The general principle of torsion angle measurements by ssNMR is based on the measurement of the relative orientations of anisotropic ssNMR parameters, which in turn can be correlated to bond angles or orientations. The most straightforward and most common example involves the relative orientations of two different dipolar couplings, since these dipolar interactions are conveniently aligned with chemical bonds. For example, one can measure the relative orientation of the dipolar coupling vector (between ¹³C and ¹⁵N) of one C-N bond to the dipolar vector associated with a second C-N bond. When these two vectors are associated with the Cα-N and C′-N bonds in a polypeptide, the angle would be equivalent to the ψ angle (Figure 1A). This is the approach behind the “NCCN” torsion angle experiments introduced in the late 1990s (Feng et al., 1997a; Costa et al., 1997).

The original implementation of NCCN measurements (Rienstra et al., 2002a, 2002b; Ladizhansky et al., 2003; Jaroniec et al., 2004; Bajaj et al., 2009; Barnes et al., 2009; van der Wel et al., 2010; Hoop et al., 2016) is as follows: a double quantum (DQ) state is generated between the directly bonded Cα and C′ carbons, e.g., via SPC5 dipolar recoupling (Figure 2A; red boxes). This DQ state is then submitted to ¹⁵N-¹³C recoupling (commonly via the REDOR approach; Figure 2A, blue box). Since each carbon has a directly attached ¹⁵N, they both experience rapid dephasing in a manner that is dominated by the directly-bonded ¹⁵N. Whilst the orientation dependence of a single ¹⁵N-¹³C REDOR recoupling experiment is masked in a typical MAS ssNMR study, these two ¹⁵N-¹³C recoupling curves display an interference effect that results in behavior that is sensitive to the N-Cα-C′-N dihedral angle (Figure 2B).

Practically speaking, a typical REDOR approach is used to measure a relaxation-corrected dephasing curve (Figure 2B). A series of datasets with varying REDOR mixing times is acquired (keeping the DQ excitation time fixed). The relaxation correction involves the measurement (at each REDOR time) for each peak of interest of a “S” signal (with active ¹⁵N REDOR pulses) and “S0” signal (without ¹⁵N REDOR pulses). The S/S0 peak intensity ratio is then plotted, yielding angle-dependent curves as shown in Figure 2B. At each time point either a 1D or 2D spectrum is acquired, with the 2D implementation shown in Figures 2A,C (Ladizhansky et al., 2003; van der Wel et al., 2010). The main benefit of the 2D version is that it allows one to resolve many signals at once (Ladizhansky et al., 2003). Naturally, it comes at the expense of signal-to-noise (per unit time), which is superior in the 1D versions. It is worth noting that the signal-to-noise can be a challenge in these experiments. This stems from the fact that a clear distinction of the different dihedral angles only occurs in those time points where already extensive dephasing has occurred (i.e., much of the signal is lost; see Figure 2B). Moreover, in this region, the differences between some of the dihedral angles can be quite small, such that a very good signal to noise may be required to narrow down a specific angle. Achieving a good signal-to-noise level is inherently a challenge due to the polarization losses associated with the multiple recoupling steps (both the ¹³C DQ recoupling and filtering, and also the heteronuclear recoupling), both due to losses inherent in the DQ filtering and relaxation processes.

Analysis and interpretation of the obtained data is done by comparing experimental data points (i.e., the S/S0 ratios) to reference curves such as the data in Figure 2B. The latter can be derived from analytical descriptions of the experiment. In our own analyses we typically employ primarily numerical simulations to generate the curves. A relevant spin system is modelled for the range of candidate dihedral angles (or molecular structures), resulting in a library of reference curves. For some types of torsion angle experiments it may also be necessary to incorporate in this “library” relaxation effects, the impact of the chemical shift anisotropy (CSA), and dynamic modulation (van der Wel et al., 2009; Hou et al., 2010; Hoop et al., 2014). Naturally, these additional free parameters increase the complexity of the analysis and reduce the expected precision of obtained results.

Additionally, it is worth noting that a dipolar recoupling experiment like REDOR reflects a through-space interaction and can therefore invoke multispin interactions in extensively or fully isotope-labeled samples. Such multispin interactions would not invalidate the dihedral angle experiment, as it is essentially based on seeing a dependence of the observed signal on the dihedral angle in question. This dependency would be modulated, but not be negated by multispin interactions, unless non-local interactions would dominate over the local interactions which are supposed to be recoupled. However, in this case the effect of “dipolar truncation”, in which strong dipolar interactions suppress or truncate the contributions from longer range interactions (Bayro et al., 2009), is actually beneficial. Directly-bonded ¹³C-¹⁵N interactions will effectively truncate the effect of long-range inter-residue and inter-molecular interactions. It is also worth noting that we benefit here from the relatively low density of ¹⁵N sites in polypeptides, which allows for the trains of ¹⁵N REDOR pulses without detrimental ¹⁵N-¹⁵N recoupling (illustrated in Figure 2A). Another important observation is that these experiments can be applied to polypeptides outfitted with uniform ¹³C and ¹⁵N labeling, without need for site- or residue-specific isotope labeling as may be needed for certain other types of torsion angle experiments. The examples in Figures 2, 3 are obtained with synthetic peptides where short stretches of residues were labeled, but the same experiments can be applied to fully labeled proteins (Ladizhansky et al., 2003).

The data in Figures 2C–F represent validation experiments in an amyloid-like, but nanocrystalline, peptide assembly. Integrated peak volumes from the SQ-DQ 2D spectrum (Figure 2C) are plotted as S/S0 ratios as a function of the REDOR recoupling time. The example curves in Figure 2D show the variable differences between residues, with some of the angles hard to distinguish (see also Figure 2B for ψ < 130°). Figure 2E shows the results of fitting the experimental curves, yielding in some cases multiple minima. This illustrates one common challenge of torsion angle experiments, which is that they often have regions of angular space where the obtained dephasing curves are essentially indistinguishable. As reviewed in earlier work (Hong, 2006; Hong and Wi, 2006; Ladizhansky, 2009), this issue can be addressed by combining multiple types of dihedral angle measurements with complementary sensitivities. Notably, the NCCN experiment minima are close to both the X-ray structure angles and the results of NCCN-based chemical shift analysis. This is illustrated in the bar graphs of Figure 2F. These results show that the NCCN experiment gives results that match the structure as known from X-ray microcrystallography (Nelson et al., 2005; van der Wel et al., 2007, 2010), consistent with earlier studies (Ladizhansky et al., 2003).

Several studies have deployed NCCN experiments to study unknown structures of amyloid-like fibrils (Jaroniec et al., 2004; van der Wel et al., 2010; Hoop et al., 2016). For the fibrils formed by the abovementioned prion-derived model peptide GNNQQNY, these experiments detected the anticipated β-sheet structures typical of amyloids, but also non-β structure as an integral part of a composite fibril structure (van der Wel et al., 2010; Lewandowski et al., 2011). Notably this feature was present in one of three conformers (monomer structures) that composed the “composite” fiber architecture (which manifest as three peaks per residue; Figure 2G). This feature made the reliance on distance measurements more difficult, due to substantial peak overlap. Thus, the complementary torsion angle measurements were particularly helpful and valuable. The three conformers’ distinct structures are clear in the torsion angle data, as illustrated for residue Asn-9 in Figure 2H. The presence of a non-β kink or turn was at first surprising, as it is absent from the nanocrystals. However, nowadays this finding is reminiscent of the common presence of kinks, turns or bends in amyloid architectures (between quite short β-strand segments) (van der Wel, 2017; Sawaya et al., 2021). Figures 2I,J shows how the experimental ssNMR data were translated to dihedral angles. The fit between the experimental data and simulated curves was evaluated as a function of the simulated dihedral angle (Figure 2I). In some cases this shows consistency (i.e., low RMSD between the experimental data and simulated data) with multiple possible angles. One approach to overcome this ambiguity, shown in Figure 2J, is to combine the optimal NCCN angles with TALOS-based chemical shift analysis, in order to obtain a unique solution not accessible through either approach alone.

A similar structural complexity was subsequently detected in ssNMR studies of polyglutamine (polyQ) amyloid structure, which was found to consistently contain two distinct types of peptide conformations (Figure 3A) (Schneider et al., 2011; Sivanandam et al., 2011; Hoop et al., 2014, Hoop et al., 2016; Isas et al., 2015; Matlahov and van der Wel, 2019). Proteins with expanded polyQ domains are of biomedical interest as they represent the molecular basis of a series of CAG repeat expansion disorders, which remain as-yet incurable and untreatable (Wetzel, 2012). The disease-associated mutant proteins are prone to aggregation and form toxic aggregates, which include fibrillar structures with all the hallmarks of typical amyloid fibrils. Unlike several other amyloid proteins (van der Wel, 2017; Sawaya et al., 2021), the 3D atomic structure of none of the pathogenic polyQ protein aggregates is as yet known (Matlahov and van der Wel, 2019). The degenerate nature of the primary sequence of these polypeptides greatly complicates structural analysis by ssNMR, which has made the deployment of torsion angle measurements valuable and indeed essential. Structural study by distance measurements is further complicated by the already noted composite nature of the amyloid core structure, presenting as two peaks for each residue (marked “a” and “b” in Figure 3). By combining NCCN measurements and TALOS-based chemical shift analysis, it was shown that these two conformers reflect two types of β-strand structures with distinct backbone conformations (Figures 3A–D) (Hoop et al., 2016). The origin of this characteristic doubled-peaks signature stems from the presence of antiparallel β-sheets in these fibrils, which contain β-hairpin motifs for the longer polyQ expansion lengths (more about polyQ structures below) (Matlahov and van der Wel, 2019).

One recent application of the HCCH experiment was also in our own work on aggregated polyglutamine (polyQ) proteins. As noted above, these protein aggregates are hard to study by distance measurements alone, requiring the application of dihedral angle measurements for structure determination. The glutamine residues feature an extended aliphatic side chain, with two methylene (CH₂) groups for the Cβ and Cγ atoms. This permitted the use of HC-CH dihedral angle measurements of the χ₁ and χ₂ angles of these amino acids. Figure 4B shows how the χ₂ angles of 60 and 180° give clearly different HCCH dephasing curves, with the dephasing at the mid-point (1/2 rotor period) varying with the χ₂ angle (Figure 4C). For a practical experiment, measuring up to this mid-point is sufficient, as shown in Figures 4D,E. These results identified the χ₂ angle for the glutamine residues in polyQ amyloid to be near 180°, in contradiction to certain prior structural models based on (low-resolution) X-ray diffraction (Sharma et al., 2005; Hoop et al., 2016). Similarly, the same experiments can be used to probe the χ₁ angle (Figure 4F), adding further structural constraints on the polyQ amyloid core structure (Figure 4G). As summarized in Figure 4H, one unfortunate aspect of the χ₁ measurements is that they were able to exclude various conformations, but did not result in a completely unambiguous single angle value. As noted above, this is not uncommon for dihedral angle measurements, which can provide a unique solution in some cases (χ₂ ∼ 180°) but only partially constrain the angle in other cases. The model in Figure 4G represents a visualization of the best-fit backbone and side chain angle results. This model shows how the two thus-obtained conformers (β-strand types a and b) are conformationally compatible, and explain the co-assembly of the polyQ amyloid core. Notably, prior structural models derived from other techniques were not consistent with the obtained ssNMR results (Hoop et al., 2016).

Another notable use of the HCCH experiments is a nice study of a small molecule substrate (glutamate) bound to a receptor protein, which predated our work on polyQ (Edwards et al., 2010). An isotopically labeled substrate was bound to the ionotropic glutamate receptor 2, which was itself unlabeled (Figure 5). Two torsion angles, defining χ ₁ and χ ₂ of the Glu (Figure 5B), were measured with HCCH-type experiments. When combined with REDOR-based distance constraints, these data defined the conformation of the receptor-bound substrate. The reliability of the method was validated by comparison to known crystal structures (Figure 5A), with the proof-of-principle ssNMR study being applied to the crystalline receptor-substrate complex, in which the glutamate substrate is uniformly labeled. The HCCH measurements represented analogous experiments to our own subsequent studies of the polyQ amyloid structure (Figure 4). The individual dihedral angle measurements were again consistent with multiple distinct solutions, but by combining the different data with a REDOR-based distance measurement, a unique structural solution was obtained. The obtained conformation matched X-ray based structures previously determined, as illustrated in Figure 5 (Armstrong and Gouaux, 2000).

Although not a focus of the current review, it is worth noting that many ssNMR distance measurements directly or indirectly constrain dihedral angles. In some cases, specific ssNMR experiments were designed with the explicit goal to determine dihedral angles via precise measurements of specific internuclear distances (Sinha and Hong, 2003; Wi and Spano, 2011; Hu et al., 2012; Pope et al., 2018). This includes for instance the so-called BARE (Backbone Recoupling) experiments that measure the distances between backbone nitrogens and carbonyls, with implications for the intervening backbone torsion angles (Hu et al., 2012).

From the above it is clear that there is an impressive library of ssNMR torsion angle measurements, many of which were developed and demonstrated in the late 1990s and early 2000s. Nonentheless, since then many ssNMR protein structures were determined without use of these types of constraints. Instead, most structural ssNMR studies currently deploy isotropic chemical shifts to estimate the residue-specific backbone torsion angles. This is facilitated by the observation that backbone chemical shifts (along with the Cβ shift) are reliable indicators of local secondary structure, and can even be used for quantitative determinations of backbone dihedral angles (Cornilescu et al., 1999; Shen and Bax, 2007; Shen et al., 2009). Although developed for (and from) solution NMR protein structures, these algorithms have been shown to be similarly effective for ssNMR chemical shifts. These results have been sufficiently reliable that several papers deploying explicit torsion angle measurements note the high degree of consistency between the two approaches (Jaroniec et al., 2004; van der Wel et al., 2010). That said, both methods have their strengths and weaknesses, and can used as complementary tools (van der Wel et al., 2010; Hoop et al., 2016). For example, chemical shift-based estimates can be used to resolve ambiguities inherent in direct dihedral angle measurements (and vice versa).

Distance-based ssNMR structure determination borrows heavily from methods perfected in solution NMR structural biology. In dissolved or crystalline globular proteins, or in membrane-associated proteins, one can assume that most NMR-detected distance constraints reflect interactions within an individual protein (Loquet et al., 2008). However, this is not always a safe assumption, for example when studying amyloid fibrils in which the predominant residue-residue interactions may be inter- rather than intra-molecular. SSNMR studies of amyloid structures have addressed this by diluting labeled monomers in an excess of unlabeled monomers, prior to the assembly process. This approach allows for the suppression of intermolecular interactions among (labeled) atoms, thus revealing intramolecular interactions that define the monomer structure within the fibril. Unfortunately, it also implies a drastic loss of sensitivity, as the sample is now only partly filled with “visible” labeled monomers. Torsion angle measurements are designed to probe the very local environment of the (bond) angle of interest. This means that they deploy relatively modest dipolar recoupling times and are largely insensitive to the presence or absence of intermolecular interactions, and that they can always be used to probe the monomer structure even in fully labeled samples. Aside from the already mentioned signal-to-noise benefits, this may also be important for (biological) assemblies that are difficult or impossible to (re)assemble in vitro from monomers, and thus are either fully labeled or fully unlabeled.

Another challenge faced in our recent studies of amyloid fibrils is that some fibrils contain the same monomer in two or three distinct configurations or conformations, as part of a complex or composite fiber architecture (van der Wel et al., 2010; Lewandowski et al., 2011; Hoop et al., 2016). This means that any single atom (or residue) yields multiple signals, which further complicates distance-based structural measurements (E.g., Figures 3, 4 show examples). Extensive signal overlap results that cannot be resolved by site-specific isotope labeling or spin-system-based spectroscopic editing. Moreover, the effective signal to noise is decreased, as you are effectively studying a system that behaves as if it is twice or thrice as large (in terms of signal to noise). Yet, unlike a larger protein sequence, this challenge cannot be resolved by residue-specific labeling, truncation of the sequence, mutations, or segmental labeling. In such a case, distance measurements become less powerful, and torsion angle measurements can become an essential tool.

It was noted above that backbone dihedral angles can be probed via the chemical shift assignments of proteins, with the results sometimes being not much “worse” than much more time-consuming torsion angle measurements (Figure 2F). However, even if the detection of extended β-strands and α-helices is quite reliable, some non-standard motifs can be more challenging. In such cases chemical shift-based analysis alone may fail to resolve a reliable backbone conformation for a combination of observed shift values (see e.g., Figure 2J). Thus, it may be helpful to deploy targeted torsion angle measurements in such cases (Franks et al., 2008; van der Wel et al., 2010). Moreover, chemical shift analysis is limited to protein backbones, while ssNMR torsion angle measurements can be applied to side chains and non-protein biomolecules.

Our work on polyQ amyloid structure illustrates one interesting class of proteins where distance constraints fail to provide a complete answer. Highly repetitive sequences such as the polyQ proteins render distance constraints more difficult to obtain, or at least interpret. Biology features many repetitive protein sequences. Protein aggregation diseases often feature proteins with low complexity sequences, which are sometimes defined as prion-like sequence elements (Franzmann and Alberti, 2019). Besides several distinct polyQ disease proteins and highly sequence-repetitive prions, much interest extends also to the repetitive dipeptide-repeats associated with ALS disease (Odeh and Shorter, 2020; Schmitz et al., 2021). The aggregated and phase-separated states of these proteins remain as yet poorly understood, with studies by ssNMR likely being important to understanding their structure, dynamics and phase behavior. However, repetitive sequences go well beyond amyloidogenic proteins. A different example is collagen, which is an essential component of the extracellular matrix (ECM), where it helps define the structural characteristics of tissues. The (mechanical) properties of the ECM are of substantial biological interest, for instance in the context of cancer research and treatments (Venning et al., 2015). Already a topic of significant ssNMR studies (Goldberga et al., 2018), collagen’s repetitive structure yields highly challenging spectra with highly overlapping signals, akin to the polyQ case study above. Dihedral angle measurements may be similarly useful for nonetheless probing the local structure in atomic detail. Similarly, the silk proteins produced by spiders and other animals are highly repetitive in sequence and have interesting structural properties that have been studied by ssNMR (van Beek and Meier, 2006; Holland et al., 2008). Also here dihedral angle measurements can be useful to probe their structures and structural transitions (van Beek and Meier, 2006).

Mostly we have focused on the study of protein samples. However, the use of dihedral angle measurements is also of interest for the study of other biomolecules (or non-biological samples). Besides repetitive protein structures, we may also consider applications to non-protein macromolecules of biological, biomedical and bioengineering interest. For instance (high-molecular weight) polysaccharides, RNA and DNA are increasingly studied by ssNMR, but present new challenges in terms of structure determination (Marchanka et al., 2015; El Hariri El Nokab and van der Wel, 2020).

However, also in small molecules (or short peptides), it can be difficult to gain sufficient information from distance constraints alone. In such a case, dihedral angle measurements can play an important role. In the context of peptides, this has now been well demonstrated (Rienstra et al., 2002b; Jaroniec et al., 2004; Bajaj et al., 2009; Barnes et al., 2009; van der Wel et al., 2010). Nice examples can also be found in the earlier literature, for example in studies of the retinal of the membrane protein rhodopsin (Feng et al., 1997b). The recent literature offers several other interesting case studies, as we have already seen above, in which torsion angle measurements probe the structure of small molecules bound to proteins, rather than the protein itself. This includes the example of a small molecule substrate bound to a receptor protein (Figure 5) (Edwards et al., 2010) and the amyloid-specific fluorescent dye congo red bound to HET-s fibrils (Gowda et al., 2017). It is likely that further applications like this can be expected in future work on, e.g., drug-protein, substrate-enzyme and other such interactions.