Cross-peaks in simple 2D NMR experiments from chemical exchange of transverse magnetization

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 characterization of underlying exchange processes.

molecules, 1 supramolecular chemistry and host/guest interactions, 2 and biomolecular function and interactions. 3 By using rf pulses to perturb the magnetisation of systems in dynamic equilibrium, the associated chemical exchange processes can be characterised with high precision across a wide range of timescales. A variety 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 1ρ relaxation dispersion. 3 All of the above experiments are based on the characterisation of exchange occurring during a specific mixing time within the pulse sequence. However, resonance lineshapes are also directly sensitive to chemical exchange processes, provided that the exchange rate, kex, is within one or two orders of magnitude of the frequency difference, ∆ω, between the exchanging resonances. Therefore, in a long-standing approach termed lineshape analysis or 'dynamic NMR', one-dimensional (1D) spectra may be fitted in a least-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. The approach has also been extended to two-dimensional (2D) lineshape analysis: the 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 a variety of biomolecular interactions. 10,11 As part of an effort to validate the accuracy of 2D lineshape analysis, we acquired a series of measurements of the small molecule N ,N -dimethyl-trichloroacetamide (DMTCA) (Fig. 1A,B), the two methyl groups of which undergo exchange by rotation about the amide bond with a rate of 125 s −1 at 298 K. 12-14 DMTCA is a simple 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 acquired a series of 2D  For very short mixing times, the propagator e Kτm reduces to the identity operator, and no exchange cross-peaks are expected. However, when such an experiment was acquired for DMTCA (with a near-zero 20 µs mixing time), cross-peaks were unexpectedly observed at frequencies (ω A , ω B ) and (ω B , ω A ), with intensities ca. 5% that of the diagonal peaks ( Fig. 1C). In contrast to the diagonal peaks, the cross-peaks were not absorption mode, but had a partially dispersive lineshape. When a non-zero mixing time was used, stronger cross-peaks were observed, as expected from the exchange of z magnetization, although a partially dispersive character could again be discerned (3 ms, Fig. 1D).
The origin of these cross-peaks can be understood through analogy with non-equilibrium stopped-flow NMR. 15 Considered in the laboratory frame, evolution of M B therefore gives rise to two dispersive signals, of opposite phases, at frequencies ω A and ω B (Fig. 2C). By symmetry, initial B spins exchanging to A during t 1 give rise to identical dispersive signals. In 1D spectra, these signals are not resolved but contribute to the frequency shifts that occur in slow/intermediate exchange, leading to coalescence. However, the dispersive signals can be resolved directly using 2D NMR, as further evolution during t 2 reveals the origin of the signals in t 1 (Fig. 2C).
Therefore, exchange-induced cross-peaks appear with frequencies (ω A , ω B ) and (ω B , ω A ), as demonstrated experimentally above (Fig. 1C). Having established a conceptual basis for these unexpected exchange cross-peaks, we sought to fit the observed spectra quantitatively, in order to verify our understanding of the process and to characterise the kinetics of the underlying exchange process. 2D lineshape fitting was performed using TITAN 9 (which in this case reduces to the numerical integration of Eq. 1), and fits of NOESY spectra with both zero and non-zero mixing times reproduced the observed spectra very closely (Fig. 1C,D), with fitted exchange rates (indicated in the figures) consistent with published results. [12][13][14] We also investigated the occurrence of exchange cross-peaks in other simple NMR experiments, and found that cross-peaks were also formed in COSY experiments, with a relative intensity of ca. 10% of the diagonal peaks (Fig. S1A). The precise form of these cross-peaks was different from those observed in the NOESY experiment (Fig. 1C), which reflects differences in the transfer of magnetisation between t 1 and t 2 evolution periods between the two pulse sequences, but again, high quality fits and measurement of the exchange rate could be obtained by 2D lineshape analysis (Fig. S1A).
We next explored the occurrence of exchange-induced cross-peaks in heteronuclear single- Coherent transverse magnetization exchange may be predicted using a simple argument (illustrated for an INEPT transfer in Fig. 3A, and assuming no change in the scalar coupling constant between states). An initial population of A spins, considered in the rotating frame of spin A, will evolve with frequency ±πJ depending on the state of the coupled heteronucleus.
Spins that chemically exchange into state B during this period will receive an additional 6 phase shift, resulting in fanning out of their magnetisation vectors. By following these spins through the rest of the sequence, it may thus be observed that spin B magnetization can be generated, with a phase shift, from the initial spin A magnetization. The extent of this exchange-mediated coherence transfer from state A to state B depends on the frequency difference between the states and the transfer time, τ INEPT = 1/2J, relative to the chemical exchange rate. Exact numerical calculations (Fig. 3B,C) show that when both the scalar coupling and frequency difference are comparable to the exchange rate (center of diagrams), a non-trivial population of B magnetization is generated with a phase shift dependant on the frequency difference, as predicted from our schematic argument above (Fig. 3A). To explore the above analysis experimentally, natural abundance 1 H, 13 C HSQC and HMQC spectra were acquired of DMTCA at 298 K and 16.44 T (Fig. 1E,F). Under these conditions, k ex τ INEPT = 0.45 and k ex /∆ωH = 0.11 (marked by an asterisk in Fig. 3B,C), which is a favourable regime to observe the predicted coherence transfer effects. Again, unexpected exchange-induced cross-peaks were observed in both experiments, with a particularly complex phase structure in the case of the HMQC experiment (Fig. 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 . In contrast, in the HMQC experiment all parts of the pulse sequence contribute to the phase of the observed magnetisation, resulting in remarkably complex lineshapes. HMQC spectra acquired at multiple fields to probe the effect of varying ∆ω clearly show that larger frequency differences are associated with larger phase distortions (Fig. S2), as predicted from Fig. 3C, while applying XY16 CPMG pulse trains 19 during HMQC coherence transfer periods suppresses the build-up of phase shifts, greatly simplifying the structure of the exchange cross-peaks ( Fig. S3). Again, high quality fits and measurement of the exchange rate could be obtained by 2D lineshape analysis (Fig. S1A). We note that coherence transfers via the three-bond scalar coupling 3 J CH (not resolved, but estimated to be ca. Finally, to illustrate a potential application of these analyses, we examined the temperature dependence of exchange within DMTCA using a series of NOESY and HMQC experiments ( Fig. 4 and S4). The HMQC 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 ± 0.6 s −1 at 278 K to 482 ± 4 s −1 at 313 K. NOESY and HMQC results both fitted well to the Eyring equation (Fig. 4), fully consistent with previous measurements (∆H ‡ = 67.4 ± 3.9 kJ mol −1 and ∆S ‡ = 15 ± 13 kJ mol −1 K −1 ). 13 We note that at higher temperatures, 2D lineshape analysis of the NOESY experiments, acquired with very short mixing times, was also required in order to characterise the exchange process (Fig. S4).
In conclusion, chemical exchange is well known to induce changes in NMR chemical shifts and intensities, which have been exploited through, for example, longitudinal magnetisation  Figure 4: Temperature dependence of chemical exchange in DMTCA, measured by 2D lineshape analysis of HMQC spectra (coloring as in Fig. 1). Measured rates were fitted to the Eyring equation as shown, together with measurements acquired using NOESY experiments (Fig. S4). in Liouville space, as previously described. 9 Supporting movies S1 and S2 were generated in Mathematica 11 (Wolfram Research Inc., Champaign, Illinois), by stochastic simulation of the evolution of 20000 spins. Two-dimensional lineshape analysis was performed in TI-TAN (v1.6) and uncertainties in the fitted exchange rates were determined by bootstrap resampling. 9 10