^{1}, Morten Bjerring

^{1}, Navin Khaneja

^{2}and Niels Chr. Nielsen

^{1,a)}

### Abstract

Dipolar truncation prevents accurate measurement of long-range internuclear distances between nuclei of the same spin species, e.g., within spin pairs in uniformly -isotope-labeled proteins, using magic-angle spinning solid-state NMR spectroscopy. Accordingly, one of the richest sources of accurate structure information is at present not exploited fully, leaving the bulk part of the experimentally derived structural constraints to less accurate long-range dipolar couplings estimated from methods based on spin diffusion through proton spins in the close environment. In this paper, we extend our previous triple-oscillating field technique [N. Khaneja and N. C. Nielsen, J. Chem. Phys.128, 015103 (2008)] for dipolar recoupling without dipolar truncation in homonuclear spin systems to a more advanced rf modulation with four independent oscillations and rotations involving nonorthogonal axes. This provides important new degrees of freedom, which are used to improve the scaling factor of the recoupled dipole-dipole couplings by a factor of 2.5 relative to the triple-oscillating field approach. This significant improvement, obtained by refocusing of otherwise defocused parts of the residual dipolar coupling Hamiltonian, may be exploited to measure much weaker dipolar couplings (and thereby longer distances) with much higher accuracy. We present a detailed theoretical description of multiple-field oscillating recoupling experiments, along with numerical simulations and experimental results on , -threonine and -ubiquitin.

We acknowledge support from the Danish National Research Foundation, the Danish Biotechnological Instrument Centre (DABIC), and the Danish Center for Scientific Computing (DCSC).

I. INTRODUCTION

II. THEORY

A. The triple-oscillating-field technique

B. Multiple-axis oscillations for improved dipolar recoupling

C. TOFU with nonorthogonal rotations

D. FOLD with variable angles

1. FOLD with -rotation:

2. FOLD with horizontal rotation:

E. Averaging of nonsecular terms

1. Analytical description

2. Supplementary conditions

III. SIMULATION AND EXPERIMENTAL DETAILS

A. Simulation details

B. Experimental details

IV. RESULTS

A. Numerical simulations

B. Experimental results

V. CONCLUSION

### Key Topics

- Chemical shifts
- 47.0
- Nuclear magnetic resonance
- 11.0
- Anisotropy
- 10.0
- Dephasing
- 10.0
- Proteins
- 7.0

## Figures

Graphical illustration of the effect of stepwise transforming the description of [(a)–(c)] TOFU and [(d)–(g)] elements into the interaction frame of the rf field, respectively. (a) The full TOFU rf field. (b) and (c) The TOFU rf field as in Eqs. (5) and (6), respectively. (d) The full rf field. The rf field after a transformation of a rotation about the (e) -axis, followed by a rotation about the (f) -axis, and succeeded by a rotation about the (g) -axis. To keep the graphics within reasonable space, [(a)–(c)] the TOFU field used and , and the [(d)–(g)] field used . These values are not the recommendable values for practical experiments (see text).

Graphical illustration of the effect of stepwise transforming the description of [(a)–(c)] TOFU and [(d)–(g)] elements into the interaction frame of the rf field, respectively. (a) The full TOFU rf field. (b) and (c) The TOFU rf field as in Eqs. (5) and (6), respectively. (d) The full rf field. The rf field after a transformation of a rotation about the (e) -axis, followed by a rotation about the (f) -axis, and succeeded by a rotation about the (g) -axis. To keep the graphics within reasonable space, [(a)–(c)] the TOFU field used and , and the [(d)–(g)] field used . These values are not the recommendable values for practical experiments (see text).

(a) Illustration of the effect of changing the phase between the horizontal modulations in a generalized TOFU experiment using and . The solid curve represents the coefficient for the normalized isotropic chemical shift in Eq. (28), while the dashed curve represents the coefficient for the recoupled dipole-dipole interaction in Eq. (33) for which and are set to unity for simplicity. (b) The solid curve represents the normalized isotropic chemical shift in Eq. (42) for a generalized experiment as a function of using , , and . The dashed curve represents the recoupled dipole-dipole coupling in Eq. (43) with , and the dashed-dotted curve represents the recoupled dipole-dipole coupling in Eq. (43) with . The recoupled dipole-dipole coupling is expressed in units of .

(a) Illustration of the effect of changing the phase between the horizontal modulations in a generalized TOFU experiment using and . The solid curve represents the coefficient for the normalized isotropic chemical shift in Eq. (28), while the dashed curve represents the coefficient for the recoupled dipole-dipole interaction in Eq. (33) for which and are set to unity for simplicity. (b) The solid curve represents the normalized isotropic chemical shift in Eq. (42) for a generalized experiment as a function of using , , and . The dashed curve represents the recoupled dipole-dipole coupling in Eq. (43) with , and the dashed-dotted curve represents the recoupled dipole-dipole coupling in Eq. (43) with . The recoupled dipole-dipole coupling is expressed in units of .

Schematic representation of the FOLD dipolar recoupling pulse sequences, with the main experiment shown in (a) and the reference experiment in (b). The refocusing pulses are of phase while the Gaussian pulses are of phase . The duration of each of the Gaussian pulses and each of the corresponding free evolution periods is in our setup four rotor periods. The overall duration of the free evolution periods bracketing the refocusing pulse is set to two rotor periods, but this choice is only intended as a guideline. The Gaussian pulses hinder refocusing of specific dipole-dipole interactions.

Schematic representation of the FOLD dipolar recoupling pulse sequences, with the main experiment shown in (a) and the reference experiment in (b). The refocusing pulses are of phase while the Gaussian pulses are of phase . The duration of each of the Gaussian pulses and each of the corresponding free evolution periods is in our setup four rotor periods. The overall duration of the free evolution periods bracketing the refocusing pulse is set to two rotor periods, but this choice is only intended as a guideline. The Gaussian pulses hinder refocusing of specific dipole-dipole interactions.

(a) Amplitude and (b) phase of the FOLD rf field element as implemented in our numerical simulations and experiments for the case , , and .

(a) Amplitude and (b) phase of the FOLD rf field element as implemented in our numerical simulations and experiments for the case , , and .

Numerical simulations of (a) -alanine, (b) -threonine, and (c) isoleucine (residue 13 in ubiquitin) FOLD (, , , or ) curves on top of an ideal Fresnel curve grid (solid lines). The numbers above/next to the Fresnel curves indicate the distance to measured in Å. The symbols represent numerically simulated curves reflecting dipole-dipole coupling to (filled circle), (open circle), or (filled square), (open square), and (filled triangle). The dashed Fresnel curves represent a distance of 2.4 Å in (b) and 3.8 Å in (c).

Numerical simulations of (a) -alanine, (b) -threonine, and (c) isoleucine (residue 13 in ubiquitin) FOLD (, , , or ) curves on top of an ideal Fresnel curve grid (solid lines). The numbers above/next to the Fresnel curves indicate the distance to measured in Å. The symbols represent numerically simulated curves reflecting dipole-dipole coupling to (filled circle), (open circle), or (filled square), (open square), and (filled triangle). The dashed Fresnel curves represent a distance of 2.4 Å in (b) and 3.8 Å in (c).

Experimental FOLD curves for -labeled samples of (a) -threonine and (b) ubiquitin plotted in a ideal Fresnel curve grid. The numbers above/next to the ideal Fresnel curves indicate the distance in Å to the nearest . Experimental data is obtained by integrating over the relevant part of the peak line shape in the detection dimension (for ubiquitin the relevant peaks are marked in Fig. 7). Data points reflecting the internuclear distance between and the nuclei in question are marked as: (filled circle), (open circle), or (filled square), (open square), and of residue 23 (filled triangle).

Experimental FOLD curves for -labeled samples of (a) -threonine and (b) ubiquitin plotted in a ideal Fresnel curve grid. The numbers above/next to the ideal Fresnel curves indicate the distance in Å to the nearest . Experimental data is obtained by integrating over the relevant part of the peak line shape in the detection dimension (for ubiquitin the relevant peaks are marked in Fig. 7). Data points reflecting the internuclear distance between and the nuclei in question are marked as: (filled circle), (open circle), or (filled square), (open square), and of residue 23 (filled triangle).

Experimental FOLD dephasing spectra obtained for -labeled ubiquitin using (a) the main experiment and (b) the reference experiment with an increasing number of FOLD blocks, (see Fig. 3), with the most intense spectrum in top representing .

Experimental FOLD dephasing spectra obtained for -labeled ubiquitin using (a) the main experiment and (b) the reference experiment with an increasing number of FOLD blocks, (see Fig. 3), with the most intense spectrum in top representing .

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