Proton Assisted Recoupling at High Spinning Frequencies

Abstract

We demonstrate the successful application of ¹³C-¹³C proton assisted recoupling (PAR) on [U-¹³C,¹⁵N] N-ƒ-MLF-OH and [U-¹³C,¹⁵N] protein GB1 at high magic angle spinning (MAS) frequencies (ω_r/2π=65 kHz). Specifically, by combining PAR mixing with low power heteronuclear decoupling (ω_1H/2π~16 kHz) and high spinning frequencies, we obtain high resolution 2D spectra displaying long range ¹³C-¹³C contacts from which distances can be extracted. These experiments therefore demonstrate the possibility of performing high resolution structural studies in the limit of high spinning frequency and low power ¹H decoupling, a regime which optimizes resolution of protein samples and preserves their integrity.

Introduction

Over the past two decades the development and refinement of magic angle spinning (MAS) ^1,2 NMR methods to perform dipolar recoupling³ have established approaches to perform spectral assignments, measure distances and torsion angles, and therefore established the foundation for de novo protein structure determination with solid state NMR.^4–23 These methods are most relevant to several classes of important biological systems – specifically, nanocrystals, amyloid fibrils, and membrane proteins^24–2828—that are inaccessible via other techniques. In addition, advances in instrumentation have contributed to the expansion of the repertoire of techniques available as structural tools. One such direction, pioneered by Samoson and coworkers, is the extension of MAS NMR spinning frequencies to the range of 50–70 kHz.^29,30 Access to MAS frequencies in this regime, which are larger than the size of ¹H-¹H dipolar couplings, has stimulated interest in the possibility of introducing a ¹H dimension into multidimensional MAS spectra, for example in direct detection experiments.^29,31–35 Concurrently, each new spinning frequency regime has required the development of new or modified methodologies in order to optimize the resolution and sensitivity of NMR experiments. For instance, the capability of performing experiments at (ω_r/2π = 20–35 kHz led to the development of a number of new tools applicable at high magnetic fields (B₀ > 15T), that promise to be valuable for protein assignments and structural studies.^{19,20,22,23,36} Two examples are cosine modulated rotary resonance (CMRR) and a version of RFDR³⁶ that do not require ¹H decoupling during mixing. Both of these approaches are less demanding of the power handling capabilities of the probe and preserve the integrity of the sample, and are therefore useful for ¹³C resonance assignments and distance measurements in proteins at high ω_r/2π. Furthermore, since CMRR it is a double quantum (DQ) experiment, the cross peaks are alternately negative and positive and provide data for unambiguous assignments²². In another advance we demonstrated new second order recoupling techniques based on the third spin assisted recoupling (TSAR) mechanism to constrain distances up to 5–7 Å in uniformly ¹³C and ¹⁵N labeled systems. In particular, in heteronuclear and homonuclear cases we employ proton assisted insensitive nuclei cross polarization (PAIN-CP)²⁰ and proton assisted recoupling (PAR)²³ experiments, respectively, which were subsequently used to calculate a de novo 3D protein structure.³⁷

There have been a few reports of experiments at spinning frequencies >30 kHz; among them, Ernst and coworkers used low power ¹H decoupling (i.e. the ¹H rf field strength applied is smaller than the spinning frequency),³⁸ and performed ¹³C-¹³C correlation experiments on proteins in the >40 kHz spinning frequency regime.^39,40 However, the repertoire of SSNMR methods that function at ω_r/2π > 40 kHz remains limited ^16,41–43and there is an especially notable paucity of techniques for measuring long distances. Thus, this is a regime that is ripe with opportunities for methodological developments.

In this manuscript we demonstrate that the homonuclear proton assisted recoupling (PAR)^23,44 pulse sequence can be successfully applied at ω_r/2π~ 65 kHz to uniformly ¹³C and ¹⁵N labeled proteins. The PAR technique, which relies on the TSAR^20,23 mechanism, is discussed in detail elsewhere for spinning frequencies ω_r/2π < 30 kHz. Here we show that, despite the fact that the technique relies on a second order recoupling mechanism, it can still be used to observe long distance ¹³C-¹³C contacts with mixing times on the order of tens of milliseconds at ω_r/2π = 65 kHz. It therefore provides an approach to provide the data necessary for protein structure determinations at high magnetic fields and spinning frequencies.

Experimental Section

Sample preparation

Preparation of [U-¹³C,¹⁵N]-N-f-MLF-OH

N-f-MLF-OH peptide was obtained by solid phase peptide synthesis (CS Bio Inc., Menlo Park, CA). The peptide was prepared with uniformly ¹³C and ¹⁵N labeled amino acids (Cambridge Isotope Laboratories, Andover, MA). It was crystallized from isopropanol and packed (~0.5 mg of peptide) in a 1.3mm Bruker rotor together with crystalline KBr.

Preparation of [U-¹³C,¹⁵N]-GB1

E. coli BL21 (DE3) cells (Invitrogen) were transformed with the T2Q mutant of GB1. Cells were grown in M9 minimal media with 3.0 g of [U-¹³C] glucose as the sole carbon source, and 1.0 g ¹⁵N ammonium chloride as the sole nitrogen source (Cambridge Isotope Laboratories, Andover, MA). Protein expression was induced with 500 µM IPTG for 4 h. The cell pellet was homogenized with a tip sonicator in phosphate buffered saline (150 mM NaCl, 50 mM KH₂PO₄/K₂HPO₄, pH 7), and the supernatant purified by heating to 80 °C for 5 minutes followed by gel exclusion chromatography (Sephadex 16/60 Hi Prep). Peak purified fractions were pooled and concentrated using Amicon Ultra-3, 500 MWCO devices. Microcrystalline samples were prepared according to ref. 45 by extensive dialysis in a total of 12L of 50 mM sodium phosphate buffer (pH 5.6) and precipitated with 3 aliquots of 2:1 MPD:IPA at a protein concentration of 25 mg/mL. The sample consisting of ~2.5 mg of protein was centrifuged into a 1.3 mm Bruker rotor.

NMR Spectroscopy

Experiments were performed on a commercial Bruker spectrometer operating at 500 MHz using a two channel ¹H-¹³C probe equipped with a 1.3 mm rotor/stator system. The spinning frequency was regulated with a Bruker MAS controller to ± 10 Hz.

a. Microcrystalline [U-¹³C,¹⁵N]-MLF-OH

Experiments on the tripeptide utilized a 1.2 ms ¹H-¹³C CP contact time with 98 kHz ¹³C irradiation and the ¹H rf field was ramped down through the n=1 Hartmann-Hahn condition (i.e. 163 kHz). Since MLF-OH is a dry peptide for which rf induced heating is less of an issue high power XiX decoupling (ω₁/2π =230 kHz) was employed during the t₁ and t₂ evolution periods with the optimal length of the XiX pulse was 60.2 µs (3.91 τ_r). The spectral width in the direct and indirect dimension was 25.2 kHz. For the spectrum presented in Fig. SI7 the acquisition times were 27.7 ms in t₂ and 5 ms in t₁ with 8 scans per t₁ point (resulting in ~52 minute 2D experiment). For the spectra presented in Fig. SI4 the acquisition times were 27.7 ms in t₂ and 12.7 ms in t₁ with 64 scans per t₁ point (resulting in 17 h per 2D experiment). The recycle delay was 1.5 s. The temperature was maintained at 284 K (at the thermocouple) using a BCU-X with 1400 L/h nitrogen flow rate.

b. Microcrystalline [U-¹³C,¹⁵N]-G_B1

The ¹H-¹³C CP contact time was 2.2 ms with 98 kHz ¹³C irradiation and ¹H power ramped down through n=1 Hartmann-Hahn condition (i.e. 163 kHz). In order to minimize the rf induced heating of the hydrated protein sample we employed low power TPPM during both the direct and indirect evolution periods. Each pulse length was set to 30.77µs (corresponding to a modulation frequency ω_C/2π equal to a quarter of the MAS spinning frequency, i.e. 16.25 kHz) and the ¹H decoupling was optimized by varying the ¹H irradiation strength ω_1H/2π (near 16.25 kHz) and the phase excursion. The optimal phase excursion was ±24°. The sweep width in the direct and indirect dimension was again 25.2 kHz. Acquisition times were 25 ms in t₂ and 10.1 ms in t₁ with 96 scans per t₁ point for the 2.5 ms spectrum (total acquisition time ~48 h) and 192 scan per t₁ point for the 10.1 ms spectrum (total acquisition time ~95.6 h). The recycle delay was 3.5 s.

c. Optimization of PAR magnetization transfer

There are at least two methods that can be used to optimize the r.f. field strengths to satisfy the matching conditions for PAR experiments. The first relies on polarization transfer following a selective excitation pulse, and the second a comparison between the polarization transfer maps and “interference” maps. In either case we recommend that these maps be generated for the spinning frequencies of interest, and, accordingly, we have included a SPINEVOLUTION ⁴⁶ script in the Supporting Information for this purpose. For reference Figure 2, Figure SI1, and Figure SI2 of this paper illustrate simulations corresponding to polarization transfer maps, and SI5 a interference maps for a simple model spin systems at ω_r/2π = 65 kHz. In our previous paper we published similar maps for ω_r/2π = 20 kHz ²³.

Figure 2.

Simulated PAR polarization transfer map obtained on the spin system shown in the inset of the figure (see also Table SI1). The contour plots represent polarization transfer between the C_α and C_β spins as a function of ¹³C and ¹H irradiation strengths in units of spinning frequency. Simulations were performed using 5 ms PAR mixing at ω_r/2π = 65 kHz and ω_0H/π =500 MHz and include chemical shifts (see Table SI1). Similar maps for C_α-C’ and C_α-C_β polarization transfer can be found in Fig. SI1. For the settings indicated by the black dashed lines the CH auto-cross term is zero. The irradiation settings employed in this study are indicated with ‘x’s (blue for N-f-MLF-OH and black for GB1).

In the first approach, which is commonly used for optimization of any ZQ sequence, we excite magnetization on one of the sites (C=O or Cα) using a selective pulse or selective CP together with a storage pulse followed by polarization transfer to a second site (e.g. Cβ’s), and the intensity of that signal is observed as a function of ¹H and ¹³C field strengths. In order to constrain the search-space we employ a PAR polarization transfer map as is shown in Figure 2 below as a guide. For the case illustrated in Figure 2, the map was calculated for polarization transfer from Cα to Cβ at ω_r/2π = 65 kHz, and indicates appreciable PAR transfer for p_C~1.12 or 1.75 (corresponding to ω_1C/2π = 72.8 kHz or 110.5 kHz) and p_H~0.3 or 1.3 (corresponding to ω_1H/2π = 19.5 kHz or 84.5 kHz), respectively. Using these parameters as a guide, we set the PAR mixing time to 2–3 ms (a period appropriate for transfer over short distances), the ¹³C r.f field to 72.8 kHz, and we scan the ¹H r.f. field (e.g. from 10 to 40 kHz) looking for optimal transfer. In general we find it essential to perform such an optimization to account for omnipresent experimental variability. For example, depending on the probe, spectrometer, etc. the powers calibrated from 180º pulses may differ as much as 10 kHz from actual nutation frequencies. Thus, it is essential to accurately calibrate the ¹³C r.f. field to use this method.

In the second approach we rely on a two step comparison between the PAR polarization transfer map and the interference map for choosing the appropriate r.f. settings. For example, in order to find the settings for a given ω_r/2π, (say 20 kHz) we start with an experiment consisting of CP followed by 2–3 ms of PAR mixing with a relatively large ¹H field -- for instance, 83 kHz or another value that is sufficiently large to avoid Hartmann-Hahn conditions between ¹H and ¹³C. We then observe the ¹³C signal as the ¹³C field strength is scanned from 15 to 50 kHz. During the scan we observe dips in ¹³C signal when the r.f. field matches ω_r/2π and 2ω_r/2π (i.e. 20 kHz and 40 kHz) due to the CSA recoupling at the rotary resonance recoupling (R³) conditions at p_C= 1 or 2 ⁴⁷, thus calibrating the ¹³C power levels. Similarly, we can use Hartmann-Hahn matching to calibrate the ¹H power relative to the ¹³C power. In particular, we fix the “calibrated” ¹³C field strength at a value that leads to appreciable transfer in the PAR polarization transfer map, and then observe the ¹³C signal as a function of the scan of the ¹H field strength. Referring to the polarization transfer map in Figure 2 of De Paepe, et. al. ²³ which was calculated for ω_r/2π = 20 kHz, we set ω_1C/2π = 52 kHz and scan the ¹H irradiation from 5 kHz to 95 kHz. We observe dips in the ¹³C signal for all the Hartmann-Hahn conditions i.e. (ω_1H ± nω_r)/2π [explicitly at 52 ± (n x 20) kHz)]. From the ω_r/2π = 20 kHz PAR map we see that in order to achieve appreciable transfer for p_C ~2.6 (ω_1C/2π = 52 kHz) we require ¹H fields with a strength slightly smaller that ¹³C field (p_H ~ 2.4 -- ω_1H/2π = 2.4*20 = 48 kHz). This can easily determined relative to the n=0 Hartmann-Hahn condition in our optimization (i.e. the dip in the ¹³C signal for ω_1H/2π = 52 kHz). All of the experiments discussed here used this procedure, rather than the first method discussed above, for adjustment of the field strengths.

Results and Discussion

Figure 1 depicts the pulse sequence used for recording 2D ¹³C-¹³C correlation spectra with the PAR mixing sequence which consists of the application of two CW fields. The irradiation strengths on the ¹H and ¹³C channels are adjusted appropriately to induce polarization transfer between ¹³C sites via the second order ¹³C-[¹H]-¹³C TSAR mechanism.^20,23 The PAR experiment is analyzed with average Hamiltonian theory (AHT) in detail elsewhere (ref. ²³). Here we outline a few of the basic steps that lead to the expressions necessary for understanding the polarization transfer process.

Figure 1.

Pulse sequence for 2D ¹³C-¹³C PAR experiments. The PAR mixing consists of simultaneous C.W. irradiation on the ¹H and ¹³C channels with the irradiation strengths chosen to produce an appreciable second order TSAR mechanism.²⁰,²³ The TSAR term is a result of a cross term between the ¹H-¹³C dipolar couplings (terms 2 and 3 in the spin system figure).

We consider a three spin system consisting of two dipolar coupled ¹³C’s - ¹³C ₁, ¹³C₂ - and an assisting ¹H spin subject to two CW rf fields of strength ω_1C and ω_1H applied to the ¹³C and ¹H spins, respectively. The internal Hamiltonian can be written as

$$\begin{array}{c}\mathrm{H}={\omega }_{C_{1}H}2T_{10}^{C_1}{T}_{10}^H+{\omega }_{H{C}_2}2T_{10}^H{T}_{10}^{C_2}+\sqrt{6}{\omega }_{C_1{C}_2}{T}_{20}^{C_1{C}_2}\hfill \\ +\mathrm{\Delta }{\omega }_{C_1}{T}_{10}^{C_1}+\mathrm{\Delta }{\omega }_{C_2}{T}_{10}^{C_2}+\mathrm{\Delta }{\omega }_H{T}_{10}^H\hfill \\ +{\omega }_{1C}\left\{\frac{-T_{11}^{C_1}+T_{1-1}^{C_1}}{\sqrt{2}}\right\}+{\omega }_{1C}\left\{\frac{-T_{11}^{C_2}+T_{1-1}^{C_2}}{\sqrt{2}}\right\}+{\omega }_{1H}\left\{\frac{-T_{11}^H+T_{1-1}^H}{\sqrt{2}}\right\}\hfill \end{array}$$ (1)

where Δω_C1, Δω_c2 and Δω_H denote the shift tensors and resonant offsets of the ¹³C (or ¹⁵N ) and ¹H nuclei, respectively, and ω_c1c2 , ω_C1H, and ω_HC2 the homonuclear and heteronuclear dipolar couplings. For simplicity, we reduce the dependence on three different averaging frequencies [ω_r/2π, the rotor frequency, and ω_ic/2π and ω_1H/2π the strength of the ¹³C and ¹H CW fields, respectively] to a single frequency dependence by assuming that these frequencies are commensurate, implying that we can find indices p_C and p_H, and integers $p_C^1,p_C^2,p_H^1,p_H^2$ such that

$$\raisebox{1ex}{${\omega }_{1C}$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.=p_C\left(\raisebox{1ex}{${\omega }_r$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.\right)=\frac{p_C^1}{p_C^2}\left(\raisebox{1ex}{${\omega }_r$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.\right)$$ (2)

$$\raisebox{1ex}{${\omega }_{1H}$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.=p_H\left(\raisebox{1ex}{${\omega }_r$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.\right)=\frac{p_H^1}{p_H^2}\left(\raisebox{1ex}{${\omega }_r$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.\right)$$ (3)

where $\frac{p_C^1}{p_C^2},\frac{p_H^1}{p_H^2}$ denote irreducible ratios. The expression for the Hamiltonian in the interaction frame can thus be obtained from Eq. (1)

$$\begin{array}{l}\tilde{\mathrm{H}}=\stackrel{1}{\overbrace{{\displaystyle \sum _{m_1=-2}^2{\displaystyle \sum _{\underset{q_{1C\ne 0}}{q_{1C}=-1}}^1{\displaystyle \sum _{\underset{q_{1H}\ne 0}{q_{1H}=-1}}^1{\omega }_{C_{1}H}^{m_1}\text{sgn}(q_{1C})\text{sgn}(q_{1H})T_{1q_{1C}}^{C_1}{T}_{1q_{1H}}^H\text{exp }\left\{-i(n{X}_1)\frac{{\omega }_{r}t}{n}\right\}}}}}}\hfill \\ +\stackrel{2}{\overbrace{{\displaystyle \sum _{m_2=-2}^2{\displaystyle \sum _{\underset{q_{2C\ne 0}}{q_{2C}=-1}}^1{\displaystyle \sum _{\underset{q_{2H}\ne 0}{q_{2H}=-1}}^1{\omega }_{H{C}_2}^{m_2}\text{sgn}(q_{2H})\text{sgn}(q_{2C})T_{1q_{2H}}^H{T}_{1q_{2C}}^{C_2}\text{exp }\left\{-i(n{X}_2)\frac{{\omega }_{r}t}{n}\right\}}}}}}\hfill \\ +\stackrel{3}{\overbrace{{\displaystyle \sum _{m_3=-2}^2{\displaystyle \sum _{q_{3C}=-2}^2{\omega }_{C_1{C}_2}^{m_3}{d}_{q_{3C}0}^2}\left(-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)T_{2q_{3C}}^{C_1{C}_2}\text{exp }\left\{-i(n{X}_3)\frac{{\omega }_{r}t}{n}\right\}}}}\hfill \\ +\stackrel{4}{\overbrace{{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{m_i=-2}^2{\displaystyle \sum _{\underset{q_{Ci\ne 0}}{q_{Ci}=-1}}^1{\omega }_{C_i}^{m_i}\left(\text{sgn}(q_{C_i})/\sqrt{2}\right)T_{1q_{Ci}}^{C_i}\text{exp }\left\{-i(n{X}_{C_i})\frac{{\omega }_{r}t}{n}\right\}}}}}}\hfill \\ +\stackrel{5}{\overbrace{{\displaystyle \sum _{m=-2}^2{\displaystyle \sum _{\underset{q_{H\ne 0}}{q_H=-1}}^1{\omega }_H^m}\left(\text{sgn}(q_H)/\sqrt{2}\right)T_{1q_H}^H\text{exp }}\left\{-i(n{X}_H)\frac{{\omega }_{r}t}{n}\right\}}}\hfill \end{array}$$ (4)

where we use the following expressions

$$\begin{array}{c}{X}_1=m_1+p_C{q}_{1C}+p_H{q}_{1H}\hfill \\ X_2=m_2+p_H{q}_{2H}+p_C{q}_{2C}\hfill \\ X_3=m_3+p_C{q}_{3C}\hfill \\ X_{Ci}=m_{Ci}+p_C{q}_{Ci}\hfill \\ X_H=m+p_H{q}_H\hfill \end{array}$$ (5)

and sgn(q) is the sign function of q.

We assume that the rf fields are chosen so that neither Hartmann-Hahn (H-H)⁴⁸ nor rotary resonance recoupling (R³)⁴⁷ conditions are matched, i.e. X_l ≠0,X₂ ≠ 0,X_Ci ≠0,X_H ≠0 and that the ¹³C-¹³C dipolar coupling is not recoupled to first order (X₃ ≠0). Under these conditions, the first order average Hamiltonian vanishes.

In order to describe the TSAR recoupling mechanism, we calculate the cross-term between terms 1 and 2 in Eq. (4)

$${\overline{\tilde{H}}}_{1x2}^{(2)}={\displaystyle \sum _{\begin{array}{c}{m}_1,q_{1C},q_{1H},\\ m_2,q_{2C},q_{2H}\end{array}}\left[\begin{array}{l}\frac{1}{2iT}{\omega }_{C_{1}H}^{m_1}\text{sgn}(q_{1C})\text{sgn}(q_{1H}){\omega }_{H{C}_2}^{m_2}\text{sgn}(q_{2H})sg(q_{2C})T_{1q_{1C}}^{C_1}{T}_{1q_{2C}}^{C_2}\left[T_{1q_{1H}}^H,T_{1q_{2H}}^H\right]\hfill \\ {\displaystyle \underset{0}{\overset{T}{\int }}d{t}_2}{\displaystyle \underset{0}{\overset{t_2}{\int }}d{t}_1\left(\text{exp}\left\{-i{\omega }_r(X_1{t}_2+X_2{t}_1)\right\}\text{exp}\left\{-i{\omega }_r(X_2{t}_2+X_1{t}_1)\right\}\right)}\hfill \end{array}\right]}$$ (6)

One of the most important solutions of Eq. (6) is obtained for for q_lc = −q_2C and m_l = −m₂ which yields ZQ terms of the form $T_{1\mathrm{m}}^{C_1}{T}_{1\pm 1}^{C_2}{T}_{10}^H$ (also providing that we concurrently avoid the DQ conditions associated with p_C={0, ±0.5, ±1, ±1.5, ±2}, H-H conditions with p_C = ±p_H ±{1, 2}, and R³ conditions with p_C = {1, 2}). With these constraints we obtain the following expression for the second order TSAR term:

$$\begin{array}{c}{\overline{\tilde{H}}}_{1x2}^{(2)}=2{\omega }_{TSAR}\left(T_{11}^{C_1}{T}_{1-1}^{C_2}\right)T_{10}^H+2{\omega }_{TSAR}^{*}\left(T_{1-1}^{C_1}{T}_{11}^{C_2}\right)T_{10}^H\hfill \\ =2{\omega }_{TSAR}{C}_1^{+}{C}_2^{-}{H}_Z+2{\omega }_{TSAR}^{*}{C}_1^{-}{C}_2^{+}{H}_Z\hfill \\ =\text{Re}\left({\omega }_{TSAR}\right)2I_{C_1{C}_2,X}^{(23)}{I}_{H,Z}+\text{Im}\left({\omega }_{TSAR}\right)2I_{C_1{C}_2,Y}^{(23)}{I}_{H,Z}\hfill \end{array}$$ (7)

and the TSAR coupling:

$$\begin{array}{c}{\omega }_{TSAR}=\left(\frac{\mathrm{Re}\left({\omega }_{C_{1}H}^1{\omega }_{H{C}_2}^{-1}\right)}{{\omega }_r}\lambda (1,p_C,p_H)+\frac{\mathrm{Re}\left({\omega }_{C_{1}H}^2{\omega }_{H{C}_2}^{-2}\right)}{{\omega }_r}\lambda (2,p_C,p_H)\right)\\ +i\left(\frac{\mathrm{Im}\left({\omega }_{C_{1}H}^1{\omega }_{H{C}_2}^{-1}\right)}{{\omega }_r}\sigma (1,p_C,p_H)+\frac{\mathrm{Im}\left({\omega }_{C_{1}H}^2{\omega }_{H{C}_2}^{-2}\right)}{{\omega }_r}\sigma (2,p_C,p_H)\right)\end{array}$$ (8)

with

$$\begin{array}{l}\lambda (m,p_C,p_H)=\left(\frac{-(p_C+p_H)}{m^2-{(p_C+p_H)}^2}+\frac{-(p_H-p_C)}{m^2-{(p_H-p_C)}^2}\right)\\ \sigma (m,p_C,p_H)=\left(\frac{m}{m^2-{(p_H+p_C)}^2}-\frac{m}{m^2-{(p_H-p_C)}^2}\right)\end{array}$$ (9)

The above expressions permit us to visualize the subspace in which the TSAR spin dynamics evolve. The TSAR subspace (Fig. SI6) can be viewed as a coupled basis described extensively in the context of solution NMR between a fictitious ZQ operator involving the two ¹³C’s and a ¹H. The proton spin is essentially a bystander spin i.e. no magnetization is sent to this proton in the spin dynamics described by Eq. (7). In this process, the dipolar couplings to the ¹H are used to create an effective transverse PAR component composed of trilinear terms of the form $C_1^{\pm }{C}_2^{\mp }{H}_Z$ that mediates polarization transfer. Another important contribution to the spin dynamics in the TSAR mechanism arises from auto-cross terms created by term 1 with itself and term 2 with itself respectively. These auto-cross terms lead to an off-resonance contribution in the TSAR subspace represented by ${\mathrm{I}}_{{\mathrm{C}}_1{C}_2,\mathrm{Z}}^{(23)}$. Note that similar longitudinal terms also arise from the chemical shift tensor with itself. Such contributions lead to a tilting of the effective recoupling axis and reduction of the PAR polarization transfer.

Figure 2 shows a numerical SPINEVOLUTION⁴⁶ simulation of the ¹³C-¹³C PAR polarization transfer at ω_r/2π = 65 kHz on a 7 spin system consisting of four ¹³C’s and three ¹H’s (see the inset of the Fig. 2 and Table SI1 for details). The contour plot illustrates the transfer from C_α to C_β (see Fig. SI1 for C_α-C’ and C_α-C_β optimization maps) after 5 ms of PAR mixing as a function of ¹³C and ¹H C.W. irradiation amplitudes in units of spinning frequency (p_C = ω_1C/ω_r and p_H =ω_1H/ω_r respectively). The carrier frequency in the simulation was set between the carbonyl and aliphatic resonances, similar to the actual experiments (see Fig. SI2 for an optimization map with the ¹³C carrier frequency set in the middle of the aliphatic region only). There are several areas in the map for which the settings lead to appreciable PAR polarization transfer. The settings employed in this study are indicated with an ‘X’, including the higher power settings used for the ¹³C-¹³C correlation spectra of N-f-MLF-OH²⁶ (see Fig. 5 and Fig. SI4) and lower power settings used for the correlation spectra of GB1.

The PAR optimization map arises from a compromise between maximizing the TSAR term and minimizing the longitudinal contribution from the auto-cross terms. The experimental irradiation settings were chosen as described in ref. 23. More specifically, we have simulated a map where we monitor ¹³C magnetization as a function of ¹H and ¹³C irradiation in a 1D experiment with 5 ms PAR mixing. Such an “interference” map (also simulated numerically in Fig. SI5) permits identification of the rf fields that lead to destructive interference and it also outlines the features of the PAR polarization transfer map. Comparing the interference maps (both simulation and experiment) and the polarization transfer map shown in Fig. 2 allows one to choose the appropriate rf settings. Note, that for both the N-f-MLF-OH and GB1 spectra presented below, we have chosen settings that favor polarization transfer between protonated ¹³C’s but also allow transfer between carbonyls and aliphatic ¹³C’s (with slightly lower efficiency). If the goal of the experiment is to maximize the magnetization transfer between carbonyls and protonated ¹³C’s, the reader should refer to Fig. SI1b to choose the appropriate rf power levels. The shift in rf settings can be explained by the variation in magnitude of the auto-cross terms involved in a PAR carbonyl to aliphatic versus PAR aliphatic to aliphatic transfer.

Combining >50 kHz spinning frequencies and low power ¹H decoupling was shown previously to yield very well resolved spectra in uniformly labeled proteins.⁴⁰ The concept of low power decoupling (introduced by Ernst et al.) was also investigated at ω_r/2π ~ 30 kHz by De Paëpe et al. in the context of the Cosine Modulation^18,51 and Two Pulse Phase Modulation⁵² irradiation scheme where an rf field strength close to the ω_r/4 and a phase angle ~20 degrees was shown to yield efficient decoupling settings.⁴⁹ This sequence was recently discussed in greater detail by Kotecha et al. at ~40 kHz spinning frequency and referred to as low power TPPM.⁵⁰ Figure 3 shows CP-MAS spectra obtained on microcrystalline [U-¹³C,¹⁵N] protein GB1 recorded at ω_H0=500 MHz, ω_r/2π=65 kHz with low power TPPM^49,50 (~ ω_1C=ω_r/4=16.25 kHz, φ=±24°) during the acquisition. Although this system has already been demonstrated to yield well resolved spectra in previous publications, it is worthwhile to point out the excellent resolution achieved in this experiment. Notably, we observe well resolved aromatic carbon resonances which appear more intense than in any SSNMR study on GB1 reported so far.^53,54 Moreover we also observe splittings due to J-coupling for most of the sites in the protein.

Figure 3.

(a) 1D ¹³C CP-MAS spectrum of microcrystalline [U-¹³C,¹⁵N]-GB1 (2.5 mg protein packed in 1.3 mm rotor) obtained at ω_r/2π = 65 kHz and ω_0H/2π = 500 MHz with low power (~16.25 kHz) TPPM decoupling⁴⁹,⁵⁰ during acquisition. (b) Expansion of the aromatic region illustrating the excellent resolution.

Several factors could contribute to the excellent resolution of the ω/2π = 65 kHz MAS spectra reported here, including very effective heteronuclear decoupling, efficient averaging of the residual anisotropic contributions (i.e. magnetic bulk susceptibility etc.), and reduction of thermal gradients across the sample compared to a larger rotor^††. This result clearly indicates that low power decoupling methods at ω_r/2π > 50 kHz provide a viable method for carrying out high resolution protein studies.

Figure 4 shows a 2D ¹³C-¹³C PAR spectrum of microcrystalline [U-¹³C,¹⁵N] protein GB1 at ω_r/2π = 65 kHz, ω_0H/2π = 500 MHz where low power TPPM^{49,50 1}H decoupling (ω_1H/27π~ 0.25 ω_r/2π = 16.25 kHz) was applied during both direct and indirect evolution. 10 ms PAR mixing was used with the irradiation settings corresponding to the black cross noted ‘X’ on Fig. 2. The spectrum shows cross peaks corresponding to both intra-residue and inter-residue medium to long distance contacts. Some representative long distance cross peaks are marked on the PAR spectrum (Fig. 4) and the corresponding contacts are shown on the rendering of the x-ray structure⁴⁵ with the distances indicated in Fig. 4c.

Figure 4.

2D ¹³C-¹³C PAR correlation spectra of microcrystalline [U-¹³C,¹⁵N]-GB1 obtained at ω_r/2 = 65 kHz and ω_0H/2 = 500 MHz with 10 ms PAR mixing. The PAR mixing employed ~73 kHz ¹³C and 19.5 kHz ¹H irradiation. Low power TPPM⁴⁹,⁵⁰ (ω_1C= ω_r/4=16.25 kHz) was applied during acquisition and t₁ evolution. The high resolution achievable with this decoupling scheme is illustrated in panels (a) and (b), which depict an expansion of a carbonyl-aliphatic region of the spectrum. The data in panel (a) were processed without linear prediction in the direct dimension (t₂=25 ms) and the data in panel (b) with liner prediction in the direct dimension. In both panels in the direct dimension we can clearly distinguish splitting due to the J-couplings for most of the cross-peaks. Panel (c) illustrates some of the representative long distance contacts that are observed in an experiment with 10 ms PAR mixing. The cross-peaks corresponding to the contacts in panel (c) are circled and marked with numbers in the spectrum: 1 - E56Cβ-G9Cα – 4.1 Å, 2 – L7Cδ2-L5Cγ2 – 4.0 Å, 3 – L7Cδ2-V54Cγ1 – 3.8 Å, 4 – N37Cβ-A34Cα – 4.8 Å.

This result demonstrates that long distance transfers can be observed even at ω/2π = 65 kHz with only 10 ms PAR mixing. This mixing time falls approximately in the same range as those used at ω/2π = 20 kHz to perform the de novo structure determination of the protein Crh and thus implies that similar work is feasible at > 50 kHz spinning frequencies.^22,23 To complement Fig. 4, a shorter PAR mixing time (2.5 ms) containing primarily one-bond and two-bond cross-peaks can be found in Fig. SI3.

Since the TSAR mechanism is based on a second order process it is not completely intuitive that the PAR mixing time required for observing a given distance transfer does not increase as the spinning frequency increases. However, as discussed in De Paepe, et. al. ²³ the higher MAS rates allow one to choose rf settings with an improved scaling factor that can compensate for lower transfer efficiencies. This feature is essential to allow practical implementations of PAR experiments at ω/2π > 30 kHz and contrasts with PDSD⁵⁶ and DARR^15,17 which would require orders of magnitude longer mixing times. Note that in a recent study by Scholz et al. already at much lower spinning frequency of 45 kHz could not observe appreciable one-bond polarization transfer using conventional PDSD or n=1,2 DARR.⁴³

The excellent resolution achieved with low power TPPM decoupling at ω_r/2π = 65 kHz is already obvious from the 1D spectrum displayed in Fig. 3, but is further illustrated in the 2D ¹³C-¹³C PAR spectrum. The inset in Fig. 4 shows expansions of selected carbonyl-aliphatic cross peaks from the spectra processed without (Fig. 4a) and with linear prediction in the direct dimension (Fig. 4b). We can clearly distinguish J-splittings for most of the resonances in the spectrum.

The mixing period and decoupling settings used for the PAR experiment on protein GB1 were chosen to optimize resolution and sensitivity while minimizing rf power requirements and thus rf induced sample heating. However, for samples where rf induced heating is less of a concern, higher power settings for both the PAR mixing and the ¹H decoupling periods can be used. As an illustration Fig. SI4 shows ¹³C-¹³C PAR spectra obtained on the dry, crystalline tripeptide [U-¹³C,¹⁵N]-f-MLF-OH with high power PAR mixing (the settings are indicated with a blue cross noted “X” in Fig. 2) and with ~230 kHz XiX^57,58 decoupling during both direct and indirect evolution periods. Note that in these data we observe clear cross-peaks between the carbons in the phenylalanine ring, which undergoes two-fold flipping motion in the temperature regimes employed for most MAS experiments.

In summary we demonstrate that the ¹³C-¹³C PAR method can be used to obtain long distance ¹³C-¹³C contacts in uniformly ¹³C labeled proteins at ω_r/2π = 65 kHz with mixing times of ~10 ms or less. Moreover, the PAR irradiation settings can be chosen to minimize rf induced heating even at such high spinning frequencies. This application opens up a venue for de novo structure determination of proteins at ω_r/2π > 50 kHz in a manner analogous to the studies performed at lower spinning frequencies. Notably, PAR provides a flexible tool for structural characterization of biomolecules over the entire range of currently available spinning frequencies and can be used alone or as a building block in more sophisticated pulse sequences.

We also demonstrate that combination of >50 kHz spinning frequency and low power decoupling provides an attractive method for carrying out high resolution protein studies by solid-state NMR. The resolution of the spectra obtained at ω_r/2π=65 kHz with low power TPPM rivals or surpasses the resolution obtainable at ω_r/2π < 30 kHz with high power decoupling, but with significantly reduced rf strength requirements.

Biographies

Józef R. Lewandowski received his B.A. in Chemistry and Theater & Dance from Amherst College (2002) and his Ph.D. in Chemistry from MIT (2008). He is currently a Marie Curie Postdoctoral Fellow at the Ecole Normale Supérieure de Lyon in France. His research is focused on the application and development of solid-state NMR methodology for protein dynamics and structure determination

Gaël De Paëpe received his B.S. and his Ph.D. in Physics from the Ecole Normale Supérieure de Lyon (2004). After spending four years as an postdoctoral pesearcher at the Massachusetts Institute of Technology, he joined the Grenoble Institute for Nanoscience and Cryogenics in 2008, where he currently holds an ANR Chair of Excellence. His research is focused on developing new methods for structure determination by solid-state NMR as well as new instrumentation and methods for high field dynamic nuclear polarization (DNP).

Matthew T. Eddy received his B.A. in Chemistry from Oberlin College (2004) and is currently a chemistry Ph.D. student in the laboratory of Prof. Robert G. Griffin at MIT. At Oberlin College, Matthew was first introduced to MAS NMR under the guidance of Professor Manish Mehta who inspired him to pursue further studies in the area. Matthew’s research interests include the development of MAS NMR methodology and applications on ion channels and fibrillar proteins.

Jochem O. Struppe received his Diploma in Physics (1992) and his Dr. re. nat. degree in Physics (1996) from Technische Universität Stuttgart. After spending 2 years as postdoctoral researcher at UCSD in the department of Chemistry and Biochemistry, he joined Bruker BioSpin, where he is currently senior scientist in the applications department. He focuses on support and development of solid state NMR experimental techniques, development of NMR probes, solid state NMR with DNP and particularly solid state NMR with biological systems.

Werner E. Maas was an undergraduate and holds a Ph.D. (1992) from the University of Nijmegen in the Netherlands. He is currently Executive Vice President for Bruker BioSpin Corp. His interests lie in the development of magnetic resonance technology and its applications in the fields of chemistry, biochemistry, materials science and clinical diagnostics

Robert G. Griffin was an undergraduate at the University of Arkansas, graduating with a BS (honors) in Chemistry (1964). He received his Ph.D. from Washington University-St. Louis (1969) with S.I. Weissman working on electron spin resonance of radical ions. He was a postdoctoral fellow (1970–72) with J.S. Waugh at MIT in the early days of high resolution solid state NMR. In 1972 he became a staff member at the Francis Bitter Magnet Laboratory where he is currently Director. In 1989 he became Professor of Chemistry at MIT. In 2007 Prof. Griffin received the Laukien Prize of the Experimental NMR Conference and the Eastern Analytical Society Award for outstanding contributions to magnetic resonance.