-
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:[email protected]
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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