- Email: [email protected]
Elimination of flip-angle effects in two-dimensional NMR spectroscopy. Application to cyclic nucleotides
JOURNAL
OF MAGNETIC
39, 399-412 (1980)
RESONANCE
Elimination of Flip-Angle Effects in Two-Dimensional NMR Spectroscopy.Application to Cyclic Nucleotides GEOFFREY Francis
Bitter
BODENHAUSEN*
National Magnet Laboratory, 170 Albany Street, Cambridge,
Massachusetts Massachusetts
Institute 02139
of Technology,
AND PHILIP Department
of Pharmaceutical
Chemistry, San Francisco,
H. BoLTON? University California
Received February
of California, 94143
San Francisco,
1, 1980
The proton and phosphorus-31 NMR spectra of 2’,3’-cyclic uridine monophosphate (2’,3’-cUMP) have been correlated by means of heteronuclear two-dimensional NMR. The spectra are found to depend on the flip angle of one of the rf pulses. This is in contrast to previously reported applications where the 3*P nucleus was coupled to a single proton only. For flip angles other than 90” the spectra cannot be readily related to conventional proton spectra and the conformational information is obscured. However, a proper linear combination of the spectra eliminates the flip-angle effect. The resulting multiplets can be analyzed by routine simulation techniques to extract proton-proton coupling constants. This information, which can be related to the molecular conformation, is often not available from conventional proton spectra.
One of the more promising new methods to emerge from the development of two-dimensional Fourier transform NMR (2, 2) is the correlation of spectra of different nuclear species (3-5). The mapping of chemical shifts and long-range couplings can be helpful in a variety of assignment problems (4-7). When the magnetization is transferred from protons to phosphorus-3 1, a simplified proton spectrum is obtained which for many biological phosphates stems only from the proton nearest to the phosphorus nucleus (8). The homonuclear proton-proton couplings, which reflect the molecular conformation through the Karplus equation (9), can be observed indirectly with a spectrometer tuned for phosphorus NMR. It is therefore possible to study the conformations of systems for which the normal proton spectra are either obscured by solvent lines or otherwise too crowded to lend themselves to spectral analysis. A previous investigation (8) showed how the two-dimensional spectra of 2’and 3’-nucleotides could be related to the conventional proton spectra of these * Current address: Laboratoriumfiirphysikalische Chemie, Eidgenossische Technische Hochschule, 8092 Zurich, Switzerland. t Current address: Department of Chemistry, Wesleyan University, Middletown, Conn. 06457. 399 ouz2-236VS0/09039P14$602.00/0 Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
400
BODENHAUSEN
AND BOLTON
compounds. In these simple cases the phosphorus-31 nucleus is coupled to a single proton only. In many cellular phosphates, however, a different situation arises because two or more protons are simultaneously coupled to the phosphorus-31 nucleus. This case is quite common, notably in 5’-phosphates where the 5’ and 5” protons form a tightly coupled ABX system with the phosphorus-31 nucleus with coupling constants J,x and JBX of similar magnitude (9). The heteronuclear two-dimensional spectra of these systems show a dramatic variation of the relative signal intensities with the flip angle of one of the proton rfpulses. This behavior, which does not arise in the simpler 2’- and 3’-nucleotides, tends to complicate the analysis of the spectra since there is no longer any simple relationship between conventional proton spectra and heteronuclear spectra obtained with an arbitrary flip angle. Cyclic phosphates such as 2’,3’-cUMP contain ABX subunits that are not too tightly coupled. At a field strength of 4.7 T, typical for modem multinuclear spectrometers, the general features of these systems can be adequately discussed in first order. The populations of the energy levels can be analyzed in terms of classical magnetization vectors. Such calculations provide insight into the origin of the flip-angle effect. In a systematic variation of the flip angle, some symmetry properties become evident. In particular, it is possible to reconstruct the spectra that would be obtained if the flip angle were adjusted to exactly 90” by linear combinations of experimental spectra obtained with an arbitrary, unknown flip angle. The heteronuclear spectra obtained in this manner have an improved signal-tonoise ratio, and lend themselves to an iterative spectral analysis because of their close relationship to conventional proton spectra. MAGNETIZATION
TRANSFER
AND CONNECTIVITY
The heteronuclear two-dimensional technique has been originally described (3) as a special case of the basic two-pulse experiment proposed by Jeener tl
‘H 90.
l-l a a
3’P
s&t,) 90.
AM ‘I
‘H 90.
3’P
nn
aa
s&t,) b
FIG. 1. (a) The normal pulse sequence appropriate for heteronuclear two-dimensional spectroscopy is altered to allow for a systematic variation of the flip angle (Y.The phosphorus free-induction decay is recorded for typically 256 regular increments of the evolution period t1 which separates the two proton pulses. (b) Equivalent sequence, which is referred to in the discussion of the AMX system. Conceptually (if not experimentally) the nonselective proton pulse may be broken down into a “cascade” of two semiselective pulses, the first affecting the A transitions only, the second applied to all M transitions.
ELIMINATION
OF FLIP-ANGLE
401
EFFECTS
(ZO), which has been treated in detail by Aue et al. (I). The probability of a transfer of magnetization between the transitions (k,Z) and (m,n) is governed by the flip angle (Yof the mixing pulse (see Fig. la). The process can be described by a unitary transformation of the density matrix a: + umn
=
[II
&k(~)~:l(&il-
In the case of N weakly coupled spins with Z = M, the matrix elements &(a) of the pulse operator can be expressed analytically in terms of the parameter A ,&, defined as the number of spins that must be flipped to connect the energy levels m and k (I ). When the magnetization is transferred between transitions belonging to different nuclei, the phase of the resulting signal is independent of the flip angle (Y. This is consistent with the discussion of Freeman (5), who suggested that the information about the magnetization (kl) is temporarily stored in the form of populations and subsequently appears in the form of an amplitude modulation of the (mn) magnetization. However, the algebraic sign associated with the transfer of magnetization from (kf) to (mn) depends on the connectivity of these transitions. Consider for example the proton transition (1,2) and the phosphorus transitions (1,l’) and (2,2’) in the energy-level diagram of the AMX system shown in Fig. 2. In terms of the definitions due to Anderson et al. (II), two cases may be distinguished: the pair of transitions (1,2) and (1,l’) is “regressively” connected, while the pair (1,2) and (2,2’) is said to be “progressively” connected. The
(PI
p(r)
r
(P)
p(r)
r
(r)
r(p)
P
P
(P)
r
(r)
P
(p) r
(r)
(r)
r
(P)
P
r
(r) p
(P)
r
(0
p
(PI
44’ 33’
22’
JMX
x
II’ hx 3
FIG. 2. The structure of 2’,3’-cyclic uridine monophosphate (2’,3’-cUMP) with the energy-level diagram appropriate to the AMX subsystem consisting of the phosphorus nucleus and the two neighboring protons. The chemical shifts of the A(H,) and M(H,) protons are 5.138 and 4.917 ppm with respect to DSS (13). The coupling constants are JAM = JAX = 6.9 Hz and Jux = 11.5 Hz. The algebraic signs of the signals in the two-dimensional spectra (see Fig. 4) are determined by the connectivity relationships between the proton and phosphorus transitions. The symbols p and r refer to progressive and regressive connections in the restricted definition (II), applicable only to transitions which have one energy level in common, while the symbols (p) and (r) refer to the broader definition due to Ernst and co-workers (I).
402
BODENHAUSEN
AND
BOLTON
distinction arises from the quantum numbers of the initial and final states spanned by either pair of transitions (22). In a heteronuclear two-dimensional experiment, the nature of the connectivity determines the algebraic sign of the signals. If we assume, for the sake of this argument, that the two proton pulses in Fig. la are applied selectively to the proton transition (1,2), their cumulative effect is to drive the longitudinal magnetization Mz12 in a periodic manner, depending on the free precession of the transverse component in the evolution period (5): Mz12(tl) = -M,&q)
~0s wldl.
PI
For a vanishingly short evolution period t 1, the two pulses simply invert the magnetization. As a result, the populations of level 1 (PI) and of level 2 (PZ) are simply interchanged, enhancing P, and depleting P, with respect to thermal equilibrium. Since none of the other populations is perturbed, it follows that the population difference across the progressively connected phosphorus transition (2,2’) has been increased, while the regressively connected phosphorus transition (1,l’) has become associated with a negative population difference. As the evolution time tl is increased, the amplitudes of both phosphorus transitions will experience a cosinusoidal modulation, their phases remaining opposite throughout the experiment. After the Fourier transformation with respect to tl, progressive and regressive pairs will generate positive and negative absorption mode signals. In the actual two-dimensional experiment, the pulses applied to the protons affect all transitions simultaneously. As a result, the transfer of information is not confined to proton and phosphorus transitions with a common energy level. This effect also occurs in the Jeener experiment (10) and has lead Ernst and co-workers (1) to broaden the original definition of the connectivity relationships (11) to encompass all pairs of transitions that are parallel to a basic pair of directly connected transitions. Thus in the energy-level diagram in Fig: 2, the pair (3,4) and (1,l’) is regressive in the broader sense, since the transitions (3,4) and (1,2) are parallel and merely differ in the polarization of the third spin. In the lower half of Fig. 2, the connectivities have been tabulated; the symbols p and r refer to the original definition (11) while the letters in parentheses (p) and (r) correspond to the broader definition (I). Progressive pairs of either variety, p or (p), give rise to positive signals in the two-dimensional spectra. THE
AMX
SYSTEM
Although the second proton pulse and the phosphorus observation pulse may be applied either simultaneously or sequentially with the same result, the latter case (Fig. la) may provide some insight into the mechanism of the information transfer (5). Whereas the discussion of symmetrical spin systems of the A,X type is straightforward, nondegenerate cases such as the AMX system and higher analogs require a more careful analysis. In order to illustrate a number of general features of the heteronuclear two-dimensional experiment, we shall concentrate our attention on 2’,3’-cyclic uridine monophosphate (Fig. 2) which has been the subject of an earlier spectral analysis (13). The H, and H3 protons and the phosphorus nucleus provide a typical AMX system (albeit with a slight ABX
ELIMINATION
OF FLIP-ANGLE
EFFECTS
403
character) while the H,. and H4 protons, which are coupled to H, and HS, are not relevant to the present argument and wilI be considered later. The first-order proton and phosphorus spectra are depicted schematically in Fig. 2. The A region features an accidental degeneracy (due to the equality JAM = JAX) but the M region has a structure typical of a nondegenerate AMX system. In the two-dimensional experiment, all eight proton magnetization vectors of the AMX system are excited simultaneously by the initial pulse. The second pulse will find the individual vectors with different phases, characteristic of their frequencies and the duration of the evolution time cl. By rotating the transverse magnetization back into the z axis of the rotating frame, the precession is effectively frozen and stored in the form of populations, leading to t,-dependent modulations of the amplitudes of the phosphorus lines. The mechanism may be clarified by the use of two conceptual aids. First, we shall assume that only one proton magnetization vector, say M12(0+), is brought into the transverse plane at the beginning of the evolution period. This could be achieved experimentally by replacing the initial proton pulse by a “Dante” pulse train (14) suitably tailored to the frequency o12. The second conceptual simplification is obtained by replacing the second proton pulse by a “cascade” of two distinct pulses, both with a flip angle cr, the first applied to the A region only, acting on (1,2) and (3,4), the second to the M transitions (1,3) and (2,4) (Fig. lb). Provided the pulse interval is short with respect to free precession and relaxation, such a pulse cascade is equivalent in all respects to a single nonselective pulse applied to all proton transitions simultaneously, but it greatly simplifies the analysis (15). Consider the fate of the magnetization vector M,, in the course of the experiment depicted in Fig. lb. At the end of the evolution period, just before the second pulse, the y component reflects the free precession: M,dG)
=
~l/12(Of)
cos
[31
qd1.
Now the A pulse, applied along the x axis of the rotating frame, will rotate this transverse component through an angle (Y, generating a longitudinal component: M,,,(tf+)
= -sin (YMUlz(t;).
This amounts to the creation of t,-dependent and 2 just after the A pulse:
[41
populations
in the energy levels 1
Pl(rf+)
= +sin (Ycos (w,,t,)AP
+ P,
[51
P,(tf+)
= -sin (Y cos (w,,t,)AP
+ I’,
161
where P = [P,(O-) + P,(O-)]/2 is the average population in thermal equilibrium and AP is defined as the difference [P,(O-) - P,(O-)]/2. At this stage, none of the populations of the other six energy levels is affected. (The A pulse can of course generate new transverse magnetization in the A region but this has no effect on the populations.) Now the it4 pulse will cause a redistribution of populations. This effect can be seen simply in terms of classical magnetization vectors (1.5). Consider, for example, the M transition (1,3) as it experiences a pulse of flip angle a:
404
[P&y+)
BODENHAUSEN
- P,(rrl’+)] = M&r?+)
AND BOLTON
= Mz13(ff+) cos ff = [Ps(r?+) - P,(rt+)]
cos a.
[7]
Since the sum of the populations must be conserved at all times, the effect of the M pulse is to cause a redistribution of the populations between the energy levels land3: P&y+)
= P,(rig+) cos2 ;
P3(ry+) = P3(tf+) cos2 f
+ P3(ff+) sin2 4 ,
F31
+ P,(tf+)
sin2 t
.
191
P2(ty+) = P2(tf+) cos2 f
+ P4(ff+) sin2 f
,
DOI
P4(fy+) = P4(ft+) cos2 I
+ P2(ft+) sin2 :
.
[ill
A similar argument applies to the (2,4) transition:
As in any other Fourier transform NMR experiment the application of a small flip angle (i.e., cos2 (a/2) + sin2 (a/2)) induces little population transfer between the energy levels. Thus, the cos alzfl modulation of the population PI, which has been generated by the A pulse according to Eq. [5], will not be significantly perturbed by an M pulse of small flip angle. In the limit of a flip angle close to 180”, however, the M pulse causes P, and P3 to interchange. As a result, the cos u&, modulation is carried over from P, to P, although energy level 3 is not directly connected with the proton transition (1,2) that is responsible for the modulation. The observation pulse applied to phosphorus-31 (Fig. 1) will detect the time dependence of PI and P2 at the phosphorus transition frequencies ollP and w221while the modulation of P3 and P4 can only affect the phosphorus signals at u33t and w44r. The implications are illustrated in Fig. 3, where the signals S(F,,F2) appearing in the two-dimensional spectra have been simulated as a function of the flip angle. For small (Y, the signal at 27rF1 = o12 (far left in all spectra), due to the proton magnetization M12, appears at 2mFz = oll, and w22, (Figs. 3a and b). With increasing flip angle, this signal gradually disappears from these spectra to reappear at the phosphorus frequencies 2rF2 = w33, and 044P (Figs. 3c and d). Thus the transfer of information does nor require the transitions to share a common energy level. It should be noted that the information is never lost, nor is the total amplitude of the signals at 2rF, = o12 attenuated by the mixing effect. At flip angles of CY= 90”, however, the information is spread over all energy levels, resulting in a larger number of signals in the two-dimensional spectrum. For this reason, the actual signal height does not necessarily increase for larger flip angles, although, in the absence of signal cancellations, the integrated intensity of the absolute value of the two-dimensional spectrum increases with sin CY.
ELIMINATION
OF FLIP-ANGLE
EFFECTS
405
d +w+--$,,-
5.2
50
C
46
5.2
5.0
46~
FIG. 3. Simulated heteronuclear spectra of the AMX subsystem of 2’,3’-cUMP for different flip angles (Y of the second proton pulse. The horizontal F, axis corresponds to the proton spectrum at 200 MHz. The four “slices” through the two-dimensional plot, corresponding to the four phosphorus transitions, are shown clockwise: (a) 27rF, = oll,, (b) 27rF, = oz2,, (c) 27rF, = We,, and (d) 2?rF, = co,,. Note the symmetry that relates the top and bottom halves of the figures.
Examination of Fig. 3 reveals a symmetry that is implicit in the discussion of the population transfer. The spectrum at 277F, = ollf, with (Y = 45” (Fig. 3a, bottom), is the exact opposite of the spectrum at 27rFz = wd4’, with (Y = 135” (Fig. 3d, top). In fact, the top and bottom halves of Fig. 3 are related by symmetry. In general, a two-dimensional spectrum obtained with a supplementary flip angle, (Y’ = 180” - cy, will not contain any new information. Spectra obtained with supplementary flip angles can be brought to superposition by reversing the algebraic signs and interchanging wll, with w44, and oz2, with ws3’. The theoretical two-dimensional spectra S(F,,F,) of the AMX system are reproduced in Fig. 4 for the flip angles Q = 45, 60, and 90”. The algebraic signs of the individual lines relate directly to the connectivity diagram in Fig. 2, progressive pairs corresponding to positive resonances. All spectra in Fig. 4 are shown on the same vertical scale (the sin (Y term in Eq. [4] has been taken into account) to emphasize the fact that the height of the individual peaks actually decreases as the flip angle approaches 90”. In the A region, the accidental degeneracy of the proton transitions (3,4) and (1’,2’), which give rise to lines of opposite algebraic signs, causes a signal cancellation for (Y = 90” which is avoided by the use of smaller flip angles. The two-dimensional spectrum for (Y = 90” has some unique features. As can be seen in Fig. 4, the spectrum S $O”(F,, F,) has an inversion symmetry with
BODENHAUSEN
406 a.459
5.2
5.0
AND BOLTON
a=60'
48
5.2
5.0
a* so'
4.6
52
5.0
4.8 ppm
FIG. 4. Simulated two-dimensional spectra of the same AMX subsystem for three different flip angles, shown on the same vertical scale. For (Y = W, the gain in signal intensity is offset by the appearance of additional lines, which lead to accidental signal cancellations in the A region at F, = 5.138 ppm. For small flip angles, only directly connected transitions p and r give rise to signals, while with increasing flip angles the indirectly Connected transitions (p) and (r) enhance the complexity of the spectra (compare Fig. 2).
respect to a line parallel to the F, axis, intersecting the F2 axis at the chemical shift of the phosphorus nucleus. Thus, the trace at 27rF, = ollt is the exact opposite of the trace at 27rFz = 044J. A similar relationship exists between the two central spectra. It has been suggested (8) that this symmetry should be used to boost the signal-to-noise ratio of experimental spectra by taking appropriate linear combinations. A more important aspect of the spectra obtained for (Y = 90” resides in their relationship to the normal proton spectra. Consider the phosphorus transition (1,l’) in the energy-level diagram in Fig. 2. In the broader definition of connectivity (I ), all four proton transitions (1,2), (1,3), (2,4), and (3,4) form regressive pairs with the (1,l’) transition, while the remaining four proton transitions between the primed energy levels are progressively connected with the (1,l’) transition. For (Y = 90”, the former will result in four negative resonances in the two-dimensional spectrum at 27rFz = w 11,,while the latter contribute four positive peaks. In terms of spectral analysis, these two groups of proton transitions correspond to the subspectra belonging to the polarizations (Y and p of the phosphorus nucleus. Thus, the trace extracted from the two-dimensional spectra at 27rFz = wllf is equivalent to the difference of the two proton subspectra (8). This relationship only holds for the phosphorus transitions wll# and 044, which connect the top and bottom states of the proton energy levels in Fig. 2. For the inner phosphorus transitions oZ2! and oS, the connectivity rules are more complicated and the amplitudes cannot be readily described in terms of proton subspectra. The spectral analysis outlined here is directly applicable only to spectra obtained with a flip angle carefully calibrated to 90”. Fortunately, it is possible to extract this information from experimental two-dimensional spectra obtained with an arbitrary flip angle (Y. Because of the symmetry properties presented in Fig. 3, a simple difference of two slices taken from the two-dimensional spectrum may be calculated
ELIMINATION
OF FLIP-ANGLE
EFFECTS
407
to correct for flip angles that differ from 90”: Ss’(F1, 27rF, = till,) = S”(F1, 27rFz = ollt) - Sn(F1, 27rF2 = oq4t).
[W
A similar difference of the traces at 27rFz = wZ2’ and wQ3!will yield reconstituted spectra corresponding to o! = 90”, whose interpretation is, however, less straightforward. The difference spectrum (Eq. [ 121) has an improved signal-to-noise ratio, largely because much of the noise is in fact correlated in the two slices and stems from the unmodulated peaks at F, = 0 (so-called t1 noise). The experimental enhancement of the sensitivity obtained in this manner is typically a factor of 2.5.
ADDITIONAL
COUPLED
PROTONS
In systems of practical interest, like 2’,3’-cyclic uridine monophosphate, the AMX system is a mere subunit among N protons. Two situations can be distinguished. When the additional protons are coupled to the phosphorus-31 nucleus, they give rise to further splittings in the Fz dimension of the two-dimensional spectrum. In this case, the energy level diagram should be extended accordingly, and the pulse cascade in Fig. lb should include as many selective pulses as there are protons. Each of these pulses flips a fraction sin2 (oJ2) of the spins to which it is applied. The population of an energy-level n is affected by a magnetization M&f;) PreCeSSing at &l according t0 the eqUatiOn P,(Q)
= sin a (cos t)““-“‘(sin
tr’cos
okltl,
1131
wherefis the number of spins that must be flipped to shift the transition (k,f) to a parallel transition that includes the energy level n. This expression is related to Eq. [51] in the work of Aue er al. (I). The sin (Yterm arises from the incomplete rotation of the magnetization as expressed in Eq. [4]. Note that the greater the number of protons N, the larger the exponents in Eq. [13], and the more critical the calibration of the flip angle (II. The simplest spectra are of course obtained for small flip angles, when all transfers withf > 0 vanish, and the modulation due to Mrkl remains confined to Pk and Pl. A much simpler case is more frequently encountered in nature. In 2’,3’-cUMP, for example, the protons of the basic AMX subunit are coupled to their next neighbors, but these do not couple directly to the phosphorus nucleus. In this case, the additional protons split the proton resonances in the A and M regions into doublets in the F, or proton dimension of the two-dimensional spectrum. Hence, a cascade pulse applied to any of the additional protons simply interconverts the two components within the F, doublet, without affecting the Fz or phosphorus dimension. Since the proton spectrum is normally excited by a nonselective pulse at the beginning of the evolution period, both components of the doublet have equal magnitudes and their interconversion does not affect the two-dimensional spectrum. The experimental two-dimensional spectrum of 2’,3’-cUMP is shown in Fig. 5 in the absolute-value mode. The spectrum was obtained with the pulse sequence in Fig. la, the flip angle of the second proton pulse being set to approximately 150”.
408
BODENHAUSEN
AND BOLTON
55
FIG. 5. Heteronuclear two-dimensional spectrum of 2’,3’-cUMP, shown in absolute-value mode, obtained at a phosphorus frequency of 81 MHz with proton pulses at 200 MHz. The flip angle LYwas set to approximately 150”. The resolution in the FZ dimension has been enhanced by multiplying the freeinduction decays with a trapezoidal function, while a Lorentzian line broadening of 0.5 Hz was applied in the F, dimension. The evolution time was incremented in 256 steps of 2.5 msec each. The 0.2 M solution contained about 60% H,O and 40% *H*O; the pH was 7.5 at a temperature of 35°C.
The decoupler frequency was positioned at the low-field side of the resonances of the 2’ and 3’ protons; the orientation of the F, axis is in agreement with the conventions of proton NMR. To prevent the “tails” of the phase-twisted lineshapes (2) from interfering in the Fz dimension, a trapezoidal resolution enhancement was applied in this domain. The four relevant slices through the two-dimensional spectra are shown in Fig. 6 in the phase-sensitive mode for two different flip angles a! = 30” and (Y = 90”. The latter proton flip angle was adjusted by using a modified version of the INEPT technique (16). The structure of these spectra is closely related to the theoretical AMX spectra in Fig. 4, except that all resonances of the H, proton are split byJ(HIHP) = 3.0 Hz and all H3 lines are split byJ(H3H4) = 5.5 Hz (13). There are a total of 16 different frequencies in theF, dimension for (Y= 90”, though only eight significant signals appear if the flip angle is small. The experimental spectra do not exhibit a perfect inversion symmetry, indicating a deviation from a! = 90”. STRONG COUPLING
So far it has been assumed that the AMX unit of 2’,3’-cUMP can be treated in first order as a weakly coupled spin system. It is possible, however, to predict the two-dimensional spectra of a strongly coupled ABX system on the basis of Eq. [l]. Using the matrix elements of the pulse operator R(a! = 90”) in the eigenbasis of the free-precession Hamiltonian and the appropriate magnitudes of the
ELIMINATION
5.2
OF FLIP-ANGLE
5.0
4.8
5.2
409
EFFECTS
5.0
48
pm
FIG. 6. Experimental “slices” extracted from two-dimensional spectra obtained approximately 30 and 90”. The algebraic signs correspond to those of the AMX Fig. 4, but the signals of the H, and H3 protons are split into doublets by J(H,H,) = = 5.5 Hz, respectively (I.?). The F, linewidths are about 2.5 Hz in all spectra, homogeneity of the B. field in a 12-mm sample tube.
with flip angles of subunit shown in 3.0 Hz andJ(H,HJ due to the limited
magnetization vectors in the evolution period 1ultl 1 = IFVkl 1 (Z7), the populations just before the phosphorus observation pulse may be calculated: PI@:) = +1/{(1
- sin 20) cos wlztl + (1 + sin 213)cos uS4fl + (1 + sin 28) cos ti13tl + (1 - sin 28) cos 014fl},
[ 141
WI
P*(C)
= -pm,
P&:)
= +?4 cos 28{-cos
M:)
= -P&:),
e&r1
-
cos
%dl
+
cos
W13rl
+
cos
024rl},
[I61 iI71
where 19= ‘/2 arctg{ JABI[aA - aB + (JAX - 5,x)/2]}.
[la
Analogous equations apply to the primed states PI . . . , etc., which are modulated by the proton frequencies 01,2, . . . , etc. The effective chemical shift difference in the expression for 8’ is [a, - aB - (JAX - J&/2]. The populations of the energy levels 2 and 3 are modulated by four proton magnetizations, all of which contribute terms of equal amplitude, regardless of the strength of the coupling, provided (Y = 90”. This remarkable feature seems to suggest that the corresponding intensities in the two-dimensional spectrum are independent of the strength of the coupling. In the ABX system, however, two additional phosphorus transitions appear at 27rF, = 023f and 02’3 which have energy levels in common with the phosphorus lines at ~TF, = w221 and oS3!. In these circumstances, none of the amplitudes of these four phosphorus lines will be simply proportional to the population differences (17).
BODENHAUSEN
410
AND BOLTON
a b
6
I
I
5
4
pm
FIG. 7. The normal proton spectrum of 2’,3’-cUMP at 200 MHz consists of the sum (d) of the two subspectra belonging to the polarizations (Yand /3 of the phosphorus nucleus (b and c). These have been calculated with the spin-simulation program NMRCAL, taking into account all six protons of the ribose system (13). The difference of the subspectra (a) should correspond to the trace extracted from an experimental two-dimensional spectrum according to Eq. [12].
In the context of spectral analysis, it is preferable to avoid these complications by concentrating attention on the phosphorus lines at 21rF~ = c.o~~,and C.Q, which reflect the population differences P, - PI, and P, - P4, in a straightforward manner regardless of the strength of the AB coupling. The modulation terms in Eq. 1141 are weighted by factors 1 k sin 28 which are identical to the characteristic “sloping” intensities of a normal AB pattern. In conventional Fourier transform spectroscopy (17) these intensities arise from F2, terms, whereas the amplitudes in the heteronuclear two-dimensional experiment are governed by Eq. [l]. If the spectra are obtained with a flip angle (I! = !W, the phosphorus transition wll’ in the ABX system will be modulated by a total of eight contributions: four negative terms with intensities -(l -+ sin 213) stemming from the proton subspectrum associated with P, and four positive terms with intensities +(l + sin 28’) which correspond to the proton subspectrum associated with the p polarization of the phosphorus nucleus. Thus, the trace extracted from the two-dimensional spectrum at 2rrFz = ollt is equivalent to the difference of the two proton subspectra AB(PJ - AB(P,). This equivalence also holds for systems of the ABMX type and higher analogs, provided the additional protons do not couple directly to the phosphorus nucleus. l EXPERIMENTAL
VERIFICATION
The simulated conventional proton spectrum of 2’,3’-cUMP at 200 MHz, shown in Fig. 7d, features the characteristic sloping line intensities due to strong coupling 1 Systems with more than two strongly coupled protons should be treated with caution, since the phosphorus transitions connecting the top and bottom energy levels are likely to overlap with other resonances. Furthermore, there is no analytical proof that the equivalence of the intensity rules in conventional and two-dimensional Fourier transform spectroscopy is applicable to ABCX systems.
ELIMINATION
OF FLIP-ANGLE
I
5.2
I
I
5.0
EFFECTS
411
I
4.8 ppm
FIG. 8. (a) The theoretical two-dimensional spectrum reproduced from Fig. 7a is in good agreement with the experimental spectrum (b) obtained by subtracting the top and bottom traces in the right-hand side of Fig. 6, which stem from an experiment with (Y = 90”. The improvement in the signal-to-noise ratio resulting from this subtraction is approximately 25fold, largely because of the cancellation of the unmodulated peak at F, = 0 which entails the suppression of much of the so-called t1 noise. The same pattern (c) is obtained by subtracting the top and bottom traces in the left-hand side of Fig. 6, showing that the same information can be obtained with a flip angle of about 30”. The reduction in signal intensity may be explained by the sin a term in Eq. [4].
between the H, and H, nuclei (6/J = 6.4), whereas the couplings to the next neighbors are first order to a good approximation. The normal proton spectrum may be decomposed into two subspectra (Figs. 7b and c) which correspond to the polarizations cz and /3 of the phosphorus nucleus, and differ in the effective chemical shifts of the Hz and H3 protons. The difference of the subspectra (Fig. 7a) is reproduced in Fig. 8a for comparison with the experimental two-dimensional spectra. The trace in Fig. 8b was obtained from an experiment with (Y = 90”, by subtracting two slices according to Eq. [12] to improve the signal-to-noise ratio. Figure 8c shows the same linear combination applied to an experiment with (Y = 30”. The equivalence of Figs. 8b and c confirms the validity of Eq. [ 121 and shows that the same pattern may be obtained with an arbitrary flip angle, thus rendering the flip-angle calibration less critical. Either spectrum b or c in Fig. 8 may be analyzed by an iterative simulation with least-squares fitting in the manner ofthe LAOCOON program (18) to extract the coupling constants between the four protons on the ribose ring. The relationship between the proton-proton coupling constants and the conformations of nucleotides has been the subject of extensive research (9) and is beyond the scope of this work. The main purpose of the methods discussed here is to provide access to molecular systems whose normal proton spectra are too crowded or otherwise obscured. It has been shown in systems of increasing complexity that a simple relationship exists between heteronuclear two-dimensional
412
BODENHAUSEN
AND
BOLTON
spectra and conventional proton NMR spectra. The elimination of the flip-angle effect is a prerequisite for the analysis of the spectra of a large class of compounds which includes cyclic phosphates, 5’-nucleotides, and oligonucleotides. ACKNOWLEDGMENTS This work was supported in part by grants from the U.S. Public Health Service, National Institutes of Health. Preliminary spectra were obtained with the kind assistance of Dr. W. W. Conover in the application laboratory of Nicolet Technology Corporation, Mountain View, California. Further experiments were performed at the Magnetic Resonance Laboratory at the University of California, Davis. The encouragement and support of Dr. L. J. Neuringer at the National Magnet Laboratory and Dr. T. L. James at the University of California, San Francisco, are gratefully acknowledged. REFERENCES 1. W. P. AUE, E. BARTHOLDI, AND R. R. ERNST, J. Chem. Phys. 64, 2229 (1976). 2. G. BODENHAUSEN, R. FREEMAN, R. NIEDERMEYER, AND D. L. TURNER, .I. Magn. Reson. 133 (1977). 3. A. A. MAUDSLEY AND R. R. ERNST, Chem. Phys. Left. 50, 368 (1977). 4. A. A. MAUDSLEY, L. MOLLER, AND R. R. ERNST, J. Magn. Reson. 28,463 (1977). 5. G. BODENHAUSEN AND R. FREEMAN, J. Magn. Reson. 28,471 (1977). 6. R. FREEMAN AND G. A. MORRIS, J. Chem. Sot. Chem. Commun. 684 (1978). 7. G. BODENHAUSEN AND R. FREEMAN, J. Am. Chem. Sot. 100, 320 (1978). 8. P. H. BOLTON AND G. BODENHAUSEN, J. Am. Chem. Sot. 101, 1080 (1979). 9. D. B. DAVIES, Progr. Nucl. Magn. Reson. Spectrosc. 12, 135 (1978). 10. J. JEENER, Ampere International Summer School, Basko Polje, Yugoslavia, 1971, unpublished. 11. W. A. ANDERSON, R. FREEMAN, AND C. A. REILLY, J. Chem. Phys. 39, 1518 (1963). 12. R. FREEMAN AND W. A. ANDERSON, .I. Chem. Phys. 31, 2053 (1962). 13. R. D. LAPPER AND I. C. P. SMITH, J. Am. Chem. Sot. 95, 2880 (1973). 14. G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29, 433 (1978). 15. G. BODENHAUSEN AND R. FREEMAN, J. Magn. Reson. 36, 221 (1979). 16. G. A. MORRIS AND R. FREEMAN, 1. Am. Chem. Sot. 101, 760 (1979). 17. S. SCHKUBLIN, A. H~HENER, AND R. R. ERNST, J. Magn. Reson. 13, 1% (1974). 18. A. A. BOTHNER-BY AND S. M. CASTELLANO, in “Computer Programs for Chemistry” (D. DeTar, Ed.), Vol. 1, Benjamin, New York, 1968.
26,
F.
OF MAGNETIC
39, 399-412 (1980)
RESONANCE
Elimination of Flip-Angle Effects in Two-Dimensional NMR Spectroscopy.Application to Cyclic Nucleotides GEOFFREY Francis
Bitter
BODENHAUSEN*
National Magnet Laboratory, 170 Albany Street, Cambridge,
Massachusetts Massachusetts
Institute 02139
of Technology,
AND PHILIP Department
of Pharmaceutical
Chemistry, San Francisco,
H. BoLTON? University California
Received February
of California, 94143
San Francisco,
1, 1980
The proton and phosphorus-31 NMR spectra of 2’,3’-cyclic uridine monophosphate (2’,3’-cUMP) have been correlated by means of heteronuclear two-dimensional NMR. The spectra are found to depend on the flip angle of one of the rf pulses. This is in contrast to previously reported applications where the 3*P nucleus was coupled to a single proton only. For flip angles other than 90” the spectra cannot be readily related to conventional proton spectra and the conformational information is obscured. However, a proper linear combination of the spectra eliminates the flip-angle effect. The resulting multiplets can be analyzed by routine simulation techniques to extract proton-proton coupling constants. This information, which can be related to the molecular conformation, is often not available from conventional proton spectra.
One of the more promising new methods to emerge from the development of two-dimensional Fourier transform NMR (2, 2) is the correlation of spectra of different nuclear species (3-5). The mapping of chemical shifts and long-range couplings can be helpful in a variety of assignment problems (4-7). When the magnetization is transferred from protons to phosphorus-3 1, a simplified proton spectrum is obtained which for many biological phosphates stems only from the proton nearest to the phosphorus nucleus (8). The homonuclear proton-proton couplings, which reflect the molecular conformation through the Karplus equation (9), can be observed indirectly with a spectrometer tuned for phosphorus NMR. It is therefore possible to study the conformations of systems for which the normal proton spectra are either obscured by solvent lines or otherwise too crowded to lend themselves to spectral analysis. A previous investigation (8) showed how the two-dimensional spectra of 2’and 3’-nucleotides could be related to the conventional proton spectra of these * Current address: Laboratoriumfiirphysikalische Chemie, Eidgenossische Technische Hochschule, 8092 Zurich, Switzerland. t Current address: Department of Chemistry, Wesleyan University, Middletown, Conn. 06457. 399 ouz2-236VS0/09039P14$602.00/0 Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
400
BODENHAUSEN
AND BOLTON
compounds. In these simple cases the phosphorus-31 nucleus is coupled to a single proton only. In many cellular phosphates, however, a different situation arises because two or more protons are simultaneously coupled to the phosphorus-31 nucleus. This case is quite common, notably in 5’-phosphates where the 5’ and 5” protons form a tightly coupled ABX system with the phosphorus-31 nucleus with coupling constants J,x and JBX of similar magnitude (9). The heteronuclear two-dimensional spectra of these systems show a dramatic variation of the relative signal intensities with the flip angle of one of the proton rfpulses. This behavior, which does not arise in the simpler 2’- and 3’-nucleotides, tends to complicate the analysis of the spectra since there is no longer any simple relationship between conventional proton spectra and heteronuclear spectra obtained with an arbitrary flip angle. Cyclic phosphates such as 2’,3’-cUMP contain ABX subunits that are not too tightly coupled. At a field strength of 4.7 T, typical for modem multinuclear spectrometers, the general features of these systems can be adequately discussed in first order. The populations of the energy levels can be analyzed in terms of classical magnetization vectors. Such calculations provide insight into the origin of the flip-angle effect. In a systematic variation of the flip angle, some symmetry properties become evident. In particular, it is possible to reconstruct the spectra that would be obtained if the flip angle were adjusted to exactly 90” by linear combinations of experimental spectra obtained with an arbitrary, unknown flip angle. The heteronuclear spectra obtained in this manner have an improved signal-tonoise ratio, and lend themselves to an iterative spectral analysis because of their close relationship to conventional proton spectra. MAGNETIZATION
TRANSFER
AND CONNECTIVITY
The heteronuclear two-dimensional technique has been originally described (3) as a special case of the basic two-pulse experiment proposed by Jeener tl
‘H 90.
l-l a a
3’P
s&t,) 90.
AM ‘I
‘H 90.
3’P
nn
aa
s&t,) b
FIG. 1. (a) The normal pulse sequence appropriate for heteronuclear two-dimensional spectroscopy is altered to allow for a systematic variation of the flip angle (Y.The phosphorus free-induction decay is recorded for typically 256 regular increments of the evolution period t1 which separates the two proton pulses. (b) Equivalent sequence, which is referred to in the discussion of the AMX system. Conceptually (if not experimentally) the nonselective proton pulse may be broken down into a “cascade” of two semiselective pulses, the first affecting the A transitions only, the second applied to all M transitions.
ELIMINATION
OF FLIP-ANGLE
401
EFFECTS
(ZO), which has been treated in detail by Aue et al. (I). The probability of a transfer of magnetization between the transitions (k,Z) and (m,n) is governed by the flip angle (Yof the mixing pulse (see Fig. la). The process can be described by a unitary transformation of the density matrix a: + umn
=
[II
&k(~)~:l(&il-
In the case of N weakly coupled spins with Z = M, the matrix elements &(a) of the pulse operator can be expressed analytically in terms of the parameter A ,&, defined as the number of spins that must be flipped to connect the energy levels m and k (I ). When the magnetization is transferred between transitions belonging to different nuclei, the phase of the resulting signal is independent of the flip angle (Y. This is consistent with the discussion of Freeman (5), who suggested that the information about the magnetization (kl) is temporarily stored in the form of populations and subsequently appears in the form of an amplitude modulation of the (mn) magnetization. However, the algebraic sign associated with the transfer of magnetization from (kf) to (mn) depends on the connectivity of these transitions. Consider for example the proton transition (1,2) and the phosphorus transitions (1,l’) and (2,2’) in the energy-level diagram of the AMX system shown in Fig. 2. In terms of the definitions due to Anderson et al. (II), two cases may be distinguished: the pair of transitions (1,2) and (1,l’) is “regressively” connected, while the pair (1,2) and (2,2’) is said to be “progressively” connected. The
(PI
p(r)
r
(P)
p(r)
r
(r)
r(p)
P
P
(P)
r
(r)
P
(p) r
(r)
(r)
r
(P)
P
r
(r) p
(P)
r
(0
p
(PI
44’ 33’
22’
JMX
x
II’ hx 3
FIG. 2. The structure of 2’,3’-cyclic uridine monophosphate (2’,3’-cUMP) with the energy-level diagram appropriate to the AMX subsystem consisting of the phosphorus nucleus and the two neighboring protons. The chemical shifts of the A(H,) and M(H,) protons are 5.138 and 4.917 ppm with respect to DSS (13). The coupling constants are JAM = JAX = 6.9 Hz and Jux = 11.5 Hz. The algebraic signs of the signals in the two-dimensional spectra (see Fig. 4) are determined by the connectivity relationships between the proton and phosphorus transitions. The symbols p and r refer to progressive and regressive connections in the restricted definition (II), applicable only to transitions which have one energy level in common, while the symbols (p) and (r) refer to the broader definition due to Ernst and co-workers (I).
402
BODENHAUSEN
AND
BOLTON
distinction arises from the quantum numbers of the initial and final states spanned by either pair of transitions (22). In a heteronuclear two-dimensional experiment, the nature of the connectivity determines the algebraic sign of the signals. If we assume, for the sake of this argument, that the two proton pulses in Fig. la are applied selectively to the proton transition (1,2), their cumulative effect is to drive the longitudinal magnetization Mz12 in a periodic manner, depending on the free precession of the transverse component in the evolution period (5): Mz12(tl) = -M,&q)
~0s wldl.
PI
For a vanishingly short evolution period t 1, the two pulses simply invert the magnetization. As a result, the populations of level 1 (PI) and of level 2 (PZ) are simply interchanged, enhancing P, and depleting P, with respect to thermal equilibrium. Since none of the other populations is perturbed, it follows that the population difference across the progressively connected phosphorus transition (2,2’) has been increased, while the regressively connected phosphorus transition (1,l’) has become associated with a negative population difference. As the evolution time tl is increased, the amplitudes of both phosphorus transitions will experience a cosinusoidal modulation, their phases remaining opposite throughout the experiment. After the Fourier transformation with respect to tl, progressive and regressive pairs will generate positive and negative absorption mode signals. In the actual two-dimensional experiment, the pulses applied to the protons affect all transitions simultaneously. As a result, the transfer of information is not confined to proton and phosphorus transitions with a common energy level. This effect also occurs in the Jeener experiment (10) and has lead Ernst and co-workers (1) to broaden the original definition of the connectivity relationships (11) to encompass all pairs of transitions that are parallel to a basic pair of directly connected transitions. Thus in the energy-level diagram in Fig: 2, the pair (3,4) and (1,l’) is regressive in the broader sense, since the transitions (3,4) and (1,2) are parallel and merely differ in the polarization of the third spin. In the lower half of Fig. 2, the connectivities have been tabulated; the symbols p and r refer to the original definition (11) while the letters in parentheses (p) and (r) correspond to the broader definition (I). Progressive pairs of either variety, p or (p), give rise to positive signals in the two-dimensional spectra. THE
AMX
SYSTEM
Although the second proton pulse and the phosphorus observation pulse may be applied either simultaneously or sequentially with the same result, the latter case (Fig. la) may provide some insight into the mechanism of the information transfer (5). Whereas the discussion of symmetrical spin systems of the A,X type is straightforward, nondegenerate cases such as the AMX system and higher analogs require a more careful analysis. In order to illustrate a number of general features of the heteronuclear two-dimensional experiment, we shall concentrate our attention on 2’,3’-cyclic uridine monophosphate (Fig. 2) which has been the subject of an earlier spectral analysis (13). The H, and H3 protons and the phosphorus nucleus provide a typical AMX system (albeit with a slight ABX
ELIMINATION
OF FLIP-ANGLE
EFFECTS
403
character) while the H,. and H4 protons, which are coupled to H, and HS, are not relevant to the present argument and wilI be considered later. The first-order proton and phosphorus spectra are depicted schematically in Fig. 2. The A region features an accidental degeneracy (due to the equality JAM = JAX) but the M region has a structure typical of a nondegenerate AMX system. In the two-dimensional experiment, all eight proton magnetization vectors of the AMX system are excited simultaneously by the initial pulse. The second pulse will find the individual vectors with different phases, characteristic of their frequencies and the duration of the evolution time cl. By rotating the transverse magnetization back into the z axis of the rotating frame, the precession is effectively frozen and stored in the form of populations, leading to t,-dependent modulations of the amplitudes of the phosphorus lines. The mechanism may be clarified by the use of two conceptual aids. First, we shall assume that only one proton magnetization vector, say M12(0+), is brought into the transverse plane at the beginning of the evolution period. This could be achieved experimentally by replacing the initial proton pulse by a “Dante” pulse train (14) suitably tailored to the frequency o12. The second conceptual simplification is obtained by replacing the second proton pulse by a “cascade” of two distinct pulses, both with a flip angle cr, the first applied to the A region only, acting on (1,2) and (3,4), the second to the M transitions (1,3) and (2,4) (Fig. lb). Provided the pulse interval is short with respect to free precession and relaxation, such a pulse cascade is equivalent in all respects to a single nonselective pulse applied to all proton transitions simultaneously, but it greatly simplifies the analysis (15). Consider the fate of the magnetization vector M,, in the course of the experiment depicted in Fig. lb. At the end of the evolution period, just before the second pulse, the y component reflects the free precession: M,dG)
=
~l/12(Of)
cos
[31
qd1.
Now the A pulse, applied along the x axis of the rotating frame, will rotate this transverse component through an angle (Y, generating a longitudinal component: M,,,(tf+)
= -sin (YMUlz(t;).
This amounts to the creation of t,-dependent and 2 just after the A pulse:
[41
populations
in the energy levels 1
Pl(rf+)
= +sin (Ycos (w,,t,)AP
+ P,
[51
P,(tf+)
= -sin (Y cos (w,,t,)AP
+ I’,
161
where P = [P,(O-) + P,(O-)]/2 is the average population in thermal equilibrium and AP is defined as the difference [P,(O-) - P,(O-)]/2. At this stage, none of the populations of the other six energy levels is affected. (The A pulse can of course generate new transverse magnetization in the A region but this has no effect on the populations.) Now the it4 pulse will cause a redistribution of populations. This effect can be seen simply in terms of classical magnetization vectors (1.5). Consider, for example, the M transition (1,3) as it experiences a pulse of flip angle a:
404
[P&y+)
BODENHAUSEN
- P,(rrl’+)] = M&r?+)
AND BOLTON
= Mz13(ff+) cos ff = [Ps(r?+) - P,(rt+)]
cos a.
[7]
Since the sum of the populations must be conserved at all times, the effect of the M pulse is to cause a redistribution of the populations between the energy levels land3: P&y+)
= P,(rig+) cos2 ;
P3(ry+) = P3(tf+) cos2 f
+ P3(ff+) sin2 4 ,
F31
+ P,(tf+)
sin2 t
.
191
P2(ty+) = P2(tf+) cos2 f
+ P4(ff+) sin2 f
,
DOI
P4(fy+) = P4(ft+) cos2 I
+ P2(ft+) sin2 :
.
[ill
A similar argument applies to the (2,4) transition:
As in any other Fourier transform NMR experiment the application of a small flip angle (i.e., cos2 (a/2) + sin2 (a/2)) induces little population transfer between the energy levels. Thus, the cos alzfl modulation of the population PI, which has been generated by the A pulse according to Eq. [5], will not be significantly perturbed by an M pulse of small flip angle. In the limit of a flip angle close to 180”, however, the M pulse causes P, and P3 to interchange. As a result, the cos u&, modulation is carried over from P, to P, although energy level 3 is not directly connected with the proton transition (1,2) that is responsible for the modulation. The observation pulse applied to phosphorus-31 (Fig. 1) will detect the time dependence of PI and P2 at the phosphorus transition frequencies ollP and w221while the modulation of P3 and P4 can only affect the phosphorus signals at u33t and w44r. The implications are illustrated in Fig. 3, where the signals S(F,,F2) appearing in the two-dimensional spectra have been simulated as a function of the flip angle. For small (Y, the signal at 27rF1 = o12 (far left in all spectra), due to the proton magnetization M12, appears at 2mFz = oll, and w22, (Figs. 3a and b). With increasing flip angle, this signal gradually disappears from these spectra to reappear at the phosphorus frequencies 2rF2 = w33, and 044P (Figs. 3c and d). Thus the transfer of information does nor require the transitions to share a common energy level. It should be noted that the information is never lost, nor is the total amplitude of the signals at 2rF, = o12 attenuated by the mixing effect. At flip angles of CY= 90”, however, the information is spread over all energy levels, resulting in a larger number of signals in the two-dimensional spectrum. For this reason, the actual signal height does not necessarily increase for larger flip angles, although, in the absence of signal cancellations, the integrated intensity of the absolute value of the two-dimensional spectrum increases with sin CY.
ELIMINATION
OF FLIP-ANGLE
EFFECTS
405
d +w+--$,,-
5.2
50
C
46
5.2
5.0
46~
FIG. 3. Simulated heteronuclear spectra of the AMX subsystem of 2’,3’-cUMP for different flip angles (Y of the second proton pulse. The horizontal F, axis corresponds to the proton spectrum at 200 MHz. The four “slices” through the two-dimensional plot, corresponding to the four phosphorus transitions, are shown clockwise: (a) 27rF, = oll,, (b) 27rF, = oz2,, (c) 27rF, = We,, and (d) 2?rF, = co,,. Note the symmetry that relates the top and bottom halves of the figures.
Examination of Fig. 3 reveals a symmetry that is implicit in the discussion of the population transfer. The spectrum at 277F, = ollf, with (Y = 45” (Fig. 3a, bottom), is the exact opposite of the spectrum at 27rFz = wd4’, with (Y = 135” (Fig. 3d, top). In fact, the top and bottom halves of Fig. 3 are related by symmetry. In general, a two-dimensional spectrum obtained with a supplementary flip angle, (Y’ = 180” - cy, will not contain any new information. Spectra obtained with supplementary flip angles can be brought to superposition by reversing the algebraic signs and interchanging wll, with w44, and oz2, with ws3’. The theoretical two-dimensional spectra S(F,,F,) of the AMX system are reproduced in Fig. 4 for the flip angles Q = 45, 60, and 90”. The algebraic signs of the individual lines relate directly to the connectivity diagram in Fig. 2, progressive pairs corresponding to positive resonances. All spectra in Fig. 4 are shown on the same vertical scale (the sin (Y term in Eq. [4] has been taken into account) to emphasize the fact that the height of the individual peaks actually decreases as the flip angle approaches 90”. In the A region, the accidental degeneracy of the proton transitions (3,4) and (1’,2’), which give rise to lines of opposite algebraic signs, causes a signal cancellation for (Y = 90” which is avoided by the use of smaller flip angles. The two-dimensional spectrum for (Y = 90” has some unique features. As can be seen in Fig. 4, the spectrum S $O”(F,, F,) has an inversion symmetry with
BODENHAUSEN
406 a.459
5.2
5.0
AND BOLTON
a=60'
48
5.2
5.0
a* so'
4.6
52
5.0
4.8 ppm
FIG. 4. Simulated two-dimensional spectra of the same AMX subsystem for three different flip angles, shown on the same vertical scale. For (Y = W, the gain in signal intensity is offset by the appearance of additional lines, which lead to accidental signal cancellations in the A region at F, = 5.138 ppm. For small flip angles, only directly connected transitions p and r give rise to signals, while with increasing flip angles the indirectly Connected transitions (p) and (r) enhance the complexity of the spectra (compare Fig. 2).
respect to a line parallel to the F, axis, intersecting the F2 axis at the chemical shift of the phosphorus nucleus. Thus, the trace at 27rF, = ollt is the exact opposite of the trace at 27rFz = 044J. A similar relationship exists between the two central spectra. It has been suggested (8) that this symmetry should be used to boost the signal-to-noise ratio of experimental spectra by taking appropriate linear combinations. A more important aspect of the spectra obtained for (Y = 90” resides in their relationship to the normal proton spectra. Consider the phosphorus transition (1,l’) in the energy-level diagram in Fig. 2. In the broader definition of connectivity (I ), all four proton transitions (1,2), (1,3), (2,4), and (3,4) form regressive pairs with the (1,l’) transition, while the remaining four proton transitions between the primed energy levels are progressively connected with the (1,l’) transition. For (Y = 90”, the former will result in four negative resonances in the two-dimensional spectrum at 27rFz = w 11,,while the latter contribute four positive peaks. In terms of spectral analysis, these two groups of proton transitions correspond to the subspectra belonging to the polarizations (Y and p of the phosphorus nucleus. Thus, the trace extracted from the two-dimensional spectra at 27rFz = wllf is equivalent to the difference of the two proton subspectra (8). This relationship only holds for the phosphorus transitions wll# and 044, which connect the top and bottom states of the proton energy levels in Fig. 2. For the inner phosphorus transitions oZ2! and oS, the connectivity rules are more complicated and the amplitudes cannot be readily described in terms of proton subspectra. The spectral analysis outlined here is directly applicable only to spectra obtained with a flip angle carefully calibrated to 90”. Fortunately, it is possible to extract this information from experimental two-dimensional spectra obtained with an arbitrary flip angle (Y. Because of the symmetry properties presented in Fig. 3, a simple difference of two slices taken from the two-dimensional spectrum may be calculated
ELIMINATION
OF FLIP-ANGLE
EFFECTS
407
to correct for flip angles that differ from 90”: Ss’(F1, 27rF, = till,) = S”(F1, 27rFz = ollt) - Sn(F1, 27rF2 = oq4t).
[W
A similar difference of the traces at 27rFz = wZ2’ and wQ3!will yield reconstituted spectra corresponding to o! = 90”, whose interpretation is, however, less straightforward. The difference spectrum (Eq. [ 121) has an improved signal-to-noise ratio, largely because much of the noise is in fact correlated in the two slices and stems from the unmodulated peaks at F, = 0 (so-called t1 noise). The experimental enhancement of the sensitivity obtained in this manner is typically a factor of 2.5.
ADDITIONAL
COUPLED
PROTONS
In systems of practical interest, like 2’,3’-cyclic uridine monophosphate, the AMX system is a mere subunit among N protons. Two situations can be distinguished. When the additional protons are coupled to the phosphorus-31 nucleus, they give rise to further splittings in the Fz dimension of the two-dimensional spectrum. In this case, the energy level diagram should be extended accordingly, and the pulse cascade in Fig. lb should include as many selective pulses as there are protons. Each of these pulses flips a fraction sin2 (oJ2) of the spins to which it is applied. The population of an energy-level n is affected by a magnetization M&f;) PreCeSSing at &l according t0 the eqUatiOn P,(Q)
= sin a (cos t)““-“‘(sin
tr’cos
okltl,
1131
wherefis the number of spins that must be flipped to shift the transition (k,f) to a parallel transition that includes the energy level n. This expression is related to Eq. [51] in the work of Aue er al. (I). The sin (Yterm arises from the incomplete rotation of the magnetization as expressed in Eq. [4]. Note that the greater the number of protons N, the larger the exponents in Eq. [13], and the more critical the calibration of the flip angle (II. The simplest spectra are of course obtained for small flip angles, when all transfers withf > 0 vanish, and the modulation due to Mrkl remains confined to Pk and Pl. A much simpler case is more frequently encountered in nature. In 2’,3’-cUMP, for example, the protons of the basic AMX subunit are coupled to their next neighbors, but these do not couple directly to the phosphorus nucleus. In this case, the additional protons split the proton resonances in the A and M regions into doublets in the F, or proton dimension of the two-dimensional spectrum. Hence, a cascade pulse applied to any of the additional protons simply interconverts the two components within the F, doublet, without affecting the Fz or phosphorus dimension. Since the proton spectrum is normally excited by a nonselective pulse at the beginning of the evolution period, both components of the doublet have equal magnitudes and their interconversion does not affect the two-dimensional spectrum. The experimental two-dimensional spectrum of 2’,3’-cUMP is shown in Fig. 5 in the absolute-value mode. The spectrum was obtained with the pulse sequence in Fig. la, the flip angle of the second proton pulse being set to approximately 150”.
408
BODENHAUSEN
AND BOLTON
55
FIG. 5. Heteronuclear two-dimensional spectrum of 2’,3’-cUMP, shown in absolute-value mode, obtained at a phosphorus frequency of 81 MHz with proton pulses at 200 MHz. The flip angle LYwas set to approximately 150”. The resolution in the FZ dimension has been enhanced by multiplying the freeinduction decays with a trapezoidal function, while a Lorentzian line broadening of 0.5 Hz was applied in the F, dimension. The evolution time was incremented in 256 steps of 2.5 msec each. The 0.2 M solution contained about 60% H,O and 40% *H*O; the pH was 7.5 at a temperature of 35°C.
The decoupler frequency was positioned at the low-field side of the resonances of the 2’ and 3’ protons; the orientation of the F, axis is in agreement with the conventions of proton NMR. To prevent the “tails” of the phase-twisted lineshapes (2) from interfering in the Fz dimension, a trapezoidal resolution enhancement was applied in this domain. The four relevant slices through the two-dimensional spectra are shown in Fig. 6 in the phase-sensitive mode for two different flip angles a! = 30” and (Y = 90”. The latter proton flip angle was adjusted by using a modified version of the INEPT technique (16). The structure of these spectra is closely related to the theoretical AMX spectra in Fig. 4, except that all resonances of the H, proton are split byJ(HIHP) = 3.0 Hz and all H3 lines are split byJ(H3H4) = 5.5 Hz (13). There are a total of 16 different frequencies in theF, dimension for (Y= 90”, though only eight significant signals appear if the flip angle is small. The experimental spectra do not exhibit a perfect inversion symmetry, indicating a deviation from a! = 90”. STRONG COUPLING
So far it has been assumed that the AMX unit of 2’,3’-cUMP can be treated in first order as a weakly coupled spin system. It is possible, however, to predict the two-dimensional spectra of a strongly coupled ABX system on the basis of Eq. [l]. Using the matrix elements of the pulse operator R(a! = 90”) in the eigenbasis of the free-precession Hamiltonian and the appropriate magnitudes of the
ELIMINATION
5.2
OF FLIP-ANGLE
5.0
4.8
5.2
409
EFFECTS
5.0
48
pm
FIG. 6. Experimental “slices” extracted from two-dimensional spectra obtained approximately 30 and 90”. The algebraic signs correspond to those of the AMX Fig. 4, but the signals of the H, and H3 protons are split into doublets by J(H,H,) = = 5.5 Hz, respectively (I.?). The F, linewidths are about 2.5 Hz in all spectra, homogeneity of the B. field in a 12-mm sample tube.
with flip angles of subunit shown in 3.0 Hz andJ(H,HJ due to the limited
magnetization vectors in the evolution period 1ultl 1 = IFVkl 1 (Z7), the populations just before the phosphorus observation pulse may be calculated: PI@:) = +1/{(1
- sin 20) cos wlztl + (1 + sin 213)cos uS4fl + (1 + sin 28) cos ti13tl + (1 - sin 28) cos 014fl},
[ 141
WI
P*(C)
= -pm,
P&:)
= +?4 cos 28{-cos
M:)
= -P&:),
e&r1
-
cos
%dl
+
cos
W13rl
+
cos
024rl},
[I61 iI71
where 19= ‘/2 arctg{ JABI[aA - aB + (JAX - 5,x)/2]}.
[la
Analogous equations apply to the primed states PI . . . , etc., which are modulated by the proton frequencies 01,2, . . . , etc. The effective chemical shift difference in the expression for 8’ is [a, - aB - (JAX - J&/2]. The populations of the energy levels 2 and 3 are modulated by four proton magnetizations, all of which contribute terms of equal amplitude, regardless of the strength of the coupling, provided (Y = 90”. This remarkable feature seems to suggest that the corresponding intensities in the two-dimensional spectrum are independent of the strength of the coupling. In the ABX system, however, two additional phosphorus transitions appear at 27rF, = 023f and 02’3 which have energy levels in common with the phosphorus lines at ~TF, = w221 and oS3!. In these circumstances, none of the amplitudes of these four phosphorus lines will be simply proportional to the population differences (17).
BODENHAUSEN
410
AND BOLTON
a b
6
I
I
5
4
pm
FIG. 7. The normal proton spectrum of 2’,3’-cUMP at 200 MHz consists of the sum (d) of the two subspectra belonging to the polarizations (Yand /3 of the phosphorus nucleus (b and c). These have been calculated with the spin-simulation program NMRCAL, taking into account all six protons of the ribose system (13). The difference of the subspectra (a) should correspond to the trace extracted from an experimental two-dimensional spectrum according to Eq. [12].
In the context of spectral analysis, it is preferable to avoid these complications by concentrating attention on the phosphorus lines at 21rF~ = c.o~~,and C.Q, which reflect the population differences P, - PI, and P, - P4, in a straightforward manner regardless of the strength of the AB coupling. The modulation terms in Eq. 1141 are weighted by factors 1 k sin 28 which are identical to the characteristic “sloping” intensities of a normal AB pattern. In conventional Fourier transform spectroscopy (17) these intensities arise from F2, terms, whereas the amplitudes in the heteronuclear two-dimensional experiment are governed by Eq. [l]. If the spectra are obtained with a flip angle (I! = !W, the phosphorus transition wll’ in the ABX system will be modulated by a total of eight contributions: four negative terms with intensities -(l -+ sin 213) stemming from the proton subspectrum associated with P, and four positive terms with intensities +(l + sin 28’) which correspond to the proton subspectrum associated with the p polarization of the phosphorus nucleus. Thus, the trace extracted from the two-dimensional spectrum at 2rrFz = ollt is equivalent to the difference of the two proton subspectra AB(PJ - AB(P,). This equivalence also holds for systems of the ABMX type and higher analogs, provided the additional protons do not couple directly to the phosphorus nucleus. l EXPERIMENTAL
VERIFICATION
The simulated conventional proton spectrum of 2’,3’-cUMP at 200 MHz, shown in Fig. 7d, features the characteristic sloping line intensities due to strong coupling 1 Systems with more than two strongly coupled protons should be treated with caution, since the phosphorus transitions connecting the top and bottom energy levels are likely to overlap with other resonances. Furthermore, there is no analytical proof that the equivalence of the intensity rules in conventional and two-dimensional Fourier transform spectroscopy is applicable to ABCX systems.
ELIMINATION
OF FLIP-ANGLE
I
5.2
I
I
5.0
EFFECTS
411
I
4.8 ppm
FIG. 8. (a) The theoretical two-dimensional spectrum reproduced from Fig. 7a is in good agreement with the experimental spectrum (b) obtained by subtracting the top and bottom traces in the right-hand side of Fig. 6, which stem from an experiment with (Y = 90”. The improvement in the signal-to-noise ratio resulting from this subtraction is approximately 25fold, largely because of the cancellation of the unmodulated peak at F, = 0 which entails the suppression of much of the so-called t1 noise. The same pattern (c) is obtained by subtracting the top and bottom traces in the left-hand side of Fig. 6, showing that the same information can be obtained with a flip angle of about 30”. The reduction in signal intensity may be explained by the sin a term in Eq. [4].
between the H, and H, nuclei (6/J = 6.4), whereas the couplings to the next neighbors are first order to a good approximation. The normal proton spectrum may be decomposed into two subspectra (Figs. 7b and c) which correspond to the polarizations cz and /3 of the phosphorus nucleus, and differ in the effective chemical shifts of the Hz and H3 protons. The difference of the subspectra (Fig. 7a) is reproduced in Fig. 8a for comparison with the experimental two-dimensional spectra. The trace in Fig. 8b was obtained from an experiment with (Y = 90”, by subtracting two slices according to Eq. [12] to improve the signal-to-noise ratio. Figure 8c shows the same linear combination applied to an experiment with (Y = 30”. The equivalence of Figs. 8b and c confirms the validity of Eq. [ 121 and shows that the same pattern may be obtained with an arbitrary flip angle, thus rendering the flip-angle calibration less critical. Either spectrum b or c in Fig. 8 may be analyzed by an iterative simulation with least-squares fitting in the manner ofthe LAOCOON program (18) to extract the coupling constants between the four protons on the ribose ring. The relationship between the proton-proton coupling constants and the conformations of nucleotides has been the subject of extensive research (9) and is beyond the scope of this work. The main purpose of the methods discussed here is to provide access to molecular systems whose normal proton spectra are too crowded or otherwise obscured. It has been shown in systems of increasing complexity that a simple relationship exists between heteronuclear two-dimensional
412
BODENHAUSEN
AND
BOLTON
spectra and conventional proton NMR spectra. The elimination of the flip-angle effect is a prerequisite for the analysis of the spectra of a large class of compounds which includes cyclic phosphates, 5’-nucleotides, and oligonucleotides. ACKNOWLEDGMENTS This work was supported in part by grants from the U.S. Public Health Service, National Institutes of Health. Preliminary spectra were obtained with the kind assistance of Dr. W. W. Conover in the application laboratory of Nicolet Technology Corporation, Mountain View, California. Further experiments were performed at the Magnetic Resonance Laboratory at the University of California, Davis. The encouragement and support of Dr. L. J. Neuringer at the National Magnet Laboratory and Dr. T. L. James at the University of California, San Francisco, are gratefully acknowledged. REFERENCES 1. W. P. AUE, E. BARTHOLDI, AND R. R. ERNST, J. Chem. Phys. 64, 2229 (1976). 2. G. BODENHAUSEN, R. FREEMAN, R. NIEDERMEYER, AND D. L. TURNER, .I. Magn. Reson. 133 (1977). 3. A. A. MAUDSLEY AND R. R. ERNST, Chem. Phys. Left. 50, 368 (1977). 4. A. A. MAUDSLEY, L. MOLLER, AND R. R. ERNST, J. Magn. Reson. 28,463 (1977). 5. G. BODENHAUSEN AND R. FREEMAN, J. Magn. Reson. 28,471 (1977). 6. R. FREEMAN AND G. A. MORRIS, J. Chem. Sot. Chem. Commun. 684 (1978). 7. G. BODENHAUSEN AND R. FREEMAN, J. Am. Chem. Sot. 100, 320 (1978). 8. P. H. BOLTON AND G. BODENHAUSEN, J. Am. Chem. Sot. 101, 1080 (1979). 9. D. B. DAVIES, Progr. Nucl. Magn. Reson. Spectrosc. 12, 135 (1978). 10. J. JEENER, Ampere International Summer School, Basko Polje, Yugoslavia, 1971, unpublished. 11. W. A. ANDERSON, R. FREEMAN, AND C. A. REILLY, J. Chem. Phys. 39, 1518 (1963). 12. R. FREEMAN AND W. A. ANDERSON, .I. Chem. Phys. 31, 2053 (1962). 13. R. D. LAPPER AND I. C. P. SMITH, J. Am. Chem. Sot. 95, 2880 (1973). 14. G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29, 433 (1978). 15. G. BODENHAUSEN AND R. FREEMAN, J. Magn. Reson. 36, 221 (1979). 16. G. A. MORRIS AND R. FREEMAN, 1. Am. Chem. Sot. 101, 760 (1979). 17. S. SCHKUBLIN, A. H~HENER, AND R. R. ERNST, J. Magn. Reson. 13, 1% (1974). 18. A. A. BOTHNER-BY AND S. M. CASTELLANO, in “Computer Programs for Chemistry” (D. DeTar, Ed.), Vol. 1, Benjamin, New York, 1968.
26,
F.