 1.1 Preface
 1.2 Interactions in Biological SolidState NMR
 1.2.1 General and Rotational Properties of NMR Interactions
 1.2.2 Chemical Shift Anisotropy
 1.2.3 Dipolar Coupling
 1.2.4 Scalar Coupling
 1.2.5 Quadrupolar Coupling
 1.2.6 Magic Angle Spinning
 1.3 Cross Polarization
 1.4 Dipolar Decoupling
 1.5 Recoupling
 1.5.1 Protein Backbone Assignment
 1.5.2 Protein Geometry Restraints
 1.5.3 Intermolecular Geometry Restraints
 1.5.4 Protein Dynamics
CHAPTER 1: Introduction to Biological SolidState NMR

Published:24 Feb 2014

Series: New Developments in NMR
M. Weingarth and M. Baldus, in Advances in Biological SolidState NMR: Proteins and MembraneActive Peptides, ed. F. Separovic and A. Naito, The Royal Society of Chemistry, 2014, pp. 117.
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This chapter summarizes the most relevant NMR interactions in solidlike biological systems and discusses the basic components of solidstate NMR pulse sequences for application to proteins and peptides.
1.1 Preface
Solidstate nuclear magnetic resonance has emerged as an established spectroscopic technique to provide atomicscale information in complex biological systems. Recent years, with the appearance of the first de novo structures of membrane proteins^{1 } and solidstate NMR spectroscopists unraveling the details of native cellular components at atomic resolution,^{2 } indeed seem to mark a watershed moment for solidstate NMR becoming a leading technique to study highly disordered or heterogeneous biological systems.
These stunning advances, however, are hardwon. What catches one's eye, especially for eyes used to the spectral resolution of liquidstate NMR, is the broadness of nonmodulated solidstate NMR signals. For liquidstate spectroscopists, such data may appear as a featureless blob, from which structural parameters are hardly deducible, let alone at atomic resolution (Figure 1.1). The broadness of the signals, which easily exceeds dozens of kHz or even several MHz, results from the presence of anisotropic interactions in solidstate NMR spectra, and a large part of solidstate NMR methodology refers to the manipulation of the Hamiltonian to dissect the anisotropic interactions or to suppress their influence on NMR spectra in a controlled manner.
In the following, we will first introduce the reader to the most relevant NMR interactions in solidlike biological systems and afterwards briefly discuss the basic components of solidstate NMR pulse sequences for biological applications.
1.2 Interactions in Biological SolidState NMR
Interactions which affect the spin system and its associated Hamiltonian can be classified as external and internal (eqn 1.1). The external Hamiltonian is directly under the control of the NMR operator and consists of the Zeeman interaction and the rf (radiofrequency) fields, while the internal Hamiltonian includes the interaction of the spins with the local electronic environment (chemical shift), with each other (dipolar and scalar couplings) and with electric field gradients (quadrupolar coupling). The internal interactions may be modulated by the spectroscopist by means of rf pulses or magic angle spinning.
1.2.1 General and Rotational Properties of NMR Interactions
Before we summarize the most relevant interactions in biological solidstate NMR, we will briefly discuss their general appearance and properties. Whether external or internal interactions, their Hamiltonians can be depicted as:
with I and S as row and column vectors, respectively, and Ã as a secondrank Cartesian tensor as a means to represent the anisotropy of an interaction, corresponding to a 3×3 square matrix. In this equation, I is a spin operator, while the other vector S represents either a second spin (dipolar or scalar coupling), or a magnetic (chemical shift) or an electric field (quadrupolar coupling). Subjected to rapid isotropic molecular motion, Ã is averaged to its trace, which is the sum of the diagonal elements:
Besides the Cartesian representation, irreducible spherical tensor operators are a particularly appropriate basis to describe transformations in solidstate NMR:
In this representation, it is easily visible that any NMR interaction can be broken down into a spatial part, represented by spherical tensor A, and a spin part, represented by spherical tensor operators T. Indices k indicate the rank of a tensor, which has 2m+1 orders (which are irreducible spherical tensors). Component A_{00} is the isotropic, A_{1−m} is the antisymmetric and A_{2−m} is the anisotropic part of an interaction. Since the antisymmetric part can usually be ignored, and since the static magnetic field B_{0} is usually orders of magnitude larger than the internal interactions (with the exception of certain quadrupolar interactions), the summation can be reduced in the socalled secular approximation to:
neglecting all terms that vanish over time and retaining only those that commute with B_{0}. Rotations of spherical tensors are carried out by Wigner rotation matrices D of the same rank, with the Euler angles α, β and γ (eqn 1.6). Details of all these operations can be found in standard references.^{3 }
NMR interactions have their simplest form in their principal axis system (PAS), in which all offdiagonal elements are zero. Yet, NMR signals are detected in the laboratory frame, into which the interactions thus have to be transformed to analyze their effect on the spectrum. Transformations between different frames obey the following relation, which is called the addition theorem of Wigner rotation matrices:
According to these rules, NMR interactions can be transformed from one frame into another. Typically, in solidstate NMR, transformations include a transformation from the PAS to a molecular frame, which is fixed to the molecular structure and in which all NMR interactions share a common orientation, followed by a transformation to a rotor frame to take into account the powder averaging, and finally into the laboratory frame (Figure 1.2).
1.2.2 Chemical Shift Anisotropy
Nuclei possessing a spin I have an associated nuclear magnetic dipole moment μ=γħI, with γ as the gyromagnetic ratio. For an isolated spin½ nucleus, subjected to a static magnetic field B_{0} along the zaxis, this gives rise to energy levels separated by ΔE=−γħB_{0}=ħω_{0}, with ω_{0} denoting the Larmor frequency. The resonance frequencies observed in NMR spectra, however, usually slightly differ from the Larmor frequency. This difference is called the chemical shift, which is caused by variations in the electron distribution around nuclei. The electron distribution around a given nucleus is rarely spherically symmetric, and the resulting anisotropic local field is termed the chemical shift anisotropy (CSA; see Figure 1.3). The CSA may be described as a Cartesian secondrank tensor, the trace of which is the isotropic chemical shift:
The chemical shift tensor can be characterized (note that there are different nomenclature conventions; here we use the Haeberlen convention with σ_{zz}−σ_{iso}≥σ_{xx}−σ_{iso}≥σ_{yy}−σ_{iso}) by its reduced anisotropy σ and anisotropy Δσ=3σ/2, which reflect the deviation from cubic symmetry, and its asymmetry η, which reflects the deviation from axial symmetry (Figure 1.3):
For an axially symmetric CSA tensor, with θ defining the direction of B_{0} in the PAS of the CSA tensor, the resonance frequency can be expressed as:
The nonzero elements of the irreducible spherical tensor operators in the PAS of the chemical shift are
1.2.3 Dipolar Coupling
The local magnetic moments of spins do not only interact with B_{0}, but also with each other. For example, in the context of the dipolar coupling, the magnetic moments interact directly through space. The interaction energy between two magnetic moments with a distance r_{12} (assuming that both dipoles have the same orientation to r_{12}; see Figure 1.4) is given by:
which after replacement of and reformulation in polar coordinates yields the dipolar Hamiltonian with the socalled dipolar alphabet:
where
The terms C–F contain singlequantum I$1\xb1$I_{2z} and doublequantum operators I$1\xb1$I$2\xb1$ and can therefore be safely neglected in the secular approximation. Note that if γ_{1}≠γ_{2} the B term is also nonsecular and can be omitted, which is the reason for the different expression for heteronuclear H^{IS} and homonuclear H^{II} dipolar Hamiltonians (eqn 1.15). Theta (θ) is defined by the direction of the internuclear vector in a coordinate system in which the B_{0} field is in the direction of the zaxis.
with as the dipolar coupling constant. The dipolar coupling tensor is axially symmetric and traceless; hence A$20PAS$ is its only PAS component in the secular approximation.
1.2.4 Scalar Coupling
Scalar couplings are indirect interactions mediated through electrons, which usually act via covalent bonds. The term “scalar” already indicates this interaction to be usually rotationalinvariant, i.e. isotropic with the scalar coupling tensor J_{jj} reduced to its trace. Scalar couplings are often small in comparison to other interactions in solidstate NMR and may be obscured by a broad powder pattern. However, they can of course be exploited in the solid state, too, especially in combination with magic angle spinning.^{4 } The homo and heteronuclear scalar coupling interactions are respectively described by:
1.2.5 Quadrupolar Coupling
Nuclei possessing a spin I > ½ have an electric quadrupole moment arising from a nonspherical distribution of the electric charge around the nucleus. This moment interacts with the electric field gradient, which arises by virtue of the distribution of other nuclei and electrons in the vicinity of the nucleus. Since quadrupolar nuclei are, except for deuterium, nonstandard nuclei for biological solidstate NMR, we will not dwell further on this NMR interaction and refer the interested reader elsewhere.^{3 }
1.2.6 Magic Angle Spinning
Solidstate NMR experiments of biological samples are usually subjected to magic angle spinning (MAS),^{5 } which is a technique to average the spatial part of secondrank interactions by making the interactions timedependent. MAS can greatly enhance spectral resolution and sensitivity. It implies spinning a rotor containing the sample at a certain angle with respect to the static B_{0} field (Figure 1.5), at which the spatial anisotropic part of the interactions, describable as secondrank tensors like the dipolar coupling or the CSA, are averaged out. The orientation dependence of these interactions is proportional to 3(cos^{2}θ−1)/2, which is the second Legendre polynomial P2(cos θ). This expression becomes zero at θ=54.74° which is, therefore, also called the magic angle θ_{m} (Figure 1.5). For axial symmetric tensors, it can be shown that MAS yields the average orientation function:
For asymmetric CSA tensors, one obtains a term depending on the orientation of all three principal axes to the spinning axis. Since this term is also multiplied by P2(cos θ) due to MAS, both axial symmetric and asymmetric CSA tensors are averaged out by fast spinning. If the MAS frequency is smaller than the magnitude of the CSA, a spinning sideband pattern is observed in the NMR spectrum, which at higher spinning frequencies collapses into one resolved peak (Figure 1.5).
For an interaction that commutes with itself over time (referred to as inhomogeneous by Maricq and Waugh^{6 }), MAS achieves complete averaging. This is, however, not the case for timedependent homogeneous interactions (which do not commute with themselves over time), like strong homonuclear dipolar proton couplings. We will refer to a handwaving explanation to illustrate the latter. As discussed above (see Section 1.2.3), unlike H$DIS$, H$DII$ exhibits a secular Bterm which contains zeroquantum operators inducing flipflop transitions among spins I at a ratio proportional to the strengths of H$DII$. This means that the spin states and thus the local dipolar fields are not constant over time (they are mixed by the Bterm), while MAS only works efficiently if the interaction is static for at least one rotor period. It is important to keep in mind that the presence of one homogeneous interaction suffices to render all internal interactions homogeneous and thus difficult to be spun out completely. The reason for this behaviour is that NMR interactions are entangled with the homonuclear couplings in higher order crossterms (see below and elsewhere^{3,6,7 } for further reading). Interestingly, the efficiency of flipflop transitions decreases with increasing chemical shift differences, i.e. with increasing B_{0}, reducing the homogenous character of interactions like dipolar proton–proton couplings. These dependencies become, for example, apparent using average Hamiltonian or Floquet theory.
Note that MAS is limited to secondrank tensors, which does not suffice to suppress interactions like quadrupolar couplings, which in addition to a P2(cos θ) orientation dependence also exhibit a P4(cos θ) dependence. An elegant means to suppress these interactions further is the double rotation (DOR)^{8 } technique, which requires spinning the sample at two angles simultaneously.
1.3 Cross Polarization
Cross polarization (CP)^{9 } is a method to enhance the sensitivity of heteronuclear NMR experiments, in which polarization is transferred from the abundant proton bath across the heteronuclear dipolar couplings to the heteronuclei. The technique, analogously to the INEPT^{10 } experiment in solution, exploits the much higher equilibrium Boltzmann polarization of the protons.
Cross polarization requires simultaneous rf irradiation of the protons (I) and the heteronuclei (S), spinlocking both species. The irradiation has to fulfill the Hartmann–Hahn matching condition ω$1S$=ω$1I$ in the static case, and ω$1S$=ω$1I$±nω_{rot} for spinning samples (with n=1,2).^{11 }
Phenomenologically (Figure 1.6), the resonance frequencies of both spins match (while the resonance frequencies are very different when subjected to the Zeeman field alone) at the Hartmann–Hahn condition so that polarization transfer can occur. In a quantum mechanical description, it can be shown that at the Hartmann–Hahn condition^{12 } (in the socalled doubly rotating frame), one obtains an effective Hamiltonian (see Section 1.5), which contains operators of the form I_{x}S_{x} and I_{y}S_{y}. Reformulating these operators in terms of raising and lowering operators I_{x}=(1/2)(I^{+}+I^{−}) and I_{y}=(1/2i)(I^{+}+I^{−}), it can be shown that, at static conditions and slow to medium spinning frequencies, CP transfer can be induced via zeroquantum flipflop transitions I^{±}S^{∓}. Note that in the spinlock field, homonuclear dipolar couplings are still active, merely scaled by a factor of −0.5. Another benefit of the CP technique is that experiments can be accumulated faster, because they depend on ^{1}HT_{1}, which is usually much shorter than ^{13}CT_{1} or ^{15}NT_{1}.
1.4 Dipolar Decoupling
As delineated in Section 1.2, internal NMR interactions can be decomposed into two factors, which are the space and the spin parts. Fast MAS averages the space part, yet not to completeness, predominately because of the homogeneous character of the homonuclear dipolar proton interactions, rendering the whole internal Hamiltonian homogeneous. To further narrow the lines and enhance sensitivity, one thus has to average simultaneously the spin part by rf pulses, which is called decoupling.
Heteronuclear dipolar decoupling in biological solidstate NMR relates to the decoupling of heteronuclear spins from the proton spins. The mode of action of strong rf pulses applied to the proton spins may be imagined as to cause a fast precession of the proton spins around the effective field, of a frequency higher than the heteronuclear dipolar couplings. In fact, the frequency also has to be larger than the homonuclear dipolar proton couplings. One should keep in mind that such a simplified description does not provide much insight into the effect of different decoupling sequences and we refer interested readers to detailed reviews on this subject.^{13 } In its simplest form, heteronuclear decoupling refers to the application of highpower continuous wave (CW) rf irradiation on the proton channel. The performance of CW decoupling, however, can be compromised by interference effects between heteronuclear dipolar couplings and the proton CSA.^{13a } This interference is significantly reduced in the case of phasemodulated sequences like twopulse phase modulation (TPPM)^{14 } and its phasecycled version, small phase incremental alternation (SPINAL).^{15 } Both are to date the standard decoupling methods at low to medium spinning frequencies. Note that there exists a large number of modifications of these two basic decoupling sequences.^{16 } With increasing spinning frequency, decoupling sequences like XiX^{17 } and PISSARRO^{18 } usually perform better than TPPM or SPINAL. One reason is the residual sensitivity of TPPM and SPINAL to interferences between heteronuclear dipolar couplings and the proton CSA, which weighs more with increasing spinning frequency (at lower spinning frequency, this interference is partly suppressed by the selfdecoupling effect induced by strong proton–proton dipolar couplings^{13a }). On the other hand, both XiX and PISSARRO are impeded by the effect of the proton–proton dipolar couplings,^{19 } which become evidently reduced with increasing spinning frequency, so that these sequences unfold their potential more at fast MAS frequencies. Besides highpower irradiation, all these heteronuclear decoupling sequences can be applied as lowpower variants at very fast spinning frequencies.^{20 } Lowpower decoupling prevents excessive sample heating, which may be of particular importance for biological samples, although slight losses in performance usually have to be accepted. Generally, the subtle differences among these decoupling pulse sequences all manifest in the higher order terms of their effective Hamiltonians (see Section 1.5), which is the reason why a description (and analysis) of these sequences is not straightforward.
Homonuclear dipolar decoupling in biological solidstate NMR usually relates to proton–proton decoupling. A powerful way to achieve additional spin space averaging of these couplings is the Lee–Goldburg (LG) irradiation technique.^{21 } It comprises tilting the effective field to the magic angle by means of offset irradiation on the proton channel, which suppresses dipolar proton–proton couplings to the lowest order, leaving the chemical shift Hamiltonian scaled by 1/√3. The LG technique is nowadays often applied in its frequency and phaseswitched versions,^{22 } which average out the proton–proton couplings to higher order (see further reading^{23 }). Besides the LG technique, sequences using continuously phasemodulated rf pulses^{24 } and sequences based on symmetry principles^{25 } also perform well to decouple proton–proton interactions.
1.5 Recoupling
Recoupling means the reintroduction of anisotropic NMR interactions, which are otherwise averaged out by MAS. Since these interactions, as delineated in Section 1.2, are orientation and/or distance dependent, their controlled reintroduction can yield structural information. Today, a plethora of recoupling pulse sequences exists and there are numerous options to reintroduce the desired interactions.^{26 } Here we will sketch some options to assign proteins and to yield structural information. Recoupling methods (and generally solidstate NMR methods) are often analyzed in terms of average Hamiltonian theory (AHT),^{7 } in which a timedependent Hamiltonian is expressed as a timeindependent effective Hamiltonian (see further reading^{3,7 }):
where is the lowest order term (the actual “average Hamiltonian”), is the first order correction term, and so on. These terms can be expressed in terms of a Magnus expansion, which shows that, while is independent of the spinning frequency ν_{rot}, higher order terms of order k scale with 1/ν$rotk\u22121$, i.e. is inversely proportional to ν_{rot}. It is advisable to understand which terms of are reintroduced by a recoupling sequence to fully exploit its potential.
1.5.1 Protein Backbone Assignment
The protein backbone reads as Cα_{(i)}–N_{(i)}–CO_{(i+1)}–Cα_{(i+1)}. Hence, to obtain sequential assignments, correlations Cα_{(i)}→ N_{(i)}→ CO_{(i+1)} (or vice versa) need to be established, while at the same time the transfer should be restrained to this pathway to avoid misleading correlations over more than one amino acid. This is usually achieved using specific ^{13}C ↔^{15}N cross polarization techniques,^{27 } which exploit a frequencyselective Hartmann–Hahn condition to reintroduce the heteronuclear dipolar ^{13}C–^{15}N couplings to lowest order, which implies that this method also works at very fast spinning frequencies (Figure 1.7). In the specific CP experiment, chemical shiftdependent transfer characteristics are introduced by using a controlled frequency offset in combination with relatively weak rf fields on the heteronuclei. Alternatively, Cα_{(i)}→ N_{(i)}→ CO_{(i+1)} correlations may be brought about with the PAIN sequence,^{28 } which is a thirdspin assisted recoupling (TSAR)based method (see below).^{29 }
The single amino acids can be identified by means of their characteristic chemical shifts and connectivity patterns, for example among carbon resonances. As mentioned above, methods to reintroduce homonuclear dipolar ^{13}C–^{13}C couplings are widespread. A popular class of experiments relies on the protonmediated reintroduction of the carbon homonuclear dipolar couplings, which is usually referred to as a longitudinal magnetization transfer by spin diffusion facilitating zeroquantum flipflop transitions among ^{13}C nuclei. In its basic form, known as proton driven spin diffusion (PDSD), it suffices to switch off the decoupling field on the proton channel (while ^{13}C magnetization is stored along the zaxis) to enhance spin diffusion. The reason why proton decoupling is switched off during the spin diffusion mixing time is to reintroduce heteronuclear dipolar couplings, making the influence of flipflop transitions more efficient. This reintroduction occurs, however, only to higher order, i.e. scales inversely with the spinning frequency. Other methods like DARR,^{30 } PARIS^{31 } or MIRROR,^{32 } therefore, outperform PDSD with increasing spinning frequency, because these methods reintroduce the heteronuclear dipolar couplings independently of the MAS frequency (Figure 1.7). However, since the flipflop transitions appear in the correction terms of the effective Hamiltonian, spin diffusion efficiency invariably decreases with increasing spinning frequency, although the transfer can still be efficient at very fast MAS.^{33 }
1.5.2 Protein Geometry Restraints
As for solidstate NMR experiments to assign protein resonances, numerous possibilities exist to collect information on protein geometry. For example, the rotational resonance (R^{2}) experiment,^{34 } which reintroduces homonuclear dipolar couplings between two nuclei to lowest order if the resonance condition nν_{rot}=ΔCS, with ΔCS being the chemical shift difference of the two nuclei, is matched. This recoupling technique is very efficient in sparsely labeled samples and, in particular, allows exact measurements of internuclear distances. However, lowest order recoupling sequences are sensitive to the socalled dipolar truncation effect, which means that in the presence of a strong homonuclear dipolar coupling, transfer S across a small coupling is significantly reduced. It has been shown^{35 } that if
i.e. if two pairs of homonuclear dipolar couplings d differ by one order of magnitude, the transfer across the small coupling is quenched by two orders of magnitude. The latter situation, with weak couplings in the presence of strong couplings, is, however, the common situation in uniformly labeled biomolecules, so that longrange distances are hardly collectable with conventional lowest order recoupling techniques. A remedy to this issue is to use frequencyselective lowest order recoupling^{36 } to reintroduce only homonuclear interactions between certain spin pairs, to selectively label the protein sample, or to resort to higher order recoupling methods like spin diffusion,^{30–32 } TSAR^{29 } (which is based on higher order crossterms of two heteronuclear couplings) or CHHC^{37 } (which reads out ^{1}H–^{1}H spin diffusion on ^{13}C nuclei), which are much less sensitive to dipolar truncation. Note that there are many other methods to obtain information on protein geometry, e.g. by analyzing the relative orientation of chemical shift tensors.^{38 }
1.5.3 Intermolecular Geometry Restraints
The measurement of distance restraints across biological molecular interfaces by solidstate NMR usually involves mixtures of differentially labeled samples (involving nucleic species X and Y in Figure 1.8a), because the direct detection of intermolecular contacts at the protein–protein interface in uniformly labeled samples is usually prohibited by spectral crowding and dipolar truncation in the presence of shortranged intramolecular interactions. A frequently applied strategy refers to the use of equimolar mixtures of ^{13}C and ^{15}N labeled proteins in combination with ^{15}N–^{13}C transfer schemes.^{39 } In general, polarization transfer across the molecular interface can be brought about either via the relatively small dipolar ^{15}N–^{13}C couplings by REDOR or TEDORbased transfer schemes^{40 } (Figure 1.8), or by involving protons in the context of NHHC^{41 } and PAIN^{28 } experiments (Figure 1.8). In principle any combination of spin½ species X and Y can be studied by the schemes shown in Figure 1.8. For example, in complexes involving nucleic acids, ^{31}P nuclei are a convenient source to establish X–Y heteronuclear transfers and can yield unambiguous intermolecular information.^{42 } Using mixtures of proteins labeled on the basis of [1^{13}C] and [2^{13}C]glucose, in combination with homonuclear ^{13}C–^{13}C recoupling sequences, has emerged as another strategy to probe intermolecular contacts of larger proteins.^{43 } Note that next to distance restraints, chemical shift perturbation studies, i.e. comparing chemical shifts before and after complex formation, are another convenient means to obtain protein binding interfaces.^{44 } For further information on the investigation of biomolecular supramolecular structure, we refer the interested reader elsewhere.^{39 }
1.5.4 Protein Dynamics
A particular advantage of solidstate NMR (and NMR in general) spectroscopy is its potential to provide information on the dynamics of biomolecules in close reference to structure and function. Rigid and mobile protein segments can readily be sitespecifically distinguished^{44,45 } by comparing spectra employing dipolar (like spin diffusion^{30–32 }) and scalar (like INEPT^{10}) transfer. Dedicated solidstate NMR experiments principally allow characterization of the dynamics of biomolecules from the nano to the millisecond timescale and beyond.^{46 } For example, the nano to millisecond timescale can readily be explored by measuring the spin–lattice relaxation in the laboratory (R_{1}) or the rotating frame (R_{1ρ}), while realtime solidstate NMR allows the course of protein refolding or of enzymatic reactions to be followed.^{47 }