {\gamma }_H^2{r}_{HH}^{-3}$; d_N = ω_NΔσ_N; d_Hx = ω_H(σ_Hxx − σ_Hzz); d_Hy = ω_H(σ_Hyy − σ_Hzz); ω_N = − γ_NB₀; ω_H = − γ_HB₀; μ₀ is the permittivity of free space; γ_N and γ_H are the gyromagnetic ratio of ¹⁵N and hydrogen nuclei; r_NH is the distance between the ¹⁵N nucleus and its neighboring ¹H nucleus; r_HH is the distance between two ¹H nuclei; Δσ_N is the chemical shift anisotropy of the ¹⁵N nucleus; σ_Hxx, σ_Hyy and σ_Hzz are the three principal components of the ¹H chemical shift tensor; B₀ is the static magnetic field.

The spectral density functions employed make the assumption of uncorrelated isotropic local and global motions. A model-free form was employed for the case of local motions on one time-scale.

$$J(\omega )=S^2\frac{{\tau }_c}{1+{\omega }^2{\tau }_c^2}+(1-S^2)\frac{{\tau }_e}{1+{\omega }^2{\tau }_e^2}$$ (A3)

where S² is the generalized order parameter, τ_c the correlation time for the overall tumbling time and τ_e is defined from the correlation time for local motions τ_loc by: ${\tau }_e^{-1}={\tau }_c^{-1}+{\tau }_{\mathit{loc}}^{-1}$. The extended model-free form was used to describe the case of the motion of a small domain in a large protein:

$$J(\omega )=S_s^2\frac{{\tau }_c}{1+{\omega }^2{\tau }_c^2}+(1-S_f^2)\frac{{\tau }_{ef}}{1+{\omega }^2{\tau }_{ef}^2}+(S_f^2-S_s^2)\frac{{\tau }_{es}}{1+{\omega }^2{\tau }_{es}^2}$$ (A4)

where $S_s^2$ and $S_f^2$ are the generalized order parameter for slow domain motions and fast local motions respectively; the time constants τ_es and τ_ef are defined from the correlation times for slow domain τ_s and fast local τ_f motions by the relationships: ${\tau }_{es}^{-1}={\tau }_c^{-1}+{\tau }_s^{-1}$ and ${\tau }_{ef}^{-1}={\tau }_c^{-1}+{\tau }_f^{-1}$. Under isotropic global, domain and local motions, the spectral density functions for cross-correlated relaxation can be derived from auto-correlated spectral density functions by the relationship:

$$J_N(\omega )=\frac{3{cos}^2({\theta }_N)-1}{2}J(\omega )$$ (A5)

$$J_{Hx}(\omega )=\frac{3{cos}^2({\theta }_{Hx})-1}{2}J(\omega )$$ (A6a)

$$J_{Hy}(\omega )=\frac{3{cos}^2({\theta }_{Hy})-1}{2}J(\omega )$$ (A6b)

where the angle θ_N describes the relative orientation of the NH vector and the principal axis of the axially symmetric ¹⁵N chemical shift anisotropy tensor; θ_Hx and θ_Hx describe the relative orientation of the NH vector and the x and y principal axes of the ¹H CSA tensor respectively.

The values employed for the above-mentioned parameters are: r_NH = 101 pm; r_HH = 290 pm; Δσ_N = 164 ppm; θ_N = 19°; σ_Hxx = 14.6 ppm; σ_Hyy = 8.2 ppm; σ_Hzz = 2.1 ppm; θ_Hx = 90°; θ_Hy = 99°.