Heteronuclear zero-quantum two-dimensional nuclear magnetic resonance as a conformational probe

JOURNAL

OF MAGNETIC

RESONANCE

57,421-446

(1984)

Heteronudear Zero-Quantum Two-Dime&d N Resonanceas a Conformatbual Probe PHILIP H. BOLTON* Wesleyan University, Chemistry Department. Middletown, Connecticut 06457 Received October 3, 1983 In many cases the proton spectrum of a sample is not interpretable due to extensive overlap of signals. Indirect detection of the proton spectml information via a heteronucleus, whose signals are resolved, can allow determination of the proton spectral parameters needed for conformational and structural investigations. Previous research has shown that the single-quantum heteronuclear two-dimensional experiment can be used to obtain proton spectral information via a heteronucleus for a wide range of samples. In this article heteronuclear zero-quantum spectroscopy is shown to be a competitive technique for indirectly detecting proton spectral information. The zero-quantum approach is distinctly different from the singlequantum technique. Among the advantages of the zero-quantum experiment is the simplicity of the result which is a spectrum directly corresponding to the heteronuclear decoupled proton spectrum. INTRODUCTION

Investigations of the conformations and structures of molecules in liquids are often based on the interpretation of proton-proton couplings and proton chemical shifts. This approach has many advantages, not the least of which is the vast literature available which details many successful applications. The straightiorward analysis of proton NMR spectra typically requires that the proton spectrum be well resolved. While this is often the case for pure solutions of small molecules it is typically not the situation for macromolecules or for small molecules in combination with protonated solvent or large molecules. Since the proton spectral information can often be directly related to the conformational and structural features of interest it is typically the information desired. However, in many instances the proton spectral information needed is not accessible by direct observation. One approach to gaining information about crowded proton spectra is to utilize a double-resonance experiment. One of the major applications of double resonance is selective decoupling in proton spectroscopy. This experiment can often be quite useful in helping to understand complicated proton spectra. While this sort of approach can be of considerable utility it typically relies on having at least part of the proton spectrum resolvable. The use of double resonance in proton spectroscopy has been essentially superseded by two-dimensional methods (Z-3) but these too rely on resolution in the proton spectrum. Another strategy is to use a heteronucleus, such as carbon- 13 or phosphorus-3 1, which has a well resolved spectrum to “spy” on the protons (4). In this tack the heteronucleus signal is used to detect the proton spectral information needed. One * Alfred P. Sloan Fellow. 427

0022-2364184 $3.00 Co~yrisht 0 1984 by Academic Press. Inc All rigJtts of reproduction in any form nservcd.

428

PHILIP

H.

BOLTON

type of experiment which has been demonstrated is heteronuclear two-dimensional spectroscopy (5-7). In this technique the proton spectrum which is obtained can be directly related to the conventional proton spectrum. The conventional proton spectrum can be considered as arising from the sum of two subspectra-one for each polarization of the heteronucleus. The two-dimensional proton spectrum corresponds to the difference between the two subspectra (4). The use of heteronucleus detection of proton spectral information by this approach overcomes, in many instances, all of the problems of signal overlap which occur in the conventional proton spectrum and allows straightforward determination of the desired proton spectral information. While the heteronuclear two-dimensional method has many admirable features it may not be the best possible way to spy on the protons. Another approach is to use heteronuclear multiple quantum spectroscopy as the means of indirect detection (8II). In the now standard singlequantum correlation experiment the frequencies detected are proton precessional frequencies and the signals have the same frequencies, but not algebraic signs, as the conventional proton spectrum. In multiple-quantum experiments the frequencies detected depend on the precessional frequencies of both the protons and the heteronucleus. The zero-quantum experiment is of special interest over other orders of multiplequantum spectroscopy for primarily two reasons. One of which is that the zeroquantum experiment has the narrowest linewidths in the dimension containing the signals which can be related to the proton-proton couplings and proton chemical shifts. A more distinct advantage lies in that in zero-quantum spectroscopy the frequencies of the signals arise from differences between proton and heteronucleus precessional frequencies rather than their sum as is the case for other orders. For this reason the zero-quantum experiment is particularly sensitive to the algebraic signs of coupling constants (28). The zero-quantum experiment of most interest is that form in which the heteronuclear coupling can be made to vanish from the dimension containing the proton spectral information. That is, the zero-quantum data corresponds to the heteronuclear decoupled proton spectrum. This capability to obtain simple proton spectra indirectly is one of the major attractions of the zero-qwmtum aperiment. As will be shown below the zero-quantum experiment is an attractive alternative to the single-quantum approach and can be applied to as wide a range of samples. There are some features of the zeroquantum method which may make it particularly suitable for investigating the conformations and structures of macromolecules and of small molecules bound to macromolecules. BASIC

PRINCIPLES

The simplest spin system for which heteronuclear zero-quantum coherence (HZQC) can be generated is an AX whose energy levels are shown in Fig. 1. A heteronuclear zero-quantum experiment consists of three parts. The fnst is the generation of HZQC which then evolves during the evolution time t, and is finally converted into observable heteronucleus singlequantum coherence which is observed. The basic pulse sequence is shown in Fig. 2. The experiment is repeated for many regular increments of the evolution time t, and the zeroquantum frequencies obtained from the Fourier trans-

HETERONUCLEAR

ZERO-QUANTUM 4

NMR

429

X._

--

A 2 I’z’

-.

A x

-1

FIG. 1. At the top is shown the energy levels for an AX spin system along with indicators of which transitions belong to which spin. The bottom is a schematic representation of a xero-quantum exper’imeut. During the preparation period the A-spin transverse magnetizations are generated and converted in HZQC which evolves during t, and is then converted into observable X-spin transverse magnetization.

form of the modulation of the intensities of the heteronucleus signals with respect to tt. The basic experiment is schematically illustrated in Fig. 1. The first proton pulse generates the two proton transverse magnetization ( 1,3) and (2,4). The simultarreous proton and heteronucleus pulses at the end of the preparation time convert the proton transverse magnetization into HZQC. The efficiency and algebraic sign of this conversion process is given by (12, 13) Z n,tu = l‘(A,-An)(sin,/2)(A”+4N)(COS,/2)(2~-An-4ru)exp(-i#(Mr - IV! - Mu + A#,)) [ 1] where A,., is the number of spins to be inverted to change state r into state t; J% is the magnetic quantum number of state r; and N is the number of weakly coupled spin- l/2 nuclei. A ‘H

IL-.,----

tP

tl

L2

2. The two pulse sequences used here for heteronuclear xeroquaritum spectroscopy are given in (A) and (B). The timing of the pulse sequences is shown in (C). The thick lines indicate 180* ptdses and the thin lines 90” pulses except for the last proton pulse which may have a flip angle other than 90”. FIG.

430

PHILIP

H.

BOLTON

Use of [l] shows that the conversions of (1, 3) and (2, 4) into HZQC (2, 3) are of equal efficiency but of opposite algebraic sign as indicated in Fig. 1. This implies that the optimal generation of (2, 3) requires that (1, 3) and (2, 4) be 180” out of phase with one another at the end of the preparation period. This condition can be obtained by having the length of the preparation time equal 1/2J*x and applying simultaneous 180” pulses to the proton and heteronucleus spins at the middle of the preparation period as shown in Fig. 2. The same analysis can be applied to the HZQC (3, 2) but only the (2, 3) case will be considered here. It is also noted that use of only the heteronucleus pulse at the end of the preparation time can be used to generate HZQC as can be shown by the use of [ 11. However, when more than one proton is coupled to the heteronucleus the use of both pulses is advantageous and is the only case considered here. Once the HZQC is generated it evolves during t, . The frequency of the HZQC (2, 3) is given by & = %, - 0, with & the offset frequency of nucleus A, which is the free precessional frequency of nucleus A - the transmitter frequency. Thus fi2,3 is simply the difference between the offset frequencies of the proton and heteronucleus. At the end of the evolution time the HZQC is converted into heteronucleus singlequantum coherence by a proton pulse as shown in Fig. 2. The efficiency and algebraic signs of the conversion of (2, 3) into (1, 2) and (3, 4) are determined by use of [l] and given in Fig. 1. The net effect of the experiment is the modulation of the intensities of the two heteronucleus signals by the same, single frequency which is &,. The modulations are of the same amplitude but of opposite algebraic sign. In the heteronuclear twodimensional experiment each heteronucleus signal is modulated by two frequencies, one for each proton subspectrum, and the two modulations are of opposite algebraic sign. Selective detection of the HZQC contribution to the modulation of the intensities of the observed heteronucleus signals can be obtained by phase cycling (II, 14, 15). Consideration of the phase of the conversion of the HZQC into heteronucleus signals given by Eq. [I] shows that the phase cycling of the two proton 90” pulses of the preparation period through X, Y, -X, -Y and keeping the phase of the last proton pulse at X gives selective detection of the HZQC contributions to the modulations of the heteronucleus signals when all of the signals are added together. Single sideband detection in both dimensions is obtained using this phase cycle in concert with keeping the phase of the receiver at X. Elimination of the signal arising from heteronucleus transverse magnetization generated by the heteronucleus pulse at the end of the preparation period is performed by cycling the phase of this pulse through X, Y, -X, - Y. Thus a four-step phase cycle allows selective detection of the HZQC frequencies, elimination of the heteronucleus transverse magnetization contribution to the observed signal, and “quadrature” detection in both dimensions. Coupling of the proton to other protons, which are not coupled to the heteronucleus, does not appreciably alter this analysis. A pulse cascade approach (16) can be used to extend the AX example (I 7). The result is simply that instead of a single HZQC frequency there will be a multiplet whose splittings are those of the directly coupled proton. This is the case for weak coupling and strong coupling is treated below. This basic theory predicts that the two-dimensional map will have signals along

HETERONUCLEAR

ZERO-QUANTUM

431

NMR

the zero-quantum axis whose frequencies are centered about the difference between the proton and heteronucleus offsets from their respective transmitters and whose multiplet structu~ is that of the heteronuclear decoupled proton spectrum. The spectra in Fig. 3 compare the experimental data with simulations for phosphothreonine and cytidine 3’-phosphate. The zero-quantum spectra are quite simple and can be readily simulated using commercially available software. Before turning to more complicated spin systems the effects of magnetic field inhomogeneity on the zeroquantum spectra will be considered. The resolution of the zeroquantum spectra ultimately depends on the dephasing which occurs during tl. The sensitivities of the HZQC and the observed signals to field inhomogeneity are different, and their ratio is given by

&= AQx

?A

-

-fX

YX

with A& = (TA - rx)ABo where Al& is a variation in the static magnetic field and AQx = yxABO. Thus, in a complete two-dimensional map for a sample in an inhomogeneous field a sloping appearance will be present as illustrated in Fig. 4. The slope arises since for a given deviation from the average field the HZQC frequency shifts less than the observed heteronucleus frequency. The smaller the slope the smaller are the linewidths (18, 19) along the zero-quantum axis, which is the cru-

3’ CMP

phosphothreonine

8

5Hz

H

FIG. 3. The spectra shown are the heteronuclear zeroquantum ones of cytidine 3’-phosphate (Y-CMP) and phosphothreonine shown at the top. The simulations, at the bottom, are of the phosphorus-3 1 decoupted proton spectra. For 3’-CMP the relevant couplings are J T,3’= 5.1 Hz, Js,,, = 6.9 Hz. For phosphotheonine the couplings are Jo7 = 7.1 Hz and J,,@ = 6.8 Hz. The figure is reprinted from Ref. (10) with permission of Academic Press.

PHILIP

432

4

12

H. BOLTON

20

28

38 Hz Fs. 4. The contour map is of the heteronuclear zero-quantum experiment on phosphothreonine using phosphorus-3 1 detection. The field homogeneity was about 8 Hz for the phosphorus-3 1 signals. The slope of the contours is discussed in the text.

cial one since it contains the proton spectral information. The slope in the zeroquantum experiment with phosphorus-3 1 as the heteronucleus is 1.5 whereas in the singlequantum (18, 19) heteronucleus two-dimensional experiment the slope is 2.5 = (?A - yx)/yx (18, 19). This implies that the zero-quantum linewidths will be less than those obtained in the heteronucleus two-dimensional experiment in any actual magnetic field. This advantage of the zeroquantum experiment becomes smaller as the gyromagnetic ratio of the detected nucleus becomes smaller. For both types of experiments the field homogeneity effects can be effectively eliminated by use of the RECYCLE procedure (19). In actual practice, linewidths of 1 Hz or less can be readily obtained in the zero-quantum dimension without recourse to RECYCLE. TWO PROTONS

COUPLED

TO THE HETERONUCLEUS

The generation of HZQC in an AMX spin system may be considered from several different viewpoints. One convenient approach is to consider the contributions of the various protons to the individual HZQCs. For simplicity the analysis will begin by considering the generation of the (2, 1’) HZQC of an AMX spin system whose energy levels are depicted in Fig. 5. The first proton pulse of the preparation period generates the eight proton transverse magnetizations as schematically shown in Fig. 5. Chemicalshift o&et dependence is removed by the simultaneous proton and heteronucleus pulses at the middle of the preparation period. At the end of the preparation time the proton and heteronucleus pulses generate HZQC with efficiency given by [ 1] and the algebraic signs of the conversions given in Fig. 5.

HETERONUCLEAR

1 +QA

2M

ZERO-QUANTUM

-- 3 -.

-.bc ---

4

-

(l’.Z’) (3’.4’) U’.3’) (2’.4’)

+ + j-E +

433

---,x

--

(1.2) (3.9) (1.3) (2.4)

NMR

’-

-. 1’

0-.

3’

4

+

(1.1’) (2.2’)

+

(3.3’) (4.4’)

+ (2.1’)

FlG. 5. The energy levels of an AMX spin system are shown at top with indicator% for the transitions of the d&rent spins. At bottom is a schematic pnsentation of the zero-quantum experiment.

The optimal generation of HZQC (2, 1’) from A-proton transverse re&res that the A spin magnetization vectors be in phase with respect to JAMand 180” out of phase with respect to JAX at the end of the preparation period. Thus, the efficiency of generation of (2, 1’) from A transverse magnetization is given by i&&J = an rtpJAx cos rtpJm with &(tp) the efficiency of generation of HZQC (2, 1’) from transverse magnetization of A as a function of tp for the case of 90’ pUlSlZS. The M-proton transverse magnetization can also be converted into (2, 1’) so long as JMx does not vanish. Conversion of the M magnetization into (2, 1’) requires that the magnetization vectors be 180” out of phase with respect to both JAx and JAM at the end of the preparation period. Thus, @,,(t,) = sin rt,,JMX sin afpJAM. The net generation of (2, 17 is simply determined by the sums of the contributions from A and M. A similar treatment can be made for (3, 1’). It is noted that the HZQCs (2, 1’) and (4, 3’) can be considered as AX type HZQCs and are generated with the same efficiency although with different algebraic signs. The same holds true for the HZQCs (3, 1’) and (4,2’) which can be labeled as MX type HZQCs. The generation efficiencies of the AX- and MX-type HZQCs are &&t,,)

= sm rtpJAx cos r&JAM + sm xt,,JMx sin r&JAM

.i$&,)

= sin rtpJMx cos a&JAM + sin zt,,JAx sin r&JAM.

It is noted that tp can be chosen such that either MX- or AX-type HZQC is generated selectively or can be picked so that both types are generated with approximately equal efficiency. The choice of t,, also determines whether or not the two types of HZQC have the same algebraic sign. When 90” pulses are used for the generation of HZQC then all of the AX HZQCs will have the same algebraic sign as will all of the MX HZQCs. As a practical consideration, as discussed below, t, is typically chosen such that the AX and MX HZQCs are generated with the same

PHILIP

434

H. BOLTON

TABLE 1 HZQC FREQUENCIES

FOR AN AMX

Pulse sequence A

(21’) (43’) (3 1’) (42’)

QA% %., Qu -

fix - (JAMfix + (JAMax - (JAMfix + (JAM-

SAMPLE

Pulse sequence B

Jr.&2 JM&~ J&d/2 JmW

QA- Jd2 % + Jd2 Qu - JAM/~ Qu + JAMI~

Note. Only one sense of the HZQCs is given with the frequency of (1’2), for example, given by the minus of the frequency of (2 1’). The phase cycling used in the pulse sequences selects for one sense of the HZQCs.

algebraic sign and magnitude. This is not difficult to obtain in most cases. After the HZQC is generated it evolves during tl . The frequencies of the HZQCs of an AMX are given in Table 1. The coupling of the A and M protons to additional protons merely results in additional splitting of the HZQC frequencies. At the end of the evolution time the HZQCs are converted into heteronucleus transverse magnetization by the final proton pulse. The flip-angle dependence of this final conversion is given in Table 2 and is obtained by use of [I]. An example of a heteronuclear AMX spin system is provided by /I-D-[ l-13C]glucose. The C, carbon is coupled to the directly bonded proton, ‘Jcn = 160 Hz, and to the proton at site 2 with ‘JCH = -6 Hz. In addition to these couplings the H1 and H2 protons are coupled together with J12 = 6 Hz while H2 and H3 are coupled with Jz3 = 7 Hz. A zeroquantum spectrum was obtained for this sample and is shown in Fig. 6. An unusual feature of the zero-quantum data is readily seen. That is, the signals centered about Q2 - Oc are split by ‘JCH - J12 + J23 and those centered about fl, - Qc are split by 2JCH - J12. These are the frequencies predicted by Table 1 and it is notable that the signals associated with H1 contain information only about 2JCH and not ’ JCH. This feature of the zero-quantum experiment is a direct result of the

TABLE 2 INTENSITIES OF SIGNALS MR AN AMX AS A FUNCTION OF THE FLIP ANGLE, p, OF THE PROTON CONVERSION PULSE

(2 1’)

(11’)

R P r 4

(22’) (33’) (44’) Note.

z... P.

R =

(431

(313

(42’)

r

R r P 4

k P P

P R P

sin /3/2 co? B/2 = -P, r = sin’ /3/2 cos 812

HETERONUCLEAR

200

ZERO-QUANTUM

400

NMR

435

HZ

Fl FIG. 6. Contour map of the data obtained from a sample of @-D-[l-‘3C]glucose in *HZO. The data was obtained using a 90” conversion pulse and a preparation time of 28 msec.

HZQC tkquencies depending on the differences between the proton and heteronuckus frequencies. Thus, in heteronuclear zeroquantum spectroscopy of AMX spin systems there is essentially an exchange of the heteronuclear coupling between the A and M spins. The flip-angle dependence of the signals predicted, given in Table 2, is verified by the data in Fig. 7. It is seen that the use of a 30” conversion puke eaiminates half of the signals from the contour map since their intensities have been reduced by about 14-fold relative to the ones which are observed. The frequencies which are not observed when a small flip-angle pulse is used are those which arise from HZQCs which do not share an energy level with the particular heteronucleus transition. It is noted that the pattern of which signals are observed with the small flip-angle pulse directly gives the relative algebraic signs of the homonuclear and heteronuclear couphngs. The spectra of Figs. 6 and 7 are readily interpretable by the use of the theory presented here. However, the experiment can be improved by removing the dependence of the HZQC frequencies on the heteronuclear couplings which can be determkied from other experiments quite easily. In addition, the sensitivity of the experiment is reduced by having the heteronuclear couplings present. The heteronuclear couplings can be eliminated by the use of pulse sequence B of Fig. 1. The application of the 180” pulse at the middle of the evohrtion time removes the heteronucleus contribution to the zero-quantum frequencies in the limit of we& coupling of the satellite signals (20) as given in Table 1. The spectra obtained using pulse sequence B correspond to the heteronuclear decoupled proton spectrum. This form of the experiment is particularly well suited for conformational studies

PHILIP

436

H. BOLTON

200

400

HZ

FIG. 7. Contour map obtained using the same conditions as for the map in Fig. 6 except that a 30” conversion pulse was used.

since the zeroquantum spectrum can be readily simulated using standard software. The zero-quantum spectrum of uridine-2’,3’-cyclic phosphate is shown in Fig. 8 along with a simulation of the proton spectrum expected. It is seen that the agreement is good. For comparison one can examine the spectrum obtained using singlequantum heteronuclear two-dimensional NMR on a cyclic nucleotide which is much more complicated. The zero-quantum spectrum exhibits the characteristic sloping pattern due to moderately strong coupling. The effects of strong coupling in zeroquantum spectroscopy will be considered next. STRONG

COUPLING:

THE ABX CASE

The effects of strong coupling in two-dimensional spectroscopy can be approached directly by the density matrix method (22-24). However, the use of density matrices often tends to obscure the physics of the experiment. Another method is to break the experiment into parts via the use of pulse cascades (16). The pulse-cascade method often gives more intuition as to the processes involved and has been applied to the case of strong coupling in singlequantum heteronuclear two-dimensional NMR (22). Even the pulse-cascade method can be rather opaque when used to consider strong coupling. A third approach is introduced here, which is also general and which is analogous to the familiar method of treating strong coupling in single-pulse experiments. In a single-pulse experiment on an AR spin system the relative intensity of a signal, P a,b 9 is given P,,, = I(Wt’L)l* w h ere Z = IA. ZB and 9, is the wavefimction of

HETERONUCLEAR 2'.3'

ZERO-QUANTUM

CUNP

T=30°

5.1

.$,,2’=3.0

437

NMR

4.9

PPN

Hz

J2t,3,=6.5

Hz

J3t,4a=5.5

Hz

T,=13

MSEC

FIG. 8. At the bottom is shown the zero-quantum spectrum of uridine 2’,3’ cyclic phosphate (Z’J-cWMP) and at top is the simulation of the phosphoxus-3 1 decoupled proton spectrum.

state n. In the presence of strong coupling there is mixing of the wavefunctions. The wavefunctions in the presence of strong coupling can be considered as linear combinations of the unperturbed wavefunctions as given by 1%) = IA) = I+ PP2) = cos 8142) + sin fil&) = cos f3la@ + sin @?cY) with lq$,l), I&) the wavefunctions in the absence of coupling and tan 28 = ~FJ,&(&, - Qa). The intensities of the signals in the case of strong coupling are obtained by using the perturbed wavefunctions to obtain the familiar result PI,* = cos? 8 t sin2 8 + 2 cos B sin 0 = 1 + sin 20.

In the zero-quantum experiment the point of interest is the conversion of the HZQC into heteronucleus transverse magnetization. The probability of such a conversion is governed by terms like P2,,,, ,, = I(~21#P1 ,,)I”. The intensities for the weakly coupled case can be directly calculated by use of [ 11. In the presence of strong coupling the wavefunctions are mixed as given by

I%,,) = 1411,) l!P21p)= cos 61\k21s)+ sin 019~).

It is noted that the states are typically divided into two classes. The pure states are

438

PHILIP

H. BOLTON

those which do not mix and the others are the mixed states. Direct substitution leads to P 21',11f = lcos ~(421~1~1~11~) + sin~(d~3~V1~,,~)1~. This expression can be solved by using Eq. [l] to determine the intensities of the individual terms which are (~2AM11~)

= (d~3~~lhh ,f) = (WV2

cos38/W2.

Thus, the intensity of the signal correlating HZQC (2, 1’) with the observed signal (1, 1’) is given by P2,,,,,. = (1 + sin 2@(sin/3/2 cos3p/2). A similar analysis can be applied to all of the HZQCs with the heteronucleus (1, 1’) and (4, 4’) transitions with the results given in Table 3. It is seen that for these two cases the intensities of the HZQC signals give the same intensity pattern as observed in the heteronuclear decoupled proton spectrum. This familiar intensity pattern is not observed for the signals correlated with the two inner heteronucleus lines. The method of calculation is the same for the inner lines and begins with substituting the perturbed wavefunctions into the probability equation to obtain + COS 8 sin ~(~31,lZlf#Q7) + COS B sin 8 l(~21’lW22’)12 = IC0s2 fl(42&/4~7) (621'1&#~33')

+

sin2

e(631'i~@33')i2

since I\k2,,) =

e14218)+ sin eld3,,)

cos

I\k22f) = cos e1422’) + sin e1433,).

This equation is solved, as above, by using [ l] to determine the values of the individual terms, yielding (421M422’)

= (4314~1433~) = itsin P/2 cos3 KY2

(621'1~1633')

=

(631'hb22')

=

bin3

/3/2

cos

8/2)1'2.

These values are then substituted into the previous equation to arrive at the result P 210,22’ = -sin b/2 cos3 p/2 + sin’ 28 sin3 /3/2 cos /3/2 P~,,,~~= 114 ~0s~ 28

when

p = 90”.

TABLE 3 INTENSITIE~OFSIGNALSFORAN

(21’) (II’) (22’) (33’) (44’)

R(1 + sin 28) P + r sin’ 20 r + P sin2 28 ~(1 + sin 28)

(43’) r(1 - sin 2@) sin2 20 sin’ 28 P(l -sin28)

p + R R + p

ABX

SAMPLE

(31’) sin 28) sin2 28 sin’ 28 p(1 +sin20)

R( 1 + r+ P P + r

(42’) r(1 -sin20) sin2 28 sin2 28 sin 28)

R + p p + R P( 1 -

Note. R, P, r, p as in Table 2. Note that when B = 90” then R = r =-P=-pandP+rsin’28=-l/4cos228.

HETERONUCLEAR

ZERO-QUANTUM

NMR

439

The intensities of all of the other HZQC correlations with the two inner lines are presented in Table 3. Thus, all of the HZQC signals correlated with the inner two heteronucleus lines have the same magnitude, when a 90” conversion pulse is used, regardless of the strength of the coupling. This is a feature not limited to zero-quantum spectroscopy but has been shown to be the case,for heteronuclear two-dimensional single experiments (17, 21) and, in fact, is a general feature of two-dimensional NMR. The mason for this feature is that the coupling which induces mixing of states 2 and 3 affects the wavefunctions of both the inner heteronucleus lines as well as the HZQCs which are converted into heteronucleus observed signals. Thus, the mixing for these cases is determined by (1 + sin 28)( 1 - sin 28) = cos* 28. When the heteronucleus transition observed is composed of pure states then the mixing appears only once and the intensities of signals are either (1 + sin 213)or (1 - sin 28). Exactly the same state of affairs occurs in homonuclear experiments for the same reasons. This treatment has completely neglected the heteronucleus combination lines since they have no appreciable intensity for the samples examined which have sin* (28 - 26v) less than 0.0 1. It has also assumed that all of the HZQCs are generated with equal efficiency. Therefore, the zero-quantum experiment can be used to investigate strongly coupled spin systems as made evident by the data obtained for uridine-2’,3’-cyclic phosphate shown in Fig. 8. The common intensity of the signals correlated with the inner lines will be demonstrated in the following example. THREE

SPINS

COUPLED

TO

THE

HETERONUCLEUS

The treatment of the case of three protons coupled to the heteronucleus is much the same as that given above for the case of two protons. The new feature is that for three protons there appears a new class of HZQC which will be referred to as class IV by analogy to the terminology used for homonuclear multiplequantum experiments (13). The class IV coherences might be thought of as involving simultaneous spin flips of three protons and one heteronucleus with no net change in the magnetic quantum number. The frequencies of the 15 HZQCs, when pulse sequence B is used, are given in Table 4 using the numbering of the energy levels shown in Fig, 9. It is seen that 12 of the HZQCs have a form the same as those seen above. However, the three class IV coherence frequencies differ in that they have no dependence on the strength of the homonuclear couplings and depend on the offset frequencies of all three protons. The class IV HZQCs are reminiscent of the combination lines observed in other experiments. The class IV coherences will be referred to as three types: A. M, or K depending on which spin’s frequency appears with the negative sign. Thus, for the three proton case there are four AX-type HZQCs of class I and one A-type HZQC of class IV. There are the same number of each class for M and K as well. The generation of the HZQCs can be considered in the same fashion as for the previous cases. During the preparation period proton transverse magnetization is generated and then converted into HZQC by the proton and heteronucleus pulses at the end of the preparation period. The algebraic signs of the conversion processes are given in Table 5 for the case of 90” conversion pulses. The results in Table 5 demonstrate that the optimal generation of AX type HZQC requires that the A

PHILIP

440

H.

BOLTON

TABLE

4

HZQC FFCEQUENCIESOFAN AMKX SAMPLE USING PULSESEQUENCE B

(211 (43’)

055’) (87’) (31’) (42’) (751

(863 (511

(623 (733 (843

52.~%A+ QAQA+

~JAM+ JAK)/~ ~(JAM- JAK)/~ ~JAM- JAKW *(JAM+ JAK)/~

QMh + %.I % +

~JAM + JMKW *(JAM- JMK)/~ ~JAM - JI.IK)/~ *(JAM+ JMK)/~

QK - ~JMK + + ?~(JMK QK ?T(JMK -

JAK)/~

JAKW JAKW QK+ ~(JMK+ JAKW

QK

(721 (633

multiplet be 180” out of phase with respect to the heteronuclear coupling JAx and in phase with respect to the homonuclear couplings JAM and JAK. This is analogous to the case for two spins. The generation of AX type HZQC from either M or K proton magnetization requires that their multiplets be out of phase with respect to their heteronuclear coupling as well as their homonuclear couplings. Thus, the efficiency of generation of AX-type HZQC is given by &&,)

= sin

uJAxtp

cos uJAMtp cos rJAKtp + sin uJMxtp sin uJ,&

cos uJMKtp

+ sin uJKxtp sin uJ,&,

cos rJMKtp.

Following the same procedure gives the equations for the generation of MX and KX type HZQCs ,&&)

= sin uJAxtp sin uJAMtp cos ?TJAKfp+ sin uJMxt,, cos uJ&, +

,&(tp) =

Sin

uJAxtp

COS

uJAMtp Sin

UJA&,

+

Sin

U&&

Sin

uJKxtp cos

cos uJMKt,,

COS ?FJA&,

?TJA&

Sin

Sil'l U&&

uJMKtp

+ sin uJKxtp cos uJAKtp cos uJMKtp.

The above results indicate that the preparation time dependence of the generation of class I HZQCs will typically exhibit rather broad maxima and minima. Approximate knowledge of the couplings will allow a reasonable choice of the preparation time for generation of the class I HZQCs. Generation of the AX, MX, and KX types with equal efficiencies will require knowledge of the couplings. The couplings can be determined from the zero-quantum data obtained with an estimate of optimal preparation time if needed.

HETERONUCLEAR

FIG. 9. The energy levels of an AMKX (15) to K, and (I 1’) to X.

ZERO-QUANTUM

NMR

441

spin system are shown. Transition ( 12) belongs to A, ( 13) to M,

The generation of the class IV HZQCs has a quite di@erent form than that for the class I cohererices. Examination of Table 5 shows that the conversion ofApsoton magnetization into A-type class IV coherence requires that the A proton be 180” out of phase with respect to all three of the A proton couplings fw transfer. The same is true for the conversion of M and K proton m these are of opposite algebraic sign from the A conversion. Therefore, the efficiency of generation of A-type class IV coherence is given by &(tp)

= sin xJ,&,, sin rJAMtp sin uJAxtp - sin rJMxtp sin rJA&, sin lrJM& - sin xJ&,

sin rJAKtp sin sJMKtp.

It should be noted that the choice of tp to have optimal generation of class I coherences of the AX type from the A proton magnetization, for example, will neccss&ly have a vanishingly small generation of A-type class IV coherence since when cos rJ,+&, cos xJMKtp is at its maximum sin rJ,&, sin rJM& is zero. A compromise choice, to have approximately equal generation of both class I and class IV, could be made TABLE 5 ALGEBRAIC

SIGNS OF CONVERSION HZQC A

(12) A

M

K

(34)

OF PROTON TRANSVERSE FOR AN AMKX SAMPLE

MAGNEIIZAIIONS

M

(21% (65’)

(3 l’), (75’)

(431, (87’)

(423, (86’)

(5 G733 (623, (84’)

IVA (72’)

IVM (63’)

-

-

-

t-

-

-

+

-

-

-

-

-

(78)

-

+ +

(13) (24) (57) (68)

+ -.

(15)

+ + -

+ -

(26)

+

(56)

(37) (48)

INTO

+

+ +

+

+

t

t .-

-

-

-

-

+ + -

-

-

+

+ +

-

+ + -

+ -

-t + -

IVK (45’)

t t t _ +

PHILIP

442

H. BOLTON

or a variable preparation time used in the manner of “accordion” (25) or “threefrequency” (26) NMR. In actual practice the class IV coherences are perhaps best suppressed since then the zero-quantum spectra will retain their correspondence to the conventional heteronuclear decoupled proton spectrum. After generation of the HZQCs they evolve during tr which is interrupted by the application of a heteronucleus 180” pulse in the middle. At the end oft, the proton pulse converts the HZQCs into heteronucleus transverse magnetizations. The efficiencies of the various conversions are given in Table 6. Examination of the results in Table 6 shows that for a flip angle other than 90” there is no linear combination of the signals correlated with the heteronucleus frequencies which will correspond to the heteronuclear proton spectrum. This arises from the nature of the four-spin system which has HZQCs which do not share an energy level with either of the outer heteronucleus lines. This implies that to obtain data suitable for conformational analysis that a 90” proton pulse must be used to the conversion of the HZQC into heteronucleus transverse magnetization. For reasons similar in kind the single-quantum heteronuclear two-dimensional experiment requires a proton 90” pulse in the magnetization transfer step (21). In some instances it may be useful to utilize the flip-angle dependence of the intensities of the signals in the two-dimensional map. When a 90” pulse is used for the conversion then the zero-quantum signals correlated with the two outer heteronucleus lines will directly correspond to the heteronuclear decoupled proton spectrum. When there is strong coupling between the protons then the sloping pattern of the normal proton spectrum will be reproduced in the zeroquantum spectrum as was the case for the ABX spin system. Using the same method of calculation as for the ABX case the ABMX signal intensities can be calculated and are given in Table 7. Cytidine S-phosphate has an ABMX spin system composed of the 5’, S’, and 4’ protons all of which are coupled to the phosphorus-31 nucleus. The zeroquantum

TABLE 6 RELATIVE SIGNAL INTENSITIESFORAN AKMX A (21’) (43’) (II’) (55’)

(22’) (66’) (33’) (771 (44’) (88’) Note.

R r P p r

M (65’)

I? 6 p P R

ii p P I?

I, p

; p

R p P

(87’) (31’)

R = -P

=

(42’)

SAMPLE K

(75’) (86’)

IV

(51’)

(62’)

(73’)

p r

P R

P R

(84’) (72’)

(63’)

(45’)

P r

L

R r

si p x r P RpPPppPPpprr p p P P

RRrrRRrrpp i

i

P p

P p

R p

p r

sin o/6/2co? 812; r = -p = sin’ p/2 cos3 812; &? = -P = sin’ p/2

L

cos

812.

HETERONUCLEAR

ZERO-QUANTUM TABLE

RELATIVE

(11’) (55’)

(43')

(65')

(87')

(31')

(42')

(75')

R+ R+

RRp P r

R+ R+ p P r

RRp P r

R+ R+ r

RRr

R+ R+ r

;-

p+

i P

i

;+

i P

p-

P+

P-

P+

P-

(66’) P (33')r

P+

P-

Note. R = -P = l;R+

= P+ =

1 + sin 28;

7

FOR AN ABMX

INTENSITIES

(21')

(22')~

(88')

SIGNAL

:+ P+

443

NMR

(86')

SAMPLE

(51’)

RRm r

R P R P L R pPPPPPRR PR PP

R- = -P- = 1 -

WHEN

fi = 90"

(62')

(73')

(84')

(72')

(63')

(45')

R P R P R

R P R P R

R P R P R

R P R R P

R P P P R

R R P R P

R P

R P

R P

R P

R P

P P

sin 28;

r = -p = cos’20.

spectrum is shown in Fig. 10 along with a simulation of the phosphorus-3 1 decoupled proton spectrum. In this example the class IV signals are not observed since the choice of t,, 15 msec, is such that very little class IV HZQC will be generated. The choice of tP was governed by having all of the class I HZQCs generated with approximately equal efficiency. It is noted that while the zero-quantum spectrum of this sample given readily interpretable results for the 5’ and 5” protons the signals associated with the 4’ proton form a poorly resolved set. In contrast to this, the heteronuclear two-dimensional spectrum of a 5’ nucleotide has good resolution in the 4’ region due to the difference spectrum nature of the data whereas the region of the 5’ and 5” protons is quite complicated though interpretable. Also shown in Fig. 10 is the spectrum constructed of the sum of the absolute-value spectra correlated with the heteronucleus (66’) and (33’) transitions. This spectrum is shown to illustrate that for the inner lines the signals all have the same intensity regardless of the strength of the coupling. This feature of the two-dimensional experiment may be useful in some cases where very strong coupling is present and Isin 26 is near vanishing which may make the AB pattern difficult to detect in the signals correlated with the outer heteronucleus lines. CONCLUDING

REMARKS

The heteronuclear zeroquantum experiment has been shown to be applicable to a variety of spin systems. In all cases spectra can be obtained which am directly related to the heteronuclear decoupled proton spectrum. Thus, the data obtained by this approach are particularly simple to interpret. The sensitivity of the zeroquantum experiment is comparable to that of the heteronuclear two-dimensional approach. In the zeroquantum experiment there is a lowering of sensitivity by half due to its having an extra proton 90” pulse. However, this loss in sensitivity is largely recovered by detection of half as many signals. Thus, there is no strong reason to decide between the two approaches on the basis of sensitivity whereas the zeroquantum method gives simpler results which contain all of the desired information. In the case of having only one proton coupled to the heteronucleus one can use

444

PHILIP

H.

BOLTON

5' CNP

I

I

4.25

4.0

Hz

J4r,5"=3.0

Hz

Jq’,5’=2.4 Hz

J3,,4,=4.4

J5,,5"=-12

Hz

PPN

FIG. 10. (A) is the simulation of the phosphorus-3 1 decoupled proton spectrum of cytidine S-phosphate (S-CMP). (B) is the zeroquantum spectrum obtained with a preparation time of 15 msec and a 90” conversion pulse and cormsponds to the difference between the spectra correlated with the phosphorus-3 1 ( 1,l’) and (8,8’) transitions. (C) is extracted from the same two-dimensional map as (B) but corresponds to the sum of the absolute-value spectra correlatexi with the phosphorus-3 1 (33’) and (66’) transitions.

heteronuclear J spectroscopy to determine the couplings of the proton (27). This method has very high resolution and typically one can determine the chemical shift of the proton by another approach. The heteronuclear J spectroscopy approach is not suitable to those samples which have more than one proton coupled to the heteronucleus or if the proton coupled to the heteronucleus is strongly coupled to another proton. Thus, the heteronuclear J spectroscopy experiment has rather limited use to AX spin systems where it works well. The emphasis of the above discussion is on the use of the two-dimensional methods to obtain high-resolution proton spectral information. A major use of heteronuclear spectroscopy is the correlation of the chemical shifts of protons and carbon- 13 without regard to the fine structure of the proton multiple& In the quest for this information the zero-quantum experiment can be used as demonstrated by .Miller (9). However,

HETERONUCLEAR

ZERO-QUANTUM

NMR

445

since the proton signals are typically decoupled from carbon- I 3, in such experiments, the z&quantum experiment will have lower sensitivity than the familiar heteronuclear singlequantum two-dimensional experiment and hence will not have much use in this particular application. However, in those cases for which the proton multiplicities are of interest the zeroquantum experiment is competitive with the singlequantum experiment in sensitivity. The real usefulness of the zeroquantum experiment lies in the examination of the conformations of molecules in liquids whose normal proton spectrum is obscured, In these cases the zeroquantum experiment will give results which can be readily interpreted to give the proton chemical shifts and couplings needed for conformational analysis. Since the zero-quantum signals are all of the same algebraic sign, where tP is chosen to make this condition so, the signals will not cancel out due to the line broadening associated with high molecular weight. Thus, the zero-quantum method can be applied to molecules in the 10,000-25,000 molecular weight range and the linewidths will be, in most cases, solely determined by the T2’s of the relevant protons. EXPERIMENTAL

All of the NMR experiments were performed using a Varian XL-200 spectrometer. The samples were all obtained from Sigma with the exception of the labeled glucose which was obtained from Merck Isotopes. The samples were dissolved in 99.8% 2H20 containing 2 mM ethylenediaminetetraacetic acid (EDTA) and 2 mM ethylene glycol his@-aminoethyl ether) N,N’-tetraacetic acid (EGTA) and the pD of the solution were adjusted to 7.0. The samples were at 30” during the experiments. ACKNOWLEDGMENTS This work was supported, in part, by the Camille and Henry Dreyfus Foundation and the Petroleum Research Fund of the American Chemical Society. REFERENCES I. R. FREEMAN AND G. A. MORRIS, Ball. Magn. Reson. 1, 5 (1979). 2. G. A. MORRIS, in “Fourier, Hadamard, and Hilbert Transforms in Chemistry” (A. G. Marshall, Ed.), p. 271, Plenum, New York, 1982. 3. R. BENN AND H. GUNTHER, Angew Chem. Int. Ed.. 22, 350 (1983). 4. P. H. BOLTON, in “Biomolecular Stereodynamics” (R. H. Sat-ma, Ed.), Vol. 2, p. 437, Adenine Press: New York, 1981. 5. A. A. MAUD~LEY AND R. R. ERNST, Chem. Phys. Lett. SO, 368 (1977). 6. A. A. MAUDSLEY, L. MOLLER, AND R. R. ERNST, J. Magn. Reson. 28,463 (1977). 7. G. BOLZNHAUSEN AND R. FREEMAN, J. Magn. Reson. 2S, 471 (1977). 8. A. MINORETII, W. AWE, M. RHEINHOLD, AND R. R. ERNST, J. Magn. Reson. 49, 175 (1980). 9. L. MULLER, J. Am. Chem. Sot. 101,448l (1979). 10. P. H. BOLTON, J. Magn. Reson. 52, 326 ( 1983). Il. G. BODENHAUSEN, Progr. NMR Spectrosc. 14, 137 (1981). 12. W. P. AUE, E. BARTHOLDI, AND R. R. ERNST, J. Chem. Phys. 64,229 (1976). 23. L. BRAUNXHWEILER, G. BODENHAUSEN, AND R. R. ERNST, kfol. Phys. 48, 535 (1983).

446

H.

BOLTON

WOKAUN AND R. R. ERNST, Chem. Phys. Lat. 52,407 (1977). A. PINES, D. WEMMER, J. TANG, AND S. SINTON, Bull. Am. Phys. Sot. 23, 21 (1978). G. B~DENHAUSENAND R.FREEMAN, J. Mugn. Reson. 36,221 (1980). G. B~DENHAUSEN AND P. H. BOLTON, J. Magn. Reson. 39, 399 (1980). A.MAUDSLEY, A. WOKAUN, AND R. R. ERNST, Chem. Phys. Lett. 55,9 (1980). P.H. B~LTONANDG.BODENHAUSEN, J. Mugn.Reson. 46, 306 (1982). P. H. BOLTON, J. Mugn. Reson. 51, 134 (1983). P. H. BOLTON, J. Mugn. Reson. 45, 239 (1981). A. BAX AND R. FREEMAN, J. Magn. Reson. 41, 507 (1980). R. FREEMAN, G. A. MORRIS, AND D. L. TURNER, J Mugn. Reson. 26, 373 (1977). G.BODENHAUSEN, R.FREEMAN,G. A.MORRIS, ANDD.L.TURNER,J. Mugn. Reson. 28, 17(1977). G.B~DENHAUSENANDR. R.ERNST, J. Am. Chem. Sot 104, 1304(1982). P. H. BOLTON, J. Mugn. Reson. 46, 343 (1982). G. B~DENHAUSEN, J. Mugn. Reson. 39, 175 (1980). G.POUZARD,S.SUKUMAR, AND L.D.HALL, J. Am. Chem. Sot 103,4209 (1981).

14. A.

15. 16. 1% 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

PHILIP