the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Rapid measurement of heteronuclear transverse relaxation rates using non-uniformly sampled *R*_{1ρ} accordion experiments

### Sven Wernersson

### Göran Carlström

### Andreas Jakobsson

Multidimensional, heteronuclear NMR relaxation methods
are used extensively to characterize the dynamics of biological
macromolecules. Acquisition of relaxation datasets on proteins typically
requires significant measurement time, often several days. Accordion
spectroscopy offers a powerful means to shorten relaxation rate measurements
by encoding the “relaxation dimension” into the indirect evolution period in
multidimensional experiments. Time savings can also be achieved by
non-uniform sampling (NUS) of multidimensional NMR data, which is used
increasingly to improve spectral resolution or increase sensitivity per unit
time. However, NUS is not commonly implemented in relaxation experiments,
because most reconstruction algorithms are inherently nonlinear, leading to
problems when estimating signal intensities, relaxation rate constants and
their error bounds. We have previously shown how to avoid these shortcomings
by combining accordion spectroscopy with NUS, followed by data
reconstruction using sparse exponential mode analysis, thereby achieving a
dramatic decrease in the total length of longitudinal relaxation
experiments. Here, we present the corresponding transverse relaxation
experiment, taking into account the special considerations required for its
successful implementation in the framework of the accordion-NUS approach.
We attain the highest possible precision in the relaxation rate constants by
optimizing the NUS scheme with respect to the Cramér–Rao lower bound of
the variance of the estimated parameter, given the total number of sampling
points and the spectrum-specific signal characteristics. The resulting
accordion-NUS *R*_{1ρ} relaxation experiment achieves comparable
precision in the parameter estimates compared to conventional CPMG (Carr–Purcell–Meiboom–Gill)
*R*_{2} or spin-lock *R*_{1ρ} experiments while saving an order of
magnitude in experiment time.

*R*

_{1ρ}accordion experiments, Magn. Reson., 2, 571–587, https://doi.org/10.5194/mr-2-571-2021, 2021.

NMR relaxation offers a powerful means to study the dynamics of proteins and other biological macromolecules (Alderson and Kay, 2020; Mittermaier and Kay, 2009; Palmer, 2004). Most commonly, relaxation experiments on proteins are acquired as a series of two-dimensional (2D) spectra, in order to resolve as many resonances as possible, wherein relaxation rates are measured via their effect on the resonance intensities in a “third dimension” obtained by parametrically varying the length of a relaxation time period or the refocusing frequency of an applied radio-frequency field, or both. Thus, relaxation experiments often involve significant time requirements and may take up to several days. An ingenious alternative to these lengthy experiments is offered by the accordion approach originally developed by Bodenhausen and Ernst (Bodenhausen and Ernst, 1981, 1982) to study chemical exchange. In accordion spectroscopy, the third dimension is incremented synchronously with the second (indirect) dimension, with the result that the relaxation decay is encoded into the interferogram of the indirect evolution period. Consequently, the total experiment time is reduced significantly. More recent implementations include the constant-time accordion experiment (Carr et al., 1998; Mandel and Palmer, 1994), from which relaxation rate constants can be extracted using either time-domain analysis of the interferogram (Mandel and Palmer, 1994) or line-shape analysis of the Fourier transformed data (Chen and Tjandra, 2009; Harden and Frueh, 2014; Rabier et al., 2001).

Non-uniform sampling (NUS) of the indirect dimensions of multidimensional NMR
data can greatly shorten the total experiment time
(Gołowicz et al.,
2020; Mobli and Hoch, 2014) and has become commonplace in the last decade.
However, most spectral reconstruction algorithms suffer from nonlinearity of
signal intensities, which limits “plug-and-play” use of NUS in quantitative
experiments and requires careful consideration of both sampling schemes and
data modeling to produce consistent results and reliable error estimates
(East
et al., 2021; Linnet and Teilum, 2016; Mayzel et al., 2017; Stetz and Wand,
2016; Urbańczyk et al., 2017). We recently introduced an approach that
avoids these problems by combining accordion spectroscopy with NUS
(Carlström et al., 2019) and analyzing the
resulting data using DSURE (damped super-resolution estimator), a sparse
reconstruction technique enabling maximum-likelihood estimation of the
time-domain signal parameters from NUS data
(Juhlin et al., 2018; Swärd
et al., 2016). We stress the point that accordion spectroscopy encodes the
desired relaxation rate constants in the interferogram of the
multidimensional dataset; hence, the analysis does not rely on measuring
intensities in multiple NUS datasets. Moreover, maximum likelihood
estimation of model parameters makes it straightforward to derive reliable
error bounds. Our approach leads to accumulated time savings through both
the accordion and NUS methods. Compared to a conventional relaxation
experiment, accordion reduces the experiment time by a factor of $M/\mathrm{2}$, where
*M* is the number of datasets included in the conventional approach, and NUS
reduces the experiment time by a factor of ${N}_{\mathrm{full}}/N$, where *N*_{full} is
the number of data points sampled in the indirect dimension of the
conventional experiment, and *N* is the number of points in the NUS scheme. We
previously demonstrated this approach by measuring longitudinal relaxation
rate constants (*R*_{1}) in proteins with time savings of up to a factor of
20 (Carlström et al., 2019). For example, using
this approach we have successfully measured *R*_{1} on protein samples with
10-fold lower concentration than normally used (Verteramo et
al., 2021).

A number of considerations are of general importance when choosing the
detailed sampling scheme for NUS, including the need to keep the total
number of increments small in order to speed up data acquisition and to sample
short *t*_{1} values to optimize sensitivity and long *t*_{1} values to
optimize spectral resolution
(Hyberts et al.,
2014; Mobli and Hoch, 2014). Various NUS schemes have been developed to
accommodate these different requirements, including the popular Poisson-gap
scheme (Hyberts et al., 2010). However, in the
context of relaxation experiments, the most important aspect is to achieve
high precision in the estimated relaxation rate constants. To this end, we
have previously developed a method to optimize the sampling scheme with
respect to the Cramér–Rao lower bound (CRLB), which yields a lower bound
on the achievable variance of the parameters, given the actual spectrum
characteristics (i.e., the total number of component signals and their
resonance frequencies, line widths, and intensities) and the number of
sampling points (Carlström et
al., 2019; Månsson et al., 2014; Swärd et al., 2018); similar
implementations have followed (Jameson
et al., 2019; Waudby et al., 2021).

Here, we introduce accordion-NUS *R*_{1ρ} pulse sequences that complement
the previously presented *R*_{1} experiment
(Carlström et al., 2019). The *R*_{1ρ}
relaxation experiment can be implemented using either on-resonance or
off-resonance spin-lock fields, making it suitable for measurement of
*R*_{2} relaxation rate constants to characterize fast dynamics, as well as
conformational/chemical exchange processes across a wide range of timescales
(Akke and Palmer, 1996). The present paper addresses several
issues concerning measurement of transverse relaxation rates in accordion
mode combined with NUS. We validate the accordion-NUS method by extensive
comparisons with data acquired using uniformly sampled accordion
experiments, as well as conventional non-accordion experiments of both the
*R*_{1ρ} and CPMG (Carr–Purcell–Meiboom–Gill) *R*_{2} types. In addition, exchange contributions to
the transverse relaxation rates were characterized using CPMG relaxation
dispersion experiments. Our results show that the accordion-NUS *R*_{1ρ}
experiment enables measurement of accurate *R*_{2} relaxation rate constants
with a relative uncertainty of only 2 %–3 % using a sampling density of
50 % in the indirect dimension. Lower sampling densities lead to
progressively reduced precision, with 5 % relative uncertainty being
obtained using less than 20 % sampling density.

## 2.1 Constant-time accordion relaxation methodology

For completeness, here we briefly outline the salient features of the
constant-time accordion method (Mandel and Palmer, 1994). Figure 1
shows the accordion *R*_{1ρ} pulse sequence. The total relaxation delay
is ${T}_{\mathit{\kappa}}=n\cdot \mathrm{4}\cdot \mathit{\tau}$, where *n* is the sampled point
number in *t*_{1}, $\mathit{\tau}=(\mathit{\kappa}\cdot \mathrm{\Delta}{t}_{\mathrm{1}})/\mathrm{4}$, Δ*t*_{1} is the dwell time, and *κ* is the accordion scaling factor.
The total constant-time delay $T={\mathit{\tau}}_{\mathrm{1}}+{\mathit{\tau}}_{\mathrm{2}}+{\mathit{\tau}}_{\mathrm{3}}$, where ${\mathit{\tau}}_{\mathrm{1}}=(T-{t}_{\mathrm{1}})/\mathrm{2}$, ${\mathit{\tau}}_{\mathrm{2}}=T/\mathrm{2}-\mathrm{\Delta}$, ${\mathit{\tau}}_{\mathrm{3}}=\mathrm{\Delta}+{t}_{\mathrm{1}}/\mathrm{2}$, $\mathrm{\Delta}=\mathrm{1}/\left(\mathrm{4}J\right)$, and *J* is the ^{1}H-^{15}N ^{1}*J*-coupling constant
(∼ 92 Hz). The constant-time evolution period leads to reduced
signal-to-noise ratio compared to the non-constant time alternative but is still
favorable due to its superior resolution of closely spaced signals
(Mandel and Palmer, 1994), which is of critical importance in
accordion experiments (see Results section). In the forward accordion experiment,
where *T*_{κ} is incremented together with *t*_{1}, the effective
relaxation rate constant of the interferogram is given by ${R}_{\mathrm{2},\mathrm{fwd}}={R}_{\mathrm{inh}}+\mathit{\kappa}\cdot {R}_{\mathrm{1}\mathit{\rho}}$, where *R*_{inh} describes
line broadening due to static magnetic field inhomogeneity. By comparison,
in a non-constant time experiment, the line width would be given by ${R}_{\mathrm{2}}+{R}_{\mathrm{inh}}+\mathit{\kappa}\cdot {R}_{\mathrm{1}\mathit{\rho}}$, i.e., essentially a factor of 2 greater than in the constant time version (Mandel and Palmer,
1994). In the reverse accordion experiment, where *T*_{κ} is
decremented as *t*_{1} is incremented, the effective relaxation rate constant
of the interferogram is given by ${R}_{\mathrm{2},\mathrm{rev}}={R}_{\mathrm{inh}}-\mathit{\kappa}\cdot {R}_{\mathrm{1}\mathit{\rho}}$. The rotating-frame relaxation rate constant is calculated as
${R}_{\mathrm{1}\mathit{\rho}}=({R}_{\mathrm{2},\mathrm{fwd}}-{R}_{\mathrm{2},\mathrm{rev}})/\left(\mathrm{2}\mathit{\kappa}\right)$. As an alternative, *R*_{1ρ} can also be determined by subtracting from *R*_{2,fwd} the
line width measured in an interferogram from a reference experiment with the
relaxation period set to 0: ${R}_{\mathrm{1}\mathit{\rho}}=({R}_{\mathrm{2},\mathrm{fwd}}-{R}_{\mathrm{ref}})/\mathit{\kappa}$. In estimating the standard error of *R*_{1ρ} by error
propagation, the factor of 2 difference in the numerator of the alternative
approaches offsets the higher signal-to-noise ratio in the reference experiment
as compared to the reverse experiment, with the result that the error bars
are actually lower in the forward–reverse accordion approach.

## 2.2 NMR sample preparation

Uniformly ^{15}N-enriched galectin-3C was expressed and purified as
described previously (Diehl et
al., 2009, 2010; Wallerstein et al., 2021). The NMR sample containing
galectin-3C in complex with the ligand
3^{′}-[4-(3-fluorophenyl)-1H-1,2,3-triazol-1-yl]-3^{′}-deoxy-*β*-D-galactopyranosyl-1-thio-*β*-D-glucopyranoside was prepared as
described (Wallerstein et al., 2021) to yield a final
protein concentration of 0.9 mM in 5 mM HEPES buffer, pH 7.3.

## 2.3 NMR relaxation experiments

All pulse sequences were based on the ^{1}H-^{15}N HSQC format
(Bodenhausen and Ruben, 1980). Conventional and accordion
^{15}N *R*_{1ρ} experiments were acquired with uniform sampling (US) on
an Agilent/Varian VNMRS 600 MHz instrument equipped with a 5 mm HCN
triple-resonance room temperature probe. To allow for comparisons between the
two different ways of estimating *R*_{1ρ} from accordion data, accordion
experiments were performed using a combination of forward and reverse
accordion modes (i.e., incrementing or decrementing the relaxation delay in
step with *t*_{1}) together with a reference experiment excluding the
accordion relaxation period but including the alignment blocks (Fig. 1).
Both the conventional and accordion *R*_{1ρ} experiments were performed
with two different methods for aligning the magnetization along the
effective spin-lock field axis: either hard pulses and delays
(Hansen and Kay, 2007) or adiabatic amplitude/frequency
ramps with a tan/tanh profile of 1.8 ms duration (Mulder et
al., 1998). In the former case, the scaling factor *δ* was set to 1.35
for optimum alignment of spins with offsets within ±*ω*_{SL}
from the spin-lock carrier frequency. The ^{15}N dimension was acquired
with a spectral width of 2006 Hz, which was sampled over 132 increments in the
accordion *R*_{1ρ} experiment utilizing adiabatic alignment, with 128
increments in the accordion experiment using hard-pulse alignment, and with
128 increments in both conventional *R*_{1ρ} experiments. The ^{1}H
dimension was acquired with a spectral width of 8446 Hz, which was sampled over 2028
complex data points, in all experiments. All accordion experiments (forward,
reverse, and reference) were acquired interleaved. Conventional *R*_{1ρ} experiments were acquired by interleaving the relaxation periods of (6,
12, 23.9, 2 × 47.9, 95.7, and 191.4) ms. All *R*_{1ρ} experiments employed a spin-lock field strength of ${\mathit{\omega}}_{\mathrm{SL}}/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{1380}$ Hz. The effective spin-lock field strength in the rotating frame
is given by ${\mathit{\omega}}_{\mathrm{eff}}=({\mathit{\omega}}_{\mathrm{SL}}^{\mathrm{2}}+{\mathrm{\Omega}}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}$, where Ω is the offset from the spin-lock carrier
(Akke and Palmer, 1996; Davis et al., 1994). The transverse
relaxation constant *R*_{2} was extracted from the *R*_{1ρ} relaxation
rates by correcting for off-resonance effects using the relationship
${R}_{\mathrm{1}\mathit{\rho}}={\mathrm{cos}}^{\mathrm{2}}\left(\mathit{\theta}\right){R}_{\mathrm{1}}+{\mathrm{sin}}^{\mathrm{2}}\left(\mathit{\theta}\right){R}_{\mathrm{2}}$, where
*θ* is the tilt angle of the spin-lock field defined by $\mathrm{tan}\left(\mathit{\theta}\right)={\mathit{\omega}}_{\mathrm{SL}}/\mathrm{\Omega}$ and the previously determined *R*_{1} rate
constant.

Conventional *R*_{1} and *R*_{2} CPMG experiments (Farrow
et al., 1995; Skelton et al., 1993) were acquired with uniform sampling on a
Bruker NEO 600 MHz instrument equipped with a 5 mm HPCN QCI cryoprobe,
using spectral widths of 2129 Hz, which was sampled over 128 increments, in the
^{15}N dimension and 9615 Hz, which was sampled over 2048 complex data points, in
the ^{1}H dimension. The *R*_{1} and *R*_{2} relaxation periods were
acquired interleaved with the *t*_{1} increments using delays of (2 × 0.04, 0.08, 0.12, 0.2, 0.4, 2 × 0.6, 0.72, 2 × 0.8, 1.0,
1.2, 1.6, and 2.0) s and (0, 42.2, 2 × 84.5, 126.7, and 169.0) ms,
respectively. The *R*_{1} experiment utilized ^{1}H WALTZ decoupling during
the relaxation period. The *R*_{2} experiment employed CPMG pulse trains
with a fixed refocusing frequency, ${\mathit{\nu}}_{\mathrm{CPMG}}=\mathrm{1}/\left(\mathrm{2}{\mathit{\tau}}_{\mathrm{CPMG}}\right)=\mathrm{625}$ Hz, where *τ*_{CPMG} is the delay between 180^{∘}
pulses in the CPMG train, and a 180^{∘} pulse length of 80 µs.
The *R*_{2} values were not corrected for off-resonance effects. We also
recorded CPMG relaxation dispersion datasets at static magnetic field
strengths of 11.7 and 14.1 T, using Agilent/Varian spectrometers equipped
with 5 mm HCN triple-resonance room temperature probes. The relaxation
dispersion experiment was run as a constant-time version
(Mulder et al., 2001) of the relaxation-compensated CPMG
pulse sequence (Loria et al., 1999), using 18 refocusing
frequencies acquired interleaved with 128 *t*_{1} increments covering
spectral widths of 1550 Hz (2006 Hz) at 11.7 T (14.1 T). The experiments at
11.7 and 14.1 T employed refocusing frequencies *ν*_{CPMG} of
(2 × 0, 2 × 50, 2 × 100, 2 × 150,
2 × 200, 250, 300, 400, 500, 600, 700, 750, and 950) Hz and
(2 × 0, 2 × 50, 2 × 100, 2 × 150,
2 × 200, 2 × 250, 350, 450, 550, 700, 850, and 1000) Hz,
respectively.

## 2.4 Non-uniform sampling schemes

NUS schemes were generated by selecting data points from the uniformly
sampled accordion *R*_{1ρ} dataset. CRLB-optimized NUS schemes were
obtained as described previously (Carlström et
al., 2019). In practice, scheme optimization can be performed using modeled
data constructed by taking known values of *A*_{k}, *ω*_{k}, and
*R*_{k} obtained from DSURE estimation of an HSQC experiment, together with
an estimate of the extra decay caused by relaxation during the accordion
period (Carlström et al., 2019). Single-column
CRLB-optimized (col-opt) (Carlström et
al., 2019; Swärd et al., 2018) and sine-weighted Poisson-gap
(Hyberts et al., 2010) sampling schemes were
implemented for the accordion *R*_{1ρ} dataset with adiabatic ramps,
using in-house MATLAB scripts. Poisson-gap sampling schemes were
generated by randomly varying the argument of the sinusoidal weighting
function between 0 and $\mathit{\pi}/\mathrm{2}$; see (Hyberts et
al., 2010). The best sampling scheme was identified as the one having the
lowest sum of the CRLB calculated over all columns containing peaks. In the
case of the col-opt approach, the selection was made among 97 different
single-column CRLB-optimized schemes corresponding to each slice of the
interferogram containing protein signals, whereas in the case of Poisson-gap
sampling 1000 different schemes were compared. The sampling scheme was
optimized individually for each of the reference, forward, and reverse
accordion experiments. In each case, we generated individual datasets
sampled with *N*=22, 27, 32, 37, 42, 47, 52, 57, 62, 66, 67, 72, 77, 82,
87, 92, 97, 102, 107, 112, 117, 122, or 127 increments in the indirect
dimension. The NUS datasets resulting from the different sampling schemes
were subsequently reconstructed using the DSURE algorithm.

## 2.5 Data reconstruction, processing, and analysis

Non-accordion (i.e., conventional) datasets were processed using NMRPipe
(Delaglio et al., 1995), with forward linear prediction to
double the number of data points, cosine-squared window apodization, and
zero-filling to twice the size rounded to the nearest power of two.
*R*_{1ρ}, *R*_{1}, and *R*_{2} relaxation rate constants were estimated
from the conventional experiments by integrating the peak volumes using PINT
(Ahlner et al., 2013; Niklasson
et al., 2017), followed by fitting mono-exponential decay functions to the
volumes using in-house MATLAB scripts. The fitted *R*_{2} values determined
by CPMG experiments were not adjusted for off-resonance effects
(Korzhnev et al., 2000). Standard errors were estimated using
jackknife resampling (Mosteller and Tukey, 1977) as implemented
in PINT. Standard errors in *R*_{2} derived from *R*_{1ρ} experiments were
estimated by Monte Carlo simulations using 10 000 samples drawn from normal
distributions with widths corresponding to the standard errors of
*R*_{1ρ} and *R*_{1} (Press et al., 1986).

The accordion datasets were processed and analyzed using the DSURE algorithm
(Juhlin et al., 2018) implemented in MATLAB (MathWorks, Inc.). DSURE reconstruction was performed on individual *t*_{1}
interferograms, as described previously
(Carlström et al., 2019). DSURE models
interferograms as sums of exponentially decaying sinusoids:

where *A*_{k}, *ω*_{k}, and *R*_{k} are the complex-valued amplitude,
frequency, and decay rate of the *k*th signal, respectively, *ε*(*t*) represents additive noise, and the sum runs over all *K* signals identified
in a given interferogram. In reconstructing accordion data, the time domain
data from the reverse mode was inverted and complex conjugated before
estimation using DSURE. Standard errors of the estimated parameters were
calculated as the CRLB, which is very close to the RMSE for statistically
efficient estimators like DSURE. Explicit comparison of the RMSE, calculated
from Monte Carlo simulations using 1000 samples, and the CRLB indicates that
the two measures are in excellent agreement and deviate by at most a factor
of 1.6 for the worst case (an interferogram containing three signal maxima)
among our 50 % NUS data.

## 2.6 Statistical analysis

To compare the performance of the different approaches for measuring
transverse relaxation rates, we used four different metrics. The relative
difference and absolute deviation between datasets *x* and *y* are defined for a
given residue *i* as ${\mathrm{\Delta}}_{\mathrm{rel}}=\mathrm{2}({x}_{i}-{y}_{i})/({x}_{i}+{y}_{i})$ and ${\mathrm{\Delta}}_{\mathrm{abs}}=|{x}_{i}-{y}_{i}|$,
respectively. The root-mean-square deviation (RMSD) between two datasets is calculated pairwise over all
residues (*N*_{res}), RMSD $=\left[\right({\sum}_{i}({x}_{i}-{y}_{i}{)}^{\mathrm{2}}/{N}_{\mathrm{res}}){]}^{\mathrm{1}/\mathrm{2}}$. The mean relative uncertainty (MRU) of a given
dataset is the mean, calculated over all residues, of the individual
uncertainty in *x*_{i} (*σ*_{xi}; 1 standard deviation, as
estimated by the DSURE algorithm) divided by *x*_{i}: MRU $=({\sum}_{i}{\mathit{\sigma}}_{xi}/{x}_{i})/{N}_{\mathrm{res}}$.

## 3.1 Pulse sequence design

The accordion-NUS *R*_{1ρ} pulse sequence (Fig. 1) is based on our
previous implementation to measure *R*_{1}
(Carlström et al., 2019), which included minor
modifications of the original constant-time accordion experiment
(Mandel and Palmer, 1994). In designing accordion-NUS versions of
transverse relaxation experiments, it is necessary to consider the interplay
between the minimum length of the relaxation block (A or B in Fig. 1), the
number of sampled *t*_{1} points, and the maximum attainable *t*_{1} value.
*R*_{2} relaxation rate measurement is typically performed in one of two
ways: using either CPMG pulse trains or a continuous spin-lock during the
relaxation period so as to maintain in-phase magnetization and avoid
significant evolution into anti-phase terms (Skelton et al.,
1993), as well as reduce chemical/conformational exchange (*R*_{ex}) and
magnetic-field inhomogeneity (*R*_{inh}) contributions to the effective
transverse relaxation rate constant (*R*_{2,eff}). Furthermore, it is
necessary to suppress the effects of cross-correlated relaxation, which
amounts to introducing additional relaxation delays
(Kay et al., 1992a; Palmer et al., 1992) and to
mitigate the effects of off-resonance effects and pulse imperfections
(Korzhnev et al., 2000), leading to extended phase cycles
(Yip and Zuiderweg, 2004). CPMG-type experiments
for measuring chemical exchange involve extended spin-echo elements to
average the relaxation rates of in-phase and anti-phase coherences
(Loria et al., 1999). All in all, these requirements
typically lead to relatively long relaxation blocks in CPMG-based
experiments. Since the accordion experiment increments (or decrements) the
relaxation period synchronously with the *t*_{1} period, the minimum
increment step for the relaxation period limits the maximum number of
points that can be acquired in the *t*_{1} dimension. In our initial testing
of CPMG-based accordion experiments to measure *R*_{2}, we found that the
maximum achievable length of the *t*_{1} dimension was 64 points, before the
duty cycle and relaxation losses became serious concerns. While the
resulting resolution in *t*_{1} might suffice in certain cases, we opted
instead for increased flexibility and designed the transverse relaxation
experiment based on a spin-lock period. This strategy allows for
significantly shorter increments of the relaxation period, and further
enables facile adaptation to off-resonance *R*_{1ρ} experiments for
conformational exchange measurement. We implemented two types of pulse
sequence elements to align the magnetization along the effective spin-lock
field: adiabatic amplitude/frequency ramps (Mulder et al.,
1998) or an element comprising hard pulses and delays
(Hansen and Kay, 2007). The hard-pulse element is shorter
than the adiabatic ramp (0.25–0.45 ms versus 1.8 ms in the present case)
and in principle reduces relaxation losses, while the adiabatic ramp
achieves superior alignment over a wider range of offsets, making it
suitable for off-resonance *R*_{1ρ} experiments used to characterize
chemical exchange processes.

## 3.2 Comparison of DSURE-modeled accordion *R*_{1ρ} relaxation data and conventional relaxation data

The ^{1}H-^{15}N 2D spectrum resulting from the accordion *R*_{1ρ}
relaxation data reconstructed using DSURE is shown in Fig. A1, together with
representative examples of DSURE models of interferograms. We compared the
performance of the accordion *R*_{1ρ} experiments acquired with the two
different alignment elements (adiabatic vs hard-pulse; see Sect. 2.3) and
also compared the results obtained using the two different combinations of
accordion modes (forward–reverse vs forward–reference; see Sect. 2.1).
To validate the accordion *R*_{1ρ} values determined using DSURE, we
first compared these with rate constants determined from the conventional
*R*_{1ρ} experiment and the *R*_{2} CPMG experiment (Fig. 2). Figure 2
shows the results obtained using the forward–reverse accordion data and
adiabatic alignment, while the corresponding data obtained using hard-pulse
alignment is highly similar and shown in Fig. A2. In general, the results
are in very good agreement, with very few residues showing statistically
significant deviations between experiments (Fig. 2a–d). Comparing the
accordion *R*_{1ρ} values with the conventional data, we obtain an RMSD
of 0.45 s^{−1}, whereas the comparison with the CPMG data yields an RMSD
of 0.59 s^{−1}. The CPMG data were not corrected for off-resonance effects
(Korzhnev et al., 2000), leading to offset-dependent systematic
errors of up to 5 % that might explain the somewhat poorer agreement in
this case. The distributions of relative differences are centered around the
mean values 0.02 and 0.00 s^{−1} and are sharper than normal
distributions (Fig. 2e, f). The means of relative differences should be
compared with the average relative uncertainties of the estimated
*R*_{1ρ} values, which are 0.14 s^{−1} for the conventional data and
0.21 s^{−1} for the accordion data, indicating that the accordion
*R*_{1ρ} experiment yields accurate data of comparable precision compared
to the conventional experiment. The slight tendency towards higher *R*_{2}
values determined from the accordion experiment reflects small but
noticeable differences for a subset of residues (viz. residues 151, 154,
181, 182, and 184). Visual inspection of these peaks indicates that these
differences are likely due to overlap problems, which are exacerbated by the
additional line broadening present in accordion spectra. In principle this
problem could be mitigated by optimizing the accordion scaling factor
*κ* or by acquiring data as a 3D experiment
(Carr
et al., 1998; Chen and Tjandra, 2009). For some residues, notably L219, the
*R*_{2} value from the reference CPMG dataset is considerably higher (Fig. 2b), reflecting the different levels of residual exchange contributions to
the transverse relaxation rate resulting from the different refocusing
frequencies. The L219 peak has an offset of −572 Hz from the ^{15}N
spin-lock carrier, which results in an effective spin-lock field of
${\mathit{\omega}}_{\mathrm{eff}}/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{1494}$ Hz, which is more than a factor of 2 greater than
the effective CPMG refocusing field (625 Hz) in the reference *R*_{2}
dataset. L219 also shows clear signatures of fast conformational exchange in
CPMG relaxation dispersion experiments (Fig. A3).

The two alignment variants yield highly similar results in the context of
the accordion experiment, and the same is true for the two combinations of
accordion modes (Fig. A4). The RMSD between the two alignment variants is
0.19 s^{−1} and the mean relative difference is 0.0 s^{−1}, and the
corresponding numbers for the two combinations of accordion modes are 0.24 and 0.0 s^{−1}, respectively. In the following presentation of
accordion-NUS experiments, we will base all analyses on the results obtained
from the forward–reverse accordion approach using data acquired with
adiabatic alignment.

## 3.3 Comparison of non-uniformly sampled and uniformly sampled accordion *R*_{1ρ} relaxation data

Next, we tested the performance of the accordion *R*_{1ρ} experiment
acquired with NUS. We have previously evaluated the performance of various
NUS schemes for the acquisition of accordion *R*_{1} data, and found that
superior results were obtained for schemes generated by column-wise
optimization directly against the CRLB (denoted col-opt in the following) or
schemes following the Poisson-gap distribution
(Carlström et al., 2019). Therefore, we
restrict our present comparisons to the performance of these two NUS
schemes. Starting from the uniformly sampled accordion *R*_{1ρ} dataset
acquired with adiabatic ramps and the forward–reverse accordion mode (from
here on denoted the US dataset), we generated two times two datasets, where
we used either col-opt or Poisson-gap sampling schemes, both optimized for
the forward accordion experiment alone (set F) or optimized individually
for each of the forward and reverse experiments (set F+R). In a real case
scenario, it is arguably more practical to perform the optimization on model
datasets constructed by taking known values of *A*_{k}, *ω*_{k}, and
*R*_{k} obtained from DSURE estimation of an HSQC experiment, together with
an estimate of the extra decay imparted by the accordion period
(Carlström et al., 2019). Thus, there is no
need to first record a US accordion dataset prior to optimizing the NUS
scheme.

Figure 3 illustrates the performance of the accordion-NUS *R*_{1ρ}
experiment acquired with different NUS schemes (red and blue symbols) and
optimization protocols (left- and right-hand columns). We compared the
*R*_{1ρ} values determined by accordion-NUS with those obtained from the
accordion-US dataset. In general, the performance decreases with decreasing
number of sampled points, as might be expected. The RMSD between the NUS and
US datasets shows a clear trend towards higher values as the number of
sampling points decrease, from less than 0.2 s^{−1} at *N*_{full} to
0.8–1.0 s^{−1} at *N*=22 or 18 % sampling density (Fig. 3a, b).
However, these plots show some degree of scatter, which reflects the random
nature of the NUS schemes, where any given scheme with a lower number of
points might yield lower RMSD than another scheme with higher number of
points. By contrast, the mean relative uncertainty (MRU) in the estimated
*R*_{1ρ} parameter shows an essentially monotonous increase with
decreasing number of points, from 2.2 % at *N*_{full} to 5 % at *N*=22
(Fig. 3c, d). The increasing uncertainty is relatively modest down to about
50 % sampling (*N*=66), where MRU is ca. 2.8 %, but beyond this point
both the RMSD and the MRU start to increase more steeply. These results are
rather similar for the Poisson-gap and col-opt optimized schemes, with a
small advantage for col-opt schemes, especially in the case of the precision
of the estimated parameters (Fig. 3c, d). However, greater improvements in
precision are expected for relaxation rate constants of signals in
interferograms whose sampling schemes have been individually optimized with
respect to the CRLB, as shown previously
(Carlström et al., 2019). Altogether, these
results indicate that our present implementation of the accordion-NUS
approach to measure *R*_{1ρ} achieves equally good precision of the
estimated relaxation rate constants as did our previously presented
accordion-NUS *R*_{1} experiment (Carlström et
al., 2019). Furthermore, the sampling schemes optimized separately for each
of the forward and reverse experiments (F+R) show a modest advantage in
performance over F for low *N*, which might be expected (compare Fig. 3a with
Fig. 3b and Fig. 3c with Fig. 3d).

## 3.4 Spectral characteristics affecting accuracy and precision of accordion-NUS *R*_{1ρ} relaxation rate constants

Next, we investigated how various spectral characteristics affect the
accuracy and precision of the estimated relaxation rate constant. We
calculated the absolute deviation (Δ_{abs}) between the US
estimate and the 50 % NUS estimate (*N*=66), as well as the relative
uncertainty, for each residue and plotted the results against signal
intensity, resonance frequency offset from the spin-lock carrier, and the
number of estimated signals present in the interferogram of the col-opt and
F+R optimized data (Fig. 4). There is no obvious relationship between
Δ_{abs} and signal intensity, although larger values of Δ_{abs} (> 0.4 s^{−1}) are not observed for the most intense
signals (Fig. 4a). However, there is a trend towards lower relative
uncertainty with higher signal intensity (Fig. 4d), where a value of 1.5 %
is observed for the strongest signals and 6 %–8 % at the other extreme. The
results further reveal that the number of signals in the interferogram has
an effect on both Δ_{abs} and the relative uncertainty, with a
trend toward slightly larger errors as the number of signals increases
(Fig. 4c, f); the relative uncertainty varies from 1.5 % for single
signals to 6 %–8 % for the worst cases among interferograms containing seven
signals. Reassuringly, the mean relative uncertainty increases only slightly
from 2 % for single peaks to 3.2 % for seven peaks. Thus, there is no
dramatic decrease in performance even at the highest number of signals. This
effect of the number of signals also explains the apparent higher Δ_{abs} and higher relative uncertainty for residues with offsets around 0
and 500 Hz, because this region of the spectrum is the most crowded (Fig. 4b, e). Furthermore, this result mirrors the observations of deviations
between the accordion and conventional data discussed above in connection
with Fig. 2.

We have described a non-uniformly sampled accordion *R*_{1ρ} experiment
that complements the previously presented accordion-NUS *R*_{1} experiment
(Carlström et al., 2019). The present
accordion-NUS *R*_{1ρ} experiment allows for accurate and precise
measurement of the transverse relaxation rate constant *R*_{2} while
reducing sampling of the indirect dimension by at least 50 %. The
combination of accordion relaxation rate measurements with NUS achieves a time
saving of an order of magnitude compared to conventional experiments, in
keeping with previous results presented for the corresponding accordion-NUS
*R*_{1} experiment (Carlström et al., 2019). In
addition to on-resonance *R*_{2} measurements, demonstrated herein, we
anticipate that this experiment will be useful for on- and off-resonance
*R*_{1ρ} experiments to characterize chemical exchange processes. The
accordion-NUS approach has broad applications in heteronuclear relaxation
studies; with suitable modifications, the pulse sequence reported here for
backbone ^{15}N spins should be applicable to many other sites, e.g.,
^{13}C spins.

Backbone chemical shift assignments have been deposited at the Biological Magnetic Resonance Bank with the accession code 50283 (https://doi.org/10.13018/BMR50283, Wallerstein et al., 2020). The NMR pulse sequences, relaxation datasets, and extracted relaxation rate constants are available at Mendeley Data together with MATLAB scripts implementing the DSURE algorithm (https://doi.org/10.17632/zyryxrgkc3.1; Wernersson et al., 2021).

SW, GC, and MA conceived the overall experimental approach and designed pulse sequences. AJ contributed the DSURE algorithm and designed the analysis protocol together with GC. SW and GC performed experiments. SW, GC, and MA analyzed data. All authors contributed to writing the article.

Mikael Akke is an associate editor of *Magnetic Resonance*.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Geoffrey Bodenhausen Festschrift”. It is not associated with a conference.

This research has been supported by the Swedish Research Council (grant no. 2018-4995) and the Swedish SRA eSSENCE (grant no. 2020 6:2).

This paper was edited by Paul Schanda and reviewed by Dominique Frueh and two anonymous referees.

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