Pulsed Third-Spin-Assisted Recoupling NMR for Obtaining Long-Range ¹³C-¹³C and ¹⁵N-¹³C Distance Restraints

Abstract

We present a class of pulsed third-spin-assisted recoupling (P-TSAR) magic-angle-spinning (MAS) solid-state NMR techniques that achieve efficient polarization transfer over long distances to provide important restraints for structure determination. These experiments utilize second-order cross terms between strong ¹H-¹³C and ¹H-¹⁵N dipolar couplings to achieve ¹³C-¹³C and ¹³C-¹⁵N polarization transfer, similar to the principle of continuous-wave (CW) TSAR experiments. However, in contrast to the CW-TSAR experiments, these P-TSAR experiments require much less radiofrequency (rf) energy and allow a much simpler routine for optimizing the rf field strength. We call the techniques PULSAR (PULSed proton Asissted Recoupling) for homonuclear spin pairs. For heteronuclear spin pairs, we improve the recently introduced ^PERSPIRATIONCP (Proton-Enhanced Rotor-echo Short Pulse IRradiATION Cross-Polarization) experiment by shifting the pulse positions and removing z-filters, which significantly broaden the bandwidth and increase the efficiency of polarization transfer. We demonstrate the PULSAR and ^PERSPIRATIONCP techniques on the model protein GB1, and found cross peaks for distances as long as 10 and 8 Å for ¹³C-¹³C and ¹⁵N-¹³C spin pairs, respectively. We then apply these methods to the amyloid fibrils formed by the peptide hormone glucagon, and show that long-range correlation peaks are readily observed to constrain intermolecular packing in this cross-β fibril. We provide an analytical model for the PULSAR and ^PERSPIRATIONCP experiments to explain the measured and simulated chemical shift dependence and pulse flip angle dependence of polarization transfer. These two techniques are useful for measuring long-range distance restraints to determine the three-dimensional structures of proteins and other biological macromolecules.

Graphical Abstract

Introduction

Magic-angle-spinning (MAS) solid-state nuclear magnetic resonance (SSNMR) spectroscopy produces high-resolution spectra by averaging orientation-dependent nuclear spin interactions such as chemical shift anisotropy (CSA), dipolar couplings, and quadrupolar couplings. MAS SSNMR is uniquely suited among spectroscopic techniques to provide atomic-resolution structural information of insoluble, non-crystalline biological macromolecules such as membrane proteins ^1–4, amyloid fibrils ^5–13, and cell walls ^14–17. Once chemical shift assignment is complete, three-dimensional structure determination predominantly relies on inter-atomic distance restraints, which are measured through internuclear dipolar couplings. Under MAS, dipolar couplings are reintroduced using tailored radiofrequency (rf) pulse sequences or a careful choice of spinning frequency ¹⁸. Heteronuclear dipolar couplings and distances can be measured using the Rotational Echo DOuble Resonance (REDOR) technique ¹⁹ and the related Transferred Echo DOuble Resonance (TEDOR) ^20–21, while homonuclear dipolar couplings and distances are typically measured qualitatively using spin diffusion techniques such as Proton-Driven Spin-Diffusion (PDSD) ²², Dipolar-Assisted Rotational Resonance (DARR) ^23–24 and COmbined R2^ν_n Driven spin diffusion (CORD) ²⁵. Higher magnetic field strengths are advantageous for obtaining higher spectral resolution and sensitivity. However, at higher fields the larger isotropic chemical shift differences weaken the spin diffusion mechanism. Moreover the larger CSA requires faster MAS to suppress, thus further weakening the dipolar couplings that drive homonuclear polarization transfer. In addition, nuclei with low gyromagnetic ratios (γ) have weak dipolar couplings, which make it difficult to measure the long distances that are especially useful for restraining three-dimensional structures. Therefore, it is important to develop improved pulse sequences for high-field fast MAS conditions to achieve efficient long-range polarization transfer ²⁶.

Third-Spin-Assisted Recoupling (TSAR) experiments are a class of techniques that transfer polarization between low-γ nuclei A and B ²⁷. This transfer occurs via cross terms in the second-order average Hamiltonian between the H–A and H–B dipolar couplings, which are generally much stronger than the direct A-B dipolar coupling. In continuous-wave TSAR (CW-TSAR) experiments, the homonuclear variant is known as Proton-Asisted Recoupling (PAR) ²⁸ whereas the heteronuclear variant is called Proton-Asissted Insensitive Nuclei Cross Polarization (^PAINCP) ^29–30. The CW irradiation in these experiments requires carefully optimized rf field strengths on both the ¹H channel and the heteronuclear channels. This is usually achieved by an extensive multidimensional search of the rf field strengths, without which the polarization transfer efficiency can easily fall well below the theoretical maximum ³¹. Moreover, mixing times of 10–30 ms are usually required to detect long-range ¹³C-¹³C and ¹⁵N-¹³C correlation peaks. The extended high-power CW irradiation on two to three channels can damage the NMR probe and heat-sensitive biological samples. These two practical drawbacks have limited the use of the PAR and ^PAINCP experiments, despite their theoretical ability for long-range spin polarization transfer, compared to the simpler spin diffusion experiments.

Here we introduce pulsed TSAR (P-TSAR) techniques that achieve efficient polarizarion transfer over long distances using low rf energy and simple optimization routines. These P-TSAR experiments rely on a pulsed spin-lock ³², in contrast to CW-TSAR experiments, which utilize a CW spin-lock. We demonstrate these techniques on a model protein, the β1 immunoglobulin binding domain of protein G (GB1), and apply them to the amyloid fibrils formed by the peptide hormone glucagon ¹². We then show by numerical simulations, average Hamiltonian Theory ³³ and an analytical model that these P-TSAR experiments have the same essential spin dynamics as CW-TSAR: they have vanishing first-order average Hamiltonians and rely on trilinear terms of the kind $H_z{C}_1^{\pm }{C}_2^{\pm }$ and $H_z{C}^{\pm }{N}^{\pm }$ in the second-order average Hamiltonian to mediate polarization transfer. We call the homonuclear experiment PULSed proton-Asissted Recoupling (PULSAR) (Fig. 1a) and the heteronuclear experiment Proton-Enhanced Rotor-echo Short Pulse IRradiATION (^PERSPIRATIONCP). The latter was introduced recently by us ³⁴ but is now significantly improved by altering the timing of the spin-lock pulses to better refocus the magnetization at the end of each rotor period (Fig. 1c, Fig. S1). This pulse timing change obviates the need for z-filters and increases the bandwidth of P-TSAR transfer. Most importantly, P-TSAR experiments use only 10–20% of the rf duty cycle of their CW-TSAR analogs on the low-frequency channels and are robust against variations in the heteronuclear rf powers.

Figure 1.

P-TSAR pulse sequences developed in this study (a, c), together with previously published CW-TSAR pulse sequences (b, d). (a) 2D ¹³C-¹³C PULSAR. (b) 2D ¹³C-¹³C PAR ²⁸. (c) 2D ¹⁵N-¹³C ^PERSPIRATIONCP with spin-lock pulses in the center of the rotor period. (d) 2D ¹⁵N-¹³C ^PAINCP ²⁸.

Materials and Methods

SSNMR experiments were conducted on Bruker Avance III HD 600 MHz (14.1 T) and Avance II 800 MHz (18.8 T) spectrometers equipped with triple-resonance ¹H/¹³C/¹⁵N 3.2 mm MAS probes. All experiments were carried out under 20 kHz MAS. ¹³C, ¹⁵N-labeled microcrystalline GB1 was expressed, purified and precipitated as previously described ^35–36. Most GB1 spectra were measured on a ¹³C, ¹⁵N-labeled GB1 that was H/D exchanged. About 20 mg protein was packed into a 3.2 mm Bruker MAS rotor. Site-specifically ¹³C, ¹⁵N-labeled glucagon peptide was purchased from Biopeptek Pharmaceuticals (Malvern, PA) and fibrillized in pH 2 solution at 21˚C for 7 days at a concentration of 8 mg/ml ¹². A total of 5 mg fibril was packed into a 3.2 mm rotor. For the anhydrous model compound formyl-Met–Leu–Phe (f-MLF) ³⁷, 2.8 mg peptide was center-packed in a 3.2 mm rotor.

PULSAR and ^PERSPIRATIONCP experiments (Appendix S4,5) were conducted at a set temperature of 278 K for GB1 and 258–263 K for f-MLF and hydrated glucagon fibrils. At 20 kHz MAS, the actual sample temperature was estimated to be 15 K higher than the set temperature due to frictional heating ³⁸. Two-Pulse Phase Modulation (TPPM) ¹H decoupling ³⁹ was applied at an rf field of 71–83 kHz during ¹³C detection. For 2D ¹³C-¹³C PULSAR experiments on GB1, the direct acquisition times were 21.5 ms and 16.1 ms for the 600 and 800 MHz spectra, respectively, whereas the indirect dimension maximum ¹³C evolution times were 9.9 ms and 7.4 ms, respectively. For 2D ¹³C-¹³C PULSAR experiments on glucagon at 800 MHz, the direct dimension acquisition time was 15.3 ms while the indirect dimension ¹³C evolution time was 7.4 ms. For the 2D ¹⁵N-¹³C ^PERSPIRATIONCP experiment on GB1 at 600 MHz, the ¹³C acquisition time was 11.9 ms and the maximum ¹⁵N evolution time was 10.2 ms. For the 2D ¹⁵N-¹³C ^PERSPIRATIONCP experiment on glucagon at 600 MHz, the ¹³C acquisition time was 11.9 ms and the ¹⁵N maximum evolution time was 6.6 ms. Most 2D experiments took 0.9–2.5 days, except for the ^PERSPIRATIONCP experiment on glucagon, which took 3.6 days. A pseudo-3D ^PERSPIRATIONCP experiment was conducted on the 800 MHz spectrometer in which one indirect dimension varied the ¹³C rf carrier frequency from 0 to 200 ppm in 2.5 ppm increments, and the other indirect dimension varied the spin-lock pulse flip angles for both ¹³C and ¹⁵N from 0 to 360º in 15º increments. For the spin-lock pulses, the ¹³C and ¹⁵N rf field strengths were 50 kHz and 38.8 kHz, respectively, while the ¹H CW rf irradiation was 56–59 kHz during PULSAR and ^PERSPIRATIONCP polarization transfer.

All 2D spectra were processed in TopSpin (Bruker BioSpin) and assigned in Sparky ⁴⁰. The distance upper limits were set to 10 Å for the ¹³C-¹³C cross peaks in the PULSAR spectra and 8 Å for the ¹³C-¹⁵N cross peaks in the ^PERSPIRATIONCP spectra. We calculated the GB1 structure using CYANA 2.1 ⁴¹. The (ϕ, ψ) torsion angles were predicted from backbone chemical shifts using the TALOS-N software ⁴². The angular uncertainty was set to twice the uncertainty given by TALOS-N. The structure calculation consisted of a total of 100 independent runs with 70,000 torsion angle steps each. The ten structures with the lowest target function were included in the final ensemble.

Numerical simulations of the PULSAR and ^PERSPIRATIONCP polarization transfer dynamics were conducted using SpinEvolution ⁴³. Analytical calculations and data analysis were performed using home-written MATLAB and Python scripts, which are provided in the Supporting Information. The magnetization trajectories and Bloch spheres were generated in MATLAB.

Results

Demonstration of PULSAR and ^PERSPIRATIONCP experiments on a microcrystalline protein

Fig. 1 and Fig. S1 show the pulse sequences for PULSAR and ^PERSPIRATIONCP and their CW-TSAR counterparts. Both P-TSAR experiments use pulses with a flip angle θ at the center of each rotor period on the low-frequency channels. CW decoupling occurs on the ¹H channel during the mixing time. As in PAR and ^PAINCP, the ¹H decoupling field strength is chosen to avoid the rotary resonance condition, ω₁= ω_r, as well as avoiding the CP matching condition with the heteronuclear channels. We conducted all experiments under 20 kHz MAS and used rf field strengths of 35–50 kHz for the θ-pulses.

We first demonstrate the PULSAR and ^PERSPIRATIONCP experiments on the model protein GB1 on a 600 MHz spectrometer (Fig. 2). We placed the ¹³C carrier frequency at 43 ppm, in the middle of the aliphatic region, and the ¹⁵N carrier frequency at 122 ppm, in the middle of the amide region. We varied the spin-lock pulse flip angle from 90° to 165°, and found 150° pulses to give maximum polarization transfer in a broadband fashion. The aliphatic ¹³C chemical shift range for which signal intensity is observed is about 9 kHz (60 ppm). Intensity is also observed in the aromatic and carbonyl regions of the spectrum. At 600 MHz, the aliphatic-carbonyl chemical shift difference is ~20 kHz (130 ppm). Smaller flip angles reduced the chemical shift range of polarization transfer, whereas larger flip angles decreased the total transferred magnetization. The optimal ¹H rf field strength during the ¹³C and ¹⁵N P-TSAR period is slightly less than $3{\omega }_r$ for both PULSAR and ^PERSPIRATIONCP experiments, consistent with numerical simulations (Fig. S2). The 2D ¹³C-¹³C PULSAR spectrum shows a large number of long-range correlations in every region of the 2D spectrum. These include aliphatic-aliphatic, aliphatic-aromatic, aliphatic-carbonyl, aromatic-aromatic correlations (Fig. 2a, b). Some of the cross peaks correspond to distances of 8 Å or more, such as T25-Y3 (Fig. 2c). Similarly, the 2D ¹⁵N-¹³C ^PERSPIRATIONCP spectrum showed many long-range correlations (Fig. 2d, e) such as V54-L7, which has a distance of 6 Å (Fig. 2f). We compared the performance of PULSAR with PAR on the 600 MHz spectrometer (Fig. S3). Relative to a 10 ms 2D PAR spectrum, the 10 ms PULSAR spectrum shows roughly equivalent sensitivity while the 20 ms PULSAR spectrum has significantly lower sensitivity than the 10 ms PAR due to relaxation. However, the 20 ms PULSAR spectrum displays many long-range cross peaks that are absent or very weak in the 10 ms PULSAR or PAR spectra. Both PULSAR experiments required less rf power than the PAR experiment.

Figure 2.

2D ¹³C-¹³C PULSAR and ¹⁵N-¹³C ^PERSPIRATIONCP spectra of GB1 measured on the 600 MHz spectrometer. The ¹³C rf carrier frequency was set at 42 ppm (orange arrow and dashed line). Positive intensities are shown in black and negative intensities are shown in blue. (a) 2D ¹³C PULSAR spectrum measured with 20 ms mixing and 150º recoupling pulses. The indirect dimension evolution time was 9.9 ms, while the direct dimension acquisition time was 21.5 ms. (b) Selected zoomed-in area and 1D cross sections of the PULSAR spectrum, showing some of the medium-range, long-range, and intermolecular correlation peaks. (c) GB1 structure, indicating one of the long-range contacts, between T25 and Y3, observed in the PULSAR spectrum. (d) 2D ¹⁵N-¹³C ^PERSPIRATIONCP spectrum measured with 15 ms mixing and 150˚ recoupling pulses. The ¹⁵N maximum evolution time was 10.2 ms while the direct dimension ¹³C acquisition time was 11.9 ms. (e) Selected zoomed-in area and 1D cross sections of the ^PERSPIRATIONCP spectrum, showing some of the medium-range, long-range, and intermolecular correlation peaks. (f) GB1 structure, indicating one of the long-range contacts, between V54 and L7, whose cross peak is observed in (d).

Because of the enhanced sensitivity and resolution provided by higher magnetic fields, we next conducted PULSAR experiments on an 800 MHz spectrometer, where the aliphatic chemical shifts span a range of 12 kHz and the aliphatic-carbonyl chemical shift separation increases to 26 kHz. To achieve the same offset-dependent polarization transfer as the 600 MHz spectra, one would need to spin the samples at 27 kHz. Since this is not feasible using 3.2 mm MAS rotors and nitrogen gas for spinning, we kept the MAS rate at 20 kHz and varied the pulse flip angle to optimize polarization transfer in the spectral region of interest. With 150º pulses, ¹³C magnetization from the aliphatic carbons gave positive transfer to other aliphatic carbons, negative transfer to aromatic carbons (110–156 ppm), and positive transfer to carbonyl carbons (172–182 ppm) (Fig. 3a, c). With 210º pulses, the transfer profile has a null in the aromatic region and negative intensities in the carbonyl region (Fig. 3b). Similarly long distances were observed as spectra measured at 600 MHz.

Figure 3.

2D ¹³C-¹³C PULSAR spectra of GB1 measured at 800 MHz to investigate the offset dependence of the experiment. (a) 2D spectrum measured using 150º spin-lock pulses. (b) 2D spectrum measured using 210˚ spin-lock pulses. The rf fields were 50 kHz for ¹³C and 56 kHz for ¹H and the mixing time was 20 ms. Both spectra were measured using a maximum evolution time of 7.4 ms and direct acquisition time of 15.3 ms. (c) Corresponding 1D cross sections illustrating the signal-to-noise ratios of the two spectra. The aromatic region was processed with Gaussian broadening to enhance the sensitivity, while the aliphatic and carbonyl regions were processed using a qsine window function to enhance the resolution. (d) 112 medium-range (green) and 176 long-range (red) correlations in GB1 obtained from these PULSAR and ^PERSPIRATIONCP spectra. (e) Lowest-energy ensemble of GB1 structures calculated based on the distance constraints obtained from the PULSAR and ^PERSPIRATIONCP spectra. The backbone ensemble shows that P-TSAR distance restraints combined with (ϕ, ψ) torsion angles are sufficient to determine the 3D structure of the protein. The M1-T11 β-strand is shown on the right to illustrate the sidechain conformations constrained by the long-range correlations.

These 2D ¹³C-¹³C and ¹⁵N-¹³C correlation spectra yielded a total of 100 sequential, 112 medium-range (2 < |i-j| < 5), and 176 long-range ( |i-j| > 5) distance restraints (Fig. 3d). Combining these restraints with (ϕ, ψ) torsion angles obtained from backbone chemical shifts, we calculated the GB1 structure. We set the upper distance limits to 10 Å for PULSAR cross peaks and 8 Å for ^PERSPIRATIONCP cross peaks. These upper limits were estimated by finding cross peaks corresponding to the longest distance in the high-resolution NMR structure of GB1 ⁴⁴ and ensuring that this upper limit does not cause restraint violations in the structure calculation. The medium-range restraints are particularly useful for determining the secondary structure, since i to i + 2 cross peaks are expected for β-strands while i to i + 3 or 4 cross peaks are expected for α-helices. The long-range restraints helped to define both the backbone fold and the sidechain conformations. The ten lowest-energy structures display a backbone RMSD of 0.4 ± 0.1 Å and a heavy-atom RMSD of 1.3 ± 0.2 Å (Fig. 3e). Therefore, these ¹³C-¹³C PULSAR and ¹⁵N-¹³C ^PERSPIRATIONCP restraints are sufficient to solve the high-resolution structure of GB1.

Application of P-TSAR to glucagon amyloid fibrils

To test the ability of the PULSAR and ^PERSPIRATIONCP experiments for structure determination of non-crystalline proteins, we applied them to the amyloid fibrils formed by the peptide hormone glucagon ¹². Glucagon is used as a pharmaceutical compound to treat severe hypoglycemia. At the high concentrations required for intravenous administration, it fibrillizes rapidly ⁴⁵ into cross-β structures. This fibrillization tendency has limited glucagon’s formulation to a dry powder, to be mixed in acidic solution immediately prior to injection. Understanding the atomic structure of glucagon fibrils is thus important for designing glucagon analogs that resist fibril formation. We measured the 2D PULSAR and ^PERSPIRATIONCP spectra of a glucagon fibril containing nine ¹³C, ¹⁵N-labeled residues (Fig. 4a). The 2D ¹³C-¹³C PULSAR spectrum was measured at 800 MHz using a 12.5 ms mixing time while the 2D ¹⁵N-¹³C ^PERSPIRATIONCP spectrum was measured at 600 MHz using a 15 ms mixing time (Fig. 4b, c). Many medium-range intramolecular cross peaks were observed, such as T5Cβ-Q3Cβ and Q24Cγ-L26Cδ₁. This is consistent with the location of i and i + 2 sidechains on the same side of the β-sheet plane. We also detected long-range correlations such as T5Cα-M27Cα and T5Cα-L26Cδ in the PULSAR spectrum and G4N-N28Cβ and W25N-M27Cε cross peaks in the ¹⁵N-¹³C correlation spectra. These long-range correlations can only be assigned to intermolecular contacts, based on additional experiments of isotopically diluted samples ¹². Therefore, the long-range correlations define the antiparallel packing of β-strands. Intra-residue backbone ¹⁵N to sidechain ¹³C correlations are also observed for bulky sidechains such as Q24N-W25Cζ₃. These restraints combine with intermolecular correlations involving the same W25 sidechain to determine the rotameric structure of the bulky sidechain (Fig. 4d, e).

Figure 4.

2D PULSAR and ^PERSPIRATIONCP spectra of glucagon fibrils for obtaining distance restraints. (a) Amino acid sequence of glucagon. ¹³C, ¹⁵N-labeled residues are shown in red. (b) 12.5 ms 2D ¹³C PULSAR spectrum, measured on the 800 MHz spectrometer using 150º ¹³C recoupling pulses. The spectrum was measured with a maximum evolution time of 2.6 ms and a direct acquisition time of 12.3 ms. (c) 15 ms 2D ¹⁵N-¹³C ^PERSPIRATIONCP spectrum, measured on the 600 MHz spectrometer using 150º ¹³C and ¹⁵N recoupling pulses. The t₁ evolution time was 6.6 ms while the direct dimension acquisition time was 11.9 ms. (d) Sequential and medium-range intramolecular correlations found in the PULSAR and ^PERSPIRATIONCP spectra. These correlations help to define the rotameric conformation of sidechains such as W25. (e) Intermolecular correlations define antiparallel hydrogen bonding of the β-strands along the fibril axis.

Numerical simulations of CW-TSAR and P-TSAR experiments

We conducted numerical simulations to understand the chemical shift offset and rf field dependences of the P-TSAR experiments compared to CW-TSAR. A 7-spin system was used for PULSAR and PAR while an 8-spin system was used for ^PERSPIRATIONCP and ^PAINCP. These spin systems were taken from amino acid Met to represent the backbone and sidechain structure of a typical residue (Fig. 5a, d). For PULSAR and ^PERSPIRATIONCP experiments, we simulated the polarization transfer as a function of ¹³C chemical shift offset of the observed (sink) spins and as a function of the pulse flip angle. For PAR and ^PAINCP experiments, we simulated the transfer efficiency as a function of the rf field strength. Representative simulation scripts are given in Supporting Information Appendix S1.

Figure 5.

Numerical simulations of the chemical shift dependence of TSAR experiments. A mixing time of 20 ms was used in all simulations. (a) 7-spin system for simulating the PAR and PULSAR ¹³C-¹³C polarization transfer. The ¹⁵N spin is excluded in these simulations. (b) PAR and PULSAR efficiencies at 600 MHz as a function of the observed ¹³C chemical shifts. The ¹³C rf carrier frequency is denoted with a black arrow while the ¹³C source spin chemical shift is denoted with a green dashed line. (c) PULSAR transfer efficiency at 800 MHz. Right: 2D contour map of the PULSAR transfer efficiency as a function of sink spin chemical shift and pulsed spin-lock flip angle. (d) 8-spin system used for simulating ^PAINCP and ^PERSPIRATIONCP ¹⁵N-¹³C polarization transfer. (e) ^PAINCP and ^PERSPIRATIONCP efficiencies at 600 MHz as a function of the observed ¹³C chemical shifts. (f) ^PERSPIRATIONCP transfer efficiencies at 800 MHz. Right: 2D contour plot of the ^PERSPIRATIONCP transfer efficiency as a function of sink-spin chemical shift and pulse flip angle.

PAR transfer is optimized when the source and sink spin have similar ¹³C chemical shift offsets. With the ¹³C rf carrier frequency at 100 ppm, the PAR polarization transfer from a CH₂ carbon at 40 ppm is efficient to aliphatic signals near the source ¹³C chemical shift and to carbonyl carbons. However, the polarization transfer is minimal at ~100 ppm (Fig. 5b, left). Over the range of ¹³C rf field strength from 62 to 42 kHz, stronger rf fields give higher polarization transfer efficiencies. ¹⁵N-¹³C ^PAINCP exhibits a similar offset dependence (Fig. 5e, left), but the transfer efficiency is much more sensitive to the ¹³C rf field strength: much higher transfer efficiency is found at a ¹³C rf field of 50 kHz than at 49 kHz, while ¹H and ¹⁵N rf fields are held constant at 59.5 kHz and 50 kHz, respectively. The same modest ¹³C rf field change also changed the offset at which maximum transfer occurs, from 60 ppm to 30 ppm. Thus, ^PAINCP is extremely sensitive to both the rf field strength and the chemical shift offset.

Compared to these CW-TSAR polarization transfer profiles, the P-TSAR experiments give more broadband transfer and are less sensitive to the rf field strength. Maximum PULSAR transfer is also found when the source and sink spin have similar chemical shift offsets (Fig. 5b, right, Fig. 5c). However, unlike PAR, there are other matching conditions that can accomplish polarization transfer. Prudent choice of pulse flip-angle and carrier frequency can allow an experiment to have a much wider range of chemical shift offsets with efficient transfer. At 600 MHz, when the ¹³C carrier frequency is set to 38 ppm, the optimal ¹³C flip angle is ~150°, at which efficient transfer is found for all three regions of the ¹³C spectra: aliphatic, aromatic, and carbonyl (Fig. 5b, right). When the flip angle increases to 210˚, the aromatic signals becomes positive while the carbonyl intensities vanish. At 800 MHz, the highest polarization transfer is found at a flip angle of 210˚, where positive transfer is observed from 0–130 ppm and negative transfer is observed for carbonyl carbons (Fig. 5c). The ¹⁵N-¹³C ^PERSPIRATIONCP transfer profiles (Fig. 5e, right, Fig. 5f) are similar to the PULSAR transfer profiles.

Average Hamiltonian analysis of P-TSAR

To further understand the mechanism of polarization transfer in PULSAR and ^PERSPIRATIONCP, we analyzed their average Hamiltonians ³³ using a similar approach to the CW-TSAR treatment ^{28, 30}. For clarity, we first analyze ^PERSPIRATIONCP; explanations on how the analysis changes for PULSAR are given in the main text and Supporting Information. We consider a three-spin system consisting of one ¹H, one ¹⁵N and one ¹³C. The ¹H spin is subjected to CW rf irradiation while δ-function pulses with flip angles θ_C and θ_N are applied at the center of each rotor period to ¹³C and ¹⁵N, respectively (Fig. 1a, c). The Zeeman-truncated internal Hamiltonian for ^PERSPIRATIONCP is

$$\begin{gathered} {\mathcal{H}}_{tot}(t)=\Delta {\omega }_C{\widehat{C}}_z+\Delta {\omega }_N{\widehat{N}}_z+\Delta {\omega }_H{\widehat{H}}_z+{\omega }_{CN}2{\widehat{N}}_z{\widehat{C}}_z+{\omega }_{HC}2{\widehat{H}}_z{\widehat{C}}_z+{\omega }_{HN}2{\widehat{H}}_z{\widehat{N}}_z \\ +{\omega }_{1,H}{\widehat{H}}_x+\left({\theta }_C{\widehat{C}}_x+{\theta }_N{\widehat{N}}_x\right)\sum _{l=0}^N\delta (t-(l+1/2){\tau }_R) \end{gathered}$$ (1)

Here $\Delta {\omega }_C$ and $\Delta {\omega }_N$ are the sum of the isotropic chemical shift offset and chemical shift anisotropy (CSA) of ¹³C and ¹⁵N, ${\omega }_{CN}$, ${\omega }_{HC}$ and ${\omega }_{HN}$ are heteronuclear dipolar couplings, and ω_1,H is the ¹H CW rf field strength. The Dirac δ-function $\delta (t-(l+1/2){\tau }_R)$ specifies the ¹³C and ¹⁵N pulses in the middle of each rotor period, ${\tau }_R$. MAS introduces time-dependence in the CSA and dipolar couplings, which is not explicitly shown in Eq. (1).

We transform this Hamiltonian into the interaction frame defined by the rf pulses for ¹H and the frame defined by both rf pulses and isotropic chemical shift offset for ¹³C and ¹⁵N. We neglect ¹H isotropic chemical shift because CW irradiation truncates chemical shift evolution. In contrast, the heteronuclear spins undergo chemical shift evolution between pulses. This interaction-frame transformation can be written as:

$${\tilde{\mathcal{H}}}_{tot}(t)=U_{C,rf+cs}^{-1}\left(t\right)U_{N,rf+cs}^{-1}\left(t\right)U_{H,rf}^{-1}\left(t\right){\mathcal{H}}_{\mathrm{int}}\left(t\right)U_{H,rf}\left(t\right)U_{N,rf+cs}\left(t\right)U_{C,rf+cs}\left(t\right)$$ (2)

Here ${\mathcal{H}}_{\mathrm{int}}$ represents all the terms in Eq. (1) except for the rf pulses and the ¹⁵N and ¹³C isotropic chemical shift. The interaction-frame transformations in Eq. (2) each only affect one spin. As such, we can separately transform each one-spin operator in the Hamiltonian, and multiply the results together to obtain the two-spin product operators. We thus calculate the effect of the interaction frame transformation on the product operators ${\widehat{H}}_z$, ${\widehat{C}}_z$, and ${\widehat{N}}_z$.

For the ¹H spin we consider the interaction frame of the ¹H CW rf irradiation. Similar to the treatment of PAR ²⁸, we rotate ${\widehat{x}}_H$ to ${\widehat{z}}_H$ so that the spin-lock direction is ${\widehat{z}}_H$. We convert product operators to spherical tensor operators:

$${\widehat{H}}_z(t)=\frac{1}{\sqrt{2}}\left\{T_{11}^H{e}^{+i{\omega }_{1,H}t}-T_{1-1}^H{e}^{-i{\omega }_{1,H}t}\right\}$$ (3)

The propagator for the ¹³C rf pulse and isotropic shift is:

$$U_{C,rf+cs}(t)=e^{-i{\Omega }_C{\widehat{C}}_z(t-n{\tau }_R)}{\left(e^{-i{\Omega }_C{\widehat{C}}_z{\tau }_R/2}{e}^{-i\theta {\widehat{C}}_x}{e}^{-i{\Omega }_C{\widehat{C}}_z{\tau }_R/2}\right)}^n,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\equiv ⌊t/{\tau }_R+1/2⌋$$ (4)

where n is the number of elapsed pulses defined by the floor function: $\hspace{0.17em}⌊x⌋$ is x rounded down to the nearest integer. Since any series of successive rotations is equivalent to a single rotation about a new axis, we define a pulsed spin-lock axis ${\widehat{x^{\prime }}}_C$ in the x-z plane, at an angle ${\psi }_C$ from the x-axis towards the z-axis, and call the net rotation about this axis in each rotor period ${\phi }_C$. Each ¹³C spin in the system has a distinct isotropic chemical shift, and thus also has a distinct pulsed spin-lock direction ${\widehat{x^{\prime }}}_C$ and net rotation per rotor period ${\phi }_C$. The propagator in Eq. (4) can then be simplified as:

$$U_{C,rf+cs}(t)=e^{-i{\Omega }_C{\widehat{C}}_z(t-n{\tau }_R)}{e}^{-in{\phi }_C{\widehat{C}}_{x^{\prime }}}$$ (5)

We next consider the interaction-frame transformation of ${\widehat{C}}_z$:

$${\tilde{C}}_z(t)=e^{+in{\phi }_C{\widehat{C}}_{x^{\prime }}}\left(\cos ({\psi }_C){\widehat{C}}_{z^{\prime }}+\sin ({\psi }_C){\widehat{C}}_{x^{\prime }}\right)e^{-in{\phi }_C{\widehat{C}}_{x^{\prime }}}$$ (6)

Similar to ¹H, we rotate ${\widehat{x^{\prime }}}_C$ to ${\widehat{z^{\prime }}}_C$ to make the pulsed spin-lock direction ${\widehat{z^{\prime }}}_C$, and convert product operators to spherical tensor operators:

$${\tilde{C}}_z(t)=\sin ({\psi }_C)T_{10}^C+\frac{\cos ({\psi }_C)}{\sqrt{2}}\left\{T_{11}^C{e}^{+in{\phi }_C}-T_{1-1}^C{e}^{-in{\phi }_C}\right\}.$$ (7)

Here, for clarity, we have dropped the primes for the rotated axes on the spherical tensor operators. The interaction-frame ¹⁵N operators are identical to Eq. (4)-(7) except with all C replaced by N: the ¹⁵N spin has its own pulsed spin-lock direction ${\widehat{x^{\prime }}}_N$, and the interaction-frame ${\tilde{N}}_z(t)$ is given by an analogous equation to Eq. (7). The interaction-frame Hamiltonian (Eq. (2)) can thus be calculated by replacing the spin operators in Eq. (1) with their interaction-frame representations, Eq. (3) and (7):

$$\begin{gathered} {\tilde{\mathcal{H}}}_{tot}=\stackrel{1}{\overbrace{\sum _{\begin{array}{l}{m}_1=-2\\ m_1\ne 0\end{array}}^2\sum _{q_{C=-1}}^1\sum _{\begin{array}{l}{q}_{H1}=-1\\ q_{H1}\ne 0\end{array}}^1{\omega }_{HC}^{m_H}{q}_{H1}\left[q_C\cos ({\psi }_{C_1})+{\delta }_{0,q_C}\sqrt{2}\sin ({\psi }_C)\right]T_{1q_{H1}}^H{T}_{1q_C}^C\exp \left(in{\phi }_C{q}_C\right)\exp \left(i[q_{H1}{\omega }_{1,I}+m_1{\omega }_R]t\right)}} \\ +\stackrel{2}{\overbrace{\sum _{\begin{array}{l}{m}_2=-2\\ m_2\ne 0\end{array}}^2\sum _{q_{N=-1}}^1\sum _{\begin{array}{l}{q}_{H2}=-1\\ q_{H2}\ne 0\end{array}}^1{\omega }_{HN}^{m_2}{q}_{H2}\left[q_N\cos ({\psi }_N)+{\delta }_{0,q_N}\sqrt{2}\sin ({\psi }_N)\right]T_{1q_{H2}}^H{T}_{1q_N}^N\exp \left(in{\phi }_N{q}_N\right)\exp \left(i[q_{H2}{\omega }_{1,H}+m_2{\omega }_R]t\right)}} \\ +\stackrel{3}{\overbrace{\sum _{S=C,N}\sum _{\begin{array}{l}{m}_3=-2\\ m_3\ne 0\end{array}}^2\sum _{q_{S=-1}}^1{\omega }_{csa,S}^{m_3}\left[q_S\cos ({\psi }_S)/\sqrt{2}+{\delta }_{0,q_S}\sin ({\psi }_S)\right]T_{1q_S}^S\exp \left(in{\phi }_S{q}_S\right)\exp \left(i{m}_3{\omega }_{R}t\right)}} \\ +\stackrel{4}{\overbrace{\sum _{m_4=-2}^2\sum _{\begin{array}{l}{q}_{H4}=-1\\ q_{H4}\ne 0\end{array}}^1\Delta {\omega }_H^{m_4}\left[q_{H4}/\sqrt{2}\right]T_{1q_{H4}}^H\exp \left(i[q_{H4}{\omega }_{1,H}+m{\omega }_R]t\right)}} \\ +\stackrel{5}{\overbrace{\sum _{\begin{array}{l}{m}_5=-2\\ m_5\ne 0\end{array}}^2\sum _{q_C=-1}^1\sum _{q_N=-1}^1\begin{array}{l}{\omega }_{CN}^{m_5}\left[q_C\cos ({\psi }_C)/\sqrt{2}+{\delta }_{0,q_C}\sin ({\psi }_C)\right]\left[q_N\cos ({\psi }_N)/\sqrt{2}+{\delta }_{0,q_N}\sin ({\psi }_N)\right]\\ \cdot T_{1q_C}^C{T}_{1q_N}^N\exp \left(in\left[{\phi }_C{q}_C+{\phi }_N{q}_N\right]\right)\exp \left(i{m}_5{\omega }_{R}t\right)\end{array}}} \end{gathered}$$ (8)

In Eq. (8) ${\delta }_{0,q_C}$ is the Kronecker delta and n is the number of elapsed pulses. Terms 1 and 2 are the ¹³C-¹H and ¹⁵N-¹H dipolar couplings, term 3 is the ¹³C and ¹⁵N CSA, term 4 is the ¹H chemical shift, and term 5 is the ¹⁵N-¹³C dipolar coupling.

To apply average Hamiltonian theory (AHT) ³³, there must be an integer N such that ${\tilde{\mathcal{H}}}_{\mathrm{int}}$ is periodic with period $N{\tau }_R$. This requires $N{\tau }_R{\omega }_{1,H}$, $N{\phi }_C$, and $N{\phi }_N$ to be integer multiples of 2π. We assume that such an integer N exists, that there is no rotary resonance recoupling, i.e. $q_H{\omega }_{1,H}+m{\omega }_R\ne 0$, and that neither the ¹³C nor ¹⁵N rf field strengths fulfill Hartman-Hahn matching conditions with the ¹H rf field, i.e. $q_H{\omega }_{1,H}{\tau }_R+q_S{\phi }_S\ne 2\pi l$, where $l$ is an integer. Under these conditions, the first-order average Hamiltonian is zero because term 4 integrates over $N{\tau }_R$ to zero, terms 3 and 5 integrate over each rotor period between pulses to zero, and terms 1 and 2 can also be proven to vanish to first order (Supporting Information 1).

For homonuclear polarization transfer by PULSAR, the internal Hamiltonian is identical to the heteronuclear Hamiltonian in Eq. (1) except for an additional “flip-flop” term, $-{\omega }_{C_1{C}_2}\left({\widehat{C}}_{1x}{\widehat{C}}_{2x}+{\widehat{C}}_{1y}{\widehat{C}}_{2y}\right)$. Numerical simulations (Fig. S4) indicate that ¹³C-¹³C dipolar couplings have minimal effect on P-TSAR transfer, therefore we neglect the “flip-flop” term in this discussion. A full treatment of the first-order average Hamiltonian of homonuclear dipolar couplings can be found in Supporting Information 2.

Because the first-order average Hamiltonian is zero, the second-order average Hamiltonian is expected to dominate. As in CW-TSAR^{28, 30}, we attribute P-TSAR transfer to the cross-term between terms 1 and 2 in Eq. (8) in the second-order average Hamiltonian:

$${\overline{\tilde{\mathcal{H}}}}_{1\times 2}^{(2)}=\sum _{\begin{array}{l}{m}_1,q_C,q_{H1}\\ m_2,q_N,q_{H2}\end{array}}\left\{\begin{array}{l}\frac{1}{2iN{\tau }_R}{\omega }_{HC}^{m_1}{q}_{H,1}\left(q_C\cos {\psi }_{C_1}+{\delta }_{0,q_C}\sqrt{2}\sin {\psi }_C\right){\omega }_{HN}^{m_2}{q}_{H,2}\left(q_N\cos {\psi }_N+{\delta }_{0,q_N}\sqrt{2}\sin {\psi }_N\right)\\ T_{1q_C}^C{T}_{1q_N}^N\left[T_{1q_{H1}}^H,T_{1q_{H2}}^H\right]\\ \underset{0}{\overset{N{\tau }_R}{\int }}d{t}_2\underset{0}{\overset{t_2}{\int }}d{t}_1\left(\begin{array}{l}\exp \left(i{n}_2{\phi }_C{q}_C+i{n}_1{\phi }_N{q}_N\right)\exp \left(i\left(q_{H1}{\omega }_{1,H}+m_1{\omega }_R\right)t_2+i\left(q_{H2}{\omega }_{1,H}+m_2{\omega }_R\right)t_1\right)\\ -\exp \left(i{n}_2{\phi }_N{q}_N+i{n}_1{\phi }_C{q}_C\right)\exp \left(i\left(q_{H2}{\omega }_{1,H}+m_2{\omega }_R\right)t_2+i\left(q_{H1}{\omega }_{1,H}+m_1{\omega }_R\right)t_1\right)\end{array}\right)\end{array}\right\}$$ (9)

Here $n_1$ and $n_2$ are numbers of elapsed pulses in t₁ and t₂, and the commutator $\left[T_{1q_{H1}}^H,T_{1q_{H2}}^H\right]$ = 0 unless $q_{H1}=-q_{H2}$, in which case $\left[T_{1q_{H1}}^H,T_{1q_{H2}}^H\right]=-q_{H1}{T}_{10}^H\propto H_z$. As we show in Supporting Information 3, the double integral is zero unless $e^{i\left({\phi }_C{q}_C+{\phi }_N{q}_N\right)}=1$, or equivalently

$${\phi }_C{q}_C+{\phi }_N{q}_N=2\pi l$$ (10)

for some integer l. Equation (10) can be satisfied in three ways:

1. If the zero-quantum matching condition, ${\phi }_C-{\phi }_N=2\pi l$ for integer $l$, is met, Eq. (10) will be satisfied with the additional condition that $q_C=-q_N=\pm 1$. The 1×2 cross term (Eq. 9) then produces terms in the second-order average Hamiltonian with spin product operators $T_{10}^H{T}_{11}^C{T}_{1-1}^N\propto H_z{C}^{+}{N}^{-}$ and $T_{10}^H{T}_{1-1}^C{T}_{11}^N\propto H_z{C}^{-}{N}^{+}$. These zero-quantum terms can induce polarization transfer from N to C and vice-versa without a change in sign, while the proton acts as a bystander.

2. If the double-quantum matching condition, ${\phi }_C+{\phi }_N=2\pi l$ for integer $l$, is met, Eq. (10) will be satisfied with the additional condition that $q_C=q_N=\pm 1$. The 1×2 cross term (Eq. 9) then produces double-quantum product operators $H_z{C}^{+}{N}^{+}$ and $H_z{C}^{-}{N}^{-}$. These double-quantum terms can induce transfer from N to C and vice-versa, with a change in sign, while the proton acts as a bystander.

3. For any ${\phi }_C$ and ${\phi }_N$, $q_C=q_N=0$ satisfies Eq. (10). The spin operator would be $T_{10}^H{T}_{10}^C{T}_{10}^N=H_z{C}_z{N}_z$. This longitudinal term commutes with both the zero-quantum and double-quantum terms from cases 1 and 2, and will not accomplish any transfer.

A direct analogy can thus be drawn between CW-TSAR and P-TSAR. In both cases, cross terms in the second-order average Hamiltonian between the H–A and H–B dipolar couplings produce double-quantum terms of the form $H_z{A}^{\pm }{B}^{\pm }$ or zero-quantum terms of the form $H_z{A}^{\pm }{B}^{\mp }$ in the interaction-frame. In CW-TSAR there are CW spin-locks on all channels, while for P-TSAR it is a pulsed spin-lock for the A and B channels. In both cases, matching conditions on the low-frequency channels dictate when TSAR transfer occurs, and whether it is positive-intensity transfer for the zero-quantum condition or negative-intensity transfer for the double-quantum condition.

Relation between chemical shift offset, pulse flip angle, and polarization transfer in P-TSAR

The matching conditions in P-TSAR are based on the time-averaged pulsed spin-lock flip angles ${\phi }_C$ and ${\phi }_N$. These flip angles are functions of the applied pulse flip angles on each channel and the isotropic offset of each spin. To best design P-TSAR experiments, we now consider how the choices of pulse flip angle and rf carrier frequencies dictate the chemical shifts that survive the pulsed spin-lock and fulfill the zero-quantum or double-quantum matching conditions.

Consider a single spin at a chemical shift offset Ω subjected to a train of rf pulses with flip angle θ about the x-axis at the center of each rotor period. This spin undergoes three successive rotations per rotor period (Fig. 6a): a rotation about the z-axis during the two free evolution periods (1 to 2 and 3 to 4), sandwiching a $\theta$ rotation about the x-axis during the pulse (2 to 3). These three successive rotations can be expressed as:

$$R_z\left(\Omega {\tau }_R/2\right)\cdot R_x\left(\theta \right)\cdot R_z\left(\Omega {\tau }_R/2\right)=R_{x^{\prime }}\left(\phi \right)$$ (11)

The net effect of the three rotations is equivalent to a single rotation of angle φ about the pulsed spin-lock axis $\widehat{x^{\prime }}$ (Fig. 6b), which can be defined as the rotation operator $R_{x^{\prime }}\left(\phi \right)$. This rotation is repeated many times during the pulsed spin-lock period (Fig. 6c), so that only magnetization along the spin-lock axis will be fully retained (Fig. 6d) while magnetization that is perpendicular to the effective spin-lock axis will precess to a different position on the Bloch sphere at the end of each rotor period (Fig. 6e). Any magnetization that is not parallel to the $\widehat{x^{\prime }}$ axis thus decays rapidly and can be discarded. Placing the spin-lock pulses in the center of each rotor period ensures that the spin-lock axis lies in the xz-plane; any initial y-magnetization is dephased during the spin-lock, thus allowing quadrature detection of the indirect dimension. However, x’ is not necessarily along x, but is instead in the xz-plane, therefore only a portion of the initial x-magnetization is properly spin-locked while the rest is dephased:

$$S_x=(\widehat{x}\cdot \widehat{x^{\prime }})S_{x^{\prime }}+(\widehat{x}\cdot \widehat{z^{\prime }})S_{z^{\prime }}\to (\widehat{x}\cdot \widehat{x^{\prime }})S_{x^{\prime }}$$ (12)

The initial x-magnetization, M_x, can be expressed in the x’ and z’ basis by projection (Fig. 6b). The time-dependent trajectory of M_x can then be expressed as two oscillating x’ and z’ components:

$$M_x\left(t\right)=\cos \psi M_{x\text{'}}\left(t\right)-\sin \psi M_{z\text{'}}\left(t\right)$$ (13)

Magnetization along z’ is perpendicular to the spin-lock axis and decays rapidly. A pulsed spin-lock will scale down magnetization by a factor of $\widehat{x}\cdot \widehat{x^{\prime }}$, with a different direction of $\widehat{x^{\prime }}$ for each isochromat (Fig. S5). This “read-in” loss is followed by a “read-out” loss in which magnetization is projected from the spin-lock direction back to the x-axis for detection (Fig. 7a). Treating the rf pulse as finite rather than a δ-function pulse gives a more accurate pulsed spin-lock profile (Fig. 7b, Supporting Information 4). For small pulse flip angles, the “read-in” and “read-out” losses are minimal for spins on resonance or at an rf offset near nω_r. For pulse flip angles near 360º, losses are minimal on resonance and at an rf offset of about 1.5ω_r. The pulsed spin-lock bandwidth increases as the nominal pulse flip angle increases from 0º to 180º or decreases from 360º to 180º.

Figure 6.

Magnetization trajectories under pulsed spin-lock. (a) Diagram of the basic pulsed spin-lock unit. The free evolution periods of 1→2 and 3→4 rotate the magnetization around the z-axis due to the isotropic chemical shift offset, while the rf pulse during 2→3 rotates the magnetization around the x-axis. (b) The pulsed spin-lock axis is tilted from the x-axis in the xz-plane. The initial x-magnetization, M_x(0), has components along the x’ and z’ axes. (c-e) Magnetization trajectories under pulsed spin-lock, calculated in MATLAB using an rf field of 50 kHz, a pulse flip angle of 90º, and a chemical shift offset of 17.5 ppm. (c) Trajectory of x-magnetization. The initial and final locations are not at the same spot on the Bloch sphere. (d) Trajectory of $M_{x\text{'}}$. This magnetization begins and ends the rotor period at the same position on the Bloch sphere, hence it is spin-locked. (e) Trajectory of $M_{z\text{'}}$. This magnetization traces out a broad trajectory, indicating rapid decay.

Figure 7.

“Read-in” and “read-out” losses associated with a pulsed spin-lock. (a) “Reading in” and “reading out” of the pulsed spin-lock causes magnetization losses due to projection of one axis onto the other. (b) Contour plot showing the “read-in” losses as a function of pulse flip angle and resonance offset. 50 kHz rf pulses were used in these calculations. (c) Calculated P-TSAR efficiencies as a function of the sink-spin offset. The source spin (C1) was placed on resonance. Positive transfer occurs at the zero-quantum condition (φ_C1 = φ_C2) while negative transfer occurs at the double-quantum condition (φ_C1 + φ_C2 = 2π).

This analytical model not only gives the pulsed spin-lock axis, but also the net rotation, $\phi$, during each rotor period. For visualization purposes, we can convert this $\phi$ into a time-averaged pulsed spin-lock field strength ω_SL:

$${\omega }_{SL}\equiv \phi /{\tau }_R$$ (14)

This time-averaged spin-lock field is analogous to the CW spin-lock field in PAR and ^PAINCP. From the average Hamiltonian analysis, the P-TSAR sequences have double- and zero-quantum matching conditions ${\omega }_{SL}^A\pm {\omega }_{SL}^B=l{\omega }_R$ for integer values of l. The zero-quantum condition leads to positive transfer intensities while the double-quantum condition leads to negative transfer intensities. This differs from typical implementations of CW-TSAR experiments, which only fulfill zero-quantum matching conditions.

To model the P-TSAR transfer, we combine the “read-in” and “read-out” losses with the matching conditions: the overall transfer efficiency ${\eta }_{A\to B}$ from spin A to B is the product of three factors: the “read-in” loss ${\widehat{x}}_A\cdot {\widehat{x^{\prime }}}_A$, projection of one spin’s spin-lock axis onto the other, ${\widehat{x^{\prime }}}_A\cdot {\widehat{x^{\prime }}}_B$, and the “read-out” loss ${\widehat{x^{\prime }}}_B\cdot {\widehat{x}}_B$. These three factors are further multiplied by Lorentzian functions centered at the zero- and double-quantum matching conditions. Finally, we include a scaling factor $(1-{\tau }_p/{\tau }_R)$ to account for the finite pulse length, which reduces the effective transfer time for long pulses:

$${\eta }_{A\to B}=\left({\widehat{x}}_A\cdot {\widehat{x^{\prime }}}_A\right)\left({\widehat{x^{\prime }}}_A\cdot {\widehat{x^{\prime }}}_B\right)\left({\widehat{x^{\prime }}}_B\cdot {\widehat{x}}_B\right)(1-{\tau }_p/{\tau }_R)\left[\frac{{\epsilon }^2}{{({\omega }_{SL}^A-{\omega }_{SL}^B)}^2+{\epsilon }^2}-\frac{{\epsilon }^2}{{({\omega }_{SL}^A+{\omega }_{SL}^B-{\omega }_R)}^2+{\epsilon }^2}\right]$$ (15)

Eq. (15) was used to calculate the polarization transfer efficiencies (Fig. 7c, 8c) with a Lorentzian width ε = 0.05ω_r. This width was chosen empirically by varying ε in the analytical model until the local bandwidth is consistent with numerical simulations and experiments. The resulting polarization transfer profile (Fig. 8c) has a nearly identical chemical shift and flip-angle dependence as that obtained from numerical simulation (Fig. 8b). Positive transfer is maximized when the source and sink spins have the same absolute offset. Negative transfer is obtained when the sink spin is ~ω_r away from the carrier for small flip angles, but splits into two separate branches for larger flip angles. A nominal flip angle of 180º gives vanishing transfer. With flip angles larger than 180º, the bandwidth for positive zero-quantum transfer is significantly larger than for smaller flip angles.

Figure 8.

Measured, simulated, and calculated ^PERSPIRATIONCP polarization transfer efficiencies as a function of the ¹³C rf carrier frequency and pulse flip angle. The experiments were conducted on an 800 MHz spectrometer under 20 kHz MAS. Calculations were conducted for the same conditions. (a) Measured ^PERSPIRATIONCP efficiencies of f-MLF as a function of ¹³C carrier frequency and ¹³C pulse flip angle. The 2D heat map is generated from 25 × 80 1D spectra measured as a function of ¹³C pulsed flip angles (0˚ to 360˚ in 15˚ increments) and rf carrier frequencies (0 to 200 ppm in 2.5 ppm increments.) The integrated Cα band intensities (46 ppm to 59 ppm) of the spectra are represented in the plot. The rf field strengths of the ^PERSPIRATIONCP block are 57 kHz, 47 kHz, and 36 kHz for ¹H, ¹³C and ¹⁵N, respectively. (b) Numerical simulations of ^PERSPIRATIONCP polarization transfer from an on-resonance ¹⁵N to ¹³Cα. (c) ^PERSPIRATIONCP transfer efficiencies calculated using the analytical model.

To test the isotropic shift dependence of P-TSAR, we carried out a pseudo-3D ^PERSPIRATIONCP experiment on the model peptide formyl-Met-Leu-Phe (f-MLF), in which we varied the ¹³C rf carrier frequency and pulse flip angle in the two indirect dimensions. Fig. 8a plots the integrated intensity of the three Cα peaks as a function of these two parameters, with representative 1D spectra shown in Fig. S6. Remarkably, the experiment gives the same qualitative features as predicted in the numerical simulations and the analytical model. Maximum positive transfer occurs when the source and sink spin chemical shift offsets are both small, while maximum negative transfer occurs when the ¹³C carrier is about ${\omega }_r$ away from the sink spins for small pulse flip angles.

Discussion

Our first implementation of the heteronuclear ^PERSPIRATIONCP experiment (Fig. S1d) ³⁴ placed the spin-lock pulses at the end of each rotor period. This timing refocuses the magnetization of spins that are on resonance or at ~nω_r from the carrier frequency for small flip angles, but does not refocus other off-resonance magnetization. Off-resonance signals have both x- and y-components at the end of the pulsed spin-lock period, which require a z-filter before and after this period to obtain clean, in-phase x-magnetization. Omitting these z-filters results in spectra with severe phase distortion. Inclusion of the z-filters leads to significant “read-in” and “read-out” losses by discarding the y-component (Fig. S7a). Spins that are ~0.5ω_r off-resonance for small flip angles do not survive the pulsed spin-lock and therefore do not contribute to polarization transfer (Fig. S7b, c). ^PERSPIRATIONCP with pulses at the end of each rotor period results in low transfer efficiency above 60 ppm and below 20 ppm and no transfer for aromatic signals at 110–140 ppm. Centering the spin-lock pulses in the rotor period refocuses magnetization along an axis in the xz-plane. This allows us to remove the z-filters, which facilitates quadrature detection and broadens the bandwidth of these experiments.

Previous studies have shown that a pulsed spin-lock can extend the coherence lifetime in solids by about 10-fold compared to a spin echo ^{32, 46–48}. These past implementations of pulsed spin-locks to solids used simple spin systems such as CaF₂ ⁴⁹, sometimes in the absence of MAS ^47–48, or dealt with amino acids with a single ¹³C label, ⁴⁶ for which spectral resolution and bandwidth were not of concern. In this study, we demonstrate the use of a pulsed spin-lock in the presence of significant isotropic shift dispersion, wherein maximizing the pulsed spin-lock bandwidth is of utmost importance to enable the detection of as many long-range correlations as possible. By placing a pulse in the center of each rotor-period, we make the sideband spacing significantly larger compared to previous MAS pulsed spin-lock experiments that pulsed only every 20 rotor periods for similar spinning frequencies ⁴⁶. Moreover, our careful choice of the pulse flip angle maximizes the breadth of the envelope of magnetization that survives the pulsed spin-lock compared to experiments that use flip angles less than 90° in order to maximize the relaxation time constant. Importantly, in contrast to previous work that uses a pulsed spin-lock to extend coherence time for greater sensitivity^{32, 46, 49–51}, here we use a pulsed spin-lock to generate distinct average Hamiltonians along the pulsed spin-lock directions that accomplish internuclear transfer.

The excellent agreement between our analytical model, numerical simulations, and measured polarization transfer profiles (Fig. 8) indicates that once a ¹H CW field strength is chosen, the main factors dictating polarization transfer are the pulse flip angle and the chemical shift offset that fulfill the zero- or double-quantum matching conditions, Eq. (10). These matching conditions are well-predicted by our analytical model, hence the ¹³C flip angle and carrier frequency can be chosen by inspecting the calculated analytical 2D map for a given magnetic field strength and MAS frequency (Appendix S3). Once the ¹³C carrier frequency and pulse flip angle are chosen, PULSAR can be optimized with only two steps. First, we optimize the ¹H CW rf field with a 1D experiment that includes a Cα selective echo before the P-TSAR block, by maximizing the non-Cα intensities (Fig. S1a). At an MAS rate of 20 kHz, we found that a ¹H CW field of ~57 kHz, which is just under 3 times the spinning speed, worked reliably for all samples and conditions. The ¹H power is briefly optimized by varying the power by 2 dB. We observed a broad maximum with multiple suitable ¹H power levels. The main power level we used would lead to a ~30% intensity loss for an error of ~0.7 dB. We then optimized the ¹³C pulse length to again maximize the transferred intensity in the desired spectral regions. ^PERSPIRATIONCP experiments require an additional step of optimizing the ¹⁵N pulse length with respect to the ¹³C pulse length, which is simple once the ¹³C flip angle has been selected. Importantly, these optimization steps are uncoupled from each other and can be run independently. Separate optimization of ¹³C and ¹⁵N powers revealed that a 0.5 dB change leads to a 30% loss of transfer efficiency, similar to cross polarization. Finally, for the ¹³C rf field strengths in the PULSAR and ^PERSPIRATIONCP experiments, as long as they are larger than the chemical shift offset, increasing the rf field strengths further do not improve the polarization transfer efficiency significantly. Instead, the pulse flip angle is the more important variable to optimize, to achieve both broadband transfer and high-efficiency transfer.

This optimization routine is much faster and easier than the multi-parameter optimization for CW-TSAR due to the extreme sensitivity of PAR and ^PAINCP’s matching conditions to rf field strengths, spin geometry, and ¹H density. Even for a given sample, different spin pairs have different optimal rf field strengths. Moreover, the rf field strengths that result in good polarization transfer for CW-TSAR experiments are inter-dependent. Such multidimensional parameter searches are very time-consuming, especially for 2D correlation experiments. For example, a recent investigation involving the 2D ¹⁵N-¹⁵N PAR experiment conducted 29 × 29 = 841 separate 2D’s to search for the optimal ¹H and ¹⁵N CW spin-lock field strengths ⁵². Simulations can facilitate this parameter search, but often show significant discrepancies from experiments ³¹. The optimized parameters can also change significantly between samples, which make it difficult to implement CW-TSAR experiments on low-sensitivity biological samples. In comparison, P-TSAR experiments do not require sample-specific optimization; the matching conditions are easily predicted using analytical theory, and the user can choose the pulse flip angle and rf carrier frequency in advance. Moreover, the ¹H CW field strength in P-TSAR experiments is only weakly coupled to other parameters, which further simplifies the optimization of P-TSAR experiments.

Conclusions

We have introduced two pulsed third-spin-assisted recoupling (P-TSAR) experiments that efficiently transfer polarization between homonuclear and heteronuclear spin pairs using a low rf duty cycle. The PULSAR and the improved ^PERSPIRATIONCP experiments are demonstrated on GB1 and glucagon amyloid fibrils and give rise to many long-range correlations that restrain the 3D structures of proteins. Using numerical simulations and analytical theory, we found that the first-order average Hamiltonian for the P-TSAR experiments is averaged to zero, apart from a homonuclear dipolar coupling term that has negligible effects on P-TSAR. We found that the second-order average Hamiltonian yields non-zero cross terms between the ¹H-¹³C₁ (¹H-¹⁵N) and ¹H-¹³C₂ (¹H-¹³C) dipolar couplings, which drive polarization transfer. These properties are the same as the PAR and ^PAINCP experiments. P- and CW-TSAR experiments give similar long-range contacts, but P-TSAR experiments use 5–10 fold less rf energy than their CW counterparts, and are much easier to optimize. The current study demonstrates these P-TSAR experiments on 600 and 800 MHz spectrometers under 20 kHz MAS, but the methodology can be extended to other magnetic field strengths, MAS frequency regimes, and spin pairs.