RIDER distortions in the CODEX experiments

CSA and dipolar CODEX experiments enable obtaining abundant quantitative information on the reorientation of the CSA and dipolar tensors on the millisecond-second time scales. At the same time, proper performance of the experiments and data analysis can often be a challenge since CODEX is prone to some interfering effects that may lead to incorrect interpretation of the experimental results. One of the most important such effects is RIDER (Relaxation Induced Dipolar Exchange with Recoupling). It appears due to the dipolar interaction of the observed X-nuclei with some other 10 nuclei, which causes an apparent decay in the mixing time dependence of the signal intensity reflecting not molecular motion but spin-flips of the adjacent nuclei. This may hamper obtaining correct values of the parameters of molecular mobility. In this contribution we consider in detail the reasons, why the RIDER distortions remain even under decoupling conditions and propose measures to eliminate them. Namely, we suggest the additional Z-filter between the cross-polarization section and the CODEX recoupling blocks, which suppresses the interfering anti-phase coherence responsible for the X-H RIDER. The 15 experiments were conducted on rigid model substances as well as microcrystalline H/N-enriched proteins (GB1 and SH3) with a partial back-exchange of labile protons. Standard CSA and dipolar CODEX experiments reveal a fast decaying component in the mixing time dependence of N nuclei in proteins, which can be interpreted as a slow overall protein rocking motion. However, the RIDER-free experimental setup provides flat mixing time dependencies meaning that the studied proteins do not undergo global motions on the millisecond time scale. 20

synchronised recoupling π-pulses applied on X-nuclei reintroduce the CSA interaction and thus, the phases Φ 1 and Φ 2 are 30 determined by the precession under the influence of the CSA interaction during the de(re)phasing periods. Potentially interfering dipolar interactions with protons are supposed to be averaged out by proton decoupling during the de(re)phasing periods. However, CODEX can be easily modified for observing motionally modulated dipolar interaction or isotropic chemical shift (i.e. chemical exchange). This can be achieved by a corresponding modification of the recoupling pulses (Krushelnitsky et al., 2013;Reichert and Krushelnitsky, 2018). 35 Figure 1. A simplified scheme of the CODEX pulse sequence. Black vertical bars denote π/2-pulses, φ CP , φ 1 , φ 2 and φ Rec are the phases of the X-channel cross-polarisation pulse, two π/2-pulses and the receiver, respectively. The COS-component is recorded 40 when the phase differences are φ CP -φ 1 =π/2 and φ 2 -φ Rec =π/2, the SIN-component corresponds to φ CP =φ 1 and φ 2 =φ Rec .
In the CODEX experiment, one can measure the signal intensity as a function of both mixing time and the length of the de(re)phasing periods NT R (T R is the MAS period and N is the number of rotor cycles in the de(re)phasing periods), which provides the information on both time scale and geometry of a molecular motion (Luz et al., 2002). Thus, the CODEX experiment enables obtaining more abundant quantitative information on molecular dynamics in comparison to standard 45 NMR relaxation studies. At the same time, CODEX is prone to some interfering effects that may distort the information on molecular dynamics and that should be taken into account in the data analysis. Two most important effects are the protondriven spin diffusion between X-nuclei and RIDER (Relaxation Induced Dipolar Exchange with Recoupling) (Saalwächter and Schmidt-Rohr, 2000). Spin-diffusion reveals itself as a signal decay in the mixing time dependence, which can be in some cases erroneously attributed to a molecular motion process. Suppressing the spin-diffusion by proton decoupling 50 during the mixing time is in principle possible, but rather difficult and not always reliable and effective (Reichert and  Preprint. Discussion started: 16 September 2020 c Author(s) 2020. CC BY 4.0 License. Krushelnitsky, 2018). The most robust way of removing the undesirable spin-diffusion effect is a spin dilution, e.g. using natural abundance 13 C or perdeuterated samples.
RIDER also leads to an additional decay in the mixing time dependence. Dipolar interaction of X-nuclei (S) with either protons or some other magnetic nuclei present in a sample (I), adds two terms of the precessing X-nuclei magnetization -in-55 phase S x cos(ωt) and anti-phase 2S y I z sin(ωt). The last term is the origin of RIDER, which can be simplistically explained as follows: flips of I z during the mixing time change the sign of the inter-nuclear dipolar interaction (for 1/2-nuclei) and thus change the sign of the dipolar contribution to the precession frequency. This in turn leads to incomplete rephasing of the Smagnetization at the end of the rephasing period and thus to decrease of the signal. Therefore, the characteristic time of the decay in the mixing time dependence due to RIDER is determined by the timescale of I z flips, that is, T 1 -relaxation of nuclei 60 I. In addition, if the homonuclear dipolar interaction between I-spins is significant, spin-diffusion (flip-flops) also contributes to the time scale of RIDER, which can be much shorter than T 1 of I-spins. The standard way of suppressing RIDER in the CODEX experiments is heteronuclear I-S decoupling during the de(re)phasing periods. For some I-nuclei with a large quadrupolar moment, e.g. 14 N, decoupling is not effective, and in this case, the only way of removing the undesirable RIDER influence is isotopic editing of a sample. 65 Our interest in the methodological problems of the CODEX experiments was stimulated by the study of the slow motions in solid proteins. Recently, it was shown by means of R 1ρ relaxometry that proteins in a solid state undergo slow overall rocking motion (Ma et al., 2015;Lamley et al., 2015;Kurauskas et al., 2017;Krushelnitsky et al., 2018). The time scale of this motion is tens of microseconds, which is the limit of the time window accessible with R 1ρ relaxation experiments. What happens in the (sub)millisecond time scale up to now remained unclear and the CODEX experiments could answer the 70 question, whether the rocking motion extends to longer correlation times or not.
We have thus conducted CSA and dipolar CODEX experiments on 15 N nuclei in 15 N, 2 H-enriched microcrystalline proteins (SH3 and GB1) with a partial back-exchange of labile protons. These experiments were conducted with a sitespecific resolution in 2D 1 H-15 N correlation spectrum using indirect proton detection of a signal (Chevelkov et al., 2006;Krushelnitsky et al., 2009). Surprisingly, all peaks in 2D spectra without exception reveal decays in the mixing time 75 dependencies as shown in Fig. 2. The amplitude of the decay and the apparent correlation time of the fast component (around 20 ms) are the same for all residues. This component cannot be due to the proton driven spin diffusion since the time scale of the spin diffusion between 15 N nuclei even in fully protonated proteins is much longer (Krushelnitsky et al., 2006). In the CSA CODEX, this could be the RIDER-effect arising due to the dipolar interaction between 15 N and abundant 2 H nuclei. On the other hand, in the dipolar CODEX experiment, we observe very similar shapes of the mixing time dependencies with 80 very similar parameters of the fast component. This was observed both for SH3 and GB1 microcrystalline proteins. In the dipolar CODEX experiment, the recoupling π-pulses are applied on the proton channel and thus, the 15 N-2 H dipolar interaction should be effectively averaged out by MAS. From this one could conclude, that the observed fast component of the mixing time dependencies is not an artefact and does report on a real overall protein motion in the millisecond time scale. This would mean that the rocking motion of a protein in a crystal has a very wide correlation times distribution, from micro-85 to milliseconds.
However, it turned out that the fast decaying component in the mixing time dependencies is actually a highly non-trivial artefact. Its nature proved to be more complicated than the simple 15

Theory
Here we consider the time evolution of spin coherences in the CSA CODEX experiments using product operator 100 formalism. It is well known that after I→S cross-polarization, appear both in-phase S x and anti-phase -2S y I z terms apprear, https://doi.org/10.5194/mr-2020-23

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Preprint. Discussion started: 16 September 2020 c Author(s) 2020. CC BY 4.0 License. see e.g. (Schmidt-Rohr and Spiess, 1994). The anti-phase term is usually neglected since in standard CP/MAS experiments it is suppressed by the heteronuclear proton decoupling during the FID acquisition. In the CSA CODEX, it is supposed to be suppressed by the proton decoupling during the de(re)phasing periods as well. However, in the CODEX pulse sequence this suppression is much less effective. The train of the X-channel recoupling π-pulses applied during the de(re)phasing periods 105 restores not only CSA, but also dipolar X-1 H interaction. Hence, the proton decoupling affects not just the residual (after MAS) dipolar interaction, but the restored value of this interaction. For this reason, the small but appreciable dipolar X-1 H interaction survives during the de(re)phasing periods, which will be demonstrated experimentally below, and we have to take it into account in our analysis. Let us consider the time evolution of the in-phase and anti-phase terms in the CSA CODEX experiment under the simultaneous influence of the CSA and (not completely suppressed) dipolar interactions during the 110 de(re)phasing periods. The phases acquired during the dephasing period under the influence of the CSA and dipolar interactions we denote as Φ CSA and Φ D , respectively. We assume for simplicity that Φ CSA remains the same for both dephasing and rephasing periods, but Φ D can change due to RIDER. Thus, for the rephasing period, the acquired phase will be denoted as Φ D +∆Φ D .
In-phase term, dephasing period: 115 The first two terms are picked up in the COS-component and two second terms -in the SIN-component. At the end of the rephasing period, we have the COS-component: 120 and the SIN-component: In Eqs. (2) and (3) we left only observable terms that correspond only to COS and SIN components, respectively.
Because of the proton decoupling, the phases Φ D and Φ D +∆Φ D are rather small. Thus, we can reasonably assume that 125 and cos(Φ D ) = cos(Φ D + ΔΦ D ) for spin I=1/2 , since ∆Φ D can be either 0 or -2Φ D .
which means that for the in-phase term, the effect of the incomplete suppression of the dipolar X-1 H interaction is almost negligible, it leads only to a small decrease of the signal, proportional to cos 2 (Φ D ).
Let us now consider the time evolution of the antiphase term. At the end of the dephasing period we have: 130 and the SIN-component is: Again, in Eqs. (7) and (8) only the observable terms are left that correspond to the COS (Eq. 7) and the SIN (Eq. 8) 140 components. It is seen from these equations that for the anti-phase term, the RIDER effect is not negligible and the inequality analogous to Eq. (4) cannot be written if Φ D is small but appreciable.
But how can the RIDER effect arising from the anti-phase term be recognized in the analysis of experimental data? This is relatively simple: one may compare the shapes of the mixing time dependence of the COS and SIN components. If these curves, namely the ratio S ∞ /S 0 (S 0 and S ∞ are the signal amplitudes at very short and very long mixing times, respectively), 145 are not similar, than RIDER is relevant. In general, the ratio S ∞ /S 0 for the COS and SIN components should be exactly the same, if only molecular motions and/or spin-diffusion are present in a sample. This can be proved as follows. Let us denote the phases acquired during the dephasing and rephasing periods as Φ and Φ+∆Φ, respectively. At short mixing times, ∆Φ=0, then the ratio S ∞ /S 0 for different cases would be as follows.
Hence, the ratio of the anti-phase-term amplitude to the in-phase-term amplitude is proportional to tan(Φ D ) or even smaller if the second terms in the parentheses in Eqs.
(2) and (7) are taken into account. The same ratio for the SIN-component, see Eqs. (3) and (8), is proportional to 2•tan(Φ D ). Thus, the contribution of the anti-phase coherence to the total signal is larger for the  The analysis presented above is valid only for an isolated I-S spin pair. For multinuclear spin systems, the description would be much more complicated since many types of multiple coherences with a complex network of homo-and heteronuclear dipolar interactions should be taken into account. Quantifying this is outside the frames of our work, still we believe that on a qualitative level, two most important points remain valid: first, the anti-phase term appearing after the crosspolarization pulses may cause RIDER distortions of the mixing time dependencies and second, the RIDER effect can be 180 recognized from the comparison of the shapes of the COS and SIN components. This will be proven experimentally below.

Samples
In our work we used four different samples. Model substances: 15  recent work on R 1ρ relaxometry (Krushelnitsky et al., 2018). Both protein samples were prepared according to the protocol ensuring 20% of the back-exchange of labile protons. However, we believe that in reality this percentage is somewhat different: in GB1 it is higher which is indicated by stronger signal and faster proton-driven spin-diffusion between 15 N nuclei 190 (see Figs. 12 and 13 below). The quantitative estimation of this difference is yet rather difficult and uncertain. Since the GB1 sample provides better signal-to-noise ratio, most of the experiments were conducted with this sample.

NMR experiments
The experiments were performed on a Bruker AVANCE II NMR spectrometer (600 MHz) with a 3.2 mm MAS probe. In the CODEX experiments with the protein samples, the integral intensity of the entire signal was determined without site-195 selective specification (except for the data shown in Fig. 2). One-dimensional double cross-polarisation ( 1 H→ 15 N→ 1 H) proton-detected spectra for SH3 and GB1 proteins were shown in (Krushelnitsky et al., 2018). For the BOC-Gly and Gly samples the direct 15 N or 13 C signal detection was employed, and for the protein samples we used indirect 1 H signal detection of the 15 N CODEX mixing time dependencies. This was implemented using back cross-polarisation section ( 15 N→ 1 H) at the end of the pulse sequence, according to the approach described earlier (Chevelkov et al., 2006;Krushelnitsky et al., 2009). 200 We have checked -the direct 15 N and indirect 1 H signal detections in the protein samples provide the same shape of the CODEX mixing time dependencies, in the latter case the signal-to-noise ratio was however better.
To exclude the effect of spin-lattice relaxation during the mixing time, each CODEX mixing time dependence was T 1normalized. For that, for each mixing time dependence two experiments were performed: measuring the mixing time dependence itself and measuring a T 1 -relaxation curve within the same time range. After that, the mixing time dependence 205 was divided by T 1 -relaxation curve. This is a routine procedure described earlier (deAzevedo et al., 1999;deAzevedo et al., 2000;Reichert et al., 2001;Reichert and Krushelnitsky, 2018). Below are shown only the T 1 -normalized mixing time dependencies for all CODEX experiments in protein samples. For BOC-Gly, the T 1 -normalization was not performed since 15 N T 1 in this sample was extremely long (800-900 s).
The pulse sequences of the CSA and dipolar CODEX are shown in Figs. 3 and 4. For measuring the mixing time 210 dependence, τ m was variable and τ r was fixed at 1 ms; for measuring T 1 -relaxation curve, τ m was fixed at 1 ms and τ r was variable. The phase cycle for both COS and SIN components consists of 64 steps: 2x spin-temperature inversion (ensuring that the signal decays to zero (Torchia, 1978)) for T 1 -relaxation, 2x spin-temperature inversion for mixing time, 4x CYCLOPS for the π/2-pulse after mixing time, 4x CYCLOPS for the π/2-pulse after τ r delay (Reichert et al., 2001). Typical values for π/2 pulse for 1 H and 15 N channels were 1.4-1.8 µs and 6.0-6.5 µs, respectively. 215 The experimental error in estimation the signal amplitude was: less than 1% for BOC-Gly, 1-2% for GB1, 2-4% for SH3 and 5-10% for natural abundance 13 C in Gly. On top of the signal noise, a certain contribution to the experimental error in the mixing time dependences comes from the instability of the MAS controller; that was however significant only for BOC-Gly. The final error of the mixing time dependencies for this sample was around 1-2%. For better visual distinguishing https://doi.org/10.5194/mr-2020-23

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Preprint. Discussion started: 16 September 2020 c Author(s) 2020. CC BY 4.0 License. between the mixing time dependencies shown in Figs. 5, 7 and 8, the adjacent averaging over 5 points filter was applied to 220 the experimental curves in these figures, which significantly decreased the noise spread of the points without the change of the overall shape of the curves. bars denote π/2 and π pulses, respectively. The mixing time τ m is an integer multiple of the MAS period, which is achieved by MAS rotor triggering before and at the end of the mixing time (see details in Reichert and Krushelnitsky, 2018) . Rotor synchronization during the τ r -delay is not necessary. Waltz decoupling in the indirect detection sequence aims to suppress only J-coupling between X and 1 H nuclei, therefore it has low amplitude (few hundred Hz). An additional initial Z-filter and 2 H-decoupling (see below) are not shown.  x, x, y, y, -x, -x, -y, -y); φ 7 =(y, -y, -x, x, -y, y, x, -x); φ 8 = (-x, -x, -y, -y, x, x, y, y, x, x, y, y, -x, -x, -y, -y) y, y, -x, -x, -y, -y, x, x, -y, -y, x, x, y, y, -x, -x) ,-x, y, -y, -x, x, -y, y, -x, x, -y, y, x, -x, y, -y) 15 N-enriched BOC-Glycine is a rigid solid sample in which we do not expect any molecular motion on the millisecond time scale. Thus, the CODEX mixing time decays can be only due to the RIDER effect since the proton-driven spindiffusion between 15 N nuclei in BOC-Gly is very slow (Krushelnitsky et al., 2006). First, we demonstrate that the anti-phase 250 term discussed above does really cause RIDER distortions in the CSA CODEX mixing time dependence. The anti-phase term appears in the course of cross-polarization; thus, its contribution to the total CODEX signal should depend on the CP contact time. The CSA CODEX mixing time dependencies at various CP times are shown in Fig. 5. These data fully confirm the qualitative theoretical analysis presented above. One may see that the amplitude of the RIDER decay depends on the CP If the heteronuclear proton decoupling during the de(re)phasing periods was effective enough, than the RIDER effect caused by the anti-phase coherence could have been of course avoided. However, this is not always possible for practical reasons because of the hardware limitations for the power of the long proton pulses. We tried to optimize the proton decoupling by the maximum signal at short mixing times. Different decoupling schemes were checked (TPPI, WALZ, 270 SPINAL) at maximum proton power around 100-130 kHz, but none of them provided much better efficiency than simple CW decoupling (which is not the case for 1 H-decoupling during FID, where CW is not the best choice). Therefore, in the experiments shown here we used CW 1 H decoupling during the de(re)phasing periods in the CSA CODEX measurements.
We do not claim that CW decoupling is the best option for this purpose. It is quite possible that some other decoupling schemes specifically designed for the case of the recoupling X-pulses can perform better. However, even having such a 275 decoupling scheme at hand, one should carefully optimize it for different MAS rates and 1 H field strengths. We suggest here another, more simple and robust way of suppressing the undesired RIDER effect.
The anti-phase term can simply be suppressed by an additional Z-filter between the CP pulses and the dephasing period, as illustrated in Fig. 6. The delay of this Z-filter should be short compared to 15  components at different delays of the Z-filter. It is clearly seen that the Z-filter fully removes the contribution of the antiphase coherence.
Still it is seen that even at long delays of the Z-filter, the mixing time dependencies are not completely flat, as they should be. The observed distortions are obviously the RIDER effect of the in-phase coherence. If the second terms in the parentheses in Eqs. (2) and (3) are not negligibly small, then the RIDER is present also in the in-phase coherence and the Z-290 filter cannot remove it. Fig. 8 presents the COS and SIN components of the mixing times dependencies at different durations of the de(re)phasing periods measured with the additional Z-filter. It is clearly seen: the longer NT R , the larger the RIDER distortions. This is reasonable since Φ D is proportional to NT R . Note that the COS component is less prone to distortions not only in the case of the "anti-phase", but also of the "in-phase" RIDER. We are not able at the moment to explain the unusual bell-shaped form of the mixing time dependencies. It is likely that the network of multi-nuclear dipolar interactions should 295 be taken into account and thus the explanation will not be simple. We also cannot exclude that transient NOE effects may play a certain role.
However, in any case this is the unwanted distortion and regardless of the exact nature of this distortion it should be maximally suppressed in real experiments. For this, the efficiency of the 1 H-decoupling during the de(re)phasing periods must be optimized as far as possible. As mentioned above, the standard heteronuclear decoupling schemes used for FID 300 detection do not help much for the case of de(re)phasing periods. This is illustrated by the example of SPINAL sequence, see Fig. 8. Still one may minimize the "in-phase" RIDER effect by keeping NT R as short as possible and by recording only the COS component of the mixing time dependence. Anyway, the "in-phase" RIDER is much smaller than the "anti-phase" one and in most real experiments it can be safely neglected, as we will see below by the example of the protein samples. At the end of this section, we demonstrate the application of the additional Z-filter to the natural abundance 13 C CSA 305 CODEX experiment performed on carbonyl carbons in 15 N-enriched glycine ( 15 N enrichment is necessary to avoid the 13 C-14 N RIDER effect). We see the same effect as in the case of 15 N CSA CODEX (Fig. 9).

Protein samples
In the protein samples we have three types of nuclei that we need to take into account -15 N, 1 H and 2 H. The direct and 325 inverse 1 H-15 N cross-polarisation sections employed in the CODEX pulse sequence ensure that in the experiment we observe only those nitrogens that have protons attached, and all 15 N's coupled to 2 H in the protein backbone remain invisible. Still, the interactions between protonated 15 N's and many remote 2 H's can be sufficient to induce RIDER-type distortions in the CODEX experiment. To demonstrate the hierarchy of the inter-nuclear interactions in our samples, we measured 15 N Hahnecho decays (Fig. 10) and the initial signal S 0 (the signal at short mixing time) in the CSA and dipolar CODEX experiments 330 as a function of NT R (Fig. 11) for various combinations of 1 H and 2 H decoupling schemes. Note that the S 0 vs NT R dependence is in fact the analogue of the Hahn-echo experiment, the only difference is that either CSA or dipolar interaction is reintroduced by means of recoupling pulses during the transverse relaxation.
The conclusions that can be deduced from these data are as follows. First, despite the proton dilution, the 15 N-1 H dipolar line broadening at the MAS frequency 20 kHz remains quite appreciable and strong proton decoupling is necessary to 335 suppress the 15 N-1 H dipolar interaction. The comparison of the S 0 vs NT R dependences of dipolar CODEX in fully protonated BOC-Glycine and the deuterated protein shows that the proton dilution reduces of course the inter-proton interaction (flip-flops) and thus, the rate of the 15 N decay: slower 1 H-flips ensure slower 15 N-1 H coupling modulation and hence, better refocusing the signal after the end of the rephasing period. Still the rate of the proton flip-flops in the protein sample remains in the millisecond range. This is an important point, which will be discussed below. This result corresponds 340 very well to the proton line width estimations made by B. Reif and co-workers (Chevelkov et al., 2006). Second, it is clearly seen that the 130 kHz CW-decoupling performs much worse in comparison to the SPINAL scheme ( Fig. 10). As mentioned above, under the influence of the 15 N recoupling π-pulses during the de(re)phasing periods, SPINAL does not provide significant advantage in comparison to CW. This confirms our previous statement that the proton decoupling efficiency under the influence of the X-channel recoupling pulses is much different in comparison to FID 345 detection.   Preprint. Discussion started: 16 September 2020 c Author(s) 2020. CC BY 4.0 License.
Third, the 15 N-2 H interaction is non-negligible. 2 H decoupling does not lead to longer the Hahn-echo decays since it is effectively (but not completely, see below) reduced by MAS even without decoupling. However, the reintroduction of the 15 N-2 H dipolar coupling by the REDOR pulse train applied on deuterons appreciably shortens the decays, see Fig. 10. In the CSA CODEX experiment, the 15 N-2 H interaction is initially reintroduced by means of REDOR pulse train applied on 15 N's.
In this case, the 2 H decoupling has the effect and makes the decay slower (Fig. 11). 360 Now the recipe for a methodologically correct CSA CODEX experiment is evident. In deuterated proteins, there are two simultaneous RIDER effects arising from the 15  This is spin diffusion between 15 N nuclei, which is easy to prove. The spin diffusion rate should not depend on temperature 390 and should depend on the MAS rate (Reichert et al., 2001;Krushelnitsky et al., 2006). We measured the mixing time dependence for the GB1 sample at two temperatures and two MAS rates, see Fig. 14. The results shown in this figure leave no doubts that this is the ordinary proton driven spin-diffusion. The rate of these decays is approximately 3-4 times slower than the spin-diffusion rate in a fully protonated protein (Krushelnitsky et al., 2006) still it is quite appreciable. Spin diffusion rate in SH3 protein is noticeably slower; we believe this is due to the lower proton density in this sample, which is 395 confirmed by a somewhat weaker signal from SH3 compared to GB1.

Dipolar CODEX
The principal problem of the dipolar CODEX is that the 15 N-1 H interaction cannot be decoupled for obvious reasons and thus the antiphase term responsible for the RIDER effect emerges explicitly during the de(re)phasing periods. To solve this problem, in our first paper on dipolar CODEX (Krushelnitsky et al., 2009) we suggested to measure only the COS 400 component of the mixing time dependence. The COS component must be RIDER-free, which directly follows from the Eq.
(2). In the dipolar CODEX Φ CSA =0, and since cos(Φ D )=cos(Φ D +∆Φ D ) (we repeat again that this is valid only for I=1/2), the COS-component of the dipolar CODEX mixing time dependence should not be affected by the 15 N-1 H RIDER. However, this is only true under the condition that we did not pay a proper attention to at that time. This condition is: the dipolar interaction must be constant during the de(re)phasing periods, i.e. the time scale of the I-spin flips should be much longer 405 https://doi.org/10.5194/mr-2020-23

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Preprint. Discussion started: 16 September 2020 c Author(s) 2020. CC BY 4.0 License. than NT R . If this is not so, then cos(Φ D )≠cos(Φ D +∆Φ D ) since Φ D and Φ D +∆Φ D are randomly modulated by I-spin flips within the de(re)phasing periods. Thus, the COS component at this condition is not RIDER-free anymore.
The comparability of NT R and the time scale of proton spins filps is exactly our case. We have estimated above the characteristic time of the protons flip-flops, which is about 10 ms (Fig. 10). The duration of the de(re)phasing periods in the CODEX experiments is usually from few hundred microseconds to several milliseconds. This is shorter than 10 ms but still 410 comparable, which is enough for the RIDER effect. From this we pessimistically conclude that the X-H dipolar CODEX experiment even in proton-diluted samples like deuterated proteins with a partial back-exchange of labile protons is not suitable for studying slow molecular dynamics -there will always be RIDER distortions. This experiment, however, can be implemented using other nuclei pairs, e.g. 13 C-15 N (McDermott and Li, 2009), ensuring that the flip-flop time is much longer than the duration of the de(re)phasing periods. 415 The last point that deserves to be discussed here is the influence of the 15 N-2 H interaction on the dipolar CODEX results.
At first sight, there should be no influence, since this interaction is not reintroduced in the dipolar CODEX and it should be simply suppressed by MAS. However, this is not the case. Fig. 15 presents the mixing time dependences in GB1 measured at different powers of the CW-2 H-decoupling during the de(re)phasing periods. The data demonstrate that in spite of MAS, the 15 N-2 H interaction has a small but well visible contribution to the short component, i.e. RIDER effect, of the mixing time 420 dependence. The residual 15 N-2 H interaction is rather small since only few kHz of CW decoupling is enough to suppress it completely. So, why MAS does not do its job alone, without the rf-decoupling? The reason is the protein mobility in the microsecond time scale. If the 15 N-2 H interaction is modulated by a molecular motion on a time scale of the MAS period (for 20 kHz it is 50 µs), then MAS cannot suppress this interaction completely, which leads to the increased linewidths of the MAS centerband (Suwelack et al., 1980). As we know, the correlation time of the protein rocking motion is few tens 425 microseconds (Krushelnitsky et al., 2018). On top of that, there can be interaction of protein nitrogens with deuterons of solvent molecules, and these molecules can also reveal a mobility in the microsecond time scale. Thus, the appearance of the residual 15 N-2 H interaction after MAS can be reasonably explained.  15 N-2 H interaction, and in the dipolar CODEX it is the 15 N-1 H interaction. As estimated above, the time constants of the two RIDER effects are similar but not the same: 2 H spin-lattice relaxation is somewhat slower than the proton flip-flop rate. Therefore, the apparent decay rate of the short component in the CSA and dipolar CODEX experiments should also be somewhat different. This is illustrated in Fig. 16, which presents the fast RIDER-components of the CSA and dipolar 440 CODEX experiments after subtraction of the spin-diffusion component and normalization of the decay amplitudes to the same value. The direct comparison of these decays is in a perfect agreement with the findings described above. Interesting to note that in SH3, the difference of the apparent correlation times of the short component for the CSA and dipolar CODEX is much smaller, see Fig. 2 (τ c as a function of the residue number). This can also be reasonably explained by the different proton density in the GB1 and SH3 samples: the lesser the proton density, the slower the flip-flop rate and thus, the smaller 445 the difference between the rates of proton spin diffusion and deuteron spin-lattice relaxation.

Conclusions.
1) The comparison of the shapes of SIN and COS components of the mixing time dependences is a simple and robust way of detecting the presence/absence of the RIDER effect in the CODEX experiments. The COS component is less prone to the RIDER distortion (appearance of the short component) and for minimising this distortion, it is advisable to record and to 455 analyse in experiments only the COS component.
2) Proton decoupling under the influence of the recoupling π-pulses applied on X-channel is not as effective as in the case of normal X-nuclei FID detection. Thus, the suppression of the antiphase coherence emerging after the crosspolarisation section can be incomplete in CSA CODEX. This may lead to the RIDER distortion in mixing time dependences. This problem can be effectively resolved by inserting additional Z-filter between the cross-polarization section and the 460 dephasing period.