Solid-state near-rotary-resonance measurements of the spin–lattice relaxation rate in the rotating frame (R ) is a powerful 1ρ NMR technique for studying molecular dynamics in the microsecond time scale. The small difference between the spin- lock (SL) and magic-angle-spinning (MAS) frequencies allows sampling very slow motions, at the same time it brings up some methodological challenges. In this work, several issues affecting correct measurements and analysis of N R data 1ρ are considered in detail. Among them are signal amplitude as a function of the difference between SL and MAS frequencies, “dead time” in the initial part of the relaxation decay caused by transient spin-dynamic oscillations, measurements under HORROR condition and proper treatment of the multi-exponential relaxation decays. The multiple N R measurements at 1ρ different SL fields and temperatures have been conducted in 1D mode (i.e. without site-specific resolution) for a set of four different microcrystalline protein samples (GB1, SH3, MPD-ubiquitin and cubic-PEG-ubiquitin) to study the overall protein rocking in a crystal. While the amplitude of this motion varies very significantly, its correlation time for all four sample is practically the same, 30–50 μs. The amplitude of the rocking motion correlates with the packing density of a protein crystal. It has been suggested that the rocking motion is not diffusive but likely a jump-like dynamic process. Keywords NMR relaxation · Protein crystals · Molecular dynamics · Rotary resonance · Magic angle spinning · GB1 · SH3 domain · Ubiquitin Introduction of fast magic angle spinning (MAS) and/or partial deutera- 1 15 tion of proteins enable obtaining well resolved H– N cor- Spin–lattice relaxation in the rotating frame is a powerful relation spectra and, as a consequence, capability to meas- NMR method used to obtain quantitative information on ure site-specific relaxation rates with a negligible spin–spin molecular dynamics in the microsecond timescale. During contribution to the incoherent relaxation (Krushelnitsky the recent decade, measurements of the N rotating-frame et al. 2010; Zinkevich et al. 2013; Good et al. 2014, 2017; relaxation rate (R = 1/T ) have been widely applied in the Lamley et al. 2015a, b; Ma et al. 2015; Smith et al. 2016; 1ρ 1ρ solid-state NMR studies of protein dynamics. Combination Kurauskas et al. 2016, 2017; Saurel et al. 2017; Lakomek et al. 2017; Gauto et al. 2017). The ability to vary the spin- lock field strength from < 1 to 40–50 kHz enables covering Electronic supplementary material The online version of this a wide frequency range of dynamics, and the simultaneous article (https ://doi.org/10.1007/s1085 8-018-0191-4) contains analysis of both the chemical-exchange contribution to R 1ρ supplementary material, which is available to authorized users. and the dipole/CSA relaxation mechanisms (Ma et al. 2014; * Alexey Krushelnitsky Lamley et al. 2015b) provides abundant data that character- email@example.com ize protein dynamics in much detail. The concept of excited states of proteins (Mulder et al. 2001; Korzhnev and Kay Martin-Luther-Universität Halle-Wittenberg, Halle, 2008), that is highly relevant to mechanisms of protein bio- Germany logical function, stresses the importance of R experiments 1ρ Institut de Biologie Structurale (IBS), Grenoble Cedex 9, since they can effectively monitor the exchange between the France ground and excited states on the microsecond timescale. Technische Universität München, Garching, Germany Present Address: Wren Therapeutics Ltd., Cambridge, UK Vol.:(0123456789) 1 3 54 Journal of Biomolecular NMR (2018) 71:53–67 The R rate constant in static samples is proportional to the Liouville-von-Neumann equation, representing a finite 1ρ the spectral density at the spin-lock frequency. When meas- number of (rotational) states accessed by a dynamic process ured under MAS, R due to the heteronuclear dipolar and by a vector of density matrices and associated Hamiltonians. 1ρ CSA relaxation mechanisms is proportional to the following The molecular dynamics is implemented in separate mix- combination of the spectral-density functions: ing steps among the density matrices on the basis of a pre- defined exchange matrix, which alternate with the quantum- R ∝ J( − 2 )+ 2J( − ) SL MAS SL MAS mechanical evolution using single-site propagators. This (1) approach was demonstrated to provide quantitative results +2J( + )+ J( + 2 ) SL MAS SL MAS once sufficiently small time steps are used, and avoids the where ω /2π and ω /2π are the MAS and spin-lock fre- exceedingly large dimension of the full Liouvillian. In the MAS SL quencies, respectively (Haeberlen and Waugh 1969; Kur- current work, the simulations always include the simplest banov et al. 2011). The dependence on the spectral density motional model: jumps between two equivalent sites with at the frequency difference between spin-lock and MAS different orientations of the N–H vector. frequencies makes it theoretically possible to measure very slow motions. If ω ~ ω or 2ω ~ ω then, according Samples MAS SL MAS SL to Eq. (1), one can measure even the zero-frequency limit of the spectral density function J(0). In practice, however, In this work four different microcrystalline protein sam- it is not feasible to benefit from this advantage since the ples were studied. These were the α-spectrin SH3 domain, conditions ω = ω and 2ω = ω correspond to the GB1, and ubiquitin in two crystal polymorphs. The SH3 MAS SL MAS SL rotary resonance (RR) phenomenon (Oas et al. 1988; Levitt domain was purified and crystallized in Bernd Reif’s labo- et al. 1988), at which the dipolar and CSA interactions are ratory (FMP Berlin) according to a protocol described in recoupled, and the spins thus evolve under these interactions. (Chevelkov et al. 2006). The crystalline GB1 sample was Spin evolution at these conditions, therefore, is governed by purchased from Giotto Biotech. The sample was prepared these coherence mechanisms, rather than molecular motions. according to the protocol described in (Franks et al. 2005). Rotary resonance is not considered in the Redfield theory The only modification of the protocol was a usage of a mix- and thus, it cannot be taken into account in the data analysis ture of 80% deuterated and 20% protonated solvents (both in a quantitative manner. Although the exact J(0) measure- organic solvents and water buffer) instead of 100% proto- ment is not possible, still sampling the slow dynamics with nated ones, since in the latter case the lines were too wide at the help of R experiments at small difference between ω 20 kHz MAS. Two different polymorphs of ubiquitin were 1ρ MAS and ω is feasible (Zinkevich et al. 2013; Ma et al. 2014; used, corresponding to the previously described “MPD-ub” SL Kurauskas et al. 2017). and “cubic-PEG-ub” forms (Ma et al. 2015). All proteins 15 2 The aim of this work is to delineate the practical chal- were uniformly N, H-labeled by recombinant protein pro- lenges and limitations of MAS R measurements and data duction in D O-based minimal medium, and the exchange- 1ρ 2 analyses in the vicinity of the RR conditions and to dem- able hydrogens were exchanged to H by placing the protein onstrate the capability of these experiments to study slow in a mixture of H O and D O-based buffer before crystal- 2 2 molecular dynamics. We use numerical computer simula- lization. The extent of the proton back-exchange was 20% tions and N MAS R measurements of four different pro- for SH3 domain and GB1 and 30% for ubiquitin. This high 1ρ tein samples. Among the problems considered in this study deuteration level enables high-resolution proton-detected 1 15 are interfering spin-dynamics effects, the homonuclear H– N experiments (Chevelkov et al. 2006) and negligible rotary resonance (HORROR) condition, and the shape of spin–spin contribution to R (Krushelnitsky et al. 2010, 1ρ the relaxation decay. We finally demonstrate the potential of 2014). R experiments by studying the rocking motion of proteins 1ρ in solid environment. NMR experiments The SH3 domain and the GB1 protein were measured at Materials and methods Halle University, while the two ubiquitin samples were measured at the IBS Grenoble. In both cases measurements Numerical simulations were performed on a Bruker 600 MHz spectrometer. 3.2 and 1.3 mm MAS probes were used with MAS rates 20 and Spin dynamics simulations were conducted using a home- 40 kHz in Halle and Grenoble, respectively. R decays were 1ρ written code described earlier (Saalwächter and Fischbach measured using routine double-CP (i.e., CP from protons to 2002). This program is based on a finite-step integration of nitrogens and then back to protons) pulse sequences with 1 3 Journal of Biomolecular NMR (2018) 71:53–67 55 proton detection of the signal described in (Krushelnitsky Results and discussion et al. 2010; Ma et al. 2014). In the middle of the N spin- lock pulse, a proton π-pulse was applied in order to exclude Experimental methodology the dipole-CSA cross-correlation effect (Kurauskas et al. 2016). In this work one-dimensional proton spectra at dif- Initial oscillations ferent durations of the N spin-lock pulse were measured and then the relaxation decay was obtained from the inte- Figures 2 and 3 present numerically simulated and experi- grated signal of the entire amide band of the H spectrum. mentally measured R relaxation decays at different spin- 1ρ One-dimensional spectra of the four proteins are shown in lock e fi lds, respectively. The initial part of the decays always Fig. 1. While this 1D approach excludes the possibility to contains coherent transient oscillations. The physical nature characterize protein dynamics site-specifically, it enables of such oscillations in R experiments has been described 1ρ conducting a large number of experiments (relaxation meas- a long time ago (VanderHart and Garroway 1979). The urements with comparably many relaxation delays at differ - exact shape of these oscillations depends on the MAS rate, ent spin-lock field strengths and temperatures, see below) strength of the dipolar and CSA interactions and spin lock within reasonable time limits. The relaxation experiments frequency. To the best of our knowledge, detailed theoretical were conducted at temperatures 13, 21.5 and 29 °C for the treatments of this process are not available, and we discuss SH3 domain and GB1, and 3, 15 and 27 °C for the two ubiq- the transient oscillations only in a phenomenological man- uitin samples. The temperature calibration was performed ner. We note, however, that these oscillations contain no using the MAS rotor with ~ 5 μl of methanol, the calibration information about molecular dynamics and they do not affect accuracy was ± 1.5 °C. the shape of the relaxation decay at longer delays, which are the focus of this work. We also note that the physical origin of these oscillations and spin–spin (coherent) contribution to the relaxation rate is not the same (VanderHart and Gar- roway 1979). In many previous experimental studies, the presence of these oscillations has been overlooked, because they can be observed only by measuring multiple relaxa- tion delays with a small time step over the initial part of the decay, which is hardly possible with a typical number (5–15) of relaxation delays in 2D R experiments. The oscillations SH3 1ρ themselves are not informative for dynamic investigations, but including a relaxation delay that falls within these oscil- lations can lead to artefacts when attempting to extract relax- ation rate constants. In fact, these oscillations play the role of a “dead time” in the R experiments; thus before running 1ρ the relaxation measurements it is advisable to determine its GB1 duration. The time span of the initial oscillations is rather dif- ferent in the simulations and experiment: in the case of experimental measurements the oscillations decay much ubiquitin faster, typically within ca. 2–3 ms, while in simulations they extend to ca. 10 ms (compare Figs. 2, 3). This obser- MPD vation has important experimental consequences and is in practice beneficial, because oscillations as long as those observed in simulations would hamper precise measure- ubiquitin ments of the relaxation rates. We ascribe the shorter life PEG time of the oscillations in experiments to B -field inhomo- geneity. The frequency of these oscillations (at least the 12 11 10 9876 5 fundamental component; they are not purely harmonic) depends on the difference between spin-lock and MAS H chem. shift, ppm frequencies. Thus, B -inhomogeneity causes a superposi- tion of frequencies which in turn renders the “dead time” Fig. 1 Proton-detected 1D spectra at room temperature of the four shorter. This is demonstrated in Fig. 4 by comparing the samples used in this study. For plotting the relaxation decays, in all cases the integral under the entire spectrum was used 1 3 56 Journal of Biomolecular NMR (2018) 71:53–67 1.0 1.0 16 kHz 20 kHz 0.9 0.9 17 kHz 21 kHz 0.8 0.8 18 kHz 22 kHz 0.7 0.7 19 kHz 23 kHz 0.6 0.6 20 kHz 24 kHz 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 0.00 0.01 0.02 0.03 0.04 0.00 0.01 0.02 0.03 0.04 Spin-lock duration / s 15 15 1 Fig. 2 Numerically simulated N R -relaxation decays for N– H 20 kHz MAS and different on-resonance spin-lock fields at different 1ρ spin pair undergoing two-site jumps with an angular amplitude 20° RF amplitude as indicated by different colours. In all cases the first −1 15 1 and an exchange rate of 500 s ( N– H dipolar coupling 11.5 kHz, point of the decays at t = 0 has an amplitude of unity CSA interaction not included). The simulations were performed at relaxation decays simulated at a single (ideally homo- Initial decay amplitude at different spin-lock fields geneous) and 10%-inhomogeneous B -field. This level and the HORROR condition of inhomogeneity is a realistic estimation for solid-state MAS probes (Haller and Schanda 2013; Tosner et al. 2017; Figures 2 and 3 demonstrate that the amplitude of the Nagashima et al. 2017). This comparison demonstrates “useful” incoherent relaxation component of the decays that a moderate level of B inhomogeneity is actually decreases upon approaching the RR condition, while the advantageous. Since the field inhomogeneity can be dif- “dead time” remains practically the same. Therefore, the RR ferent for different probes (coils) and samples, the “dead condition seems to mainly affect the signal-to-noise ratio of time” in R -experiments can also be somewhat different, the relevant later part of the decay but does not lead to seri- 1ρ however, we believe it is always in the range of few ms. ous systematic variation in the measured R . This should 1ρ Of note, the B -inhomogeneity renders the exact theoreti- now be demonstrated in more detail. cal description of the initial transient oscillations rather Figure 5a presents the experimental and simulation difficult for practical purposes, since the B -profile over dependencies of the amplitude of the decay as a function of a sample is not known in general case and is difficult to the spin-lock field. Experimentally only one point at spin- measure. lock duration 3 ms was measured, being the point where It is worth mentioning that the B -field inhomogeneity the initial oscillations have practically vanished. In simula- can be a limiting factor for the minimal value of the differ - tions, the amplitude of the decay was determined by a single- ence between the spin-lock and MAS frequencies. If this exponential fit of the initial (4–6 ms) part of the decay. This difference is smaller than the width of the distribution of fit provides the middle line of the oscillating decays since the spin-lock frequencies, then the quantitative analysis of upper and lower half-periods of the oscillations compensate the relaxation data using Eq. (1) might be ambiguous for each other, and its extrapolation to zero time determines the same reason—the exact shape of the ω distribution in the amplitude plotted in Fig. 5. The coincidence between SL general case is not known. However, the more significant the simulated and experimental dependences is quite good. limiting factor for this difference is a sensitivity problem This figure demonstrates that R can be measured almost at 1ρ that is discussed below. all spin-lock fields except the very narrow range around the rotary resonance points where the usable signal decreases drastically. Thus, if very slow dynamics is not the aim of the study, it is advisable to keep the difference between 1 3 intensity, a.u. Journal of Biomolecular NMR (2018) 71:53–67 57 MAS and spin-lock frequencies as high as possible (at least time resolution 5 s more than 3–5 kHz) for a better signal-to-noise ratio. Other- wise, one should seek for a compromise between the small 1.0 frequency difference and the signal amplitude, which can 9.4 kHz be specific for different samples and experimental condi- 32 kHz tions. Figure 5b shows that the relaxation signal amplitude 0.8 depends only on the difference between the spin-lock and 14.2 kHz MAS frequencies, but the absolute values of these frequen- 25 kHz cies do not matter. 0.6 A very interesting feature in the experimental dependence in Fig. 5 is the drop of the amplitude at spin-lock 10 kHz, which is exactly half of the MAS frequency. This behaviour 0.4 15 1 cannot be reproduced in the simulations of the N– H pair. However, when simulating the 4-spins structure shown in Fig. 6, the shape of the relaxation decay at 10 kHz becomes 0.2 20.5 kHz more complicated, revealing additional slower oscillations of the amplitude. This corresponds to the HORROR con- 20 kHz dition (Nielsen et al. 1994) which of course arises from 0.0 15 15 15 15 homonuclear dipolar N– N interactions. The N– N interaction in the 4-spins structure (Fig. 6) is rather weak (56 Hz), however, this is sufficient for an appreciable distor - -0.2 tion of the shape of the relaxation decay. Thus, in spite of the fact that the dominating relaxation mechanisms in N 1 15 R experiments in proteins are the heteronuclear H– N 1ρ 0.00.5 1.01.5 2.02.5 dipole–dipole and N CSA mechanisms, the HORROR con- spin-lock duration / ms dition should be also avoided as well as rotary resonance condition, otherwise the observed decay contains also the Fig. 3 Experimentally measured N R -relaxation decays (initial 15 15 1ρ coherent evolution due to the dipolar N– N (at the HOR- part) of the integral signal in GB1 protein at 20 kHz and different 1 15 15 ROR condition) and the H– N dipole–dipole and N CSA spin-lock fields as indicated. All decays were normalized to unity at mechanisms. the initial point at 5 μs Non-exponential shape of the relaxation decays Figure 7 presents typical examples of the experimental and simulated R relaxation decays on a semilogarithmic 1ρ 1.0 scale. It is seen that the decays are not straight lines, i.e. they cannot be described by single-exponential functions. 0.9 Hence, there is a distribution of relaxation rate constants, which manifests the inhomogeneity of the spin system. We stress that this inhomogeneity is observed not only for the 0.8 integral signal of a protein (which should be since a pro- tein is a rather inhomogeneous structure with a relatively 0.7 wide distribution of relaxation rates for different sites, see 15 1 below) but also for a single ideal N– H spin pair, the latter being relevant for the analysis of the site-specific relaxa- 0.6 tion decays. The reason of the inhomogeneity in the case 15 1 of isolated N– H pair is a powder averaging: the internu- 01 23 4 clear dipolar coupling and hence the relaxation rate depend spin-lock duration / ms on the orientation of the internuclear vector with respect to the B field. Thus, in solids the relaxation decays, both R 0 1 15 1 Fig. 4 Simulated R decays for the N– H pair undergoing two-site 1ρ and R , are always non-exponential if powder averaging is 1ρ jumps at a single spin-lock field of 31.5 kHz (black line) and super - relevant and if there is no fast spin diffusion (Torchia and position of the decays simulated at B -fields from 30 to 33 kHz with a Szabo 1982; Giraud et al. 2005; Schanda and Ernst 2016). In step of 250 Hz (red line). MAS rate 20 kHz 1 3 intensity, a.u. intensity, a.u. 58 Journal of Biomolecular NMR (2018) 71:53–67 1.0 1.0 A B 0.8 0.8 0.6 0.6 0.4 0.4 ν = 20 kHz MAS 0.2 0.2 ν = 40 kHz MAS 0.0 0.0 51015202530354045 -35-30 -25-20 -15-10 -5 0 ν / kHz (ν -ν ) / kHz SL SL MAS Fig. 5 a Experimental (solid circles) and simulation (open triangles) lated at MAS rates 20 and 40 kHz plotted as a function of the differ - amplitude of the relaxation decay at 20 kHz MAS as a function of the ence between the spin-lock and MAS frequencies. Experiments were spin-lock field. The experimental dependence was arbitrarily normal- conducted with the GB1 sample. For the simulations, the model of −1 ized in order to get a coincidence with the simulation dependence at 2º-jumps with the jump rate 5000 s was used low spin-lock fields. b The amplitudes of the relaxation signal simu- 1.0 parameter from such a non-exponential relaxation decay. This can be done in two ways: one may define the mean 0.9 relaxation time constant or the mean relaxation rate constant. spin-lock The former is provided by fitting the whole decay using the 0.8 single-exponential fitting function, while the latter is repre- sented by the initial slope of the decay (see Supp. Info to ref. 0.7 10 kHz Krushelnitsky et al. 2014), as demonstrated in Fig. 8. While both procedures are mathematically correct, only the mean 9 kHz 0.6 relaxation rate constant has a physical meaning, as shown a NN N long time ago (Kalk and Berendsen 1976). The relaxation 0.5 10 kHz rate constant is directly proportional to the spectral density function R ~ J(ω), hence the mean relaxation rate con- 1,1ρ spin-lock duration / ms stant is proportional to the mean spectral density function: <R > ~ <J(ω)>. This is not the case for the mean relaxa- 1,1ρ Fig. 6 Three R decays simulated for 20 kHz MAS, different spin- 1ρ tion time constant. The correlation function obtained from 15 1 lock fields and different spin systems (indicated in the figure). N– H 15 15 the analysis of the initial slope of the relaxation decays is distance in both cases was 1.02 Å, N– N distances in 4-spins struc- ture was 2.8 Å, which corresponds to the typical distance between formally identical to that obtained from solution-state relax- backbone nitrogen atoms of neighbouring residues in a protein. In all ation times analyses (Torchia and Szabo 1982). 15 1 cases two-site jumps of the N– H bond with angular amplitude 5° Although these issues have been known for a long time, −1 were considered at a jump rate of 5000 s solid-state relaxation decays have often been analyzed in terms of mean relaxation time (not rate) constants, which in some cases may lead to incorrect conclusions, one of such static solids, the decays are strongly multi-exponential; MAS causes a partial averaging of the dipolar couplings, which is examples is described in (Smith et al. 2018). Practically, the initial slope can be determined from fit- still not complete, and the decays remain non-exponential although this is not always clearly seen in the experiments ting the decay with a sum of few exponential functions or using any phenomenological distribution function for the due to the insufficient signal-to-noise ratio. The exact shape of the relaxation decay cannot be calculated for a general relaxation rates. From such fittings, the mean relaxation rate can be readily defined. According to our experience, fitting case since it depends on the angular amplitude of motion. For a quantitative analysis in terms of the motional corre- functions with a minimal number of independent param- eters such as a double exponential (two R , one relative lation function, one usually employs a model-free approach 1ρ or its modifications in order to estimate a single relaxation amplitude, amplitude normalization coefficient) or a decay 1 3 intensity, a.u. intensity, a.u. Journal of Biomolecular NMR (2018) 71:53–67 59 Fig. 7 Typical examples of 1 1 experimental (a) and simulated A B 0.9 (b) R decays on a semi- 1ρ 0.9 logarithmic scale. Dashed red 0.8 lines indicate single-exponen- 0.7 tial decays for comparison. 0.8 Experiment: SH3 sample, MAS 20 kHz, spin-lock field 15 kHz; 0.6 simulations: jump amplitude 0.7 −1 20°, rate 500 s 0.5 0.6 0.4 0.5 0.3 0.4 01020304050 010203040 spin-lock duration / ms only possible when the “dead time” is considerably shorter than the inverse value of the relaxation rate, enabling a stable minimal-parameter fit as discussed above at times beyond the dead time. mean relaxation time constant Whole‑protein rocking motion The 1D-integral experiments discussed above do not allow obtaining detailed site-specific information on slow con- mean relaxation formational protein dynamics. On the other hand, there rate constant is a kind of molecular motion for which the site-specific spectral resolution is not critical, which is the overall rock- TIME ing motion of proteins in a solid environment, as reported recently both experimentally and computationally (Ma et al. Fig. 8 Schematic presentation of the multi-exponential experimental 2015; Lamley et al. 2015b; Kurauskas et al. 2017). Such a relaxation decay (circles), initial slope (dashed line) and fitting the global process assumes that all parts of a protein undergo whole decay using a single-exponential function (red solid line) the same motion and the correlation functions of all N–H bonds have a component with the same correlation time. The parameters of this common component can be obtained based upon a single-mode log-normal rate distribution (one median R , one distribution width parameter, amplitude from 1D experiments and site-specific resolution for this is 1ρ not necessary. The amplitude of this overall motion for dif- normalization coefficient) are sufficient practically in all cases (Roos et al. 2015). Some practical hints for the proper ferent N–H bonds can be however different. For example, if the rocking motion is a restricted rotation around an axis analysis of non-exponential decays are described in ESM. It must be mentioned that in the case of the R experi- then the amplitude (order parameter) depends on the angle 1ρ between this axis and N–H bond. Thus, the rocking motion ments, the initial slope of the decays is not seen due to the initial oscillations considered above, i.e. “dead time”. At the amplitude obtained from 1D experiments is a mean ampli- tude over all N–H bonds. As discussed below, the overall moment we are not in a position to suggest a truly robust and reliable way to decompose oscillations and relaxation motion in general case can be overlapped with the internal conformational motion that has similar time scale. In this contributions at the beginning of the decay. Thus, the correct quantitative analysis of the non-exponential R decays is case, the unambiguous discrimination between these two 1ρ 1 3 Log (INTENSITY) intensity, a.u. 60 Journal of Biomolecular NMR (2018) 71:53–67 types of molecular mobility from 1D data is impossible and the collection of such an extensive data set at multiple RF site-specific measurements are necessary. field strengths, three different temperatures and four different Previous analyses have shown that protein rocking occurs protein samples. on the microsecond timescale (Lamley et al. 2015b; Kuraus- Below we present the analysis of the R rates measured 1ρ kas et al. 2017), thus near-rotary-resonance R measure- at a wide range of differences (ω − ω ) and at three 1ρ MAS SL ments should provide critically important information on the temperatures for each sample. The range of temperatures parameters of the rocking motion. In this part of the work was limited so that the proteins remain native and that the we apply N R experiments for a comparative study of the microcrystalline samples would not freeze. In addition to 1ρ rocking motion in a set of four different solid protein sam- R ’s, we also measured R ’s at the same temperatures. R ’s 1ρ 1 1 ples, taking into account all the methodological problems are not useful for studying rocking motions since these discussed above. Since in all four cases the experiments and relaxation rates are sensitive to much faster dynamics. At the data analyses were performed in the same manner, the the same time, while fitting the data we need to take into results can be compared directly. account faster motions as well since without knowledge of Interpreting relaxation measurements in terms of molecu- the amplitude of the fast motions, the order parameter of the lar motion is generally challenged by the fact that the spec- rocking motion cannot be determined precisely. tral density function is sampled usually only at few frequen- Figure 9 presents typical examples of the R relaxation 1ρ cies. When only small number of relaxation parameters are decays measured at different spin-lock frequencies. Simi- measured, the fitted time constants and amplitudes may be lar decays for one of the ubiquitin samples measured at subject to a large uncertainty (Smith et al. 2017). To resolve, 40 kHz are shown in ESM, Figs. S1 and S2. Even without at least partially, these ambiguities in determining motional any numerical analysis it is clearly seen that the relaxation time scales, we use relaxation measurements at multiple becomes faster upon approaching spin-lock frequency to temperatures. Even when measured within a narrow tem- the MAS frequency. This is an unambiguous indication of perature range, the temperature dependence of a relaxation the fact that the protein undergoes dynamics on the micro- rate, i.e. its slope—even the sign of the slope (positive or second timescale, otherwise the R versus (ω − ω ) 1ρ MAS SL negative)—is very informative for the determination of the dependence would be flat (Krushelnitsky et al. 2014; Ma motional correlation time. Varying both the RF field strength et al. 2014). and the temperature and analysing all the data simultane- The initial oscillations were excluded from the analysis ously renders the set of the relaxation data effectively “two- and the fitting was performed over all points beyond 3.5 ms dimensional”, which significantly improves the accuracy spin-lock pulse duration. Figure 10 shows the result of and precision of the fit results. The 1D approach allowed the fitting of a typical relaxation decay using mono- and Fig. 9 Experimental R decays 3.5 ms 1ρ measured for GB1 at 21.5 °C 5.0 Spin-lock frequencies: and 20 kHz MAS for five 4.5 different spin-lock fields as 8 kHz indicated (a). The panel b is 4.0 12 kHz a zoomed region indicated by 0.95 the rectangle in the panel a. All 15 kHz 3.5 decays were normalized to unity 17 kHz by the point at 3.5 ms (“dead 3.0 0.9 18 kHz time”), where the initial oscilla- 2.5 tions have practically vanished. The experimental error can be 0.85 2.0 assessed from the scatter of the points in the decays 1.5 0.8 1.0 0.5 0.75 0.0 -0.5 0.7 -1.0 AB 0.01 0.11 10 100 01020304050 spin-lock duration / ms spin-lock duration / ms 1 3 intensity, a.u. Journal of Biomolecular NMR (2018) 71:53–67 61 ( , , ) is the distribution function that is parameterized by two parameters— , the centre of the distribution and β, the distribution width parameter. Without a distribution of 0.95 the correlation times for the fast motion, good data fitting 0.9 was unachievable. As for the distribution of the fast-motion correlation times, we used two phenomenological models, a 0.85 log-normal and a modified non-symmetric Fuoss-Kirkwood distribution (Schneider 1991), for details see the ESM. The 0.8 analysis assumes that the order parameters and the shape of the distribution function are the same at all temperatures. 0.75 While in reality these are obviously temperature-dependent, the temperature dependence is likely rather weak, otherwise 0.7 the proteins and crystals would not be rigid and stable. Tak- ing the temperature dependence of these parameters by some spin-lock duration / ms phenomenological function into account is in principle pos- sible, but this would make the fitting less certain and in any Fig. 10 A typical example of the R relaxation decay with barely case would not lead to any significant change of the final 1ρ noticeable multi-exponentiality, measured on GB1 at 21.5 °C, results. 20 kHz MAS and a spin-lock field of 17 kHz. The dashed blue The slow (rocking) motion likely has a distribution of and solid red lines are single- and bi-exponential fits, respectively. correlation times that can be caused either by inhomogeneity The relaxation rates obtained from the single- and bi-exponen- −1 tial fits are 6.5 ± 0.25 and 8.0 ± 1.0 s , respectively. In the lat- of a sample (e.g. defects in crystal packing) or an intrinsi- ter case, the mean relaxation rate (i.e. initial slope) was obtained as cally complicated shape of the rocking-motion correlation R = p ⋅ R +(1 − p) ⋅ R , where p and R are the relative 1 1a 1b 1ρa,b function arising from a possible inter-correlation of motion amplitude of one of the components and relaxation rate constants of neighbouring proteins in a crystal. We tried to fit the data of two components of the decay, respectively (see Eqs. S1–S4 of ESM). The fitting values of these parameters are: p = 0.74 ± 0.5; assuming a distribution for the slow motion as well. How- −1 −1 R =2.8 ± 5 s ; R =23 ± 20 s 1ρa 1ρb ever, in all cases the inclusion of one more fitting parameter (distribution width for the slow-motion correlation times) bi-exponential fitting functions. This is merely a demonstra- leads to a practically negligible improvement of the fitting tion of the difference between the meaningless mean relaxa- quality (results not shown). At the same time, the depend- tion time (from the single-exponential t) fi and the physically ence of the main parameters of the slow motion, the cor- relevant mean relaxation rate (see above) using a typical relation time and the order parameter, on the width of the experimental relaxation decay. The non-exponential shape distribution is rather weak. Thus, following the principle of of the decays is barely seen, tempting one to neglect the non- Occam’s razor, we include in the fitting model only a single exponentiality and to fit the decay with a single exponential. correlation time of the slow rocking motion. However, as it is shown in the figure, the values of the mean As expected, the type of the distribution function has an time and the mean rate are appreciably different. effect on the parameters of the fast motion, but the slow- After determining the set of the relaxation rates for all motion parameters are practically insensitive to it. Since we samples, one has to fit the data using a specific motional analyze the relaxation times at different temperatures, we model. We suggest using a model with two motional assume an Arrhenius dependence of the correlation times modes—fast internal conformational motion on the nano- on temperature, second timescale and slow overall protein rocking motion. Since we analyse the overall signal from the whole protein, f ,S 1 1 = ⋅ exp − (3) f ,S f ,S we introduce a distribution of correlation times of the fast R T 293K internal motion that takes into account the internal dynamic heterogeneity of a protein. Thus, a spectral density function where are the correlation times at temperature 20 °C, and f ,S reads E are activation energies of the fast and slow motions. f ,S Thus, the total number of fit parameters is seven: two order 2 2 2 J()=(1 − S ) (, , ) d + S (1 − S ) parameters, two correlation times, two activation energies f f S 2 2 1 +() 1 +( ) (for the fast and slow motions) and the distribution width (2) parameter for the fast motion. At the same time, the number where S and are the order parameters and the correla- of experimental points (relaxation rates) was 18–24 for each f ,S f ,S sample. Further details of the fitting procedure are laid out tion times of the fast and slow motions, respectively; 1 3 intensity, a.u. 62 Journal of Biomolecular NMR (2018) 71:53–67 100 100 in the ESM. The parameters of the fast motion may be poorly SH3 GB1 determined, since only R rate constants at a single reso- nance frequency contain information on the fast dynamics, which is obviously not enough for a reliable determination of the nanosecond timescale part of the motional correlation function. Also, the activation energies are not determined 10 10 very precisely since the temperature range of the experi- ments was quite narrow. Fortunately, the uncertainty of these parameters is a minor problem since the main goal of this work is to assess the timescale of the slow rocking motion and its order parameter, which could be reliably determined from the data since the abundant R rates sample the micro- 1ρ 1 1 68 10 12 14 16 18 20 68 10 12 14 16 18 20 second range of molecular dynamics in sufficient detail. All 1000 1000 the fit parameters for two types of the distribution functions ubiq. MPD ubiq. PEG are presented in ESM, see Tables S1 and S2, and below in Table 1 we collect only the most important and relevant results—the order parameters and the correlation times of the slow rocking motion for the four different protein sam- ples. Table 1 also presents the angular amplitudes assuming 100 100 the motional model to be jumps between two equivalent sites, which gives an impression of the angle of the whole protein reorientation on the microsecond time scale (this of course should not be necessarily the 2-site jumps). The last row in Table 1 is a solvent content in the protein crystals which will be discussed below. Figures 11 and 12 present 10 10 510152025303540 510152025303540 the experimental relaxation rate constants and the fit curves. spin-lock / kHz The detection of slow motion in all crystals suggests, as mentioned above, the presence of an overall motional Fig. 11 R relaxation rates as a function of spin-lock frequency for 1ρ process, such as rigid-body rocking (Ma et al. 2015; Lam- four different samples at three different temperatures. Circles—exper - ley et al. 2015b; Kurauskas et al. 2017). Alternatively, the imental data, solid lines—fitting curves. Blue dashed, black solid and observed very low amplitude of the slow motion in the non- red dotted lines correspond to the temperatures 13, 21.5 and 29 °C for SH3 and GB1 and 3, 15 and 27 °C for two ubiquitin samples, respec- selective experiment could also be explained by an internal tively conformational motion that affects only a (possibly rather small) fraction of residues. For the case of cubic-PEG-ub, the rigid-body nature of the motion has been confirmed by correlation function with the same correlation time for all N–H bonds in a protein. To provide evidence for the global site-resolved measurements (Ma et al. 2015). For other pro- teins, especially revealing higher S , the existence of the character of the motion, we performed site-resolved R 1ρ experiments in 2D-mode for the GB1 sample, i.e. the sample whole-body motion is so far not established. A criterion of such overall motion is a common component of the motional having the highest S out of all four samples studied in the Table 1 Order parameters, GB1 SH3 Ubiq. MPD Ubiq. PEG angular amplitudes (assuming 2-site jumps model) and the 2 S 0.9995 ± 0.00006 0.9915 ± 0.005 0.9965 ± 0.001 0.987 ± 0.003 correlation times of the rocking Ang.ampl. 2-site jumps 1.5° 6.1° 3.9° 7.6° motion at 20 °C for four 41 ± 5 46 ± 4 30 ± 7 52 ± 7 /μs samples Solvent content in crystal/% 33.6 49.8 49.6 56 The numbers are averaged values of the fit results obtained using two different models for the τ-distribution functions for the fast motion (see Tables S1 and S2 in ESM for details). The solvent content in the crystals was determined using the molecular weight and unit cell parameters in the PDB entries 2GI9 (GB1), 1U06 (SH3), 3ONS (MPD-ub) and 3N30 (cubic-PEG-ub); the data for all these crystal structures have been col- lected at 100 K. Note that for a different crystal form of GB1, grown under very similar conditions (MPD, pH 4.5), corresponding to PDB entry 2QMT, the solvent content is also rather low, 45%, with a similar intermolecular β-sheet 1 3 -1 relaxation rate / s Journal of Biomolecular NMR (2018) 71:53–67 63 spin-lock frequency 8 kHz 17 kHz SH3 05 10 15 20 25 30 35 40 45 50 ubiq. PEG -5 ubiq. MPD 05 10 15 20 25 30 35 40 45 50 peak number GB1 Fig. 13 Relaxation rates for different peaks in 2D-spectrum measured 0.1 in GB1 at 21.5 °C and MAS 20 kHz with spin-lock frequencies of 8 and 17 kHz (a), and difference R (17 kHz)–R (8 kHz) (b), both 1ρ 1ρ 05 10 15 20 25 30 35 plotted vs the peak number (which does not correspond to the pri- o mary structure of the protein) T / C assignment is not crucial, in particular as we find indeed a Fig. 12 R relaxation rates as a function of temperature for four dif- ferent samples as marked. Circles—experimental data, solid lines— very similar behaviour for all residues. fitting curves Except for one peak (labelled as peak no. 6), all resi- dues reveal a positive difference, R (17 kHz)—R (8 kHz), 1ρ 1ρ current work. Since collecting the large set of temperatures which indicates that all parts of the protein undergo the same and RF field strengths that we covered by the 1D experi- type of motion on the μs timescale. Although the error bars ments would be too time consuming, we performed only two in Fig. 13b are rather large, the decays themselves, even R experiments at two spin-lock frequencies, 8 and 17 kHz, without the analysis, leave no doubt about the uniform R 1ρ 1ρ at a single temperature 21.5 °C. Having only two relaxation spin-lock frequency dependence, see Fig. S4 (ESM). The rate constants does not allow for any reliable quantitative absolute value of this difference for the majority of peaks −1 estimation of the order parameters and correlation times, is very similar, around 5 ± 2 s , which is in good agree- but the difference between the relaxation rates measured at ment with the corresponding value from the 1D relaxation −1 two spin-lock fields unambiguously reveals the presence of experiment, 4.3 s (Fig. 11). This finding fits much better μs motion. If the motion is faster or slower in comparison to the overall protein rocking motion rather than internal −6 −4 to a 10 –10 s correlation time range, then the difference conformational dynamics since in the latter case only a part is zero, otherwise the difference has an appreciable value of residues would reveal such a dependence of R on the 1ρ proportional to the amplitude of the microsecond motion. frequency difference ω − ω . MAS SL Figure 13 presents the relaxation rates R measured at Several residues in the protein have elevated relaxation 1ρ two spin-lock fields as well as their difference. In this analy - rates in comparison to the mean level, e.g. those correspond- sis, we have only used arbitrary peak numbers, rather than ing to the peak numbers 3, 4, 5,12, 14, 15, 43, 47 and few residue numbers. The spectrum of our sample shown in Fig. others (Fig. 13a). These residues obviously undergo μs time S3 (ESM), obtained from similar crystallization conditions scale motion and this motion is attributed to the internal as previously reported ones (see “Materials and methods” conformational dynamics since only a small portion of resi- section) differs from the previously published spectra to a dues reveal such a behaviour. This motion for most peaks point where unambiguous assignment of peaks to residues is however much faster (the correlation time presumably is possible only for 2–3 peaks. For the present analysis, the around 1 μs or shorter) than the overall rocking motion 1 3 -1 relaxation rate / s -1 -1 R / s ΔR / s 1ρ 1ρ 64 Journal of Biomolecular NMR (2018) 71:53–67 observed in the 1D experiments because the R difference in excellent agreement with the value found in the present 1ρ (Fig. 13b) reveals no correlation with the absolute values of study. The motional amplitude had previously been esti- the relaxation rates. It follows from the fact that R versus mated to be ca. 3°–5°, corresponding to an order parameter 1ρ (ω ω ) dependence for the motions with a correlation ca. 0.98. Independent MD simulations had estimated an MAS − SL time around 1 μs and shorter is flat, see Fig. 1 in (Krushelnit- order parameter of overall rocking motion of ca 0.95–0.98. sky et al. 2014). Thus, this internal motion does not contrib- These values are also in a satisfactory agreement with the ute to the rocking motion parameters obtained from the 1D value found here, S = 0.987. data. The only exception is the peak no. 43: for this signal, The availability data of four different protein crystals pro- the elevated rates reveal also the elevated difference between vides interesting insight into the nature of rocking motion the rates, which indicates an additional internal motion with and allows us to suggest an explanation why the rocking −1 a correlation time of the order of (ω − ω ) . motion amplitudes in these samples are so different, vary - MAS SL We need to admit that the available 2D data do not prove ing by a factor of 5 when expressed as an effective rocking- that there are no additional internal conformational motion motion angle. The solvent content of these crystals varies with a correlation time around 30–50 μs, i.e. the motion from 33% (GB1) to 56% (cubic-PEG-ub), see Table 1, with that undergoes only a part of residues in the protein. The a good qualitative correlation with the amplitude of the rock- accuracy of these data is rather low and probably this hypo- ing motion. For the case of GB1, which shows the lowest- thetical motion is “hidden” within the error bars in Fig. 13. If amplitude overall motion, it is interesting to note that the this is true, the rocking motion amplitude has a contribution proteins form inter-molecular β-sheets, extending through from this internal motion. On the other hand, the 2D data the entire crystal, with three NH···O=C hydrogen bonds con- demonstrate that all the protein structure is dynamic on the necting the outermost β-strands of neighbouring molecules, μs timescale, and that the level of this mobility is roughly the see Fig. S5 (ESM). This tight interaction and packing of same across the different parts of the protein. At the same neighbouring molecules obviously limits the slow overall time, the presence of the overall rocking motion does not motion, providing a plausible explanation for the observed necessarily mean that the protein itself remains fully rigid low amplitude. on the μs timescale. The rocking motion is possibly associ- While the overall-rocking motion amplitude die ff rs signif - ated with a bend or a twist of the whole structure, and thus icantly between the different protein crystals, the previously discriminating between the whole protein reorientation and determined fast local motional amplitudes of NH bonds global structural plasticity is not a simple problem. Thus, the appear to be more similar for GB1 (Mollica et al. 2012), SH3 term “rocking motion” in some cases may imply rather some (Chevelkov et al. 2009) and ubiquitin (Haller and Schanda global dynamics than the restricted rigid body rotation only. 2013; Ma et al. 2015). The observation that the local motion Based on the analysis of MD simulations and NMR data, it is less influenced by crystal packing is not surprising as it has been suggested that local microsecond motion may be has been established that the internal motion amplitudes in coupled to overall rocking motion (Kurauskas et al. 2017). proteins are dictated by the local density of the internal pro- Specifically, the reorientation of molecules within the crys- tein structure (Halle 2002). Local structural density in a pro- tal lattice is accompanied by the breaking and formation of tein and packing density of protein globules in a crystal of a different set of inter-molecular interactions, which likely course should not be necessarily correlated, however, in both influence also the interconversion of different internal local cases they seem to be a crucial factor affecting the amplitude conformations. Nonetheless, while we cannot exclude the of the fast internal and slow overall motions, respectively. presence of additional local motions having a correlation Despite the very different rocking amplitudes, the time time close to that of global protein rocking, the uniformity scales for all samples are rather uniform from ca. 30 to 50 µs, of the R (17 kHz)–R (8 kHz) difference strongly suggests without correlation to the amplitude. While we can only 1ρ 1ρ an overall/global motion, even for an order parameter as high speculate about the origins of this similarity of time scales, as 0.9995. this observation may indicate that the rocking motion is not a As a further support for the global character of the diffusive but rather a jump-like process. For a restricted dif- detected motion, the two ubiquitin samples used in this study fusive motional model, e.g. wobbling in a cone or rotational have previously been studied in a site-resolved manner, and, diffusion around an axis, the apparent correlation time would similarly to the case of GB1, a RF-field dependence of R be dependent on the amplitude of motion, assuming that 1ρ has been observed over all residues in the cubic-PEG-ub the diffusion coefficient is the same. Such a dependence has sample (Kurauskas et al. 2017). A global fit of the site-spe- been observed e.g. in numerical simulations of the motional cific near-rotary-resonance R relaxation dispersion profiles correlation function in the Supporting Information to ref. 1ρ of 22 available well-resolved amide sites in cubic-PEG-ub (Zinkevich et al. 2013). For jumps between a small number had been performed, and the obtained motional time con- of sites, the correlation time does not depend on the ampli- stant was in the range of tens of microseconds, which is tude. Thus, the rocking motion is likely a jump-like process 1 3 Journal of Biomolecular NMR (2018) 71:53–67 65 between different molecular orientations. These different somewhat larger than the width of ν -distribution due SL orientations or protein molecules in the crystal certainly to B -field inhomogeneity. Otherwise, the quantitative differ in the pattern of inter-molecular interactions. Indeed, analysis of the relaxation data would be too uncertain. long MD simulation of different ubiquitin crystals with up (b) During the first few milliseconds of the R decay, the 1ρ to 48 molecules have revealed a pattern of fluctuating inter - useful relaxation signal is subject to initial coherent molecular salt bridges and hydrogen bonds, and have sug- oscillations. The proper analysis of these oscillations is gested that different orientational states of the protein are complex and difficult, and we suggest to consider this stabilized each by a different set of interactions (Kurauskas initial part of the decay as an effective “dead time” and et al. 2017). Thus, each set of interactions defines a certain not to include it in the analysis. energy minimum (a certain protein orientation). However, (c) One should avoid not only rotary resonance but also we are not yet sure that the life time of these inter-molec- HORROR (ω = ω /2) conditions. Homonuclear SL MAS 15 15 ular interactions and hence, the overall-motion time scale N– N interaction in proteins is weak, but it is still is a universal value for all protein crystals, this obviously capable of distorting the relaxation decays significantly. requires further experimental evidences. This distortion is even more difficult to handle because Lastly, we note that the very high order parameter of the of its low frequency, thus precluding simple time aver- rocking motion can be not only the result of a small angular aging; it thus affects the apparent initial decay rate the amplitude between similarly populated states, but could also is the most relevant quantity. be due to skewed populations of states which have a larger (d) R and R relaxation decays in solids are always non- 1 1ρ difference in their relative orientations. Previously observed exponential. Quite often this non-exponentiality is high order parameters of the microsecond motion in SH3 hardly seen in experiments, still, the decays are non- were ascribed to a concept of low-populated (excited) states exponential. For correct quantitative analysis of the that implies jumps between conformational states with non- relaxation data in terms of the correlation function for- equal probabilities (Zinkevich et al. 2013). We now think malism, one needs to determine not the mean relaxation that this microsecond motion is not internal but overall rock- time but the mean relaxation rate, the latter being the ing motion, still the latter can be associated with the jumps initial slope of the relaxation decay. between non-equal probability states. However, the available data do not allow making definite conclusions on this issue. Making use of the fact that the 1D mode of the relaxation experiments allows much faster measurement, multiple N R ’s at different spin-lock frequencies and temperatures 1ρ Conclusions were measured for four different microcrystalline protein samples in order to determine the parameters of the over- In this work we explored the capabilities and practical limi- all rocking motion. The results show that the correlation tations of studying slow molecular motions by means of N time of the rocking motion is almost the same for all sam- R experiments in the vicinity of the rotary resonance con- ples (30–50 μs), however, the amplitudes are very different. 1ρ ditions. A small difference between the spin-lock and MAS The absolute values of the order parameter of the rocking frequencies allows expanding the frequency range of the motion are rather high; the most rigid protein, GB1, has sampled molecular motions towards rather low frequencies. an order parameter of 0.9995. The R experiments at two 1ρ At the same time, practical problems should be properly han- different spin-lock fields performed for this sample in 2D dled while conducting the experiments. The most important mode show that the very small amplitude corresponds to the methodological issues are as follows. overall motion of the whole protein, not internal conforma- tional dynamics of only a (possibly small) part of the protein (a) Upon approaching to the rotary resonance condition, structure. Thus, the rocking motion seems to be a general the amplitude of the useful relaxation signal decreases feature of proteins in a crystal. The rocking motion ampli- significantly. Consequently, one should seek a compro- tude of the four samples correlates with the solvent content mise between a small difference between spin-lock and in the protein crystal suggesting that the amplitude depends MAS frequencies (if the slow dynamics is the target on the packing density of a protein crystal. There are indica- of study) and acceptable signal to noise ratio. When tions that the rocking motion is likely a jump-like dynamic performing a series of measurements at different spin- process, this however should be confirmed in future studies. lock RF field strengths, the measurement time may be Acknowledgements The authors are grateful to Prof. Matthias Ernst distributed such that the sensitivity drop at near-rotary- for very useful and stimulating discussions in the course of the prepara- resonance conditions is compensated. If the signal is tion of this work. Prof. Bernd Reif is thanked for his support in prepa- very strong and sensitivity is not a limiting factor, then ration of the SH3 sample. The authors acknowledge the funding from one should keep the ω − ω difference at least the Deutsche Forschungsgemeinschaft (DFG), collaborative research MAS SL 1 3 66 Journal of Biomolecular NMR (2018) 71:53–67 center SFB TRR 102 (project A08), and the European Research Coun- MAS-dependent N rotating-frame NMR relaxation. J Magn cil (ERC-StG-311318). This work used the platforms of the Grenoble Reson 248:8–12 Instruct Center (ISBG; UMS 3518 CNRS-CEA-UJF-EMBL) with Kurauskas V, Weber E, Hessel A, Ayala I, Marion D, Schanda P (2016) support from FRISBI (ANR-10-INSB-05-02) and GRAL (ANR-10- Cross-correlated relaxation of dipolar coupling and chemical-shift LABX-49-01) within the Grenoble Partnership for Structural Biology anisotropy in magic-angle spinning R NMR measurements: 1ρ (PSB). application to protein backbone dynamics measurements. J Phys Chem B 120:8905–8913 Kurauskas V, Izmailov SA, Rogacheva ON, Hessel A, Ayala I, Wood- Open Access This article is distributed under the terms of the Crea- house J, Shilova A, Xue Y, Yuwen T, Coquelle N, Colletier tive Commons Attribution 4.0 International License (http://creat iveco J-P, Skrynnikov NR, Schanda P (2017) Slow conformational mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- exchange and overall rocking motion in ubiquitin protein crys- tion, and reproduction in any medium, provided you give appropriate tals. Nat Commun 8:145 credit to the original author(s) and the source, provide a link to the Kurbanov R, Zinkevich T, Krushelnitsky A (2011) The nuclear Creative Commons license, and indicate if changes were made. magnetic resonance relaxation data analysis in solids: general R /R equations and the model-free approach. 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Published: May 30, 2018
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