Rotor-driven spin diffusion in natural-abundance 13C spin systems

Continuous Floquet theory in solid-state NMR

This article presents the application of continuous Floquet theory in solid-state nuclear magnetic resonance (NMR). Continuous Floquet theory extends the traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by the traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first- and second-order effective Hamiltonians, streamlining integration with the traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.

Serial Polarization Transfer by Combination of Cross-Relaxation and Rotational Resonance for Sensitivity-Enhanced Solid-State NMR

Methods which induce site-specificity and sensitivity enhancement in solid-state magic-angle spinning NMR spectroscopy become more important for structural biology due to the increasing size of molecules under investigation. Recently, several strategies have been developed to increase site specificity and thus reduce signal overlap. Under dynamic nuclear polarization (DNP) for NMR signal enhancement, it is possible to use cross-relaxation transfer induced by select dynamic groups within the molecules which is exploited by SCREAM-DNP (Specific Cross Relaxation Enhancement by Active Motions under DNP). Here, we present an approach where we additionally reintroduce the homonuclear dipolar coupling with rotational resonance (R2 ) during SCREAM-DNP to further boost the selectivity of the experiment. Detailed analysis of the polarization buildup dynamics of 13 C-methyl polarization source and 13 C-carbonyl target in 2-13 C-ethyl 1-13 C-acetate provides information about the sought-after and spurious transfer pathways. We show that dipolar-recoupled transfer rates greatly exceed the DNP buildup dynamics in our model system, indicating that significantly larger distances can be selectively and efficiently hyperpolarized.

Reducing the effects of weak homonuclear dipolar coupling with CPMG pulse sequences for static and spinning solids

The Carr-Purcell/Meiboom-Gill (CPMG) pulse sequence, initially introduced for measuring transverse relaxation time constants (T₂), can provide significant signal enhancements for solid-state NMR (SSNMR) spectra. The proper implementation of CPMG for acquiring spectra influenced by chemical shift anisotropies (CSAs), first and/or second order quadrupolar interactions, or paramagnetic broadening has been well documented to date, as have the effects of heteronuclear dipolar coupling on CPMG echo trains and T₂ lifetimes. Homonuclear dipolar coupling can also impact T₂ lifetimes and CPMG echo trains; these effects have been thoroughly investigated for spectra of homonuclear dipolar coupled spin-1/2 nuclei typically acquired under static conditions that are predominantly influenced by dipolar broadening (e.g., ¹H, ¹⁹F, etc.). In particular, it has been shown that short refocusing pulses with small flip angles can extend the effective T₂ (T₂^eff, the observed T₂ constant as impacted by experimental conditions) measured by CPMG sequences for strong homonuclear dipolar coupled spin-1/2 pairs under static conditions. To date, these effects have not been explored for (i) spin-1/2 nuclei that have significant CSAs and simultaneously feature weak homonuclear dipolar couplings, (ii) for quadrupolar nuclei that are also weakly homonuclear dipolar coupled, and (iii) for either of these cases under magic-angle spinning (MAS) conditions. Herein, we demonstrate that short refocusing pulses that cause small flip angles can reduce the attenuation of signal in CPMG echo trains resulting from dipolar dephasing caused by the weak homonuclear dipolar couplings. For both spin-1/2 and quadrupolar nuclei, this can lead to significant extensions in T₂^eff and signal enhancements of up to three times compared to conventional CPMG in favourable cases. These phenomena can occur under both static and magic-angle spinning (MAS) conditions, in the latter of which homonuclear couplings are reintroduced by rotational resonance (R²) recoupling. Experimental examples of ¹³C (I = 1/2), ²H (I = 1), ⁸⁷Rb (I = 3/2), ²³Na (I = 3/2), and ³⁵Cl (I = 3/2) NMR under static and MAS conditions, as well as simulations of these phenomena, are shown and discussed.

Solid-State NMR: Methods for Biological Solids

In the last two decades, solid-state nuclear magnetic resonance (ssNMR) spectroscopy has transformed from a spectroscopic technique investigating small molecules and industrial polymers to a potent tool decrypting structure and underlying dynamics of complex biological systems, such as membrane proteins, fibrils, and assemblies, in near-physiological environments and temperatures. This transformation can be ascribed to improvements in hardware design, sample preparation, pulsed methods, isotope labeling strategies, resolution, and sensitivity. The fundamental engagement between nuclear spins and radio-frequency pulses in the presence of a strong static magnetic field is identical between solution and ssNMR, but the experimental procedures vastly differ because of the absence of molecular tumbling in solids. This review discusses routinely employed state-of-the-art static and MAS pulsed NMR methods relevant for biological samples with rotational correlation times exceeding 100's of nanoseconds. Recent developments in signal filtering approaches, proton methodologies, and multiple acquisition techniques to boost sensitivity and speed up data acquisition at fast MAS are also discussed. Several examples of protein structures (globular, membrane, fibrils, and assemblies) solved with ssNMR spectroscopy have been considered. We also discuss integrated approaches to structurally characterize challenging biological systems and some newly emanating subdisciplines in ssNMR spectroscopy.

Floquet theory in magnetic resonance: Formalism and applications

Floquet theory is an elegant mathematical formalism originally developed to solve time-dependent differential equations. Besides other fields, it has found applications in optical spectroscopy and nuclear magnetic resonance (NMR). This review attempts to give a perspective of the Floquet formalism as applied in NMR and shows how it allows one to solve various problems with a focus on solid-state NMR. We include both matrix- and operator-based approaches. We discuss different problems where the Hamiltonian changes with time in a periodic way. Such situations occur, for example, in solid-state NMR experiments where the time dependence of the Hamiltonian originates either from magic-angle spinning or from the application of amplitude- or phase-modulated radiofrequency fields, or from both. Specific cases include multiple-quantum and multiple-frequency excitation schemes. In all these cases, Floquet analysis allows one to define an effective Hamiltonian and, moreover, to treat cases that cannot be described by the more popularly used and simpler-looking average Hamiltonian theory based on the Magnus expansion. An important example is given by spin dynamics originating from multiple-quantum phenomena (level crossings). We show that the Floquet formalism is a very general approach for solving diverse problems in spectroscopy.

Biomolecular solid-state NMR spectroscopy at 1200 MHz: the gain in resolution

Progress in NMR in general and in biomolecular applications in particular is driven by increasing magnetic-field strengths leading to improved resolution and sensitivity of the NMR spectra. Recently, persistent superconducting magnets at a magnetic field strength (magnetic induction) of 28.2 T corresponding to 1200 MHz proton resonance frequency became commercially available. We present here a collection of high-field NMR spectra of a variety of proteins, including molecular machines, membrane proteins, viral capsids, fibrils and large molecular assemblies. We show this large panel in order to provide an overview over a range of representative systems under study, rather than a single best performing model system. We discuss both carbon-13 and proton-detected experiments, and show that in ¹³C spectra substantially higher numbers of peaks can be resolved compared to 850 MHz while for ¹H spectra the most impressive increase in resolution is observed for aliphatic side-chain resonances.

Can proton-proton recoupling in fully protonated solids provide quantitative, selective and efficient polarization transfer?

Dipolar recoupling sequences have been used to probe spatial proximity of nuclear spins and were traditionally designed to probe rare spins such as ¹³C and/or ¹⁵N nuclei. The multi-spin dipolar-coupling network of the rare spins is weak due to smaller couplings and large chemical shift dispersion. Therefore, the recoupling approaches were tailored to design offset compensated or broadband sequences. In contrast, protons have a substantially stronger dipolar-coupling network and much narrower chemical shift range. Broadband recoupling sequences such as radio-frequency driven recoupling (RFDR), back-to-back (BABA), and lab frame proton-proton spin diffusion have been routinely used to characterize the structures of protein/macromolecules and small molecules. Recently selective ¹H-¹H recoupling sequences have been proposed that combine chemical shift offset of the resolved proton spectrum (at fast MAS) with first- and second-order dipolar recoupling Hamiltonians to obtain quantitative and qualitative proton distances, respectively. Herein, we evaluate the performances of broadband and selective proton recoupling sequences such as finite pulse RFDR (fp-RFDR), band-selective spectral spin diffusion (BASS-SD), second-order cross-polarization (SOCP), and selective recoupling of proton (SERP) in terms of the selectivity and efficiency of ¹H-¹H polarization transfers in a dense network of proton spins and explore the possibility of measuring ¹H-¹H distances. We use theoretical considerations, numerical simulations, and experiments to support the distinct advantages and disadvantages of each recoupling sequence. Experiments were performed on L-histidine.HCl.H₂O at a MAS frequency of 71.43 kHz. This study rationalizes the proper selection of ¹H-¹H recoupling sequences when working with fully protonated solids.

The origin of negative cross-peaks in proton-spin diffusion spectrum of fully protonated solids at fast MAS: Coherent or incoherent effect?

Spin-diffusion (SD) is amongst the first methods proposed to spatially transfer polarization between dipolar-coupled nuclear spins. Lab-frame SD has proved particularly useful in structural characterization of a large variety of molecules. During SD, the rate of magnetization transfer between the two nuclei depends on the square of the dipolar coupling and the zero-quantum lineshape of the two spins. The relative sign of the diagonal and cross-peaks is determined by the spin part of the dipolar Hamiltonian. Practically, SD experiments are used in two ways: (a) SD transfer amongst only the protons (known as proton spin-diffusion or PSD) and b) SD amongst rare nuclei, coupled to a strong proton bath, known as proton driven spin-diffusion (PDSD). It is well established that the diagonal and cross-peaks have the same sign during SD based polarization transfer. 2D PSD experiments recorded on Histidine.HCl.H₂O sample at fast magic angle spinning (MAS) show that some of the cross-peaks in the 2D spectrum are negative with respect to the diagonal peaks. Cross-relaxation due to stochastic motion is generally believed to give rise to such negative peaks. Herein, we use theoretical calculations, numerical simulations and experiments to show that the origin of the negative cross-peaks in PSD spectrum is due to coherent interactions. The origin of negative peaks can be specifically ascribed to a four spin, double-flip-double flop term, in the third-order Hamiltonian. These terms become the dominant terms at fast spinning when additional - conditions are satisfied.

Probing the Surface Structure of Semiconductor Nanoparticles by DNP SENS with Dielectric Support Materials

Surface characterization is crucial for understanding how the atomic-level structure affects the chemical and photophysical properties of semiconducting nanoparticles (NPs). Solid-state nuclear magnetic resonance spectroscopy (NMR) is potentially a powerful technique for the characterization of the surface of NPs, but it is hindered by poor sensitivity. Dynamic nuclear polarization surface enhanced NMR spectroscopy (DNP SENS) has previously been demonstrated to enhance the sensitivity of surface-selective solid-state NMR experiments by 1-2 orders of magnitude. Established sample preparations for DNP SENS experiments on NPs require the dilution of the NPs on mesoporous silica. Using hexagonal boron nitride (h-BN) to disperse the NPs doubles DNP enhancements and absolute sensitivity in comparison to standard protocols with mesoporous silica. Alternatively, precipitating the NPs as powders, mixing them with h-BN, and then impregnating the powdered mixture with radical solution leads to further 4-fold sensitivity enhancements by increasing the concentration of NPs in the final sample. This modified procedure provides a factor of 9 improvement in NMR sensitivity in comparison to previously established DNP SENS procedures, enabling challenging homonuclear and heteronuclear 2D NMR experiments on CdS, Si, and Cd₃P₂ NPs. These experiments allow NMR signals from the surface, subsurface, and core sites to be observed and assigned. For example, we demonstrate the acquisition of DNP-enhanced 2D ¹¹³Cd-¹¹³Cd correlation NMR experiments on CdS NPs and natural isotropic abundance 2D ¹³C-²⁹Si HETCOR of functionalized Si NPs. These experiments provide a critical understanding of NP surface structures.

Rapid Structure Determination of Molecular Solids Using Chemical Shifts Directed by Unambiguous Prior Constraints

NMR-based crystallography approaches involving the combination of crystal structure prediction methods, ab initio calculated chemical shifts and solid-state NMR experiments are powerful methods for crystal structure determination of microcrystalline powders. However, currently structural information obtained from solid-state NMR is usually included only after a set of candidate crystal structures has already been independently generated, starting from a set of single-molecule conformations. Here, we show with the case of ampicillin that this can lead to failure of structure determination. We propose a crystal structure determination method that includes experimental constraints during conformer selection. In order to overcome the problem that experimental measurements on the crystalline samples are not obviously translatable to restrict the single-molecule conformational space, we propose constraints based on the analysis of absent cross-peaks in solid-state NMR correlation experiments. We show that these absences provide unambiguous structural constraints on both the crystal structure and the gas-phase conformations, and therefore can be used for unambiguous selection. The approach is parametrized on the crystal structure determination of flutamide, flufenamic acid, and cocaine, where we reduce the computational cost by around 50%. Most importantly, the method is then shown to correctly determine the crystal structure of ampicillin, which would have failed using current methods because it adopts a high-energy conformer in its crystal structure. The average positional RMSE on the NMR powder structure is ⟨r_av⟩ = 0.176 Å, which corresponds to an average equivalent displacement parameter U_eq = 0.0103 Ų.

Quantitative 1H-1H Distances in Protonated Solids by Frequency-Selective Recoupling at Fast Magic Angle Spinning NMR

Nuclear magnetic resonance (NMR) spectroscopy of protons in protonated solids is challenging. Fast magic angle spinning (MAS) and homonuclear decoupling schemes, in conjunction, with high magnetic fields have improved the proton resolution. However, experiments to quantitatively measure ¹H-¹H distances still remain elusive due to the dense proton-proton dipolar coupling network. A novel MAS solid-state NMR pulse sequence is proposed to selectively recouple and measure interproton distances in protonated samples. The phase-modulated sequence combined with a judicious choice of transmitter frequency is used to measure quantitative ¹H-¹H distances on the order of 3 Å in l-histidine·HCl·H₂O, despite the presence of other strongly coupled protons. This method provides a major boost to NMR crystallography approaches for structural determination of pharmaceutical molecules by directly measuring ¹H-¹H distances. The band-selective nature of the sequence also enables observation of selective ¹H-¹H correlations (e.g., H^N-H^N/H^N-H^α/Η^Ν-Η^Methyl) in peptides and proteins, which should serve as useful restraints in structure determination.

Partial solid-state NMR 1H, 13C, 15N resonance assignments of a perdeuterated back-exchanged seven-transmembrane helical protein Anabaena Sensory Rhodopsin

Anabaena Sensory Rhodopsin (ASR) is a unique photochromic membrane-embedded photosensor which interacts with soluble transducer and is likely involved in a light-dependent gene regulation in the cyanobacterium Anabaena sp. PCC 7120. We report partial spectroscopic ¹H, ¹³C and ¹⁵N assignments of perdeuterated and back-exchanged ASR reconstituted in lipids. The reported assignments are in general agreement with previously determined assignments of carbon and nitrogen resonances in fully protonated samples. Because the back-exchange was performed on ASR in a detergent-solubilized state, the location of detected residues reports on the solvent accessibility of ASR in detergent. A comparison with the results of previously published hydrogen/exchange data collected on the ASR reconstituted in lipids, suggests that the protein has larger solvent accessible surface in the detergent-solubilized state.

Fast Magic-Angle-Spinning 19F Spin Exchange NMR for Determining Nanometer 19F-19F Distances in Proteins and Pharmaceutical Compounds

Internuclear distances measured using NMR provide crucial constraints of three-dimensional structures but are often restricted to about 5 Å due to the weakness of nuclear-spin dipolar couplings. For studying macromolecular assemblies in biology and materials science, distance constraints beyond 1 nm will be extremely valuable. Here we present an extensive and quantitative analysis of the feasibility of 19F spin exchange NMR for precise and robust measurements of interatomic distances up to 1.6 nm at a magnetic field of 14.1 T, under 20-40 kHz magic-angle spinning (MAS). The measured distances are comparable to those achievable from paramagnetic relaxation enhancement but have higher precision, which is better than ±1 Å for short distances and ±2 Å for long distances. For 19F spins with the same isotropic chemical shift but different anisotropic chemical shifts, intermediate MAS frequencies of 15-25 kHz without 1H irradiation accelerate spin exchange. For spectrally resolved 19F-19F spin exchange, 1H-19F dipolar recoupling significantly speeds up 19F-19F spin exchange. On the basis of data from five fluorinated synthetic, pharmaceutical, and biological compounds, we obtained two general curves for spin exchange between CF groups and between CF3 and CF groups. These curves allow 19F-19F distances to be extracted from the measured spin exchange rates after taking into account 19F chemical shifts. These results demonstrate the robustness of 19F spin exchange NMR for distance measurements in a wide range of biological and chemical systems.

Beyond Structural Biology to Functional Biology: Solid-State NMR Experiments and Strategies for Understanding the M2 Proton Channel Conductance

In terms of structural biology, solid-state NMR experiments and strategies have been well established for resonance assignments, leading to the determination of three-dimensional structures of insoluble membrane proteins in their native-like environment. It is also known that NMR has the unique capabilities to characterize structure-function relationships of membrane-bound biological systems beyond structural biology. Here, we report on solid-state NMR experiments and strategies for extracting functional activities on a sub-millisecond time scale. Specifically, we use the His37-labeled full length M2 (M2FL) protein of the Influenza A virus embedded in synthetic lipid bilayers as an example to characterize the proton conduction mechanism and kinetics. The integral membrane M2 protein assembles as a tetrameric bundle to form a proton-conducting channel that is activated by low pH and is essential for the viral lifecycle. Our results present convincing evidence for the formation of imidazolium-imidazole hydrogen bonds in the His37 tetrad at low pH and that these hydrogen bonds have a low barrier that facilitates the proton conduction mechanism in the M2FL protein. Moreover, it has been possible to measure hydronium ion exchange between water and the protons in the His37 NH bonds based on chemical exchange spectroscopy with minimized spin diffusion. The results identify an exchange rate constant of ∼4000 s^-1 for pH 5.8 at -10 °C.

Moderate MAS enhances local (1)H spin exchange and spin diffusion

Proton NMR spin-diffusion experiments are often combined with magic-angle spinning (MAS) to achieve higher spectral resolution of solid samples. Here we show that local proton spin diffusion can indeed become faster at low (<10 kHz) spinning rates as compared to static conditions. Spin diffusion under static conditions can thus be slower than the often referred value of 0.8 nm(2)/ms, which was determined using slow MAS (Clauss et al., 1993). The enhancement of spin diffusion by slow MAS relies on the modulation of the orientation-dependent dipolar couplings during sample rotation and goes along with transient level crossings in combination with dipolar truncation. The experimental finding and its explanation is supported by density matrix simulations, and also emphasizes the sensitivity of spin diffusion to the local coupling topology. The amplification of spin diffusion by slow MAS cannot be explained by any model based on independent spin pairs; at least three spins have to be considered.

Solid state field-cycling NMR relaxometry: instrumental improvements and new applications

The paper reviews recent progress in field cycling (FC) NMR instrumentation and its application to solid state physics. Special emphasis is put on our own work during the last 15years on instrumentation, theory and applications. As far as instrumentation is concerned we report on our development of two types of electronical FC relaxometers, a mechanical FC relaxometer and a combination of FC and one-dimensional microimaging. Progress has been achieved with respect to several parameters such as the accessible field and temperature range as well as the incorporation of sample spinning. Since an appropriate analysis of FC data requires a careful consideration of relaxation theory, we include a theory section discussing the most relevant aspects of relaxation in solids which are related to residual dipolar and quadrupolar interactions. The most important limitations of relaxation theory are also discussed. With improved instrumentation and with the help of relaxation theory we get access to interesting new applications such as ionic motion in solid electrolytes, structure determination in molecular crystals, ultraslow polymer dynamics and rotational resonance phenomena.

Direct observation of ¹⁷O-¹⁸⁵/¹⁸⁷Re ¹J-coupling in perrhenates by solid-state ¹⁷O VT MAS NMR: temperature and self-decoupling effects

(17)O MAS NMR spectra recorded at 14.1T and room temperature (RT) for (17)O-enriched samples of the two perrhenates, KReO4 and NH4ReO4, exhibit very similar overall appearances of the manifold of spinning sidebands (ssbs) for the satellite transitions (STs) and the central transition (CT). These overall appearances of the spectra are easily simulated in terms of the usual quadrupole coupling and chemical shift interaction parameters. However, a detailed inspection of the line shapes for the individual ssbs of the STs and, in particular, for the CT in the spectrum of KReO4 reveals line-shape features, which to our knowledge have not before been observed experimentally in 1D MAS NMR spectra for any quadrupolar nucleus, nor emerged from simulations for any combination of second-order quadrupolar interaction and chemical shift anisotropy. In contrast, such line-shape features are not observed for the corresponding ssbs (STs and CT) in the 14.1T RT (17)O MAS NMR spectrum of NH4ReO4. Considering the additional interaction of a combination of residual heteronuclear (17)O-(185/)(187)Re dipolar and scalar J coupling between this spin pair of two quadrupolar nuclei, spectral simulations for KReO4 show that these interactions are able to account for the observed line shapes, although the expected (1)J((17)O-(185/)(187)Re) six-line spin-spin splittings are not resolved. Low-temperature, high-field (21.1T) (17)O VT MAS NMR spectra of both KReO4 and NH4ReO4 show that full resolution into six-line multiplets for the centerbands are achieved at -90°C and -138°C, respectively. This allows determination of (1)J((17)O-(187)Re)=-268Hz and -278Hz for KReO4 and NH4ReO4, respectively, i.e., an isotropic (1)J coupling and its sign between two quadrupolar nuclei, observed for the first time directly from solid-state one-pulse 1D MAS NMR spectra, without resort to additional 1D or 2D experiments. Determination of T1((187)Re) spin-lattice relaxation times, observed indirectly through a 2D (17)O EXSY experiment for NH4ReO4 at several low temperatures, show that the dynamics observed for the ReO4(-) anion in the (17)O VT MAS NMR spectra at low temperatures are caused by self-decoupling of (1)J((17)O-(187)Re). The (1)J((17)O-(187)Re) values determined here for ReO4(-) from solid-state (17)O MAS NMR, along with literature (1)J((17)O-M) values for oxoanions (M being a quadrupolar nucleus) obtained from liquid-state NMR, have allowed correlations to be established between the reduced coupling constant (1)K((17)O-M)=2π(1)J((17)O-M)/(γ17OγMℏ) and the atomic number of M.

Proton-driven spin diffusion in rotating solids via reversible and irreversible quantum dynamics

Proton-driven spin diffusion (PDSD) experiments in rotating solids have received a great deal of attention as a potential source of distance constraints in large biomolecules. However, the quantitative relationship between the molecular structure and observed spin diffusion has remained obscure due to the lack of an accurate theoretical description of the spin dynamics in these experiments. We start with presenting a detailed relaxation theory of PDSD in rotating solids that provides such a description. The theory applies to both conventional and radio-frequency-assisted PDSD experiments and extends to the non-Markovian regime to include such phenomena as rotational resonance (R(2)). The basic kinetic equation of the theory in the non-Markovian regime has the form of a memory function equation, with the role of the memory function played by the correlation function. The key assumption used in the derivation of this equation expresses the intuitive notion of the irreversible dissipation of coherences in macroscopic systems. Accurate expressions for the correlation functions and for the spin diffusion constants are given. The theory predicts that the spin diffusion constants governing the multi-site PDSD can be approximated by the constants observed in the two-site diffusion. Direct numerical simulations of PDSD dynamics via reversible Liouville-von Neumann equation are presented to support and compliment the theory. Remarkably, an exponential decay of the difference magnetization can be observed in such simulations in systems consisting of only 12 spins. This is a unique example of a real physical system whose typically macroscopic and apparently irreversible behavior can be traced via reversible microscopic dynamics. An accurate value for the spin diffusion constant can be usually obtained through direct simulations of PDSD in systems consisting of two (13)C nuclei and about ten (1)H nuclei from their nearest environment. Spin diffusion constants computed by this method are in excellent agreement with the spin diffusion constants obtained through equations given by the relaxation theory of PDSD. The constants resulting from these two approaches were also in excellent agreement with the results of 2D rotary resonance recoupling proton-driven spin diffusion (R(3)-PDSD) experiments performed in three model compounds, where magnetization exchange occurred over distances up to 4.9 Å. With the methodology presented, highly accurate internuclear distances can be extracted from such data. Relayed transfer of magnetization between distant nuclei appears to be the main (and apparently resolvable) source of uncertainty in such measurements. The non-Markovian kinetic equation was applied to the analysis of the R(2) spin dynamics. The conventional semi-phenomenological treatment of relxation in R(2) has been shown to be equivalent to the assumption of the Lorentzian spectral density function in the relaxatoin theory of PDSD. As this assumption is a poor approximation in real physical systems, the conventional R(2) treatment is likely to carry a significant model error that has not been recognized previously. The relaxation theory of PDSD appears to provide an accurate, parameter-free alternative. Predictions of this theory agreed well with the full quantum mechanical simulations of the R(2) dynamics in the few simple model systems we considered.

Fractional deuteration applied to biomolecular solid-state NMR spectroscopy

Solid-state Nuclear Magnetic Resonance can provide detailed insight into structural and dynamical aspects of complex biomolecules. With increasing molecular size, advanced approaches for spectral simplification and the detection of medium to long-range contacts become of critical relevance. We have analyzed the protonation pattern of a membrane-embedded ion channel that was obtained from bacterial expression using protonated precursors and D(2)O medium. We find an overall reduction of 50% in protein protonation. High levels of deuteration at H(α) and H(β) positions reduce spectral congestion in ((1)H,(13)C,(15)N) correlation experiments and generate a transfer profile in longitudinal mixing schemes that can be tuned to specific resonance frequencies. At the same time, residual protons are predominantly found at amino-acid side-chain positions enhancing the prospects for obtaining side-chain resonance assignments and for detecting medium to long-range contacts. Fractional deuteration thus provides a powerful means to aid the structural analysis of complex biomolecules by solid-state NMR.

Broad-band homo-nuclear correlations assisted by 1H irradiation for bio-molecules in very high magnetic field at fast and ultra-fast MAS frequencies

We propose a new broadband second-order proton-assisted (13)C-(13)C correlation experiment, SHANGHAI. The (13)C-(13)C magnetization transfer is promoted by (1)H irradiation with interspersed four phases super-cycling. This through-space homo-nuclear sequence only irradiates on the proton channel during the mixing time. SHANGHAI benefits from a large number of modulation sidebands, hence leading to a large robustness with respect to chemical shift differences, which permits its use in a broad MAS frequency range. At ultra-fast MAS (ν(R) 60 kHz), SHANGHAI is only efficient when the amplitude of (1)H recoupling rf-field is close to half the spinning speed (ν(1) ≈ ν(R)/2). However, at moderate to fast MAS (ν(R)=20-35 kHz), SHANGHAI is efficient at any rf-power level larger than ν(1) ≈ 10 kHz, which simultaneously permits avoiding excessive heating of bio-molecules, and using large sample volumes. We show that SHANGHAI can be employed at the very high magnetic field of 23.5 T and then allows the observation of correlation between (13)C nuclei, even if their resonance frequencies differ by more than 38 kHz.

A master-equation approach to the description of proton-driven spin diffusion from crystal geometry using simulated zero-quantum lineshapes

Measurements of proton-driven carbon-13 spin diffusion (PDSD) by NMR spectroscopy are a central component of structural analyses of biomolecules in the solid-state. However, the quantitative link between experimental PDSD data and structural information is difficult to make. Here we observe that a master-equation approach can be used to model full PDSD dynamics accurately in polycrystalline (13)C-labelled L-histidine·HCl·H(2)O under magic-angle spinning. In the master-equation approach, PDSD rates and effective dipolar couplings are related by a function of the carbon-carbon zero-quantum lineshapes; we find that numerical simulations of the zero-quantum lineshapes are sufficiently accurate so as to allow the calculation of PDSD rates that are in good agreement with the measured rates, directly from crystal geometry and with no adjustable parameters. Finally, using carbon-carbon internuclear distances we illustrate the potential of the master-equation approach for structural studies. Generalisation of these results to proton-driven carbon-13 spin diffusion in more complex molecular systems is readily envisaged.

Characteristics of zero-quantum correlation spectroscopy in MAS NMR experiments

Zero-quantum coherence generation and reconversion in magic-angle spinning solid-state NMR is analyzed. Two methods are discussed based on implementations using symmetry-based pulse sequences that utilize either isotropic J couplings or dipolar couplings. In either case, the decoupling of abundant proton spins plays a crucial role for the efficiency of the zero-quantum generation. We present optimized sequences for measuring zero-quantum single-quantum correlation spectra in solids, achieving an efficiency of 50% in ubiquitin. The advantages and disadvantages of zero-quantum single-quantum over single-quantum single-quantum correlation spectroscopy are explored, and similarities and differences with double-quantum single-quantum correlation spectroscopy are discussed. Finally, possible application of zero-quantum single-quantum experiments to polypeptides, where it can lead to better spectral resolution is investigated using ubiquitin, where we find high efficiency and high selectivity, but also increased line widths in the MQ dimension.

Structural characterization of GNNQQNY amyloid fibrils by magic angle spinning NMR

Several human diseases are associated with the formation of amyloid aggregates, but experimental characterization of these amyloid fibrils and their oligomeric precursors has remained challenging. Experimental and computational analysis of simpler model systems has therefore been necessary, for instance, on the peptide fragment GNNQQNY7−13 of yeast prion protein Sup35p. Expanding on a previous publication, we report here a detailed structural characterization of GNNQQNY fibrils using magic angle spinning (MAS) NMR. On the basis of additional chemical shift assignments we confirm the coexistence of three distinct peptide conformations within the fibrillar samples, as reflected in substantial chemical shift differences. Backbone torsion angle measurements indicate that the basic structure of these coexisting conformers is an extended β-sheet. We structurally characterize a previously identified localized distortion of the β-strand backbone specific to one of the conformers. Intermolecular contacts are consistent with each of the conformers being present in its own parallel and in-register sheet. Overall the MAS NMR data indicate a substantial difference between the structure of the fibrillar and crystalline forms of these peptides, with a clearly increased complexity in the GNNQQNY fibril structure. These experimental data can provide guidance for future work, both experimental and theoretical, and provide insights into the distinction between fibril growth and crystal formation.

Operator-based Floquet theory in solid-state NMR

This article reviews the application of operator-based Floquet theory in solid-state NMR. Basic expressions for calculating effective Hamiltonians based on van Vleck perturbation theory are reviewed for problems with a single frequency or multiple incommensurate frequencies. Such a treatment allows calculation of effective Hamiltonians for resonant and non-resonant problems. Examples from literature are given for single-mode to triple-mode Floquet problems, covering a wide range of applications in solid-state NMR under magic-angle spinning and radio-frequency irradiation of a single nucleus or multiple nuclei.

Homonuclear dipolar recoupling techniques for structure determination in uniformly 13C-labeled proteins

In solid-state NMR magic angle spinning is often used to remove line broadening associated with anisotropic interactions, such as chemical shift anisotropy and dipolar couplings. Dipolar recoupling refers to sequences of pulses designed to reintroduce dipolar interactions that are otherwise averaged by magic angle spinning. One of the key applications of homonuclear (and heteronuclear) dipolar recoupling is for the purpose of protein structure determination. Recoupling experiments, originally designed for applications in spin-pair labeled samples, have been revised in recent years for applications in samples with extensive or uniform incorporation of isotopic labels. In these samples multiple internuclear distances can in principle be probed simultaneously, but the dipolar truncation effects (i.e. attenuation of the effects of weak couplings by strong ones) circumvent such measurements. In this article we review some of the recent developments in homonuclear recoupling methods that allow overcoming this problem.

Targeted 13C-13C distance measurements in a microcrystalline protein via J-decoupled rotational resonance width measurements

Rotational resonance width (R(2)W) magic-angle spinning (MAS) NMR experiments are performed to measure (13)C-(13)C distances in the hydrophobic core of the microcrystalline model protein G(Beta1). Such inter-residue distances are of particular value in NMR structure determinations. The experiments are done at a Larmor frequency of 750 MHz (1)H where the contribution of (13)C chemical shift anisotropy (CSA) to the R(2) transfer mechanism is significant. To minimize line broadening in the 2D spectra, we employ a combination of even/odd isotopic labeling with [1,3-(13)C] glycerol, and J-decoupling in the indirect dimension. This results in high-precision distance measurements between aromatic side chains of three tyrosine residues and distant methyl groups in the hydrophobic core of the protein. Even in the absence of information on the relative orientation of the shift tensors, we obtain relatively high precision data, which can be further improved by additional constraints on the tensor orientations.

(13)C-(13)C distance measurements in U-(13)C, (15)N-labeled peptides using rotational resonance width experiment with a homogeneously broadened matching condition

In this publication, we introduce a version of the rotational resonance width experiment with a homogeneously broadened matching condition. The increase in the bandwidth is achieved by the reduction of the proton decoupling power during mixing, which results in the reduction of zero-quantum relaxation, and broadens the rotational resonance condition. We show that one can achieve recoupling of the carbonyl-aliphatic side chain dipolar interactions band selectively, while avoiding the recoupling of strongly interacting C'-Calpha and C'-Cbeta spin pairs. The attenuation of the multi-spin effects in the presence of short zero-quantum relaxation enables a two-spin approximation to be employed for the analysis of the experimental data. The systematic error introduced by this approximation is estimated by comparing the results with a three-spin simulation. The experiment is demonstrated in [U-(13)C,(15)N]N-acetyl-L-Val-L-Leu dipeptide, where 11 distances, ranging from 2.5 to 6 A, were measured.

Broadband rotational resonance in solid state NMR spectroscopy

A new technique for restoring nuclear magnetic dipole-dipole couplings under magic-angle spinning (MAS) in solid state nuclear magnetic resonance (NMR) spectroscopy is described and demonstrated. In this technique, called broadband rotational resonance (BroBaRR), the coupling between a pair of nuclear spins with NMR frequency difference close (but not necessarily equal) to the MAS frequency is restored by the application of a train of weak radio-frequency pulses at a carrier frequency close to the average of the two NMR frequencies. Phase or amplitude modulation of the pulse train at half the MAS frequency splits the carrier into sidebands close to the two NMR frequencies. The pulse train then removes offsets from the exact rotational resonance condition, leading to dipolar recoupling over a bandwidth controlled by the amplitude of the pulse train. (13)C NMR experiments on uniformly (15)N,(13)C-labeled L-valineHClH(2)O powder validate the theoretical analysis. BroBaRR will be useful in studies of molecular structures by solid state NMR, for example in the detection of long-range couplings between carbons in uniformly labeled organic and biological materials.

Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion

Gaussian spherical quadrature methods in the guise of the Lebedev sampling grids are highly efficient for some orientational ("powder") averaging problems in solid state NMR. However, their applicability is currently restricted, as the sets of orientations are derived analytically and because they are not well adapted to simulate the broad peakshapes encountered, for example, in the NMR on static powders or on half-integer quadrupolar spins subject to second order quadrupolar interactions under magic-angle spinning conditions. We remedy these problems by (i) introducing the recursive procedure regularized octahedral symmetry expansion (ROSE), to which any existing Lebedev set may be subjected. Each recursive step gives a 9-fold enlarged set of orientations. (ii) We demonstrate that ROSE-expanded grids, in conjunction with spectral interpolation, is well suited for calculating broad peakshapes. These advances combine into the apparently most efficient general-purpose two-angle orientational averaging technique proposed to date for solid state NMR applications.

(1)H-(13)C INEPT MAS NMR correlation experiments with (1)H-(1)H mediated magnetization exchange to probe organization in lipid biomembranes

Two-dimensional (1)H-(13)C INEPT MAS NMR experiments utilizing a (1)H-(1)H magnetization exchange mixing period are presented for characterization of lipid systems. The introduction of the exchange period allows for structural information to be obtained via (1)H-(1)H dipolar couplings but with (13)C chemical shift resolution. It is shown that utilizing a RFDR recoupling sequence with short mixing times in place of the more standard NOE cross-relaxation for magnetization exchange during the mixing period allowed for the identification and separation of close (1)H-(1)H dipolar contacts versus longer-range inter-molecular (1)H-(1)H dipolar cross-relaxation. These 2D INEPT experiments were used to address both intra- and inter-molecular contacts in lipid and lipid/cholesterol mixtures.

Solid-state NMR yields structural constraints on the V3 loop from HIV-1 Gp120 bound to the 447-52D antibody Fv fragment

Solid-state NMR measurements were performed on the complex of an 18-residue peptide derived from the V3 loop sequence of the gp120 envelope glycoprotein of the HIV-1 MN strain with Fv fragments of the human anti-gp120 monoclonal antibody 447-52D in a frozen glycerol/water solution. The peptide was uniformly (15)N- and (13)C-labeled in a 7-residue segment containing the conserved GPGR motif in the epitope. (15)N and (13)C NMR chemical shift assignments for the labeled segment were obtained from two-dimensional (13)C-(13)C and (15)N-(13)C magic-angle spinning NMR spectra. Reductions in (13)C NMR line widths and changes in chemical shifts upon complex formation indicate the adoption of a well-defined, antibody-dependent structure. Intramolecular (13)C-(13)C distances in the complex, which constrain the peptide backbone and side chain conformations in the GPGR motif, were determined from an analysis of rotational resonance (RR) data. Structural constraints from chemical shifts and RR measurements are in good agreement with recent solution NMR and crystallographic studies of this system, although differences regarding structural ordering of certain peptide side chains are noted. These experiments explore and help delineate the utility of solid state NMR techniques as structural probes of peptide/protein complexes in general, potentially including membrane-associated hormone/receptor complexes.

13C nuclear overhauser polarization-magic-angle spinning nuclear magnetic resonance spectroscopy in uniformly 13C-labeled solid proteins

A recently introduced (13)C polarization technique based on the nuclear Overhauser effect in rotating solid (nuclear Overhauser polarization-magic-angle spinning, NOP-MAS) (Takegoshi, K.; Terao, T. J. Chem. Phys. 2002, 117, 1700-1707) is applied to uniformly (13)C, (15)N-labeled proteins. NOP enhancement factors per scan of 1.5 approximately 2.0 are obtained, while that by cross polarization (CP) is less than 1.0. We show that uniform enhancement of all (13)C signals by CP is difficult to attain, while it is easily achieved by NOP, thus enabling quantitative comparison of signal intensities. NOP is easy to carry out under fast MAS and works well even for somewhat mobile molecules, for which CP does not work. Moreover, in labeled protein samples containing nonlabeled additives, NOP can eliminate the latter signals. For these features, NOP is superior to CP in many uniformly (13)C labeled proteins.

Uniform-sign cross-peak double-quantum-filtered correlation spectroscopy

We detail the uniform-sign cross-peak double-quantum-filtered correlation spectroscopy (UC2QF COSY) experiment, a new through-bond correlation method for disordered solids. This experiment is a refocused version of the popular double-quantum-filtered correlation spectroscopy experiment in liquids. Its key feature is that it provides in-phase and doubly absorptive line shapes, which renders it robust for chemical shift correlation in solids. Both theory and experiment point to distinct advantages of this protocol, which are illustrated by several experiments under challenging conditions, including fast magic-angle spinning (30kHz), anisotropic molecular motion, and (13)C correlation spectroscopy at the natural abundance isotope level.

Rotational-resonance distance measurements in multi-spin systems

It is demonstrated that internuclear distances can be evaluated from rotational-resonance (RR) experiments in uniformly (13)C-labelled compounds. The errors in the obtained distances are less than 10% without the need to know any parameters of the spin system except the isotropic chemical shifts of all spins. We describe the multi-spin system with a simple fictitious spin-1/2 model. The influence of the couplings to the passive spins (J and dipolar coupling) is described by an empirical constant offset from the rotational-resonance condition. Using simulated data for a three-spin system, we show that the two-spin model describes the rotational-resonance transfer curves well as long as none of the passive spins is close to a rotational-resonance condition with one of the active spins. The usability of the two-spin model is demonstrated experimentally using a sample of acetylcholine perchlorate with labelling schemes of various levels of complexity. Doubly-, triply-, and fully labelled compounds lead to strongly varying RR polarization-transfer curves but the evaluated distances using the two-spin model are identical within the expected error limits and coincide with the distance from the X-ray structure. Rotational-resonance distance measurements in fully labelled compounds allow, in particular, the measurement of weak couplings in the presence of strong couplings.

Rotational resonance in uniformly 13C-labeled solids: effects on high-resolution magic-angle spinning NMR spectra and applications in structural studies of biomolecular systems

We describe investigations of the effects of rotational resonance (R(2)) on solid state (13)C NMR spectra of uniformly (13)C-labeled samples obtained under magic-angle spinning (MAS), and of the utility of R(2) measurements as structural probes of peptides and proteins with multiple uniformly labeled residues. We report results for uniformly (13)C-labeled L-alanine and L-valine in polycrystalline form, and for amyloid fibrils formed by the 15-residue peptide A beta(11-25) with uniform labeling of a four-residue segment. The MAS NMR spectra reveal a novel J-decoupling effect at R(2) conditions that may be useful in spectral assignments for systems with sharp (13)C MAS NMR lines. Pronounced dependences of the apparent isotropic (13)C NMR chemical shifts on MAS frequency near R(2) conditions are also observed. We demonstrate the feasibility of quantitative (13)C-(13)C distance determinations in L-valine, and qualitative determinations of inter-residue (13)C-(13)C contacts in A beta(11-25) fibrils. Finally, we demonstrate a "relayed" R(2) technique that may be useful in structural measurements on systems with poorly resolved (13)C MAS NMR lines.

Multiple-spin analysis of chemical-shift-selective (13C, 13C) transfer in uniformly labeled biomolecules

Chemical-shift-selective (13C, 13C) polarization transfer is analyzed in uniformly labeled biomolecules. It is shown that the spin system dynamics remain sensitive to the distance of interest and can be well reproduced within a quantum-mechanical multiple-spin analysis. These results lead to a general approach on how to describe chemical-shift-selective transfer in uniformly labeled systems. As demonstrated in the case of ubiquitin, this methodology can be used to detect long-range distance constraints in uniformly labeled proteins.

13C−13C Rotational Resonance Width Distance Measurements in Uniformly 13C-Labeled Peptides

The rotational resonance width (R2W) experiment is a constant-time version of the rotational resonance (R2) experiment, in which the magnetization exchange is measured as a function of sample spinning frequency rather than the mixing time. The significant advantage of this experiment over conventional R2 is that both the dipolar coupling and the relaxation parameters can be independently and unambiguously extracted from the magnetization exchange profile. In this paper, we combine R2W with two-dimensional 13C-13C chemical shift correlation spectroscopy and demonstrate the utility of this technique for the site-specific measurement of multiple 13C-13C distances in uniformly labeled solids. The dipolar truncation effects, usually associated with distance measurements in uniformly labeled solids, are considerably attenuated in R2W experiments. Thus, R2W experiments are applicable to uniformly labeled biological systems. To validate this statement, multiple 13C-13C distances (in the range of 3-6 A) were determined in N-acetyl-[U-13C,15N]l-Val-l-Leu with an average precision of +/-0.5 A. Furthermore, the distance constraints extracted using a two-spin model agree well with the X-ray crystallographic data.

Enhanced triple-quantum excitation in 13C magic-angle spinning NMR

We describe a new method for exciting triple-quantum coherences in 13C-labelled powder samples under MAS. The proposed method combines selective double-quantum excitation with rotational resonance and frequency-selective composite pulses. The spin dynamics of this new method are described theoretically. Numerical calculations of the spin dynamics are compared to experimental results on fully 13C-labelled L-alanine. The observed triple-quantum filtering efficiency is around 10% for the most intense spectral peak. The method is also demonstrated on other fully 13C-labelled compounds, including a uniformly 13C-labelled amino acid.