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A systematic approach to the suppression of J cross peaks in 2D exchange and 2D NOE spectroscopy
JOURNAL
OF MAGNETIC
RESONANCE
62, 497-5 10 (1985)
A Systematic Approach to the Suppressionof J Cross Peaks in 2D Exchangeand 2D NOE Spectroscopy M. RANCE,*,~~# G. BODENHAUSEN,~ G. WAGNER,* K, W~~THRICH,* AND R. R. ERNST? *fnstitut fir Molekularbiologie and Biophysik, Eidgeniissische Technische Hochschule, CH-8093 Zurich, Switzerland, and tLaboratorium fir Physikalische Chemie, Eidgeniissische Technische Hochschule, CH-8092 Zurich. Switzerland Received November 26. 1984 An optimization procedure is described for NMR techniques which require signal averaging of experiments with a variable time delay, either for the suppression of artifacts or for the optimization of coherence transfer over a given frequency range. The procedure is applied to 2D exchange spectroscopy, where artifacts due to zero-quantum coherence evolving in the mixing period (.I cross peaks) can be canceled by coaddition of signals from experiments with a refocusing pulse inserted at suitably chosen points in the mixing interval. It is shown that experiments must be averaged over a sizeable number of ri values to obtain satisfactory cancellation. Furthermore, suppression schemes may fail in systems with more than two spins if two or more of the chemical shifts are nearly degenerate. 0 1985 Academic press, IX. INTRODUCTION
A wide variety of one- and two-dimensional pulse experiments involve fixed delays 7i that must be of the order of the reciprocal of a frequency parameter, such as an offset, a chemical-shift difference, or a scalar, dipolar, or quadrupolar coupling clonstant. Such delays, which may be referred to as “matched intervals,” can only be optimized for a single value of the frequency parameter. In many cases, it is desirable to combine several experiments with different 7i intervals such as to maximize the response over a wide range of offsets or couplings. This situation arises in multiple-quantum excitation (l-5), in relayed magnetization transfer (610), in multiplequantum filtering (2, II), n-spin filtering (2), and related coherencetransfer experiments. In other cases, it is essential to minimize the response in a specified range of frequencies. This situation is typical for J filters (12), which are used, for example, for the simplification of heteronuclear relayed spectra, and for z filters (13), which may be employed for removing phase and multiplet anomalies in various methods. Such a procedure is also used for the suppression of J cross peaks due to zero-quantum coherence (14-I 7) in 2D exchange spectroscopy. A desired frequency response can be achieved by variation of a time parameter 7i from experiment to experiment with coaddition of the resulting 1D or 2D spectra. $ Resent address: Department of Molecular Biology, Research Institute of Scripps Clinic, 10666 North Torrey Pines Road, La Jolla, Calif. 92037. 497
0022-2364/85 $3.00 Copyright Q 1985 by Academic Press, Inc. All rigbu of reproduction in any form reserved.
RANCE
498
ET AL.
This variation leads to a modulation of the signals in such a way that the response can be made uniform or canceled over a suitable range of frequencies, depending on the filter characteristics required in the particular experiment. The crucial problem is the judicious choice of the values of the Tj intervals, such as to give the best possible approximation to the desired frequency response with a minimum number of experiments and a minimum loss of sensitivity. In many cases, it is sufficient to design a low-pass filter which eliminates all frequencies above a certain cutoff frequency. However, more general fdter characteristics may be desirable, such as band-reject filters which eliminate artifacts in 2D NOE (NOESY) spectra that are associated with a given range of chemical-shift differences. In this paper, we focus attention on the optimum choice of the zero-quantum precession intervals Tj for the suppression of J cross peaks in 2D exchange spectroscopy. The filtration procedures discussed are closely related to so-called t, filters in 2D spectroscopy (18). An attractive application of NOESY is the study of the spatial structure and intramolecular interactions of biological macromolecules, since a single 2D experiment can provide information about a network of through-space proton-proton connectivities which extends over the entire molecular structure (19, 20). NORSY plays a fundamental role in recently developed procedures for obtaining sequence-specific resonance assignments in proteins (21-23) and DNA fragments (24, 25), and for the determination of the secondary structure of polypeptides (26, 27). The use of NOESY distance constraints as input for distance geometry calculations (28, 29) enables the determination of the tertiary structure of small noncrystalline proteins by NMR in solution (30, 32). Further progress in the use of NMR in this area will undoubtedly depend on improved quantitation of ‘H-‘H distance measurements by NOESY (31). The NOESY cross-peak intensities may be calibrated with respect to known internuclear distances between protons that belong to a rigid fragment such as methylene or phenyl groups. Since such protons are often scalar coupled, the suppression of J cross peaks is a prerequisite for obtaining a reliable calibration of the cross relaxation rates. It may be sufficient for the purpose of such calibrations to eliminate J cross peaks in limited spectral regions. ZERO-QUANTUM
INTERFERENCE
IN TWO-SPIN
SYSTEMS
Before discussing the suppression of artifacts associated with zero-quantum coherence, we analyze the evolution of the spin systems in the course of the NOESY experiment shown in Fig. 1 in terms of products of Cartesian operators (32). The components responsible for the NOE cross and diagonal peaks remain inphase throughout the entire experiment. In coupled systems, however, the transverse magnetization may dephase in the course of the ti period such as to generate antiphase magnetization terms of the form 2ZhZ,=. The second pulse therefore creates various orders of multiplequantum coherence, which may be transferred to the coupling partners by the last pulse. These purely coherent transfer processes lead to so-called J cross peaks in the 2D spectrum, even in the absence of exchange or cross-relaxation processes (14). Most of the undesired coherence-transfer pathways may be blocked by phase cycling (14, 33-35), but zero-quantum coherence evolving
SUPPRESSION
499
OF J CROSS PEAKS IN 2D SPECTRA
FIG. 1. (a) Basic pulse sequence for 2D exchange spectroscopy (NOESY), and (b) sequence with a A pulse inserts in the mixing period to restrict the effective zero-quantum precession to ri by refocusing the chemical shifts in the remaining intervals 7j = (7, - 732.
be separated by these procedures from the longitudinal magnetization that leads to the NOE cross peaks. For a weakly coupled two-spin system, the undesirable part of the density operator at the beginning of the mixing period is
in 7, cannot
u(tI, T, = 0) = {ZQC},(cos
[II
f2,dI - cos Q&i)sin rJk,t,,
where flk and $ are the offset frequencies (chemical shifts) of the two coupled spins, and {ZQC}, = 2(ZbZlY - ZkyZ,-Jrepresents the y component of the zero-quantum coherence involving the two spins k and 1. The zero-quantum coherence evolves under the chemical-shift difference frequency (!& - $23 in the mixing period. To suppress zero-quantum coherence, a ?r pulse is inserted within a constant mixing period T, at a time 7i + 7j = (T, + Ti)/2 (Fig. 1b). The effective precession of zero-quantum coherence under the chemical shifts is restricted to the Ti interval, since the precession in the two T; periods that precede and follow the ?r pulse is canceled ( 14). Neglecting transverse relaxation, one obtains nkrizkz GWQf---
9
{ZQC],
cos(Qk - Qr)~i - {ZQC},
sin&
- fi$ri,
[2]
where {ZQC], = 2(ZhZh + ZkyZl,,)represents the x component of the zeroquantum coherence. If the last pulse in the sequence is accurately set to /3 = 7r/2, this (ZQC}, term cannot lead to observable magnetization, and only the {ZQC}, term needs to be retained. The relevant part of the density operator at the end of the mixing period is then a(t, , 7,) = -[2ZhZ,,, - 2ZkyZ,Jcos(& - &)~dcos f&t, - cos !@,)sin uJklf, .
]31
After the last pulse, the observable part of the density operator is a(t , , TV,t2 = 0) =
-[L?~J&
-
21&&Os(~~
-
@)T.,{COS
f&t1
-
cos
i$tl)Sin
u&tI.
[4]
500
RANCE
ET
AL.
These terms lead to cross and diagonal peaks at (wi, w2) = (a,, Q), (Q, Q,), (&, a,), and (Ql, Q,). The cosinusoidal dependence on 7i is essential for the elimination of the zero-quantum terms as will be discussed below. The relative amplitudes of the NOE cross and diagonal peaks are not affected by the position of the r pulse. For this reason the scheme of Fig. 1b is to be preferred over techniques that involve a variation of the overall duration T, of the mixing period (14, 17). SYSTEMS
WITH
MORE
THAN
TWO
SPINS
In larger spin systems, the evolution of zero-quantum coherence involving two active spins k and I is also affected by scalar couplings to passive spins n. Thus in a three-spin system, one may obtain antiphase zero-quantum coherence at the end of the mixing period:
[21/J/y- 2Z/J/xICosdJ/cn - J/n)~mCOS(Q~- Q/)Ti - [4hA&
+ GJ~,&&in
- [2ZbZk +
~Z~~Z,JCOS
4Jkn - JI,)~,cos(Q~ - &hi
?T(J~,,- J&msin(Qk
- Ql)Ti
- [4ZhZl,,Z,, - 4Zk,ZhZl,,]sin r(Jkn - J&,sin(& - Q/)Ti. [51 Note that the relevant time parameters are Ti for the chemical shifts and 7, for the coupling terms, since the former are refocused by the x pulse of Fig. lb while the latter remain unaffected. The first term is converted by the last 7r/2 pulse into observable magnetization and leads to J cross peaks analogous to those found in a two-spin system. The second term in Eq. [5] can also be converted into observable magnetization by the last pulse. Depending on the phase of the latter, one obtains:
The third and fourth terms in Eq. [5] cannot lead to observable magnetization if accurate 7r/2 pulses are used. Equation [6] expresses a coherence transfer of zeroand double-quantum coherence of spins k and 1 to single-quantum coherence of the passive spin n and leads to J cross peaks in NOESY spectra with frequency coordinates (wr , 02) = (&, Q,) and (a,, Q,). Since it is also possible to excite terms of the type 4ZhZhZ,,, and 4Z&,Z,, at the beginning of the mixing period, the peaks appear in pairs symmetrically disposed about the diagonal. The essential feature of these signals is that they are associated with a zeroquantum modulation frequency (& - Q,) that is not related to the frequency coordinates of the cross peaks at (&, a,) and (Q,, Q,). In a three-spin system where (& - Q,) is too small for the signals to be removed by variation of the effective precession interval Ti, it is therefore possible to have large undesired cross peaks that appear far from the diagonal, for example if fik N Q, $ Q,,. This situation can be encountered in virtually all amino acids, starting with the NH-CH”Hd fragment in glycine.
SUPPRESSION PROCEDURES
501
OF J CROSS PEAKS IN 2D SPECTRA
FOR THE SUPPRESSION
OF J CROSS PEAKS
The cosinusoidal dependence on 7i of the zero-quantum terms in Eqs. [4] and [5] can be utilized to eliminate the resulting J cross peaks. Two procedures have been suggested previously for the variation of ri in a sequence of experiments: (i) Random variation of Ti in a sequence of experiments with coaddition of the resulting signals in time or in frequency domain (24). For a sufficient number of randomly selected 7i values, it is likely that all relevant difference frequencies (Qk - Q) are suppressed. In practical applications however, the number of Ti values has to’ be kept small (typically 4 to 10 values) to avoid excessively long experiments, si:nce 7i should not be incremented before the phase cycle is completed. This renders th.e randomization questionable. (ii) Systematic incrementation of Ti. If the time parameter 7i is incremented systematically in proportion to t, , Ti = Tp + Xt),
thle zero-quantum peaks are shifted in frequency along the wl axis by +(& - O,)xtr , and the 2D spectrum looses its mirror-image symmetry with respect to the diagonal (16). This procedure is closely related to accordion spectroscopy (36, 37). The zeroquantum signals can then be eliminated by symmetrization (38, 39). Symmetrization may fail to cancel signals if unrelated peaks appear accidentally in symmetric positions. For example, in crowded spectral regions, such as the methylene proton region in protein spectra, care must be taken that shifted zero-quantum peaks do not overlap with genuine NOE cross peaks. In this paper, we propose an experiment which combines improved versions of the procedures (i) and (ii). Instead of a random variation of $, a discrete set of n) is used, which are optimized to achieve the best possible values#(i= 1,2,..., suppression of zero-quantum coherence over a specified frequency range. The actual delays ri are derived from these rp values by systematic incrementation in proportion to t, according to Eq. [7]. Finally, the delay ri may be varied randomly from one t, value to the next, 7i = ‘$ + X~I +
PI
Trandom(tl),
in order to spread any remaining J cross peaks along a line parallel to the w, axis. DESIGN OF THE FILTERING
PROCESS BY JUDICIOUS
SELECTION
OF 7: VALUES
Coadding signals recorded for different 79 values is equivalent to a filtering process in frequency domain. The problem boils down to finding the best set of T: values for a given filter frequency characteristics. In many cases, we have some knowledge about the range of zero-quantum frequencies AQ = (& - Q) that are likely to olccur, and the required filter must have good suppression in a range AQmin < AL! <: A!&,.,,,. If two-spin zero-quantum coherence is transferred to one of the two spins involved, there is a direct relation between the position of a J cross peak and its zero-quantum precession frequency, and a band-reject filter leads to suppression of J cross peaks in two strips parallel to the diagonal of the 2D spectrum, as shown in Fig. 2. If however the zeroquantum coherence has been transferred to a passive
R4NCE
ET AL.
FIG. 2. Schematic NOESY spectrum with suppression of zeroquantum signals of two-spin systems by filtering in a range [AI&l < IAQl < lAQ,l. This does not apply to systems with more than two spins, as may be seen from the discussion of Eqs. [5] and [6].
spin in a larger spin system, this simple correspondence no longer holds (see Eqs.
PI andPI). To determine the optimum set of TY values, we first compute the response of zeroquantum coherence with a frequency AQ to a filtering process consisting of the averaging over n values $, T!, . . . , 71), combined with tr-proportional 7i incrementation. We introduce the response function A(AQ, tr) of the zero-quantum frequency AQ, A(AQ, t,) = ,-’
5 cos AQ(&’ + xt,) i=l
= Re(exp(iA&1r)H(AQ)),
[91
with the frequency-response function H(AQ) = n-* i exp{iAQ$} i=l
= R(AQ) + iI
WI
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
503
where R(AQ) = .-I 5 cos A%: i=l Z(AQ) = n-l 5 sin ART:.
i=I
1111
With Eq. [4] we obtain the average density operator at the beginning of the detection period: u(t,, t2 = 0) = -[2ZLZ,z - 2ZkzZ&(AQ, tl)(cos i&t, - cos @f&in aJkltl.
[ 121
After Fourier transformation, one finds four zero-quantum satellites in the cross sections at w2 = Qk and w2 = II,, which appear at w1 = Qk I+_xAQ and w1 = fi, 3: xAQ with amplitudes and phases determined by the frequency-response function ZY(AQ). In a 2D exchange spectrum recorded with x = 0, i.e., without shifting the
I
-
’ \ y-, / /’ \ OI/’ \ ,\j” I I’ ‘.___’
I
1 ' /
6 I'\ 0
\
' 1:--i;'
/'
Xc
/
_,
I'\
1 ;\ 0
\I, !
: \
IF-4
i‘r.,,'
Lx--'
\ /
I c+ ',.,'
,
/
,: ,I\
I
500
looo
AQ 2n
FIG. 3. Residual amplitude IH( (-), and real and imaginary componexks R(AQ) (-. -. ) and I(AQ) (---) of the filter function in Eq. [l I] for a number of ~7 values n = 2, 3, 4, and 5, optimized by a gridxarch procedure for A&J& = 0.5. The mnge of’zeroquantum frequencies fi=om5OOtolMiOHz rbr which the optimization was carried out is indicated by an arrow at the top of the figure. The maximum residual amplitudes iH( (Eq. [ 131) in this frequency range are listed in Table 1.
RANCE ET AL.
504
signals, the zero-quantum signals appear in pure 2D dispersion if the NOE cross peaks are phase corrected for pure 2D absorption. Upon shifting, there are two satellites with dispersion lineshapes proportional to R(AS2) with equal phases, superimposed upon two absorption peaks with opposite phases that are proportional to Z(AQ). To determine the optimum set of 7: (i = 1, 2, . . . , n) values for a band-reject filter in the selected interval AC&i, < AQ < A&,, a grid search has been performed to minimize the maximum of the residual amplitude function ]ZZ(AQ)] = [{R(AQ))’
+ {Z(AQ))2]“2
[I31 in this interval. Typical examples of the residual amplitude functions are shown in Fig. 3. The solid curves represent IZZ(AQ)], the dashed curves show Z(AQ), and the curves indicated by (-*-a) represent R(AQ). The optimized values rp in Table 1 have been determined for a standard frequency range AQ,d(27r) = 1 kHz and AQmiJ(2r) = 900, 750, 500, 250, and 100 HZ. The 77 values in Table 1 are appropriate for a typical proton spectrum of 10 ppm width at 100 MHz. For a different Iarmor frequency fO, expressed in MHz, the optimum values are TABLE 1 Best 7: Values for Minimizing the Filter Function ]H(Aw)l (Eq. [ 131) in the Interval An,,,, -c AQ < AI&,,= Obtained by a Grid Search”
AfL Best 7: values (ps)
Maximum residual amplitude IHI
Unax
n
0.9
2 3 4 5
1950, 2450 850, 1200, 1550 850, 1350, 1400, 1900 750, 1150, 1350, 1550, 1950
0.16 0.068 0.0055 0.018
0.75
2 3 4 5
1100, 1650 1050, 1450, 1850 700, 1200, 1350, 1850 1700,2200, 2250, 2600, 2950
0.27 0.20 0.042 0.034
0.5
2 3 4 5
50, 700 200, 850, 1500 850, 1400, 1750, 2300 400, 900, 1150, 1550, 1950
0.52 0.33 0.15 0.14
0.25
2 3 4 5
loo, 900 900, 1550,255o 50, 850, 1650, 2450 500, 1200, 1850, 2500, 3150
0.81 0.52 0.27 0.24
0.1
2 3 4 5
loo, loo0 550, 1300, 3300 500, 1300, 3450, 5ooo 200,1000,1800,3400,5800
0.96 0.75 0.52 0.43
a The values ~7 were varied in increments of 50 p from 0 to 6 ms. AQ,,,,J(2*) is set to 1 kHz, appropriate for a 10 ppm proton spectrum at 100 MHz.
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
T:(&) = $( 100 MHz) loo . Al
505 iI41
For example, all T: values in Table 1 must be divided by a factor 5 for a spectrometer operating at 500 MHz. If it is sufficient to suppress zero-quantum artifacts in a frequency band as shown in Fig. 2, the maximum zero-quantum frequency may be expressed in ppm by a parameter A,,,,, and the renormalized optimum ~7 values are 7p(f0, a,,,) = $( 100 MHz, 10 ppm) loo - 6,,, . fo 10
[I51
Table 1 shows that the suppression becomes more difficult if the ratio An,,/ W-II,, is small, i.e., if the suppression must be effective over a wide range. In this case, the number of required 77 values must be increased to obtain satisfactory suppression. For a larger number of 7: values, however, the numerical optimization is quite tedious, and it is often more convenient to employ approximate procedures. A rather satisfactory approximation to the desired filter in the range A&” < AQ < Wmx can be obtained without grid search by using n equidistant TVvalues with 70I = iAT El61 wherei=O, l,...,nland ATO = 2*(AQ2,,, + AQmin)-’
1171
using the weights hi= 1,
fori&
1.
[I81
Instead of employing unequal weights, it is of advantage to record twice as many scans for i > 1 as for i = 0 to optimize the sensitivity. The resulting filter characteristics are shown in Fig. 4 for A&ax/2~ = 1000 Hz and A& J2r = 100 H[z, with AT’ = 0.9 ms, according to Eq. [ 171, and different numbers n of delays as indicated in the figure. It is obvious that the filter characteristics are improved the larger the number n of 7: values is. In a typical NOESY experiment with an eight-step phase cycle (35) it is possible to use about 12 different 73 delays in an experiment lasting 24 h with 5 12 ti values, 7, = 0.1 s, an acquisition time of 0.3 s, and a relaxation delay of 1 s between individual scans. Figure 5 shows a comparison of the filter characteristics obtained by grid search and by equidistant increments of 7:. In both cases, n = 4 delays ~9 were used. For a narrow range of zero-quantum frequencies, the grid-search procedure yields a significantly better suppression of J cross peaks than the equidistant incrementation of the T: values. For a wide frequency range, equidistant incrementation yields a good suppression in the center of the frequency range while the grid-search procedure shows its superiority only at the edges of the frequency range. Fig. 6 compares the maximum residual amplitude IH( (Eq. [ 131) in a given range A&i, < AQ < A&, as a function of the number of delays n. The suppression achieved by grid search (crosses in Fig. 6) could be further improved if noninteger coefficients were allowed. In the case of systematic increments (dots in Fig. 6) the
506
RANCE ET AL.
n
6
lo
20
89 2s FIG. 4. Residual amplitudes jJZ(AQ)l (-), R(AQ) (-e-e), and Z(AQ) (---) of the filter function obtained in Eq. [ 111 with equidistant incrementation of ~7 according to F.q. [16], with ry = z&r(AQ, + A&,$‘, i = 0,l , . . . . n - 1. n is indicated on the right-hand side of the figure. A&,, = 100 Hz and AQ, = 1000 Hz are indicated in the bottom curve. The second maximum appears at AQ = (AQ,, + At&.).
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
507
FIG. 5. Comparison of the residual amplitude IH( (-), R(AQ) (-.-a), and 1(AQ) (---) of the filter function in Eq. [l I] for n = 4 different 7: values obtained by the grid-search procedure (A) and by equidistant incrementation of T: according to Eq. 116) (B). The frequency ranges for which the delays were optimized are indicated with arrows: from top to bottom, A&./A&,, = 0.75, 0.5, and 0.1, respectively.
fact that the signal obtained for & = 0 is weighted by ho = 4 (Eq. [ 18)) explains that the suppression can exceed the performance of procedures optimized by grid search. If all weights for the systematic increments were set to unity, the dots in Fig. 6 would all lie above the crosses. The curves in Fii. 3-6 describe upper limits of the expected amplitudes of J cross peaks due to zero-quantum coherence. The actual amplitudes will be reduced, since zero-quantum relaxation is faster than the T1 decay of the NOE cross peaks (1’4, 16, 17). EXPERIMENTAL
VERIFICATION
To demonstrate the efficiency of the suppression of zero-quantum peaks, NOESY spectra were obtained from the protein basic pancreatic trypsin inhibitor (BPTI) in *I120. The phase-sensitive cross sections shown in Fig. 7 were taken from 2D NOE spectra obtained with incrementation of 7i according to Eq. [7] with x = 0.25 to shift the signals. In (a) and (c) the zeroquantum peaks were not suppressed (n = 1, ~1’ = 0) while in the corresponding sections (b) and (d) ten equidistant 7: values were coadded with AT’ = 0.9 ms to obtain adequate suppression (compare Fig. 4).
508
RANCE ET AL.
H a5
0 0.25 a5
0
a5 I
--
0
f
t
a75
0.5
0
i
a5
0
i
&my 5
10
i 15
20
n
FIG. 6. Maximum residual absolute-value amplitudes IH( (Eq. [ 131) in the zero-frequency range A&, < An < AQmsr for AQ,,/AQ,, = 0.1, 0.25, 0.5, 0.75, and 0.9. The abscissa represents the number n of delays used. The values IH( calculated for equidistant incrementation of 7: or obtained with the delays from Table 1 are identified with dots and crosses, respectively.
The refocusing in the r, interval was effected by a composite pulse of the type (?r/2)Aa),,(a/2), (41). A 32-step phase cycle was employed as described elsewhere (42), and time-proportional phase incrementation (TPPI) was employed in conjunction with a real Fourier transformation with respect to tl to obtain pure phase lineshapes (35, 43). The residual zero-quantum signals are reduced to less than 10%. CONCLUSIONS
In summary, we recommend that J cross peaks due to zero-quantum coherence in 2D exchange spectra should be eliminated by coadding signals for well selected 7: values using the sequence in Fig. 1b. In addition, the 7i values should be
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
509
b I
6
I
I
8
4
I
2 (4
1
I
0
hm)
FIG. 7. Cross sections parallel to U, taken from 360 MHz NOESY spectra of a 0.02 M solution of Bl?TI in *H20, 36’C, p*H 4.6, T, = 20 ms. All labile protons were exchanged by deuterium prior to the eqeriment. (a) Zero-quantum signals associated with the C” and CB protons of Ala 58 (6(C”H) = 4.01 ppm, S(C%I,) = 1.30 ppm), and the C8 and CT protons of Thr 32 (fi(C@I-I)= 4.03 ppm, G(C?Hs) = 0.59 ppm), which were shifted (x = 0.25) but not suppressed. (b) Same shift but suppression with n = 10 values rp as in Fig. 4. (c and d) Same as (a) and (b), but for the C”H-@H3 cross peak of Ala 25 (8(C”H) = 3.76 ppm, 6(C%) = 1.55 ppm.
incremented in concert with tl according to Eq. [7] to shift residual signals. Finally, to “smear” the remaining peaks in the w1 domain, the 7i delay may be varied stochastically for different tl values (14) (Eq. [S]). This last step should, however, be used only as a supplementary remedy as it tends to enhance the tl noise. Good suppression can only be achieved with a large number n of 7: values (typically n 3r lo), which can be taken in regular increments according to Eq. [ 171. If one is limited to using a very small number of 7: values (n < 5), the values in Table 1, which were optimized by a grid-search procedure, are recommended, particularly if J’ cross peaks must be suppressed within a narrow frequency band. Special attention must be paid to systems with three and more coupled spins, where J cross peaks with a low precession frequency may appear far from the diagonal. ACKNOWLEDGMENTS This research was supported in part by the Swiss National Science Foundation (Projects 3.284.82 and 2.44 l-0.82) and by the Kommission zur Fijrderung der wissenschaftlichen Forschung (Project 1120).
510
RANCE
ET
AL.
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Mugn.
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Progr. NMR Spectrosc. 14, 137 (1981). 5. D. P. WEITEKAMP, A& Magn. Reson. 11, 111 (1983). 6. G. W. EBX, G. B~DENHAUSEN, AND R. R. ERNST, J. Am. Chem. Sot. 104, 3731 (1982). 7. P. H. BOLTON AND G. B~DENHAUSEN, Chem. Phys. Lett. 89, 139 (1982). 8. P. H. BOLTON, 3. Mugn. Reson. 48, 336 (1982). 9. G. WAGNER, .I Magn. Reson. 55, 151 (1983). 10. G. KING AND P. E. WRIGHT, J. Mugn. Reson. 54,328 (1983). II. H. KESSLER, H. OSCHKINAT, 0. W. WRENSEN, H. KOGLER, AND R. R. ERNST, J Mum.
Reson. 55,
329 (1983). 12. H. KOGLER, 0. W. WRENSEN, G. B~DENHAUSEN, AND R. R. ERNST, J. Magn. Reson. 55, 157 (1983). 13. 0. W. WRENSEN, M. RANCE, AND R. R. ERNST, J. Magn. Reson. 56,527 (1984). 14. S. MACIJRA, Y. HUANG, D. SUTER, AND R. R. ERNST, J. Mugn. Reson. 43,259 (1981). IS. ANIL KUMAR, G. WAGNER, R. R. ERNST, AND K. WUTHRICH, J. Am. Chem. Sk. 103,3654 (1981). 16. S. MACVRA, K. W~THRICH, AND R. R. ERNST, J. Magn. Reson. 46,269 (1982). 17. S. MACURA, K. WUTHRICH, AND R. R. ERNST, J. Magn. Reson. 47,351 (1982). 18. K. NAGAYAMA, 25th Exper. NMR Conference, Wilmington, Delaware, April 1984. 19. ANIL KUMAR, R. R. ERNST, AND K. W~THRICH, B&hem. Biophys. Res. Commun. 95, 1 (1980). 20. ANIL KUMAR, G. WAGNER, R. R. ERNST, AND K. WUTHRICH, Biochem. Biophys. Res. Commun. 96, 1156 (1980). 21. G. WAGNER, ANIL KUMAR, AND K. WUTHRICH, Eur. J. B&hem. 114, 375 (1981). 22. M. BILLETER, W. BRAUN, AND K. WUTHIUCH, .T. Mol. Biol. 155, 321 (1982). 23. G. WAGNER AND K. WUTHRICH, J. Mol. Biol. 155, 347 (1982). 24. R. M. SCHEEK, N. Russo, R. BOELENS, R. KAPTEIN, AND J. H. VAN BOOM, .I. Am. Chem. Sot. 105, 2914 (1983). 25. D. R. HARE, D. E. WEMMER, S. H. CHOU, G. DROBNY, AND B. R. REID, J. Mol. Bioi. 171, 319 (1983). 26. E. R. P. ZUIDERWEG, R. KAFTEIN, AND K. WOTHRICH, Proc. Nat/. Ad. Sci. USA. 80, 5837 (1983). 27. M. WILLIAMSON, D. MARION, AND K. WUTHRICH, J. Mol. Biol. 173, 341 (1984). 28. W. BRAUN, C. B&cH, L. R. BROWN, N. Go, AND K. WOTHRICH, Biochim. Biophys. Acta 667, 377 (1981). 29. T. HAVEL, I. 0. KUNZ, AND G. M. CRIPPEN, Bull. Math. Biol. 45, 665 (1983). 30. K. WOTHRICH, G. WIDER, G. WAGNER, AND W. BRAUN, J. Mol. Biol. 155, 3 11 (1982). 31. T. HAVEL AND K. WOTHRICH, Bull. Math. Biol., 46,673 (1984). 32. 0. W. SORENSEN, G. W. EICH, M. H. LEVIS, G. B~DENHAUSEN, AND R. R. ERNST, Progr. NMR Spectrosc. 16, 163 (1983). 33. G. B~DENHAUSEN AND R. R. ERNST, Mol. Phys. 47, 319 (1982). 34. A. D. BAIN, J. Mugn. Reson. 56, 418 (1984). 35. G. BODENHAUSEN, H. KOGLER, AND R. R. ERNST, J. Magn. Reson. 58, 320 (1984). 36. G. BODENHAUSEN AND R. R. ERNST, J. Magn. Reson. 45, 367 (1981). 37. G. B~DENHAU~EN AND R. R. ERNST, J. Am. Chem. Sot. 104, 1304 (1982). 38. R. BAUMANN, ANIL KUMAR, R. R. ERNST, AND K. WUTHRICH, J. Magn. Reson. 44, 76 (1981). 39. R. BAUMANN, G. WIDER, R. R. ERNST, AND K. WUTHRICH, J. Mum. Reson. 44, 402 (1981). 40. S. MACURA AND R. R. ERNST, Mol. Phys. 41,95 (1980). 41. R. FREEMAN, S. P. KEMPSELL, AND M. H. LEVITT, J. Magn. Reson. 38,453 (1980). 42. G. BODENHAUSEN, M. RANCE, G. WAGNER, 0. W. SI~RENSEN, K. WOTHRICH, AND R. R. ERNST, J. Magn. Reson. 59,542 (1984). 43. D. MARION AND K. WOTHRICH, B&hem. Biophys. Res. Commun. 113, 967 (1983).
OF MAGNETIC
RESONANCE
62, 497-5 10 (1985)
A Systematic Approach to the Suppressionof J Cross Peaks in 2D Exchangeand 2D NOE Spectroscopy M. RANCE,*,~~# G. BODENHAUSEN,~ G. WAGNER,* K, W~~THRICH,* AND R. R. ERNST? *fnstitut fir Molekularbiologie and Biophysik, Eidgeniissische Technische Hochschule, CH-8093 Zurich, Switzerland, and tLaboratorium fir Physikalische Chemie, Eidgeniissische Technische Hochschule, CH-8092 Zurich. Switzerland Received November 26. 1984 An optimization procedure is described for NMR techniques which require signal averaging of experiments with a variable time delay, either for the suppression of artifacts or for the optimization of coherence transfer over a given frequency range. The procedure is applied to 2D exchange spectroscopy, where artifacts due to zero-quantum coherence evolving in the mixing period (.I cross peaks) can be canceled by coaddition of signals from experiments with a refocusing pulse inserted at suitably chosen points in the mixing interval. It is shown that experiments must be averaged over a sizeable number of ri values to obtain satisfactory cancellation. Furthermore, suppression schemes may fail in systems with more than two spins if two or more of the chemical shifts are nearly degenerate. 0 1985 Academic press, IX. INTRODUCTION
A wide variety of one- and two-dimensional pulse experiments involve fixed delays 7i that must be of the order of the reciprocal of a frequency parameter, such as an offset, a chemical-shift difference, or a scalar, dipolar, or quadrupolar coupling clonstant. Such delays, which may be referred to as “matched intervals,” can only be optimized for a single value of the frequency parameter. In many cases, it is desirable to combine several experiments with different 7i intervals such as to maximize the response over a wide range of offsets or couplings. This situation arises in multiple-quantum excitation (l-5), in relayed magnetization transfer (610), in multiplequantum filtering (2, II), n-spin filtering (2), and related coherencetransfer experiments. In other cases, it is essential to minimize the response in a specified range of frequencies. This situation is typical for J filters (12), which are used, for example, for the simplification of heteronuclear relayed spectra, and for z filters (13), which may be employed for removing phase and multiplet anomalies in various methods. Such a procedure is also used for the suppression of J cross peaks due to zero-quantum coherence (14-I 7) in 2D exchange spectroscopy. A desired frequency response can be achieved by variation of a time parameter 7i from experiment to experiment with coaddition of the resulting 1D or 2D spectra. $ Resent address: Department of Molecular Biology, Research Institute of Scripps Clinic, 10666 North Torrey Pines Road, La Jolla, Calif. 92037. 497
0022-2364/85 $3.00 Copyright Q 1985 by Academic Press, Inc. All rigbu of reproduction in any form reserved.
RANCE
498
ET AL.
This variation leads to a modulation of the signals in such a way that the response can be made uniform or canceled over a suitable range of frequencies, depending on the filter characteristics required in the particular experiment. The crucial problem is the judicious choice of the values of the Tj intervals, such as to give the best possible approximation to the desired frequency response with a minimum number of experiments and a minimum loss of sensitivity. In many cases, it is sufficient to design a low-pass filter which eliminates all frequencies above a certain cutoff frequency. However, more general fdter characteristics may be desirable, such as band-reject filters which eliminate artifacts in 2D NOE (NOESY) spectra that are associated with a given range of chemical-shift differences. In this paper, we focus attention on the optimum choice of the zero-quantum precession intervals Tj for the suppression of J cross peaks in 2D exchange spectroscopy. The filtration procedures discussed are closely related to so-called t, filters in 2D spectroscopy (18). An attractive application of NOESY is the study of the spatial structure and intramolecular interactions of biological macromolecules, since a single 2D experiment can provide information about a network of through-space proton-proton connectivities which extends over the entire molecular structure (19, 20). NORSY plays a fundamental role in recently developed procedures for obtaining sequence-specific resonance assignments in proteins (21-23) and DNA fragments (24, 25), and for the determination of the secondary structure of polypeptides (26, 27). The use of NOESY distance constraints as input for distance geometry calculations (28, 29) enables the determination of the tertiary structure of small noncrystalline proteins by NMR in solution (30, 32). Further progress in the use of NMR in this area will undoubtedly depend on improved quantitation of ‘H-‘H distance measurements by NOESY (31). The NOESY cross-peak intensities may be calibrated with respect to known internuclear distances between protons that belong to a rigid fragment such as methylene or phenyl groups. Since such protons are often scalar coupled, the suppression of J cross peaks is a prerequisite for obtaining a reliable calibration of the cross relaxation rates. It may be sufficient for the purpose of such calibrations to eliminate J cross peaks in limited spectral regions. ZERO-QUANTUM
INTERFERENCE
IN TWO-SPIN
SYSTEMS
Before discussing the suppression of artifacts associated with zero-quantum coherence, we analyze the evolution of the spin systems in the course of the NOESY experiment shown in Fig. 1 in terms of products of Cartesian operators (32). The components responsible for the NOE cross and diagonal peaks remain inphase throughout the entire experiment. In coupled systems, however, the transverse magnetization may dephase in the course of the ti period such as to generate antiphase magnetization terms of the form 2ZhZ,=. The second pulse therefore creates various orders of multiplequantum coherence, which may be transferred to the coupling partners by the last pulse. These purely coherent transfer processes lead to so-called J cross peaks in the 2D spectrum, even in the absence of exchange or cross-relaxation processes (14). Most of the undesired coherence-transfer pathways may be blocked by phase cycling (14, 33-35), but zero-quantum coherence evolving
SUPPRESSION
499
OF J CROSS PEAKS IN 2D SPECTRA
FIG. 1. (a) Basic pulse sequence for 2D exchange spectroscopy (NOESY), and (b) sequence with a A pulse inserts in the mixing period to restrict the effective zero-quantum precession to ri by refocusing the chemical shifts in the remaining intervals 7j = (7, - 732.
be separated by these procedures from the longitudinal magnetization that leads to the NOE cross peaks. For a weakly coupled two-spin system, the undesirable part of the density operator at the beginning of the mixing period is
in 7, cannot
u(tI, T, = 0) = {ZQC},(cos
[II
f2,dI - cos Q&i)sin rJk,t,,
where flk and $ are the offset frequencies (chemical shifts) of the two coupled spins, and {ZQC}, = 2(ZbZlY - ZkyZ,-Jrepresents the y component of the zero-quantum coherence involving the two spins k and 1. The zero-quantum coherence evolves under the chemical-shift difference frequency (!& - $23 in the mixing period. To suppress zero-quantum coherence, a ?r pulse is inserted within a constant mixing period T, at a time 7i + 7j = (T, + Ti)/2 (Fig. 1b). The effective precession of zero-quantum coherence under the chemical shifts is restricted to the Ti interval, since the precession in the two T; periods that precede and follow the ?r pulse is canceled ( 14). Neglecting transverse relaxation, one obtains nkrizkz GWQf---
9
{ZQC],
cos(Qk - Qr)~i - {ZQC},
sin&
- fi$ri,
[2]
where {ZQC], = 2(ZhZh + ZkyZl,,)represents the x component of the zeroquantum coherence. If the last pulse in the sequence is accurately set to /3 = 7r/2, this (ZQC}, term cannot lead to observable magnetization, and only the {ZQC}, term needs to be retained. The relevant part of the density operator at the end of the mixing period is then a(t, , 7,) = -[2ZhZ,,, - 2ZkyZ,Jcos(& - &)~dcos f&t, - cos !@,)sin uJklf, .
]31
After the last pulse, the observable part of the density operator is a(t , , TV,t2 = 0) =
-[L?~J&
-
21&&Os(~~
-
@)T.,{COS
f&t1
-
cos
i$tl)Sin
u&tI.
[4]
500
RANCE
ET
AL.
These terms lead to cross and diagonal peaks at (wi, w2) = (a,, Q), (Q, Q,), (&, a,), and (Ql, Q,). The cosinusoidal dependence on 7i is essential for the elimination of the zero-quantum terms as will be discussed below. The relative amplitudes of the NOE cross and diagonal peaks are not affected by the position of the r pulse. For this reason the scheme of Fig. 1b is to be preferred over techniques that involve a variation of the overall duration T, of the mixing period (14, 17). SYSTEMS
WITH
MORE
THAN
TWO
SPINS
In larger spin systems, the evolution of zero-quantum coherence involving two active spins k and I is also affected by scalar couplings to passive spins n. Thus in a three-spin system, one may obtain antiphase zero-quantum coherence at the end of the mixing period:
[21/J/y- 2Z/J/xICosdJ/cn - J/n)~mCOS(Q~- Q/)Ti - [4hA&
+ GJ~,&&in
- [2ZbZk +
~Z~~Z,JCOS
4Jkn - JI,)~,cos(Q~ - &hi
?T(J~,,- J&msin(Qk
- Ql)Ti
- [4ZhZl,,Z,, - 4Zk,ZhZl,,]sin r(Jkn - J&,sin(& - Q/)Ti. [51 Note that the relevant time parameters are Ti for the chemical shifts and 7, for the coupling terms, since the former are refocused by the x pulse of Fig. lb while the latter remain unaffected. The first term is converted by the last 7r/2 pulse into observable magnetization and leads to J cross peaks analogous to those found in a two-spin system. The second term in Eq. [5] can also be converted into observable magnetization by the last pulse. Depending on the phase of the latter, one obtains:
The third and fourth terms in Eq. [5] cannot lead to observable magnetization if accurate 7r/2 pulses are used. Equation [6] expresses a coherence transfer of zeroand double-quantum coherence of spins k and 1 to single-quantum coherence of the passive spin n and leads to J cross peaks in NOESY spectra with frequency coordinates (wr , 02) = (&, Q,) and (a,, Q,). Since it is also possible to excite terms of the type 4ZhZhZ,,, and 4Z&,Z,, at the beginning of the mixing period, the peaks appear in pairs symmetrically disposed about the diagonal. The essential feature of these signals is that they are associated with a zeroquantum modulation frequency (& - Q,) that is not related to the frequency coordinates of the cross peaks at (&, a,) and (Q,, Q,). In a three-spin system where (& - Q,) is too small for the signals to be removed by variation of the effective precession interval Ti, it is therefore possible to have large undesired cross peaks that appear far from the diagonal, for example if fik N Q, $ Q,,. This situation can be encountered in virtually all amino acids, starting with the NH-CH”Hd fragment in glycine.
SUPPRESSION PROCEDURES
501
OF J CROSS PEAKS IN 2D SPECTRA
FOR THE SUPPRESSION
OF J CROSS PEAKS
The cosinusoidal dependence on 7i of the zero-quantum terms in Eqs. [4] and [5] can be utilized to eliminate the resulting J cross peaks. Two procedures have been suggested previously for the variation of ri in a sequence of experiments: (i) Random variation of Ti in a sequence of experiments with coaddition of the resulting signals in time or in frequency domain (24). For a sufficient number of randomly selected 7i values, it is likely that all relevant difference frequencies (Qk - Q) are suppressed. In practical applications however, the number of Ti values has to’ be kept small (typically 4 to 10 values) to avoid excessively long experiments, si:nce 7i should not be incremented before the phase cycle is completed. This renders th.e randomization questionable. (ii) Systematic incrementation of Ti. If the time parameter 7i is incremented systematically in proportion to t, , Ti = Tp + Xt),
thle zero-quantum peaks are shifted in frequency along the wl axis by +(& - O,)xtr , and the 2D spectrum looses its mirror-image symmetry with respect to the diagonal (16). This procedure is closely related to accordion spectroscopy (36, 37). The zeroquantum signals can then be eliminated by symmetrization (38, 39). Symmetrization may fail to cancel signals if unrelated peaks appear accidentally in symmetric positions. For example, in crowded spectral regions, such as the methylene proton region in protein spectra, care must be taken that shifted zero-quantum peaks do not overlap with genuine NOE cross peaks. In this paper, we propose an experiment which combines improved versions of the procedures (i) and (ii). Instead of a random variation of $, a discrete set of n) is used, which are optimized to achieve the best possible values#(i= 1,2,..., suppression of zero-quantum coherence over a specified frequency range. The actual delays ri are derived from these rp values by systematic incrementation in proportion to t, according to Eq. [7]. Finally, the delay ri may be varied randomly from one t, value to the next, 7i = ‘$ + X~I +
PI
Trandom(tl),
in order to spread any remaining J cross peaks along a line parallel to the w, axis. DESIGN OF THE FILTERING
PROCESS BY JUDICIOUS
SELECTION
OF 7: VALUES
Coadding signals recorded for different 79 values is equivalent to a filtering process in frequency domain. The problem boils down to finding the best set of T: values for a given filter frequency characteristics. In many cases, we have some knowledge about the range of zero-quantum frequencies AQ = (& - Q) that are likely to olccur, and the required filter must have good suppression in a range AQmin < AL! <: A!&,.,,,. If two-spin zero-quantum coherence is transferred to one of the two spins involved, there is a direct relation between the position of a J cross peak and its zero-quantum precession frequency, and a band-reject filter leads to suppression of J cross peaks in two strips parallel to the diagonal of the 2D spectrum, as shown in Fig. 2. If however the zeroquantum coherence has been transferred to a passive
R4NCE
ET AL.
FIG. 2. Schematic NOESY spectrum with suppression of zeroquantum signals of two-spin systems by filtering in a range [AI&l < IAQl < lAQ,l. This does not apply to systems with more than two spins, as may be seen from the discussion of Eqs. [5] and [6].
spin in a larger spin system, this simple correspondence no longer holds (see Eqs.
PI andPI). To determine the optimum set of TY values, we first compute the response of zeroquantum coherence with a frequency AQ to a filtering process consisting of the averaging over n values $, T!, . . . , 71), combined with tr-proportional 7i incrementation. We introduce the response function A(AQ, tr) of the zero-quantum frequency AQ, A(AQ, t,) = ,-’
5 cos AQ(&’ + xt,) i=l
= Re(exp(iA&1r)H(AQ)),
[91
with the frequency-response function H(AQ) = n-* i exp{iAQ$} i=l
= R(AQ) + iI
WI
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
503
where R(AQ) = .-I 5 cos A%: i=l Z(AQ) = n-l 5 sin ART:.
i=I
1111
With Eq. [4] we obtain the average density operator at the beginning of the detection period: u(t,, t2 = 0) = -[2ZLZ,z - 2ZkzZ&(AQ, tl)(cos i&t, - cos @f&in aJkltl.
[ 121
After Fourier transformation, one finds four zero-quantum satellites in the cross sections at w2 = Qk and w2 = II,, which appear at w1 = Qk I+_xAQ and w1 = fi, 3: xAQ with amplitudes and phases determined by the frequency-response function ZY(AQ). In a 2D exchange spectrum recorded with x = 0, i.e., without shifting the
I
-
’ \ y-, / /’ \ OI/’ \ ,\j” I I’ ‘.___’
I
1 ' /
6 I'\ 0
\
' 1:--i;'
/'
Xc
/
_,
I'\
1 ;\ 0
\I, !
: \
IF-4
i‘r.,,'
Lx--'
\ /
I c+ ',.,'
,
/
,: ,I\
I
500
looo
AQ 2n
FIG. 3. Residual amplitude IH( (-), and real and imaginary componexks R(AQ) (-. -. ) and I(AQ) (---) of the filter function in Eq. [l I] for a number of ~7 values n = 2, 3, 4, and 5, optimized by a gridxarch procedure for A&J& = 0.5. The mnge of’zeroquantum frequencies fi=om5OOtolMiOHz rbr which the optimization was carried out is indicated by an arrow at the top of the figure. The maximum residual amplitudes iH( (Eq. [ 131) in this frequency range are listed in Table 1.
RANCE ET AL.
504
signals, the zero-quantum signals appear in pure 2D dispersion if the NOE cross peaks are phase corrected for pure 2D absorption. Upon shifting, there are two satellites with dispersion lineshapes proportional to R(AS2) with equal phases, superimposed upon two absorption peaks with opposite phases that are proportional to Z(AQ). To determine the optimum set of 7: (i = 1, 2, . . . , n) values for a band-reject filter in the selected interval AC&i, < AQ < A&,, a grid search has been performed to minimize the maximum of the residual amplitude function ]ZZ(AQ)] = [{R(AQ))’
+ {Z(AQ))2]“2
[I31 in this interval. Typical examples of the residual amplitude functions are shown in Fig. 3. The solid curves represent IZZ(AQ)], the dashed curves show Z(AQ), and the curves indicated by (-*-a) represent R(AQ). The optimized values rp in Table 1 have been determined for a standard frequency range AQ,d(27r) = 1 kHz and AQmiJ(2r) = 900, 750, 500, 250, and 100 HZ. The 77 values in Table 1 are appropriate for a typical proton spectrum of 10 ppm width at 100 MHz. For a different Iarmor frequency fO, expressed in MHz, the optimum values are TABLE 1 Best 7: Values for Minimizing the Filter Function ]H(Aw)l (Eq. [ 131) in the Interval An,,,, -c AQ < AI&,,= Obtained by a Grid Search”
AfL Best 7: values (ps)
Maximum residual amplitude IHI
Unax
n
0.9
2 3 4 5
1950, 2450 850, 1200, 1550 850, 1350, 1400, 1900 750, 1150, 1350, 1550, 1950
0.16 0.068 0.0055 0.018
0.75
2 3 4 5
1100, 1650 1050, 1450, 1850 700, 1200, 1350, 1850 1700,2200, 2250, 2600, 2950
0.27 0.20 0.042 0.034
0.5
2 3 4 5
50, 700 200, 850, 1500 850, 1400, 1750, 2300 400, 900, 1150, 1550, 1950
0.52 0.33 0.15 0.14
0.25
2 3 4 5
loo, 900 900, 1550,255o 50, 850, 1650, 2450 500, 1200, 1850, 2500, 3150
0.81 0.52 0.27 0.24
0.1
2 3 4 5
loo, loo0 550, 1300, 3300 500, 1300, 3450, 5ooo 200,1000,1800,3400,5800
0.96 0.75 0.52 0.43
a The values ~7 were varied in increments of 50 p from 0 to 6 ms. AQ,,,,J(2*) is set to 1 kHz, appropriate for a 10 ppm proton spectrum at 100 MHz.
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
T:(&) = $( 100 MHz) loo . Al
505 iI41
For example, all T: values in Table 1 must be divided by a factor 5 for a spectrometer operating at 500 MHz. If it is sufficient to suppress zero-quantum artifacts in a frequency band as shown in Fig. 2, the maximum zero-quantum frequency may be expressed in ppm by a parameter A,,,,, and the renormalized optimum ~7 values are 7p(f0, a,,,) = $( 100 MHz, 10 ppm) loo - 6,,, . fo 10
[I51
Table 1 shows that the suppression becomes more difficult if the ratio An,,/ W-II,, is small, i.e., if the suppression must be effective over a wide range. In this case, the number of required 77 values must be increased to obtain satisfactory suppression. For a larger number of 7: values, however, the numerical optimization is quite tedious, and it is often more convenient to employ approximate procedures. A rather satisfactory approximation to the desired filter in the range A&” < AQ < Wmx can be obtained without grid search by using n equidistant TVvalues with 70I = iAT El61 wherei=O, l,...,nland ATO = 2*(AQ2,,, + AQmin)-’
1171
using the weights hi= 1,
fori&
1.
[I81
Instead of employing unequal weights, it is of advantage to record twice as many scans for i > 1 as for i = 0 to optimize the sensitivity. The resulting filter characteristics are shown in Fig. 4 for A&ax/2~ = 1000 Hz and A& J2r = 100 H[z, with AT’ = 0.9 ms, according to Eq. [ 171, and different numbers n of delays as indicated in the figure. It is obvious that the filter characteristics are improved the larger the number n of 7: values is. In a typical NOESY experiment with an eight-step phase cycle (35) it is possible to use about 12 different 73 delays in an experiment lasting 24 h with 5 12 ti values, 7, = 0.1 s, an acquisition time of 0.3 s, and a relaxation delay of 1 s between individual scans. Figure 5 shows a comparison of the filter characteristics obtained by grid search and by equidistant increments of 7:. In both cases, n = 4 delays ~9 were used. For a narrow range of zero-quantum frequencies, the grid-search procedure yields a significantly better suppression of J cross peaks than the equidistant incrementation of the T: values. For a wide frequency range, equidistant incrementation yields a good suppression in the center of the frequency range while the grid-search procedure shows its superiority only at the edges of the frequency range. Fig. 6 compares the maximum residual amplitude IH( (Eq. [ 131) in a given range A&i, < AQ < A&, as a function of the number of delays n. The suppression achieved by grid search (crosses in Fig. 6) could be further improved if noninteger coefficients were allowed. In the case of systematic increments (dots in Fig. 6) the
506
RANCE ET AL.
n
6
lo
20
89 2s FIG. 4. Residual amplitudes jJZ(AQ)l (-), R(AQ) (-e-e), and Z(AQ) (---) of the filter function obtained in Eq. [ 111 with equidistant incrementation of ~7 according to F.q. [16], with ry = z&r(AQ, + A&,$‘, i = 0,l , . . . . n - 1. n is indicated on the right-hand side of the figure. A&,, = 100 Hz and AQ, = 1000 Hz are indicated in the bottom curve. The second maximum appears at AQ = (AQ,, + At&.).
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
507
FIG. 5. Comparison of the residual amplitude IH( (-), R(AQ) (-.-a), and 1(AQ) (---) of the filter function in Eq. [l I] for n = 4 different 7: values obtained by the grid-search procedure (A) and by equidistant incrementation of T: according to Eq. 116) (B). The frequency ranges for which the delays were optimized are indicated with arrows: from top to bottom, A&./A&,, = 0.75, 0.5, and 0.1, respectively.
fact that the signal obtained for & = 0 is weighted by ho = 4 (Eq. [ 18)) explains that the suppression can exceed the performance of procedures optimized by grid search. If all weights for the systematic increments were set to unity, the dots in Fig. 6 would all lie above the crosses. The curves in Fii. 3-6 describe upper limits of the expected amplitudes of J cross peaks due to zero-quantum coherence. The actual amplitudes will be reduced, since zero-quantum relaxation is faster than the T1 decay of the NOE cross peaks (1’4, 16, 17). EXPERIMENTAL
VERIFICATION
To demonstrate the efficiency of the suppression of zero-quantum peaks, NOESY spectra were obtained from the protein basic pancreatic trypsin inhibitor (BPTI) in *I120. The phase-sensitive cross sections shown in Fig. 7 were taken from 2D NOE spectra obtained with incrementation of 7i according to Eq. [7] with x = 0.25 to shift the signals. In (a) and (c) the zeroquantum peaks were not suppressed (n = 1, ~1’ = 0) while in the corresponding sections (b) and (d) ten equidistant 7: values were coadded with AT’ = 0.9 ms to obtain adequate suppression (compare Fig. 4).
508
RANCE ET AL.
H a5
0 0.25 a5
0
a5 I
--
0
f
t
a75
0.5
0
i
a5
0
i
&my 5
10
i 15
20
n
FIG. 6. Maximum residual absolute-value amplitudes IH( (Eq. [ 131) in the zero-frequency range A&, < An < AQmsr for AQ,,/AQ,, = 0.1, 0.25, 0.5, 0.75, and 0.9. The abscissa represents the number n of delays used. The values IH( calculated for equidistant incrementation of 7: or obtained with the delays from Table 1 are identified with dots and crosses, respectively.
The refocusing in the r, interval was effected by a composite pulse of the type (?r/2)Aa),,(a/2), (41). A 32-step phase cycle was employed as described elsewhere (42), and time-proportional phase incrementation (TPPI) was employed in conjunction with a real Fourier transformation with respect to tl to obtain pure phase lineshapes (35, 43). The residual zero-quantum signals are reduced to less than 10%. CONCLUSIONS
In summary, we recommend that J cross peaks due to zero-quantum coherence in 2D exchange spectra should be eliminated by coadding signals for well selected 7: values using the sequence in Fig. 1b. In addition, the 7i values should be
SUPPRESSION
OF J CROSS PEAKS IN 2D SPECTRA
509
b I
6
I
I
8
4
I
2 (4
1
I
0
hm)
FIG. 7. Cross sections parallel to U, taken from 360 MHz NOESY spectra of a 0.02 M solution of Bl?TI in *H20, 36’C, p*H 4.6, T, = 20 ms. All labile protons were exchanged by deuterium prior to the eqeriment. (a) Zero-quantum signals associated with the C” and CB protons of Ala 58 (6(C”H) = 4.01 ppm, S(C%I,) = 1.30 ppm), and the C8 and CT protons of Thr 32 (fi(C@I-I)= 4.03 ppm, G(C?Hs) = 0.59 ppm), which were shifted (x = 0.25) but not suppressed. (b) Same shift but suppression with n = 10 values rp as in Fig. 4. (c and d) Same as (a) and (b), but for the C”H-@H3 cross peak of Ala 25 (8(C”H) = 3.76 ppm, 6(C%) = 1.55 ppm.
incremented in concert with tl according to Eq. [7] to shift residual signals. Finally, to “smear” the remaining peaks in the w1 domain, the 7i delay may be varied stochastically for different tl values (14) (Eq. [S]). This last step should, however, be used only as a supplementary remedy as it tends to enhance the tl noise. Good suppression can only be achieved with a large number n of 7: values (typically n 3r lo), which can be taken in regular increments according to Eq. [ 171. If one is limited to using a very small number of 7: values (n < 5), the values in Table 1, which were optimized by a grid-search procedure, are recommended, particularly if J’ cross peaks must be suppressed within a narrow frequency band. Special attention must be paid to systems with three and more coupled spins, where J cross peaks with a low precession frequency may appear far from the diagonal. ACKNOWLEDGMENTS This research was supported in part by the Swiss National Science Foundation (Projects 3.284.82 and 2.44 l-0.82) and by the Kommission zur Fijrderung der wissenschaftlichen Forschung (Project 1120).
510
RANCE
ET
AL.
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