Journal of Biomolecular NMR 27: 101–113, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Perspective

New methods for fast multidimensional NMR

Ray Freemana,∗ & Eriks Kupceb

aJesus College, Cambridge, U.K.; bVarian, Inc, Eynsham, Oxford, U.K.

Received 29 April 2003; Accepted 5 May 2003

Key words: filter diagonalization method, GFT-NMR, Hadamard spectroscopy, multidimensional, projection-reconstruction, single-scan two-dimensional NMR

Abstract

Considerable excitement has been aroused by recent new methods for speeding up multidimensional NMR exper-iments by radically modifying the normal time-domain sampling protocols. These new schemes include the filterdiagonalization method, GFT-NMR, the single-scan two-dimensional technique, Hadamard spectroscopy, and aproposal based on projection-reconstruction of three-dimensional spectra. All these methods deliver appreciableimprovements in the speed of data acquisition and show promise for speeding up multidimensional NMR ofproteins. This perspective aims to describe these important new procedures in simple terms and to comment ontheir advantages and possible limitations.

Introduction

When two-dimensional spectroscopy was first intro-duced (Jeener, 1971; Aue et al., 1976) it was shown tohave virtually the same sensitivity as one-dimensionalmeasurements performed in the same total time, sinceall the NMR signals are gathered in each and everyscan. The same considerations also apply to multi-dimensional spectroscopy. Recently attention has fo-cused on the rather less obvious point that such spectrado not enjoy optimum efficiency in terms of speed.Systematic exploration of the indirect evolution di-mensions one step at a time has the advantage that nosignal frequencies are overlooked, but this is not thefastest mode of data acquisition for relatively sparsespectra. Time is actually wasted examining regions offrequency space where there are no signals, only noise.This realization has excited a recent flurry of activityto see whether multidimensional experiments could bespeeded up, provided that the inherent sensitivity isalready high enough. Since it is clear that multidimen-sional NMR spectroscopy of proteins can be seriously

∗To whom correspondence should be addressed. E-mail:rf110@cus.cam.ac.uk

handicapped by the speed factor, these are importantdevelopments.

To fix ideas, imagine a three-dimensional array oftime-domain data. The traditional approach would beto examine each and every element of this array ina systematic fashion, by following the evolution ofthe NMR signal as a function of t1, t2 and t3 inde-pendently. By contrast, all the new methods seek tosimplify this process by cutting down the number ofindividual measurements, either in the time domain orin the frequency domain. Sensitivity is reduced, but formany samples in high-field spectrometers, sensitivityis already more than adequate; it is speed that is thecritical factor. These new methods go far beyond theusual compromises adopted in multidimensional spec-troscopy – less-than-optimum resolution, the reduc-tion or elimination of phase cycling, linear predictionof the tails in the evolution dimensions, and the ac-ceptance of aliasing where this can be tolerated. Suchshort-cuts do help, but normally do not go far enoughin reducing the duration of high-dimensional NMRspectroscopy; an expensive spectrometer can still betied up for days on a single investigation.

The new methods reviewed here are all quite dis-tinct, but because these ideas have been ‘in the air’

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for some time, one mode of attack on the speed prob-lem tends to suggest alternatives. All have advantagesand drawbacks, and it is not yet apparent that a clearwinner has emerged. It is hoped that this Perspectivemay go some way to demystifying these new ideas,without doing too much mischief to the underlyingbasic theory.

The filter diagonalization method

The filter diagonalization method (Mandelshtam,2000, 2001; Hu et al., 2000; Chen et al., 2000) wasproposed as an alternative to the ubiquitous Fouriertransform methodology, but initially not as a methodfor speeding up multidimensional spectra. It is a data-processing technique related to the better-known linearprediction method (Barkhuijsen et al., 1985), and itstarts from the premise that the inherent NMR sensit-ivity is high enough that the effects of noise can beneglected, and that the resonance lines are Lorent-zian in shape. The former assumption is common toall the fast multidimensional methods, and the latteris not a really serious restriction, since, if necessary,an arbitrary lineshape could be constructed from asuperposition of Lorentzians.

Linear prediction assumes that the NMR time-domain signal is a superposition of complex sinusoidsthat decay exponentially with time. In an ideal world,where there is only one resonance line in the spectrumand no noise, only two complex data acquisitions areenough to define the frequency of that line and itswidth. In real situations the presence of nearby res-onances, non-Lorentzian lineshapes and the existenceof random noise can render the method rather unreli-able. Furthermore the number of component sinusoids(called the ‘filter length’) has to be assumed; if it is toolarge the calculation is too slow; if it is too small thefitting becomes unreliable. So linear prediction is gen-erally used with caution, usually to extrapolate the tailof a time-domain signal to avoid truncation artifacts,or to predict the first few points of a free inductionsignal that might have been falsified by transmitterbreakthrough. In the presence of noise, linear pre-diction can have quite unpredictable behaviour, thusrefuting the all-too-common misconception that thisprocessing technique can improve sensitivity.

Filter diagonalization shares with linear predictionthe invaluable property that linear algebra is used, andit attains the same goal, but it does so by fitting theexperimental time-domain data as if in the frequency

domain. That is to say, it takes into account the undeni-able fact that lines that are reasonably well separated infrequency show little or no interference effects. Filterdiagonalization uses a matrix representation of an op-erator, in which interference effects between differentcomponents are represented by off-diagonal elements.It aims to derive a representation of the NMR spec-trum by a procedure that does not involve Fouriertransformation at all.

Our concern here is what the filter diagonalizationmethod can accomplish, not primarily with the math-ematical details. The key theoretical paper (Wall andNeuhauser, 1995) is entitled ‘Extraction, through fil-ter diagonalization, of general quantum eigenvaluesor classical normal mode frequencies from a smallnumber of residues or short-time segments of a sig-nal’. An operator Û represents a time autocorrelationfunction that has the effect of computing the next time-domain data point each time it is applied. By repeatedapplication of this operator, a model for the NMRtime-domain signal can be created which may thenbe fitted to the actual experimental signal. If Û canbe diagonalized, its powers can be written in termsof powers of the eigenvalues. The eigenvalues of Ûthen give the line frequencies and widths, while theeigenvectors give amplitudes and phases. If the fit-ting process addressed all the time-domain data atonce, the diagonalization problem would be immensebut the decomposition into small segments renders ittractable.

The crucial innovation is in the choice of basis – aset of ‘frequency-like’ basis functions that are roughlylocalized in frequency space, for example, in a setof segments uniformly distributed across the NMRspectrum. This breaks down a potentially enormousgeneralized eigenvalue problem into a set of muchsmaller problems where unimportant matrix elementslying far from the diagonal can be safely neglected.Only a ‘local’ fitting procedure is then required, neg-lecting basis functions outside the particular segmentsunder consideration, leaving much smaller ‘filtered’matrices for diagonalization. Each frequency-domainsegment is multiplied by a shaped window function,usually a cosine-squared envelope with 50% overlapbetween adjacent windows. Intense or very broad linesoutside the window may require special treatment,and spectral windows containing a strong solvent peakare normally excluded from the process. The fittingprocedure is not bedevilled by the usual problems of‘false’ minima because linear algebra is employed, buteven a perfect fit to the experimental NMR signal is

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no absolute guarantee that the derived spectrum is re-liable. This is particularly serious if the basis is toosmall, so it is necessary to fit the segments severaltimes, using varying basis sizes. The segments are thenassembled to give the full NMR spectrum.

In the extension to two frequency dimensions thebasis is much larger, being determined by the productof the number of points in each dimension. Hereinlies the great strength of the filter diagonalizationmethod applied to two-dimensional spectroscopy – theresolving power depends essentially on the area inthe two-dimensional time domain rather than on theindividual lengths in the two time dimensions takenseparately. Thus a very large number of experimentaldata points recorded in the t2 dimension (acquiredwithout a significant time penalty) can compensate fora much smaller number of data points in the t1 di-mension (where time is of the essence). The minimumrequirement is that there should be substantially morebasis functions in the two-dimensional window thanthere are NMR peaks in that area of the spectrum.This offers a considerable economy in instrument timecompared with the conventional sampling regime. Theonly change to the traditional two-dimensional exper-iment is the reduction in the number of samples takenin the evolution dimension; all the rest is a data pro-cessing operation. Analogous considerations apply tohigher-dimensional spectra.

Unfortunately the application to multidimensionalspectroscopy introduces some serious new problems,some spectroscopic and some numerical. These diffi-culties include the phase-twist line shape (which posesmore fitting problems than the Lorentzian shape as-sumed for one-dimensional spectra), the presence oft1 noise, phase shifts induced by low-level instru-mental noise, an inherent lack of correlation betweenthe F1 and F2 co-ordinates of a given spectral feature,large changes in the computed spectrum generatedby quite small perturbations of the input data, andcomplications when there are exactly degenerate fre-quencies, as in COSY multiplets. These problems arepartly addressed by the introduction of a ‘regulariz-ation’ procedure (Chen et al., 2000), broadly relatedto the method proposed by Tikhonov (1963). A reg-ularization parameter q2 is introduced and increasedfrom zero until the calculation becomes stable. Oneof the consequences of regularization is a broadeningof noise components (and small signals comparablewith the noise) to the point where they disappearfrom the computed spectrum. Some spectroscopistswould regard this kind of non-linearity as unsatisfact-

ory, even dangerous, but it has the useful side effect ofimproving the presentation of the contour map.

The filter diagonalization method has been ap-plied to the two-dimensional constant-time (Powerset al., 1991) heteronuclear single-quantum correlation(HSQC) spectrum of ubiquitin (Chen et al., 2003). Theprincipal aim of these measurements was to demon-strate an improvement in spectral resolution comparedwith conventional Fourier transform experiments un-der the same conditions. However it also permits asignificant reduction in the constant-time parameter,which in the general case would improve sensitivityby reducing relaxation losses. It further speeds updata gathering by reducing the number of samplesin the evolution dimension by a factor of approxim-ately six. Figure 1 compares the traditional Fouriertransform HSQC results on one part of the ubiquitinspectrum obtained with a constant-time parameter of26.4 milliseconds with those from the filter diagonal-ization method recorded with this parameter reducedto 4.25 milliseconds. The latter spectrum, obtainedin less than 4 min, corresponds very closely to theconventional Fourier transform spectrum except in thevery crowded region. Ubiquitin has uncharacteristic-ally slow spin-spin relaxation; for large proteins ingeneral it may not be feasible to follow the free induc-tion signals for long enough to obtain fine digitizationin the direct frequency dimension.

The filter diagonalization scheme is a method ofspectral estimation that offers an alternative to thestandard Fourier transform methodology. It can speedup the data-gathering stage of multi-dimensional spec-troscopy by concentrating the sampling operations inthe direct detection dimension, thereby compensatingfor deliberately sparse sampling in the indirect evol-ution dimensions. The fitting process is not alwayscompletely reliable and may require some trial-and-error adjustment of the number of basis functionsand of the ‘regularization parameter’, so it cannot besaid to be a ‘push-button’ procedure at the presenttime. The time needed for the computation is gener-ally much longer than that of the conventional Fouriertransform protocol, although it can be reduced by con-fining the calculations to signal-bearing regions. Paral-lel processing is expected to reduce the computationaltime to a more acceptable level.

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Figure 1. Part of the 500 MHz constant-time HSQC spectra of a 1 mM D2O solution of isotopically labelled ubiquitin, showing the car-bon-13-proton correlations. (a) The conventional Fourier transform spectrum with the constant time parameter CT = 26.4 ms. (b) The spectrumobtained by the filter diagonalization method with CT = 4.25 ms, and joint processing of N and P type data sets. Spectrum (b) required lessthan 4 min of data acquisition, but note the slight loss of resolution in the very crowded region. Spectra courtesy of A.J. Shaka.

GFT- spectroscopy

This technique attacks the speed problem of multi-dimensional NMR by incrementing several evolutionperiods jointly, either exactly in step with one anotheror with suitable scaling factors between them. Thisis clearly much quicker than incrementing these timedimensions independently. The ‘reduced dimensional-ity’ idea has evolved over quite a long period (Szyper-ski et al., 1993a, b, 1995, 1998, 2002; Brutscher et al.,1994, 1995; Simorre et al., 1994; Löhr and Rüterjans1995; Ding and Gronenborn, 2002). Locking severalevolutions together is the easy part; there remains theproblem of disentangling the different chemical shiftfrequencies that evolve in the various ‘locked’ dimen-sions. The chemical shifts appear in the experimentaltime-domain data set as linear combinations, and theyare extracted by a procedure based on a G-matrix (Kimand Szyperski, 2003).

The extraction technique is most easily understoodin the frequency domain. Consider the simple exampleof an N-dimensional correlation experiment where theaim is to reduce the investigation to N − 2 dimensionsby incrementation of the first three evolution periodsjointly and in step. In the simplest case where no scal-ing is used, this involves setting t1 = t2 = t3 = t .At first sight this appears to be counter-productive,

conflating signals from three different dimensions, butin fact the exploration of these dimensions can berendered independent again by the GFT procedure.Let a typical chemical shift frequency in the first di-mension be �1 so that the normalized signal intensitytransferred to the second evolution period is given bycos(�1t), neglecting relaxation losses for simplicity.After evolving at the new chemical shift frequency �2,it passes on a signal proportional to cos(�1t) cos(�2t)to the third evolution period where it evolves at an-other new frequency �3. At the end of the third evolu-tion period the transferred signal is given by cos(�1t)cos(�2t) cos(�3t). Further evolution stages, t4, t5,. . . , etc., are monitored by the conventional methodwhere the various time developments are followedindependently.

Now comes the crucial trick. The entire sequenceis repeated with a 90◦ phase shift of the radiofrequencypulse that initiates the second evolution period, so thatthe corresponding transfer function becomes sin(�2t)rather than cos(�2t). This stratagem is repeated in asuccession of four ‘scans’ in which the radiofrequencyphase shifts are systematically changed, keeping afixed phase in the first evolution period:

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scan (1) 0◦ 0◦ 0◦,

scan (2) 0◦ 90◦ 0◦,

scan (3) 0◦ 0◦ 90◦,

scan (4) 0◦ 90◦ 90◦.This introduces cosine or sine modulation of the fourtransfer functions according to the products:

cos �1t cos�2t cos�3t ,

cos �1t sin�2t cos�3t ,

cos �1t cos�2t sin�3t ,

cos �1t sin�2t sin�3t .These four equations describe the time developmentfor a typical chemically-shifted site; analogous equa-tions can be written for all the other sites.

Fourier transformation of these time-domain sig-nals with respect to t gives the correspondingfrequency-domain representation. In order to make allthe responses absorptive, the receiver phase is shiftedby n times 90◦ where n is the number of sine modula-tions. Standard trigonometric functions for sums anddifferences of angles indicate that cosine modulationtransforms as an in-phase doublet whereas a sine mod-ulation transforms as an antiphase doublet. Schemat-ically the frequency-domain patterns from this singlesite can be represented as stick spectra (Figure 2).

Without prejudice to the argument, it can be as-sumed that �3 is larger than �2, so that the ‘sum anddifference’ frequencies A through D are given by:

Figure 2. Schematic spectra for the GFT method from a singlechemical site. The different cosine and sine modulations in four suc-cessive scans create these different patterns in the frequency domain.The individual frequencies can be extracted by adding or subtractingthese spectra with the appropriate sign combinations.

A = +�1 + �2 + �3,

B = +�1 − �2 + �3,

C = +�1 + �2 − �3,

D = +�1 − �2 − �3.It is clear that the three frequencies can be separ-ated by combining the columns with the appropriateplus or minus signs (compare section on Hadam-ard spectroscopy). In this way, although the threefrequency dimensions have been explored jointly in‘lock-step’, the requisite chemical shift informationcan be disentangled.

These four simultaneous equations would be suf-ficient to obtain all three chemical shifts in an idealsituation where there is no accidental overlap of anyof the responses; indeed the problem is slightly over-determined. But in the real world, and particularlyin large molecules, this is unlikely, and further stepsmust be taken to resolve ambiguities caused by de-generacies in the frequencies. This entails furthermeasurements in which the evolution in certain dimen-sions is inhibited. Additional scans are performed witht3 = 0, giving the transfer functions:

cos �1t cos �2t ,

cos �1t sin �2t .After Fourier transformation, the frequency-domainspectra consist of in-phase and antiphase doubletsrespectively, with frequencies given by

E = +�1 + �2,

F = +�1 − �2.Finally, if both t2 and t3 are set to zero there is justa single peak at the ‘centre of gravity’ of the previousspectra, G = �1. This last piece of information can bevery useful for discriminating between the subspectralresponses from different chemical sites.

Synchronization of more than three evolution peri-ods leads to correspondingly more complicated ex-pressions, with larger G-matrices and larger numbersof repeated ‘scans’ but the principle remains the same.For example, joint evolution in four dimensions re-quires 15 scans, with correspondingly more complexG-matrices. The reduction in measurement time im-proves exponentially as the number of locked dimen-sions increases, despite the increase in number ofmandatory scans. However, the technique sacrificessensitivity for each stage of dimensionality reduction,and higher-dimensional experiments necessarily incurincreased signal loss through relaxation, so eventuallya point is reached where the sensitivity, rather than theduration, becomes the limiting factor.

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The GFT method has been applied to the 600 MHzspectra of the aptly-named 76-residue protein ubi-quitin (Kim and Szyperski, 2003). The sample wasdoubly-labelled in carbon-13 and nitrogen-15, andconsisted of a 2 mM solution in 90%H2O/10%D2O.The conventional five-dimensional HACACONHNexperiment was reduced to only two effective dimen-sions by incrementing the H(α), C-13(α), C-13′ andN-15 evolution dimensions together and in step. Thisentailed 2 h 18 min of data gathering. After assignmentof the frequencies, the chemical shifts were extrac-ted and measured by a least-squares fitting procedure.It was calculated that the equivalent conventional ex-periment would have required an unacceptably longduration of almost 6 days of spectrometer time.

The strength of the GFT method lies in thepossibility of reducing the dimensionality of high-dimensional spectra – those that cannot be contem-plated by conventional Fourier transform methods be-cause of impossibly long data-gathering times. Forexample, pulse sequences for five-dimensional NMRcan be readily devised but are rarely implemented bythe traditional methodology. The limitations of theGFT method might be expected to become evidentin cases of severe overlap and in situations wheresensitivity is also a problem.

Single-scan multidimensional spectroscopy

Frydman et al. (2002, 2003) have demonstrated aningenious ‘single-scan’ scheme that collapses evolu-tion and acquisition into a single real-time dimension.The new method involves the application of a select-ive radiofrequency pulse in the presence of a magneticfield gradient, so that at any one instant only one thinslice of the sample is excited. Typically a z-gradientis used, acting along the long axis of the standard cyl-indrical sample, but in principle gradients in all threedirections could be employed. The selectivity in thefrequency domain is low enough in terms of chemicalshift dispersion that the entire spectral width is excitedby each pulse, but nevertheless sufficiently selectivethat the slice thickness is small in comparison withthe active length of the sample. The gradients mustbe very intense. Typically Gaussian-shaped selectivepulses are employed. This excitation cycle is repeatedN1 times with a suitable increments in the frequencyof the selective pulse so that N1 discrete slices areexcited. In a typical example N1 might be 40–60.

The conventional scan-by-scan exploration of t1that has proved so profligate with spectrometer timeis now replaced by a procedure that maps out theentire spin evolution within a single scan. A slowserial measurement has been converted into a paral-lel measurement completed in a much shorter time.This spatial encoding procedure can be thought of ascreating N1 little ‘boxes’ in which the signals fromdifferent spin packets are temporarily stored, ratherlike mail sorted into different pigeon-holes. In practicethese ‘boxes’ are slices through the sample normal tothe z axis. These N1 spatially-resolved sub-ensemblesof nuclear spins evolve independently. Since the ex-citation of each slice is initiated at a slightly latertime, the evolution time t1 is automatically decremen-ted at the same rate (Figure 3). This is the key feature– we can associate a specific t1 value with each ex-cited slice. Free precession due to a chemical shift isseparated from the dispersal in the applied gradientby the application of a second equal gradient of op-posite polarity; the chemical shift precession persiststhroughout. The repeated matched gradient pairs haveno cumulative effect, and the undesirable influenceof molecular diffusion is minimized. In practice thenumber of active slices (and the number of evolutionincrements N1) is limited by the available gradientstrength compared with the spectral width under in-vestigation, suggesting that the method is best suitedto proton spectroscopy.

For simplicity we consider the well-knownTOCSY sequence, but a COSY experiment would fol-low similar lines. At this point the operation deviatesfrom conventional methodology and becomes quitesubtle. Consider the evolution of spins in one repres-entative slice with NMR chemical shifts �A, �B and�C . This slice (a) is associated with its characteristicevolution time (t1)a , so at the end of the excitationstage this particular spin packet has accumulated pre-cession angles that depend on �A(t1)a , �B (t1)a and�C(t1)a . At this point there is no applied gradient, andall the slices experience the same magnetic field, beingdistinguished only by their different t1 values.

Next there is an isotropic mixing stage which isnot spatially selective. Now consider the interferenceeffects between spin magnetization vectors from allthe active slices. These vectors have been twisted intothree distinct helices with pitches in the ratio �A, �B

and �C (Figure 4). The first stage of the acquisitionprocess employs a positive magnetic field gradientpulse that progressively unwinds the three helices,bringing them to a focus one at a time, as the three

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Figure 3. Schematic representation of the single-scan two-dimensional experiment. In the applied field gradient, selective radiofrequencypulses with linearly incremented frequencies (νa , νb, νc , etc.) define successive slices (a, b, c, etc.) in the z direction. The pulses are strongenough to excite the entire spectrum, and because they are applied with decreasing time delays, they excite spectra with increasing evolutiontimes (t1)a , (t1)b, (t1)c, etc, thus exploring the entire evolution dimension in a single scan. Each slice is associated with a specific t1 value.

Figure 4. In the single-scan method for two-dimensional spectroscopy the excitation stage spreads magnetization vectors from different slicesof the sample into helices, the pitch increasing in proportion to the chemical shifts �A, �B and �C . During the acquisition stage the ‘read’gradient unwinds these three helices, but because the initial pitches are different, three successive echoes are formed. They represent the NMRspectrum in the F1 dimension.

different ‘handicaps’ are compensated one after theother. This crucial refocusing effect relies on the linearrelationship between the initial t1 evolution and the z-

ordinates, as each slice experiences just the right localmagnetic field to compensate for the evolution that oc-curred during the excitation stage. It refocuses (rather

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than twisting the helices more tightly) because an ap-plied magnetic field rotates magnetization vectors ina direction opposite to the sense of nuclear preces-sion (in the same way that a linearly-increasing fieldsweep traces out a spectrum in the order of decreas-ing NMR frequency). This is a rather special kind ofgradient-recalled echo where the initial dispersal dueto free evolution is refocused by an actual field gradi-ent. Regular sampling during this interval detects threesuccessive echoes, representing the three peaks of theNMR spectrum in the F1 dimension. Paradoxically,the high intensity of the field gradient is a favorablefeature, ensuring that the three echoes are very sharp.Note that no Fourier transformation is involved inderiving this ‘spectrum’.

The positive gradient pulse is followed by amatched gradient pulse of opposite polarity. This hasthe effect of refocusing the three echoes once again,generating an F1 spectrum that is reversed in direction.To improve sensitivity these signals are processed sep-arately and later reversed and combined with the firstset. The module consisting of a pair of opposed gradi-ent pulses is then repeated and data collected at regularintervals as before. A sequence of N2 modules is used,where N2 is normally considerably larger than N1. Asa result of the progressively increased delay betweenmixing and acquisition, the intensities in the later ver-sions of the F1 spectrum are modified, as coherenceis transferred between sites through the TOCSY ef-fect. This imposes a t2 dependence on the successiveF1 spectra. Eventually the signal decays through theusual instrumental and relaxation processes. In thismanner each F1 spectrum is associated with its cor-related time-domain response S(t2), and after a singlestage of Fourier transformation with respect to t2, thetwo-dimensional spectrum f (F1, F2) is obtained.

For those of us accustomed to the normal rulesof two-dimensional spectroscopy where two stages ofFourier transformation are mandatory, this is a quitesurprising feature. Only the transformation with re-spect to t2 is needed; the F1 spectra arise automaticallythrough sequential echo refocusing. The F1 ‘evolu-tion’ dimension is confined to quite short time inter-vals (typically 480 microseconds) embedded withina much longer sequence (typically 128 milliseconds)where signals are acquired as a function of t2. Theentire measurement is completed in a fraction of asecond.

It would be easy to fall into the trap of thinking thatthe sensitivity of this single-scan technique is com-promised because, at any one instant, slice selection

limits the number of nuclear spins being excited. Butremember that each step of this repetitive cycle gath-ers information about the entire spectrum, and signalsfrom all the active slices combine in the acquisitionstage. Slice selection is simply acting as a way oftemporarily introducing a new ‘storage’ dimension;all the available nuclear spins contribute to the finalspectrum. Although the signal strength is comparablewith that of a conventional free induction decay of thesame duration, the noise level is higher. Comparedwith conventional two-dimensional spectroscopy thesensitivity is reduced because the total duration ofdata gathering is so much shorter; there is always thistrade-off between speed and sensitivity. Furthermore,the dwell time has to be short, typically 2 micro-seconds, and the receiver bandwidth has to be broad,thus letting in more noise. The receiver duty cycleis not optimum, and there are certain non-idealitiesin the selective pulses and artifacts generated by therapidly-switched field gradients. There is however onedistinct advantage of any single-scan technique – itavoids the partial saturation that occurs in most multi-scan methods. As in the other fast multidimensionalmethods outlined here, it is assumed that sensitiv-ity is not the main consideration. The limitations ofthe single-scan technique are more likely to lie indifficulties associated with the generation of suitablyintense, reproducible, and rapidly-switched gradientpulses – technical features that may well be remediedin due course. Whatever the eventual outcome, onehas to admire the ingenuity behind this remarkableexperiment.

The method has been illustrated (Frydman et al.,2003) for the single-scan two-dimensional 500 MHzproton TOCSY spectrum of the dipeptide AspAla,using a 1.5 s water presaturation pulse and a 120millisecond DIPSI sequence (Shaka et al., 1988) forisotropic mixing. There were N1 = 58 selective pulsesapplied in a gradient strength of 65 G per cm. Thedetection stage involved N2 = 256 repeated gradientpairs. The resulting two-dimensional TOCSY spec-tra show the expected correlations (Figure 5). Notethat, neglecting water presaturation, the spectrum wasacquired in only 0.28 s, far faster than any of thecompeting ‘fast’ two-dimensional methodologies de-scribed here. The duration is so short that the extensionto three- or even four-dimensional experiments couldbe contemplated even if the additional evolution peri-ods are explored by conventional Fourier transformmethods.

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Figure 5. The single-scan two-dimensional method applied to the500 MHz phase-sensitive TOCSY spectrum of protons in thedipeptide AspAla (the two Hα resonances have the same chemicalshift). Apart from the water presaturation period, the measurementwas completed in only 0.28 s. Spectrum courtesy of Lucio Frydman.

Hadamard spectroscopy

This approach to the speed problem exploits the sparsenature of most multidimensional NMR spectra by at-tacking the frequency domain directly, thereby avoid-ing the usual requirement for uniform sampling in theevolution dimensions (Kupce and Freeman, 2003a, b).It is predicated on the assumption that the relevantNMR frequencies can be measured from exploratoryone-dimensional experiments carried out in a rathershort time. This information is used to set the frequen-cies of an array of simultaneous selective radiofre-quency pulses, one for each distinct chemical site. Onemultiplex array is used for the initial excitation stageof a multidimensional experiment and further arraysare employed for the subsequent coherence transferstages. Acquisition of the free induction decays isconventional and employs Fourier transformation toderive the spectra.

The key to this method is an encoding schemebased on Hadamard matrices (Hadamard, 1893) madeup of plus and minus signs. To fix ideas, consider the

simple example of 15 chemical sites irradiated with 15different radiofrequencies simultaneously. A Hadam-ard matrix of order 16 would be employed, modulatingthe individual radiofrequencies according to the signsin each row of the matrix. Each column correspondsto one of the irradiation channels.

+ + + + + + + + + + + + + + + ++ − + − + − + − + − + − + − + −+ + − − + + − − + + − − + + − −+ − − + + − − + + − − + + − − ++ + + + − − − − + + + + − − − −+ − + − − + − + + − + − − + − ++ + − − − − + + + + − − − − + ++ − − + − + + − + − − + − + + −+ + + + + + + + − − − − − − − −+ − + − + − + − − + − + − + − ++ + − − + + − − − − + + − − + ++ − − + + − − + − + + − − + + −+ + + + − − − − − − − − + + + ++ − + − − + − + − + − + + − + −+ + − − − − + + − − + + + + − −+ − − + − + + − − + + − + − − +For practical instrumental reasons the first ‘all

plus’ column would not normally be used. For the ini-tial excitation stage, the plus and minus signs define+90◦ and −90◦ radiofrequency pulses, whereas forthe transfer stages they designate ‘on’ and ‘off’ 180◦pulses; in either case the modulation reverses thesign of the final NMR signal. The measurements arerepeated until 16 scans have been carried out, the mod-ulation code being changed in each successive scanaccording to the relevant row of the H-16 matrix. All16 scans must be completed. The free induction decayrecorded at the end of each scan is a composite of theresponses from all 15 chemical sites, some positiveand the others negative in sense.

This encoding procedure serves as a means fordisentangling the individual responses, rather like theoperation of a telephone scrambling device. For ex-ample, subtraction of the sum of the free inductionsignals from the even-numbered scans from the sumof those from odd-numbered scans gives the ‘pure’response from the first site, the one that employedcolumn two of the matrix; all other responses can-cel, being made up of equal numbers of positive andnegative contributions. Formally this operation can berepresented as a Hadamard transformation.

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There is of course a slight mismatch between thenumber of chemical sites to be irradiated and the orderof the nearest Hadamard matrix, but the effect is small,as matrices exist for any order N = 4n where n is aninteger. Common values for this type of experimentwould be N = 8, 12, 16, 20, 24, 28, 32, etc. Thenumber of ‘wasted’ columns of the encoding matrixis therefore quite small, and it is useful to be able toomit the ‘all plus’ column on the grounds that it couldprove rather more sensitive to residual spectrometerimperfections.

Two or more encoding operations can be cas-caded, allowing the Hadamard method to be applied insuccessive frequency dimensions. Note that the intro-duction of each new dimension multiplies the requirednumbers of scans. Thus a typical three-dimensionalexperiment might employ (for example) Hadamardmatrices of order 16 in the first dimension and 12 inthe second dimension, with the usual acquisition ofthe free induction decays in the third dimension. Therewould be 192 scans. At first sight the number of man-datory scans might seem to constitute a real limitationon the application of the Hadamard method. However,it replaces a traditional multidimensional experimentthat systematically explores the corresponding evol-ution dimensions with a much larger total numberof scans. A conventional three-dimensional experi-ment, equivalent to the example outlined above, mightinvolve 128 increments in the t1 dimension and 64 in-crements in t2 to cover the desired frequency rangeswhile maintaining only marginal resolution in these di-mensions, hence 8192 scans. This is a speed advantageof almost 43 in favour of the Hadamard mode. Further-more, direct multiplex irradiation in the frequency do-main defines the radiofrequencies unequivocally; thereis no sign ambiguity and no question of aliasing, sothere is no need for quadrature phase detection, savinganother factor of two. Certain types of spectrometerimperfections are reduced as a result of the Hadamardmodulation scheme, thus diminishing the necessity forphase cycling. When all these factors are taken intoaccount there is a speed advantage of between twoand three orders of magnitude for high-dimensionalHadamard spectroscopy.

Clearly the Hadamard experiment is completedmore quickly the fewer the number of chemically dis-tinct sites chosen for irradiation. This can be turnedto good advantage in applications to protein chem-istry, where non-specific enrichment in carbon-13 andnitrogen-15 is a popular stratagem. In multidimen-sional Hadamard spectroscopy the operator has the

option of selecting only the most interesting sites forirradiation, essentially converting a globally-enrichedsample into one that has been specifically labelled inthe important carbon and nitrogen locations. This abil-ity to generate a subspectrum from specific residues isillustrated for the 700 MHz three-dimensional HNCOspectrum of 0.3 mM agitoxin, a 4 kDa protein glob-ally enriched in carbon-13 and nitrogen-15 (Kupce andFreeman, 2003c). Figure 5a shows the projection ontothe proton-carbon-13 plane derived from the full three-dimensional conventional spectrum with two scans perincrement, recorded in 20 h 43 min. This projec-tion does not discriminate with respect to nitrogen-15frequencies. Seven selected responses, indicated by ar-rows, were then recorded by the Hadamard technique(Figure 5b) with a time saving of more than 200.

The distinguishing feature of direct-irradiationHadamard spectroscopy is the need for prior measure-ments to determine the NMR frequencies. For smalland medium-sized molecules these preparatory ex-periments are readily implemented and cost little inspectrometer time. However, some new problems canarise with large proteins because of the very complex-ity of the spectra, poor resolution and low inherentsensitivity, making it difficult to attack an ‘unknown’protein entirely from scratch. Unless assignment in-formation is available from earlier work, it may bedifficult to select frequencies in the second and sub-sequent dimensions that correlate with those chosenin the first dimension. The real power of the Hadam-ard method for biological macromolecules emergesin studies of protein binding or folding, where a fullmultidimensional spectrum has already been acquiredand where the interest lies in the time development ofthe spectra. The ability to focus attention on residuesclose to the active site is a distinct advantage in thisapplication.

Projection-reconstruction

The internal structure of a three-dimensional objectcan be reconstructed from projections of the X-rayabsorption measured in several different directions,an idea that has been successfully exploited in X-ray tomography (Hounsfield, 1973). An analogousreconstruction of a three-dimensional NMR spectrumfrom its plane projections should present a rathereasier problem because the absorption sites are dis-crete and normally well-resolved. So the questionnaturally arises, could all the three-dimensional NMR

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Figure 6. The 700 MHz three-dimensional HNCO spectrum of agitoxin projected onto the proton-carbon-13 plane. (A) The conventionalFourier transform spectrum with seven selected residues indicated by arrows. (B) The Hadamard subspectrum where only these seven responseswere excited. The speed advantage of the Hadamard mode was more than 200.

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Figure 7. A small selected region of the 500 MHz proton HNCO spectrum of ubiquitin, labelled with carbon-13 and nitrogen-15. (a) Projectiononto the proton-nitrogen-15 plane, obtained by setting t1 = 0. Two responses at the bottom right overlap. (b) Projection onto a plane thatsubtends an angle of 10◦ with respect to the proton-nitrogen-15 plane. This displaces all the responses in a vertical direction by differentamounts, separating the two overlapping peaks. In the tilted projection the vertical scale cannot be expressed in ppm.

information be extracted from a small number of two-dimensional projections measured in a relatively shorttime? If F1 and F2 are the evolution dimensions andF3 the direct acquisition dimension, then the projec-tions onto the orthogonal F1F3 and F2F3 planes arereadily recorded by setting t2 = 0 and t1 = 0, re-spectively. These are sometimes referred to as the ‘firstplanes’ of the three-dimensional spectrum and can bemeasured quite quickly. In an ideal situation wherenot too many peaks contribute to the intensity of asingle response in the projection, and where the meas-ured NMR intensities in both projections are reliable,the entire three-dimensional spectrum can be extrac-ted from information on these two planes. In actualpractice, ambiguities are to be expected and must beresolved if the full three-dimensional spectrum is tobe recovered.

A projection is made up of intensities summedalong the various parallel ‘rays’, and if a given rayintersects several peaks, there is a potential ambigu-ity in the relative intensities of these peaks, whichmay or may not be resolved by consideration of in-tensities in the orthogonal projection. One solution isto measure a skew projection intermediate between

the F1F3 and F2F3 planes by incrementing t1 and t2jointly, with some suitable scaling factor. Judiciouschoice of this new projection direction should separ-ate any overlapping peaks. In extreme cases furtherskew projections could be envisaged. Even if tiltedprojections were taken at every 10◦ interval between0◦ and 90◦, the measurement would still be completedmuch faster than the full three-dimensional spectrum.Skew projections are obtained by replacing the normalevolution-time increments �t1 and �t2 with �t1sinα

and �t2cosα where α is the desired projection angle.The linewidths in the skew projection are related tothose in the two orthogonal projections and to the pro-jection angle α. In a favorable case, as few as threeplane projections measured in a short time can map outthe entire three-dimensional spectrum. The final resultmight well be a complete set of three-dimensional co-ordinates along with the associated intensities. Sincethe two orthogonal ‘first-plane’ projections are lesssusceptible to relaxation losses, the method is wellsuited to studies of large proteins, but being faster thanthe full three-dimensional protocol, it sacrifices someof the multiplex sensitivity advantage.

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Resolution of ambiguity is demonstrated for500 MHz proton spectra taken from a small selec-ted region of the HNCO spectrum of ubiquitin (la-belled in carbon-13 and nitrogen-15). The projectiononto the proton-nitrogen-15 plane shows two overlap-ping responses (Figure 7a). A skew projection ontoa plane tilted through 10◦ with respect to this planedisplaces all these responses in the vertical dimensionby varying amounts, neatly separating the two peaksin question (Figure 7b). Note that this tilted projectionmethod assigns a trajectory to each peak.

Conclusions

The time factor stands out as the most serious curbto the application of multidimensional spectroscopyto proteins and other macromolecules of biochemicalinterest, and it is clear that the new methodologiesoutlined above go a long way to addressing this prob-lem. Such gains naturally involve some penalties. Inmost cases the inherent NMR sensitivity is assumed tobe so high that the effects of instrumental noise canbe neglected, but as the performance of commercialspectrometers continues to improve, this may becomea less demanding consideration. The key advantageof the filter diagonalization method is that it requiresno operational changes at all, but the sophisticateddata processing can prove time-consuming; it is notyet a ‘turn-key’ operation. The strength of GFT spec-troscopy is that it can tackle very-high-dimensionalspectra without sacrificing resolution, and the G-matrix transform is fast, but it may run into overlapproblems in very crowded spectra. Frydman’s single-scan technique is by far the fastest of all, acquiringa two-dimensional data set in less than a second, butit relies on high-intensity rapidly-switched magneticfield gradients, a technology that may need to be im-proved for the method to gain wide acceptance. Directirradiation Hadamard spectroscopy is predicated onthe prior acquisition of the chemical shift frequencies,and some correlations may also be required to set upan experiment on a large protein. However, it offersthe opportunity to focus attention on the resonances ofprime interest, for example those close to the activesite of a protein. The proposal for reconstructing athree-dimensional spectrum from a small number ofplane projections promises similar improvements inspeed. It is conceivable that hybrid combinations ofthese techniques may serve to combine some of therespective advantages, or diminish any drawbacks.

Acknowledgements

The authors are indebted to A.J. Shaka, ThomasSzyperski, and Lucio Frydman for providing manu-scripts prior to publication and for helpful adviceabout their respective techniques, and to A.J. Shakaand Lucio Frydman for permission to reproduce fig-ures from published work.

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