Analysis of conformational exchange processes using methyl-TROSY-based Hahn echo measurements of quadruple-quantum relaxation

Abstract Transverse nuclear spin relaxation is a sensitive probe of chemical exchange on timescales on the order of microseconds to milliseconds. Here we present an experiment for the simultaneous measurement of the relaxation rates of two quadruple-quantum transitions in 13 CH 3 -labelled methyl groups. These coherences are protected against relaxation by intra-methyl dipolar interactions and so have unexpectedly long lifetimes within perdeuterated biomacromolecules. However, these coherences also have an order of magnitude higher sensitivity to chemical exchange broadening than lower order coherences and therefore provide ideal probes of dynamic processes. We show that analysis of the static magnetic field dependence of zero-, double- and quadruple-quantum Hahn echo relaxation rates provides a robust indication of chemical exchange and can determine the signed relative magnitudes of proton and carbon chemical shift differences between ground and excited states. We also demonstrate that this analysis can be combined with established Carr–Purcell–Meiboom–Gill (CPMG) relaxation dispersion measurements, providing improved precision in parameter estimates, particularly in the determination of 1 H chemical shift differences.


Introduction
Solution NMR spectroscopy is a powerful tool for the characterisation of macromolecular dynamics over a range of timescales relevant to biological function (Sekhar and Kay, 2019): from backbone and sidechain disorder on ps-ns timescales, characterised by S 2 order parameters (Frederick et al., 2007;Stetz et al., 2019;Sun et al., 2011); rotational diffusion and domain motions, characterised by rotational correlation times, τ c (Ryabov et al., 2009;Waudby et al., 2021); through to real-time measurements of kinetics on timescales of seconds and beyond, following rapid mixing, temperature or pressure jumps (Charlier et al., 2018;Franco et al., 2017;Waudby et al., 2018). NMR is also particularly well suited to the analysis of reversible chemical exchange on timescales of the order of microseconds to milliseconds, via lineshape analysis across a titration series (Stadmiller et al., 2020;Waudby et al., 2016Waudby et al., , 2020 or using sophisticated pulse sequences such as ZZ-exchange spectroscopy, chemical exchange saturation transfer (CEST) and Carr-Purcell-Meiboom-Gill (CPMG) and R 1ρ relaxation dispersion (Alderson et al., 2020;Boswell and Latham, 2019).
An NMR-active spin in chemical exchange between two conformations experiences an additional contribution, R ex , to its observed transverse relaxation rate: where represents the chemical shift difference, normalised by the particular parameters of the exchange process. The exchange term therefore in principle contains information on thermodynamic, kinetic and structural aspects of the chemical exchange process, and a variety of methods have been developed to extract this information, adapted to varying functional groups, molecular weights, populations and timescales (Alderson et al., 2020;Sekhar and Kay, 2019). Although the overall R 2 (Eq. 1) is readily measured by Hahn echo (HE) experiments (provided only that the timescale of exchange is much shorter than the echo duration), for the analysis of exchange it is necessary to quantify the exchange term R ex , either by direct modulation through application of radio frequency (rf) fields or by determination of the exchange-free relaxation rate, R 2,0 . CPMG experiments, and related R 1ρ measurements, are a popular example of the former approach, in which the application of a train of refocusing pulses, with frequency ν CPMG k ex , reduces the magnitude of the exchange contribution. Analysis of the static field and frequency dependence of this effect can be used to quantify chemical shift differences and the populations and timescales of exchange (Millet et al., 2000). CPMG experiments have now been developed for a variety of spin systems, enabling the analysis of dynamics in protein backbone and sidechains, as well as nucleic acids (Hansen et al., 2008;Juen et al., 2016;Korzhnev et al., 2004aKorzhnev et al., , 2005Loria et al., 1999;Tugarinov et al., 2020;Yuwen et al., 2019;Yuwen and Kay, 2019).
The analysis of methyl groups is particularly interesting, as these groups are well suited to the observation of highmolecular-weight systems through the combination of 13 CH 3 labelling against a perdeuterated background, reducing dipolar relaxation pathways, with methyl-TROSY (transverse relaxation optimised) pulse sequences, such as the HMQC (heteronuclear multiple quantum coherence), which select for slowly relaxing coherences within the complex energy levels of the spin system ( Fig. 1) (Tugarinov et al., 2003).
Given this, the multiple quantum (MQ) CPMG experiment, based on the analysis of zero-quantum (ZQ) and doublequantum (DQ) coherences, provides the greatest sensitivity, although the analysis is complex as dispersions depend on both 1 H and 13 C chemical shift differences (Korzhnev et al., 2004b, a). In other experiments, pulse imperfections can cause unwanted mixing between energy levels, requiring the use of compensation elements (e.g. 1 H SQ) , while in addition, back-transfer from C z coherences is inefficient, so 13 C SQ measurements can be challenging without the use of complex pulse sequences or labelling schemes (Lundström et al., 2007;Skrynnikov et al., 2001;Tugarinov et al., 2020;Weininger et al., 2012). Higher order coherences can also be exploited within fully 13 CH 3labelled methyl groups, such as 1 H DQ or triple-quantum (TQ) CPMG experiments, which provide exquisite sensitivity to small 1 H chemical shift differences as the effective frequency difference, ω, is magnified 2-or 3-fold, resulting in 4-to 9-fold increases in R ex (Eq. 2) Yuwen et al., 2016). However, the sensitivity of these experiments is somewhat low, particularly for large systems, due to rapid relaxation through C-H dipolar interactions during the constant-time (CT) CPMG relaxation period.
An alternative approach to the detection of chemical exchange is the analysis of the relaxation and cross-correlated relaxation of multiple quantum coherences, R MQ and R MQ , defined as the sum and difference of ZQ and DQ relaxation rates ( Fig. 1): Originally measured in amide spin systems, multiple quantum cross-correlated relaxation in particular was identified as being sensitive to chemical exchange through correlated fluctuations in the component proton and nitrogen frequencies, as well as being sensitive to correlated fluctuations between anisotropic chemical shifts due to rotational diffusion (Kloiber and Konrat, 2000;Tessari and Vuister, 2000). CPMG pulse trains may be combined with these measurements to identify the contributions to R MQ due exclusively to chemical exchange (Dittmer and Bodenhausen, 2004). Leveraging the multi-spin nature of MQ coherences, measurements of MQ cross-correlated relaxation have since been applied to other extended spin systems, such as adjacent Cα and Cβ spins, adjacent Cα (i) and Cα (i−1) spins and 15 N spins separated by hydrogen bonds, to detect long-range conformational exchange processes (Chiarparin et al., 2001;Früh et al., 2001;Lundstrom et al., 2005). Within methyl groups, multiple quantum relaxation and cross-correlated relaxation rates can be determined indirectly, by measurement of ZQ and DQ relaxation rates using methyl-TROSY HE experiments that incorporate a filter to select only the slowly relaxing inner lines (Gill and Palmer, Figure 1. Energy level diagram for an isolated 13 CH 3 spin system. The multiple quantum transitions analysed in this work are indicated with arrows. Energy levels are indexed |C, H according to 13 C and 1 H spin states, where symmetrised combinations of 1 H spins are indicated in the inset box (Tugarinov and Kay, 2006). 2011). CPMG pulse trains may again be employed to distinguish the effects of cross-correlations in chemical shift anisotropy (CSA) from chemical exchange (Toyama et al., 2016). Alternatively, deviations from an empirical correlation established between R MQ and measurements of crosscorrelated dipole-dipole relaxation between methyl protons (η HH , which is proportional to S 2 axis τ c ) may be used to identify methyl groups involved in chemical exchange processes (Gill et al., 2019). However, a more detailed analysis may also be carried out, wherein the magnetic field dependence of R MQ and R MQ rates is analysed and the field-dependent contributions from the 1 H and 13 C CSAs measured and subtracted to determine the contributions due to chemical exchange (Toyama et al., 2017) (re-cast from the original according to Eq. 3): These expressions may be used to compare the calculated R MQ,ex and R MQ,ex terms with the chemical shift differences to some known reference state, assuming that all spins are part of the same exchange process and so have identical excited state populations and exchange rates (Toyama et al., 2017). However, as the expressions are symmetric in ξ C and ξ H , these parameters themselves cannot be determined unambiguously. While it is possible to determine the relative sign of ξ C and ξ H from the sign of R MQ,ex , additional approaches based on R 1ρ measurements, or the analysis of peak positions between single and multiple quantum correlation experiments or between fields, are required to determine their absolute signs (Auer et al., 2010;Bouvignies et al., 2010;Skrynnikov et al., 2002).
In this paper, we extend these analyses to higher order coherences accessible within methyl spin systems and specifically to the quadruple-quantum (QQ) and four-spin doublequantum (DQ ) transitions (Fig. 1). In this context, it is helpful to reframe the earlier analysis of the magnetic field dependence of R MQ and R MQ rates in terms of the ZQ and DQ rates that are directly observed in HE experiments. The magnetic field dependences of these rates all have similar functional forms (Table 1), but the DQ and QQ rates have significantly higher sensitivity to 1 H chemical shift differences (akin to the 1 H TQ CPMG experiment; Yuwen et al., 2016). We show that the four-spin relaxation rates R DQ and R QQ can be measured with high sensitivity and that analysis of the magnetic field dependence of these rates together with R ZQ and R DQ breaks the degeneracy of solutions and allows ξ C and ξ H to be determined unambiguously up to an overall sign. We also report improved experiments for control measurements of 1 H and 13 C CSAs and finally demonstrate the simultaneous analysis of field-dependent HE relaxation rates with CPMG relaxation dispersion measurements.

Theory
Transverse relaxation rates of methyl ZQ, DQ, DQ and QQ transitions were calculated using RedKite (Bolik-Coulon et al., 2020), in the macromolecular limit assuming rapid rotation about the 3-fold methyl symmetry axis and including 1 H and 13 C CSAs and interactions with external protons and deuterons (Table 1). We observe that for both four-spin coherences, DQ and QQ, relaxation due to intra-methyl dipolar interactions is zero, in common with the ZQ and DQ coherences traditionally selected in methyl-TROSY experiments. This contrasts with 1 H TQ coherences used previously in Table 1. Theoretical transverse relaxation rates of selected multiple quantum methyl transitions. Expressions for relaxation rates are divided into dipolar, CSA and exchange contributions, and transitions are labelled as indicated in Fig. 1. Dipolar contributions are summed over external protons and deuterons, H ext and D ext , where b AB is the dipolar coupling between a methyl spin A and an external proton or deuteron B, b AB = µ 0 γ A γ B 4πr 3 AB P 2 (cos θ B ), and P 2 (x) = 1 2 (3x 2 − 1). Exchange contributions are calculated in the fast exchange limit, where the normalised chemical shift differences ξ are defined in Eq. (3).
Frequency Dipolar contribution CSA contribution Exchange contribution (Yuwen et al., 2016), which are relaxed strongly by intra-methyl dipolar interactions between 1 H and 13 C spins. However, in common with 1 H TQ coherences, the DQ and QQ transitions remain sensitive to relaxation from dipolar interactions with external spins, and so to minimise these contributions in this work, we have used only perdeuterated, selectively methyl-labelled samples.
Contributions of chemical exchange to relaxation rates have been calculated in the limit of fast exchange (Table 1). As observed previously for the relaxation and crosscorrelated relaxation of MQ coherences (Toyama et al., 2017), total relaxation rates therefore have a quadratic dependence on the static field strength, B 0 , and this may be measured to determine the sum of the CSA and exchange terms. Therefore, control measurements of 1 H and 13 C CSA values, together with the effective rotational correlation time, S 2 axis τ c , are required to isolate the contribution due only to chemical exchange.
Exchange contributions to relaxation are written in the form of squared linear combinations of normalised 1 H and 13 C chemical shift differences (Table 1). As discussed earlier, the symmetry between ξ C and ξ H in ZQ and DQ coherences precludes determination of their absolute values, even up to the overall ambiguity in sign arising from the squares. However, the four-spin DQ and QQ coherences are substantially (effectively 9-fold) more sensitive to 1 H chemical shift differences. This not only provides greater sensitivity to small chemical shift perturbations, as exploited in TQ CPMG measurements (Yuwen et al., 2016), but by breaking the symmetry between 1 H and 13 C terms, it becomes possible to determine ξ C and ξ H unambiguously (up to an overall sign) from a combined analysis of the field dependence of ZQ, DQ, DQ and QQ relaxation rates.

A Hahn echo super-experiment for the measurement of relaxation rates of four-spin coherences
We have developed a Hahn echo (HE) pulse sequence for the measurement of transverse relaxation rates DQ and QQ coherences ( Fig. 2a). This is adapted from the methyl-TROSYoptimised 1 H TQ CPMG experiment (Yuwen et al., 2016), with the omission of 13 C 90 • pulses flanking the relaxation period such that the magnetisation at the beginning of this period is 8C . .} indicates summation over cyclic permutations. This is a mixture of coherences from which DQ ± and QQ ± transitions may be isolated by phase cycling of ψ 1 and ψ 2 (Fig. 2a). As demonstrated for the simultaneous acquisition of HZQC (heteronuclear zero quantum coherence) and HDQC (heteronuclear double quantum coherence) correlation experiments , by storing individual increments of the phase program and applying receiver phase cycling post-acquisition, it is possible to acquire P-and N-type pathways for both DQ and QQ relaxation measurements within a single super-experiment (Schlagnitweit et al., 2010), without loss of sensitivity.
The sequence is demonstrated here using a sample of [ 2 H, 13 CH 3 -ILV]-labelled FLN5, the fifth immunoglobulin domain from the Dictyostelium discoideum filamin protein, comprising 108 residues. Measurements of relaxation rates were carried out at four static field strengths (600, 700, 800 Figure 2. Hahn echo measurement of R 2 relaxation rates for methyl DQ and QQ coherences. (a) Pulse sequence and coherence transfer pathways for the simultaneous measurement of DQ and QQ Hahn echo relaxation rates. Individual steps of the phase cycles ψ 1 = (0, 51.4, 102.9, 154.3, 205.7, 257.1, 308.6) 3 , ψ 2 = 0 • 7 , 120 • 7 , 240 • 7 , are stored and processed post-acquisition to select P-and Ntype DQ and QQ coherence transfer pathways. T represents the relaxation time, and the delay δ = 1/4J = 2 ms. Shaded grey pulses represent consecutive 5 kHz 2 ms H x and 3 ms H y purge pulses before the recycle delay, T rec (with low-power presaturation applied, if required). φ 1 = x, −x; receiver phase = x. Trapezoidal gradients were applied with the following powers and lengths: g 0 11 G cm −1 , 1 ms; g 1 14 G cm −1 , 400 µs; g 2 11 G cm −1 , 200 µs; g 4 −14 G cm −1 , 300 µs; g 4 −27 G cm −1 , 500 µs; g 5 −44 G cm −1 , 700 µs. (b) HE relaxation measurements of four-spin DQ and QQ coherences for methyl groups in ILV-labelled FLN5 (800 MHz, 283 K). Measurements of ZQ and DQ HE relaxation rates are also shown and were acquired using established methods (Gill and Palmer, 2011). Values of fitted relaxation rates are shown in the legends, with uncertainties indicating the standard error derived from fitting. and 950 MHz), and high-quality correlation spectra were obtained (Fig. S1 in the Supplement) and fitted to determine DQ and QQ relaxation rates (Tables S1 and S2). ZQ and DQ relaxation rates were also measured using established methods (Gill and Palmer, 2011) (Tables S3 and S4). Measurements for representative methyl groups are plotted in Fig. 2b. Fitted relaxation rates span a wide range of values, from approximately 10 to 300 s −1 . We observe no fixed ordering of the various relaxation rates, and indeed in some cases, fourspin relaxation rates are slower than ZQ or DQ rates (e.g. for DQ relaxation in L701CD1, Fig. 2b, right-hand panel).

Measurement of methyl group dynamics and 13 C chemical shift anisotropy
We next sought to determine the CSA contribution to the observed relaxation rates (Table 1), in order to isolate the exchange contribution for further analysis. Methyl 13 C CSA values can be determined by analysis of cross-correlated relaxation between the 13 C CSA and the 1 H / 13 C dipolar in-teraction in 13 C SQ transitions, which results in differential relaxation between inner or outer lines within the 13 C quartet (Liu et al., 2003;Tugarinov et al., 2004). 13 C CSA values have previously been measured in isolated methyl groups using constant-time 1 H-coupled HSQC (heteronuclear single quantum coherence) experiments, in which the length of the constant-time delay is varied, and the relaxation of inner or outer lines fitted to exponential functions to determine the cross-correlated relaxation rate, η C , from which the 13 C CSA, σ C , can be derived (Tugarinov et al., 2004): While this measurement strategy is effective, overlap of multiplets between adjacent methyl resonances can limit the ability to resolve the relaxation of individual transitions. A pseudo-4D variant has been developed in which the coupling is evolved in a third frequency dimension (Toyama et al., 2017), but this can require substantial measurement time.
Here we suggest an alternative approach, in which an additional relaxation delay is incorporated into a non-constanttime 1 H-coupled HSQC experiment (Fig. 3a). Based on our analysis of the standard 1 H-coupled HSQC experiment (Waudby et al., 2021), and fully incorporating the effects of relaxation and cross-correlated relaxation throughout the pulse sequence, the entire multiplet lineshape may be expressed as a function of the 1 H and 13 C chemical shifts (determined from a regular HMQC spectrum), 1 H and 13 C relaxation rates, the scalar coupling 1 J CH ≈ 125 Hz and the parameters of interest, S 2 axis τ c and σ C . This expression may be used to fit 2D spectra obtained at multiple relaxation delays, to estimate the parameters S 2 axis τ c and σ C simultaneously. By avoiding constant-time evolution periods, this approach provides higher sensitivity, while the parametric estimation strategy allows analysis even of highly overlapped resonances, in contrast to measurements of the decay of individual multiplet components.
Measurements were acquired for ILV-labelled FLN5 at 800 MHz, using four relaxation delays from 1 to 150 ms. Clusters of overlapping multiplets were detected, and the pseudo-3D data for each cluster were then fitted to analytical expressions for the lineshape to determine S 2 axis τ c and σ C . Our observations fitted theoretical expectations closely, and for well-resolved resonances, fits can be visualised as one-dimensional cross-sections (Fig. 3b), while fits of multiple overlapping resonances are shown in Fig. S2. Values of S 2 axis τ c determined in this manner were in good agreement with control measurements of the cross-correlated relaxation-induced build-up of 1 H TQ magnetisation (Sun et al., 2011) (R 2 = 0.97, with gradient = 0.861 ± 0.096 and offset = 0.93 ± 0.98 not significantly different from 1 and 0 respectively; Fig. S3). In line with previous reports (Tugarinov et al., 2004), 13 C CSA values we determined varied depending on residue type (Fig. 3c), with mean values (± SD) for isoleucine of 19.2 ± 3.4 ppm, leucine of 39.9 ± 4.7 ppm and valine 32.4 ± 5.2 ppm. Full results, including S 2 axis τ c , are presented in Table S5.

Measurement of methyl 1 H chemical shift anisotropy
The methyl 1 H CSA, σ H , can be determined from the differential relaxation of 1 H doublets due to cross-correlated relaxation between the 1 H CSA and the 1 H / 13 C dipolar interaction. This can be measured using an HMQC experiment incorporating a 1 H Hahn echo and a filter to exclude the outer lines of the 13 C multiplet (Tugarinov et al., 2004), but in common with measurements of the 13 C CSA, overlap between multiplets can be a problem. While this can again be avoided using a pseudo-4D experiment (Toyama et al., 2017), here we present a simple alternative that uses an IPAP filter to isolate 13 C spin states (Fig. 4a). This allows measurement of their relaxation rates, R α and R β , from which the crosscorrelated relaxation rate η H and the 1 H CSA may be deter-mined (Eq. 7).
Relaxation measurements were acquired for ILV-labelled FLN5 at 950 MHz, and high-quality sub-spectra were obtained corresponding to 13 C |α and |β spin states (Fig. 4b), from which the relaxation rates R α and R β could be measured (Fig. 4c). We note that the use of high magnetic field strengths is particularly important for these measurements given the small size of the 1 H CSA, resulting in differences in relaxation rates typically less than 1 s −1 (Fig. 4c). Combining these measurements with earlier measurements of S 2 axis τ c , 1 H CSA values could be determined (Eq. 7) with a precision of ca. 0.2 ppm (Fig. 4d, Table S5). 1 H CSA values also depended on the residue type, although not as strongly as for the 13 C CSA, with mean values (± SD) of 0.61 ± 0.55 ppm for isoleucine, 0.79 ± 0.33 ppm for leucine, and 0.29 ± 0.28 ppm for valine.

Analysis of chemical exchange through field-dependent HE relaxation rates
Having carried out measurements of ZQ, DQ, DQ and QQ HE relaxation rates in FLN5 at multiple magnetic field strengths, together with control measurements of CSA, we sought to analyse the data quantitatively in terms of chemical exchange. Assuming fast exchange, we performed linear regression of HE relaxation rates vs B 2 0 (Fig. 5a, b), from which measured CSA contributions were then subtracted in order to provide estimates of (ξ C ± ξ H ) 2 and (ξ C ± 3ξ H ) 2 (Table 1). Each of these four measurements represents a pair of lines, corresponding to the positive and negative roots, in (ξ H , ξ C ) parameter space (Figs. 5c,d and S4). The points at which all four lines intersect indicate the values of ξ H and ξ C consistent with the complete set of relaxation data. The linearity of the observed relaxation rates vs B 2 0 (Fig. 5a, b), together with the closeness of the intersection between all four lines (Fig. 5c,  d), validates our assumption of a fast exchange process underlying the observed relaxation behaviour. We note that exchange involving a third state, in which exchange-induced fluctuations in 1 H and 13 C chemical shift are no longer correlated, would also result in lines that do not intersect (discussed further in the Supplement and Fig. S5).
To evaluate the uncertainty in the position of intersection points, χ 2 surfaces can be calculated as a function of ξ H and ξ C (Fig. 5c, d, black contours). The location of the minimum on this surface for a given methyl group indicates the optimal values of ξ H and ξ C , but in the absence of additional information, it is not possible to deconvolute structural information (i.e. the chemical shift perturbations δ H and δ C ) from thermodynamic and kinetic terms. However, if several resonances are involved in the same exchange process, then ξ H and ξ C can be compared, at least in relative terms, to provide structural insight. T represents the relaxation time, and the delay δ = 1/4J = 2 ms. Shaded grey pulses represent consecutive 5 kHz 2 ms H x and 3 ms H y purge pulses before the recycle delay, T rec (with low-power presaturation applied, if required). φ 1 = x, −x; φ 2 = x, x, −x, −x; receiver phase = x, −x, −x, x. Trapezoidal gradients were applied with the following powers and lengths: g 0 25 G cm −1 , 1 ms; g 1 7 G cm −1 , 1 ms; g 2 9 G cm −1 , 1 ms; g 3 22 G cm −1 , 300 µs; g 4 18 G cm −1 , 1 ms; g 5 16 G cm −1 , 1 ms. States-TPPI quadrature detection in t 1 was achieved by incrementation of φ 1 . (b) Cross-sections through 13 C multiplets of isolated FLN5 methyl groups measured according to the pulse sequence in panel (a), with relaxation delays as indicated, acquired at 283 K, 950 MHz ( 1 H Larmor frequency). Cross-sections through the fitted spectra are also shown by dashed lines, with the fitted values of S 2 axis τ c and σ C indicated. (c) Measured values of methyl 13 C CSA in FLN5, 283 K, coloured according to the residue type. Error bars indicate the standard error determined from fitting.

Global analysis of HE relaxation and CPMG relaxation dispersion measurements
To resolve the ambiguity in chemical shift changes, thermodynamics and kinetics inherent in ξ values, we explored the joint analysis of field-dependent HE relaxation together with MQ and 1 H SQ CPMG relaxation dispersion measurements (Korzhnev et al., 2004b;Yuwen et al., 2019) (Fig. 6a,  b). While large MQ dispersions were observed for several methyl resonances, 1 H SQ dispersions were generally small, which indicated the potential utility of DQ and QQ data in providing increased sensitivity to small 1 H chemical shift differences. Inspection of individual CPMG traces indicated clearly that two separate groups of methyl resonances, located in distinct regions of the molecule, were undergoing exchange with different rates and populations (e.g. I743CD1 located in the G strand and L701CD1 located in the C/D hairpin; Fig. 6a, b). CPMG and HE data for multiple residues were fitted simultaneously to determine 1 H and 13 C chemical shift differences, the population of the intermediate state and the exchange rate (Table S6). Given the additional kinetic information available from the CPMG experiment, we may relax our earlier assumption of fast chemical exchange (Table 1). Interleaved experiments are acquired using IP/AP blocks as indicated, and the sum and difference are then calculated to isolate 13 C |α and |β spin states. Delays = 1/2J = 4 ms, δ = 1/4J = 2 ms, ξ = 1/8J = 1 ms. Shaded grey pulses represent consecutive 5 kHz 2 ms H x and 3 ms H y purge pulses before the recycle delay, T rec (with low-power presaturation applied, if required). φ 1 = x, −x; φ 2 = (y) 4 , (−y) 4 ; φ 3 = y, y, −x, −x, −y, −y, x, x; φ 4 = x, x, y, y, −x, −x, −y, −y; φ 5 = x; receiver phase = x, −x, −x, x. States-TPPI quadrature detection in t 1 was achieved by incrementation of φ 5 . Trapezoidal gradients were applied with the following powers and lengths: g 0 31 G cm −1 , 1 ms; g 1 3.9 G cm −1 , 1 ms; g 2 −22 G cm −1 , 300 µs; g 3 16 G cm −1 , 300 µs. (b) Spectra of FLN5, 283 K, acquired at 950 MHz ( 1 H Larmor frequency) using the pulse sequence in panel (a), and a relaxation delay of 2 ms. Spectra corresponding to 13 C |α and |β spin states are plotted in red and blue respectively. Light colours indicate negative, folded signals originating from isoleucine methyl groups. (c) Spin-stateselective relaxation of representative resonances, with fitted rates as indicated. (d) Measured 1 H CSA values for FLN5, 283 K, coloured by residue type (Table S5). Error bars indicate the standard error propagated from fits to Eq. (7).  Table 1 assuming fast exchange and subtracting measured CSA contributions. Shading indicates the standard error propagated from linear regression analysis and CSA measurements. Black contours indicate 68 % and 95 % confidence intervals in ξ H and ξ C , based on all four HE measurements and assuming two-state fast exchange. Red symbols indicate ξ H and ξ C parameters derived from global fitting of HE and CPMG data (Fig. 6).
Exchange contributions to HE measurements were therefore calculated directly from the dominant eigenvalue of the Liouvillian superoperator (Palmer and Koss, 2019), and data are plotted against B 0 directly rather than B 2 0 . In general, we note that the comparison of multiple quantum HE relaxation rates will provide useful additional information in all exchange regimes, except when all transitions are in very slow exchange, at which point the exchange contribution to relaxation rates R ex = k AB (lifetime line broadening), irrespective of the frequency difference (Wang and Palmer, 2002).
Good-quality fits were obtained for all resonances, indicating the mutual consistency of HE and CPMG measurements (Figs. 6a-d and S6). We suggest that residual oscillations in some CPMG traces (e.g. Fig. 6b) are likely to reflect the impact of pulse imperfections and off-resonance effects, which in the future could be handled by more detailed calculations. HE and CPMG data for resonances within the first cluster of exchanging methyl groups, located between the A and G strands (Fig. 6e, red-yellow colouring), fitted closely to a two-state exchange model with an intermediate population of 2.9 ± 0.2 % and an exchange rate of 872 ± 41 s −1 (Fig. 6a, c). Exchange within the second cluster, located around the C/D hairpin (Fig. 6e, cyan-magenta colouring), was more rapid, with a fitted exchange rate of 6900 ± 360 s −1 (Fig. 6b, d). As expected for resonances in fast exchange (Palmer and Massi, 2006), the fitted chemical shift differences and intermediate state population were strongly correlated, which results in the large uncertainties reported. Nevertheless, for both clusters, ξ values back-calculated from the fit results were consistent with our previous analysis of HE data alone (Fig. 5c, d, red symbols).

Discussion
The study of methyl groups is of fundamental importance to modern biomolecular NMR spectroscopy, both in proteins and in labelled nucleic acids (Abramov et al., 2020;Kerfah et al., 2015;Rosenzweig and Kay, 2014). Methyl 13 CH 3 spin systems provide a complex set of energy levels (Fig. 1), and in this work we have utilised the maximum range available, describing the first applications of four-spin double-andquadruple-quantum coherences to the study of biomolecular dynamics.
Building on earlier analyses of relaxation and crosscorrelation relaxation of multiple quantum coherences (Gill et al., 2019;Gill and Palmer, 2011;Kloiber and Kon-Figure 6. Joint analysis of field-dependent HE and CPMG measurements in FLN5. (a, b) MQ and 1 H SQ CPMG relaxation dispersion measurements (Korzhnev et al., 2004b;Yuwen et al., 2019) for (a) I743CD1 and (b) L701CD1 resonances. Error bars indicate standard errors derived from the spectrum noise. Solid lines show global fits in conjunction with HE measurements. (c, d) Magnetic field dependence of multiple quantum HE relaxation rates of (c) I743CD1 and (d) L701CD1 resonances. Error bars indicate standard errors determined from curve fitting. Solid lines show global fits to Eq. (11) in conjunction with CPMG measurements, with best fit parameters as shown; the sign of δ C has been assumed to be positive. (e) Stereoview projection of fitted methyl chemical shift differences, δ CH = ( δ CH /4) 2 + δ 2 H , on the crystal structure of FLN5 (1qfh, McCoy et al., 1999). rat, 2000Toyama et al., 2017), by measuring the dependence of ZQ, DQ, DQ and QQ HE relaxation rates on the static magnetic field strength, we have demonstrated the ability to unambiguously determine the relative chemical shift perturbations of sparsely populated intermediate states (up to an overall sign). The analysis of DQ and QQ relaxation also provides increased sensitivity to small 1 H chemical shift differences. This is particularly important as these tend to be smaller than associated 13 C chemical shift differences: 1 H chemical shifts are determined primarily by magnetic anisotropy and ring current effects (Li and Brüschweiler, 2012), while 13 C chemical shifts are additionally and strongly affected by rotamer distributions through the γ -gauche effect . We have also demonstrated the joint analysis of HE and CPMG relaxation dispersion measurements (O'Connell et al., 2009), which provides greater detail on the thermodynamics and kinetics of the exchange process, and indeed on the existence of multiple excited states, than can be determined from analysis of HE data alone. We suggest that the combination of HE and MQ CPMG measurements may be a particularly useful pairing to provide precise estimates of both 1 H and 13 C chemical shift differences while avoiding insensitive DQ or TQ 1 H CPMG experiments Yuwen et al., 2016). However, HE measurements alone may also be useful in extending the analysis of dynamics in methyl groups beyond the timescales accessible to CPMG experiments.
It is interesting to observe that the four-spin DQ and QQ transitions, in common with the inner ZQ and DQ methyl-TROSY transitions (Fig. 1), have zero relaxation due to intra-methyl dipolar interactions (Table 1). While the fourspin transitions are certainly more sensitive to relaxation through dipolar interactions with protons from other methyl groups, this effect nevertheless leads to an unexpectedly low exchange-free relaxation rate, particularly in comparison to 1 H DQ or TQ transitions previously used in CPMG experiments Yuwen et al., 2016), which in turn leads to greater sensitivity to the effects of chemical exchange.
We have described a new super-experiment (Schlagnitweit et al., 2010) for the simultaneous measurement of these DQ and QQ relaxation rates (Fig. 2a). This approach, in which individual phase cycle steps are stored during acquisition and recombined during processing, was also recently applied to the simultaneous measurement of HZQC and HDQC correlation spectra . In both instances, the total time required to acquire the pair of experiments is halved without any compromise to the quality of the individual measurements. Indeed, the IPAP scheme we have described here for measurement of the 1 H CSA (Fig. 4a) can also be regarded as a form of super-experiment, containing two individual experiments for the measurement of spin-state-dependent 1 H relaxation rates. The traditional onthe-fly implementation of receiver phase cycling during acquisition is simply a relic of the high cost of data storage in the past, with no practical benefits, and we would echo calls for the storage of individual free induction decays for processing post-acquisition to become more systematically implemented in modern software (Schlagnitweit et al., 2012).
A number of experiments have been reported for the measurement of the 13 C CSA, σ C (Toyama et al., 2017;Tugarinov et al., 2004). Here, building on our previous analysis of cross-correlated relaxation in 1 H-coupled HSQC experiments (Waudby et al., 2021) together with developments in two-dimensional lineshape fitting , we have described an alternative approach based on pseudo-3D lineshape fitting of a relaxation-weighted 1 H-coupled HSQC to simultaneously determine S 2 axis τ c and σ C (Fig. 3). S 2 axis τ c is encoded in the relative intensities of inner and outer lines in the 13 C quartet, while σ C is encoded by the relative intensity of up-and down-field lines. We have shown previously that S 2 axis τ c can be measured accurately up to correlation times of at least 100 ns (Waudby et al., 2021). Moreover, since peak positions can be accurately determined from a fully decoupled 2D spectrum, and 1 J CH scalar couplings are largely uniform, we have found that the analysis of overlapping resonances or crowded regions of a spectrum is in practice not as severe a problem as might be expected. Nevertheless, in larger systems, it may be useful to measure S 2 axis τ c independently using alternative methods (Sun et al., 2011) and use lineshape fitting to determine only σ C or indeed to employ pseudo-4D experiments to resolve the quartet structure separately (Toyama et al., 2017).
Notwithstanding the above discussion, it is useful to examine the necessity of measuring 13 C and 1 H CSA values for individual methyl groups at all. Considering a methyl group in exchange with δ C = 1.5 ppm, δ H = 0.2 ppm, p B = 0.1, k ex = 5000 s −1 and S 2 axis τ c = 50 ns, given the mean and standard deviation of 13 C and 1 H CSA values determined for valine residues above (34.2 ± 5.2 and 0.29 ± 0.28 ppm respectively), the impact of the 13 C and 1 H CSAs on relaxation rates is 10.7 ± 1.6 % and 0.7 ± 0.7 % respectively relative to the exchange terms. Therefore, while it is clearly important not to neglect the impact of the 13 C CSA, the use of an average value contributes an uncertainty only on the order of a single percent. In practical terms, this indicates that for most purposes it is likely sufficient to use average values for the CSAs and measure only S 2 axis τ c , thus simplifying the application of the methods described in this paper.
Quadruple-quantum coherences have been utilised previously in biomolecular NMR for the identification of methyl resonances in 1 H-13 C correlation spectra, via heteronuclear quadruple-quantum correlation (HQQC) and quadruplequantum-filtered HMQC experiments (Diercks et al., 1998;Kessler et al., 1991). Their effective high gyromagnetic ratio (3γ H + γ C ) has also been used to increase the sensitivity of stimulated echo diffusion measurements (Zheng et al., 2009). However, we can envisage and are currently exploring further applications of these coherences to the analysis of dynamics. Given the favourable relaxation properties of these transitions (at least, in perdeuterated molecules), HQQC experiments may prove useful in enhancing the sensitivity of titration measurements to chemical exchange, complementing our earlier work on the two-dimensional lineshape analysis of HSQC, HMQC, HZQC and HDQC experiments . Similarly, we expect that the four-spin analogue of the HMQC, in which a mixture of DQ and QQ coherences are evolved during t 1 , may provide a useful complement to HMQC and HSQC experiments, as well as 13 C-detected SQ, DQ and TQ experiments, in determining the absolute sign of the chemical shift differences to sparsely populated intermediate states Skrynnikov et al., 2002).
Finally, we anticipate further developments will emerge in the analysis of field-dependent HE relaxation rates. We have demonstrated the joint analysis of HE data with CPMG measurements for residues undergoing two-state exchange (Fig. 6) (O'Connell et al., 2009), but applications to more complex multi-state mechanisms should be possible using the same approach, based on analysis of the Liouvillian superoperator. Analysis in combination with other experiment types should also be possible, for example, adiabatic relaxation dispersion (Chao et al., 2019). Further, while in this work we have focussed only on the gradients obtained from regression of relaxation rates with respect to B 2 0 , the intercepts (i.e. at zero field) also contain information on relaxation 788 C. A. Waudby and J. Christodoulou: Quadruple-quantum Hahn echo relaxation rates in the absence of exchange that could be used to constrain analyses of CPMG or similar data (O'Connell et al., 2009;Phan et al., 1996;Wang et al., 2001). Even in systems such as amides restricted to two-spin coherences, ZQ and DQ HE data provide restraints that we have now demonstrated can be applied in conjunction with CPMG measurements. SQ HE measurements, which we have not explored in this work, could also provide additional restraints and remove the ambiguity of multiple solutions illustrated in Fig. 5c and d. Together, these methods may provide new approaches for the analysis of correlated motions in extended systems, for example, across nucleic acid base pairs (Chiarparin et al., 2001), within protein sidechains (Früh et al., 2001) or between adjacent residues in polypeptide backbones (Lundstrom et al., 2005).

NMR spectroscopy
NMR measurements were acquired using Bruker Avance III HD spectrometers running Topspin 3.5pl6, equipped with cryoprobes and operating at 1 H Larmor frequencies of 500, 600, 700, 800 and 950 MHz. Data were acquired at 283 K, which was calibrated between spectrometers using a sample of d4-methanol (Findeisen et al., 2007). 1 H, 13 C correlation spectra were typically acquired with a sweep width of 16 ppm and acquisition time of ca. 100 ms in the direct dimension and a sweep width of 15 ppm (or, if isoleucine resonances are folded, 8 ppm) and acquisition time of ca. 35 ms in the indirect dimension, centred at offsets of 0.4 and 16.7 ppm respectively. The complete set of measurements across all field strengths was acquired in 3 d, or 4.5 d including CPMG experiments (Table S7). Data were processed on NMRbox (Maciejewski et al., 2017) using nmrPipe (Delaglio et al., 1995), viewed using Sparky (Lee et al., 2015 and analysed using FuDA (https://www.ucl.ac.uk/hansen-lab/fuda/, last access: 20 October 2021) and Julia 1.5 (Bezanson et al., 2017) using the NMRTools.jl package (Waudby, 2021). The assignment of FLN5 methyl resonances was obtained from the BMRB (entry 15814) (Hsu et al., 2009).

Measurement of chemical shift anisotropies
Measurements of methyl 13 C CSA and S 2 axis τ c were acquired at 800 MHz using the pulse sequence described in Fig. 3a. A recycle delay of 1.5 s was used, and two scans were acquired, with a 56 ms acquisition time and 15 ppm sweep width for the indirect dimension and relaxation delays of 1.1, 50, 100 and 150 ms. Data were processed with a cosine-squared window function in the direct dimension and linear prediction and 10 Hz exponential line broadening in the indirect dimension. A list of peak positions was prepared from a 1 H, 13 C HMQC experiment and parsed to determine clusters of nonoverlapping resonances based on a 0.05 ppm strip width in the 1 H dimension and a 2.4 ppm strip width in the 13 C dimension. For each peak within a cluster, 2D spectra of 13 C quartets were simulated as recently described (Waudby et al., 2021), incorporating the additional relaxation period, T , shown in Fig. 2a: y (ω H , ω C , T ) = A · L ω H ; ω 0,H , R 2,H · I outer exp (−R ααα T ) L ω C ; ω 0,C + 3π J CH , R ααα +I inner exp −R ααβ T L ω C ; ω 0,C + π J CH , R ααβ +I inner exp −R αββ T L ω C ; ω 0,C − π J CH , R αββ +I outer exp −R βββ T L ω C ; ω 0,C − 3π J CH , R βββ , (8) where L(ω; ω 0 , R) describes a Lorentzian signal with resonance frequency ω 0 and relaxation rate R observed at a frequency ω; I outer = 3 + 3 and I inner = 3 − ; = e −4η HHHH τ cosh 2η HHHC τ ; τ = 1/(2J CH ); axis τ c . Multiplets within each cluster were fitted as a function of the 1 H and 13 C relaxation rates, S 2 axis τ c , σ C and the scalar coupling J CH . S 2 axis τ c were also measured at 500 MHz via the build-up of 1 H TQ magnetisation (Sun et al., 2011), with delays of 2,6,12,20,30,40,50 and 60 ms used for both SQ relaxation and TQ build-up measurements. Peak amplitudes were measured using FuDA and fitted as described (Sun et al., 2011) to determine S 2 axis τ c values. 1 H CSA measurements were acquired at 950 MHz using the pulse sequence described in Fig. 4a. A recycle delay of 1 s was used, and eight scans were recorded for both IP and AP experiments, before looping over relaxation delays and then 13 C chemical shift evolution. Measurements were acquired for relaxation times of 2,5,10,15,20,30,50,75,100 and 150 ms. Peak amplitudes were fitted using FuDA to determine the spin-state-selective relaxation rates R α and R β , from which σ H was then determined according to Eq. (7).

CPMG relaxation dispersion
Multiple quantum CPMG relaxation dispersion experiments (Korzhnev et al., 2004b) were acquired at 800 and 950 MHz. A 2 s relaxation delay was used, with a 40 ms relaxation period comprising 19.2 kHz 13 C pulses applied at 28 CPMG frequencies from 25 Hz to 2 kHz. Measurements were interleaved at the single scan level, alternating between high-and low-power CPMG pulse trains. A 1 H SQ CPMG measurement  was acquired at 800 MHz, with a 40 ms relaxation period comprising 10.4 kHz 1 H pulses applied at 28 CPMG frequencies from 25 Hz to 2 kHz. For both experiments, peak amplitudes were fitted using FuDA in order to determine effective relaxation rates as a function of CPMG frequency.

Data analysis
Measurements of the HE relaxation rates at multiple magnetic fields were fitted by weighted linear regression as a function of B 2 0 to determine the combination of CSA and exchange contributions to the relaxation rate, assuming fast chemical exchange (Table 1, Fig. 5a, b): The CSA contribution, β CSA , was then subtracted, based on the measurements above, to determine the pure exchange contribution, β ex . Errors were propagated using standard methods and used to restrain (ξ H , ξ C ) parameter space, ξ C + nξ H = ± √ β ex , where n = −3, −1, +1 or +3 depending on the multiple quantum coherence being analysed (Fig. 5c, d). χ 2 surfaces (Fig. 5c, d) were computed as a function of ξ H and ξ C as the sum across ZQ, DQ, DQ and QQ coherences of the sum of the squares of residuals to fits of Eq. (9) as a function of R 2,0 , using values of β ex calculated from the specified ξ H and ξ C assuming fast chemical exchange (Table 1).
The joint analysis of HE and CPMG relaxation dispersion measurements was performed to determine values of δ C and δ H for each methyl group observed, together with the parameters k ex and p B reflecting the kinetics and thermodynamics of the exchange process. HE relaxation rates were fitted as a function of the magnetic field strength: where the CSA contribution was calculated as before (Table 1), and the exchange contribution was calculated from the major eigenvalue of the Liouvillian superoperator (Palmer and Koss, 2019). In the case of two-state exchange analysed here, where ω is the (field-dependent) frequency difference of the multiple quantum coherence under consideration (Palmer and Koss, 2019). CPMG data were fitted to numerical simulations of the propagation of magnetisation through CPMG elements, assuming no pulse imperfections, with basis spaces ZQ − , ZQ + , DQ − , DQ + and H x , H y for MQ and 1 H SQ CPMG experiments respectively. While for the SQ CPMG experiment this approach is equivalent to an established analytical solution (Baldwin, 2014), to maintain generality and allow for the analysis of pulse imperfections and offresonance effects in the future, we have preferred to use a numerical approach throughout. Residuals from all HE and CPMG datasets across multiple residues were weighted by their standard error and coupled to a Levenberg-Marquardt algorithm for fitting. Uncertainties in fitted parameters were determined from the curvature of the χ 2 surface. 790 C. A. Waudby and J. Christodoulou: Quadruple-quantum Hahn echo relaxation

Conclusions
Quadruple-quantum and four-spin double-quantum coherences in 13 CH 3 -labelled methyl groups can be easily generated from equilibrium magnetisation and are protected against relaxation by intra-methyl dipolar interactions, leading to unexpectedly low exchange-free transverse relaxation rates in perdeuterated macromolecules. In contrast however, these high-order coherences are highly sensitive to relaxation through chemical exchange processes and particularly to 1 H chemical shift perturbations. The combination of these effects means that these coherences provide near-ideal probes of conformational exchange, which we have investigated in this study using a newly developed suite of pulse sequences. Analysis of the magnetic field dependence of multiple quantum relaxation rates provides a sensitive indicator of chemical exchange, and we have shown that the combined analysis of multiple such measurements can accurately determine relative 1 H and 13 C chemical shift perturbations, up to an overall sign. We have further demonstrated that this analysis may be combined with established CPMG relaxation dispersion measurements, providing increased confidence in chemical shift perturbations together with kinetic and thermodynamic descriptions of the exchange process. Indeed, the combination of field-dependent Hahn echo relaxation measurements with relaxation dispersion measurements that we have demonstrated here for the first time may have more general applicability beyond methyl groups, improving the unique ability of NMR spectroscopy to characterise conformational exchange processes involving sparsely populated intermediate states.
Author contributions. Measurements and analyses were conceived and performed by CAW, and the paper was written by CAW and JC.
Competing interests. The contact author has declared that neither they nor their co-author has any competing interests.

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