Article

Sensitivity enhancement in NMR of macromolecules by applicationof optimal control theory

Dominique P. Frueha, Takuhiro Itob, Jr-Shin Lic, Gerhard Wagnera, Steffen J. Glaserd

& Navin Khanejac,*aDepartment of Biological Chemistry and Molecular Pharmacology, Harvard Medical School, Boston, MA,02115; bRIKEN Genomic Sciences Center, 1-7-22 Suehiro-cho, Tsurumi, Yokohama, 230-0045Japan;cDivision of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138; dDepartment ofChemistry, Technische Universität München, 85747 Garching, Germany

Received 6 December 2004; Accepted 16 February 2005

Key words: cross-correlation, GroEL, optimal control, macromolecular NMR, relaxation

Abstract

NMR of macromolecules is limited by large transverse relaxation rates. In practice, this results in lowefficiency of coherence transfer steps in multidimensional NMR experiments, leading to poor sensitivity andlong acquisition times. The efficiency of coherence transfer can be maximized by design of relaxationoptimized pulse sequences using tools from optimal control theory. In this paper, we demonstrate that thisapproach can be adopted for studies of large biological systems, such as the 800 kDa chaperone GroEL.For this system, the 1H–15N coherence transfer module presented here yields an average sensitivityenhancement of 20–25% for cross-correlated relaxation induced polarization transfer (CRIPT) experi-ments.

Abbreviations: CSA – Chemical Shift Anisotropy; CRIPT – Cross-Relaxation Induced PolarizationTransfer; SQC – Single Quantum Coherence; TROPIC – Transverse Relaxation Optimized Polarizationtransfer Induced by Cross-correlation effects.

Introduction

Nuclear Magnetic Resonance (NMR) is a widelyused technique for structural, kinetic and dynamicstudies of biological molecules (Ferentz andWagner, 2000; Abelson and Simon, 2001). Appli-cation of NMR to very large macromolecularsystems is limited, however, by the low sensitivityof the technique, rapid transverse relaxation andspectral crowding. Important technological devel-opments, such as cryogenic probes, high-fieldmagnets, or advanced pulse sequences have alle-viated some of these problems. However, there is a

fundamental limitation of NMR spectroscopywith very large systems because transverse relax-ation is enhanced with increasing molecular size.This leads to rapid decay of signals during andafter pulse sequences applied to derive structuralinformation. In particular, transverse relaxationleads to poor efficiency of coherence transfer inmultidimensional NMR experiments. In contrastto transverse spin order, relaxation of longitudinalspin order components decreases with increasingmolecular size, a phenomenon we want to takeadvantage of for the design of coherence transfermodules described in this manuscript.

The goal of the work described here is to utilizethe slow longitudinal relaxation to improve the

*To whom correspondence should be addressed. E-mail:navin@hrl.harvard.edu

C ( ), T ( )

Journal of Biomolecular NMR (2005) 32: 23–30 � Springer 2005DOI 10.1007/s10858-005-3592-0

sensitivity of multiple-resonance experiments. Wecan significantly improve efficiency of coherencetransfer by storing spin order as much as possiblein its slowly relaxing longitudinal components.The experiment described here is based on recentlydeveloped concepts of relaxation optimized pulsesequences designed using optimal control theory(Khaneja et al., 2003a, b, 2004). We consider two-dimensional experiments that correlate the signalsof two atoms that are covalently bound, e.g. anamide proton and its corresponding nitrogenatom. In the standard HSQC (HeteronuclearSingle-Quantum Correlation spectroscopy) exper-iment (Bodenhausen and Ruben, 1980), this isachieved by transferring the magnetization from aproton to its attached nitrogen via the scalarcoupling between the two spins. However, even forthis simple experiment, the size of the moleculesthat can be studied is greatly limited by transverserelaxation. This effect leads to a decay of the signalover the course of the experiment, sometimes tothe extent that no signal can be detected. Majorimprovements have been made in reducing theselosses by using interference effects (or cross-correlated relaxation) between relaxation mecha-nisms involving the chemical shift anisotropy(CSA) of one spin and the dipole–dipole interac-tion (DD) of that spin with a nearby, second spin.Cross-correlated relaxation slows the decay of thesignal during encoding and detection periods as isexploited in Transverse Relaxation OptimizedSpectroscopy (TROSY) (Pervushin et al., 1997).Moreover, the interference between 1H protonCSA and amide proton–nitrogen 1H–15N DD canbe used to transfer magnetization from proton tonitrogen (Brüschweiler and Ernst, 1991; Dalvit,1992)

Hx ����!R

CSA=DD

H=HN �2HxNz ð1Þ

where RCSA=DDH=HN is the relevant cross-correlation

rate, acting on the single-quantum coherence(SQC) Hx. The efficiency of this transfer dependson the ratio of the cross-correlation rate to theautorelaxation rate and is largely independent ofthe size of the molecule (Riek et al., 1999). Thus, inthe case of large systems, scalar couplings aresupplemented or even supplanted by cross-correlated relaxation to achieve magnetizationtransfer in, respectively, the CRINEPT (Cross-correlated Relaxation Enhanced Polarization

Transfer) (Riek et al., 1999) and CRIPT (Cross-Relaxation Induced Polarization Transfer) exper-iments (Brüschweiler and Ernst, 1991). In spite ofthese improvements, until recently the maximumtransfer efficiency for a molecule with givenrelaxation rates was unknown, and no experimenthad been designed that would achieve such amaximum transfer (Khaneja et al., 2003a). In thispublication, we make use of optimal control the-ory to maximize the efficiency of coherence trans-fers mediated solely by cross-correlation effects.We illustrate the method by recording protonnitrogen correlation maps of the 800 kDa homo-tetradecameric protein GroEL. This protein waspreviously investigated by Wüthrich and cowork-ers by using the [15N, 1H]-CRIPT-TROSY exper-iment (Riek et al., 2002), which we take as areference. We show that the technique presentedhere enhances coherence transfer efficiency by 20–25% when compared to the best and optimizedpre-existing CRIPT coherence transfer method.

Material and methods

In the reference CRIPT experiment shown inFigure 1c (Brüschweiler and Ernst, 1991; Riek etal., 2002), the transfer depicted in Equation (1) isachieved by applying a 90� pulse on proton fre-quencies followed by a mixing time TC. The evolu-tion under chemical shifts and scalar couplings isrefocused during the mixing time. The anti-phaseterm 2HxNz resulting from cross-correlated relax-ation is then converted into longitudinal two-spinorder 2HzNz. The efficiency of the transfer is usuallyoptimized by adjusting TC, based on the ratio of thecross-correlation rate to the transverse auto-relax-ation rate. Recently developed methods relying onoptimal control theory exploit the fact that therelaxation rates of longitudinal operators Hz and2HzNz aremuch smaller than those of the transverseoperators Hx and 2HxNz. Thus, storing part of thespin order in a longitudinal state can further mini-mize signal losses during the transfer pulse sequencemodule. The resulting relaxation optimized experi-ment involves gradual rotation of the magnetiza-tion from Hz to Hx, where the latter is converted to–2HxNz through cross-correlated relaxation. Foran isolated spin system, this can be achieved byusing a selective pulse applied on-resonance withthe unique proton frequency. Throughout the

24

optimal transfer process the ratio of the expecta-tion values of the two transverse operators ismaintained constant to a value that depends ontheir relaxation rates (Khaneja et al., 2003a):

h�2HxNzihHxi

¼ g; ð2Þ

where g ¼ n�ffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 � 1q

, with

n ¼ R2ðHxÞR

CSA=DDH=HN ðHxÞ

; ð3Þ

and R2(Hx) and RCSA=DDH=HN ðHxÞ are the auto- and

cross-correlated relaxation rates, respectively,affecting the proton SQC. This transfer techniquecan be made broadband by discretely flipping themagnetization toward the transverse plane. This isachieved by successive application of hard radio-frequency pulses with small flip angles separated bydelays and refocusing pulses during which the in-phase coherence is converted to the desired anti-phase coherence (Khaneja et al., 2004). Theresulting experiment then is comprised of n evolu-tion periods (n = 4 in the current experiment)where each small flip angle pulse simultaneouslyflips proton magnetization toward the transverseplane and the antiphase component toward thelongitudinal axis. The flip angles and the lengths ofthe evolution periods can be determined based onmethods of dynamic programming in subject ofoptimal control (Khaneja et al., 2004). A 180� pulseon protons in the center of each evolution periodrefocuses evolution under chemical shifts and sca-lar couplings. We refer to this new experiment asTROPIC (Transverse Relaxation OptimizedPolarization transfer Induced by Cross-correlationeffects). Computation of the flip angles and lengthsof the evolution periods for the TROPIC experi-ment is discussed in detail in the Supportingmaterial. The program for computing TROPICpulse sequence can be downloaded from the web-site ‘http://eecs.harvard.edu/� shin’. The optimalnumber of evolution periods can be estimatedthrough simulation. When n = 1, no relaxationoptimization is obtained and the experiment isidentical to the original CRIPT technique. Whenthe transverse period is subdivided into periods wherethe magnetization is only partially oriented toward thetransverse plane, relaxation losses decrease. For thecurrent case, with the rates R2ðHxÞ ¼ 446 s�1

y

1H

15N

Gz g1 g3

y y -y y -y y y y

t2

t1

-x

1H

15N

Gz g1 g3

t2

t1

-x

g2 g2

-x

Hz

Hx2HzNz

-2HxNz

TT

TC

(a)

(b)

(c)

Figure 1. (a) Evolution of the source density operator elementsHx and Hz (top) and the target terms, )2HxNz and 2HzNz(bottom). The lengths and orientations of the vectors weresimulated for a transfer optimized for rates with magnitudesR2ðHxÞ ¼ 446 s�1 and RCSA=DDH=HN ðHxÞ ¼ 326 s�1. (b) Pulse se-quence obtained with optimal control theory. Narrow and largeblack rectangles indicate 90 and 180� pulses, respectively. Smallsolid rectangles indicate 90� rectangular water selective pulsesof length 600 ls with a bandwidth of 0.55 ppm (416.6 Hz).Narrow open rectangles represent hard pulses with flip angles biand small broad open rectangles indicate water selective pulseswith the same flip angles bi but with an opposite phase, )y. Fora transfer optimized for the rates given above, b1 = 18.4�,b2 = 22.3�, b3 = 23.5�, b4 = 22.2�, and b5 = 18.3�. Thecorresponding delays are s1 = 0.3511 ms, s2 = 0.2594,s3 = 0.2595 ms, and s4 = 0.3525 ms with a totalTT = 2.44 ms. /1 = x, )x; /2 = 4(x), 4()x); /rec = )x, x,)x, x, x, )x, x, )x. The power of the flip-back pulses wereadjusted experimentally leading to the following values of theflip angles: 20.4�, 19.6�, 23.2�, 21.9� and 22.8�, respectively forb1 to b5. (c) CRIPT pulse sequence (Riek et al., 2002)D1 = 0.455 ms for the rates mentioned above, TC = 0.91 ms./3 = 2(x), 2()x); /rec = )x, x, x, )x, x, )x, )x, x. Phasesensitivity was achieved by applying the States-TPPI techniqueto phase /1. A second pair of experiments were recorded for therates R2ðHxÞ ¼ 578 s�1 and RCSA=DDH=HN ðHxÞ ¼ 422 s�1: TT =1.66 ms (s1 = 0.2718 ms, s2 = 0.1918 ms, s3 = 0.1726 ms,and s4 = 0.1916 ms), b1 = 23.4�, b2 = 28.8�, b3 = 31.5�,b4 = 31.5�, and b5 = 41.4�. The length of the flip-back pulsehad to be reduced to 450 ls, corresponding to a bandwidth of0.74 ppm (555.5 Hz). The values of the flip-angles of the flip-back pulses were adjusted experimentally to 24.9�, 24.4�, 29.9�,29.9� and 49.5�, respectively for b1 to b5; TC = 0.7 ms, D1 =0.35 ms.

25

and RCSA=DDH=HN ðHxÞ ¼ 326 s�1 (see below), the theo-

retical improvement over the standard CRIPTexperiment is 10% for n=2, 17% for n=3, 21% forn=4 and 22% for n=5. Although the maximumimprovement would be obtained with an infinitenumber of subperiods, in practice, finite pulsedurations and imperfections deteriorate thisimprovement when increasing the value of n. As thenumber of evolution periods increases, so does thenumber of refocusing pulses. Since each 180� pulsehas non-zero duration in which the magnetizationpasses through the transverse plane, an increasednumber of such pulses contributes to relaxationlosses. In general, there is an optimal number ofevolution periods for given relaxation rates, whichcan be readily determined by simulations. In thecurrent case, no substantial gain is obtained whenincreasing n from 4 to 5. In addition, four evolutionperiods allow for incorporation of a supercycle onthe 180� refocusing pulses. The [15N, 1H]-TROPIC-TROSY pulse sequence resulting from optimalcontrol theory (see above) is displayed in Fig-ure 1b. During the period TT, the proton magne-tization Hz is slowly flipped toward the transverseplane, where the single-quantum coherence Hx canbe converted into anti-phase coherence )2HxNz.This is accomplished by dividing the total periodTT in four periods of lengths 2si in which hardpulses with small flip angles bi (denoted by narrowopen rectangles labeled with the corresponding flipangles bi) are used to control the orientation of themagnetization. The generated term is concomi-tantly stored as longitudinal two-spin order 2HzNz.This is achieved by calculating the values of thedelays si and the flip angles bi for a given set of therelaxation rates R2(Hx) and R

CSA=DDH=HN ðHxÞ, which

were estimated as described below. A simulation ofthe magnetization transfer for a given set ofrelaxation rates is depicted in Figure 1a. After atransfer period of total length TR, a z-filter is ap-plied to suppress unwanted coherences once allproton magnetization has been converted to lon-gitudinal two-spin order. Following a 90� pulse,indicated by a solid rectangle, nitrogen SQC isallowed to evolve during the encoding time t1.Finally, the anti-phase proton SQC 2HxNz is de-tected. For comparison, the pulse sequence of [15N,1H]-CRIPT-TROSY as used in Riek et al. (2002) isdisplayed in Figure 1c with the same scale as inFigure 1b. As mentioned in Riek et al. (2002)

particular care must be taken to ensure that thewater is maintained along the z-axis during exper-iments to avoid cross-saturation between water andamide proton spins. For the [15N, 1H]-TROPIC-TROSY the water flip-back pulse (Grzesiek andBax, 1993) lengths were calculated in order tocompensate the small flip angles of each accom-panying hard pulse. Thus, five water-selective pul-ses with flip angles bi and phases opposing thephases of the accompanying hard pulses were in-serted in the period TT. These pulses are denoted bybroad open rectangles in Figure 1b. In bothexperiments, each water-selective pulse was thenadjusted individually, not only to minimize theresidual signal of water protons, but also simulta-neously to maximize the signals of the protein. Todesign the experiments, the relaxation rates R2(Hx)

and RCSA=DDH=HN ðHxÞ were first estimated assuming a

rigid molecule (see for example Goldman, 1984;Boyd et al., 1991; Brüschweiler and Ernst, 1991;Cavanagh et al., 1996) including dipolar interac-tions with remote protons and using the protonCSA as determined by Bax and Cornilescu (2000).This enables the estimation of the ratio n. Thetransfer in the CRIPT experiment was then opti-mized by changing the delay TC as described inRiek et al. (2002), while focusing on signals withsmall intensities. The relaxation rates correspond-ing to an optimum transfer with that delay werethen calculated by using

RCSA=DDH=HN ðHxÞ ¼

coth�1ðnÞpTC

ð4Þ

and R2(Hx) was deduced from Equation (3). Fi-nally, these values were used to generate the cor-responding TROPIC experiment.

2H, 15N-labeled GroEL was over-expressed inE. coli BL21 strain transformed by the plasmidpG-Tf3 (Nishihara et al., 2000) using M9 mediaenriched with 15NH4Cl and

2H2O. The over-expressed protein was purified with Q sepharoseFast Flow anion exchange and HiLoad 16/60superdex 200 gel filtration column chromatogra-phies (Amersham Biosciences). The protein wasconcentrated to a final concentration of 0.08 mM(1.1 mM per monomer) at pH = 6.0 with 25 mMpotassium phosphate and 20 mM KCl.

Each spectrum was obtained in 17 h byrecording 72 · 1024 complex points in 15N and 1Hdimensions, respectively (1024 scans, 0.3 s recy-

26

cling delay). The spectral widths were 16.033 ppmfor 1H and 35 ppm for 15N. The measurementswere effected at 35� on a Bruker 750 MHzAVANCE spectrometer equipped with a quadru-ple resonance probe. The spectra were processed asdescribed in Riek et al. (2002).

Results and discussion

We first recorded a pair of CRIPT and TROPICexperiments with TC = 0.9 ms and TT = 2.4 ms,respectively. The CRIPT transfer time had beenchosen to be shorter than in the experiment ofRiek et al. (2002) (where TC = 1.4 ms) since thefocus here was on signals with large relaxationrates. The resulting spectra are displayed inFigure 2a (CRIPT) and b (TROPIC). Signals withslow relaxation rates, such as those of residueslocated in loops or those of side-chains, are farfrom the optimized conditions and will not bediscussed in the analysis of the results. Theyappear as strong antiphase signals, since theexperiments detect proton SQC in antiphase withnitrogen. For less mobile protons the negativecomponent of the antiphase doublet is broadenedbeyond detection due to its relaxation, which is

enhanced by the CSA/DD cross-correlation effect.A number of peaks corresponding to these morerigid residues appear in the TROPIC spectrum dueto the predicted enhancement (see circled peaks inFigure 2b). However, the enhancement is far frombeing uniform. The magnitude of the relaxationrates involved in both types of transfers is expectedto vary from residue to residue, so that a non-uniform enhancement will be observed. The cross-correlation rate depends on the magnitude of theproton CSA tensor and on its orientation with re-spect to the dipole–dipole interaction (Goldman,1984; Boyd et al., 1991). In addition, internal mo-tions occurring on a time scale which is faster thanthe tumbling of the molecule affect relaxation rates,especially if these motions are anisotropic (Fischeret al., 1997; Brutscher et al., 1998; Lienin et al.,1998; Frueh, 2002). Both auto- and cross-corre-lated relaxation rates also vary with the orientationof each interaction vector with respect to the axesof the rotational diffusion tensor of the molecule(Woessner, 1962; Brüschweiler et al., 1995). Con-formational exchange and chemical exchange willaffect the detected signals by enhancing the mag-nitude of the autorelaxation rate and possiblyaffecting the cross-correlation rate. To determinethe effect of all of these variations, we simulated the

10 9 8 71H (ppm)

10 9 8 71H (ppm)

130

120

110

15N (ppm

)

8

1

234

5

67

9

1011

12

1314 15

1617

18 19 2021 222324 25

2627

28 2930

32 3331 34

35

130

120

110

15N (ppm

)

36

(a) (b)

(c) (d)

Figure 2. TROPIC (b and d) and CRIPT (a and c) spectra of GroEL recorded with TC = 0.91 ms (a), TT = 2.44 ms (b),TC = 0.7 ms (c), and TT = 1.66 ms (d).

27

conversion of magnetization during a transferperiod optimized for given values of the auto- andcross-correlation rates, R2ðHxÞ ¼ 446 s�1 andR

CSA=DDH=HN ðHxÞ ¼ 326 s�1 with the resulting ratio

n)1 = 0.73. The relaxation rates were then variedto measure the change of efficiency of both exper-iments under non nominal conditions. The resultsof these simulations are illustrated in Figure 3.Each curve in Figure 3a corresponds to a fixedratio of the two rates while their magnitude ischanged, with the solid lines depicting the case

n)1 = 0.73 for which the pulse sequence wasoptimized. As expected, the yield is reduced forboth experiments when the conditions are awayfrom the nominal case. The ratio of the efficienciesof the TROPIC to CRIPT experiments (Figure 3b)shows that the enhancement is increased as therates deviate together from the optimal conditions.Thus, even if both experiments suffer losses whenthe rates are different from the values used tooptimize the transfer, smaller losses are found inthe TROPIC experiment.

Under optimal conditions, i.e., whenR2ðHxÞ ¼ R02ðHxÞ, the gain is calculated to bearound 21%. This simulation was repeated for fourvalues of the ratio of the two relaxation rates.Whenthis ratio is different than the nominal value, thegain in efficiency of TROPIC over CRIPT may re-duce. The predicted robustness of the TROPICexperiment suggests that it may be beneficial tooptimize the sequence for fast relaxing signals.Losses for residues with smaller relaxation rates willaffect signals of larger intensities. As shown above,these losses are expected to be smaller in theTROPIC experiment than in the CRIPT experi-ment. Thus, we repeated the two experiments withtransfer times of TC = 0.7 ms and TT = 1.66 ms,i.e., for a transfer optimum for signals with largerelaxation rates. The resulting spectra are shown inFigure 2c and d. A few peaks are now emerging inthe TROPIC spectrum and low intensity peaks areenhanced while most other peaks are little affected.No new peaks are observed in the CRIPT spectrum.This suggests that these signals can be detectedbecause the transfer ismore efficient in the TROPICexperiment for relaxation rates that are differentthan those the experiments were optimized for. Thesignal enhancement is best seen in the traces shownin Figure 4a, particularly peak 35 which is absent inthe two CRIPT experiments and peak 36, whichbarely emerges from the noise in the CRIPT spec-tra. The intensities of 34 peaks uniformly distrib-uted in the spectrum were measured. Other signalsthat were too close to the noise in the CRIPTexperiment (such as peaks 35 and 36) or which werenear large antiphase signals belonging to slowrelaxing residues were discarded. The ratio of theTROPIC signal intensities to the CRIPT signalintensities are shown in Figure 4b (TC = 0.9 msand TT = 2.4 ms) and c (TC = 0.7 ms andTT = 1.6 ms). In both pairs of experiments, a fewsignals are actually decreased in the TROPIC

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

R2(Hx)/R2(Hx) (-)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

(b)

(a)

Figure 3. (a) Simulations of the transfer yields (g) of theTROPIC (thick lines) and CRIPT (thin lines) transfers inexperiments optimized for the rates R2ðHxÞ ¼ 446 s�1 andR

CSA=DDH=HN ðHxÞ ¼ 326 s�1 (TC = 0.91 ms, TT = 2.44 ms). The

efficiency was calculated as a function of the change inmagnitudes of both the cross-correlation rate and the autore-laxation rate from their nominal value, denoted by R02ðHxÞ forthe latter. The simulations were effected for various values ofthe fixed ratio n)1 = 0.5 (dashed line), 0.6 (doted line), 0.7(nominal value, solid line), 0.8 (dashed and dotted) and 0.9(dash – two dots). (b) Ratio of the curves obtained in (a) tocompare the robustness of the two types of experiments.

28

spectra as can also be observed in Figure 2. This isobserved for 9 peaks (decreased on average byabout 19% with a standard deviation of 14%) whenTC = 0.9 ms and TT = 2.4 ms, and for 6 peaks(decreased on average by about 32% with a stan-dard deviation of 17%) when TC = 0.7 ms andTT = 1.6 ms. This can be explained in part by pulseimperfections that will penalize the TROPICexperiment because of its many small flip angle

pulses. This effect can be added to reductions in theoptimal enhancement due to variations in therelaxation rates, as previously mentioned (seeFigure 3). More importantly the preservation ofwater along the positive longitudinal axis is ofprime importance in the current experiment. Whena water presaturation scheme is used instead ofwater flip-back pulses, about 75% of the signal islost in both experiments, in agreement with what isreported in Riek et al. (2002). Maintaining thewater along the positive z-axis is harder to achievein the TROPIC experiment with a number of smallflip-back pulses. Moreover the water spends moretime along the negative axis in the long TROPICpulse sequence than in the shorter CRIPT experi-ment. In TROPIC with TT = 1.6 ms, the 6 peaksthat are reduced (peaks 1, 5, 9, 16, 21, and 22) allhave lower frequencies than the others and are thuscloser to the water frequency. Since this experimentrequires many short flip-back pulses (due to theshorter delays si), the less selective pulses affectsignals with frequencies close to the water, as foundin simulations. For sequences shorter than thoseutilized in the current work, it might be preferableto control the water trajectory by using only onepulse before the TROPIC transfer period and an-other pulse after this period. The length of the waterselective pulse, which is limited by the delays si,would otherwise correspond to a bandwidth thatwould affect amide proton frequencies. In spite ofits more difficult experimental challenges, theTROPIC experiment has a significantly better sen-sitivity. The average enhancement is about 30%(calculated on 25 peaks) with a standard deviationof 19% for TT = 2.4 ms and TC = 0.9 ms, whilean enhancement of 37% (calculated on 28 peaks)with a standard deviation of 25% was determinedfor the second pair of experiments. If one includesthe peaks that are decreased, the overall enhance-ments are 19% and 25% for TT = 2.4 ms andTT = 1.6 ms, respectively. This is in agreementwith the 21% expected theoretically, if one accountsfor variations in the magnitudes and the ratio of thetwo relaxation rates involved in the transfer.

Conclusions

Using methods of optimal control theory, we wereable to develop an experiment that enhances thesensitivity by 20–25% over the CRIPT method.

36

35

30

2427 22

28 25

7 810 9 1H (ppm)

0 5 10 15 20 25 30 34arbitrary peak number

0

1

3

2

ratio

(-)

ratio

(-)

0

1

2 (b)

(c)

(a)

Figure 4. (a) slices extracted from the spectra of Figure 2d(red) and c (black). The corresponding peak numbers areindicated. (b) Ratio between the signal intensities of theTROPIC and the CRIPT experiments for TT = 2.44 ms andTC = 0.91 ms. (c) Same as (b) for TT = 1.66 ms andTC = 0.70 ms. Peak numbers are defined in Figure 2.

29

The TROPIC experiment was successfully imple-mented although only a crude estimation of therelaxation rates was used, and despite challengesrelated to water suppression. The technique hasbeen shown to be more robust than CRIPT inpresence of variations in relaxation rates, whichmaximizes the information content of the spectra.This practical demonstration of the principle ofusing optimal control theory for pulse sequencedesign opens the venue to a number of applica-tions. For small molecules, optimal control theorycan be used to optimize INEPT transfers, or even acombination of cross-correlated and scalar cou-pling mediated transfers (Khaneja et al., 2004).For larger molecules, such as GroEL, we showedthat the TROPIC pulse sequence becomes thetechnique of choice for cross-correlation driventransfers – provided that the water is carefullymaintained longitudinal during the experiment.Furthermore these methods of optimally manipu-lating dynamics of quantum mechanical systems inpresence of decoherence are expected to haveapplications in the whole field of coherent spec-troscopy and quantum information.

Supporting material available in electronic form(at http://dx.doi.org/10.1007/s10858-005-3592-0):Dynamic programming method for finding opti-mal flip angles and delays for the TROPIC pulsesequence.

Acknowledgements

We thank Dr Dmitri Ivanov for useful discussions.This research was supported by grants from NIH(GM 47467 and RR00995 to GW) and NSF (NSF0133673 and AFOSR FA9550-04-1-0427 to NK).SJG acknowledges support from the DeutscheForschungsgemeinschaft (Gl 203/4-2).

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