Broadband measurement of true transverse relaxation rates in systems with coupled protons: application to the study of conformational exchange

Accurate measurement of transverse relaxation rates in coupled spin systems is important in the study of molecular dynamics, but is severely complicated by the signal modulations caused by scalar couplings in spin echo experiments. The most widely used experiments for measuring transverse relaxation in coupled systems, CPMG and PROJECT, can suppress such modulations, but they also both suppress some relaxation contributions, and average relaxation rates between coupled spins. Here we introduce a new experiment which for the first time allows accurate broadband measurement of transverse relaxation rates of coupled protons, and hence the determination of exchange rate constants in slow exchange from relaxation measurements. The problems encountered with existing methods are illustrated, and the use of the new method is demonstrated for the classic case of hindered amide rotation and for the more challenging problem of exchange between helical enantiomers of a gold(i) complex.


Scalar relaxation effects in the intermediate exchange regime
If the inverse 1/T 1 X of the X spin-lattice relaxation time is not small compared to 2πJ AX , a small modulation is superimposed on the exponential relaxation of transverse magnetization. This is because X spin transitions only cause decoherence of A spin coherence that is antiphase with respect to spin X, since when the A spin doublet components are in phase exchanging them has no effect. The rate of loss of coherence in the AX example therefore oscillates between 1/T 1 A and 1/T 1 A + 1/T 1 X , with period J AX , as the doublet components go in and out of phase. The oscillation between these can be observed directly both in spin-selective spin echo experiments and in the TRUE experiment, which is a broadband analogue of the selective spin echo. It is important to distinguish between the small oscillations caused by scalar relaxation and the much larger oscillations, in which signals can change sign, caused by classical J modulation, as for example in the simple (Carr-Purcell Method A) spin echo. The net effect of scalar relaxation is an equal mixture of "in-phase relaxation" and "antiphase relaxation". For small modulation index (1/T 1 X < J AX ) the average magnetization decay is well approximated by an exponential with decay constant 1/T 1 A + 1/(2T 1 X ), but in extreme cases significant deviations from exponential decay can be seen.
One consequence of scalar relaxation is that real multiplets in experimental spectra are not simply composed of absorption mode Lorentzian lines, as is almost universally assumed (including in standard spectral simulation software). As is (moderately) well known, the bandshape of two chemically exchanging resonances in intermediate exchange (i.e. around coalescence) is not just the sum of two absorption mode Lorentzians, but rather of two Lorentzians with equal and opposite phase deviations from pure absorption mode. It is the combined effect of the two phase shifts that gives rise to the familiar appearance in which the signal amplitude between the maxima is higher, and that outside the peaks lower, than is expected for two overlapping Lorentzian peaks. Exactly the same thing happens with scalar relaxation: the doublet of spin A in an AX spin system shows the same type of bandshape as in chemical exchange, as the magnitude of the scalar relaxation contribution made by spin-lattice relaxation of spin X to the spin-spin relaxation of spin A approaches J AX . Figure S1 shows the result of a Spinach simulation of the A doublet of an AX spin system with J AX = 5 Hz, T 1 A = 1 s and T 1 X = 0.1 s, with the calculated spectrum fitted with two absorption mode Lorentzians (a,b) and with two Lorentzians with equal and opposite phase shifts (c,d). As can be seen, the residuals show significant errors in the former case but a perfect fit in the latter. In this example the parameters were chosen to give a particularly strong effect, but particularly in high dynamic range spectra the lineshape changes caused by scalar relaxation are a potential limiting factor in global spectral fitting software. The impact of homonuclear scalar relaxation on multiplet bandshapes is almost always neglected, but its effect has been noted in macromolecular structure determination (G. S. Harbison, J. Am. Chem. Soc., 1993, 115, 3026-3027). It is much more commonly recognised in the heteronuclear case, especially where quadrupolar relaxation leads to short T 1 s (see e.g. R.K. Harris, Nuclear Magnetic Resonance Spectroscopy, Pitman, London, 1983).

Fig. S1
Doublet of spin A from a conventional 1D 1 H NMR spectrum simulated using Spinach for an AX spin system with J AX = 5 Hz,  A - X = 500 Hz, T 1 A = 1.0 s, T 1 X = 0.1 s. In (a) and (c) the simulated spectrum, fitted curve and individual peaks are shown in blue, red and dashed grey respectively; in (b) and (d), the simulated spectrum and residuals are shown in blue and orange respectively. Fitting used the sum of two pure absorption mode Lorentzian lines of equal amplitudes in (a) and (b); in (c) and (d) equal and opposite phase shifts of the two Lorentzian lines were allowed.

Fig. S2
Relaxation data (apparent T 2 ) for selected well-resolved protons of azithromycin in DMSO-d 6 . Data shown as red, blue and green bars are the results of CPMG, PROJECT, and TRUE T 2 experiments respectively. The error bars indicate the errors estimated in exponential fitting, and assume a normal error distribution. The outliers in the CPMG data are the result of residual J modulation, which makes exponential fitting unreliable for these sites.

Study of amide rotation in N,N-diethylacetamide
To test the consistency between kinetic data obtained with TRUE experiments and with conventional methods, a full analysis of the kinetics of exchange between the methyl resonances of N,N-diethylacetamide (DEA) was performed using the new TRUE method, bandshape analysis, and selective inversion recovery (the Hoffman-  with the frequency scale in Hz centred on the average chemical shift. Experimental data, fitted curve and residuals are shown as blue, red, and orange respectively, with fits using estimated instrumental linewidths of 0.3 Hz (left column) and 2 Hz (right column). An impurity peak at +22 Hz was excised from the experimental spectra before fitting. The sample temperature was regulated at nominal temperatures of 50, 60, 70, 80, 90, 100, and 110 °C, respectively, from top to bottom.

Fig. S8
Arrhenius plot as in Fig. S5, but with an empirical correction of 0.7 s -1 applied to the methylene TRUE data to allow for the effects of unresolved couplings. Linear regression of all the data gave an estimated activation energy of 78.5 ± 1.2 kJ mol -1 .

Study of conformational exchange in [Au 2 (µ-xantphos) 2 ](NO 3 ) 2 complex
CPMG, TRUE and inversion recovery experiments were performed on a sample of [Au 2 (µ-xantphos) 2 ](NO 3 ) 2 (PAuP) in CD 2 Cl 2 as described in section 3 above and in the main text. Fitted variable temperature 1D 1 H NMR spectra and results of Hoffman-Forsén selective inversion experiments are shown in Figs. S10 and S11 respectively, and results are summarised in Tables S5 to S10.      1.0 ± 0.3 0.5 ± 0.3 0.34 ± 0.14 -60 0.6 ± 0.5 0.0 ± 0.5 0.04 ± 0.14    S12 gCOSY spectrum of [Au 2 (µ-xantphos) 2 ](NO3) 2 in CD 2 Cl 2 at -45°C. The cross-peaks between the equatorial and axial methyl protons, and the equatorial methyl and closest aromatic protons of the xanthene ring (X1 and X8) are due to long-range proton J-coupling which is not resolved in the methyl proton signals but contribute to their linewidth significantly. The experiment was optimised to observe such long-range couplings using a 160 ms fixed evolution time prior to t 1 . 2 scans and 108 increments were collected using a spectral width of 5 kHz. Sinebell apodisation, zero filling to 4096 in both dimensions and symmetrisation were applied in magnitude mode 2D FT processing.  ;sp12(p12):wvm:kp_soft:f1 rsnob(cnst50 Hz; NPOINTS=1000)