Deuteron Rotating Frame Relaxation for the Detection of Slow Motions in Rotating Solids

Abstract

We demonstrate experimental and computational approaches for measuring ²H rotating frame NMR relaxation for solid samples under magic angle spinning (MAS) conditions. The relaxation properties of the deuterium spin-1 system are dominated by the reorientation of the anisotropic quadrupolar tensors, with the effective quadrupolar coupling constant around 55 kHz for methyl groups. The technique is demonstrated using the model compound dimethyl-sulfone at MAS rates of 10 and 60 kHz as well as for an amyloid fibril sample comprising an amyloid-β (1–40) protein with a selective methyl group labeled in the disordered domain of the fibrils, at an MAS rate of 8 kHz. For both systems, the motional parameters fall well within the ranges determined by other techniques, thus validating its feasibility. Experimental and computational factors such as i) the probe’s radio frequency inhomogeneity profiles, ii) rotary resonances at conditions for which the spin-lock field strength matches the half- or full-integer of the MAS rate, iii) the choice of MAS rates and spin-lock field strengths, and iv) simulations that account for the interconversion of multiple coherences for the spin-1 system under MAS and deviations from the analytical Redfield treatment are thoroughly considered.

Graphical Abstract

Introduction

Internal motions in proteins on μs-ms time scale range are essential for multiple biological functions.[1–3] Following solution NMR approaches, relaxation dispersion approaches for proteins in the solid state are in the active stage of development.[2, 4–15] Most of the currently developed techniques cover spin ½ nuclei such as ¹⁵N, ¹³C, and ¹H.[11, 15–17] Rotating frame relaxation (R_1ρ) experiments in solids have the capacity to probe both the fluctuations of the isotropic chemical shift component and the fluctuations of the anisotropic tensor due to conformational changes.[15] Under magic angle spinning (MAS) conditions, the latter has been shown to modulate the width of rotational resonances.[13, 14]

Deuterium relaxation is a sensitive tool for measuring the dynamics in a variety of systems.[18–21] Deuterium is a spin-1 nucleus and its relaxation is governed primarily by the quadrupole interaction.[20] Further, as the typical quadrupolar interaction is of the order of 150–200 kHz, their magnitude is considerably larger than dipolar and chemical shift anisotropy interactions.[20–22] On the one hand, this larger magnitude allows us to focus on the quadrupolar interaction as a major source of relaxation, while neglecting minor dipolar contributions. On the other hand, it requires modifications of established experimental and computational approaches developed for spin ½ nuclei governed by smaller magnitudes of the interactions. ¹⁵N and ¹³C R_1ρ studies in the solid state often focus on changes in the isotropic chemical shifts upon conformational changes, similar to solution NMR approaches. In our case, focus is placed on the anisotropic quadrupolar tensor reorientations due to stochastic processes. We previously developed the R_1ρ experiment for ²H nuclei under static conditions [23] and applied it to model compounds and amyloid fibril samples.[24–27] In many applications, MAS conditions are desirable for site-specific resolution[2]. For the case of deuterium, site-specificity in multiply labeled samples will require polarization transfer steps to nuclei that offer a larger chemical shift dispersion, such as ¹³C.[28] Coupling to MAS also offers an advantage of increased signal to noise ratio, as an alternative to static multiple echo acquisition detection. [29]

The goal of this work is to demonstrate experimental and computational approaches for conducting deuterium R_1ρ measurements under MAS conditions. We employ the same systems used to develop static measurements to allow the direct comparison of the results (Figure 1). Dimethyl-sulfone (DMS-D₆) is a well-studied model compound undergoing 2-site flips around the C₂ symmetry axis (Figure 1A), which fall into the approximate range of 5·10² to 4·10⁴ s⁻¹ at elevated temperatures between about 45°C and 85°C.[9, 30–34] As usual with methyl groups, the additional mode of fast methyl jumps at the picosecond time scale narrows the quadrupolar tensor threefold, yielding an effective quadrupolar coupling constant of 55 kHz.[20] The second system is a test of feasibility in proteins, especially in amorphous powder samples. This comprises hydrated amyloid-β fibrils in the threefold symmetric polymorph formed by the Aβ_1–40 peptide [35, 36] (Figure 1B). Aβ fibrils, hallmarks of Alzheimer’s disease, exist in multiple morphological forms, of which the 3-fold symmetric polymorph is the more toxic.[36] The disordered N-terminal domain (residues 1–16) has been found to be mobile by multiple studies and has been implicated in the aggregation control of Aβ.[37–41] The fibrils employed in this work are labeled with deuterium at a single site of the alanine-CD₃ group, located at position 2 of the flexible disordered N-terminal domain. This domain undergoes a complex conformational exchange process, which we have previously characterized by static ²H NMR approaches and ²H CEST under MAS measurements.[24–27] Here, we assess the motions using the on-resonance R_1ρ relaxation dispersion technique and compare the results with those of previous approaches to cross-validate and assess the limits of applicability of the technique. In addition to the experimental data on DMS and the fibrils, we perform simulations involving a simple two-site exchange model (Figure 1C) to understand the behavior of the spin-1 system under spin-lock conditions in the presence of coherent time evolution due to MAS.

Figure 1.

A) Structure of DMS indicating the 180° flipping motion around the C₂ axis, corresponding to θ=106°. An additional motional mode of 3-site methyl jumps is a part of the model but is not shown explicitly. B) Slow time scale motional modes of the A2-CD₃ methyl group of the N-terminal domain of hydrated Aβ_1–40 fibrils.[27] The disordered N-terminal domain (residues 1–16, curved line) transiently interacts with the structured C-terminal domain (blue rectangle). In the free state, the N-terminal domain undergoes large-scale diffusive motion, as represented by the gray sphere, which is absent in the bound state. The position of the A2 residue is shown as an orange dot. C) Theoretical 2-site exchange model for two axially symmetric tensors with quadrupolar coupling constants denoted by C_q,1 and C_q,2 and the jump angle denoted by θ.

Experimental

Materials

DMS-D₆ was purchased from Cambridge Isotope Laboratories, Inc. (MA) and packed as a powder into rotors. The Aβ_1–40 fibrils labeled at the A2-CD₃ site were prepared as previously described in the 3-fold symmetric polymorph.[24, 27] The monomeric sequence of the Aβ_1–40 peptide was D{A-CD₃)EFRHDSGYEVHHQKLVFFAEDVGSNKGAIIGLMVGGVV. The lyophilized powder was hydrated to 200% by weight with deuterium-depleted water using direct pipetting and equilibrating at room temperature for 5 days. The hydrated sample was then packed into a 2.5 mm rotor.

NMR Spectroscopy

The R_1ρ measurements at MAS rates of 8 kHz (fibrils) and 10 kHz (DMS) were taken using a 17.6 T Bruker Avance I spectrometer equipped with a Bruker 2.5 mm HXY probe (College of William and Mary). The measurements at a 60 kHz MAS rate for DMS were taken using a 14.1 T Bruker NEO console spectrometer equipped with a Bruker 1.3 mm HXY probe (NHFML). High-power 90° radio frequency (RF) pulses corresponded to 2 μs. The number of scans acquired was between 12 and 32 at a 10 kHz MAS rate for DMS, 4 and 16 at a 60 kHz MAS rate for DMS, and 1792 and 4096 at an 8 kHz MAS rate for the fibril sample. For spectral processing, the exponential line-broadening function was employed with 100 Hz for DMS at the 10 kHz MAS rate, 10 Hz for DMS at the 60 kHz MAS rate, and 50 Hz for the fibril sample.

Modeling

The evolution under MAS was modeled using the numerical integration of the Liouville–von Neumann equation (Eq. (SI-5)) with the quadrupolar frequency evolution given by Eq. (SI-4). The time dependence of the Liouvillian matrix was modeled by 20 points equally spaced over a single MAS rotation period. An internal Matlab function for matrix exponentiation was used to approximate the evolution operator over the corresponding time span for each of the 20 points. The matrix of the evolution operator for the full MAS period rotation was then calculated by multiplying by the evolution matrices for the individual time intervals. No approximation involving separation of the coherent and incoherent evolution was made. The sufficiency of 20 time steps was established by confirming the stability of the results after increasing the number of time steps to 100. The evolution matrix for a time period during the spin-lock was obtained by first calculating the evolution matrix for the integer number of rotor periods within the selected time of evolution. The evolution matrix for the integer part was calculated by taking an appropriate power of the evolution matrix for a single rotor period. The evolution matrix during the remaining fractional time of the rotor period is calculated separately and is included as an additional matrix multiplication. The FID evolution was calculated in a likewise manner but with a simplification owing to the possibility of the separation of the transverse coherence evolution for the linear combinations of individual coherences ${\widehat{S}}_x$, ${\widehat{S}}_y$, ${\widehat{J}}_x$, ${\widehat{J}}_y$ from the rest of the basis listed in Eq. (SI-1) in the absence of the RF field. The time interval for the individual step for generating FID was taken as the smaller value between 1/20^th of the MAS period and the dwell time. The above procedure presents the calculations for an individual crystallite of the solid powder. The powder averaging was performed according to the ZCW scheme.[42, 43] We used 1596 crystallite orientations for the DMS calculations and 233 orientations for the fibril simulations.

The simulations of the jump matrices necessary to model the motional mechanisms of Figure 1 closely followed the procedures outlined in prior work for the cases of the static measurements. In particular, the simulation of the spherical diffusion for the dominant free state of Aβ fibrils (depicted in Figure 1B) coupled to the conformational exchange was performed in the following manner. The exchange matrix K_ij of Eq. (SI-5) combines components responsible for the isotropic diffusion and the conformational exchange between the free diffuse ion state and the bound state, according to the following scheme:

The block of the exchange matrix K_ij responsible for the diffusion, is represented by nearest neighbor jumps with sites of equal populations distributed (almost) uniformly on the surface of the sphere and with identical exchange rates between the sites, K_ij = k_D. We used the program DistMesh,[44] which creates the uniform distribution of sites by assuming a linear repulsive force and solving for the equilibrium. The inclusion of N_D = 192 sites is sufficient to adequately represent the isotropic diffusion process and leads to the following range of jump angles in the crystal-fixed frame between the sites: 13.7–18.7°. Due to the spin 1 symmetry properties, only the second-order spatial spherical harmonics give rise to the spectral densities. Thus, the second-order eigenfunction of the Smoluchowski diffusion equation encodes the spatial reorientation and the corresponding eigenvalue is related to the diffusion coefficient D as Dl(l + 1) = 6D. As expected, the simulated eignevalues of K_ij appear in groups of 2l + 1, corresponding to the eigenvectors with the angular momentum l of the continuous limit of the diffusion equation. The five eigenvalues λ₂ corresponding to l = 2 are then averaged to obtain the numerical value of D. This establishes the correspondence between D and k_D as k_D = 6D/〈λ₂ 〉. To include the exchange process, one additional site with a fixed angular position corresponding to the bound state was added to yield a total of N_D +1 sites. The exchange was modeled by jumps between every site describing the spherical diffusion and the bound-state site with an arbitrary fixed angular position with the rate k_ex/N_D. Modeled in this fashion, the relative tensor orientation in the free and bound states is averaged over the uniform spherical distribution. Simulated decays utilized time points identical to those used in the experiment. The simulations were performed on an x86_64 Intel Xenon Silver dual core CPU node. Each node had 16 Intel Xenon Silver dual core CPUs and 16 GB of memory.

The RF inhomogeneity profiles for both probes (depicted in Figure 2B) can be approximated using discrete profiles around each selected RF strength value.[45] The chosen discretization coarseness depends on a number of factors such as proximity to the rotary resonances requiring a finer grid and the overall time of each simulation, which limits the number of points for computationally intensive motional models. Once the grid is determined, partially relaxed spectra are calculated for each RF value of the grid and the weighted average is constructed with the appropriate weights of the distribution profile to obtain the final simulated spectrum at each <ω_RF > value. For DMS, a grid of 0.1 kHz was used, yielding 12–50 discretization points. For the computationally intense simulations of the fibrils, for each simulated average <ω_RF > value, a five-point grid was used. These points corresponded to the RF values at 0.25, 0.5, 0.75, 1, and 1.25 multiples of the average frequency with the corresponding relative weights of 0.065, 0.097, 0.103, 0.236, and 0.499 for the 2.5 mm probe profile.

Figure 2.

A) Pulse sequence for the ²H solid-state R_1ρ measurements under MAS. The heat compensation block (max-T) is followed by the inter-scan delay d₁ and the preparation 90° pulse, followed by a variable spin-lock delay (T), directly proceeded by the FID collection. The phase cycle corresponds to ϕ₀=x, ϕ₁= −y,y, and receiver= −y,y. The duration of d₁ varied between 1 and 3.5 s to maintain a constant temperature. B) Inhomogeneity profiles of the two probes as a function of ω_RF /<ω_RF > obtained for the weighted averages <ω_RF > in the range of 2.8–40 kHz for the 1.3 mm Bruker HXY probe using DMS-D₆ (at 14.1 T) and 5–31 kHz for the 2.5 mm Bruker HXY probe using hexamethylbenzene-D₁₈ (at 17.6 T).[45]

Results and Discussion

The R_1ρ experiment

The R_1ρ measurements were performed according to the scheme in Figure 2A, which utilizes a heat compensation block that ensures the same amount of heating for all the relaxation delays. Transverse magnetization was created by a high-power 90° pulse, followed by the spin-lock evolution and detection periods. The phase cycle was analogous to that of Vega.[46] The inter-scan delay d₁ was set between 1 and 3.5 s to minimize the RF-induced heating for all the spin-lock field strengths and maintain a constant temperature among them. As DMS longitudinal relaxation times (T₁) are very sensitive to temperatures in the 40–85°C range, we used them as an internal calibration standard to obtain the actual temperature in the sample.[23] T1 values in the presence of a “dummy” spin-lock heat compensation block were also used to assess the inter-scan delay necessary to keep the effective T₁, and thus temperature, the same across all the spin-lock field strengths. The same optimized parameters were applied to the fibril sample, assuming as a first approximation that the RF induced-heating would be relatively similar in the two samples.

As will be elaborated below, the choices of MAS rates and spin-lock powers depend on multiple factors such as the magnitude of the quadrupolar tensor of the system and tolerance to the RF-induced heating of the probe and sample. For each spin-lock field strength, magnetization decay curves need to be sampled with a sufficient number of points to capture the details of the decay. The RF inhomogeneity factor can be rather significant and must be taken into account for quantitative interpretation of the results. We have previously assessed the inhomogeneity of the profiles of the two probes (Bruker 2.5 mm HXY and Bruker 1.3 mm HXY probes) using ²H nutation experiments on either DMS or hexamethyl-benzene-D₁₈. As the latter has a narrower tensor [47], it is more suitable for calibrations at slow MAS conditions. The profiles (Figure 2B) are approximately proportional to the average RF frequency (Figure S1) and thus can be plotted as a function of ω_RF /<ω_RF >, in which <ω_RF > is the weighted average over the distribution. The profiles’ shape and width, with relatively long low-field tails and a sharper high-field edge, are consistent with studies of ¹³C nuclei. [48] We report <ω_RF > as the effective RF field strength in all the data below.

Qualitative insights from the 2-site jump simulations

Rotating frame relaxation under MAS for spin ½ nuclei involved in heteronuclear and homonuclear dipolar interactions, CSA interactions, and isotropic chemical shift changes has been considered in multiple recent works and reviews.[2, 4–15] Most works adopt the Redfield approximation when applicable for ease of interpretation; however, the limitations of the Redfield treatment has been noted in many cases,[2] especially for conditions in the vicinity of rotary resonances at which the spin-lock RF frequency ω_RF matches the integer or half-integer multiples of the MAS frequency ω_MAS. For the proton–proton homonuclear dipolar case, the Redfield approximation can significantly overestimate relaxation rates, even far from the rotary resonance conditions for large-angle fluctuations and/or strong collision limit as well as in the intermediate motional regime.[12] A large magnitude of quadrupolar tensors compared with dipolar interactions shrink the limit of the validity of the Redfield theory even further for a ²H spin-1 nucleus. In the static case, the limit of its validity is given by the condition that the conformational exchange rate constant k_ex is much higher than the typical quadrupolar frequency ω_Q for different crystallite orientations.[23] For DMS methyl groups, the typical value of ω_Q is about 1.3 ·10⁵ rad/s (calculated from ω_Q =2π 20 kHz), while the large-angle flip rate constant is in the 0.2–1 ·10⁵ s⁻¹ range, thus the Redfield limit does not hold.[23] In the case of MAS, the situation becomes even more complex due to the coherent averaging of the quadrupolar frequency (considered in more detail below).

We thus perform all the subsequent simulations using the full Liouvillian treatment and below we demonstrate the main insight into relaxation behavior according to the 2-site jump model with the parameters of the quadrupolar tensors typical for methyl groups. A summary of the theory is presented in the SI1 and is similar to approaches in previous works.[45] [23]

The coherent dephasing of the transverse magnetization for spin-1 nuclei under MAS in the presence of the spin-lock field occurs at the rotary resonance conditions for which ${\omega }_{RF}=\frac{n}{2}{\omega }_{MAS}$, where n is an integer (Figure 3). They arise due to the loss of lock for the transverse coherence ${\widehat{S}}_x$ at these matching conditions. The situation is analogous to that described for homonuclear dipolar interactions,[12, 14, 17] with the half-integer rotary resonance referred to as the HORROR condition.[49]

Figure 3.

Mean value of the transverse deuteron ${\widehat{S}}_x$ coherence over a 5 ms spin-lock period in the absence of motions vs ${\omega }_{RF}/{\omega }_{MAS}$ for the four values of ω_MAS shown directly in the panel. An axially symmetric tensor with C_q=55 kHz was used with powder averaging over 1596 crystallite orientations and a static magnetic field strength of 17.6 T.

When the value of C_q is greater than the MAS rate, one can expect apparent shifts in these resonance conditions from the above equations (see Figure 3 and Figure S2, where the effect is elaborated for individual crystallite orientations). The apparent shift occurs due to presence of coherence oscillations of ω_Q at both the ω_MAS and the 2ω_MAS frequencies:

$${\omega }_Q(t)=\frac{3\pi }{4}{C}_q\left(\sqrt{2}\sin 2\beta \sin \left({\omega }_{MAS}t+\alpha \right)-{\sin }^2\beta \cos \left(2{\omega }_{MAS}t+2\alpha \right)\right)$$ (1)

where β and α are the polar and azimuthal angles with respect to the axis of rotation, respectively. For high MAS frequencies with respect to the C_q value, the second- and higher-order effects beyond n=2 are greatly diminished (Figure 3, black and red lines).

Motions lead to the incoherent contribution of ${\widehat{S}}_x$ evolution by stochastic changes in the quadrupolar tensor parameters by changing either the magnitude of the quadrupolar coupling constant in different conformational states or the orientation of the tensor, both of which affect the value of ω_Q and thus lead to relaxation. As discussed above, the conventional and simplest treatment of relaxation can be performed according to the Redfield formalism. The Hamiltonian corresponding to the quadrupolar interaction of spin-1 nucleus has the same form as the homonuclear dipolar Hamiltonian for two spins ½, due to the same symmetry. The analytical expression for rotating frame relaxation for the homonuclear case in the presence of MAS has been derived previously[12, 17, 50]. This expression can be applied to the quadrupolar spin-1 case, at least for an axially symmetric quadrupolar tensor:

$$\begin{gathered} R_{1\rho }=\frac{3{\pi }^2}{2}{C}_q^2 \\ \left[\frac{1}{2}\left(J_0\left({\omega }_{MAS}-2{\omega }_{RF}\right)+J_0\left({\omega }_{MAS}+2{\omega }_{RF}\right)\right)+\frac{1}{4}\left(J_0\left(2{\omega }_{MAS}-2{\omega }_{RF}\right)+J_0\left(2{\omega }_{MAS}+2{\omega }_{RF}\right)\right)+\frac{5}{2}{J}_1\left({\omega }_L\right)+J_2\left(2{\omega }_L\right)\right] \end{gathered}$$ (2)

where J₀, J₁, and J₂ are the spectral densities, defined in detail for deuteron nuclei in [20], and ω_L is the Larmor frequency. For static solid powders, the spectral densities depend on the orientation of crystallites in addition to the details of molecular motions. However, in the presence of MAS, a simplified approach is often employed in which orientation dependence is averaged, leading to the same functional form of the spectral densities.[2, 50].

For a 2-site exchange model (Figure 1C) with axially symmetric tensors, these orientation-averaged spectral densities can be calculated from the static spectral densities given in [20, 51]

$$J_0(\omega )=J_1(\omega )=J_2(\omega )=\frac{3}{5}{p}_1{p}_2{\sin }^2\theta \frac{kex}{ke{x}^2+{\omega }^2},$$

where p₁ and p₂ are the populations of the two sites, θ is the jump angle, and k_ex = k₁ + k₋₁ is the sum of the forward and backward jump rate constants. Figure 4 compares the relaxation rates calculated using either the Redfield approach or the Liouvillian treatment and employing the two-site exchange model with the quadrupolar coupling constant fixed at both exchange sites at 55 kHz, typical of the effective tensor of methyl groups. We choose two conditions away from the rotary resonances, with ω_RF/2π = 40 kHz and two MAS rates of 60 and 200 kHz as well as one condition approaching the half-integer rotary resonance conditions with ω_RF/2π = 36 kHz and ω_MAS/2π = 60 kHz. Two cases of jump angles, 10° and 90°, are considered, with p₁ = p₂ = 0.5. As expected, the deviations between the two approaches are larger for the larger value of the jump angle as well as in the vicinity of the rotary resonances.[12] There is an additional source of deviations when k_ex ≈ C_q, originating from the anisotropic loss of intensity due to the intermediate motional regime. Consequently, an expression based on the uniform averaging of the relaxation rates is inadequate in this case. When k_ex ≫C_q, the deviations between the two treatments are the least pronounced. Further, when ω_MAS/2π≫ C_q and for small-angle fluctuations the deviations become negligible for high values of spin-lock fields away from the rotary resonances, in agreement with homonuclear proton–proton rotating frame relaxation.[12] For deuterated methyl groups with a tensor magnitude of the order of 55 kHz, the condition ω_MAS/2π≫ C_q implies the necessity of spinning at a rate of about 200 kHz.

Figure 4.

Ratio of the ²H NMR relaxation rates R_1ρ calculated by applying the Liouvillian treatment to the Redfield treatment (Eq. (2)) as a function of k_ex for the values of ω_RF/2π and ω_MAS/2π shown directly in the panels. The 2-site exchange model in Figure 1C was used with C_q,1 = C_q,2= 55 kHz, p₁= p₂= 0.5, and two values of the jump angle θ. The dotted line indicates the condition C_q= k_ex.

We now focus on the details of the relaxation rates behavior for large-angle jumps in different motional regimes and employ the Liouvillian approach in the subsequent simulations. Figure 5A displays the simulated relaxation behavior for θ = 90° and a range of k_ex rates and several values of ω_MAS at a fixed value of the spin-lock field of 24 kHz. At conditions close to ω_MAS = ω_RF, 2ω_RF elevated relaxation rates are observed with broadened maxima, pointing to efficient relaxation, as expected qualitatively based on Eq. (2); here, the frequencies of the two major rotary resonance conditions, ${\omega }_{RF}=\frac{1}{2}{\omega }_{MAS}$ and ω_RF = ω_MAS, act as analogs of the conditions of J(0). At higher values of ω_MAS, the relaxation rates fall and the maxima shift to higher values of k_ex. When C_q ≫ ω_MAS (see the black line in Figure 5A with ω_MAS/2π = 7 kHz), the qualitative picture of rotary resonance-driven maxima in the relaxation rates appears to break down.

Figure 5.

A) ²H relaxation rates R_1ρ as a function of k_ex for ω_RF/2π of 24 kHz using the 2-site exchange model (Figure 1C) with equal populations, C_q,1 = C_q,2= 55 kHz, and θ = 90°. The MAS rates and corresponding ω_MAS/ω_RF ratios are shown directly in the panels. The static magnetic field strength was taken as 17.6 T. The rates were obtained using the mono-exponential fits to the magnetization decay curves starting with the dead time of 100 μs. B) Examples of normalized magnetization decay curves M(t) on the semilog scale for the maxima positions of part A). The solid lines represent mono-exponential fits starting with 100 μs dead time. Normalization was performed to the t=0 value. Note the residual coherent oscillations in the vicinity of the rotary resonance conditions. The errors in the relaxation rate in Panel B were obtained using the inverse covariance matrix method.

Although in general the decay of magnetization during the spin-lock period is not expected to behave in a single-exponential fashion, the single-exponential approximation has been widely applied because of its ease of use. Examples of the magnetization decay curves at the maxima of relaxation rates (of Figure 5A) are shown in Figures 5B and 5C for the sum of all the side bands obtained for spin-lock times of 0 to 2 ms. The coherent oscillations are superimposed onto the decay due to motions and are especially apparent in the vicinity of the rotary resonances as expected. Due to these oscillations, we arbitrarily choose the “dead time” of the fits of 100 μs, as this likely corresponds to the minimal time at which the spin-lock can be established in most experimental conditions. In the theoretical simulations of Figure 5 the sampling density is very high and the resulting relaxation times are practically independent of the choice of the dead time. The concept of dead time has been also used for fitting ¹⁵N R_1ρ relaxation data in proteins.[14] The duration is dependent on the system and extent of motions. The semilog plot emphasizes the deviations from the single-exponential approximation, which are apparent when the magnetization decays to 5% or below its initial value, justifying the use of the approximation. For low k_ex rates, a decay of magnetization below 5% may require long spin-lock times. In the experimental section, we provide a practical approach for dealing with incomplete decays when sampling time is limited by probe heating considerations.

Sensitivity to the exchange rate is shown in another way when we plot the simulated relaxation times $T_{1\rho }=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$R_{1\rho }$}\right.$ either as a function of MAS rate at a fixed value of the spin-locking field of 30 kHz (Figure 6) or as a function of the spin-lock field strength at a fixed value of MAS rate of 60 kHz. As pointed out for spin ½ nuclei R_1ρ relaxation,[5] the sensitivity of the k_ex rate constant is similar in the two approaches; however, from the experimental standpoint, it is easier to take the measurements in a controlled fashion by varying the RF field strength rather than the MAS rate.[5] Sensitivity to the value of k_ex is observed across the range shown. However, in the fast exchange limit, the dependence of the relaxation rate on either the MAS rate or the spin-lock field strength disappears in accordance with the extreme narrowing limit. The T_1ρ lines for the different k_ex rate constants can intersect, and the most direct way to resolve the ambiguity in the motional parameters is to measure the relaxation rates at either different spin-lock field strengths or different MAS conditions. It is preferable to perform these dispersion experiments outside the region of the half and full rotary resonance conditions, where dispersion behavior is non-monotonous, appears somewhat diminished, and can be partially masked by experimental errors. Additionally, as usual for dynamics studies, probing the temperature dependence of the motions is useful not only for resolving the ambiguities in the k_ex values but also for refining the motional mechanisms and determining the activation energies of the motional modes. While all the major rotary resonances (n=1 or 2) are visible for all the k_ex rates, higher-order rotary resonances for n>2 become less pronounced and are not visible for k_ex > ω_MAS due to the interference of motions with the coherent evolution. The rotary resonances remain visible for small values of k_ex (red and blue lines, Figure 6A) at low MAS conditions. Figure S3A displays an analog of simulations shown in Figure 6B for small-angle fluctuations of 10°, with the qualitative features of the dispersion profiles remaining similar to those of the large-angle fluctuations. Figure S3B focuses on the analog of Figure 5A for the 10° amplitude fluctuations.

Figure 6.

Simulated ²H T_1ρ profiles (semilog scale) as a function of A) ω_MAS/2π with the fixed value of ω_RF/2π =30 kHz and B) ω_RF/2π with a fixed value of ω_MAS/2π = 60 kHz for the 2-site exchange model (Figure 1C) with equal populations, equal axially symmetric quadrupolar tensors in the two sites with C_q,1 = C_q,2 = 55 kHz, and the jump angle θ = 90°. The different values of the conformational exchange rate constants k_ex are indicated directly in the panel legend. The relaxation times were obtained using a single-exponential function M(t) = Ae^−t/T1ρ with 52 spin-lock times in the 200 μs to 50 ms range, except for the limit of very slow relaxation for which additional large delays up to 5 s were included. The sampling of ω_MAS/2π in part A and ω_RF/2π in part B was performed with a step of 2 kHz, except in the vicinity of rotary resonances where the 1 kHz step was used. Powder averaging was performed with 1596 crystallite orientations and MAS evolution included at least 20 steps per rotor period. The static magnetic field strength was taken as 17.6 T.

Results for DMS-D₆ validate the experimental and computational approaches

We now describe the experimental results for DMS-D₆. The situation is conceptually similar to the theoretical case of the 2-site exchange with large-angle jumps described above, with the important distinction of the addition of fast 3-site methyl jumps (Figure 1A). In the theoretical case, the C_q values of 55 kHz corresponded to the fast averaging of methyl jumps, which has a non-averaged C_q value of 166 kHz for DMS.[31] However, in principle, the fast jump mode itself can contribute to the experimental relaxation rates. At low temperatures at which the large-angle fluctuations are too slow to have an appreciable effect on the relaxation, we observe dispersion profiles dominated by coherent oscillations, most pronounced in the vicinity of the rotary resonance conditions. Figure S4 presents the experimental data at 277 K and a 25 kHz MAS rate. Away from the resonance conditions, the relaxation times obtained using the single-exponential approximation are consistent with those of the simulations performed using the corresponding 3-site jump rate constant (k₃) at these low temperatures. As elaborated above, in all our results, the x-axis is given by 〈ω_RF〉, representing the average over the distributions of the spin-lock amplitudes (Figure 2). Since the distributions themselves are highly asymmetric with a large left skew, the modal values of the spin-lock amplitudes are appreciably larger than the average values (1.5 times for the 1.3 mm probe and 1.25 times for the 2.5 mm probe). Thus, the rotary resonances shift to the apparent values 〈ω_RF〉, which are effectively lower than those given by the $\frac{n}{2}{\omega }_{MAS}$ condition. The R_1ρ relaxation data for DMS were collected at two MAS rates: the relatively low 10 kHz MAS rate that allows us to use spin-lock fields with magnitudes away from the rotary resonance conditions and the higher MAS rate of 60 kHz that is more suitable for samples for which site-specific resolution is desired. At the 10 kHz MAS rate, the main data were collected at a high temperature of 85°C at which the flip motions are pronounced; however, at the 60 kHz MAS rate, we performed the experiments at 55°C due to probe limitations. The static magnetic field was different in the two cases, however it has a minor effects on the rotating frame ²H relaxation in the presence of slow motions which occur on the time scales much slower than the inverse of the Larmor frequency.

The partially relaxed spectra examples (i.e., spectra obtained as a result of rotating frame relaxation at a fixed value of the relaxation delay) obtained using the pulse sequence in Figure 2A are shown in Figure 7A. For the 10 kHz MAS rate, the major intensity concentrates at the first side bands at ±10 kHz, and we thus used these side bands in the main analysis. At the 60 kHz MAS rate, most of the intensity is refocused into the central band. The residual coherent oscillations for small spin-lock fields (for the 10 kHz MAS rate) or those near rotary resonance conditions (for the 60 kHz MAS rate) are evident in the magnetization curves in Figure 6B. Thus, the decay curves (Figure 7B and Figure S5) were sampled using 100 to 200 μs “dead times”. The experimentally accessible length of the sampling of magnetization decay curves can be limited by probe duty cycle considerations and the sensitivity of the sample to RF-induced heating. For the 1.3 mm probe, we did not sample beyond 8 ms. With this upper limit on the sampling, the crystallite orientations with slower relaxation rates create an apparent baseline in some of the decay patterns. Figure S6 shows an example of a histogram of simulated R_1ρ relaxation rates for individual crystallites of the powder pattern. For consistency, all the magnetization decay curves were thus fitted with the function $M\text{(}t\text{)}=A{e}^{-t/T_{1\rho }}+B$. When limiting experimental sampling times, it is important to perform the simulations with the same spin-lock durations as used in the experiment in order not to artificially skew the simulated distributions compared with the experimental sampling. This approach thus emphasizes the faster relaxing components of the overall magnetization decay. One of alternative approaches would be to focus on the average relaxation rate over the distribution, given by $-{\frac{1}{M(0)}\frac{dM}{dt}|}_{t=0}\approx \frac{1}{T_{1\rho }}\frac{A}{A+B}$. The introduction of the baseline parameter is similar to the bi-exponential fit sometimes used in the fitting of ¹⁵N MAS R_1ρ decay curves[8]. The focus on the T_1ρ parameter is, therefore, analogous to the selection of the fast relaxing component of the bi-exponential fitting function, especially for the situation when the slow-relaxing component is not well defined within the experimental precision. The fast and slow relaxing components were clearly visible in the static ²H R_1ρ measurements on DMS, for which the sampling time stretched to 50 ms.[23] It is in principle possible to obtain the selection of the model parameters by direct comparison of the experimental and simulated decay curves. However, the model parameters obtained this way will still depend on the exact choice of the relaxation delays, but the qualitative information encoded in the relaxation times will be lost. On the other hand, once the fits are obtained, it could be useful to cross-check the correspondence between the experimental decay curves and those obtained using the best-fit parameters (Figure S7). The detailed analysis of the parametrization of the decay curves is presented in Figure S8.

Figure 7.

A) Example of the partially relaxed spectra of DMS-D₆ for the R_1ρ measurements at the 60 kHz MAS rate, 14.1 T, and 55°C and the 10 kHz MAS rate, 17.6 T and 85°C. The spin-lock strength and relaxation delay values are indicated directly in the panels. B) Examples of the magnetization decay curves for the 60 kHz MAS rate central band intensities and 10 kHz MAS rate first side-band intensities. The lines represent the fits to the mono-exponential function of the form $M(t)=A{e}^{-t/T_{1\rho }}+B$. The insert for the 10 kHz MAS condition enlarges the initial magnetization decay curve time points to demonstrate the presence of the remaining coherent oscillations.

The resulting relaxation dispersion profiles (Figure 8) demonstrate several distinct features. For the 10 kHz MAS rate, most of the employed spin-lock fields are higher than the first integer rotary resonance condition and a strong extent of the dispersion is observed at the high temperature of 85°C. For the 60 kHz MAS rate and 55°C, the dispersion is only observed for low values of ω_RF, as most of the ω_RF range falls between the first half-integer and full-integer resonance conditions. The RF inhomogeneity inclusion affects visual features of the dispersion profiles in comparison to the theoretical graphs. We next assess the sensitivity of the profiles to the k_flip rate constant and obtain the best-fitting values according to the model in Figure 1A. The values of the k₃ rate constants for each temperature were obtained from the fits to the experimental longitudinal relaxation rates determined by the inversion recovery method at each temperature. The longitudinal relaxation times were 48 ms at 85°C and the 10 kHz MAS rate and 26 ms at 55°C and the 60 kHz MAS rate. For both MAS conditions, the profiles are sensitive to the value of k_flip, as shown in detail in Figure S9 for the different side bands at the 10 kHz MAS rate and in Figure S10 for the central band at the 60 kHz MAS rate. The best-fitting parameters (black bands in Figure 8) are k_flip = 3.5 ·10⁴ s⁻¹ for the 10 kHz MAS rate at 85°C and k_flip = 1.6 ·10³ s⁻¹ for the 60 kHz MAS rate at 55°C. These values fall well within the ranges of parameters found in previous assessments of motions in DMS.[9, 23, 30–34] It appears that improving commercially available probes to minimize RF inhomogeneity factors would benefit these types of measurements.

Figure 8.

T_1ρ times vs ω_RF/2π in DSM-D₆ at the 10 kHz MAS rate, 17.6 T, and 85°C for the first side bands at a ±10 kHz spectral frequency (top panel) and at the 60 kHz MAS rate, 14.1 T, and 55°C for the central band (bottom panel). The black bands represent the best-fit simulations according to the model in Figure 1A with k₃ = 5.7 ·10⁹ s⁻¹ and k_flip = 3.5 ·10⁴ s⁻¹ for the 10 kHz MAS rate/85°C data and k₃ = 3.2 ·10⁹ s⁻¹ and k_flip = 1.6 ·10³ s⁻¹ for the 60 kHz MAS rate/55°C data. The effect of RF inhomogeneity with the inhomogeneity profiles in Figure 2 was included in the simulations as described in the text and the reported ω_RF/2π values stand for the weighted averages over the distributions of the RF values. The error limits for both the experimental and the simulated values are obtained from the fits of the magnetization decay curves to the mono-exponential function of the form $M(t)=A{e}^{-t/T_{1\rho }}+B$.

The temperature dependence of the dynamics probes the details of the motional mechanisms and cross-validates the methodologies. Thus, we also assessed k_flip at different temperatures for DMS at a 10 kHz MAS rate. For this purpose, we applied a spin-lock field strength range well away from the first full major resonance and strong enough to ensure the sufficient locking of all the crystallite orientations, namely, 24 to 44 kHz. In the temperature range between 85°C and 51°C, the flipping motion is well pronounced and is expected to follow Arrhenius behavior with an activation energy range of 63 to 92 kJ/mol.[9, 23, 30–34] This range was determined by a variety of solid-state NMR techniques performed over different temperature ranges depending on the sensitivity of each technique. We have performed ²H R_1ρ measurements for five temperatures (Figure 9A) and fitted the relaxation times of the first ±10 kHz side bands to the model in Figure 1A. The k₃ rate constants were fixed at values consistent with the T₁ measurements at each temperature (Table S1). The resulting Arrhenius plot (Figure 9B) yields an activation energy E_a of 94 ± 4 kJ/mol, which is at the high end of the previously determined ranges. The Arrhenius prefactor lnA is 42.1± 1.5, again falling in the upper range of the previously determined values, which is expected, as E_a and A have a positive correlation.

Figure 9.

A) T_1ρ vs ω_RF/2π in DSM-D₆ obtained from the mono-exponential fits of the first side-band intensities, collected at the 10 kHz MAS rate and 17.6 T, with the temperatures indicated directly in the panel. The error bars smaller than the sizes of the symbols are not shown. B) Semilog plot of the fitted values of the k_flip rates for the T_1ρ times shown in the top panel vs 1000/T. The resulting activation energy value is E_a= 94 ± 4 kJ/mol.

Aβ fibrils: An example of application to complex protein systems

We now assess the feasibility of the measurements for protein samples using the hydrated Aβ_1–40 fibrils labeled with a single methyl group at the A2 residue (Figure 1B). Based on our previous measurements, the simplest model describing the dynamics of the disordered N-terminal domain is a two-site conformational exchange model with unequal populations.[27] The major state (referred to as the “free” state) has about a 92% population for the A2 residue and undergoes extensive large-scale rearrangements approximated as isotropic diffusion. These large-amplitude motions lead to a dramatic narrowing of the tensor, with an effective quadrupolar coupling constant of around 3 kHz.[27] By contrast, due to intra- and inter-molecular contacts, this diffusive motion is entirely quenched in the minor bound state, which retains the effective tensor typical of alanine methyl groups of 55 kHz. Because of the spectral dominance of the narrowed form (Figure 10A) as well as the preference to employ low RF fields to avoid RF-induced heating and dehydration, an MAS rate of 8 kHz was chosen. With this MAS rate, a spin-lock field strength ranging between 10 and 27 kHz is sufficient for locking the major transverse component and this avoids the first half- and full-integer rotary resonance conditions. The magnetization decay curves (Figure 10B, shown for the central band intensities) were sampled with nine points between 0.2 and 3.9 ms. The resulting relaxation dispersion profiles (Figure 10C) indicate a moderate degree of dispersion. The fitting parameters of the motional model are the population of the bound state (p_bound), which was fixed at 8% based on the results of prior ²H static line shape measurements,[27] the diffusion coefficient D for the large-scale rearrangement of the free state, and the conformational exchange rate constant k_ex between the two states. In principle, the population can be introduced as a free parameter; however, the three-variable parameter grid coupled with a long computational time is difficult to optimize. On our server, for each subset (p_bound, D, k_ex), the computational time for the R_1ρ simulations under the full Liouvillian approach is 10 hours without the inclusion of inhomogeneity modeling and 72 hours with the inclusion of inhomogeneity modeling. Thus, a true full grid search was not feasible even for the (D, k_ex) space with a fixed value of p_bound. Alternative computational approaches can involve trajectory-based random walk methods.[52]

Figure 10.

Results of the R_1ρ measurements for the Aβ_1–40 fibrils in the hydrated state and in the threefold symmetric polymorph, labeled at the A2-CD₃ site. The data were collected at an 8 kHz MAS rate, 17.6 T, and 37°C. A) An example of partially relaxed spectra at ω_RF/2π = 22 kHz and relaxation delays indicated directly in the panel. The line-broadening function of 50 Hz was applied. B) Corresponding magnetization decay curve of the central band intensities. The line represents the fit to the mono-exponential function form $M(t)=A{e}^{-t/T_{1\rho }}+B$. C)T_1ρ times vs ω_RF /2π obtained from the mono-exponential fits of the central band. Uncertainties were obtained using the covariance matrix method. The line represents the best-fit according to the model in Figure 1B with D = 1.9 ·10⁶ rad² ·s⁻¹, k_ex = 1 ·10⁵ s⁻¹, and p_bound = 0.08.

We varied the values of D and k_ex around the wide subset of previously defined ranges for this system in an iterative manner to determine the best-fit set: the value of D was previously assessed to be in the range of 1.4 to 2.2·10⁶ rad² ·s⁻¹ and k_ex dependence was analyzed to be between 1 ·10⁴ and 2 ·10⁵ s⁻¹. Both the values of the relaxation times and the extent/pattern of the dispersion profiles are clearly sensitive to the fitting parameters within these ranges (Figure S11). The value of D has a major effect on the magnitude of T_1ρ, while k_ex variations have a more pronounced effect on the dispersion pattern. The best-fit set of simulations to the experimental T_1ρ curve is D = 1.9 ·10⁶ rad² ·s⁻¹ and k_ex = 1 ·10⁵ s⁻¹, which is right within the range found by the ²H MAS CEST measurements, for which the best-fitting parameters were D = 1.9 ·10⁶ rad² ·s⁻¹ and k_ex = 0.2–1 ·10⁵ s⁻¹.[45] The combination of static ²H R_1ρ, quadrupolar CPMG relaxation, and static CEST data indicated the presence of multiple free states of the N-terminal domain in exchange with a single bound state.[24, 26, 27]. The ensemble is characterized by clusters of D values around 1–3·10⁶ rad²/s and 1·10⁸ rad²/s, with corresponding k_ex values clustered at 0.1–1·10⁵ and 1–3·10⁶ s⁻¹. Here, we constraint ourselves to the simple two-state model, as the current data and modeling grid are not sufficient to introduce a higher level of complexity. Nonetheless, the best-fitting parameters in this simplified model fall within one of these main clusters. The fact that we can match the data with the same set of parameters as previously determined from other techniques provide further confirmations of the model. This is especially important for the case of the fibrils, which clearly have a very complex dynamics in the disordered N-terminal domain.

A question remains whether the ²H R_1ρ rates would be sensitive to motional parameters at higher spinning speeds, which would be desirable if we probed the dynamics of the fibrils in a multiply labeled sample. While we have not performed the experiments under these conditions, we have simulated the relaxation rates at a 60 kHz MAS rate for the best-fitting parameters (D = 1.9 ·10⁶ rad² ·s⁻¹, k_ex = 1 ·10⁵ s⁻¹) and a k_ex range of 0.5 to 2 ·10⁵ s⁻¹ (Figure S12). We conclude that the measurements are expected to retain measurable sensitivity to the motional parameters and that the overall extent of dispersion is not reduced.

We expect the technique to be useful for a variety of biological systems from globular protein to amorphous aggregates. Identifying the best ways of motional modeling is the key for extracting meaningful information from these measurements. In cases when the validity of the Redfield approximation is expected, the so-called “model-free” approach may be appropriate and simplify the data analysis.[53]

Conclusion

Our results demonstrated the feasibility of the ²H R_1ρ relaxation technique under MAS conditions for studies of dynamics, particularly for protein methyl groups. Special care must be taken to select a range of MAS rates and spin-lock conditions that retain the sensitivity of the quadrupolar tensor reorientations to motions. When selecting these conditions, one has to be aware of the rotary resonances at ${\omega }_{RF}=\frac{n}{2}{\omega }_{MAS}$ that are often significantly broadened due to the RF inhomogeneity factors of the coil. The RF inhomogeneity needs to be assessed and included explicitly in the simulations if its contribution is significant. The durations of spin-lock times are often limited by probe safety limits and/or sample tolerance to RF-induced heating. When the sampling window is limited, it is important to conduct simulations with spin-lock times matching those of the experiment.

In our modeling approach, we provided examples of explicit motional modeling, which allows for confirmation of motional mechanisms and its parameters. In particular, for DMS a relatively simple 2-site rotameric exchange model and known ranges of parameters across the 85–51°C temperature range were confirmed. For the A2-CD₃ methyl group in the disordered domain of amyloid-β fibrils, the results confirmed the large-scale fast diffusive motions of the N-terminal domain coupled with a slow conformational exchange between the free and bound states of the domain at k_ex = 1 ·10⁵ s⁻¹. Large magnitudes of quadrupolar tensors in general preclude analytical Redfield treatment for MAS rates less than or comparable to the quadrupolar coupling constant, especially for large-angle fluctuations that are typical of rotameric interconversions. Instead, a numerical integration of the Liouville–von Newman equation is suggested for the computation. These computations can be time consuming and require optimization of computational strategies and computer hardware.

The rotating frame deuterium relaxation data are well suited for the assessment and confirmation of the motional mechanisms, particularly for the large-angle fluctuations of protein side-chains. It is also complementary to other solid-state NMR techniques such as laboratory frame relaxation focused on faster time scales, line shape analysis, and Q-CEST measurements. The latter in principle has the capacity to probe even slower time scales in the slow motional regime with respect to the effective field.[22, 24, 45] Possible extensions of the approach include a) off-resonance measurement to increase the range of effective fields available and thus access a wider range of time scales without adding RF-induced heating; and b) coupling ²H R_1ρ measurements to ¹³C polarization transfer approaches [28, 54] to enable site-specific resolution in a two-dimensional fashion for multiply labeled protein samples. These extensions may require the use of adiabatic full passage pulses for the optimization of the cross-polarization conditions as well as adiabatic ramp pulses for the alignment of the magnetization along the effective field for the off-resonance irradiation.[55, 56] In terms of applications, the technique is not limited to methyl groups but can encompass a range of side chain reorientation dynamics such as phenylalanine, histidine, and proline rings as well as polar side chain sites.

• 2H rotating frame NMR relaxation measurements for dynamics studies in rotating solids

• Experimental systems are DMSO-D6 and amyloid-beta fibrils

• Theoretical/computational considerations within 2-site jump models are presented