cpmg nmr experiment

CPMG relaxation dispersion NMR experiments measuring glycine 1 H α and 13 C α chemical shifts in the ‘invisible’ excited states of proteins

• Published: 25 March 2009
• Volume 45 , pages 45–55, ( 2009 )

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• Pramodh Vallurupalli 1 ,
• D. Flemming Hansen 1 ,
• Patrik Lundström 1 &
• Lewis E. Kay 1  

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Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion NMR experiments are extremely powerful for characterizing millisecond time-scale conformational exchange processes in biomolecules. A large number of such CPMG experiments have now emerged for measuring protein backbone chemical shifts of sparsely populated (>0.5%), excited state conformers that cannot be directly detected in NMR spectra and that are invisible to most other biophysical methods as well. A notable deficiency is, however, the absence of CPMG experiments for measurement of 1 H α and 13 C α chemical shifts of glycine residues in the excited state that reflects the fact that in this case the 1 H α , 13 C α spins form a three-spin system that is more complex than the AX 1 H α – 13 C α spin systems in the other amino acids. Here pulse sequences for recording 1 H α and 13 C α CPMG relaxation dispersion profiles derived from glycine residues are presented that provide information from which 1 H α , 13 C α chemical shifts can be obtained. The utility of these experiments is demonstrated by an application to a mutant of T4 lysozyme that undergoes a millisecond time-scale exchange process facilitating the binding of hydrophobic ligands to an internal cavity in the protein.

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Acknowledgments

This work was supported by funds from the Canadian Institutes of Health Research (CIHR) in the form of a research grant to LEK and postdoctoral fellowships to DFH and PL (Protein Folding Training Grant). LEK holds a Canada Research Chair in Biochemistry.

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Pramodh Vallurupalli, D. Flemming Hansen, Patrik Lundström & Lewis E. Kay

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Vallurupalli, P., Hansen, D.F., Lundström, P. et al. CPMG relaxation dispersion NMR experiments measuring glycine 1 H α and 13 C α chemical shifts in the ‘invisible’ excited states of proteins. J Biomol NMR 45 , 45–55 (2009). https://doi.org/10.1007/s10858-009-9310-6

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Received : 12 January 2009

Revised : 18 February 2009

Accepted : 26 February 2009

Published : 25 March 2009

Issue Date : September 2009

DOI : https://doi.org/10.1007/s10858-009-9310-6

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Breakdown of Carr-Purcell Meiboom-Gill spin echoes in inhomogeneous fields

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Nanette N. Jarenwattananon , Louis-S. Bouchard; Breakdown of Carr-Purcell Meiboom-Gill spin echoes in inhomogeneous fields. J. Chem. Phys. 28 August 2018; 149 (8): 084304. https://doi.org/10.1063/1.5043495

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The Carr-Purcell Meiboom-Gill (CPMG) experiment has been used for decades to measure nuclear-spin transverse ( T 2 ) relaxation times. In the presence of magnetic field inhomogeneities, the limit of short interpulse spacings yields the intrinsic T 2 time. Here, we show that the signal decay in such experiments exhibits fundamentally different behaviors between liquids and gases. In gases, the CPMG unexpectedly fails to eliminate the inhomogeneous broadening due to the non-Fickian nature of the motional averaging.

Previously, we found that the decay of nuclear induction signal in a magnetic field gradient differs fundamentally in gases compared with liquids, as manifested in the temperature dependence of the nuclear magnetic resonance (NMR) linewidth. 1,2 In the conventional description of NMR, the spectral lines should broaden in a gradient of the magnetic field as temperature increases, 3 which results in a larger diffusion coefficient. However, we found experimentally that in gases, the NMR linewidth instead decreases with temperature, which is consistent with a motional narrowing effect. The importance of this motional narrowing effect was not predicted by the conventional theory. 3,4 This conventional theory 4 was extended by Herzog and Hahn 5 by modeling of the random fluctuations in the resonance frequency (rather than the phase shift) as a Markovian process. An extensive discussion of the exponential time-cubed echo decays can be found in Ref. 6 . In the case of a liquid, the original Hahn result 4 is obtained as a special case. However, the Markovian approach yields an incorrect temperature dependence for gases. In Ref. 2 , we explain that the stochastic process describing the particle velocity process in gases must follow a generalized Langevin equation with memory kernel rather than a memoryless (Markovian) process.

In this article, we demonstrate that this difference in motional averaging between gases and liquids also manifests itself in the signal decay of Carr-Purcell Meiboom-Gill (CPMG) spin echoes. In liquids, a series of spin echoes in the limit of short interpulse spacings minimizes signal decay effects due to diffusion in a gradient (as expected from the conventional theory 3 ). For gases, however, we find that the CPMG is unable to eliminate this signal decay in the limit of short interpulse spacings. This result implies that any T 2 - or diffusion-weighted NMR measurements of gases made in the presence of magnetic susceptibility gradients or applied field gradients are potentially compromised.

In a 90° − τ − 180° − τ spin-echo experiment, the free induction signal in the presence of a magnetic field gradient is given by Hahn’s famous result, 3,4

in which 2 τ is the evolution time (total length of the spin-echo experiment), T 2 is the intrinsic spin-spin relaxation time, γ is the nuclear gyromagnetic ratio, D is the self-diffusion coefficient, and g is the magnetic field gradient ( g = ∂H z / ∂r , where r is a spatial direction). In the absence of a large magnetic field inhomogeneity and large diffusivity, the expression collapses to exp(−2 τ / T 2 ). The result (1) assumes that the Einstein-Fick limit holds, implying that we can make the approximation to the position autocorrelation function (PAF), ⟨ x ( t ) x (0)⟩ ≈ ⟨ x ( t ) x ( t )⟩ = 2 Dt ; i.e., the PAF is approximated by the mean square displacement.

According to the same conventional theory, the signal in the CPMG experiment, which features a train of n echoes (Fig. 1 ), decays according to 3,7,8

where n 2 τ is the total duration of the sequence, which is held fixed. In practice, an upper bound on 2 nτ is imposed by the relaxation time of the sample. In the limit of short echo spacing ( τ → 0, holding n 2 τ constant), the CPMG minimizes the effect of molecular self-diffusion on nuclear spin decoherence in inhomogeneous magnetic fields. Under these conditions, the contribution of the inhomogeneous term becomes negligible, recovering exp(− n 2 τ / T 2 ).

Measurement of T 2 in an inhomogeneous field (gradient, g ) using a CPMG sequence. A 90° radio frequency (r.f.) pulse tips the magnetization, which is refocussed by a series of n 180° pulses at intervals of 2 τ .

However, the expression (2) only holds for substances whose diffusional properties obey the Einstein-Fick limit, which mainly applies to liquids. Gases typically lie outside the Einstein-Fick limit. We have derived in prior work an expression for signal decay in gases, 1,2

where κ is a term that depends on the PAF of diffusing molecules. It depends, among other things, on temperature and viscosity. We note that the time dependence is τ , not τ 3 . This difference in exponents has important consequences. Namely, the application of a CPMG sequence with n echoes

no longer eliminates the second term describing inhomogeneous field decay in the limit of short interpulse spacing τ → 0 (2 nτ is fixed). Thus, any measurement of T 2 in a gas using a CPMG sequence will yield an apparent T 2 value that is affected by diffusivity effects in the inhomogeneous magnetic field. This could include, for example, unwanted weightings due to temperature, viscosity, external hardware, pulse sequence design, magnetic susceptibility, and pore geometry. This is in contrast to the case of liquids, where diffusion effects can be mitigated by extrapolation to extract the true (intrinsic) T 2 time.

Consider the CPMG experiment (Fig. 1 ) with a 90° broadband pulse and a series of 180° pulses to refocus the magnetization at intervals of 2 τ . The following phases were applied in the CPMG sequence: 90 ° x − τ − ( 180 ° y − τ − τ ) n ⁠ , where the 180° pulse is repeated n times. For each τ value, the echo envelope was acquired in a single-shot experiment. An external magnetic field gradient during the course of the experiment creates an inhomogeneous magnetic field. For gas-phase experiments, a sealable 5 mm diameter J. Young NMR tube was filled with liquid and freeze-pump-thawed to evacuate excess air. The NMR tube was heated by the NMR spectrometer’s variable temperature unit until the tube was vaporized. For liquid-phase experiments, a solution of 0.5% weight/volume tetramethylsilane (TMS) in acetone- d 6 was degassed and flame-sealed in an NMR tube. Measurements were performed on a 14.1 T vertical bore Bruker AV 600 MHz NMR spectrometer equipped with a 5 mm broadband probe with a z -gradient. The receiver was operated in qsim mode (forward Fourier transform, quadrature detection). The pulses were hard pulses whose lengths were 16 μ s and 32 μ s for the 90° and 180° pulses, respectively. The size of the sensitive RF region is less than 1 cm. All NMR signals were analyzed in magnitude mode and decay functions included baseline subtraction.

We take an explicit look at the time decay of the CPMG echo train, which, the theory [cf. Eqs. (1) and (3) ] predicts, should exhibit fundamentally different behavior ( t 3 vs t 1 , respectively). A direct verification is obtained by plotting the NMR signal in the CPMG experiment versus time along the echo train (see Fig. 2 ). The normalized NMR signal decay of the CPMG spin echo (with an interpulse spacing of 5 ms) for liquid-phase TMS is plotted in Fig. 2(a) . The normalized NMR signal decay of the CPMG spin echo (with an interpulse spacing of 5 ms) is plotted in Fig. 2(b) for gas-phase TMS. For gas, the normalized NMR signal decays exponentially, according to exp(− t / T 2 ) exp(− t / b ), in agreement with our revised theory of the NMR linewidth. We note that neither alteration of the phase cycling scheme to 90 ° x − ( 180 ° y − 180 ° − y ) n nor replacement of the 180° pulse with a 90° x /180° y /90° x composite pulse affected the results.

Direct verification that signal decay in the presence of a magnetic field gradient follows a time dependence of the form exp(− t / T 2 ) exp(− t / b ) for the gas and exp ( − t / T 2 ) exp ( − ( t / b 2 ) 3 ) for the liquid (solid lines, fit; dots, data). Here, we show sample CPMG decay curves for τ = 5 ms and g = 0, 0.01, 0.05, and 0.5 G/cm. (a) Liquid-phase TMS ( t 1 ). Number of scans (ns) = 1. (b) Gas-phase TMS ( t 3 ). ns = 8. In both liquid and gas cases, we scanned multiple acquisitions (ns = 1 and 8 for liquid and gas, respectively), and a T 2 value with experimental error bars was derived for Fig. 3 . The fits to the respective models are excellent. Fits to the converse equation ( t 1 ↔ t 3 ) do not yield acceptable fits (not shown here).

Measurements of the CPMG echo train signal decay as a function of interpulse spacing τ for TMS in the liquid phase are shown in Fig. 3(a) . Figure 3(b) shows the corresponding experiment in the gas phase. TMS is a liquid at room temperature but a gas at 26 °C; thus, a modest temperature increase enables comparison of the same substance in two different phases. TMS was also chosen due to its long relaxation times in both liquid and gas phases, enabling us to apply a large number of refocusing pulses even at long τ values. For liquids, as the interpulse spacing decreases, the measured T 2 value approaches a single value irrespective of the applied gradient strength g , as if there was no external gradient [Figs. 3(a) and 3(c) ]. This corresponds to the limit τ → 0 in Eq. (2) . Extrapolation of T 2 to the limit τ → 0 is the most commonly used method to extract true T 2 times in the presence of magnetic field inhomogeneities (from external or internal fields). For gases, however, the T 2 values in the limit τ → 0 do not approach a single value [Figs. 3(b) and 3(d) ] but instead converge to different values depending on the applied gradient strength g . This fundamentally different behavior implies that the inhomogeneous field decay term is still present, as predicted by Eq. (4) .

T 2 relaxation time of tetramethylsilane (TMS) vs interpulse spacing τ under conditions of magnetic field inhomogeneity (field gradient g , in G/cm). (a) Liquid-phase T 2 values approach the limit of no applied gradient as τ → 0. T 2 values shown range from 50 ms to 20 s. τ values shown range from 1 ms to 100 ms. Inset: Expansion of red boxed region. (b) Gas-phase T 2 values do not converge to a single value as τ → 0. T 2 values shown range from 18 ms to 1 s. τ values shown range from 1 ms to 100 ms. Inset: Expansion of red boxed region in A. (c) Data from (a) plotted on a linear scale. The straight lines are linear extrapolations as τ → 0. The g values are the same as in (a). (d) Data from (b) plotted on a linear scale. The straight lines are linear extrapolations as τ → 0. The g values are the same as in (b).

In this study, we have confirmed the t 1 dependence in the NMR signal decay function of gases in the presence of an external gradient [Eqs. (3) and (4) ]. In our prior work, we had verified the temperature dependence of the linewidth. 1 The verification of the t 1 time dependence can be considered the missing part of the puzzle, which now unambiguously confirms the validity of the revised linewidth theory presented in Ref. 1 . The g 2 dependence has already been verified in our previous paper. 1  

The fundamentally different motional averaging behavior of the NMR experiment in gases has important implications for several experiments. This behavior has previously led to the development of a novel non-invasive method for mapping temperatures of gases. 9 Gas-phase MRI experiments that utilize frequency-encoding gradients could be affected; gradients during readout affect measurements of T 2 or diffusion, introducing an apparent coupling between them. This means that a quantitative interpretation of these parameters would need to account for the non-Fickian nature of the diffusion. Dynamic decoupling schemes such as the Uhrig sequence, 10 which aim at generating the longest possible coherence times, are also expected to break down in the case of gases because short τ values no longer guarantee the elimination of environmental factors. Finally, the interpretation of restricted diffusion results 11–19 in porous media and other confined geometries may require new theoretical developments that model the signal decay in restricted environments in light of the new theory of signal decay.

This work was partially funded by a Beckman Young Investigator Award and the National Science Foundation through Grant Nos. CHE-1153159 and CHE-1508707.

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