Cross‐Peaks in Simple Two‐Dimensional NMR Experiments from Chemical Exchange of Transverse Magnetisation

Abstract Two‐dimensional correlation measurements such as COSY, NOESY, HMQC, and HSQC experiments are central to small‐molecule and biomolecular NMR spectroscopy, and commonly form the basis of more complex experiments designed to study chemical exchange occurring during additional mixing periods. However, exchange occurring during chemical shift evolution periods can also influence the appearance of such spectra. While this is often exploited through one‐dimensional lineshape analysis (“dynamic NMR”), the analysis of exchange across multiple chemical shift evolution periods has received less attention. Here we report that chemical exchange‐induced cross‐peaks can arise in even the simplest two‐dimensional NMR experiments. These cross‐peaks can have highly distorted phases that contain rich information about the underlying exchange process. The quantitative analysis of such peaks, from a single 2D spectrum, can provide a highly accurate characterisation of underlying exchange processes.

NMR spectroscopy is an exceptionally powerful technique for the label-free analysis of intramolecular dynamics and chemical exchange,w ith ar ange of applications to fluxional molecules, [1] supramolecular chemistry and host/guest interactions, [2] and biomolecular function and interactions. [3] By using radiofrequencyp ulses to perturb the magnetisation of systems in dynamic equilibrium, the associated chemical exchange processes can be characterised with high precision across aw ide range of timescales.Avariety of experiments have been developed towards this end, including NOESY (also referred to in this context as EXSY), [4] ZZ-exchange spectroscopy, [5] chemical exchange saturation transfer (CEST), [6] and CPMG/R 11 relaxation dispersion. [3] All of the above experiments are based on the characterisation of exchange occurring during as pecific mixing time within the pulse sequence.However,resonance lineshapes are also directly sensitive to chemical exchange processes, provided that the exchange rate, k ex ,i sw ithin one or two orders of magnitude of the frequency difference, Dw,between the exchanging resonances.T herefore,i nal ong-standing approach termed lineshape analysis or "dynamic NMR", onedimensional (1D) spectra may be fitted in al east-squares sense to solutions of the Bloch-McConnell or Liouville-von Neumann equations [7,8] that govern the evolution of magnetisation, in order to characterise the chemical exchange process.T he approach has also been extended to twodimensional (2D) lineshape analysis:t he fitting of 2D NMR spectra, by direct simulation of the relevant pulse sequence. [9] This approach, and the associated TITAN analysis software, has since found applications to av ariety of biomolecular interactions. [10,11]  As part of an effort to validate the accuracy of 2D lineshape analysis,weconducted aseries of measurements of the small molecule N,N-dimethyl-trichloroacetamide (DMTCA)( Figure 1A,B), the two methyl groups of which undergo exchange by rotation about the amide bond with arate of 125 s À1 at 298 K. [12][13][14] DMTCAisasimple molecule, with no resolved homonuclear scalar couplings,but serves to illustrate fundamental principles that will be equally applicable to more complex molecules and exchange processes.
We first performed as eries of 2D 1 H, 1 HN OESY (aka EXSY) experiments to characterise chemical exchange within DMTCA ( Figure 1C,D). An essentially complete mathematical description of the NOESY experiment for ao ne-spin system in the presence of exchange has been given by Jeener et al., [4] describing the observed magnetisation M + as af unction of the evolution periods t 1 and t 2 ,a nd am ixing time, t m [Eq. (1)].
Here K, W, R,a nd L are the superoperators describing chemical exchange,c hemical shift evolution, transverse relaxation, and longitudinal relaxation, respectively.T he observed magnetisation M + is cosine modulated in t 1 ;s ine modulation can be obtained by taking the imaginary component in Equation (1). TheNOESY pulse sequence is designed to select z magnetisation that is present in the mixing time t m , and we will consider short mixing times such that longitudinal relaxation (or cross-relaxation) can be neglected. In conventional applications of the experiment, in-phase cross-peaks are generated as ar esult of chemical exchange of z magnetisation during the mixing time,a nd the experiment thus creates areadily interpreted "map" of the propagator e Kt m for the exchange superoperator. Forv ery short mixing times,t he propagator e Kt m reduces to the identity operator, and no exchange cross-peaks are expected. However,w hen such an experiment was acquired for DMTCA (with anear-zero 20 msmixing time), cross-peaks were unexpectedly observed at frequencies (w A , w B )and (w B , w A ), with intensities ca. 5% that of the diagonal peaks ( Figure 1C). In contrast to the diagonal peaks,t he crosspeaks were not absorption mode,b ut had ap artially dispersive lineshape.W henanon-zero mixing time was used, stronger cross-peaks were observed, as expected from the exchange of z magnetisation, although ap artially dispersive character could again be discerned (3 ms,F igure 1D). We note that zero-quantum artifacts cannot account for these cross-peaks,a sz ero-quantum coherences formed during t m give rise to antiphase dispersive signals that will be strongly suppressed because the exchange-broadenedl inewidth (ca. 20 Hz) is much larger than the 4 J HH coupling constant (estimated to be ca. 0.4 Hz by analogy with ar elated molecule [15] ).
Theorigin of these cross-peaks can be understood through analogy with non-equilibrium stopped-flow NMR. [16,17] Following Christianson and Landis, [17] we consider first only "A" spins present at the beginning of the t 1 evolution period, and work in the rotating frame of spin B. As the Aspins precess, those that chemically exchange to state Bhave zero frequency in this frame and do not precess further (barring further chemical exchange). In other words,t he Am agnetisation vector can be envisaged as "dropping" Bspins behind it as it precesses ( Figure 2A,M ovies S1 and S2 in the Supporting Information). As illustrated by this schematic, the total magnetisation of these Bs pins, M B ,p recesses with offset W A = w A Àw B about apoint displaced from the origin. This can be represented ( Figure 2B)a st he vector sum of magnetisation, M' B ,along + y,and magnetisation, M'' B ,that is initially along Ày and precesses with offset W A .C onsidered in the laboratory frame,evolution of M B therefore gives rise to two dispersive signals,o fo pposite phases,a tf requencies w A and w B ( Figure 2C). By symmetry,initial Bspins exchanging to A during t 1 give rise to identical dispersive signals.In1Dspectra, these signals are not resolved but contribute to the frequency shifts that occur in slow/intermediate exchange,l eading to coalescence.H owever,t he dispersive signals can be resolved Origin of cross-peaks due to chemical exchange during chemical shift evolution periods. A) Evolution of an initial population of Aspins (blue), undergoingexchange to spin B, depicted in the rotating frame of spin B. Individual Bs pins are shown in pink, and the net magnetisation vector in magenta. Trajectories followed by magnetisation vectors are shown with dashed lines. Amore detailed analysis of the trajectory of spin Bmagnetisation is shown in (B). Adapted from Christiansona nd Landis. [17] C) Components of 2D lineshapes in w 1 and w 2 frequency domains arising from chemical exchange between spins Aa nd Bd uring consecutive chemical shift evolution periods, t 1 and t 2 .Solid arrows show the pathway followed by spins that do not undergo exchange in the indicated time period, while dashed arrows show the pathway of exchanging spins. The final column illustrates the resulting contribution from each pathway to absorption, dispersion, and double-dispersion components of the observed 2D lineshape. directly using 2D NMR techniques,a sf urther evolution during t 2 reveals the origin of the signals in t 1 ( Figure 2C). Therefore,e xchange-induced cross-peaks appear with frequencies (w A , w B )and (w B , w A ), as demonstrated experimentally above ( Figure 1C). These cross-peaks will be observable provided R 2,0 9 k ex 9 Dw,that is,when exchange is comparable to or faster than the decay of transverse magnetisation in the absence of exchange broadening (R 2,0 ), and up to the coalescence point, beyond which only as ingle resonance is observed.
Having established ac onceptual basis for these unexpected exchange cross-peaks,w es ought to fit the observed spectra quantitatively,inorder to verify our understanding of the process and to characterise the kinetics of the underlying exchange process.2 Dl ineshape fitting was performed using TITAN [9] [which in this case reduces to the numerical integration of Equation (1)],a nd fits of NOESY spectra with both zero and non-zero mixing times reproduced the observed spectra very closely ( Figure 1C,D), with fitted exchange rates (indicated in the figures) consistent with published results. [12][13][14] We note that while in principle crosspeaks formed at zero mixing times are dispersive and integrate to zero,i np ractice the long dispersive tails render this impractical and lineshape fitting is therefore recommended.
We also investigated the occurrence of exchange crosspeaks in other simple NMR experiments,a nd found that cross-peaks were also formed in COSY experiments,w ith ar elative intensity of ca. 10 %o ft he diagonal peaks ( Figure S1A). Thep recise form of these cross-peaks was different from those observed in the NOESY experiment ( Figure 1C), which reflects differences in the transfer of magnetisation between t 1 and t 2 evolution periods between the two pulse sequences,b ut again, high-quality fits and measurement of the exchange rate could be obtained by 2D lineshape analysis ( Figure S1A).
We next explored the occurrence of exchange-induced cross-peaks in heteronuclear single-and multiple-quantum coherence experiments (HSQC and HMQC,r espectively). Heteronuclear experiments are typically more complex pulse sequences,c ontaining extended coherence transfer periods and zz filters,which create multiple possibilities for chemical exchange to affect the spectra. Chemical exchange broadening is well understood to reduce the efficiency of coherence transfer in DEPT and INEPT experiments near intermediate exchange regimes, [18,19] but the possibility of coherent exchange of transverse magnetisation between states (outside of the fast exchange regime) does not appear to have been recognised.
Coherent transverse magnetisation exchange may be predicted using as imple argument (illustrated in Figure 3A for the 1/4 JÀp(I x + S x )À1/4 J sequence that forms part of an INEPT transfer, and assuming no change in the scalar coupling constant between states). An initial population of Aspins,considered in the rotating frame of spin A, will evolve with frequency AE pJ depending on the state of the coupled heteronucleus.S pins that chemically exchange into state B during this period will receive an additional phase shift, resulting in the fanning out of their magnetisation vectors.By following these spins through the rest of the sequence,itmay thus be observed that spin Bmagnetisation can be generated, with aphase shift, from the initial spin Amagnetisation. The extent of this exchange-mediated coherence transfer from state At os tate Bd epends on the frequencyd ifference between the states and the total evolution time,2 t = 1/2 J, relative to the chemical exchange rate, k ex .E xact numerical calculations ( Figure 3B,C) show that when both the scalar coupling and frequency difference are comparable to the exchange rate (center of diagrams), anontrivial population of Bmagnetisation is generated with aphase shift dependent on the frequency difference,a sp redicted from our schematic argument above ( Figure 3A).
To explore the above analysis experimentally,n aturalabundance 1 H, 13 CH SQC,a nd HMQC spectra of DMTCA were acquired at 298 Ka nd 16.44 T( Figure 1E,F). Under these conditions,2k ex t = 0.45 and k ex /Dw H = 0.11 (marked by an asterisk in Figure 3B,C), which is af avourable regime to observe the predicted coherence transfer effects.A gain, unexpected exchange-induced cross-peaks were observed in both experiments,with aparticularly complex phase structure in the case of the HMQC experiment ( Figure 1F). This may be rationalised since in the HSQC experiment zz filters suppress phase distortions from the first INEPT transfer and some phase distortions arising from exchange during t 1 .I n contrast, in the HMQC experiment all parts of the pulse sequence contribute to the phase of the observed magnetisation, resulting in remarkably complex lineshapes.H MQC spectra acquired at multiple fields to probe the effect of varying Dw clearly show that larger frequency differences are associated with larger phase distortions ( Figure S2), as predicted from Figure 3C,w hile applying XY16 CPMG pulse trains [20] during HMQC coherence transfer periods suppresses the build-up of phase shifts,greatly simplifying the structure of the exchange cross-peaks ( Figure S3). Again, high-quality fits and measurement of the exchange rate could be obtained by 2D lineshape analysis ( Figure S1A). We note that coherence transfers via the three-bond scalar coupling 3 J CH (not resolved, but estimated to be ca. 3.5 Hz by analogy with ar elated molecule [15] )w ill have ca. 0.01 %o ft he efficiency of the one-bond transfer and cannot account for the amplitudes of the observed cross-peaks.
Finally,t oi llustrate ap otential application of these analyses,w ee xamined the temperature dependence of exchange within DMTCAu sing as eries of NOESY and HMQC experiments (Figure 4a nd Figure S4). TheH MQC experiment was selected, as this was established above to be the most sensitive experiment for observing phase distortions. Exchange rates were determined from 2D fitting of individual spectra, and varied from 19.0 AE 0.6 s À1 at 278 Kto482 AE 4s À1 at 313 K. NOESY and HMQC results both fitted well to the Eyring equation (Figure 4), fully consistent with previous measurements (DH°= 67.4 AE 3.9 kJ mol À1 and DS°= 15 AE 13 kJ mol À1 K À1 ). [13] We note that at higher temperatures,2 D lineshape analysis of the NOESY experiments,acquired with very short mixing times,w as also required in order to characterise the exchange process ( Figure S4).
In conclusion, chemical exchange is well known to induce changes in NMR chemical shifts and intensities,w hich have been exploited through, for example,l ongitudinal magnetisation exchange,lineshape analysis,and relaxation dispersion experiments. [3] In this work, we have shown that chemical exchange may also give rise to detectable phase shifts and coherent transfers of transverse magnetisation. Fors uitable exchange regimes,this provides rapid and accurate characterisation of exchange in asingle 2D spectrum. We have focussed our examples on the simple molecule DMTCA, which does not contain any large homonuclear scalar couplings,b ut the presence of scalar couplings (although not yet implemented within TITAN) is not expected to fundamentally alter this picture.T he approach is also expected to be applicable to more complex multistep reactions.T he analysis of transverse magnetisation exchange is free from complicating effects of cross relaxation, and may be aparticularly useful complement to NOESY measurements where the exchange rate approaches the slow/intermediate exchange regime.M ore generally,o ur work highlights the complexity of spectral features that can arise from exchange and points to the importance of quantitative analysis in terms of the fundamental spin dynamics.
Simulations of exchange during 1/4 JÀp(I x + S x )À1/4 J and XY16-CPMG scalar coupling evolution sequences were performedi n MATLAB (R2016b,T he MathWorks,I nc.), by numerical propagation of density operators in Liouvillespace,aspreviously described. [9] Movies S1 and S2 in the Supporting Information were generatedi n Mathematica 11 (Wolfram Research Inc., Champaign,I llinois), by stochasticsimulationofthe evolutionof20000 spins freely precessing in the xy plane,i nt he absence of relaxation. Tw o-dimensional lineshape analysis was performed in TITAN (v1.6) and uncertainties in the fitted exchange rates were determinedb yb ootstrap resampling. [9] Figure 4. Temperaturedependence of chemical exchange in DMTCA, measured by 2D lineshape analysis of HMQC spectra (coloring as in Figure 1). Measuredr ates were fitted to the Eyring equation as shown, together with measurements acquired using NOESY experiments (Figure S4).