Abstract

A scheme enabling the complete sampling of multidimensional NMR domains within a single continuous acquisition is introduced and exemplified. Provided that an analyte's signal is sufficiently strong, the acquisition time of multidimensional NMR experiments can thus be shortened by orders of magnitude. This could enable the characterization of transient events such as proteins folding, 2D NMR experiments on samples being chromatographed, bring the duration of higher dimensional experiments (e.g., 4D NMR) into the lifetime of most proteins under physiological conditions, and facilitate the incorporation of spectroscopic 2D sequences intoin vivo imaging investigations. The protocol is compatible with existing multidimensional pulse sequences and can be implemented by using conventional hardware; its performance is exemplified here with a variety of homonuclear 2D NMR acquisitions.

Methods

The general scheme for the experiment that we propose follows the canonical multidimensional spectroscopy outline given in Eq.1 for the case of 2D NMR. By contrast to Eq.1, however, our proposal does not involve a homogeneous initial excitation of all spins but a heterogeneous one, with the selectivity being achieved via the introduction of an inhomogeneous spectral distribution within the sample.¶ Once this artificial spectral heterogeneity has been imposed, the sample can be viewed as composed of independent subensembles, the evolution of which can be manipulated selectively. Selective manipulations can in turn be used toward the complete multiplexing of multidimensional NMR acquisitions, if exploited as illustrated in Fig.2. Such a scheme assumes a single-axisB_o field gradient that is the source of spectral heterogeneities and a generic 2D NMR sequence that we summarize as

An important difference between this scheme and Eq.1 is that spins here are excited not with a single but with a train of consecutive pulses, each spectrally selective. As a result the sample is effectively partitioned intoN₁ subensembles, each characterized by an individualt₁ evolution time. For the purpose of Fourier processing, it is convenient to space out the resulting train of slice-selective excitation pulses in constant intervals Δt₁ and then conclude theN₁ individual evolution times with a conventional mixing sequence that affects all spins in the sample simultaneously.∥ A complete 2D NMR signal set can be retrieved from this encoded information if the spin evolution duringt₂ is then collected in a spatially discriminated fashion. Of the several alternatives available for achieving such a goal with high resolution and within a single scan (23,24), we assayed here the echo planar chemical-shift imaging technique. With it, the overall protocol succeeds in carrying out the acquisition of a complete 2D data set within one continuous signal scan while still relying entirely on FT principles.

The approach in Fig.2 involves two imaging-type modules: one containing an initial spatially dependent evolution and another involving a final spatially discriminated detection. Both of these modules are achieved with the aid of external field gradients, the goal of which is to endow different locations within the sample with separate NMR frequencies. Yet the overall information being sought by the protocol as a whole is not spatial but spectroscopic. This in turn requires a complete refocusing of all spatially dependent spin precessions while leaving unaffected the intrinsic frequencies and correlations imparted by the spin Hamiltonians throughout the evolution, mixing, and acquisition periods. Of the several approaches that could be used to achieve such an objective, the one illustrated in Fig.3 was used in this study. It consists of a train of inverting gradients ±G_e that enables the desired slice-selective initial excitation, and a final spatially resolved acquisition based on alternating ±G_a gradients. Because of their oscillatory nature, no cumulative effects are imparted by these gradients in either thet₁ ort₂ dimensions.

Results and Discussion

Fig.4 illustrates further the potential of this scheme, with results obtained on a model organic sample using three homonuclear NMR experiments chosen for the distinct cross-peak patterns that they yield. These experiments include (i) a basic 2D sequence with an initial excitation pulse but no mixing event that only results in peaks along the main diagonal, (ii) a COSY sequence involving a π/2 mixing pulse meant to establish cross-peaks among directly J-coupled nuclei, and (iii) a TOCSY sequence that establishes cross-peaks via an isotropic mixing among all protons within a J-coupled network. The agreement between the results afforded by these ultrafast 2D NMR experiments and theira priori expectations is excellent, an encouraging observation given the ample room available for further improving both the hardware as well as the pulse sequences supporting our acquisition principle.

An unusual feature of the approach that was just described is that it not only provides 2D NMR data sets within a single scan but may also do so without requiring the typical 2D numerical Fourier transformation involved in conventional acquisitions. Indeed the use of a slice-selective encoding duringt₁ followed by a slice-selective decoding duringt₂ amounts to a built-in FT along the indirect domain. This feature can be appreciated from the mathematical derivation of the signal arising from the schemes illustrated in Figs.2 and3. Because the physical collection of data during such an experiment is carried out while implementing a spatially resolved EPI-type acquisition, its resulting signal can be described as depending on two independent extraction variables: a timet₂ encoding spectral frequencies ν₂ during the acquisition time, and a variablek = ∫G_a(t)dt encoding the spatial positionsz of spins within the sample. Considering as well that spins at differentz positions were endowed with differentt₁ evolution times, this signal can be summarized as

HereP(z, ν₁, ν₂) is the joint probability distribution that spins at a coordinatez will have precessed at a particular set of spectral frequencies ν₁ duringt₁ and ν₂ duringt₂. Assuming for simplicity staticz slices that are being homogeneously excited leads to a joint spatial/spectral distributionP(z, ν₁, ν₂) ≈ρ(z)⋅I(ν₁, ν₂), whereI(ν₁, ν₂) is the 2D NMR spectrum being sought, andρ(z) is a shape function. For an ideal cylinder of constant composition this function will be simply a constantρ. Fourier analysis along thek dimension then leads to

If it is now assumed that, as was actually done for acquiring the data,t₁ was incremented linearly with thez position of the spins,t₁(z)= C⋅(z −z_o), and it is clear that 2D FT ofS(z, t₂) will yield the desiredI(ν₁, ν₂) 2D NMR spectrum. Yet because the Fourier transformation is a self-reciprocal operation, it also follows that a second FT along thez dimension will simply revert the signal back into its originalk-space distribution. A single 1D FT of theS(k, t₂) data alongt₂ is thus sufficient to retrieve the complete 2DI(ν₁, ν₂) spectrum.

Equations aside, this explanation gives further physical insight into the origin and shape of peaks in this type of experiment. To appreciate this it is illustrative to consider the line shapes expected after subjecting a typicalS(k, t₂) imaging/spectroscopic signal tot₂ Fourier transformation: peaks will then be observed along ν₂ at the appropriate Ω₂ frequencies present in the sample, whereas sharp echoes centered atk = 0 and possessing point-spread functions proportional to ∫_zP(z)e^ikzdzwill be encoded along the gradientk domain. If, however, spins have been initially subject to an excitation of the type illustrated in Figs.2 and3 witht₁(z)= C⋅(z −z_o), an additionalz-dependent dephasing proportional to Ω₁t₁= Ω₁⋅C⋅z will also become part of the total gradient encoding. After FT alongt₂ signals will then be shifted from their originalk = 0 positions to coordinates (k, ν₂)= (−C⋅Ω₁,Ω₂), thus simultaneously defining both the Ω₁ and the Ω₂ frequencies for each chemical site.

These arguments also clarify the line shapes defining this type of 2D spectroscopy. Multidimensional peaks will be dictated here by the conventionalabsorptive +i ⋅dispersive combination along the direct ν₂ domain, and by the FT of the sample's excited profile along thek-encoded axis. Assuming an ideal excitation of contiguous slices and the usual (rectangular) sample profile of conventional NMR tubes, real-only sinc-type point-spread functions thus emerge for the line shapes along thek axis. It follows that purely absorptive 2D peaks are an inherent feature of this acquisition mode. These arguments allow us as well to derive the Nyquist relations associated to this acquisition scheme. Sampling conditions along the ν₂ dimension will be as for a conventional EPI-shift experiment: denoting the length of thek-axis scan asT_a (Fig.3), consecutive points along thet₂ axis will be spaced by 2T_a dwell times and result in 2N₂⋅T_a totalt₂ acquisition times andSW₂= (2T_a)⁻¹ spectral widths. Nyquist conditions along the ν₁ dimension will be dictated by a combination of the EPI acquisition parameters and the selective excitation conditions used for the initial encoding. Referring again to the simplestt₁(z)= C⋅(z −z_o) sequential excitation scheme, we conclude that under ideal conditionsSW₁ will depend on the magnetogyric ratios γ_e and γ_a of the species being excited and detected, the gradient strengthsG_e andG_a used to encode and decode positions during the excitation and acquisition, the 2T_a dwell time alongt₂, the temporal delay Δt₁ between the initial excitation pulses, and the constant ΔO =  |[O_i+1 −O_i]| offset increment characterizing this pulse train. The resulting spectral span then can be summarized bySW₁= ΔO⋅[γ_aG_aT_a]/[γ_eG_eΔt₁], an expression of practical quantitative use for evaluating this type of experiments.

Also worth commenting on is the S/N expected for this type of experiments. From the slice-selective considerations in Fig.2 one could assume that this class of experiments will be feasible only if, for a particular chemical site, each of the excited slices has enough S/N to afford a complete 2D NMR spectrum. This argument, however, would ignore that in the final scheme signals from all slices end up contributing to every (Ω₁, Ω₂) peak in the spectrum. One could also expect translational motions to count as important factors in attenuating these gradient-based experiments, yet the fact that gradient echoes are a feature of both the excitation and acquisition portions of the sequence alleviates considerably the potentially destructive effects of molecular diffusion. In fact it appears as if, except for potential dissipative processes occurring throughout the sequences (T₁ andT₂ relaxation, storage processes, etc.), the integrated signal intensity arising from diagonal and cross-peaks at a given Ω₂ frequency will be comparable, in principle, to theI(Ω₂) observed in a one-scan 1D spectrum. This does not mean, however, that the S/N observed throughout our measurements were as good as in their optimized 1D NMR counterparts. Some of these signal losses could be traced to nonidealities in the selective pulses and in the fast-switching gradients that were used. Yet even if these preventable losses were to be removed via further optimizations, a noise penalty would have to be paid for the sake of rapidly sampling both dimensions of the experiment in a single scan. Indeed if ≈N₁ points are to be collected during the course of eachT_a gradient application, the filter width of the spectrometer needs to be opened to accommodateca. N₁/T_a spectral widths. As a result there will be a noise increase ofvis-à-vis the noise that would characterize a conventional 1D acquisition. Consequently, although capable of affording a 2D NMR spectrum within a single scan, the S/N per unit time of these ultrafast acquisition experiments will be lower in general than that of their conventional counterparts.** Because of the reliance of ultrafast 2D NMR on a fast digitization of the signal along both spectral axes, it is hard to envision routes by which this penalty could be avoided.

Conclusion

We conclude by remarking the convenience of introducing at the current stage of NMR's maturity a fully multiplexed format for the acquisition of multidimensional NMR spectra. Indeed developments in probehead and magnet technologies have increased NMR's sensitivity by almost an order of magnitude during the last decade (25), thus limiting the acquisition times of many multidimensional NMR experiments to signal digitization rather than to S/N considerations. Research areas where this bonus in signal could be combined with the advantages of single-scan 2D acquisitions include the structural analysis of rapidly changing dynamic systems (e.g., folding proteins, ref.26), the characterization of analytes subject to flow (as in emerging chromatography-NMR techniques, ref.27), the rapid survey of large numbers of chemicals such as those made available by new combinatorial chemistry techniques (28), and the speeding up of quantum-computing algorithms based on multidimensional NMR (29). Ultrafast methods could also be of importance for shortening up a variety of NMR experiments that demand high spectral dimensionalities. These experiments include structural elucidations of very large biopolymers, systems with complexities that will eventually require ≥4D NMR for achieving sufficient spectral resolution but at the same time are generally incapable of withstanding the long times hitherto associated with such acquisitions. They also include hybrid imaging/spectroscopy techniques where 2D¹H-X spectral correlations are combined with spatial localization methods (30),in vivo experiments that are usually discouraged by their relatively long acquisition times. In fact, extending the scheme in Eq.2 to shortennD experimental times into those of (n − 1)D acquisitions is straightforward. But the opportunity also arises to reduce 3D and 4D NMR acquisitions to just a few signal scans via the proper use of multiplex-,y-, andz-gradient encodings. Progress along these various research avenues will be detailed elsewhere.

Acknowledgments

We are grateful to Professor M. Fridkin (Weizmann Institute of Science) for his gift of the hexapeptide sample. This work was supported by The Philip M. Klutznick Fund for Research.

Abbreviations

• FT, Fourier transform

• S/N, signal-to-noise ratio

• EPI, echo planar imaging

• TOCSY, total correlation spectroscopy

References