Zero-quantum frequency-selective recoupling of homonuclear dipole-dipole interactions in solid state nuclear magnetic resonance

Abstract

We describe a method for measuring magnetic dipole-dipole interactions, and hence distances, between pairs of like nuclear spins in a many-spin system under magic-angle spinning (MAS). This method employs a homonuclear dipolar recoupling sequence that creates an average dipole-dipole coupling Hamiltonian under MAS with full zero-quantum symmetry, including both secular and flip-flop terms. Flip-flop terms are then attenuated by inserting rotor-synchronized periods of chemical shift evolution between recoupling blocks, leaving an effective Hamiltonian that contains only secular terms to a good approximation. Couplings between specific pairs of nuclear spins can then be selected with frequency-selective π pulses. We demonstrate this technique, which we call zero-quantum shift evolution assisted homonuclear recoupling, in a series of one-dimensional and two-dimensional ¹³C NMR experiments at 17.6 T and 40.00 kHz MAS frequency on uniformly ¹³C-labeled L-threonine powder and on the helix-forming peptide MB(i+4)EK, synthesized with a pair of uniformly ¹³C-labeled L-alanine residues. Experimental demonstrations include measurements of distances between ¹³C sites that are separated by three bonds, placing quantitative constraints on both sidechain and backbone torsion angles in polypeptides.

INTRODUCTION

Solid state NMR has become a powerful tool for investigating the molecular structures of amorphous and polycrystalline biological solids. With high-resolution spectra obtained from magic-angle spinning (MAS) and heteronuclear decoupling, relatively simple solid state NMR techniques can be used to determine protein structures from a large number of qualitative structural constraints.^{1, 2, 3, 4} Quantitative structural constraints on internuclear distances and torsion angles can be obtained from more sophisticated techniques involving dipolar recoupling methods,^{5, 6, 7, 8, 9} in principle allowing molecular structures to be determined more completely and with higher precision. Quantitative structural techniques that are applicable to uniformly ¹⁵N,¹³C-labeled samples are particularly desirable,^{10, 11, 12} because in principle uniform isotopic labeling provides a faster route to structure determination than does labeling of specific sites in a large set of samples.^{13, 14} However, even if the spectral resolution for a given sample of interest is sufficiently high to permit uniform isotopic labeling, the presence of many-like spins leads to “dipolar truncation” effects, in which the quantum mechanical noncommutivity of pairwise magnetic dipole-dipole couplings causes the stronger one-bond and two-bond couplings to effectively shut off the weaker, longer-range couplings. Since the weaker dipole-dipole couplings contain the most important structural information (e.g., constraints on backbone torsion angles, sidechain conformations, and tertiary contacts), dipolar truncation prevents standard dipolar recoupling techniques from being useful as quantitative structural techniques for uniformly labeled samples. Dipolar truncation is particularly problematic for homonuclear recoupling techniques, e.g., techniques for measuring ¹³C–¹³C dipole-dipole couplings under MAS.^{15, 16, 17, 18}

Frequency-selective dipolar recoupling techniques represent one approach to overcoming dipolar truncation. Such techniques have the effect of switching on dipole-dipole couplings only between pairs of spins with specific NMR chemical shifts under MAS. The earliest frequency-selective homonuclear recoupling techniques were based on the rotational resonance (RR) effect^{6, 19, 20} and its extension to recoupling under continuous wave radio-frequency (rf) irradiation, through RR in the tilted rotating frame (RRTR).^{10, 11} The RR and RRTR techniques are limited to relatively low MAS frequencies or spin pairs with relatively large chemical shift differences. More recent alternatives involve simultaneous recoupling of both dipole-dipole couplings and isotropic or anistropic chemical shifts.^{21, 22} Truncation of the recoupled homonuclear dipole-dipole couplings by chemical shift differences within the recoupling period (unlike the approach described below) produces couplings that can be manipulated by frequency-selective π pulses. These alternative approaches are limited by practical difficulties of implementing the simultaneous dipole-dipole∕chemical shift recoupling techniques at higher MAS frequencies and removing residual heteronuclear couplings to abundant proton spins.

Recently, Paravastu and Tycko²³ introduced the shift evolution assisted selective homonuclear recoupling(SEASHORE) technique for frequency-selective double-quantum homonuclear recoupling. The SEASHORE technique consists of alternating blocks of nonselective double-quantum recoupling and blocks of free spin precession under isotropic chemical shifts, with the length of each block being a multiple of the MAS rotation period τ_R. For homonuclear couplings with double-quantum symmetry, the chemical shift precession periods have the effect of averaging out all pairwise couplings except those for which the net precession angle in each period is an integral multiple of 2π. In principle, the SEASHORE technique is quite general, as the desired frequency selectivity can be achieved for any spin system with any set of chemical shifts by adjusting the lengths of the chemical shift precession periods. In practice, the double-quantum SEASHORE technique depends on having a nonselective dipolar recoupling pulse sequence that produces pure double-quantum couplings at an appropriate MAS frequency over the experimentally relevant range of chemical shifts. The experiments of Paravastu and Tycko²³ employed the POST-C7 recoupling sequence of Hohwy et al.²⁴ at MAS frequencies in the 6–10 kHz range and at a 9.4 T magnetic field, but other double-quantum recoupling sequences could be employed at higher MAS frequencies and higher magnetic fields.

In this paper, we describe an approach to frequency-selective zero-quantum (ZQ) homonuclear recoupling that employs the SEASHORE principle of separating dipolar recoupling and chemical shift precession into discrete periods within a rf pulse sequence. Starting with homonuclear couplings with full ZQ symmetry, chemical shift precession periods (and, in some cases as described below, dipolar truncation itself) produce secular ZQ couplings that can be manipulated by frequency-selective π pulses, as suggested²³ or demonstrated^{21, 22} in earlier work. Zero-quantum recoupling is achieved with the finite-pulse radio-frequency-driven recoupling (fpRFDR) sequence,^{9, 25} which is effective at high MAS frequencies over large chemical shift ranges and produces adequate proton decoupling without proton irradiation at sufficiently high MAS frequencies.²⁵ Thus, the ZQ-SEASHORE approach has important practical advantages over previous approaches to frequency-selective homonuclear recoupling.

THEORY

A general ZQ recoupling sequence leads to an average Hamiltonian for a many-spin system under MAS of the form

$$H_{\text{ZQ}}=\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}+b_{ij}{I}_i^{+}{I}_j^{-}+b_{ij}^{\ast }{I}_i^{-}{I}_j^{+},$$ (1)

where a_ij=2s_ijω_d and b_ij=s_ijω_d∕2, with the orientation-dependent scaling factor S_ij and ${\omega }_d=-\left({\gamma }^2\hslash ∕r_{ij}^3\right)$. The distance between spins i and j is r_ij. I_iz is the z-component of spin angular momentum for spin i, and $I_i^{\pm }\equiv I_{ix}\pm i{I}_{iy}$. If a recoupling period of length mτ_R is followed by a free chemical shift evolution period of length nτ_R (during which dipolar interactions are averaged to 0 by MAS), the evolution operator for the spin system becomes

$$U=\text{exp}\left(-in{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left(-i{H}_{\text{ZQ}}m{\tau }_R\right),$$ (2)

where ω_i is the NMR frequency of spin i relative to the rf carrier frequency, i.e., the resonance offset. After N repetitions, the total evolution operator for the ZQ-SEASHORE sequence becomes

$$U_{\text{tot}}=U^N=\text{exp}\left(-in{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left(-i{H}_{\text{ZQ}}m{\tau }_R\right)\text{exp}\left(-in{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left(-i{H}_{\text{ZQ}}m{\tau }_R\right)\dots \text{exp}\left(-in{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left(-i{H}_{\text{ZQ}}m{\tau }_R\right)=\left[\prod _{k=1}^N\text{exp}\left(-i{\tilde{H}}_{\text{ZQ}}^{\left(k\right)}m{\tau }_R\right)\right]\text{exp}\left(-iNn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right),$$ (3)

where

$${\tilde{H}}_{\text{ZQ}}^{\left(k\right)}=\text{exp}\left(-ikn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)H_{\text{ZQ}}\text{\hspace{0.17em}exp}\left(ikn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)=\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}+b_{ij}\text{\hspace{0.17em}exp}\left[-ikn{\tau }_R\left({\omega }_i-{\omega }_j\right)\right]I_i^{+}{I}_j^{-}+b_{ij}^{\ast }\text{\hspace{0.17em}exp}\left[ikn{\tau }_R\left({\omega }_i-{\omega }_j\right)\right]I_i^{-}{I}_j^{+}.$$ (4)

If the dipole-dipole couplings are sufficiently weak relative to the MAS frequency, Eq. 3 can be approximated as

$$U_{\text{tot}}\approx \text{exp}\left(-i\sum _{k=1}^N{\tilde{H}}_{\text{ZQ}}^{\left(k\right)}m{\tau }_R\right)\text{exp}\left(-iNn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right).$$ (5)

Note that $1∕N\mid {\sum }_{k=1}^N\text{exp}\left[\pm ikn{\tau }_R\left({\omega }_i-{\omega }_j\right)\right]\mid$ diminishes with increasing N, as long as nτ_R(ω_i−ω_j) is not a multiple of 2π. Therefore, the total evolution operator for N repetitions of the ZQ recoupling and chemical shift precession blocks is approximately

$$U_{\text{trunc}}\left(N\right)\approx \text{exp}\left(-iNm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-iNn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right),$$ (6)

where the full ZQ dipole-dipole coupling Hamiltonian is replaced by a truncated Hamiltonian that contains only products of z-components of spin angular momenta, i.e., the secular terms in the ZQ coupling Hamiltonian.

If a selective π pulse covering frequencies ω_s and ω_r is applied in the middle of the ZQ-SEASHORE sequence, as shown in Fig. 1a, spins s and r are decoupled from all other spins but remain coupled to one another, as shown by

$$U_{\text{trunc}}^{\prime }\left(N\right)=U_{\text{trunc}}\left(\frac{N}{2}\right)\text{exp}\left[i\pi \left(I_{sx}+I_{rx}\right)\right]U_{\text{trunc}}\left(\frac{N}{2}\right)=\text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left[i\pi \left(I_{sx}+I_{rx}\right)\right]\times \text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)=\text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\times \text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{g}_i{g}_j{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{g}_i{\omega }_i{I}_{iz}\right)\text{exp}\left[i\pi \left(I_{sx}+I_{rx}\right)\right]=\text{exp}\left[-iNm{\tau }_R\left(a_{rs}{I}_{rz}{I}_{sz}+\sum _{i,j\ne r,s}{a}_{ij}{I}_{iz}{I}_{jz}\right)\right]\text{exp}\left(-iNn{\tau }_R\sum _{i\ne r,s}{\omega }_i{I}_{iz}\right)\text{exp}\left[i\pi \left(I_{sx}+I_{rx}\right)\right],$$ (7)

where g_i≡(1−2δ_ir−2δ_is).

Figure 1.

(a) General form for a ZQ-SEASHORE pulse sequence. Periods of zero-quantum (ZQ) dipolar recoupling alternate with periods of spin precession at isotropic chemical shift (CS) frequencies, producing a truncated ZQ dipole-dipole Hamiltonian containing only secular terms. A frequency-selective π pulse in the middle of the ZQ-SEASHORE sequence decouples the selected spins from others. (b) In this paper, ZQ recoupling is achieved with the fpRFDR sequence, (Refs. ⁹, ²⁵) consisting of one π pulse per MAS rotor period τ_R, with XY-8 phases. (Ref. ²⁷)

The net chemical shift precession of spins other than r and s in Eq. 7 can be removed by applying both a nonselective π pulse and a selective π pulse in the middle of the ZQ-SEASHORE sequence, as shown by

$$U_{\text{trunc}}^{″}\left(N\right)=U_{\text{trunc}}\left(\frac{N}{2}\right)\text{exp}\left[i\pi \left(I_{sx}+I_{rx}\right)\right]\text{exp}\left(i\pi \sum _i{I}_{ix}\right)U_{\text{trunc}}\left(\frac{N}{2}\right)=\text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left(i\pi \sum _{i\ne r,s}{I}_{ix}\right)\text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)=\text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{I}_{iz}{I}_{jz}\right)\text{exp}\left(-\frac{i}{2}Nn{\tau }_R\sum _i{\omega }_i{I}_{iz}\right)\text{exp}\left(-\frac{i}{2}Nm{\tau }_R\sum _{i<j}{a}_{ij}{g}_i{g}_j{I}_{iz}{I}_{jz}\right)\text{exp}\left(\frac{i}{2}Nn{\tau }_R\sum _i{g}_i{\omega }_i{I}_{iz}\right)\text{exp}\left(i\pi \sum _{i\ne r,s}{I}_{ix}\right)=\text{exp}\left[-iNm{\tau }_R\left(a_{rs}{I}_{rz}{I}_{sz}+\sum _{i,j\ne r,s}{a}_{ij}{I}_{iz}{I}_{jz}\right)\right]\text{exp}\left[-iNn{\tau }_R\left({\omega }_r{I}_{rz}+{\omega }_s{I}_{sz}\right)\right]\text{exp}\left(i\pi \sum _{i\ne r,s}{I}_{ix}\right).$$ (8)

MATERIALS AND METHODS

L-threonine with 27% uniformly ¹⁵N,¹³C-labeled molecules was prepared by recrystallization of a mixture of labeled and unlabeled L-threonine from de-ionized water. The 17-residule helical peptide MB(i+4)EK (N-acetyl-AEAAAKEAAAKE AAAKA-NH₂) was synthesized using standard FMOC solid-phase methods as previously described,²⁶ with uniformly ¹⁵N,¹³C-labeled L-alanine at the Ala9 and Ala10 positions. Both samples (approximately 3 mg each) were finely ground and center-packed in a 1.8 mm MAS rotor. NMR experiments were performed at 17.6 T (188.04 MHz ¹³C NMR frequency), using a Varian Infinity spectrometer console and a three-channel high-speed MAS probe built by the group of Dr. Ago Samoson (National Institute of Chemical Physics and Biophysics, Tallinn,Estonia). Rf pulse sequences are shown in Figs. 25. Zero-quantum ¹³C–¹³C dipolar recoupling was performed with fpRFDR,²⁵ using 7.0 μs π ¹³C pulses at 40.00 kHz MAS frequency, one π pulse per rotor period, and XY-8 phases.²⁷ Pulse sequences were actively synchronized to an MAS tachometer signal, with synchronization occurring at the beginning of each rotor period in the recoupling blocks. The π pulses were applied either at the beginning of a rotor period (A blocks), or after delays of τ_R∕3 (B blocks) or 2τ_R∕3 (C blocks), as discussed below. Proton decoupling fields were 120 kHz, with two-pulse phase modulation²⁸ (TPPM) during chemical shift precession and signal detection periods, except as noted below. No proton rf was applied during fpRFDR periods, as the fpRFDR sequence on ¹³C spins was found to provide adequate decoupling of ¹³C from protons. The ¹³C rf carrier frequency was set to the average of the CO and C_α chemical shifts during ZQ-SEASHORE periods in all experiments, except in Fig. 6a where the carrier frequency was set to the average of C_α and C_β chemical shifts. ¹³C NMR chemical shifts in all spectra are relative to tetramethylsilane. Other details of NMR measurements are given below or in the figure captions.

Figure 2.

Pulse sequences used in experiments for Figs. 34. (a) Sequence for 2D ¹³C–¹³C spectroscopy with longitudinal mixing and without frequency selectivity. ¹³C spin polarization is prepared initially by ¹H–¹³C cross-polarization. Zero-quantum recoupling is applied during the mixing period for time 8Nτ_R. TPPM decoupling is applied during t₁ and t₂ periods. For quadrature detection in t₁, signals are acquired with ϕ=x and ϕ=y. (b) Sequence for 2D ¹³C–¹³C spectroscopy with longitudinal mixing under truncated ZQ dipole-dipole couplings. (c) Sequence for 2D ¹³C–¹³C spectroscopy with transverse mixing under truncated ZQ dipole-dipole couplings. (d) Sequence for 2D ¹³C–¹³C spectroscopy with frequency-selective transverse mixing under truncated ZQ dipole-dipole couplings. Under this sequence, transverse polarization transfers can occur within groups of spins that are inverted by the frequency-selective π pulses, or within groups that are unaffected by the frequency-selective π pulses. The length of these pulses is determined by the integer q. Vertical dashed lines indicate the beginning of each 2qτ_R period.

Figure 5.

Pulse sequences for ¹³C–¹³C distance measurements by constant-time frequency-selective ZQ recoupling. (a) Sequence in which spins that are inverted by the frequency-selective π pulse are decoupled from other spins. Distances among spins that are unaffected by the frequency-selective π pulse are determined from the dependence of their NMR signals on the integer k. (b) Sequence for determining distances among spins that are inverted by the frequency-selective π pulse. (c) Sequence for indirect detection of internuclear distances by 2D spectroscopy. Transverse polarization of spins that are inverted by the frequency-selective π pulse (e.g., backbone ¹³CO spins in a polypeptide) dephases under frequency-selected ZQ recoupling for a variable time 24kτ_R, and is then transferred to directly bonded spins (e.g., backbone ¹³C_α spins) during a longitudinal mixing period of length 8k^′τ_R. After a t₁ evolution period, a second longitudinal mixing period of length 8k^″τ_R allows additional polarization transfers (e.g., to ¹³C_β spins). Distances are extracted from the dependence of crosspeak volumes on k. Initial polarization is selected by taking the difference between signals acquired with ζ=x from signals acquired with ζ=y. For quadrature detection in t₁, signals are acquired with ϕ=x and ϕ=y. ξ is adjusted empirically from its ideal value (ξ=x) to maximize signals. (d) Definition of A, B, and C blocks, which differ only by displacement of the ¹³C π pulses in the fpRFDR sequence by multiples of τ_R∕3 relative to beginning of each MAS rotor period. According to symmetry principles, (Ref. ³³) concatenation of the A, B, and C blocks results in cancellation of fpRFDR recoupling, so that ideally the “N-k” periods in a, b, and c do not contribute to the dephasing of transverse polarization.

Figure 6.

Constant-time frequency-selective dipolar dephasing data for uniformly ¹⁵N,¹³C-labeled L-threonine powder, at 188.04 MHz ¹³C NMR frequency and 40.00 kHz MAS frequency. (a) Dephasing of ¹³CO signals by intramolecular coupling to ¹³C_γ spins, using the pulse sequence in Fig. 5a with the frequency-selective π pulse applied to ¹³C_α and ¹³C_β spins. Experimental data were obtained with n=7 and N=14, 11, and 9 (squares, circles, and triangles, respectively). Solid lines are ideal simulations for ¹³C spin pairs with purely secular dipole-dipole couplings, with internuclear distances from 2.5 to 3.5 Å in increments of 0.1 Å. Heavy dashed line is a simulation for the full pulse sequence and the full coupled four-spin ¹³C system of L-threonine with realistic geometry and NMR parameters. (b) Dephasing of ¹³CO signals under the pulse sequence in Fig. 5b with frequency-selective π pulses applied to ¹³CO spins. Experimental data were obtained with n=1 and N=14, 11, and 9 (squares, circles, and triangles, respectively). Solid lines are ideal simulations for ¹³C spin pairs with purely secular dipole-dipole couplings. Heavy dashed line is a full simulation for the four-spin system.

RESULTS

Spin polarization transfers under truncated and frequency-selective ZQ recoupling

Two-dimensional (2D) ¹³C–¹³C NMR spectra of polycrystalline L-threonine were acquired with pulse sequences in Fig. 2. The 2D spectrum in Fig. 3a was acquired with untruncated fpRFDR recoupling during the mixing period and with longitudinal mixing [i.e., spin polarization transfers between z-components of angular momentum; pulse sequence in Fig. 2a]. Strong crosspeaks that connect isotropic chemical shifts of directly bonded ¹³C sites are observed. Weaker two-bond crosspeaks (and even weaker three-bond crosspeaks) are also observed. The 2D spectrum in Fig. 3b was acquired with truncated fpRFDR recoupling during the mixing period and with longitudinal mixing [pulse sequence in Fig. 2b]. All crosspeaks are strongly suppressed, as expected if the effective coupling Hamiltonian contains only secular terms as in Eq. 6. If nonsecular terms are absent, the z-component of angular momentum of each individual spin is conserved, preventing longitudinal polarization transfers.

Figure 3.

2D ¹³C–¹³C spectra of uniformly ¹⁵N,¹³C-labeled L-threonine powder (27% labeled molecules diluted in unlabeled molecules by recrystallization) obtained with pulse sequences in Fig. 2, at 188.04 MHz ¹³C NMR frequency and 40.00 kHz MAS frequency. ¹³C π pulses in fpRFDR blocks were 7.0 μs. (a) Spectrum obtained with sequence in Fig. 2a, withN=12. (b) Spectrum obtained with sequence in Fig. 2b, with N=8 andn=7. (c) Spectrum obtained with sequence in Fig. 2c, with N=8 andn=7. (d) Spectrum obtained with sequence in Fig. 2d, with N=64 andn=7, and with frequency-selective π pulses (250 μs pulse length) applied to C_α and C_β spins to decouple them from CO and C_γ spins. 1D slices at NMR frequencies of CO, C_α, C_β, and C_γ (at 170.0, 60.2, 65.4, and 18.9 ppm, respectively) are shown to the right of each 2D spectrum.

The 2D spectrum in Fig. 3c was acquired with truncated fpRFDR recoupling during the mixing period and with transverse mixing [i.e., spin polarization transfers between x- or y-components of angular momentum; pulse sequence in Fig. 2c]. Note that the pulse sequence in Fig. 2c includes a π∕2 pulse in the center of the mixing period, phase-shifted by π∕2 relative to the π∕2 pulses at the beginning and end of the mixing period. This central pulse is required to produce transverse polarization transfers under a truncated ZQ coupling Hamiltonian, as is well known from simple product-operator rules.²⁹ Note also that “relayed” polarization transfers (i.e., transfers between spin pairs that are not directly coupled, mediated by their couplings to a common third spin) should not occur under the secular coupling. Crosspeak patterns in Fig. 3c are similar to those in Fig. 3a, except that two-bond crosspeaks are weaker, as expected in the absence of relayed polarization transfers.

The 2D spectrum in Fig. 3d was acquired with frequency-selective, truncated fpRFDR recoupling during the mixing period, and with transverse mixing [pulse sequence in Fig. 2d]. The frequency-selective, Gaussian-shaped π pulses in Fig. 2d were applied with the ¹³C rf carrier frequency between the isotropic shifts of the C_α and C_β sites, so that only the ¹³C_α and ¹³C_β spins experience the π rotation. According to Eq. 8, this should leave the ¹³C_α and ¹³C_β spins coupled to one another and the ¹³CO and ¹³C_γ spins coupled to one another, but ¹³C_α–¹³CO, ¹³C_β–¹³C, ¹³C_α–¹³C_γ, and ¹³C_β–¹³C_γ couplings should be absent. Note that two frequency-selective π pulses are required, one in each half of the mixing period, with a π∕2 pulse in the center of the mixing period as discussed above. As expected, crosspeaks between CO and C_γ chemical shifts are observed in Fig. 3d, while C_α∕CO, C_β∕CO, C_α∕C_γ, and C_β∕C_γ crosspeaks are suppressed. C_α∕C_β crosspeaks and diagonal peaks for C_α and C_β are also attenuated, due to the relatively short T₂ relaxation times for these sites.

Results in Fig. 3 demonstrate that the ZQ-SEASHORE technique can be implemented successfully at high MAS frequencies and high fields, that this technique does indeed create secular ZQ couplings, and that frequency selectivity can be achieved with frequency-selective π pulses. Figure 4 compares experimental crosspeak volumes (plots on left side) with simulated crosspeak volumes (plots on right side) as functions of the mixing time for the four types of 2D experiments in Figs. 23. Numerical simulations were performed with the SPINEVOLUTION program³⁰ and include the four ¹³C spins in L-threonine with the geometry from the reported crystal structure,³¹ isotropic chemical shifts and chemical shift anisotropies (CSAs) reported for L-threonine,³² and the same ¹³C rf pulse sequence as in experiments (except for the initial ¹H–¹³C cross-polarization step). Proton spins were not included in simulations. Agreement between experiments and numerical simulations is generally good. Comparison of Figs. 4d, 4h illustrates the effects of T₂ relaxation, which attenuates the C_α∕C_β crosspeaks and (to a lesser extent) the C_γ∕CO crosspeaks in experimental measurements at long mixing times.

Figure 4.

Dependence of experimental [(a)–(d)] and simulated [(e)–(h)] crosspeaks on mixing time, from 2D spectra as in Figs. 3a, 3b, 3c, 3d, respectively. Crosspeak volumes from experimental spectra are scaled to the CO diagonal peak volume at the shortest mixing time in each panel. Crosspeak volumes from simulations (i.e., polarization transfer amplitudes) are scaled to the initial spin polarization at a single site. Simulations include four ¹³C spins with the geometry and NMR parameters of L-threonine in crystalline form.

Distance measurements by constant-time frequency-selective recoupling

Figures 5a, 5b show pulse sequences for quantitative measurements of ¹³C–¹³C distances by frequency-selective ZQ recoupling. As discussed above, rotor-synchronized, finite-length π pulses (in groups of eight with XY-8 phases, and with each pulse starting τ_R∕3 after the beginning of a rotor period, comprising B blocks) are applied alternately with chemical shift precession periods of length nτ_R, creating dipole-dipole couplings that contain only the secular terms to a good approximation. A frequency-selective π pulse [Fig. 5b] or both a frequency-selective and a nonselective π pulse [Fig. 5a] are applied in the middle of the ZQ-SEASHORE period, decoupling spin pairs of interest from other spins. To extract internuclear distances, the dipolar dephasing time (determined by k in Fig. 5) is incremented, and the decay of ¹³C NMR signals is measured and compared with simulations.

When the dephasing time extends over many milliseconds, the ¹³C NMR signals also decay due to T₂ relaxation and incomplete proton decoupling. These extraneous contributions to signal decay can be minimized by performing the measurements in a “constant-time” manner, as previously demonstrated in related contexts.^{25, 33, 34} In Fig. 5, constant-time recoupling sequences are created by adding ABC blocks separated by chemical shift precession periods of length 3nτ_R before and after the ZQ-SEASHORE period, where the A and C blocks differ from the B blocks only by displacements of the finite-length π pulses by ±τ_R∕3. As previously shown, an ABC block produces no net dipolar recoupling, in the limit that the MAS frequency is large compared with chemical shift differences.³³ Therefore, the ABC blocks do not contribute to the net dipolar dephasing time. The dephasing time can be incremented without changing the overall pulse sequence duration and without changing the number of rf pulses, by incrementing k from 0 to N while keeping N constant.

Figure 6 shows experimental measurements of the dephasing of CO signals from uniformly ¹⁵N,¹³C-labeled L-threonine. For Fig. 6a, the pulse sequence in Fig. 5a was used, with the frequency-selective π pulse applied to ¹³C_α and ¹³C_β spins. The signal decay should then result from ¹³CO–¹³C_γ dipole-dipole couplings. Comparison with numerical simulations of signal decay for ¹³C spin pairs with ideal secular dipole-dipole couplings shows good agreement at a 3.1 Å internuclear distance, corresponding to the 3.09Å intramolecular CO–C_γ distance in the L-threonine crystal structure.³¹ These simulations assume the ideal ZQ average Hamiltonian for fpRFDR,^{25, 33} truncated to secular terms. The experimental data also agree well with numerical simulations that include the full rf pulse sequence with MAS and the four-spin system of dipole-coupled ¹³C nuclei in L-threonine [heavy dashed lines in Fig. 6a]. Experiments were performed with total constant-time recoupling periods (equal to 48Nτ_R, not including chemical shift precession periods) of 10.8, 13.2, and 16.8 ms. As shown in Fig. 6a, experimental decay curves (normalized to the signals at zero dephasing time) are independent of the constant-time period.

For Fig. 6b, the pulse sequence in Fig. 5b was used, with the frequency-selective π pulse applied to ¹³CO spins. This should decouple ¹³CO spins from ¹³C_α, ¹³C_β, and ¹³C_γ spins. Signal decay is significantly slower than in Fig. 5a, agreeing approximately with simulations for 5 Å internuclear distances. In crystalline L-threonine, each carboxylate carbon has two nearest neighbor carboxylates, at 3.845 Å distances.³¹ Given the dilution of isotopically labeled molecules in our sample, approximately 53% of ¹³CO spins have no nearest neighbor ¹³CO at this distance. Next-nearest neighbors are at 5.110 Å. Thus, data in Fig. 6b are in rough agreement with expectations based on the L-threonine crystal structure.

Data in Fig. 6a are corrected for natural-abundance ¹³C contributions to the CO signals by subtracting a constant from the experimental signal amplitudes, equal to 22% of the signal at zero dephasing time for data at 48Nτ_R=16.8 ms. This correction is larger than the 3% correction that would be based on the ratio of natural-abundance carboxylate ¹³C nuclei to ¹³C labels because the transverse spin relaxation rate for carboxylate ¹³C nuclei in uniformly labeled L-threonine molecules is greater than the relaxation rate for natural-abundance carboxylate ¹³C nuclei by a factor of 5.

Figure 7 shows constant-time frequency-selective recoupling data for lyophilized MB(i+4)EK, with uniform ¹⁵N,¹³C-labeling of Ala9 and Ala10. As shown in Fig. 7d, this sample contains a mixture of peptides with an α-helical conformation and peptides with an extended conformation,²⁶ in an approximate 7:1 ratio. Figure 7a, obtained with the pulse sequence in Fig. 5b and with frequency-selective π pulses applied to ¹³CO spins (∼15 ppm excitation bandwidth), shows measurements of inter-residue ¹³CO–¹³CO dipole-dipole couplings. The decay of helical CO signals indicates a CO–CO distance of 2.9 Å. The decay of nonhelical CO signals indicates a CO–CO distance of 3.3 Å. These distances are in good agreement with distances expected in α-helical (backbone torsion angle ϕ=−50°) and extended (ϕ=100°) conformations at Ala10. Note that the nonhelical MB(i+4)EK molecules are expected to adopt a distribution of conformations, so that the apparent CO–CO distance for these molecules represents an average over the distribution. The distribution of ϕ values for α-helical molecules is expected to be in the range from −45° to −70°, based on earlier molecular dynamics simulations,²⁶ corresponding to distances in the range from 2.86 to 3.06 Å.

Figure 7.

Constant-time frequency-selective dipolar dephasing data for MB(i+4)EK with uniform ¹⁵N,¹³C-labeled L-alanine at Ala9 and Ala10, at 188.04 MHz ¹³C NMR frequency and 40.00 kHz MAS frequency. (a) Dephasing of ¹³CO signals by ¹³CO–¹³CO couplings, using the pulse sequence in Fig. 5b with frequency-selective π pulses applied to ¹³CO spins. Data for helical (squares, up triangles, and plus symbols) and nonhelical (circles, down triangles, and asterisks) signal components are shown, obtained n=1 and N=9 (squares, circles), N=7 (up and down triangles), and N=5 (plus symbols and asterisks). Lines are ideal two-spin simulations for internuclear distances from 2.6 to 3.6 Å, in increments of 0.1 Å. (b) Dephasing of helical ¹³C_β signals, using the pulse sequence in Fig. 5b with frequency-selective π pulses applied to ¹³C_β spins. Data were obtained with n=1 and N=9, 7, and 5 (circles, squares, and triangles, respectively). (c) Dephasing of helical ¹³CO signals by ¹³CO–¹³C_β couplings, using the pulse sequence in Fig. 5a with frequency-selective π pulses applied to ¹³C_α spins. Data were obtained with n=1 and N=9, 7, and 5 (circles, squares, and triangles, respectively). (d) 1D ¹³C NMR spectrum, showing helical (strong peaks) and non-helical (asterisks) signal components.

Figure 7b, obtained with the pulse sequence in Fig. 5b and with frequency-selective π pulses applied to ¹³C_β spins, shows measurements of inter-residue ¹³C_β–¹³C_β dipole-dipole couplings. The slow signal decay indicates distances greater than 4.5 Å, as expected in both helical and extended conformations. The slow signal decay confirms that ¹³C_β spins are well decoupled from ¹³CO and ¹³C_α spins under these experimental conditions, despite the relatively short intra-residue C_α–C_β and CO–C_β distances. Figure 7c, obtained with the pulse sequence in Fig. 5a and with frequency-selective π pulses applied to ¹³C_α spins, shows measurements of ¹³CO–¹³C_β dipole-dipole couplings. The decay of both helical and nonhelical CO signals indicates a CO–C_β distance of 2.5 Å, in good agreement with the expected intraresidue distance (2.51 Å, independent of backbone conformation).

In experiments for Figs. 7b, 7c, proton rf fields of 120 kHz were applied for proton decoupling during chemical shift precession periods, but not during fpRFDR periods (A, B, or C blocks in Fig. 5). In experiments for Fig. 7a, proton rf fields were not applied during both the chemical shift precession periods and the fpRFDR periods, as proton couplings to ¹³CO spins are sufficiently weak that rapid MAS (40.00 kHz rotation frequency) provides adequate proton decoupling during the chemical shift precession periods.

Experimental data points in Figs. 67 were obtained from peak areas in one-dimensional (1D) ¹³C NMR spectra, recorded at various values of the dephasing time. In most applications to biomolecular systems, the resolution in 1D spectra would be insufficient to permit measurements of signal amplitudes for individual sites and their dependence on the dephasing time. In particular, resolution in the CO region of 1D solid state ¹³C NMR spectra of proteins and peptides is generally poor, necessitating higher-dimensional spectroscopy. Figure 5c shows a 2D version of the technique in Fig. 5b, intended for measurements of site-specific backbone CO–CO distances in peptides and proteins with multiple uniformly labeled residues. After the constant-time frequency-selective ZQ-SEASHORE period, a period of longitudinal mixing under the fpRFDR sequence allows CO–C_α polarization transfers. After a subsequent t₁ evolution period, a second period of longitudinal mixing allows C_α–C_β polarization transfers before ¹³C NMR signals are detected in the t₂ period. In this way, the decay of transverse CO polarization under ¹³CO–¹³CO dipole-dipole couplings (which depend on backbone conformation) can be monitored through the amplitude of C_α∕C_β crosspeaks in a series of 2D spectra, obtained with different dephasing times [i.e., different values of k in Fig. 5c]. Depending on the spectral resolution and the longitudinal mixing conditions, CO∕C_α, CO∕C_β, or other crosspeaks can also be used.

2D spectra of the MB(i+4)EK sample with dephasing times of 1.2 and 8.4 ms are shown in Figs. 8a, 8b. These spectra illustrate the more rapid decay of crosspeak (and diagonal) signals associated with helical MB(i+4)EK molecules, compared with signals associated with nonhelical molecules. The dependences of signal amplitudes on dephasing time in Fig. 8c, determined from C_α∕C_β crosspeak volumes, are nearly identical to the dependences from 1D spectra in Fig. 7a, for both helical and nonhelical signals. This result demonstrates the feasibility of obtaining quantitative constraints on backbone conformation from ¹³CO–¹³CO dipole-dipole couplings in measurements on multiply or uniformly labeled samples.

Figure 8.

[(a) and (b)] 2D ¹³C–¹³C NMR spectra of MB(i+4)EK obtained with the pulse sequence in Fig. 5c at 188.04 MHz ¹³C NMR frequency and 40.00 kHz MAS frequency. Frequency-selective π pulses were applied to ¹³CO spins, n=1, N=7, k^′=14, and k^″=14. The ¹³C rf carrier frequency was set to 115 ppm during the ZQ-SEASHORE dephasing period, 115 ppm during the first longitudinal mixing period, and 33 ppm during the second longitudinal mixing period. Spectra are shown for k=1 (a) and k=7 (b). 1D slices at the helical (i) and nonhelical (ii) ¹³C_α chemical shift values are shown to illustrate the more rapid dephasing of helical signal components, due to the shorter ¹³CO–¹³CO distances in helical molecules. (c) Comparison of experimental C_α∕C_β crosspeak volumes for helical (filled squares) and nonhelical (filled circles) signal components with ideal two-spin simulations for indicated internuclear distances. Open symbols are CO signal amplitudes from 1D experiments, as in Fig. 7a.

In Fig. 8, signals originating from CO polarization were selected by phase cycling of the frequency-selective π pulse in Fig. 5c (i.e., difference between signals acquired with ζ=x and ζ=y). In addition, the phase ξ in Fig. 5c was adjusted empirically to maximize the signals, accounting for residual phase shifts under the frequency-selective ZQ-SEASHORE sequence.

Finally, because the chemical shift difference between labeled CO sites is quite small for helical MB(i+4)EK, it may be surprising that the experimental data in Figs. 7a, 8c agree well with simulations for ¹³C spin pairs with the expected helical internuclear distance, given that the simulations assume purely secular ZQ dipole-dipole couplings. In fact, truncation of ¹³CO–¹³CO couplings to secular terms is brought about primarily by one-bond ¹³CO–¹³C_α couplings, which are stronger than and do not commute with the longer-range ¹³CO–¹³CO couplings. Thus, CO chemical shift differences are not required in these experiments. To support this analysis, Fig. 9 shows simulations of frequency-selective ZQ-SEASHORE recoupling with the full pulse sequence in Fig. 5b for a six-spin system, representing all ¹³C sites in two sequential alanine residues in a helical peptide. Simulations with and without a CO chemical shift difference are nearly identical, and agree well with two-spin simulations that assume a fully truncated coupling (solid line). Two-spin simulations with an untruncated ZQ coupling show a more rapid signal decay (dashed line).

Figure 9.

Comparison of full numerical simulations of ¹³CO signals from a six-spin ¹³C system, representing all carbon sites in two sequential alanine residues in an α-helical peptide (up and down triangles), with two-spin simulations that include only the CO sites (solid and dashed lines). All simulations use the pulse sequence in Fig. 5b with n=1 and N=12 at 188.04 MHz ¹³C NMR frequency and 40.00 kHz MAS frequency. The CO–CO distance is 2.97 Å. In six-spin simulations, the chemical shift difference between the two CO sites is either 0 Hz (up triangles) or 400 Hz (down triangles). In two-spin simulations, the chemical shift difference is either 0 Hz (dashed line, corresponding to dephasing under full, untruncated ZQ dipole-dipole couplings) or 20.000 kHz (solid line, corresponding to truncated, secular ZQ dipole-dipole couplings). In six-spin simulations, truncation of ¹³CO–¹³CO couplings to secular terms results primarily from noncommuting ¹³CO–¹³C_α couplings.

DISCUSSION

The theory presented above shows that effective homonuclear dipole-dipole couplings containing only secular ZQ terms can be created by alternation of ZQ dipolar recoupling periods (which create full ZQ couplings) and chemical shift precession periods (which truncate the couplings to secular terms). This is the ZQ-SEASHORE concept. Frequency-selective π pulses can then be used to refocus undesired couplings, allowing selected couplings of interest to be measured from dephasing curves (as in Figs. 67) or in principle from polarization transfer curves under transverse mixing (as in Figs. 34). Compared with previously proposed approaches to the creation of secular ZQ couplings,^{21, 22} the ZQ-SEASHORE concept has the advantage of flexibility. Any ZQ recoupling sequence can be used, so the recoupling sequence can be adapted to suit the relevant chemical shift range, MAS frequency, magnetic field strength, available rf field strengths, or other experimental parameters. The relative magnitudes of dipole-dipole couplings and chemical shift differences, which determine the efficiency of truncation to secular terms, can be adjusted by varying the lengths of the recoupling and the chemical shift precession periods.

Experiments described above used the fpRFDR recoupling sequence^{9, 25} because of its simplicity, compatibility with high fields and high MAS frequencies, and relatively good proton decoupling efficiency (either without proton rf irradiation at high MAS frequencies or with proton rf irradiation at lower MAS frequencies). In addition, constant-time recoupling techniques are readily constructed with the A, B, and C variants of fpRFDR as shown in Fig. 5, based on symmetry principles as previously described.³³ Constant-time techniques are necessary if experimental data are to be compared directly with numerical simulations to extract internuclear distances, as previously demonstrated.^{25, 33, 34}

Experiments in Figs. 34678 were performed with MAS at 40.00 kHz. This relatively high MAS frequency is required in ¹³C NMR experiments on uniformly labeled peptides and proteins to ensure that the dipolar recoupling sequence averages out isotropic and anisotropic chemical shifts (which are on the order of 20 kHz at 17.6 T), producing a pure dipole-dipole Hamiltonian during the recoupling periods. At lower fields, somewhat lower MAS frequencies could be used. However, the MAS frequency must still be high enough that each dipolar recoupling period in the ZQ-SEASHORE sequence (minimally 4τ_R for fpRFDR with XY-4 phases) is short compared with the dipole-dipole coupling time scale (approximately 500 μs for one-bond ¹³C–¹³C couplings). Thus, the MAS frequency can not be much lower than 20 kHz in ¹³C NMR experiments on uniformly labeled systems, even at lower fields.

The optimal MAS frequency for ZQ-SEASHORE is also dictated by the requirement that the precession angles for chemical shift differences during the nτ_R periods in Figs. 125 not be close to integer multiples of 2π, especially for ¹³C pairs that are separated by only one or two bonds. In peptides and proteins, CO–C_α and CO–C_β shift differences (typically 105–130 ppm and 100–160 ppm, respectively) demand MAS frequencies greater than 30 kHz at 17.6 T ifn=1. When C_α–C_β shift differences (or other relevant shift differences) are small, larger values of n may be required to ensure adequate truncation of one-bond C_α–C_β couplings. For typical sets of ¹³C chemical shifts in peptides and proteins, we find that MAS frequencies near 40 kHz generally allow adequate truncation of the relevant ¹³C–¹³C couplings in experiments at 17.6 T.

As demonstrated in Figs. 7a, 8, the ZQ-SEASHORE concept allows quantitative constraints on backbone ¹³CO–¹³CO distances to be obtained for peptides and proteins that are uniformly ¹³C-labeled, or are prepared with two or more sequential uniformly labeled residues. Backbone ¹³CO–¹³CO distances depend on backbone ϕ torsion angles (although ψ torsion angles affect the angles between successive ¹³CO–¹³CO internuclear vectors and therefore must also be considered when analyzing data for uniformly labeled samples). We have previously shown that constraints on backbone ¹⁵N–¹⁵N distances, which depend on ψ torsion angles, can be obtained for uniformly ¹⁵N,¹³C-labeled peptides and proteins with the PITHIRDS-CT technique.^{14, 33} Thus, the ZQ-SEASHORE and PITHIRDS-CT techniques provide a route to obtaining full sets of quantitative backbone conformational constraints for uniformly labeled samples. As shown in Fig. 8 and in our earlier work,^{14, 33} constraints on ¹³CO–¹³CO or ¹⁵N–¹⁵N distances can be obtained even when the ¹³CO or ¹⁵N NMR lines are poorly resolved, by transferring ¹³CO or ¹⁵N polarization to directly bonded ¹³C_α sites and employing multidimensional spectroscopy.

As demonstrated in Figs. 6a, 7c, the ZQ-SEASHORE concept can also be used to measure distances between backbone and sidechain carbons, in principle placing constraints on sidechain conformations. Conformational information is contained in distances between backbone carbonyl and sidechain methyl sites (for methionine, isoleucine, leucine, valine, and threonine residues), between backbone α-carbon and sidechain methyl sites (for methionine, isoleucine, and leucine residues), between backbone carbonyl and sidechain carbonyl or carboxylate sites (for asparate, asparagine, glutamate, and glutamine residues), between backbone α-carbon and sidechain carbonyl or carboxylate sites (for glutamate and glutamine residues), between backbone carbonyl and sidechain aromatic carbon sites (for phenylalanine, tyrosine, tryptophan, and histidine residues), and between backbone carbonyl or α-carbon sites and sidechain δ- or ε-carbon sites (for lysine and arginine residues). Conformational constraints from intraresidue carbon-carbon distances are complementary to constraints from intraresidue nitrogen-carbon distances that have been obtained with frequency-selective heteronuclear recoupling methods.^{12, 35}

Finally, it may also be possible to obtain quantitative constraints on tertiary contacts or on intermolecular distances in supramolecular structures from ZQ-SEASHORE measurements. Together with quantitative constraints on backbone and sidechain conformations, such measurements would represent a significant improvement on the more qualitative approaches that have been employed in recent structural studies of proteins by solid state NMR.^{1, 2, 3, 4} In particular, we note that information about tertiary contacts or intermolecular contacts derived from proton-driven spin diffusion¹⁶ or proton-assisted three-spin effects³⁶ is useful for establishing proximities of amino acid sidechains or other aspects of protein structures, but that such measurements do not yield precise distances between specific pairs of nuclei.