Accurate Sampling of High-Frequency Motions in Proteins by Steady-State ¹⁵N-{¹H} Nuclear Overhauser Effect Measurements in the Presence of Cross-Correlated Relaxation

Abstract

The steady-state {¹H}-¹⁵N NOE experiment is used in most common NMR analyses of backbone dynamics to accurately ascertain the effects of the fast dynamic modes. We demonstrate here that, in its most common implementation, this experiment generates an incorrect steady state in the presence of CSA/dipole cross-correlated relaxation leading to large errors in the characterization of these high-frequency modes. This affects both the quantitative and qualitative interpretation of ¹⁵N backbone relaxation in dynamic terms. We demonstrate further that minor changes in the experimental implementation effectively remove these errors and allow a more accurate interpretation of protein backbone dynamics.

Dynamics are essential for protein function. Fast motions that are important for protein interactions¹ can also be linked to slower motions involved in enzyme catalysis.² The measurement of backbone ¹⁵N relaxation rates in NMR has become a routine procedure^3,4 to characterize fast local motions.^5,6 These measurements can be complemented by other relaxation data to obtain a more refined view of backbone dynamics.⁷ Usually, the ¹⁵N relaxation rates employed to characterize fast protein dynamics are the longitudinal relaxation rate R₁, the transverse relaxation rate R₂ and the dipolar cross-relaxation rate between the ¹⁵N and ¹H nuclei.⁸ The last of these rates is essential for the accurate estimation of the spectral density function at high frequency^5,9 and crucial for the identification of fast backbone motions.^10,11 This rate is derived from what is commonly called the ‘NOE experiment’ where the ratio between the steady-state longitudinal ¹⁵N polarization under ¹H irradiation or at equilibrium without any perturbation, is measured.^8,12 The saturation of ¹H’s is more difficult to achieve than previously assumed, and cross-correlated relaxation¹³ affects the steady-state ¹⁵N polarization during ¹H irradiation. We quantify these errors for the ¹H irradiation scheme¹² commonly used in the ‘NOE experiment’ and show that they can be suppressed in a straightforward manner, leading to more accurate measurements of spectra density at high frequency.

Several schemes have been proposed for ¹H irradiation in steady-state NOE experiments. Although composite pulse decoupling¹⁵ can be used, the most common procedure applies a series of pulses and delays.^12,14 Pulses with a variety of flip angles have been advocated,¹⁴ and schemes with 120° pulses are the most popular, based on empirical observations.¹⁶ Surprisingly, theoretical and experimental studies of irradiation schemes are sparse.^14,17 We have recently shown that the symmetry of the saturation scheme is important.¹¹ Here, we reexamine the accuracy of the experiment as a function of the ¹H saturation scheme utilizing the Homogeneous Liouville Equation formalism. ^17,18

All experiments were performed on a 600 MHz Bruker Avance spectrometer equipped with a room-temperature probe, using a sample of perdeuterated and ¹⁵N-labelled human ubiquitin (0.5 mM, in 50 mM ammonium acetate, 300 mM NaCl, pH 4.8). We measured ¹⁵N{¹H} Overhauser effects using a symmetric proton irradiation scheme composed of repeated elements [delay τ/2 – β pulse – delay τ/2]_n. Experiments were performed with flip angles β = 90°, 120° and 180°. In Figure 1a, we show differences, Δ(β₁,β₂), in the NOE ratio (of intensities with and without ¹H irradiation) obtained with different flip angles β. Deviations are significant with β = 120° and maximal for β = 90°. As highlighted by color in Figure 1a (e.g. blue points appear on a blue background), the deviations are a periodic function of the ¹H carrier frequency, with a period of 200 Hz, which corresponds to the inverse of the interval between two pulses τ. A 100 Hz shift of the ¹H carrier leads to an inversion of all deviations, i.e., to a phase shift π of the periodic function. This observation is rationalized by numerical simulations (Fig. 1b).

Figure 1.

(a) Experimentally observed deviations of the NOE ratio, Δ(β,180) for rigid NH pairs measured with β = 120° (o); β = 90° (o); and β = 120° (o) with the ¹H carrier frequency shifted by 100 Hz from 2820 to 2720 Hz. The blue boxes illustrate the periodic oscillation with a 200 Hz period. (b) Numerical calculations of the deviations of the NOE ratio for a rigid NH pair (solid lines) and a more mobile NH pair (dotted lines), taking CSA/DD cross-correlation into account, color coded as in (a). The delay τ = 5 ms in the experiments with β ≠ 180° and τ = 50 ms for β = 180°. See Supporting Information (SI) for details about the pulse sequence and numerical calculations.

Figure 1b shows that the calculated steady state ¹⁵N polarization depends on the ¹H carrier frequency, with a period equal to the inverse of the interpulse delay: 1/(τ) = 200 Hz for τ = 5 ms. The oscillations can only be reproduced when the cross-correlation of the ¹⁵N chemical shift anisotropy (CSA) and the ¹⁵N¹H dipolar coupling is included in simulations.

The oscillations seen in Figure 1 stem from a superposition of two separate effects. The dominant one is largest when the evolution under the offset from the ¹H carrier and the heteronuclear scalar coupling leads to the conversion H_y → 2H_yN_z during the interpulse delay τ. If β = 90°, the sequence β – τ – β leads to the conversion H_z → −2H_zN_z. Subsequent cross-correlated cross-relaxation between 2H_zN_z and N_z, results in an effective cross-relaxation pathway between the longitudinal polarization operators H_z and N_z, that perturbs the steady state. The second effect is most observable when the dominant effect vanishes, i.e. when the delay τ is a multiple of 1/|¹J_NH|. The average component H_z^av vanishes if the ¹H polarization precesses an integer number of full rotations during the delay τ, i.e. at the DANTE condition.¹⁹ Otherwise, the polarization H_z is not effectively saturated (H_z^av ≠ 0) so that the ¹⁵N polarization does not evolve towards the desired steady state.

To illustrate the validity of this analysis, consider the experiment described in Figure 2, a “spin-temperature jump”, demonstrating that a scheme like (A) creates a steady state different from the steady state obtained with 180° pulses. Intuitively, one might suppose that proton irradiation with any scheme would lead to proper saturation if applied long enough with sufficient power, so that both irradiation schemes would lead to the same steady state. However, as shown in Figure 2, under a sequence of 120° pulses, the polarization N_z evolves from the ‘ideal’ ¹⁵N steady state (achieved after a 4s irradiation with 180° pulses) towards the same state as the one obtained with scheme A. Deviations measured under scheme A correspond to a different steady state and not to a transient effect.

Figure 2.

Spin-temperature jump experiment using a ¹H irradiation scheme with β = 120° pulses. (A) starting right after the FID, and (B) starting from a steady state obtained with β = 180° pulses. (C) Correlations of deviations in the NOE ratios obtained using schemes (A) and (B) with those using β = 180° pulses for 4 s.

Figure 3 shows how systematic errors due to imperfect saturation propagate to the derived spectral density function at high frequency.¹² This value is essential to the quantitative analysis of relaxation data.^3,20 Deviations in the NOE ratios are amplified, so that large deviations are observed (with a maximum of 47% for Gln49, see SI). When an improper irradiation scheme is used, qualitative interpretations may be biased; for example, high-frequency motions seem to vary significantly in the β-hairpin comprising residues 9–11, while they appear to be uniform with the correct saturation scheme.

Figure 3.

Apparent spectral density function J(0.87ω_H), ω_H being the ¹H Larmor frequency, for backbone NH groups in human ubiquitin derived from longitudinal relaxation rates and different sets of NOE’s.

In conclusion, we have shown that the steady state ¹⁵N polarization in NOE experiments depends on the flip angle β employed in the saturation sequence. Widely used trains of 120° pulses do not lead to the ideal steady state because of CSA/DD cross-correlated relaxation. Flip angles β should be set to 180° and delays τ to integer multiples of the inverse of the scalar coupling ¹J_NH. Improper irradiation leads to significant errors in the estimate of the spectral density functions at high frequency. This effect may be comparable to experimental errors in many studies with low-precision NOE ratios and it occurs whether a protein is deuterated or not (SI).