Solid state NMR structural characterization of self-assembled peptides with selective ¹³C and ¹⁵N isotopic labels.

Abstract

For the structural characterization methods discussed here, information on molecular conformation and intermolecular organization within nanostructured peptide assemblies is discerned through analysis of solid state NMR spectral features. This chapter reviews general NMR methodologies, requirements for sample preparation, and specific descriptions of key experiments. An attempt is made to explain choices of solid state NMR experiments and interpretation of results in a way that is approachable to a non-specialist. Measurements are designed to determine precise NMR peak positions and line widths, which are correlated with secondary structures, and probe nuclear spin–spin interactions that report on 3-dimensional organization of atoms. The formulation of molecular structural models requires rationalization of data sets obtained from multiple NMR experiments on samples with carefully chosen ¹³C and ¹⁵N isotopic labels. The information content of solid state NMR data has been illustrated mostly through the use of simulated data sets and references to recent structural work on amyloid fibril-forming peptides and designer self-assembling peptides.

Introduction

This chapter aims to be a practical guide to solid state nuclear magnetic resonance (NMR) based structural measurements of self-assembled peptide systems. We focus on well-established experiments with detection of ¹³C or ¹⁵N NMR signals, performed on isotopically labeled peptide samples produced by chemical synthesis (Figure 1). These nuclei have low natural abundances (1% for ¹³C and 0.4% for ¹⁵N), such that isotopic labeling at specific sites makes it possible to obtain localized structural information. This information is extracted from observed effects of structure on NMR spectra and nuclear spin dynamics. Our descriptions of solid state NMR methods do not assume in-depth understanding of NMR theory. However, we do not attempt to explain basic operation of an NMR spectrometer. NMR spectroscopy involves: 1) placement of samples into large static magnetic fields (~10 Tesla); 2) excitation of nuclear spins (nuclei with magnetic moments) by resonant radio frequency magnetic fields delivered via induction by a coil around the sample; and 3) detection of current induced in the coil by time-dependent nuclear magnetization. We will describe a toolkit consisting of a series of NMR experiments, each with radio frequency excitations designed to obtain specific structural information from samples with deliberate patterns of isotopic labeling. We will also mention important complementary methods, including sample preparation, computational molecular modeling, and nuclear spin simulations.

Figure 1:

a) Schematic representation of a polypeptide chain. The ϕ and ψ torsion angles specify secondary structure and molecular fold. b) Uniform ¹³C/¹⁵N labeled valine. c) Selectively ¹³C labeled alanine, labeled at C_β or CO. Sites labeled with ¹³C are highlighted in grey ovals.

Solid state NMR is often the method of choice for structural biology of self-assembling peptides because such samples usually lack the solubility and crystalline order necessary for solution state NMR spectroscopy and X-ray crystallography, respectively. Solid state NMR refers to NMR techniques that are specially designed for analysis of samples with highly constrained molecular motions, i.e., with correlation times for molecular orientation that are long on the measurement timescale (milliseconds per scan). The non-crystalline nature of self-assembled peptides further restricts experiments to those designed for random distributions of molecular orientation (“powder” samples). Solid state NMR experiments involve hardware and methods designed to compensate for orientation-dependent effects and selectively manipulate local nuclear magnetic interactions. Observables provide information on backbone torsion angles (see Figure 1a), proximity maps for different amino acids (residues), and 3-dimensional (3D) arrangements of isotopically labeled sites. The direct observables are peak positions, peak line widths, and decays in intensity under the influence of nuclear spin-spin dipolar couplings.

In practice, data obtained by other techniques are used to complement information which is uniquely attainable by solid state NMR. While the implementation of these techniques is beyond the scope of this chapter, understanding them can be essential to the rationalization of NMR data. Molecular dynamics computer simulations can be instrumental in understanding NMR-derived structural constraints by identifying reasonable molecular structures (1-3). These simulations can be performed with artificial force fields inspired by NMR results (4-8). Electron microscopy and atomic force microscopy (4,5,7,9-11) can quantify structural features at larger length scales (nanometers) than are typically accessible by NMR. Quantitative measurements of fibril mass have been reported using scanning transmission electron microscopy (4,11-14) and tilted-beam transmission electron microscopy (15). Secondary structure (β-strands and α-helices) may be probed by Fourier transform infrared spectroscopy (16-19) and circular dichroism (20-22). Fluorescent dyes, i.e., Thioflavin-T (23-26) and Congo Red (27-29), are effective indicators for monitoring β-sheet formation. Distinction of quaternary structure by cryogenic electron microscopy has been reported in the characterization of peptide nanofibers (13,30-32). Molecular folding and intermolecular packing (e.g., β-sheet stacking) can be measured using many diffraction-based techniques (33-37), H/D exchange (38,39), electron paramagnetic resonance (EPR) (40-44), and proteolysis (45,46).

Important background and insights are revealed by previous scientific contributions based on solid state NMR of self-assembled peptides. For each experiment presented here, we reference precedent applications. A majority of these techniques were originally developed to study naturally-occurring peptide self-assembly, which is most often associated with pathological amyloid deposition (47-52). The study of designer peptides presents a promising new area of application for solid state NMR (5,7,53). For non-natural self-assembling peptides, solid state NMR plays the indispensable role of “closing the design loop” by providing a means of comparing hypothesized structures with those that are actually produced. The following sections review general solid state NMR methodologies, requirements for sample preparation, and specific descriptions of key experiments. The NMR data presented in this chapter are illustrated using simulated data sets based on previously published results.

Overview of Solid State NMR Structural Investigations

Unique Characteristics of Solid State NMR Equipment

The experiments discussed here require an NMR system (magnet, spectrometer, and probe) that is equipped for solid state NMR measurements. As illustrated in Figure 2, the NMR magnet establishes a uniform static magnetic field (${\overrightarrow{B}}_0$) throughout the volume of the sample, while the NMR probe produces a time-dependent radio frequency magnetic field (${\overrightarrow{B}}_1$). The NMR probe contains a resonant radio frequency circuit to induce ${\overrightarrow{B}}_1$ by means of a current delivered through a coil surrounding the sample; the coil is the only part of the probe that is drawn in Figure 2. The experiments described here require magic angle spinning (MAS) of samples and double-resonance (simultaneous excitation of ¹H and ¹³C) or triple-resonance (simultaneous excitation of ¹H, ¹³C, and ¹⁵N) NMR probes. In addition, solid state NMR experiments tend to require higher powers for radio frequency excitation compared to most solution state NMR experiments.

Figure 2:

Depiction of the sample (cylinder), coil, and magnetic fields involved in the NMR experiment. The sample is orientated at the magic angle relative to the static magnetic field ${\overrightarrow{B}}_0$. A tuned radio frequency probe circuit, to which the coil is connected, produces a time-dependent ${\overrightarrow{B}}_1$ field and detects the NMR signal.

MAS refers to sample rotation about an angle $\theta ={\cos }^{-1}\frac{1}{\sqrt{3}}\approx {54.7}^{\circ }$ relative to ${\overrightarrow{B}}_0$ (Figure 2) (54). Typical MAS rotation frequencies range from 5 kHz to 30 kHz. MAS improves spectral resolution (narrows NMR peaks) by eliminating molecular orientation-dependent effects on spin dynamics. NMR probes designed for MAS require that samples be placed inside special sample holders, called rotors, which are designed to withstand the forces involved in sample spinning. Specific probes are designed to work with specific rotors, with typical rotor diameters ranging between 2.5 mm and 4 mm. Probes designed for larger MAS rotor sizes make it possible to perform experiments on larger sample volumes, producing stronger NMR signals. However, larger rotors correspond to lower maximum MAS rotation rates, which in turn limit the capacity to attenuate line-broadening mechanisms.

The high radio frequency power requirements for solid state NMR are necessitated by two special experimental features: cross polarization and decoupling. Cross polarization is a signal enhancement method in which the polarization of ¹H is transferred to ¹³C or ¹⁵N (55). Decoupling is the application of high power radio frequency excitation of ¹H in order to eliminate ¹H-¹³C or ¹H-¹⁵N dipolar couplings that are not fully removed by MAS (56). Decoupling is often necessary when applying pulses to excite ¹³C or ¹⁵N spin states and when detecting ¹³C or ¹⁵N NMR signals. Probes with larger coils (to accommodate larger rotors) have lower limits on radio frequency pulse powers.

Requirements for Samples

Preparation of peptides with selective ¹³C and ¹⁵N isotopic labels is accomplished by chemical peptide synthesis using isotopically labeled amino acids that are available commercially (57). Individual amino acids could be uniformly labeled with ¹³C and ¹⁵N (Figure 1b), or selectively labeled (Figure 1c). The choice of isotopic label depends on the planned NMR experiments. A widely used toolkit for solid-phase peptide synthesis is based on fluorenyl-9-methoxycarbonyl (FMOC) chemistry (58). Peptide synthesis produces practical yields for most peptides with up to ~50 amino acids (59). Required steps following peptide synthesis include chemical cleavage from the solid support resin, precipitation of product from the cleavage solvent (e.g., cleavage by trifluoroacetic acid followed by precipitation in diethyl ether), and purification. Purification is usually carried out using high performance liquid chromatography. Product quality can be tested using matrix-assisted laser desorption or electrospray ionization mass spectrometry (60-62).

Preparation of samples with well-defined molecular structures is critical to the success of a solid state NMR structural investigation. The phenomenon of peptide self-assembly is complex, often promoting structural inhomogeneity in samples. Naturally-occurring amyloidogenic peptides are known to assemble along multiple energetically similar pathways to distinct amyloid fibril structures (4,63). The formation of multiple fibril structures by self-assembly of a single peptide is often called polymorphism, because fibrils with different molecular structures exhibit distinct morphologies (nanofiber width, width modulation, and protofilament sub-structure) when observed by electron and atomic force microscopy (9,41). Amyloidogenic peptides are known to also form non-fibrillar species that have been described as oligomers and protofibrils (64-66). There have been fewer studies of assembly pathways for designer peptides, but it is likely that they are similarly capable of self-assembly along multiple pathways. Therefore, researchers must take special care to control molecular structure during sample preparations.

Experimental protocols designed to prepare structurally homogeneous self-assembled peptide samples rely on a combination of controlled self-assembly conditions and seeded assembly. Fibril structures in samples are known to depend on solution conditions during self-assembly, such as pH, ionic strength, access to surfaces, and solution agitation (67-70). Fibrils also exhibit seeded propagation, such that dissolved peptide molecules can be induced to self-assemble by addition of fibril fragment solutions aliquoted from sonicated fibril hydrogels (71-73). Furthermore, seeding results in propagation of molecular structure, such that fibrils produced by seeding have structures that match those of the fibril fragments used as seeds. When seeded self-assembly is applied repeatedly to polymorphic fibril samples, fibril polymorphs that propagate most rapidly can be purified (11,4,74). In contrast to fibril studies, structural analysis of non-fibrillar aggregates of amyloid-forming peptides (oligomers and protofibrils) have not relied on seeded assembly. Instead, achievement of structural homogeneity has relied on careful production of monomeric peptide and control of subsequent assembly conditions. Since non-fibrillar aggregates are often distinguished based on their sizes, size exclusion chromatography has been used to separate specific aggregates from monomers and fibrils (64,75). Environment-dependent self-assembly has been reported for the RADA16-I designer peptide (76), and evidence of polymorphism has been observed in solid state NMR of MAX8 designer peptide nanofibers (53). It is possible that the relative simplicity of designer peptides facilitates establishment of thermodynamic equilibrium structures rather than kinetically trapped states (77,78).

From the spectroscopic point of view, structural stability and uniformity of the sample affect NMR sensitivity, resolution, and data interpretation. Stability is important because NMR measurements often require days to weeks of repeated scanning. Unstable samples may be especially affected by radio frequency fields that elevate sample temperature (79). Decoupling fields tend to be the largest contributor to sample heating because of their relatively high power and long duration. Structural uniformity is also important because samples containing multiple molecular structures would exhibit multiple NMR peaks for each labeled site. Such heterogeneity would result in more NMR peaks overall, promoting reduced spectral resolution due to spectral overlap. Furthermore, since peak intensity is approximately proportional to the number of atoms contributing to each NMR signal, increased structural homogeneity promotes stronger signals. Finally, the presence of multiple distinct molecular structures in a sample complicates interpretation of experiments based on measurement of spin dynamics (57,53).

General Description of Solid State NMR Measurements

NMR is a spectroscopic technique used to probe nuclei with magnetic moments (nuclear spins) in a magnetic field (${\overrightarrow{B}}_0$). The magnetic moment is a fundamental property of each NMR-active nucleus, such that all nuclei of a specific isotope possess the same magnetic moment. The magnetic moment for each nuclear isotope has been measured precisely and is proportional to the gyromagnetic ratio $\left(\frac{MHz}{T}\right):{\gamma }_{{}^{1_H}}=42.576$, γ_13C = 10.705, and γ_15N = −4.316 (80). Populations of nuclear spins are perturbed by radio frequency excitation, and subsequent spin dynamics are then measured by magnetic induction (see Figure 3). Measured current in the coil as a function of time is a decaying waveform referred to as the free induction decay (FID). Observed signals depend on local magnetic interactions and thus report on structure. Local interactions include the chemical shift and nuclear magnetic dipolar couplings. Chemical shift is a change in the effective strength of ${\overrightarrow{B}}_0$ experienced by each NMR-active nucleus, due primarily to magnetic field-induced distortion of orbitals of paired electrons local to each nucleus. For example, NMR signals from ¹³C atoms at carbonyl sites are distinguishable (different frequencies) from NMR signals from ¹³C atoms at methyl sites because of the significant differences in the local electronic orbital structures. Nuclear magnetic dipolar couplings are analogous to interactions between classical magnetic dipoles, which depend on relative position and the magnetic moments involved. If nuclei are separated by ~1 nm or less, their spin dynamics will be measurably affected by dipolar couplings.

Figure 3:

Diagram of a single scan in an NMR experiment. Time-varying heights of radio frequency pulses or Free Induction Decay waveforms correspond to time-dependent voltage or current in the coil (not drawn to scale).

NMR experiments involve the following general steps shown Figure 3:

1. An initial delay, d₁, allows nuclear spin populations in the sample to reach thermal equilibrium population distributions in the ${\overrightarrow{B}}_0$ field. At equilibrium, nuclear magnetization is collinear with ${\overrightarrow{B}}_0$ and (by convention) the z-axis.¹ The initial delay d₁ must be sufficiently long to achieve thermal equilibrium and is set based on the relaxation time T₁ of the peptide sample $(d_1\cong 5\times T_1)$.

2. Radio frequency pulses of current are applied to the coil, producing a time-varying magnetic field ${\overrightarrow{B}}_1(t)$ inside the sample volume (see Figure 2). The transverse component of ${\overrightarrow{B}}_1(t)$ (on the xy-plane) induces changes in quantum mechanical nuclear spin states, resulting in detectable transverse nuclear magnetization (81-83). A resonance condition occurs such that ${\overrightarrow{B}}_1(t)$ induces a change in nuclear spin state if it oscillates close to the resonant or Larmor frequency of the target nucleus.^{3, 4} NMR signals also depend on the pulse duration, power, and phase for ${\overrightarrow{B}}_1(t)$. Radio frequency pulses are often described by the angle of deflection (tip angle) of nuclear magnetization they induce. For example, applying an “on-resonance” $\frac{\pi }{2}$ pulse to a sample initially at thermal equilibrium would deflect magnetization from the z-axis onto the xy-plane.⁶ We express pulse powers in units of frequency, such that pulse power corresponds to $\frac{1}{{\tau }_{2\pi }}$, where τ_2π is the duration of 2π pulse.⁷

3. The measured FID is a time-dependent current induced in the coil. Magnetic induction occurs because the nuclear magnetization component on the xy-plane will precess about the z-axis at the Larmor frequency (due to quantum interference effects). NMR signals decay because stochastic atomic-scale fluctuations in nuclear spin interactions destroy bulk-averaged phase coherence in NMR signal throughout the sample (81,82). Fourier transform of the FID produces the NMR spectrum.

4. Since NMR experiments tend to produce weak signals relative to typical noise levels in detection circuitry (85), a single application of Steps 1-3 is often insufficient to obtain desirable signal-to-noise ratios. Therefore, experiments typically require repeated scanning: Steps 1-3 are repeated and measured signals from successive scans are added together. If n repeated scans are applied for an experiment, the signal-to-noise ratio would be proportional to $\sqrt{n}$ (83).

5. NMR experiments employ phase cycling (86), since NMR is a phase-sensitive spectroscopic technique. Phase cycling is the systematic modulation of pulse and signal detection (receiver reference, ϕ_Receiver) phases between scans so that co-addition of data from successive scans minimizes effects of experimental imperfections (e.g., spatial inhomogeneity of $\mid {\overrightarrow{B}}_0\mid$) or selects for specific signal components (e.g., ¹³C signal exclusively corresponding to transferred magnetization from ¹H).

The following is a list of special experimental features for solid state NMR experiments:

• MAS is employed to eliminate most effects of chemical shift anisotropy and magnetic dipolar couplings that do not involve ¹H (87).

• High power ¹H decoupling can be used to eliminate the effects of dipolar couplings between ¹³C or ¹⁵N and ¹H atoms. The effects of couplings involving ¹H are not usually eliminated by MAS alone at typical rotation frequencies (88-91).

• Some radio frequency excitations are designed to take advantage of the non-random nature of molecular reorientation under MAS. In contrast to liquid state molecules, which undergo uncorrelated rotations, molecules in solid samples reorient in synchrony as the sample rotates. As a result, NMR signals from ¹³C or ¹⁵N can be enhanced through transfer of polarization from the ¹H spin system (cross polarization) (92,93). In addition, techniques involving application of pulses timed in synchrony with sample rotation can temporarily reintroduce orientation-dependent spin interactions such that structural information can be extracted.

The solid state NMR methods to probe self-assembled peptide molecular structure depend primarily on the following characteristics of nuclear spins:

• Each ¹H, ¹³C, or ¹⁵N nucleus is a spin-$\frac{1}{2}$ particle (total angular momentum quantum number $l_n=\frac{1}{2}$). Its behavior is described by two energy levels (quantum mechanical eigenstates) (81,82,94).

• In the presence of a static magnetic field ${\overrightarrow{B}}_0$, energy differences between spin eigenstates will scale in proportion to the strength of the magnetic field. At a specific magnitude of ${\overrightarrow{B}}_0$, the Larmor frequency of each type of nucleus can be predicted accurately. For example at $\mid {\overrightarrow{B}}_0\mid$ = 11.75 Tesla, the Larmor frequencies of ¹H, ¹³C, and ¹⁵N are 500 MHz, 126 MHz, and 51 MHz, respectively.⁸

• Radio frequency excitation near the Larmor frequency is required to induce nuclear spin transitions. Consequently, a single radio frequency pulse can only directly affect the spins of a single nuclear isotope (e.g., ¹³C). However, heteronuclear (e.g., ¹³C-¹⁵N) spin-spin couplings could lead to effects on other spin systems.

• Long lifetimes (related to durations of FIDs) of NMR signals (ms) allow for accurate measurements of NMR frequencies. NMR signals from nuclear spins in chemically inequivalent sites are usually distinguishable by chemical shift.

• Molecular conformation and intermolecular packing affect the chemical shift. Thus, precise NMR peak positions, especially for atoms near peptide backbones, can report on secondary structure. NMR line width is also an indicator of structural order within a sample.

• Dipolar couplings between adjacent (proximities within ~1 nm) nuclei with magnetic moments perturb nuclear spin energy levels and affect spin dynamics. Dipolar couplings can therefore report on 3D arrangements of NMR-detectable atoms.

• Dipolar couplings depend strongly on distances between nuclei with magnetic moments: these couplings scale with r⁻³, where r is the distance between a pair of nuclear spins. When more than two nuclear spins are separated by multiple inter-atomic distances less than ~1 nm, spin dynamics tend to be dominated by the smallest inter-atomic distance(57,5).

Formats for Data Sets

Figures 4-6 illustrate three different formats of data sets that could be obtained from NMR the experiments described in this chapter. These figures are intended to provide a basis for understanding what types of information can be extracted from NMR experiments. The radio frequency pulse sequences used to obtain these data sets will be discussed in more detail subsequently.

Figure 4:

(a) Simulated ¹³C NMR spectrum of RADA16-I nanofiber sample labeled at R9, A10 and D11. (b) All-atom depiction of R, A, and D residues connected by peptide bonds.

Figure 6:

a) Simulated dephasing curves for PITHIRDS-CT experiments. The vertical axis corresponds to integrated intensity of an NMR peak corresponding to a selectively ¹³C-labeled sample (e.g., with ¹³C only at a single carbonyl or methyl site). The solid and dashed lines represent predicted decays in peak intensity as a function of time under the influence of a pattern of radio frequency pulses designed to reintroduce ¹³C-¹³C dipolar couplings that are normally averaged away by MAS. The dephasing curves are the results of numerical simulations of spin dynamics for 7 ¹³C atoms arranged in a straight line separated by the constant inter-atomic distance, d. The arrangement of atoms is illustrated in (b).

The most basic NMR data set is a spectrum that relates intensity to frequency, such as that shown in Figure 4. This is a simulated ¹³C NMR spectrum from the nanofibers of an isotopically labeled RADA16-I peptide sample (5). For Figure 4, all ¹³C labeled sites are assumed to produce an NMR peak with equal intensity and a 1 part per million (ppm)⁹ line width (full width at half maximum). The peak positions were taken from experimental data (5).¹⁰ By convention, ¹³C NMR frequencies are reported relative to the peak position for ¹³C within tetramethylsilane (TMS) (95). NMR signals from aliphatic ¹³C nuclei appear in the 10-70 ppm region, aromatic ¹³C signals appear in the 100-150 ppm region, and ¹³C signals from atoms with double bonds to electronegative atoms (O or N) appear in the 160-180 ppm region. The 65-150 ppm region of Figure 4 has been omitted because there are no aromatic groups in the RADA16-I peptide. Although each of the peaks in Figure 4 is labeled with the corresponding ¹³C nucleus, this spectrum alone is insufficient to determine correspondence between NMR peaks and ¹³C-labeled sites (spectral assignments). Assignment of NMR spectra usually requires multidimensional NMR spectroscopy.

Multidimensional NMR spectroscopy is employed to probe exchange of nuclear magnetization between atoms that correspond to different NMR frequencies. Such an exchange would require magnetic interactions between atoms, and therefore spatial proximity. In multidimensional NMR spectroscopy, the number of dimensions refers to the number of frequency axes in a spectrum. Each frequency axis corresponds to the Fourier transform of NMR intensity as a function of a corresponding time domain in the experiment. Every NMR spectrum includes a direct dimension, which is the frequency determined by Fourier transform for a detected FID. Multidimensional NMR spectra are characterized by additional indirect dimensions, which correspond to Fourier transforms of NMR intensities as a function of variable time delays within radio frequency excitations. The scope of this chapter does not include NMR experiments with more than two dimensions. Figure 5 shows a sample 2-dimensional (2D) exchange ¹³C-¹³C NMR spectrum, in which symbols emulate contours that would define intensity relative to two axes, both corresponding to ¹³C NMR frequency. By convention, the horizontal frequency axis corresponds to the direct dimension. Off-diagonal peaks (crosspeaks) that correspond to distinct frequencies on the horizontal and vertical axes, are generated through magnetization exchange between distinct ¹³C atoms. In this particular spectrum, crosspeaks are observed between ¹³C atoms that are directly bonded to one another, allowing determination of spectral assignments through comparison of crosspeak patterns to configurations of atoms within each labeled amino acid. Different 2D NMR techniques produce different patterns of crosspeaks, including “homonuclear” correlation experiments for which both frequency axes correspond to a single type of nucleus and “heteronuclear” correlation experiments that probe interactions between different NMR-detectable nuclei (e.g., ¹⁵N and ¹³C).

Figure 5:

A simulated 2D fpRFDR spectrum for RADA16-I nanofiber sample labeled at R9, A10 and D11. The chemical shifts values shown in the table on the right are adopted from (5). Single-bond correlations are marked by dashed lines. Solid circles, solid squares, and empty triangles indicate positions of crosspeaks within the D11, A10, and R9 residues, respectively. Since crosspeaks are expected only for directly bonded ¹³C atoms, crosspeak patterns (highlighted with different dashed lines) are unique to each amino acid. Different dashed lines highlight crosspeak patterns that are unique to each residue. A solid line highlights the diagonal.

An additional scheme for solid state NMR experiments is the quantitative measurement of decays in NMR signal intensity under the influence of dipolar recoupling pulses. By applying dipolar recoupling techniques for variable durations, one can measure a decay in NMR intensity that depends on the 3D arrangements of NMR-active atoms in the sample (¹³C or ¹⁵N). Decays can be simulated by calculating nuclear spin dynamics based on postulated atomic positions (96,97). As an example, Figure 6 shows a set of simulated decay curves for normalized NMR signal intensity (peak integral) for a sample labeled with ¹³C at a single site. The simulations employed consider seven ¹³C atoms arranged in a straight line and separated by a constant distance (98). For these experiments, samples are selectively ¹³C or ¹⁵N labeled so that measureable dipolar couplings provide specific information about molecular structures.

Framework for Radio Frequency Pulse Sequences

NMR experiments are described in terms of pulse sequence diagrams such as that shown in Figure 7. This “single pulse” experiment is the simplest pulsed NMR experiment. Symbols representing different parts of the experiment are arranged on a horizontal time line, which is labeled to specify a nucleus (¹H). Radio frequency pulses (rectangles) and signal detection periods (decaying waveform symbol) are performed at a carrier frequency that is set equal to the Larmor frequency of the nucleus specified to the left of the horizontal line. Although we normally employ the single pulse experiment to detect ¹H signal for our samples, the experiment can be used on any NMR-active nucleus. The rectangle marked p_H,1 represents a radio frequency pulse. The power, phase, and duration of the pulse are specified in Table 1. The pulse is marked with $‘‘\frac{\pi }{2}"$ because optimal signal is obtained when the pulse corresponds to a $\frac{\pi }{2}$ tip angle. The delay d₁ is necessary for the establishment of thermal equilibrium magnetization when multiple scans are performed. When multiple phases are listed in correspondence with a pulse, the list specifies how the pulse phase should be changed with each application (between scans or when a pulse appears more than once in a pulse sequence). When multiple receiver phases are listed, the list corresponds to receiver phase to be applied for successive scans. The phase lists are cyclic: phases should return to those specified at the beginning of a list after the list is completed.

Figure 7:

Pulse sequence diagram for single pulse NMR with ¹H signal detection.

Table 1:

Typical parameters for single pulse NMR experiment on ¹H.

MAS rate: 25 kHz (rotor period, τr = 40 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1(π/2)	110	2.27	x,y,−x,−y
ϕReceiver	y,−x,−y,x
Delay	Duration
d1	2 s

An example of a double-resonance NMR experiment is shown in Figure 8. We refer to this experiment as a ¹H-¹³C cross polarization magic angle spinning (CPMAS) experiment. The “CP” or cross-polarization pulses are aligned vertically because they require simultaneous excitation at ¹H and ¹³C Larmor frequencies. These pulses induce transfer of ¹H magnetization to the ¹³C spin system. The wide rectangle marked with “dec” indicates ¹H decoupling during acquisition of ¹³C NMR signal (continuous wave radio frequency input to ¹H or a pulse sequence, depending on the decoupling method), to eliminate broadening of ¹³C NMR peaks by ¹H-¹³C dipolar couplings. The dotted “flip-back” pulse is only used for calibration of ¹³C pulse powers; signal intensity is maximized in the absence of this pulse. Detailed parameters are specified in Table 2. Spectra such as that shown in Figure 4 would normally be obtained using this experiment because of the enhanced ¹³C NMR sensitivity relative to single-pulse ¹³C experiments. Cross polarization from ¹H to ¹³C produces more ¹³C NMR signal per scan because each ¹H atom has a larger magnetic moment than each ¹³C, and ¹H atoms outnumber ¹³C atoms within each amino acid. Furthermore, the use of the ¹H spin system as the source of detected magnetization allows more rapid scanning, since T₁ is usually shorter for ¹H than ¹³C in peptide samples.

Figure 8:

The ¹H-¹³C CPMAS pulse sequence.

Table 2:

Typical parameters for a ¹H-¹³C cross-polarization experiment.

MAS rate: 25 kHz (rotor period, τr = 40 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1(π/2)	110	2.27	y,−y
pH,2	Linear ramp: 50 to 100	2,000	x
pC,1	50	2000	x,x,y,y,−x,−x,−y,−y
pC,2	0*	5	y,y,−x,−x,−y,−y,x,x
ϕReceiver		x,−x,y,−y,−x,x,−y,y
Delay	Duration
d1	2 s

Decoupling period	Power (kHz)	Method
dec1	100	TPPM
comments	*zero except during calibration

More complex NMR experiments include 2D NMR experiments and measurements of decay under dipolar coupling. Simplified versions of these pulse sequences are shown in Figure 9, with details provided when specific 2D and dipolar coupling experiments are discussed later. The 2D NMR experiment requires acquisition of a set of FIDs corresponding to series of delay values (t₁) in the pulse sequence (see Figure 9a). The frequency indicated by the vertical axis in Figure 5 corresponds to Fourier transform of NMR peak intensity as a function of t₁, and is referred to as the “indirect dimension.” The frequency indicated by the horizontal axis of the 2D NMR spectrum in Figure 5 (direct dimension) corresponds to the Fourier transform of each detected FID. For phase sensitivity in the indirect dimension, two FIDs must be collected for each value of t₁, each corresponding to a different phase for the pulse following the t₁ delay (99). Dipolar recoupling pulses are used in “mixing” periods between the t₁ periods and the signal acquisition periods in order to induce spin-spin couplings necessary to produce crosspeaks.¹¹ Figure 9b illustrates the general scheme for pulse sequences of experiments that measure decay of NMR peaks under the influence of dipolar couplings. The vertical axis of the simulated data in Figure 6a corresponds to integrated peak intensities for acquired NMR signals. The horizontal axis corresponds to effective time for spin dynamics under the effects of dipolar couplings. Dipolar evolution time does not necessarily correspond directly with a delay in the recoupling pulse sequence: different evolution times could be achieved by applying a variable number (N) of repeated blocks of dipolar recoupling pulses, or by varying positions of pulses within the recoupling period. Dipolar recoupling experiments that involve variation of effective dipolar coupling times without changing the overall duration of applied pulses are called “constant time” pulse sequences. Constant time pulse sequences are designed to compensate for time-dependent loss of NMR coherences during application of the pulse sequence.

Figure 9:

General schemes for the 2D (a) and dipolar recoupling (b) NMR pulse sequences presented in this chapter.

Strategy for Structural Characterization

The process of structure investigation generally follows an iterative procedure, such as that depicted in Figure 10, in which NMR experiments and corresponding isotopic labels are chosen to each probe specific structural features. Formulation of NMR-constrained structural models for self-assembling peptides thus requires a series of experiments on samples with different isotopic labels. As sample preparation can be expensive and time consuming, it is desirable to obtain maximal structural information from each isotopically labeled sample. The largest contributors of sample costs include costs of isotopically labeled materials and peptide synthesis costs.¹² Choice of isotopic labeling requires a reasonable hypothesized structure (or set of candidate structures), which can be revised as more experimental data are obtained. For 2D NMR experiments, we desire minimal overlap of peaks and informative crosspeaks. Overlap of NMR peaks can lead to ambiguity in spectral assignments and interpretation of crosspeaks.¹³ Informative crosspeaks tend to be those between amino acids that are nonadjacent in the primary structure but adjacent in the molecular structure. For dipolar recoupling measurements, isotopic labels are chosen so that decays in NMR signal intensity inform on unknown interatomic distances. Choice of labels (and interpretation of NMR data) can be facilitated significantly by computer modeling, which includes molecular modeling (constrained molecular dynamics simulations) and modeling of nuclear spin dynamics during dipolar recoupling experiments (96,97).

Figure 10:

Flowchart showing an overall approach for assignment and structure determination in peptide/protein structure characterization by solid state NMR.

NMR observables that can be interpreted in terms of structure include positions and line widths of NMR peaks, crosspeaks in 2D NMR spectra, and peak decay curves for dipolar recoupling experiments (see Figure 11). Peak positions and line widths report on secondary structure (backbone conformation) and structural order, respectively. Crosspeaks indicate sets of spatially proximate atoms. Dipolar recoupling decay curves provide more quantitative information about 3D arrangements of atoms. Our nucleus of choice for detection of NMR signals is usually ¹³C. Samples can be selectively isotopically labeled with ¹³C, and spectral resolution (separation of different peaks) can be achieved with MAS and ¹H decoupling (see Figure 12). Although ¹H NMR spectroscopy is more sensitive (larger magnetic moment, higher natural abundance, shorter T₁), ¹H NMR spectra of typical samples do not exhibit resolved peaks from different sites (Figure 13). Poor ¹H resolution makes it difficult to extract structural information from ¹H spectra, but we do collect ¹H spectra when it is important to monitor levels of sample hydration (100). Signals from ¹⁵N also contain important structural information, but lower NMR sensitivity for ¹⁵N (smaller magnetic moment than ¹³C) often makes it preferable to measure ¹⁵N signals indirectly via interactions with ¹³C. The choice of isotopic labeling scheme for a sample depends on the intended NMR experiment. Samples studied by 2D NMR tend to have uniform ¹³C and ¹⁵N isotopic labels at a small number (~7 or less) of amino acids (see Figure 1b). Isotopic labeling schemes chosen for dipolar recoupling experiments tend to avoid more than one labeled site per amino acid (Figure 1c). Multiple labeled sites within a single amino acid would be undesirable because inter-amino acid atomic distances tend to be longer, and decays in dipolar recoupling curves tend to be dominated by the shortest distances between labeled sites.

Figure 11:

An overall summary of spectroscopic and structural information that can be obtained from different solid state NMR techniques.

Figure 12:

¹H-¹³C cross polarization NMR spectra of amyloid fibrils of the 40-residue isoform of the Alzheimer’s amyloid β-peptide, acquired on a 400 MHz magnet. The sample was uniformly ¹³C- and ¹⁵N- labeled at seven amino acids: F19, V24, G25, A30, I31, L34, and M35. a) A ¹³C NMR spectrum acquired without MAS or ¹H decoupling. b) A spectrum acquired with 20 kHz MAS and no ¹H decoupling. c) A spectrum acquired with 20 kHz MAS and 110 kHz ¹H decoupling.

Figure 13:

¹H NMR Spectra comparison between dry (solid) and hydrated (dashed) sample.

Commonly employed 2D NMR experiments include 2D finite pulse radio frequency driven recoupling (fpRFDR) (91), 2D dipolar assisted rotational resonance (DARR) (101), 2D CHHC (102), and 2D transferred echo double resonance (TEDOR) (103). Spectral assignments for ¹³C peaks are normally determined using 2D fpRFDR (Figure 5) or 2D DARR with a short mixing period (~10 ms), for which crosspeaks would correspond mostly to bonded ¹³C atoms.¹⁴ An advantage of 2D DARR over 2D fpRFDR is that long (up to 1.5 s) mixing periods are possible, because high decoupling powers are not required during these periods. Longer mixing periods make it possible to observe inter-amino acid crosspeaks. The 2D CHHC experiment is advantageous for the analysis of antiparallel β-sheets. These 2D spectra can exhibit specific C_α-C_α crosspeaks because of a unique structural feature: when distinct amino acids are brought in close proximity by hydrogen bonding within antiparallel β-sheets, their H_α atoms (H covalently linked to C_α) are separated by only 0.2 nm. When interpreting inter-amino acid crosspeaks for any 2D NMR technique, it can be unclear whether these crosspeaks are between atoms within the same molecule, or atoms that reside on different molecules. This ambiguity can be resolved by preparing samples that contain mixtures of isotopically labeled and unlabeled peptide (e.g., where 30%-50% of the molecules in a sample are labeled) and observing the effects of isotopic dilution on crosspeak intensity(104,4,6). Attenuation of crosspeak intensity relative to diagonal signals and intra-amino acid crosspeaks indicates magnetic interactions between atoms on adjacent molecules.

Dipolar recoupling NMR experiments can be used to probe 3D arrangements of isotopically labeled sites more quantitatively. Samples labeled for these experiments tend to be more selectively labeled than samples prepared for 2D NMR experiments. Samples are usually characterized by a small number of NMR peaks, for which spectral assignments are unambiguous. We are interested in measuring peak intensity decay as a function of time under the influence of dipolar coupling. The scheme in Figure 9b illustrates how this measurement could be accomplished. However, this figure does not show experimental features that are necessary to compensate for relaxation-induced loss of NMR coherence that would depend on the overall time of application of a dipolar recoupling sequence. Our method of choice for measuring ¹³C-¹³C dipolar couplings is PITHIRDS-CT (not an acronym) (57). For probing ¹³C-¹⁵N dipolar coupling, we use rotational echo double resonance (REDOR) (105). Frequency selective REDOR (fsREDOR) (106) has the added feature of frequency-selective pulses that are used to eliminate the effects of intra-amino acid dipolar couplings to sites with peaks that are isolated in frequency. This experiment can be used to measure dipolar couplings between sidechain carbonyl ¹³C and sidechain ¹⁵N atoms.

Solid State NMR Experiments

Calibration

It is necessary to calibrate precise frequencies of NMR peaks, set the sample rotation angle to the magic angle, and establish the relationship between radio frequency pulse powers and ${\overrightarrow{B}}_1$ field strength. Our descriptions of calibration experiments assume that the reader has a basic understanding of NMR spectrometer operation and possesses the skills listed below.

1. Knowledge of how to tune an NMR probe.

2. Shim the ${\overrightarrow{B}}_0$ magnetic field so that it is homogeneous over the volume of the sample. Homogeneity of ${\overrightarrow{B}}_0$ could be assessed by measuring ¹³C line widths for adamantane, which should be much less than the narrowest lines expected from the samples being studied (86).

3. The ability to measure the necessary duration for a pulse of a specific tip angle, or set the power of a pulse if a specific duration and tip angle are required. For a fixed tip angle, pulse duration would scale linearly with $\mid {\overrightarrow{B}}_1\mid$ and proportional to the square root of pulse power (83).

4. An understanding of the conventions for reporting NMR frequencies (referenced ppm scale) and setting spectrometer frequencies (83).

5. The ability to measure spin-lattice relaxation time T₁ and set the initial delay for each scan (d₁) appropriately (83,85).

Magnet shimming, or the adjustment of current through electromagnets in order to tune the homogeneity of ${\overrightarrow{B}}_0$, is usually performed on liquid samples (107,86): ²H NMR could be performed on D₂O, or ³¹P NMR could be performed on phosphoric acid (108).

NMR peak frequency calibration of NMR signal from any nucleus (¹H, ¹³C and ¹⁵N) is equivalent to precise measurement of the ${\overrightarrow{B}}_0$ field strength. Once NMR frequency is calibrated for a single nucleus, frequency calibrations necessary for other nuclei studied on the same magnet can be determined by calculations (109). Frequency calibrations are typically performed through measurements on standard samples that exhibit NMR signals with known and stable frequencies (chemical shifts in ppm) (95). TMS is a common standard sample for ¹³C NMR, with a single ¹³C peak set at 0 ppm. A single-pulse experiment would be sufficient to measure the NMR spectrum of TMS. However, this liquid is not routinely used for frequency calibration with solid state NMR spectrometers because of its volatility and the inconvenience of loading it into sample rotors. Instead, a more common solid state ¹³C standard is adamantane, which reliably exhibits peaks at 38.48 ppm and 29.45 ppm. The NMR signal of adamantane can be measured using a single pulse or ¹H-¹³C CPMAS experiment (Figure 8). The ¹³C NMR signal is strongest for a ¹H-¹³C CPMAS experiment when the ¹H carrier is at the central frequency of the ¹H NMR spectrum; this spectrum can be measured using a single-pulse experiment.

Calibration of the sample rotation angle is readily performed using samples with NMR line shapes that are sensitive to this angle. The magnitude of quadrupolar spinning sidebands for ⁷⁹Br NMR KBr powder are maximized for sample rotation at the magic angle; this measurement can be performed with single-pulse NMR (86,110). The sample rotation angle can also be calibrated using ¹H-¹³C CPMAS NMR of polycrystalline Gly selectively labeled with ¹³C at the carbonyl site (¹³CO Gly). The ¹³C NMR line width of this sample is minimized at the magic angle (111). The same sample could be used for peak frequency calibration, as long as the ¹³CO peak frequency from this sample is calibrated against adamantane. When using an amino acid to calibrate spectrometer frequency, it is important to be aware that peak positions are sensitive to the possible co-existence of multiple crystal structures with sample-dependent abundances (111). It is also common practice to use an NMR rotor containing a mixture of adamantane and KBr powers, so that the same sample could be used to calibrate both NMR peak frequency and the sample rotation angle.

The powers of radio frequency fields can be calibrated using CPMAS experiments. With the ¹H-¹³C CPMAS experiment, the ¹H radio frequency field can be calibrated by varying the power of the p_H,1 pulse. Calibration of ¹H pulse power is most easily performed by doubling the duration of this pulse so that it corresponds to a π pulse at the desired power level; the desired input power would correspond to the lowest power that would yield no detected ¹³C signal. The optional flip-back pulse could be used to calibrate the power of the ¹³C pulses in a similar manner. When multiple ¹³C peaks are observed in a spectrum, the ¹³C carrier frequency should be placed on resonance with a peak that is strong and relatively isolated (such as a carbonyl ¹³C peak); off-resonant effects could lead to inaccurate measurements of pulse tip angle. A $\frac{\pi }{2}$ flip-back pulse would eliminate observed signal for a peak that is on resonance. Analogously, the ¹H-¹⁵N CPMAS experiment could be employed to calibrate ¹⁵N pulse powers.

When working with a self-assembled peptide sample, our laboratory routinely performs the calibration experiment in the following order:

1. Insert a standard adamantane/KBr sample. Spin the sample at the speed that is expected for the experiment to be performed.

2. Tune the probe for ⁷⁹Br NMR. Employ single-pulse ⁷⁹Br NMR to measure the ⁷⁹Br NMR signal. Adjust the sample rotation angle to maximize the ⁷⁹Br quadrupolar sidebands.

3. Tune the probe for a double-resonance experiment (pulses on ¹H and ¹³C, with detection of ¹³C NMR).

4. Employ a single-pulse ¹H NMR experiment to determine the ¹H carrier frequency that is centered on the adamantane ¹H spectrum. Use a single pulse experiment to measure the ¹H NMR spectrum (Figure 7). This experiment does not require precise calibration of the ¹H radio frequency field strength. We usually use a short pulse (e.g., 2 μs) and an input power within the limit specified by the probe manufacturer. We typically use a delay d₁ of 5 s. Use this spectrum to determine the ¹H carrier frequency, which should be at the center of the ¹H spectrum.

5. Employ CPMAS experiments to calibrate radio frequency pulse powers and NMR peak frequencies. Before the ¹H pulse powers are calibrated, use a conservative ¹H pulse power (particularly for decoupling).

   1. Calibrate the ¹H pulse power by varying the power (or duration) of the p_H,1 pulse. Once the ¹H pulse power calibration is known, the ¹H pulse powers can be set. Our laboratory typically employs 110 kHz powers for ¹H pulses, including decoupling pulses. For cross polarization, polarization transfer efficiency is typically maximized when the average power of the simultaneous ¹H and ¹³C pulses differ in power by the sample rotation rate (112,55). For example, if the sample is spinning at 25 kHz and a 50 kHz ¹³C pulse is used for cross polarization, then the average power for the ¹H cross-polarization pulse should be 75 kHz. In our experience, the use of a ramped ¹H pulse during cross polarization ensures little sensitivity of the polarization transfer efficiency to the average ¹H pulse power.
   2. Calibrate the ¹³C pulse power using the ¹H-¹³C CPMAS experiment. By setting the ¹³C carrier frequency to the frequency of the left ¹³C peak, it is possible to calibrate ¹³C pulse power by adjusting the power or duration of p_C,2 flip-back pulse. Use the obtained ¹³C pulse calibration to set the powers of the ¹³C pulses. Our laboratory typically uses 50 kHz pulse power for ¹³C pulses (5 μs $\frac{\pi }{2}$ pulses).
   3. Set the frequency calibration so that the adamantane ¹³C NMR peaks are at their known NMR frequencies. Record the ¹³C carrier frequency that corresponds to 100 ppm. This is the carrier frequency that we typically employ for ¹³C NMR measurements because it is approximately at the center of the chemical shift range of ¹³C signals.
   4. If the intended experiment involves ¹⁵N NMR, calibrate the ¹⁵N pulse power using a ¹H-¹⁵N CPMAS experiment on a Gly or comparable sample (adamantane and KBr do not contain ¹⁵N atoms).

6. Insert and spin the sample to be studied. Retune the probe. Recalibrate the radio frequency field strengths using CPMAS experiments. It will not be necessary to recalibrate NMR peak frequencies, since these depend on $\mid {\overrightarrow{B}}_0\mid$ and not the sample.

The pulse sequence in Figure 14 (typical parameters in Table 3) shows how to calibrate a frequency selective Gaussian ¹³C π pulse. The Gaussian pulse is drawn with a dashed line because it appears in the pulse sequence every second scan. The ¹³C carrier frequency of the experiment (or just the Gaussian pulse) should be on-resonance with the peak to be selectively inverted. Alternation of the receiver phase would ensure that any ¹³C NMR peaks not inverted by the Gaussian pulse would be averaged away after even numbers of scans. The duration and power of the Gaussian pulse can be optimized to maximize the intensity of the selectively inverted peak and achieve the desired frequency selectivity. For the selective pulse, a trade-off should be expected between selectivity (increased for longer pulses), and sensitivity, which is reduced by relaxation effects for pulses that are too long.

Figure 14:

Pulse sequence for calibration of a frequency selective ¹³C pulse.

Table 3:

Typical parameters for calibration of frequency selective pulse.

MAS rate: 10 kHz (rotor period, τr = 100 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1(π/2)	110	2.27	(x)8, (−x)8
pH,2	Linear Ramp between 40 to 80	2000	y
pC,1	50	2000	y
pC,2 (π/2)	50	5	x,−x
pC,3 (π, Gaussian)	0.5*	1000	x
pC,4 (π/2)	50	5	x,x,y,y,−x,−x,−y,−y
ϕReceiver	x,−x,y,−y,−x,x,−y,y,−x,x,−y,y,x,−x,y,−y
Delay	Duration
d1	2 s

Decoupling period	Power (kHz)	Method
dec1	110	TPPM
comments	*The power designated for the Gaussian pulse is the average power.

2D Experiments

Multidimensional NMR methods are unified by a single concept: multidimensional spectra are correlation maps that indicate the presence of interactions between nuclear spins that produce NMR peaks at distinct frequencies. In the solid state NMR methods discussed here, these interactions are eliminated (or greatly attenuated) by MAS and ¹H decoupling. To produce informative crosspeaks, dipolar recoupling pulse sequences are applied during mixing periods to temporarily reintroduce spin-spin interactions. Spectral frequencies are encoded during time periods t₁ and t₂, during which nuclear magnetization precesses about the z-axis under minimal influence of orientation-dependent or spin-spin interactions. This scheme maximizes spectral resolution while providing a means of identifying couplings that inform on structure. The 2D methods described here all involve direct detection of ¹³C NMR signals, so all the horizontal frequency axes correspond to ¹³C NMR frequency. The ¹³C-¹³C 2D NMR experiments described here involve precession of ¹³C magnetization during t₁ and report on couplings between ¹³C nuclei. The ¹³C-¹³C 2D NMR experiments described here have peaks on the diagonal with intensities that depend on frequency in a way that approximately resembles ¹H-¹³C CPMAS, such as that shown in Figure 4. There are ¹³C-¹³C 2D methods that do not exhibit peaks on the diagonal, but we do not discuss them here(113-116). Heteronuclear (¹⁵N-¹³C) 2D NMR spectra do not have diagonal signals.

Choice of MAS speed is important for 2D NMR experiments. Atomic sites with large anisotropies (orientation dependences) in chemical shift (large line widths without MAS) exhibit spinning sidebands, or satellite peaks, under MAS. Spinning sidebands will occur at frequencies of the primary NMR peak plus and minus the sample rotation rates. It is possible to detect crosspeaks between spinning sidebands and other coupled atoms. In practice, sidebands are most easily detected for carbonyl ¹³C signals and are also readily detectable for aromatic ¹³C peaks. Care must be taken to avoid overlap between spinning sidebands and crosspeaks of interest. If such overlap is unavoidable, desired information can be obtained by performing a 2D NMR experiment at multiple spinning speeds. Furthermore, polarization transfer efficiencies for recoupling methods used during mixing periods can be highly dependent on MAS speed.

2D Finite Pulse Radio Frequency Driven Recoupling (2D fpRFDR):

Figure 15 and Table 4 specify the pulse sequence and typical experimental parameters for the 2D fpRFDR experiment (91). This experiment produces ¹³C-¹³C 2D exchange spectra with signals on the diagonal and crosspeaks that are approximately symmetric about the diagonal. This symmetry occurs because polarization transfers can occur in both directions between coupled pairs of spins: for example, Figure 5 includes crosspeaks at coordinates (53, 32.2) and (32.2, 53) because polarization transfers from R9 C_α to R9 C_β occur at the same rate as transfers from R9 C_β to R9 C_α during the fpRFDR mixing period. At mixing times near 1.28 ms (N = 32), crosspeaks are observed primarily between directly bonded ¹³C atoms (e.g., between ¹³C labeled C_α and C_β sites), as with the simulated spectrum depicted in Figure 5. Weak crosspeaks could also be observed between ¹³C atoms connected by two bonds (e.g., between ¹³C labeled C_α and C_γ sites). The 2D fpRFDR technique has been used to obtain ¹³C spectral assignments and precise NMR peak positions for each labeled site for self-assembled peptides labeled with uniform ¹³C at select amino acids. Analysis of secondary chemical shifts (values relative to the same atoms on the same amino acids within random coil peptides) for ¹³C atoms near the peptide backbone (carbonyl, C_α, C_β) has been used to assess secondary structure (117-119). Analysis of line widths is a basis for evaluating structural order, and this analysis has been used to identify unstructured regions of peptide assemblies (120,121). In addition, comparisons of crosspeak positions and line shapes have been used as a basis for evaluating structural variation between different samples (73,104,122).

Figure 15:

2D Finite Pulse Radio Frequency Driven Recoupling (2D-fpRFDR) pulse sequence.

Table 4:

Typical parameters for 2D-fpRFDR experiment.

MAS rate: 25 kHz (rotor period, τr = 40 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1	110	2.27	(x)8, (−x)8
pH,2	Linear ramp between 50 and 100	2,000	y
pC,1	50	2,000	y
pC,2 (π/2)	50	5	x,−x and y,−y for each t1
pC,3 (π)*	37.5	$\frac{{\tau }_r}{3}=13.33$	x,y,x,y,y,x,y,x,−x,−y,−x,−y,−y,−x,−y,−x
pC,4 (π/2)	50	5	x,x,y,y,−x,−x,−y,−y
ϕReceiver	x,−x,y,−y,−x,x,−y,y,−x,x,−y,y,x,−x,y,−y
Delay	Duration (s)
d1	2
t1	varied for 2D acquisition
Decoupling period	Power (kHz)		Method
dec1	110		Two-pulse phase modulation (TPPM)
comments	N = 32

2D Dipolar Assisted Rotational Resonance (2D DARR):

The 2D DARR technique produces ¹³C-¹³C 2D exchange spectra that are similar to those produced by the 2D fpRFDR experiment, but without the need for high power ¹H decoupling during the mixing period. Consequently, the experiment can be performed with a wider range of possible mixing times. At mixing times of ~10 ms or less, 2D DARR spectra exhibit crosspeaks that correspond mostly to ¹³C atoms within a single amino acid (123). As mixing times are increased to 50 ms, additional crosspeaks can be observed that correspond to adjacent amino acids within the primary structure. With mixing times at 500 ms or above, crosspeaks include those that report on amino acids brought into close proximity by molecular folding. Typical parameters for setting up a 2D DARR experiment are shown in Table 5 and the corresponding pulse sequence is shown in Figure 16.

Table 5:

Typical parameters for 2D-DARR experiments.

MAS rate: 10 kHz (rotor period, τr = 100 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1	110	2.27	(x)8, (−x)8
pH,2	Linear ramp between 40 and 80	2,000	y
pC,1	50	2,000	y
pC,2 (π/2)	50	5	x,−x and y,−y for each t1
pC,3	50	5	x,x,y,y,−x,−x,−y,−y
ϕReceiver	x,−x,y,−y,−x,x,−y,y,−x,x,−y,y,x,−x,y,−y
Delay	Duration (s)
d1	2
t1	varied for 2D acquisition
Mixing time	Duration (ms)
τ	10-1500
Decoupling period	Power (kHz)		Method
dec1	110		TPPM
Comments	During mixing, 1H continuous wave radio frequency should be applied with power equal to the MAS rotation rate (10 kHz).

Figure 16:

2D Dipolar Assisted Rotational Resonance (2D-DARR) pulse sequence.

Figure 17a shows a simulated 2D DARR spectrum (for ~500 ms mixing) detailing the expected crosspeaks (bicolored circles) corresponding to contacts between I32 and V40 brought together by antiparallel arrangement of β-sheets (Figure 17b) (6). To further determine if the contact is intramolecular or intermolecular, an isotopically diluted sample could be probed using the same technique. If the inter-amino acid crosspeaks are between atoms on different molecules, the crosspeaks shown in the bi-colored circle positions will be attenuated in the isotopically diluted sample when the spectra are normalized to the intra-amino acid crosspeaks. At longer mixing times, crosspeak patterns in 2D DARR spectra can become crowded. For this reason, isotopic labeling of aromatic amino acids (e.g., F) is advantageous, because aromatic ¹³C NMR peaks are spectrally isolated from carbonyl and aliphatic signals. Concerns about spectra overlap with spinning sidebands are also relevant to 2D DARR spectra. In addition, magnetization transfer can be enhanced when the spinning sidebands are near crosspeaks of interest. Crosspeaks corresponding to longer distances (above ~0.5 nm) are also more difficult to detect when MAS speeds are too fast (above ~20 kHz).

Figure 17:

a) A schematic of the expected 2D-DARR spectrum (aliphatic region) for oligomers of the 42-residue isoform of the Alzheimer’s amyloid-β peptide (6). The sample is uniformly ¹³C-labeled at the I32 and V40 residues. Empty diamonds and empty circles represent on-diagonal peak positions as well as intra-sidechain crosspeaks within I32 and V40, respectively. The cross marks represent expected inter-residue crosspeaks between the I32 and V40. b) Schematic picture showing I32 and V40 are brought into proximity by an anti-parallel β-sheet arrangement.

2D CHHC:

An antiparallel β-sheet structure will place H_α atoms in close proximity (< 0.3 nm) for specific pairs of amino acids (102). This configuration predicts a strong ¹H-¹H dipolar coupling between these H_α atoms. The presence of this coupling can be indirectly detected using the 2D CHHC measurement through the detection of C_α-C_α crosspeaks. Typical parameters for setting up a CHHC experiment are shown in Table 6 and the pulse sequence is shown in Figure 18. Figure 19 is a simulated spectrum from a 2D CHHC experiment corresponding to results obtained previously for an Aβ42 oligomer sample (6). Expected C_α-C_α crosspeaks are shown in bicolored circles. The labeling scheme for 2D CHHC experiment should be carefully chosen to avoid spectral overlap. While spectral overlap is a concern with all 2D NMR experiments, the 2D CHHC experiment is particularly sensitive to this issue because all C_α sites have similar chemical shifts. As a result, expected 2D CHHC crosspeaks may become undetectable due to overlap with the diagonal. For example, in Figure 19 chemical shifts for C_α sites in M35 and G37 are better separated and crosspeaks show less overlap with the diagonal than those for I32 and V40.Also, it is important to use short cross-polarization pulses before and after the mixing period (p_C,3 and p_H,3) to achieve selective polarization transfers between C_α and H_α atoms. The efficiency of cross-polarization transfer tends to decrease as MAS rate is increased above 10 kHz.

Table 6:

Typical parameters for 2D-CHHC experiments.

MAS rate: 10 kHz (rotor period, τr =100 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1	110	2.27	(y)16, (−y)16
pH,2	Linear ramp: 40 to 80	2,000	x
pH,3	Linear ramp between 66.67 and 53.33	150	x,x,y,y,−x,−x,−y,−y
pH,4	110	2.27	y,y,−x,−x,−y,−y,x,x
pH,5	110	2.27	−y,−y,x,x,y,y,−x,−x,y,y,−x,−x,−y,−y,x,x
p,H,6	Linear ramp between 66.67 and 53.33	150	x,x,y,y,−x,−x,−y,−y
pC,1	50	2,000	x
pC,2 (π/2 pulse)	50	5	y
pC,3	50	5	−y,y and −x,x for each t1
pC,4	50	5	x
pC,5	50	5	x,x,y,y,−x,−x,−y,−y
ϕReceiver	x,−x,y,−y,−x,x,−y,y,−x,x,−y,y,x,−x,y,−y,−x,x,−y,y,x,−x,y,−y,x,−x,y,−y,−x,x,−y,y
Delay	Duration (ms)
d1	2
td (z-filter)	3
t1	varied during 2D acquisition
Mixing time	Duration (μs)
τ	2 τr = 200
Decoupling period	Power (kHz)		Method
dec1	110		TPPM

Figure 18:

2D-CHHC pulse sequence.

Figure 19:

Positions of diagonal peaks (solid/empty diamonds and circles) and crosspeaks (half solid diamonds and circles) for α-carbons for the 2D-CHHC experiment of oligomers of the 42-residue isoform of the Alzheimer’s amyloid-β peptide (6). The crosspeaks between I32 and V40 and between M35 and G35 are anticipated because of antiparallel arrangements of β-strands; these residues were uniformly labeled with ¹³C and ¹⁵N. In the experimental data, we were unable to clearly resolve the I32 C_α-V40 C_α crosspeaks because of the close peak positions observed for the two sites.

2D Transferred Echo Double Resonance (2D TEDOR):

The 2D TEDOR (103) experiment produces a ¹⁵N-¹³C 2D exchange spectrum. For this experiment, ¹³C is normally the directly detected nucleus (horizontal axis) because of higher sensitivity. Figure 20 shows a recent version of this experiment (103). Typical parameters for setting up a 2D TEDOR experiment are shown in Table 7. This experiment has been used to establish inter-sidechain proximity, providing information similar to that offered by 2D DARR experiments (see Figure 21) (4). The technique can also be used for spectral assignments (124). By varying the mixing time, it is possible to analyze heteronuclear dipolar interactions between ¹³C and ¹⁵N in order to assess ¹³C-¹⁵N distances (125,126).

Figure 20:

2D Z-filtered Transferred echo double resonance (2D TEDOR) pulse sequence.

Table 7:

Typical parameters for 2D ZF-TEDOR experiments.

MAS rate: 9 kHz (rotor period, τr =~110 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1 (π/2)	110	2.27	(x)16, (−x)16
pH,2	Linear ramp: 39.33 to 78.67	2,000	x
pC,1	50	2,000	x
pC,2 (π)	50	10	x,x,y,y
pC,3 (π/2)	50	5	x
pC,4 (π/2)	50	5	y,y,−y,−y
pC,5 (π/2)	50	5	(x)4,(y)4,(−x)4,(−y)4
pN,1 (recoupling)	50	10	−y,y,y,−y,x,−x,−x,x,y,−y,−y,y,−x,x,x,−x,y,−y,−y,y,−x,x,x,−x,−y,y,y,−y,x,−x,−x,x
pN,2 (π/2)	50	5	x,−x
pN,3 (π/2)	50	5	x,−x
ϕReceiver	−y,y,y,−y,x,−x,−x,x,y,−y,−y,y,−x,x,x,−x,y,−y,−y,y,−x,x,x,−x,−y,y,y,−y,x,−x,−x,x
Delay	Duration (s)
d1	2
t1	varied for 2D acquisition
τzf	0.0002
Decoupling period	Power (kHz)		Method
dec1	110		TPPM
dec2	110		TPPM
dec3	110		CW
comments	Two pN,1 pulses are applied per rotor period. The number of pN,1 applied at a time depends on the choice of tmix, which is typically between 1 and 20 ms.

Figure 21:

Simulated 2D-TEDOR data for amyloid fibrils of the 40-residue isoform of the Alzheimer’s amyloid-β peptide (4). The peptide was uniformly ¹³C- and ¹⁵N-labeled at I31 and V39 and other amino acids for which signals are not shown. The solid diamonds and solid squares indicate positions of the intra-residue ¹³C-¹⁵N crosspeaks for I31 and V39, respectively. The cross marks correspond to the inter-residue ¹³C-¹⁵N crosspeaks.

Dipolar Recoupling Experiments

Dipolar recoupling NMR measurements refer to the measurement of peak intensity as a function of evolution time under the influence of dipolar couplings. These measurements are sometimes called NMR distance measurements, but the shapes of measured decay curves are not determined solely by distance when interactions are more complex than what would be expected for coupled pairs of spins. Since magnetic dipolar couplings are strongly dependent on distance (r⁻³), decays are influenced most significantly by the spacing between the most proximate pairs of spins. When groups of more than two spins are arranged at similar inter-atomic distances, the shapes of observed decay curves are influenced by the 3D arrangements of atoms (56,127,128). Data from dipolar recoupling measurements are most readily interpreted when they are compared to predictions of computer simulations of nuclear spin dynamics. Commonly used spin nuclear simulation software packages include SIMPSON (97) and SPINEVOLUTION (96). Since a dipolar recoupling measurement is usually only concerned with only one NMR peak, NMR sensitivity to this peak may be enhanced by pulsed spin-locking (129).

A common challenge with dipolar recoupling experiments is encountered because coherences that underlie detectable signals are lost over time. The loss of signal could be caused by relaxation phenomena or imperfections in the pulse sequence. As a result, NMR signal could decay as dipolar recoupling pulses are applied for varying length of time, even if the nuclear spins being probed are not coupled to one another. For the heteronuclear (¹⁵N-¹³C) dipolar recoupling techniques REDOR and fsREDOR, data are plotted as a ratio of measured ¹³C intensity (S) scaled to the peak intensity measured for the same experiment conducted without the ¹⁵N pulses (S₀). This method of data collection assumes that any mechanism for loss of ¹³C intensity that occurs during application of the pulse sequence would affect S and S₀ in the same way. For the homonuclear dipolar recoupling method PITHIRDS-CT, effective time for evolution of ¹³C-¹³C dipolar couplings is varied by changing placement of pulses within a pulse sequence that does not change in overall duration.

Rotational Echo Double Resonance (REDOR) and Frequency-Selective REDOR (fsREDOR):

Rotational Echo Double Resonance (REDOR) reports on the strength of the heteronuclear dipolar coupling and the dipolar evolution time to infer spatial proximities (105,130). This technique has been used to measure distances for samples that are selectively labeled with different nuclei (¹³C and ¹⁵N). The pulse sequence of REDOR is shown in Figure 22 and typical parameters for setting up a REDOR experiment is shown in Table 8 (130). Figure 23 shows simulated REDOR curves for different atomic distances. REDOR experiments have been used to determine the register of constituent β-strands within amyloid fibrils (131-133). The technique has also been used to detect the presence β-hairpin conformation in the MAX1 nanofiber structure (7).

Figure 22:

Rotational Echo Double Resonance (REDOR) pulse sequence.

Table 8:

Typical parameters for REDOR experiment.

MAS rate: 10 kHz (rotor period, τr = 100 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1(π/2)	110	2.27	y,−y
pH,2	Linear ramp: 40 to 80	2,000	x
pC,1	50	2,000	x,x,y,y,−x,−x,−y,−y
pC,2 (π)	50	10	x,x,y,y,−x,−x,−y,−y
pN,1 (π)	50	10	x,y
ϕReceiver	x,−x,y,−y,−x,x,−y,y
Delay	Duration
d1	2 s

Decoupling period	Power (kHz)	Method
dec1	110	CW
dec2	110	TPPM
comments	Loop parameter N: 1, 2, 3,…
	The experiment is run as a pseudo-2D sequence, where instead of t1 the loop parameter n is increased from 1 to 2, 3…

Figure 23:

Simulated REDOR dephasing curves indicating different atomic distances between a pair of ¹³C and ¹⁵N atom as shown by different solid and dashed lines.

Frequency-selective REDOR (fsREDOR) (134) introduces a frequency selective spin echo to selectively recouple the ¹³C-¹⁵N dipolar interaction samples with multiple ¹³C and ¹⁵N isotopic labels (Figure 24 and Table 9). The frequency selective pulses can be used to isolate pairs of spectrally isolated ¹³C and ¹⁵N atoms by eliminating the effects of interactions involving other nearby ¹³C and ¹⁵N atoms. This feature extends the utility of REDOR to samples with uniformly ¹³C and ¹⁵N and amino acids with functional sidechains. Sidechain carboxylic acid ¹³C and ¹⁵N atoms often correspond to NMR peaks at frequencies that are distinct from signals from other labeled sites. The fsREDOR technique has been applied on uniformly ¹³C/¹⁵N labeled Aβ fibril samples to probe the interaction between the positively-charged sidechain ¹⁵N (e.g., K28) and negatively-charged sidechain carbonyl ¹³C atoms (e.g., D23), which can form salt bridges in Aβ fibrils (135,4,8).

Figure 24:

Frequency Selective Rotational Echo Double Resonance (fsREDOR) pulse sequence.

Table 9:

Typical parameters for fsREDOR experiments.

MAS rate: 10 kHz (rotor period, τr = 100 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1 (π/2)	110	2.27	y
pH,2	Linear ramp: 40 to 80	2000	x
pC,1	50	2000	(x)4,(y)4,(−x)4,(−y)4
pC,2 (Gaussian, π)	0.5	1000	x,y,−x,−y,y,−x,−y,x,−x,−y,x,y,−y,x,y,−x
pN,1 (π)	50	10	x,y,x,y,y,x,y,x,−x,−y,−x,−y,−y,−x,−y,−x
pN,2(Gaussian, π)	0.5	1000	x
ϕReceiver	x,−x,x,−x,y,−y,y,−y,−x,x,−x,x,−y,y,−y,y
Delay	Duration
d1	2 s

Decoupling period	Power (kHz)	Method
dec1	110	TPPM
comments	Loop parameter N:1,2,3…

PITHIRDS-CT:

PITHIRDS-CT (57) is a constant-time homonuclear (¹³C-¹³C or ¹⁵N-¹⁵N) dipolar recoupling technique that has been used to probe 3D arrangements of ¹³C or ¹⁵N atoms in selectively labeled samples (see Figure 25 and Table 10). For PITHIRDS-CT experiments on ¹³C, samples are usually labeled selectively at carbonyl or methyl sites. Since carbonyl and methyl sites have very different ¹³C chemical shifts, it is possible to label a sample with ¹³C at a carbonyl site and a methyl site (98,5). If such a labeling scheme is used, the labeled carbonyl site should be sufficiently far from the labeled methyl (> 1 nm) so that it can be assumed that there is no dipolar interaction between carbonyl and methyl ¹³C atoms.

Figure 25:

PITHIRDS-CT pulse sequence

Table 10:

Typical parameters for PITHIRDS-CT experiments.

MAS rate: 20 kHz (rotor period, τr = 50 μs)
Pulse	Power (kHz)	Duration (μs)	Phase
pH,1 (π/2)	110	2.27	y
pH,2	Linear ramp: 46.67 to 93.34	2,000	x
pC,1	50	2,000	x,x
pC,2 (π)*	30	τr/3 = 16.67	x,y,x,y,y,x,y,x,−x,−y,−x,−y,−x,−y,−x,−y
ϕReceiver	x
Delay	Duration
d1	2 s

Decoupling period	Power (kHz)	Method
dec1	110	TPPM
comments	*The xy-16 phase cycle is applied to pC,2.
	The constants k1 =4, k2 + k3 = 32, for 38.4 ms total recoupling time (57).

The PITHIRDS-CT experiment is particularly useful for identifying in-register parallel β-sheets. In such a molecular configuration, equivalent backbone sites on adjacent molecules are arranged within 0.5 nm of one another. PITHIRDS-CT data can also provide a basis for studying samples with structures that deviate from in-register parallel arrangements of β-strands, including those with registry shifts, antiparallel β–sheets, and structural heterogeneities (98,5,7,53). As an example, Figure 26 shows simulated PITHIRDS-CT curves for a sample of MAX8 nanofibers with ¹³C at two carbonyl sites. The two labeled carbonyl sites produce NMR peaks at the same frequency. The curves show the expected behavior for molecules in β-hairpin conformations (solid line) and the effect of the presence of molecules in conformations that do not correspond to measurable ¹³C-¹³C dipolar couplings (dashed curves). PITHIRDS-CT technique can also be used to measure backbone torsion angles (57).

Figure 26:

a) The solid curve shows PITHIRDS-CT simulated data for eight ¹³C nuclei at the V3 CO and V18 CO positions within MAX8 β-sheets as shown in b) (53).Dashed curves show how the curve changes upon incorporation of the indicated percentages of MAX8 molecules in minor conformations that do not exhibit ¹³C-¹³C dipolar couplings.