Spin Diffusion Driven by R-Symmetry Sequences: Applications to Homonuclear Correlation Spectroscopy in MAS NMR of Biological and Organic Solids

Abstract

We present a family of homonuclear ¹³C-¹³C magic angle spinning spin diffusion experiments, based on R2_n^v (n = 1 and 2, v = 1 and 2) symmetry sequences. These experiments are well suited for ¹³C-¹³C correlation spectroscopy in biological and organic systems, and are especially advantageous at very fast MAS conditions, where conventional PDSD and DARR experiments fail. At very fast MAS frequencies the R2₁¹, R2₂¹, and R2₂² sequences result in excellent quality correlation spectra both in model compounds and in proteins. Under these conditions, individual R2_n^v display different polarization transfer efficiency-dependencies on isotropic chemical shift differences: R2₂¹ recouples efficiently both small and large chemical shift differences (in proteins these correspond to aliphatic-to-aliphatic and carbonyl-to-aliphatic correlations, respectively), while R2₁¹ and R2₂² exhibit the maximum recoupling efficiency for the aliphatic-to-aliphatic or carbonyl-to-aliphatic correlations, respectively. At moderate MAS frequencies (10–20 kHz), all R2_n^v sequences introduced in this work display similar transfer efficiencies, and their performance is very similar to that of PDSD and DARR. Polarization transfer dynamics and chemical shift dependencies of these R2-driven spin diffusion (RDSD) schemes are experimentally evaluated and investigated by numerical simulations for [U-¹³C,¹⁵N]-alanine and the [U-¹³C,¹⁵N] N-formyl-Met-Leu-Phe (MLF) tripeptide. Further applications of this approach are illustrated for several proteins: spherical assemblies of HIV-1 U-¹³C,¹⁵N CA protein, U-¹³C,¹⁵N enriched dynein light chain DLC8, and sparsely ¹³C/uniformly ¹⁵N enriched CAP-Gly domain of dynactin. Due to the excellent performance and ease of implementation, the presented R2_n^v symmetry sequences are expected to be of wide applicability in studies of proteins and protein assemblies as well as other organic solids by MAS NMR spectroscopy.

Introduction

With the recent developments in decoupling and recoupling techniques, solid-state NMR has become a powerful tool for determining molecular structure and dynamics in biological solids, ranging from amyloid fibrils to membrane proteins to intact viruses to protein assemblies.^1–10 Magic angle spinning (MAS) recoupling methods, both dipolar- and scalar-based, have emerged as an essential tool for structural and dynamics characterization of uniformly and extensively enriched biopolymers.^11–24 With the recent breakthroughs in probe technology, very fast MAS frequencies of the order of 40–70 kHz are now accessible to the experimentalist,^25–27 and these fast MAS conditions result in greatly enhanced spectral resolution due to efficient averaging of the homonuclear ¹H-¹H dipolar interactions at spinning rates above 30 kHz, and due to efficient heteronuclear ¹H-X decoupling that can be achieved at high rotation frequencies. The narrow ¹H lines under very fast MAS also permit proton detection, and indeed, several investigators have recently demonstrated the benefits of proton-detected experiments in proteins, both in terms of greatly enhanced sensitivity and the additional information that can be gained by incorporating the proton dimension into the spectra.^28–31 In the heteronucleus-detected experiments, the narrow lines attained at very fast MAS frequencies also give rise to significant sensitivity enhancements, and therefore, very fast MAS conditions are in principle advantageous across the board for structural and dynamics studies of proteins and protein assemblies. However, implementation of fast-MAS protocols for protein structure determination requires the development of modified homo- and heteronuclear recoupling methods,^26,32–36 because many of the recoupling techniques that are commonly employed at MAS frequencies below 20 kHz do not work above 30 kHz.

Types of experiments that do not work in their original implementation under very fast MAS conditions comprise homonuclear spin-diffusion-based recoupling methods, such as proton-driven spin diffusion (PDSD)^37,38 and dipolar-assisted rotational resonance (DARR)^20,39 or RF-assisted diffusion (RAD).⁴⁰ These techniques are ubiquitously used in structural analysis of proteins and protein assemblies at moderate MAS frequencies (10–20 kHz), both for resonance assignments and for deriving long-range distance constraints as well as for quantitative CP/MAS measurements.^{1,6,7,11,12,22,23,41–57} These sequences are some of the very few homonuclear recoupling methods (along with PAR^34,58,59 and CHHC^60,61) that do not suffer from dipolar truncation^{36,44,62–64} and are thus particularly advantageous for extracting distance information in uniformly and extensively enriched proteins.

In PDSD experiments, the recoupling efficiency depends strongly on the line broadening due to the residual ¹H-X dipolar couplings and on the spinning frequency, and in some cases it is hard to achieve broadband homonuclear correlation spectra. This limitation is alleviated when a weak radio frequency (rf) field irradiation is applied on the protons during the mixing time, such as in DARR (RAD) experiments. For DARR (RAD) irradiation, the ¹H rf field strength is ω_1H=nω_r, satisfying the rotary resonance matching condition and resulting in broadband rotational resonance recoupling. During the mixing time, ¹³C-¹³C polarization transfer is driven by the reintroduced ¹H-¹³C and ¹H-¹H dipolar couplings, and the transfer efficiency for the coupled ¹³C spins is somewhat less dependent on MAS frequency than in the conventional PDSD experiment. While dependence of polarization transfer efficiency on the MAS frequency in DARR (RAD) experiments is not as dramatic as in the PDSD sequence, the dependence on the chemical shift difference between the coupled spins is very strong.⁶⁵ This chemical shift difference dependence is particularly severe at very fast MAS rates (>30 kHz), resulting in lower and non-uniform polarization transfer efficiencies across the spectrum, and at frequencies of 40 kHz or above DARR/RAD would not be a method of choice for ¹³C correlation spectroscopy.

Recently, several amplitude- and phase-modulated DARR/RAD experiments were developed to improve the spin diffusion efficiency under very fast MAS conditions.^36,65,66 Second-order rotational resonance conditions, ω₁ = nω_r ± Δω_iso, were introduced by Ernst et al.³⁶ Broadband homonuclear dipolar recoupling at very fast MAS frequencies can be performed by various amplitude-modulated ¹H rf field irradiation schemes, referred to as mixed rotational and rotary-resonance (MIRROR) technique. Tekely et al. proposed a phase-alternated recoupling irradiation (PARIS)^66,67 scheme to obtain more efficient polarization transfer at very fast MAS, and the phase alternated DARR irradiation compensates efficiently for rf field inhomogeneity and improves the magnetization transfer rate. Nevertheless, the chemical shift offset sensitivity still remains an issue in these sequences, and it is difficult to achieve uniform broadband polarization transfer.

In this work, we present a family of spin diffusion experiments in which the magnetization transfer is driven by rotor-synchronized R2_n^v symmetry-based recoupling. Such symmetry-based approaches in MAS recoupling pulse sequences were originally described by Levitt in a series of elegant studies that established the general design principles for these sequences as well as classified them as CN_n^v and RN_n^v schemes depending on the rotation properties of the spin angular momenta during the rotor-synchronized train of rf pulses.^68–71 RN_n^v -type sequences, employed in this work, are designed to produce a net rotation of the spin angular momenta by π around the x axis of the rotating frame.⁶⁸ Following the Levitt nomenclature, n and v are small integers and are referred to as the symmetry numbers of the pulse sequence. In the R2_n^v symmetry-based spin diffusion experiments introduced here, all pairwise combinations of n = 1, 2 and v = 1, 2 are employed to produce four sequences, R2₁¹, R2₁², R2₂¹, and R2₂². The pulse element of the R2 symmetry sequences can be either a basic π pulse or composite pulses such as 90°_x90°_·y90°_x, 90°_x270°_-x or 90°_x180°_y90°_x. In these R2 sequences, the rf field amplitude and phase alternation are defined according to the R-type symmetry. It should be noted that the basic R2₁¹ symmetry scheme has the same pulse type as the rotor synchronized PARIS⁶⁶ scheme, and that the basic R2₁² symmetry scheme has the same pulse type as the conventional DARR/RAD scheme. We demonstrate that these different R2-driven spin diffusion schemes (RDSD) exhibit similar transfer efficiencies at moderate MAS rates (<20 kHz) except that R2₁¹ and POST-R2₂¹ are advantageous for the recoupling of coupled spins with small chemical shift differences. At very fast MAS frequencies, the individual R2 irradiation schemes display distinct dependencies of the polarization transfer efficiencies on the chemical shift differences between coupled carbon spins. Experiments and numerical simulations in [U-¹³C,¹⁵N]-alanine and [U-¹³C,¹⁵N] N-formyl-Met-Leu-Phe (MLF) tripeptide indicate that under very fast MAS the polarization transfer in these R2_n^v spin diffusion experiments is driven by the broadened second-order rotational resonance condition (ω₁ ± nω_r - K_scω_DD ≤ Δω_iso ≤ ω₁ ± nω_r + K_scω_DD), and we present the theoretical treatment of these sequences that correctly accounts for the polarization transfer dynamics and for the observed chemical shift dependence of the individual R2_n^v schemes. Our results further indicate that due to these distinct chemical shift dependencies it is advantageous to employ a specific R2 sequence for a given correlation experiment, e.g., CACB vs. CACO.

We further demonstrate that these R2_n^v spin diffusion sequences perform very efficiently in proteins and protein assemblies, using three examples: spherical assemblies of U-¹³C,¹⁵N-enriched HIV-1 CA protein, U-¹³C,¹⁵N-enriched dynein light chain DLC8, and sparsely-¹³C/U-¹⁵N-labeled dynactin CAP-Gly domain. We anticipate that the approach presented here for R2_n^v spin diffusion under fast-MAS conditions will be broadly applicable to studies of proteins and protein assemblies by MAS NMR spectroscopy, which is a rapidly growing area of research.

Experiments and Methods

Samples

[U-¹³C,¹⁵N]-alanine and [U-¹³C,¹⁵N] N-formyl-Met-Leu-Phe (MLF) tripeptide were purchased from Cambridge Isotope Laboratories. Both powder samples were used in the subsequent NMR experiments without any further purification or recrystallization. The sample of spherical assemblies of U-¹³C,¹⁵N-enriched HIV-1 capsid protein (CA) was prepared according to established procedures.⁶ CAP-Gly sparsely enriched in ¹³C and uniformly labeled with ¹⁵N was prepared from E. coli, grown in minimal medium containing 1,3-¹³C glycerol (Cambridge Isotopes) as the sole carbon source and ¹⁵NH₄Cl as the sole nitrogen source, as described previously.^72–74 The solid-state NMR sample of CAP-Gly was prepared by controlled precipitation from polyethylene glycol,^36,44,62,63 slowly adding a solution of 30% PEG-3350 to the solution of 24.3 mg of CAP-Gly (38.5 mg/mL), both dissolved in 10 mM MES buffer (10 mM MgCl₂, pH 6.0), as described previously.⁷ U-¹³C, ¹⁵N-enriched dynein light chain protein (DLC8) was expressed in E.coli and purified as described previously.⁷⁵ The solid-state NMR sample of DLC8 was prepared by controlled precipitation, slowly adding a solution of 30% PEG-3350 to the solution of 11 mg of DLC8 (30 mg/ml). Prior to the precipitation step, both PEG-3350 and DLC8 were dissolved in 10 mM MES buffer (10 mM MgCl₂, pH 6.0) and doped with 50 mM EDTA-Cu(II). The protein samples were packed in 1.8 mm MAS rotors for solid-state NMR experiments.

Solid-State NMR Spectroscopy

All NMR experiments were carried out on a Varian InfinityPlus solid-state NMR spectrometer operating at a Larmor frequency of 599.8 MHz for ¹H, and 150.8 MHz for ¹³C. Solid-state NMR experiments were performed using either a 3.2 mm Varian triple-resonance T3 probe (spinning rates of up to 25 kHz) or a 1.8 mm MAS triple-resonance probe developed in the Samoson laboratory (spinning rates up to 50 kHz).

The pulse schemes for RDSD 2D experiments are shown in Figure 1. During the mixing period, a series of R2_n^v symmetry pulses is applied on the ¹H channel, as illustrated in Figure 1(b)–(e) for each specific R2_n^v experiment. For R2₁^v symmetry irradiation, the rf field strength equals the MAS frequency, and one rotor period contains two π pulses. For R2₂^v symmetry irradiation, the rf field strength equals half the MAS frequency, and one rotor period contains one π pulse. For POST-R2_n^v symmetry irradiation, the basic π pulse is replaced by a composite pulse, (π/2)₀(3π/2)₁₈₀, and the rf field strength is twice that of the R2_n^v symmetry irradiation, as shown in Figure 1(f) for POST R2₂¹.

Figure 1.

(a) The general pulse sequence for 2D ¹³C-¹³C R2_n^v homonuclear correlation experiments. An RF field irradiation is applied on proton spins during the mixing time. R irradiation represents rotor-synchronized R2-type symmetry pulses: (b) R2₁² (ω_rf = ω_r), (c) R2₁¹ (ω_rf = ω_r), (d) R2₂² (ω_rf = ½ ω_r), (e) R2₂¹ (ω_rf = ½ ω_r), and (f) POST R2₂¹ (ω_rf = ω_r).

NMR experiments on the MLF sample packed into a 3.2 mm rotor were recorded at room temperature and MAS frequency of 16 kHz. Typical 90°· pulse lengths were 2.8 µs for ¹H and 4.0 µs for ¹³C. The cross-polarization contact time was 1.4 ms, and the recycle delay was 3.0 s. Two-pulse phase-modulation (TPPM)^{76 1}H decoupling with an rf field strength of 96 kHz was performed during the acquisition, and continuous wave (CW) ¹H decoupling with the same rf field strength was applied during the t₁ evolution time. ¹H irradiation with an rf field strength of 16 and 8 kHz during the mixing time was applied for 2D R2₁^v and R2₂^v (v = 1, 2) correlation experiments, respectively. The 2D spectra were collected as (1000 × 256) (complex x real) matrices with 8 scans; TPPI (time proportional phase incrementation) scheme was used for phase-sensitive detection in the indirect dimension.⁷⁷ A series of experiments with different mixing times ranging between 1 and 50 ms were performed.

NMR experiments on an alanine sample packed into a 1.8 mm rotor were performed at the MAS frequency of 40 kHz. To reduce sample heating during fast MAS spinning, nitrogen gas at 0°C was used for cooling, resulting in a final sample temperature of 20°C. The rf field strength on the ¹H channel during the mixing time was 40 and 20 kHz for R2₁^v and R2₂^v experiments, respectively. CW ¹H decoupling with rf field strength of 9 kHz was employed in the t₁ and t₂ dimensions. The 2D spectra were collected as (2000 × 512) (complex x real) matrices with 8 scans; TPPI scheme was used for phase-sensitive detection in the indirect dimension. A series of experiments with different mixing times ranging between 5 and 500 ms were performed.

NMR experiments on protein samples packed into a 1.8 mm rotor were performed at the MAS frequencies of 10 and 40 kHz. To reduce the sample heating due to fast MAS spinning at 40 kHz, a stream of cooling nitrogen gas at −20°C, −25 °C and −35°C was used for the HIV-1 CA, CAP-Gly and DLC8 samples, resulting in final sample temperatures of 0°C, −5°C and −15°C, respectively. For all 2D R2_n^{v 13}C-¹³C correlation experiments, TPPI scheme was used for phase-sensitive detection in the indirect t₁ dimension. The other parameters were the same as those used for the experiments on alanine. For all RDSD NMR experiments conducted at the MAS frequency of 10 kHz, a 96 kHz ¹H TPPM decoupling was applied during t₁ and t₂. Detailed acquisition and procession parameters are shown in the Supporting Information.

Numerical Simulations

All numerical simulations were performed in SIMPSON.⁷⁸ 168 REPULSION (α, β) angles and 16 γ angles were used to generate a powder average. The atomic coordinates for the model spin systems employed in the simulations were taken from the SSNMR structure of the leucine residue in the N-f-MLF-OH tripeptide (PDB ID 1Q7O).⁵⁰ These atomic coordinates are generally regarded as a valid representation for spin systems of other amino acids. For C_α-C′ spin diffusion simulations, a spin cluster containing two carbons and one proton was used. For C_α-C_β spin diffusion simulations, a spin cluster containing two carbons and three protons was employed. The one-bond dipolar coupling constants for ¹H-¹³C and ¹³C-¹³C were 22,690 and 2,251 Hz, respectively. In all simulations, all possible pairwise dipolar couplings were taken into account. J couplings were ignored as their effects are negligible given their small size. Other parameters used in simulations were the same as in the corresponding experiments.

Results and Discussion

Symmetry-Based Spin Diffusion Experiments at Moderate MAS Frequencies

R2_n^v symmetry-based pulse irradiation on protons can be used to reintroduce hetero- and homonuclear dipolar couplings, which in turn allow for spin diffusion among carbon spins. The space-spin selection rules for the symmetry-allowed first-order average Hamiltonian terms have been reported elsewhere.⁶⁸ Figure 2 demonstrates space-spin selection diagrams for the interaction Hamiltonians in the R2₁¹ sequence. It can be seen that both the ¹H-¹³C heteronuclear dipolar couplings and the ¹H-¹H homonuclear dipolar couplings are reintroduced by the R2₁¹ symmetry irradiation, with six heteronuclear dipolar coupling terms H_IS⁽¹⁾ and ten homonuclear dipolar coupling terms H_II⁽¹⁾ of the first-order average Hamiltonian being symmetry-allowed. Even though spin diffusion between coupled ¹³C spins is primarily determined by the ¹³C-¹³C coupling strength and the first-order rotational resonance conditions (nω_r = Δω_iso), the reintroduced ¹H-¹³C and ¹H-¹H dipolar couplings represent an essential contribution, resulting in greatly broadened RR matching conditions. Furthermore, these dipolar couplings also contain the second-order average Hamiltonian terms. Even though the recoupled second-order average Hamiltonian terms are much smaller than the first-order contributions, they can also drive the ¹H-¹³C and ¹H-¹H cross relaxation and assist the ¹³C-¹³C spin diffusion. Table 1 summarizes the number of the first- and second-order average Hamiltonian terms in the R2_n^v symmetry sequences. These recoupled hetero- and homo-nuclear dipolar Hamiltonian terms would drive the polarization transfer (see the Supporting Information for details on the recoupled terms for each R2_n^v sequence). The first-order rotational resonance conditions would be broadened greatly by R2_n^v re-introduced dipolar interactions, and these can be expressed as, nω_r - K_scω_DD ≤ Δω_iso ≤ ± nω_r + K_scω_DD, where K_scω_DD denotes the size of the recoupled dipolar interactions by R2_n^v and K_sc is the corresponding scaling factor.

Figure 2.

Space-spin selection diagrams for the first-order average dipolar Hamiltonian in the R2₁¹ symmetry sequence for recoupling of I-S heteronuclear terms (a) and I-I homonuclear terms (b).

Table 1.

The total number of symmetry-allowed first- and second-order average Hamiltonian terms in the R2_n^v (n, v = 1 and 2) symmetry sequences.

Symmetry sequences	Number of symmetry-allowed terms
	${\overline{H}}_{IS}^{(1)}$	${\overline{H}}_{II}^{(1)}$	${\overline{H}}_{I\times IS}^{(2)}$	${\overline{H}}_{II\times IS}^{(2)}$
R212	6	10	96	208
R211	6	10	96	208
R222	0	20	96	208
R221	8	12	96	208

It is obvious from Table 1 that the number of the symmetry-allowed Hamiltonian terms is different for each recoupling sequence, except for R2₁² and R2₁¹. Even though for R2₁² and R2₁¹ the same number of first- and second-order Hamiltonian terms are symmetry-allowed, the nature of these recoupled Hamiltonian terms {l, m, λ, μ} as well as the corresponding scaling factors are totally different (see the Supporting Information), suggesting that different polarization transfer dynamics might be observed in the corresponding spin diffusion experiments. To examine the ¹³C-¹³C transfer rates in each specific sequence at moderate MAS frequencies, we have recorded a series of two-dimensional R2_n^v (n, v = 1 and 2) correlation spectra with varying mixing times in the MLF tripeptide. The ¹³C-¹³C polarization transfer build-up curves acquired at the MAS frequency of 16 kHz are illustrated in Figure 3a for the C_α-C_β correlation of Leu residue, and in Figure 3b for the C_α-C′ correlation of Met residue.

Figure 3.

Polarization transfer in the 2D R2_n^v-driven (n, v = 1 and 2) spin diffusion experiments as a function of the mixing time for (a) C_α-C_β of Leu residue and (b) C_α-C′ of Met residue in [U-¹³C,¹⁵N] N-formyl-Met-Leu-Phe (MLF) tripeptide. The sample was spun at a MAS frequency of 16 kHz. The cross-peak intensities in each spectrum were normalized to those of the corresponding diagonal peaks in the following way: $I_{norm}=\frac{2I_{AB}}{I_{AA}+I_{BB}}$, where I_norm is the normalized cross peak intensity, I_AB is the non-normalized cross peak intensity, I_AAand I_BB are the non-normalized intensities of the corresponding diagonal peaks. The build-up curves corresponding to the different recoupling sequences are color coded as follows: R2₁¹, black; R2₁², blue; R2₂¹, red; R2₂², green; and POST-R2₂¹, purple.

The results demonstrate that, despite the different number of symmetry-allowed Hamiltonian terms in each of the R2_n^v symmetry-based schemes, all of these sequences exhibit similar polarization transfer efficiencies and polarization transfer rates for C_α-C_β correlations at moderate spinning frequencies. On the other hand, for the C_α-C′ correlation, the R2₁¹ scheme shows a slightly higher polarization transfer rate than the other sequences. We also note that, as expected, the POST versions of sequences in which the basic π-pulse of the R-element is replaced with composite (90₀270₁₈₀) pulses, display improved transfer efficiency in the ¹³C-¹³C correlation experiments, especially for coupled ¹³C spins with large chemical shift differences.

Symmetry-Based Spin Diffusion Experiments at Very Fast MAS Frequencies: Alanine

Either proton-driven or dipolar-assisted spin diffusion can be used to perform broadband homonuclear correlation experiments, but both of these recoupling methods display low tolerance to large resonance frequency offsets, which are most pronounced at fast and very fast MAS rates (>30 kHz). Since we found that at moderate MAS frequencies the performance of the R2_n^v symmetry-driven sequences is largely independent of the chemical shift offset between the coupled spins, we have proceeded with examining the spin diffusion behavior of these sequences at very fast MAS frequencies. Two idealized models of dipolar-coupled spin networks were constructed that represent the smallest coupled C-C spin systems in proteins, using the atomic coordinates of the leucine residue in N-f-MLF-OH, as shown in Figure 4. We used these models in the numerical simulations, to evaluate the ¹³C_α-¹³C_β and ¹³C_α-¹³C′ polarization transfer dynamics.

Figure 4.

Spin models for numerical simulations of the polarization transfer dynamics in the R2_n^v symmetry-driven sequences. (a) C_α-C_β spin system (three protons and two carbons), (b) C_α-C′ spin system (one proton and two carbons). The atomic coordinates were obtained from the SSNMR structure of MLF⁵⁰. The interatomic distances are indicated in the figure. The oxygen atom was not taken into account in the simulations.

Figure 5 demonstrates the ¹³C-¹³C polarization transfer dynamics by basic (solid curves) and POST (dotted or dashed curves) R2_n^v symmetry sequences simulated for the above model spin systems at the MAS frequency of 40 kHz. The build-up curves for the C_α-C_β and C_α-C′ correlations are shown in Figure 5(a) and 5(b), respectively. As can be appreciated, under the very fast MAS conditions, the various R2_n^v symmetry driven schemes exhibit very different polarization transfer dynamics, depending greatly on the chemical shift difference between the coupled ¹³C spins. For the C_α-C′ polarization transfer by the basic R2_n^v, the R2₂² sequence with an rf field irradiation of 20 kHz exhibits the fastest polarization transfer rate. On the other hand, there is almost no C_α-C′ polarization transfer observed for the R2₁^v (v = 1, 2) sequences with rf field irradiation of 40 kHz. At the same time, for the C_α-C_β recoupling, the R2₁¹ symmetry scheme yields the most efficient polarization transfer. Even though the C_α-C_β polarization transfer can also be accomplished with the R2₂² sequence, the transfer rate is significantly slower. Confirming the experimental results, the current numerical simulations indicate that DARR (referred to as R2₁² symmetry sequence here) is inefficient for polarization transfer at high MAS frequencies; furthermore, it is very sensitive to the chemical shift differences, rendering DARR experiments impractical under these conditions.

Figure 5.

Simulated polarization build-up curves for different R2_n^v and POST R2₂^v recoupling sequences for (a) C_α-C_β and (b) C_α-C′ correlations. Spin models shown in Figure 4(a) and (b) were used for C_α-C_β and C_α-C′ simulations, respectively. The ¹H Larmor frequency was 599.8 MHz. The MAS frequency was 40 kHz. The solid build-up curves corresponding to the different basic R2_n^v recoupling sequences are color coded as follows: R2₁¹, red; R2₁², black; R2₂¹, blue; and R2₂², green. The dotted and dashed curves corresponding to POST-type symmetry sequences with composite pulses 90°_x270°_−x (POST1), and 90°_x90°_y90°_x (POST2), respectively.

As discussed above, the replacement of the basic π pulse by composite pulses, gives rise to the various POST R2_n^v symmetry sequences, and some of these are expected to yield higher transfer efficiency with the appropriate selection of composite pulses. For example, for C_α-C_β polarization transfer, POST R2₂¹ symmetry sequence consisting of (90_x90_y90_x) composite pulses, exhibits transfer efficiency much higher than the basic R2₁¹ scheme. In the case of C_α-C′ recoupling, the POST R2₂² symmetry sequence, consisting of (90_x270_−x) composite pulses, exhibits a much higher transfer efficiency than the basic R2₁¹ scheme. It should be noted, however, that not all POST R2_n^v sequences perform better than the basic R2_n^v schemes. For instance, then POST R2₂¹ symmetry sequence, consisting of (90_x270_−x) composite pulses, has a lower transfer efficiency than the basic R2₂¹ scheme for the C_α-C_β polarization transfer.

In order to understand the behavior of the R2_n^v symmetry driven sequences as a function of the MAS frequency and the chemical shift difference, one needs to further consider the various factors important in the polarization transfer dynamics. For example, at low or moderate MAS frequencies, DARR (or R2₁² symmetry sequence) can be regarded as a broadband rotary resonance (RR) based recoupling technique, since the first-order RR matching conditions are broadened greatly by the reintroduced ¹H-¹³C dipolar couplings. Following this logic, one might think that at very fast MAS frequencies, the transfer efficiency in the DARR experiments may be poor since the partly recoupled heteronuclear dipolar interactions are not strong enough to satisfy the first-order RR matching condition. To investigate this possibility, we have examined the extent of dipolar broadening by DARR and by other R2_n^v symmetry sequences at the MAS frequency of 40 kHz. Figure 6 shows the experimental ¹³C CP/MAS spectra of alanine spun at 40 kHz, recorded with the ¹H rf field irradiation during the acquisition period by (a) R2₁² (DARR), (b) R2₁¹, (c) R2₂¹, and (d) R2₂² symmetry pulses. The results show that ¹³C NMR lines are strongly broadened in the 1D R2₁^v experiments (Figure 6 (a) and (b)), and that the line shapes for these two cases are similar because both sequences have the same number of symmetry-allowed Hamiltonian terms (see Table 1). Interestingly, even though the R2₂¹ symmetry scheme retains the largest number of symmetry-allowed ¹H-¹³C and ¹H-¹H Hamiltonian terms, the broadening of the ¹³C NMR resonances by the R2₂¹ irradiation is much smaller than that by the R2₁¹ and R2₁² sequences. This is the result of the smaller scaling factor for recoupling of the heteronuclear dipolar couplings by R2₂¹. Since no first-order average ¹H-¹³C dipolar Hamiltonian terms are allowed in the R2₂² symmetry scheme (see Table 1), the ¹³C MAS spectrum exhibiting the least broadening in the presence of the ¹H rf field is obtained.

Figure 6.

1D ¹³C CP/MAS spectra of [U-¹³C,¹⁵N]-alanine spun at 40 kHz. These spectra were recorded with (a) R2₁², (b) R2₁¹, (c) R2₂¹, and (d) R2₂² irradiation applied to the protons during the acquisition period.

It is clear from the above discussion that ¹³C MAS spectra acquired with the various R2_n^v irradiation sequences are broadened to different degrees, and in the case of R2₂² irradiation, no broadening from the recoupled first-order average Hamiltonian interactions ensues. However, these re-introduced dipolar couplings cannot broaden efficiently the first-order rotational resonance conditions (nω_r = Δω_iso) under very fast MAS, and the polarization transfer profile depends to a greater extent on the second-order rotational resonance conditions (ω₁ = nω_r ± Δω_iso).³⁶ The strength of the rf field irradiation and the size of the recoupled dipolar couplings would jointly dominate the polarization transfer. The degree of broadening in the different R2_n^v symmetry sequences, therefore, does not directly correspond to the polarization transfer efficiencies, indicating that the dipolar broadening is not the dominant factor in spin diffusion experiments.

Figure 7 provides the experimental ¹³C-¹³C polarization transfer dynamics in the various R2_n^v symmetry-driven spin diffusion experiments for U-¹³C, ¹⁵N-alanine, spun at 40 kHz. The C_α-C_β and C_α-C′ build-up curves are plotted in Figure 7 (a) and (b), respectively. The results demonstrate that the R2₁¹sequence exhibits the most efficient polarization transfer for the C_α-C_β spin diffusion experiments. For the C_α-C′ polarization transfer, the R2₂^v sequences yields much higher transfer rates than the R2₁^v symmetry sequences. This finding is consistent with the simulation results shown in Figure 5 except for the differences in the transfer rates, which can be attributed to relaxation effects. Various cross-relaxation times, such as T₁^HH, T₁^HC and T₁^CC contribute to the magnetization transfer process, and were not taken into account in SIMPSON simulations. However, disregarding these relaxation times in the numerical simulations does not affect the comparison of the performances of the different R symmetry sequences, as demonstrated experimentally on protein samples (see below). Despite the difference in the absolute transfer rates between the experimental and simulated data the relative polarization transfer dynamics between the different R2 sequences is captured correctly in the simulations.

Figure 7.

Experimental build-up curves for polarization transfer in U-¹³C, ¹⁵N alanine: (a) C_α-C_β, and (b) C_α-C′, by R2₁², R2₁¹, R2₂¹, and R2₂² symmetry-based sequences. The sample was spun at a MAS frequency of 40 kHz. The cross-peak intensities in each spectrum were normalized to those of the corresponding diagonal peaks in the following way: $I_{norm}=\frac{2I_{AB}}{I_{AA}+I_{BB}}$, where I_norm is the normalized cross peak intensity, I_AB is the non-normalized cross peak intensity, I_AA and I_BB are the non-normalized intensities of the corresponding diagonal peaks.

It should be noted, that although the R2₁² scheme (DARR or RAD) resulted in the largest broadening of the ¹³C lines (Figure 6 and discussed above), the DARR polarization transfer efficiency at very fast MAS frequencies is very low, especially for the C^α-C′ correlation experiments, since neither the first-order (nω_r = Δω_iso) nor the second-order (ω₁ = nω_r ± Δω_iso) rotary resonance conditions are fulfilled at the MAS frequency of 40 kHz. In other words, the ¹H-¹³C heteronuclear dipolar couplings are no longer the primary factor in creating uniform ¹³C-¹³C polarization transfer at very fast MAS frequencies, and the polarization transfer profile depends on the rf field strength, the chemical shift difference and the size of the recoupled dipolar interactions as well. On the other hand, even though the other R2_n^v sequences also depend significantly on the chemical shift differences of coupled carbons, each of these experiments exhibits high recoupling efficiency for either C_α-C_β or C_α-C′ correlations, suggesting that with a pair of experiments, R2₁¹ and R2₂², both types of correlations can be recorded. Alternatively, if a single experiment is desirable to achieve both C_α-C_β and C_α-C′ correlations, the R2₂¹ symmetry sequence can be used. Although this experiment does not exhibit the highest transfer efficiencies for either C_α-C_β or C_α-C′ correlations, the transfer efficiency is still high enough to attain high sensitivity and uniform excitation for both spectral regions.

To further evaluate the experimental results, we examined the dependence of chemical shift difference on the polarization transfer dynamics at fast and moderate MAS frequencies by numerical simulations. Figure 8 illustrates the simulated dependence of the ¹³C-¹³C polarization transfer efficiency on the chemical shift difference for R2₁², R2₁², R2₂², and R2₂¹ sequences at the MAS frequency of 40 kHz. It can be noted that the R2₁¹ symmetry scheme exhibits relatively high polarization transfer efficiency for the coupled ¹³C spins with small chemical shift differences (of less than 47 ppm). This indicates that the R2₁¹ symmetry sequence is suitable for recording C_α-C_β, C_β-C_γ, and other sidechain correlations in spin diffusion experiments. On the other hand, essentially no polarization transfer is expected for the coupled ¹³C spins with chemical shift differences larger than 50 ppm in the R2₁¹ experiment, while DARR (the R2₁² sequence) results in very weak polarization transfer. At the same time, the R2₂² and R2₂¹ symmetry sequences give rise to much higher transfer efficiencies for correlations between coupled carbons with large chemical shift differences, such as C_α-C′ spin pairs whose chemical shift differences are of the order of 120 ppm, even multi-bond C_β/γ-C′ spin pairs with chemical shift differences of 120–170 ppm. This simulated behavior agrees well with the experimental results presented in Figures 5 and 7.

Figure 8.

Simulated dependence of the ¹³C-¹³C polarization transfer efficiency on the isotropic chemical shift difference for various R2_n^v symmetry sequences: (a) R2₁¹, (b) R2₁² (DARR), (c) R2₂¹, and (d) R2₂¹. The simulations were performed in SIMPSON and the model spin systems contained 2 protons and 2 carbons. The MAS frequency was 40 kHz, and the mixing time was 4 ms. Note, that the simulated resonance frequency offsets are ±25 kHz covering the entire spectral range for ¹³C at the magnetic field of 14.1 T.

It is obvious from the above that the polarization transfer under very fast MAS can be driven by the R2_n^v recoupled dipolar interactions when the second-order rotational resonance conditions are fulfilled, ω₁ ± nω_r -.K_scω_DD ≤ Δω_iso ≤ ω₁ ± nω_r + K_scω_DD, where K_scω_DD denotes the scaled dipolar couplings re-introduced by R2_n^v symmetry sequences. However, even when these conditions are fulfilled, it is still difficult to obtain a true broadband recoupling. The transfer efficiency is dependent on the chemical shift difference, and it is therefore anticipated that polarization transfer profile will be dependent on the magnetic field strength. Even though we have not yet performed R-type spin diffusion experiments at magnetic field strengths other than 14.1 T, the magnetic field dependence of the performance of these sequences can be understood by the analysis of Figure 8. The simulated dependence of the chemical shift difference reveals the conditions where strong vs. no correlations would be observed for each R2_n^v sequence. For instance, at the MAS frequency of 40 kHz, for the R2₂¹ sequence there will be no correlations for coupled ¹³C nuclei with chemical shift differences of 9–14 kHz. Translating the CS difference into ppm, we realize that no correlations are expected between the coupled ¹³C spins with the chemical shift differences of 90–140 ppm at 9.4 T, of 60–90 ppm at 14.1 T (as confirmed experimentally in this work), and of 40–60 ppm at 21.1 T. In other words, and considering that the R2-driven spin diffusion is accomplished by the second-order rotational resonance conditions, the above considerations imply that the correlation information for the desired chemical shift differences, i.e., aliphatic-to-carbonyl or aliphatic-to-aliphatic regions in proteins, can be obtained at each magnetic field by an appropriate setting of the MAS frequency for each R2_n^v sequence, i.e. 30 kHz at 9.4 T, 40 kHz at 14.1 T, and 60 kHz at 21.1 T.

Indeed, numerical simulations of the dependence of the polarization transfer efficiencies on the chemical shift differences for each R2_n^v sequence, conducted for the magnetic field strength of 21.1 T and the MAS frequency of 60 kHz demonstrate that the transfer profiles (see Figure S4 of the Supporting Information) are in agreement with the above analysis, which was made on the basis of the simulations at 14.1 T and 40 kHz.

At moderate MAS frequencies (<20 kHz), the numerical simulations indicate that the polarization transfer dynamics is only mildly dependent on the chemical shift differences, except for the rotational resonance conditions (see Supporting Information). Therefore, broadband homonuclear recoupling can be easily achieved by each of the four R2_n^v schemes, confirming our experimental observations. This behavior is expected to hold at moderate MAS frequencies for any magnetic field strength practically accessible in a modern NMR spectrometer.

Symmetry-Based Spin Diffusion Experiments at Very Fast MAS Frequencies: Applications to Proteins and Protein Assemblies

As discussed in the previous sections, at very fast MAS frequencies where DARR (R2₁²) experiment fails, the other R2_n^v symmetry sequences are expected to result in efficient polarization transfer and therefore should be useful for ¹³C-¹³C homonuclear correlation spectroscopy of proteins. In Figure 9, we provide the 14.1 T ¹³C-¹³C correlation spectra using the various R2_n^v symmetry schemes for spherical assemblies of U-¹³C,¹⁵N enriched HIV-1 CA capsid protein, spun at the MAS frequency of 40 kHz. We note that it is of great interest to further characterize the structure and dynamic properties of this morphology since it relates to the assembly in the immature HIV-1 virions. At the same time, the inherently limited spectral resolution of the spherical assemblies at moderate MAS frequencies (see Supporting Information showing a 10 kHz DARR spectrum) so far made their solid-state NMR studies challenging.

Figure 9.

2D ¹³C-¹³C correlation spectra of spherical assemblies of U-¹³C,¹⁵N-enriched HIV-1 CA, recorded at 14.1 T with the different R2_n^v sequences: (a) R2₁¹, (b) R2₁², (c) R2₂¹, and (d) R2₂². A mixing time of 250 ms was used to record each spectrum. The sample was spun at a MAS frequency of 40 kHz. CW ¹H decoupling (ω₁ = 9 kHz) was employed in the t₁ and t₂ dimensions.

As illustrated in Figure 9 (b), at 40 kHz the DARR spectrum exhibits only few cross peaks, even at the mixing time as long as 250 ms. Notably, even cross peaks for directly bonded carbons are not observed in the spectrum, confirming that DARR fails under very fast MAS conditions in proteins. On the other hand, the R2₁¹ correlation spectrum displayed in Figure 9 (a) contains a large number of cross peaks in the aliphatic region. However, almost no aliphatic-to-carbonyl correlations are seen, illustrating the inefficient recoupling of the carbon spin pairs with large chemical shift differences by this sequence. Conversely, Figure 9 (d) illustrates that the R2₂² symmetry irradiation results in very efficient recoupling between the carbonyl and aliphatic carbons with large chemical shift differences, while the polarization transfer within the aliphatic carbons is very weak. The most uniform polarization transfer is achieved by the R2₂¹ irradiation, and the corresponding spectrum is provided in Figure 9 (c), exhibiting cross peaks over the entire spectral range. At the same time, the polarization transfer efficiency for spin diffusion within the aliphatic carbons is about 70% of that in the R2₁¹ symmetry driven experiment. This is consistent with the numerical simulations presented in Figure 8 and with the model experiments on alanine presented in Figure 7, that show the second highest polarization transfer efficiency for the R2₂¹ symmetry scheme among all R2_n^v sequences, for both aliphatic-to-aliphatic and carbonyl-to-aliphatic carbons. Therefore, the combined results provided in Figure 9 demonstrate that the R2₂², R2₂¹, and R2₁¹ symmetry-driven spin diffusion experiments are efficient for ¹³C-¹³C correlation spectroscopy in proteins and can be used in place of DARR at very fast MAS frequencies.

Another observation made on the basis of the experimental ¹³C-¹³C spin diffusion spectra of spherical HIV-1 CA assemblies acquired at 40 kHz and at 10 kHz is that the 40 kHz R2-based spectra exhibit higher resolution (as illustrated in Figure S6 of the Supporting Information). To further test this experimental observation, we carried out R2-based experiments for U-¹³C,¹⁵N-enriched DLC8 protein. The resulting 2D ¹³C-¹³C correlation spectra acquired with MAS frequencies of 10 kHz (DARR) and 40 kHz (R2₁¹) are shown in Figure 10. As can be appreciated, the R2₁¹ correlation spectrum recorded at ω_r=40 kHz (Figure 10 (b)) exhibits significantly higher resolution without any loss of the correlation information, compared to the DARR spectrum at 10 kHz. The 1D slices extracted along the ω₁ and ω₂ frequency dimensions of the 2D spectra (Figure 10 (c) and (d)), clearly demonstrate that the lines are narrower along both frequency dimensions at the MAS frequency of 40 kHz, even when low-power CW ¹H decoupling with the rf field strength of 9 kHz was performed. This resolution gain is also obvious in the comparison of the frequency domains of the indirect (ω₁) dimensions of the 40 and 10 kHz R2 and DARR spectra (see Supporting Information).

Figure 10.

Comparison of the aliphatic region of the 2D ¹³C-¹³C correlation spectra recorded on U-¹³C,¹⁵N-enriched DLC8. (a) R2₁² (DARR) spectrum of the sample spun at 10 kHz, high-power TPPM ¹H decoupling with (ω₁ = 96 kHz) was employed in the t₁ and t₂ dimensions. (b) R2₁¹ spectrum of the sample spun at 40 kHz, low-power CW ¹H decoupling (ω₁ = 9 kHz) was employed in the t₁ and t₂ dimensions. Mixing times of 20 and 200 ms were used to record the DARR spectrum at 10 kHz and the R2₁¹ spectrum at 40 kHz, respectively. In (c) and (d), 1D slices extracted along the ω₂ (I-III) and ω₁ (IV-VI) frequency dimensions of the 2D ¹³C-¹³C correlation spectra. The first contour in all spectra is set at 5xσ (σ is the noise rmsd) with the multiplier of 1.2.

R2_n^v driven spin diffusion experiments were also conducted on the sparsely-¹³C, U-¹⁵N-enriched CAP-Gly domain of dynactin in which only few neighboring ¹³C atoms are enriched (for enrichment pattern, see^72–74). Figure 11 displays the 2D ¹³C-¹³C correlation spectra with the various R2_n^v symmetry schemes at the MAS frequency of 40 kHz. As can be appreciated, excellent resolution is seen, and the correlation information as well as the transfer efficiencies using the various R2_n^v sequences for the sparsely labeled CAP-Gly sample are consistent with those obtained for HIV-1 CA protein assemblies and DLC8. Interestingly, in the case of the sparsely enriched CAP-Gly protein, no appreciable resolution enhancements were noted for the MAS frequency of 40 kHz compared to the 10 kHz spectra (see Supporting Information), suggesting that considerable contributions to the linewidths in the 10 kHz spectra of the uniformly enriched proteins are due to ¹³C-¹³C dipolar and ¹³C-¹⁵N scalar couplings.

Figure 11.

2D ¹³C-¹³C correlation spectra of sparsely-¹³C, U-¹⁵N-enriched CAP-Gly domain of dynactin. The spectra were acquired at 14.1 T with the different R2_n^v sequences: (a) R2₁¹, (b) R2₁², (c) R2₂¹, and (d) R2₂². Each spectrum was recorded with a mixing time of 300 ms and the sample was spun at a MAS frequency of 40 kHz. The 2D spectra were collected as (1000 × 200) (complex x real) matrices with 128 scans; TPPI scheme was used for phase-sensitive detection in the indirect dimension. Low-power CW ¹H decoupling (ω₁ = 9 kHz) was employed in the t₁ and t₂ dimensions. The first contour in all spectra is set at 5xσ (σ is the noise rmsd) with the multiplier of 1.2.

Overall, the above results on model compounds and on three different protein samples demonstrate that R2₂², R2₂¹, and R2₁¹ experiments perform well at very fast MAS rates, while DARR and PDSD sequences fail. These R2₂², R2₂¹, and R2₁¹ experiments not only have high transfer efficiencies, but also display greatly enhanced spectral resolution for uniformly enriched samples under very fast MAS conditions. Both of these features are critically important in multidimensional experiments for resonance assignments and measurements of distance restraints in large proteins and protein assemblies, in both uniformly labeled and sparsely or selectively labeled samples.

Conclusions

In summary, we describe a family of rotor-synchronized R2_n^v symmetry sequences for homonuclear correlation spectroscopy in rotating solids, in which R2₁¹, R2₁², R2₂¹, or R2₂² irradiation is applied to the proton spins during the mixing time to assist ¹³C-¹³C polarization transfer. Three of these R2-driven spin diffusion sequences (RDSD), R2₁¹, R2₂¹, and R2₂², work efficiently at both moderate and fast MAS frequencies. Our results indicate that RDSD sequences exhibit similar transfer efficiencies at moderate MAS rates (up to 20 kHz), and that R2₁¹ and POST R2₂¹ sequences are superior for the recoupling of coupled spins that possess large chemical shift differences, such as carbonyl-to-aliphatic carbons. At very fast MAS frequencies (40 kHz), where the PDSD and DARR (R2₁²) experiments fail, the R2₂², R2₂¹, and R2₁¹ sequences exhibit high polarization transfer efficiencies. Since the broadened second-order rotational resonance conditions (ω₁ ± nω_r - K_scω_DD ≤ Δω_iso ≤ ω₁ ± nω_r+ K_scω_DD) are dominated by the rf field strength and the size of the re-introduced dipolar interactions, these symmetry-based experiments exhibit different dependencies on the chemical shift differences between coupled spins. The R2₁¹ sequence is most efficient for recoupling of the carbons with small chemical shift differences, while the R2₂² sequence shows highest polarization transfer efficiencies for coupled spins with large chemical shift differences. The R2₂¹ symmetry sequence displays high transfer efficiency and uniform excitation for both aliphatic-to-aliphatic and carbonyl-to-aliphatic correlations (see the summary figure in Supporting Information). With the replacement of the basic π pulse by various composite pulses, the appropriate POST R2_n^v symmetry sequences with appropriate are expected to give faster transfer rate and higher transfer efficiency. Given the advantageous properties of the R2_n^v symmetry-based spin diffusion schemes introduced here, we believe that they will become indispensible for resonance assignments and structure determination of proteins and protein assemblies.