Magic Angle Spinning Effects on Longitudinal NMR Relaxation: 15 N in L-Histidine

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Introduction
Determining the structure and dynamics of proteins and peptides is key to understanding their function in biological systems.Solid-state Nuclear Magnetic Resonance (ssNMR) is a unique technique that can reveal such complex information by probing C !# , N !" , H ! , and H $ nuclei in biological samples through harnessing the power of magic angle spinning (MAS), cross polarization, proton decoupling, and advanced recoupling techniques.Isotopic labelling, high magnetic fields, and faster spinning rates further increase the sensitivity of ssNMR such that the structure of many proteins and peptides have been determined.The molecular structure is usually obtained from signal assignments and structural constraints while information on fast molecular motion may be obtained especially from relaxation 1,2 .
Longitudinal relaxation is an incoherent process that returns the longitudinal component of nuclear magnetization to its thermal equilibrium through internal dynamics of the molecule 3 .It has previously been pointed out that spin diffusion may influence the measured relaxation rate constants 4,5 , as the apparent longitudinal relaxation rate constant  !* determines the combination of both coherent and incoherent contributions to the change of nuclear magnetization. 3,6 he incoherent, or stochastic, contribution to signal decay contains information on the time-dependent fluctuation of bond vectors and samples the spectral density function 7 .The spectral density function is the Fourier transform of the time-correlation function that informs of the dynamics of a molecule and hence displays the intensity of dynamics at different frequencies and is largely independent of external factors like the sample spinning rate 8 .The coherent contribution to signal decay, however, depends on several external factors and could thus be affected by the measurement method.
Understanding the dynamics of complex biological systems requires a clear understanding, and untangling of, these two contributing factors to the relaxation so that only the desired relaxation processes occur in the relaxation period and the measurement method itself does not interfere with the desired measurand.
While higher magnetic fields provide better signal separation and thus provide more details than experiments at lower fields, we here report that it may also create new challenges in the measurement and interpretation of relaxation due to rotational resonance.Rotational resonance 9 is a phenomenon that occurs when the chemical shift frequency difference ∆ &' =  & −  ' between two dipolar coupled nuclei  and  matches an integer multiple of the MAS rate  ( , i.e., Without chemical shift anisotropy  is limited to 1 and 2; while  can assume higher integer values in the presence of chemical shift anisotropy.At the rotational resonance condition, nuclear magnetization is transferred between the two nuclear sites due to the recoupling of homonuclear dipolar coupling 9 .This coherent process is independent of the dynamics of the molecule -the stochastic part of relaxation -and promotes spin diffusion. 10 have performed measurements of longitudinal relaxation rate constants  !* in isotopicallyenriched biological materials with N !" in two model compounds of L-histidine hydrochloride monohydrate and glycine at different spin rates ( ( ).We have observed that the N !" longitudinal relaxation time of the two nitrogen nuclei in the imidazole ring is reduced by almost three orders of magnitude at the condition of rotational resonance with the amine.
MAS rate effects on the apparent longitudinal relaxation rate constant has far-reaching implications for ssNMR in biophysics and materials science.Some important examples include (a) the proper measurement of dynamics in peptides, (b) time-efficient ssNMR experiments 11 , (c) selective transfer in dynamic nuclear polarization 12 , and (d) spin-diffusion experiments 13 .Similar processes are also relevant for C !# , similar to that of N !" 14 .

NMR Measurements
NMR spectra were acquired on a Bruker 950 MHz spectrometer with an Avance III HD console equipped with 1 H-13 C-15 N-2 H 1.9 mm MAS probe with radiofrequency (rf) field strengths for the hard pulses and decoupling of 83.3 kHz, 28.4 kHz, and 10.9 kHz for 1 H, 79 Br, and 15 N, respectively.A crosspolarization recovery pulse sequence (from Torchia 15 , similar to CPXT1 in TopSpin) measured 15 N longitudinal relaxation with the phase of the contact pulse and receiver inverted in alternate scans to ensure an experimental setup where the observed magnetization returns to zero.Twenty three spinning rates in the range of 5.0 to 41.3 kHz were scanned for L-histidine.SPINAL-64 proton decoupling was employed. 16For each relaxation experiment, 64 different delay times  ) in the range of 10 s up to 10000 s were used.The FIDs were recorded using 2880 time-domain points with a spectral width of 28.8 kHz and deadtime of 6.5 s before acquisition.Similar measurements were performed for glycine at five spinning rates of {6, 14.5, 23, 31.5, 40} kHz.
All spectra were acquired at an internal sample temperature of 25 °C with the maximum deviation of 2.2 °C.Single-exponential  !

Materials
Rotors were packed with polycrystalline amino acids and KBr with a ~0.8 0.2 ⁄ mass ratio.Amino acids include L-histidine hydrochloride monohydrate (with 98% uniform 15 N labelling) from Cambridge Isotope Laboratories (Andover, MA, USA) and glycine (98% 15 N labeling) from Merck (Saint Louis, MO, USA).8][19] Its molecular structure consists of two groups of nearly coplanar atoms and its crystal structure belongs to the space group P2 ! 2 ! 2 ! .L-histidine hydrochloride monohydrate is only referred to as L-histidine in the rest of the manuscript. 15N in the imidazole ring of L-histidine has chemical shifts of N * :  + = 189.6 ppm (closest to branch) and N , :  -= 176.2 ppm.The amine 15 N chemical shift is  .= 47.3 ppm in L-histidine and  /01 = 32.9ppm in glycine.All spectra were referenced indirectly using the chemical shift of 39.3 ppm for solid N !" H 2 Cl at 25°C. 20mperature Control Accurate internal sample temperature control was essential to the data of this work.For all MAS rates, the sample temperature was set to 25°C by measuring the 79 Br  !relaxation of KBr, which serves as internal thermometer in the rotor. 21Calibration curves for the difference between the set and actual temperatures were established by K 79 Br  ! for different spin rates as prior information.
The sample temperature was adjusted and kept with the average and maximum deviations of 1 °C and 2.2 °C from the target temperature of 25°C as assessed by the 79 Br  !relaxation time of KBr.At the highest MAS rate of 41.3 kHz, the difference between the probe set temperature and that of the internal sample was a staggering 51.5 °C.The magic angle was set by maximizing the intensity of the second sideband to the main peak of 79 Br in KBr 22 .

Results and Discussion
Longitudinal 15   ) match the rotational resonance conditions and corresponding minima for  !,* * and  !,, * .In contrast to L-histidine, glycine (Gly) exhibits a constant  !,/01 * =  !,/01 = 0.375 s for all MAS rates.Lines between points are eye guides and do not have a theoretical meaning.Figure 1a and 1b show the same data with linear and logarithmic -axis, respectively.

Cross-Relaxation Rate Constants
It is our hypothesis that the  !* values for L-histidine may be explained by a classical magnetizationexchange model between the three sites, and with exchange/cross-relaxation rate constants being defined by incomplete averaging of the homonuclear dipole-dipole interactions.For three sites with magnetizations and solution I( ) ) = exp(L  ) ) I(0); where exp is the matrix exponential and  ) is the longitudinal relaxation delay. = 0 indicates no cross relaxation, thus no coupling between two nuclei.
A direct-search method 31 varied log !?  .*, log !?  :, , and log !?  ,* in the range of [−8, +1], and log !? g >,* ( ) = 0)h, log !? g 9,, ( ) = 0)h, and log !? g 9,: ( ) = 0)h in the range of [6.8, 7.8], to minimize the difference between the model values and integrals of each chemical shift peak at all longitudinal relaxation delay times.Constant  !-values corresponding to 1/ !values from Region (A) of Figure 1a were used for all three sites ( !,* = 3.145 • 10 @2 s @! ,  !,, = 3.704 • 10 @2 s @! , and  !,: = 0.6061s @! .Optimization operations were undertaken for each MAS rate to inform on the transfer of magnetization between 15 N sites (see Figure 2).Seven points (out of 69) from predicted  &' values were regarded as outliers and were removed from results because of the bounds of optimization (see Table S2).No conditions were directly imposed on the computation of rate constants to achieve partial magnetization exchange.Experiments and simulations display several interesting features including multiexponential decay (see Figure 2c,  >,: at 12 kHz), and especially an increase and then decrease in magnetization (see Figure 2d, inset of  >,* at 10 kHz, bottom left).The multiexponential decay close to the first rotational resonance condition has also been observed by Kubo and McDowell 10 who recognized that the exchange of magnetization cannot be described by a single spin-diffusion time constant between the first and second rotational resonance conditions in their experiments.Although the decays are multiexponential in some cases, the dominant decay rate constant of magnetization in multiexponential analysis follows the same behavior as shown in Figure 1. Figure 3 shows three cross-relaxation parameters of  .*,  :, , and  ,* as functions of  ( (data in  (see Figure 1).

Numerical Simulations
The classical treatment described above does not establish a connection between the mechanism of magnetization transfer and the cross-relaxation rate constants.In relaxation experiments of this work, however, MAS-dependent recoupling of the homonuclear dipolar interaction of 15 N is an obvious source of the magnetization transfer.The cross-relaxation rate constant is proportional to the square of the dipolar coupling constant where  &' is the internuclear distance between nuclei  and ,  is the gyromagnetic ratio, and  ? is the vacuum magnetic permeability. &' ( ( ) is a function of zero-quantum line shapes and hence a function of the spinning rate 10,13,14 .
In contrast to the more phenomenological cross-relaxation rate constants introduced above, the evolution of the density matrix provides mechanistic information on the mechanisms of magnetization transfer through various field and interaction effects.We have used numerical simulations to calculate the magnetization transfer between three 15 N sites in L-histidine to obtain the effectiveness factor  as a function of MAS rate.The effectiveness factor reports on the maximum transfer of longitudinal magnetization of one nucleus to another and depends on the MAS rate.In the simulations, only chemical shift anisotropy and dipolar coupling parameters were considered for the three 15 N atoms; scalar coupling parameters and protons were neglected.The effectiveness factor is analogous -although not directly comparable -to the cross-relaxation rate constant and determines the possibility of magnetization transfer between two spins where 0 and 1 denote no and complete magnetization transfer, respectively.No restrictions were imposed in numerical simulations to achieve effectiveness factors between 0 and 1.

MAS Rate Effects in Longitudinal Relaxation ssNMR Measurements
Andrew et al. 33 provided the first account of the effect of rotational resonance on the longitudinal relaxation.They observed an equal  !* s for two P #! sites with distinct  ! at static conditions, if the MAS rate matched that of the frequency between the two sites.Other studies have shown the surprising effects of MAS on the longitudinal relaxation rate constants.For example, Roos et al. 34 observed that moderate MAS enhances 1 H spin diffusion; while Krushelnitsky et al. 35 recognized that 15 N relaxation times in uniformly-labelled proteins can be significantly distorted by spin diffusion effects that are MAS-rate dependent.Similar dependency in 2 H longitudinal relaxation has been observed (see Cutajar et al. 36 , their Tables 1 and 2).
We observed a similar behavior of longitudinal relaxation in two other series of measurements: for (1)

Effects in Common and Emerging Measurements
Manipulation of the longitudinal relaxation time by MAS can be used to reduce experiment time.In a similar setup, band-selective pulses 11 enabled faster 2D measurements such as SOFAST-HMQC 37 by exploiting fast relaxation of specific species.Spin diffusion ssNMR DNP techniques are two other measurements that may be affected by processes described in this work.Spin diffusion experiments are commonly employed in structural measurements and employ virtually the same pulse sequence scheme as that of the longitudinal relaxation measurement 38 .Rotational resonance recoupling provides a very specific transfer mechanism allowing long-range magnetization transfer (for example see 39 ) that is a central component of structural analysis in biomolecules by ssNMR 13 .
In 13 C-13 C spin diffusion measurements on 13 C-labelled L-histidine, Dumez and Emsley 13 observed a complex non-monotonic influence of the spinning rate on the rate constant of magnetization transfer -analogous to the cross-relaxation rate constants measured here.They recognized that the spinning frequency dependence is associated with the chemical-shift differences between sites and is dominated by the proximity to rotational resonance.Despite their measurements at only three spinning rates of {10, 15, 20} kHz at 16.44 T, the 13 C NMR data of Dumez and Emsley 13 show similar  ( dependence as what we observe in our present experiments (see their Figure 4).We envision new amino acid type-selective schemes 40,41 in ssNMR based on rotational resonance at few spinning rates could be established to complement common assignment pulse sequences.Such experiments would remove ambiguity or eliminate the need for selective labeling procedures for large protein systems 42 .
The significant influence of MAS rate on the longitudinal relaxation rate constants observed here raises questions about how rotational resonance may effect magnetization transfer between 15 N-15 N, 13 C-13 C, and 2 H-2 H 14,36 in uniformly isotope-labeled samples and hence whether all measurements are directly or indirectly affected by it.Functional dependence of the longitudinal relaxation on the spinning rate for different spin pairs guides 15 N, and 13 C, spin diffusion experiments by (a) optimizing magnetization transfer between a specific spin pair, (b) reducing experiment time, (c) careful selection of the magnetic field and spinning rates, and (d) promoting long-range magnetization transfer.Even in natural-abundance systems, the MAS rate affects spin diffusion in proximity to rotational resonance conditions as shown in an early study 43 (see their Figure 3).
A correct understanding of the effect of rotational resonance on the longitudinal relaxation is important also in DNP experiments -where the focus is on how to transfer magnetization to a desired nuclear site.While amine 15 N and methyl 13 C nuclei have short relaxation times and act as magnetization sinks in longitudinal relaxation measurements, they serve as entry points in DNP of biological samples 44 .For example, it was recently observed that a single methyl-nucleotide contact could be responsible for most of the DNP transfer to RNA 45 .Furthermore, it has recently been shown by Biedenbänder et al. 12 that spinning speeds close to the rotational resonance conditions expedite magnetization transfer in SCREAM-DNP 44 .A better understanding of magnetization transfer by rotational resonance offers more efficient pathways in DNP.Sweeping spinning speeds could be another strategy to expedite magnetization transfer between many sites 46 .

Perspective
The recoupling of measurable dipolar coupling requires being almost exactly on the rotationalresonance condition -as evidenced by the very narrow resonance conditions observed in Figure 4.
In fact, the rotational-resonance width experiment 47,48 demonstrates that the width of the resonance is of similar size as that of the dipolar coupling.In this case, the dipolar couplings leading to the dramatic change in longitudinal relaxation are only in the order of 10-20 Hz.However, when the rotational resonance is to enhance subtle effects like relaxation, this requirement for proximity to the rotational-resonance condition is relieved, thereby allowing a more flexible experimental setup.The fact that proper manipulation of a weak dipolar coupling may lead to a change of three orders of magnitude for another observable parameter is very encouraging for designing new experiments to probe long-range effects in solid-state NMR.If it were possible to transfer the same types of measurements to 1 H and measure effects of 1 H- relaxation.This work concludes that although not exactly at rotational resonance conditions, many ssNMR experiments may be affected by it.This phenomenon may be employed in promoting magnetization transfer between specific nuclei and utilizing shorter repetition times in experiments.
N Relaxation at 22.3 T The imidazole N !" sites of N * ( + = 189.6 ppm) and N , ( -= 176.2 ppm) exhibit variations in  !The local minima of the longitudinal relaxation for N * and N , exactly match the rotational resonance conditions; they occur at  ( = 13702 Hz, 6851 Hz (corresponding to the  = 1, 2 rotational resonance condition between N : and N * ),  ( = 12412 Hz, 6206 Hz (corresponding to the  = 1, 2 rotational resonance condition between N : and N , ).Close examination of Figure 1b reveals multi-site magnetization transfers of N , ↔ N * ↔ NH # ; (visible in Figure 1b at 13.702 kHz and 6.851 kHz) and N * ↔ N , ↔ NH # ; (visible in Figure 1b at 12.412 kHz and 6.206 kHz) especially at the exact rotational resonance conditions.Giraud et al. 4 observed similar multi-step magnetization transfers in 15 N spin diffusion experiments on a uniformly-15 N-labelled catabolite repression HPr-like protein.In their work employing 10 kHz spinning speed at 11.7 T, magnetization transfer was between linked residues in the order according to the protein sequence.

Figure 1 :
Figure 1: 15 N apparent longitudinal relaxation time constants  !* of N * , N , , and N : ( = NH # ; ) in Lhistidine and glycine as a function of MAS rate on linear (a) and log (b) scales at 25°C.For all cases, the  !* values are determined from monoexponential fits.For  ( > 25 kHz,  !* s are approximately

Figure 2
Figure 2 displays the results of such parameter-estimation simulations at  ( = 10 and 12 kHz as illustrative examples.In all cases, the evolution of magnetization  > ( ) ) at the three sites is matched by simulations (results not shown for all MAS rates).It was possible to match all three decay curves at all MAS rates with the single set of  !,* ,  !,, , and  !,, values corresponding to  !,& = 1/ !,& regarded as true incoherent longitudinal relaxation times and with MAS-rate-dependent cross relaxation rate constants  .*( ( ),  :, ( ( ), and  ,* ( ( ).

Figure 4
Figure 4 demonstrates the results of numerical experiments with SIMPSON for ( ( ) with  ( ∈[200, 42000] Hz (see Data and Codes).Rotational resonance was observed for at least six rotational resonance conditions (vertical lines in Figure4) due to the presence of chemical shift anisotropy.The close similarity of Figure4to Figure3indicates the soundness of simulations and of rotational resonance recoupling of the homonuclear 15 N-15 N dipole-dipole coupling as the source of magnetization transfer.No protons were considered in simulations, indicating that it is the direct magnetization transfer between the three 15 N sites that determines the functional dependence of longitudinal relaxation to MAS rate.Previously, it was observed that 1 H does not have effects on 15 N-15 N and 13 C-13 C spin diffusion as shown by deuterated uniformly-labeled solid protein 4 and peptide 14 samples (also see Agarwal et al.32 ).

Figure 4 :
Figure 4: The effectiveness factor  of magnetization transfer between the three 15 N nuclei in Lhistidine as a function of MAS rate  ( on linear (a) and logarithmic (b) scales.Magnetization transfer is simulated using SIMPSON at 22.3 T by considering chemical shift anisotropy and dipolar coupling for the three 15 N atoms.The peaks indicate rotational resonance conditions.The vertical lines mark expected rotational resonance conditions of the peaks with the same color.Experimental magnetization transfer (---) shows the relative ratio of the transferred longitudinal magnetization at  ) = 4.5 s which shows a strong correlation with the effectiveness factor.

Table S2 )
. A remarkable feature of  .(( = 13698 Hz) = 0.91 and  :, ( ( = 12413 Hz) = 0.90 coincided with the  = 1 rotational resonance condition; while at the second rotational resonance condition, crossrelaxation parameters are approximately an order of magnitude smaller with  .*( ( = 6849 Hz) = 0.12 and  :, ( ( = 6206 Hz) = 0.076. ,* is generally stronger than other cross-relaxation parameters, both due to the stronger dipolar coupling between these nuclei and the smaller chemical shift difference.The cross-relaxation rate constant  , * ( ( ) and  :, ( ( ) is that even minute cross-relaxation values are strong enough to quench  !,* * and  !,, * and draw them closer to  !,: .Strongest cross-relaxation parameters of  .** increases steadily in the direction of smaller  ( values; spin rates around the rotational resonance condition (1290 Hz) is out of reach of the MAS probe employed in this work.
14N in15N-L-histidine at 400 MHz and 25°C, and (2)13C and15N in15N-13C-L-histidine at 950 MHz without temperature control (both series of experiments have a lower resolution; results not shown, data available in research data).Although at much lower resolution of spinning rate and without internal sample temperature control, Fry et al.14independently observed similar  !* trends for both of13C and15N in labelled Glycyl-Alanyl-Leucine 3H2O (GAL) at 800 MHz.Fry et al.14associated this behavior not directly to the rotational resonance, but to the overlap of CSA-governed spinning sidebands with another main peak.In our model compound L-histidine, rotational resonance occurs whenever a sideband of N * or N , coincides with the NH # ; resonance.Magnetization transfer is efficient if two peaks coincide, with the first rotational resonance condition being the most efficient.Although Fry et al.14only considered spin pairs, this picture may be extended to more complex spin systems as shown in this work.
1H homonuclear couplings down to 10-20 Hz -e.g. by deuteration and selective back-substitution of 1 H -it would be possible to probe distances up to approximately 20 Å, thus radically changing current ways to perform solid-state NMR structural studies!ConclusionWe demonstrated the effect of spin rate on the longitudinal relaxation rate constant in isotopicallyenriched biological materials with N!" in two model compounds of L-histidine hydrochloride monohydrate and glycine.Rotational resonance, and mere proximity to it, significantly affected N