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Observation of multi-exponential longitudinal and transverse relaxation in two-dimensional NMR
CHEMICAL PHYSICS LETTERS
Volume 131, number 3
OBSERVATION LONGITUDINAL Norbert Insiitut
OF MULTI-EXPONENTIAL AND TRANSVERSE RELAXATION
IN ‘IWO-DIMENSIONAL
7 November 1986
NMR
MijLLER
fiuChemie, Johannes Kepler Unrversrttit, A-4040 Linz, Austria
Received 21 August 1986
In a spin system that exhibits multi-exponential longitudinal relaxation, multiplequantum coherence may be created by a n/2 pulse acting on a state of partially recovered polarization. This, together with recently discovered violations of COherence-transfer selection rules through multiexponential transverse relaxation, allows one to design a new twodimensional NMR method. The shapes of the peaks situated on the skew diagonal of the spectrum are determined by longitudinal and transverse relaxation functions along the wt and wp axes, respectively.
1. Introduction Multi-exponential relaxation has long been known to occur in systems with equivalent spins [l] or quadrupolar nuclei [2]. Coherence-transfer selection rules [3] have recently been shown to break down because of unequal decay of degenerate coherences [4-61. This has stimulated efforts to apply two-dimensional (2D) multiplequantum NMR for the detection of multi-exponential relaxation [6,7]. Such an approach should be superior to one-dimensional (1D) NMR methods such as inversion recovery and spin echo techniques in cases where signal overlap prevents complete interpretation of 1D spectra, for example in spectra of large molecules. The correlation times that become accessible through the relaxation parameters may provide insight into molecular dynamics [6-81.
2. Theory The new pulse sequence shown in fig. 1 may be interpreted in two different ways, First, it may be regarded as a 2D correlation spectroscopy (COSY) pulse sequence [9], preceded by inversion and partial relaxation. In contrast to ordinary correlation spectroscopy, however, the experiment is designed to monitor the evolution of multiple-quantum coherence, which in this case is not created through other coher218
Fig. 1. (a) Pulse sequence and (b) coherence transfer pathways for multiexponential relaxation spectroscopy (MERCY). The phases rpand up’are cycled together with the receiver reference phase to select multiplequantum coherence of order p. The special case of three-quantum order selection is shown in (b). The variation of 7 is described by eq. (12).
ences but, as is seen below, from types of longitudinal spin order other than Zeeman order. Second, the sequence can be regarded as a modified inversion recovery experiment, where a multiple-quantum evolution time t1 and an observation pulse p’, that converts multiplequantum into onequantum coherences, have been added. The transformations occurring under the influence of this pulse sequence will be discussed with particular attention to X, spin systems and implications for AX, 0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CHEMICAL PHYSICS LETTERS
Volume 131, number 3
spin systems (neglecting cross-relaxation between the A and X spins for simplicity). The experiment is, however, applicable to any system that exhibits both longitudinal and transverse multi-exponential relaxation. (In the presence of resolved spin-spin couplings non-exponential decay of longitudinal magnetization is sufficient for certain peaks to occur, as will be seen below.) An extensive account on related methods applied to quadrupolar nuclei is given by Jaccard et al.
7 November 1986
j$‘(~ = 0) = 1, while the other relaxation functions consist of linear combinations of exponentials with amplitudes such that @(7 = 0) = 0 for I > 1 [4,6,7]. The tensor terms of second and third ranks (I > 1) are transferred into coherences of orders Ip I = 1, .... I by the radio-frequency pulse /3 of phase cpin fig. 1, while their ranks I are conserved. Using the elements dLo of the Wigner rotation matrices [ 1 l] * this transformation is described by [6]
171. The three magnetically equivalent protons of a methyl group can be described in a symmetry-adapted basis [IO], where the spin system is represented as a superposition of three group spins, two with S = l/2 and one with S = 312. In the absence of symmetry breaking cross-relaxation, only the latter need be considered if one focuses on multiplequantum coherences [6]. The transformations of the density matrix u will be discussed in terms of irreducible tensor operators T$ that have recently been found convenient for handling the effects of multi-exponential relaxation [6,7]. After the initial inversion pulse one can describe the displacement from equilibrium by u((7 = 0) - ueq = -20’122$,
(1)
which represents inverted Zeeman order. Under the influence of a relaxation operator IQ=O) the rank 1 of the tensor changes during the time interval 7 while the coherence order p is conserved. In the notation of refs. [6,7] this is
r(O)7
Go-
c
By a suitable cycle of the phase p the desired coherence level can be selected among the various orders p [ 121. In conventional inversion-recovery experiments Tf _1 is detected at this point in the pulse sequence. Ih a COSY experiment on the other hand, +l quantum coherences would be selected in the evolution period by the phase cycle. Murali and Kumar [13] have recently attributed spurious peaks in COSY spectra to multiple-quantum coherences created from incompletely recovered polarization by similar pathways. In the experiment discussed here multiplequantum coherence is allowed to precess during t 1. For methyl groups the relevant terms comprise only two- and three-quantum coherences. The latter can only be created from third-rank tensor terms, while twoquantum coherence is generated both from second- and third-rank tensors, as can be seen from the explicit forms of eq. (3) (neglecting terms with Ip I <2):
T$,$;tO,
0<1<3
where Tfo and Tfo represent different forms of longitudinal spin order and are analogous to quadrupolar and octupolar orders in I = 3/2 nuclei. The functions @(r) describe the build-up and decay of the tensor terins T&. In order to keep the discussion as general as possible - but excluding symmetry breaking crossrelaxation -we will not assume any particular relaxation mechanism but treat an arbitrary 4 X 4 relaxation matrix I’(o) with three different negative eigenvalues RI, R,, and Rg (one eigenvalue,Ro, is always zero). The functionsfif) will therefore generally be triexponential. In many cases bi-exponential behaviour is found to a good approximation [6,7]. The coefficients of all exponentials in #(T) have equal signs and
P GO
-
T&2
(3/8)1j2 si&,
P
@O -
Tf, * 2 (1 5/8)li2 sin20 cos fl T T&3
(5/16)112 sin3&
(5)
Selecting threequantum coherence during tl allows one to focus on third-rank tensor operators. However, while it is possible to select twoquantum coherences of second rank by setting 0 equal to n/2, there is no obvious way to select Tf k2 without interference from Tf f 2. Zeeman prec&sion changes neither the rank nor * The relevant elements in this context have been compiled in ref. [6]. 219
CHEMICALPHYSICS LETTERS
Volume I3 1, number 3
the coherence order of the tensors [6]. During tl the Hamiltonian simply leads to the transformation n&S1
Ti$-
T& exp(-ipaStL)
t other terms.
(6)
This precession labels the signals with the multiplequantum frequency pS$ in the tl domain. The last pulse /3’converts multiplequantum coherence (among others) into observable +l quantum coherence, i.e. in analogy to eq. (3)
P’ 1
T,f,2 T&ld!_ Q@‘) exp[-i(-1
-PM].
(7)
7 November 1986
The relaxation functionsf#(t2) describe the multiexponential behaviour of one-quantum coherences in an analogous manner to the way the functions fif)(r) in eq. (2) describe multi-exponential longitudinal relaxation. In a 2D NMR spectrum obtained via the pulse sequence of fig. 1 the above transformations give rise to peaks situated at the coordinates (wl = pas, a2 = as), i.e. on the skew diagonal of a multiple-quantum spectrum of order p. The existence of such a peak requires that both longitudinal and transverse relaxation be multi-exponential. Therefore the acronym MERCY (multi-exponential relaxation spectroscopy) is sugges-
A(Z = 2, *2 + -1) = sin p’ cos p’,
(8a)
ted, which may be extended to pQ MERCY to indicate the order Ip I of multiplequantum coherence selected. Note that this method applies primarily to singlets without scalar, dipolar or quadrupolar splittings, which were previously considered a prerequisite for the generation of multiplequantum coherence. In order to obtain quantitative information about the multi-exponential longitudinal relaxation functions #(r), the delay r can be varied synchronously with the incrementation of tL, in a similar manner to the accordion experiment [ 141:
@1=2,+2-t-l)=sin$,
(8b)
T=Tmax-Xll.
A(I = 3, +2 + -1) = (S/2)li2 sin /3’cos /I’,
(9a)
If the experiment is run with a constant delay 7, it can only serve to prove the existence of multiexponential longitudinal and transverse relaxation. If r is, however, varied according to eq. (12), the functions .#(r) influence the lineshape along ol. This lineshape function is a convolution of the Lorentzian shape imposed by the exponential decay of the multiplequanturn coherence of order p during tl (with time constant Tip)) and the multi-Lorentzian shape caused by multi-exponential behaviour during the delay 7. The functions@)(r) are scaled with respect to tl by the factor x, which serves to adjust for the different lifetimes of longitudinal spin order and multiplequantum coherence. In the w2 domain the lineshape is given by the Fourier transform of the relaxation functions@(t2) (eq. (1 l)), which are characteristic of multi-exponential transverse relaxation of degenerate onequantum coherences. The lineshape along w2 consists therefore of a superposition of at least two Lorentzians of different signs and widths, which correspond to the eigenvalues of the relaxation operator F(l) [6,7].
In order to obtain pure phase peakshapes, coherence transfer pathways from tp and -p quantum coherences must be selected simultaneously by the phase cycle [ 121, For the particular case of interest one obtains the following transfer coefficients A(Z, kp + - 1) andD(I, &p + -1) for the absorptive and dispersive components of the 2D peaks. By substituting the Wigner matrix elements into eq. (7) and using the procedure outlined in ref. [6], one obtains
D(I = 3, +2 --f -1) = (5/8)Lj2 sin 0 (3 cos2$ - l), (9b) A(1=3,+3+-1)=(15/16)1/2sin2/3’,
(lOa)
D(I=3,+3+-1)=-(15/16)1~2sin2~‘cos~’.
(lob)
From these equations it can be seen that if p’ = a/2 the two-quantum peaks obtained from second-rank tensor terms (eq. (8b)) will be of opposite phase to those stemming from third-rank tensors (eq. (9b)). A flip angle of n/2 also enables one to obtain pure phase peak shapes. The one-quantum tensor operator terms Tf_l (eq. (7)) precess with their characteristic Larmor fiequency during ~2. However, as their ranks exceed one, they are not observable unless conversion to first-rank terms is effected by multi-exponential transverse relaxation (described by the operator F(P=L)) as explained in ref. [6] : I-
q-1 220
(‘)tz
T;,-I~?(Q-
(11)
(12)
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CHEMICAL PHYSICS LETTERS
The two-dimensional time envelope of the free induction decay in such an experiment with threequanturn selection for an uncoupled S = 312 group spin is accordingly:
S(Q) t2) = fg(Tmax - xfl) exp(-tllT~3))fl13((t2).
7 November 1986
3. Experimental verification Multi-exponential transverse relaxation within the S = 3/2 subsystem in a system of three magnetically equivalent spins with I = l/2 can only occur outside the fast-motion limit 161. Therefore experimental evidence was sought by investigating a sample of a small protein (basic pancreatic trypsin inhibitor, BPTI, molecular weight a6000). Fig. 2 shows partial contour plots of a 3Q MERCY spectrum (i.e. three-quantum coherence is selected during fl). In order to suppress po-
(13)
Examples of theoretical lineshapes for the case of biexponential decay can be found in refs. [4,6,7].
r
1
1 ,,P 1 T3; )’
2.0
I18 U”II9119 IOHz
3.98 ppm
30Hz _---__
--Ii
-
$6
A48 4.0
w
A27 A40 A.58
I 9
T’l 1
5.0
A25 T54 6.0
7.0
t 2.4
2.2
a 2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
PPm
-9
Fig. 2. Partial contour plots of a 2D NMR spectrum of BPTI (36”C, 18 mM in *HaO, p*H = 4.7) obtained with the pulse sequence given in fig. la, with p = p’ = n/2 and X = 20 (eq. (12)). Threequantum coherence is selected during tl. The peaks on the skew diagonal (which appears at a slope of 1 because the plot scale ratio is 3: 1) in (a) stem from aII methyl groups in BPTI and are Iabelled accordingly. In (b) an enlarged off-diagonal multiplet stemming from coherence transfer between the methyl- and ar-protons of Ala-58 is shown. The signs of these multiplet components are shown in (c), where only the lowest contour levels have been drawn and painted black for positive signs. The arrows in (a) and (b) indicate positions of cross sections shown in fig. 3. The spectrum was recorded on a Bruker AM-400 spectrometer equipped with a digital phase shifter. 144 scans for each of 800 t 1 values were accumulated, using time-proportional phase incrementation in steps of 30” to separate positive and negative frequencies with the carrier in the center. The spectral width was 10 kHz in wi. For the contour plots the data were multiplied by Gauss-Lorentz window functions in both dimensions (LB1 = -20, GBl = 0.3, LB2 =. -10, GB2 = 0.2) before zero filling to 4K X 4K and twodimensional real Fourier transformation.
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CHEMICAL PHYSICS LETTERS
tential artifacts that might obscure the weak peaks that are expected, several precautions were taken. In addition to the phase cycle for selection of three-quantum coherence, the phase of the inversion pulse (which actually was a (r/2), --(n& -(n/2), sandwich [ 151) was cycled in six steps of IT/~ while the receiver reference phase was kept constant to suppress any spurious coherences of orders lower than six during the interval r. This delay was also varied randomly within ~5% for each t1 value to suppress zero-quantum coherences 1161. All peaks centered on the skew diagonal of the spectrum in fig. 2a can be assigned to methyl groups of the amino acid residues alanine, threonine, valine , leucine , isoleucine and methionine. All methyl groups in BPTI give rise to such “skew diagonal peaks”. They are labelled according to their assignments as given by Chazin et al. [17]. An in-phase doublet structure along w2 is evident for the Ala, Thr, Val and Leu peaks. Cross sections of the Met-52 singlet along the two axes are shown in figs. 3a and 3b. No window functions have been applied to the raw data in this case to obtain undistorted peakshapes. Sidelobes with signs opposite to the central lobe, indicating bi-Lorentzian lineshape [4,6,7], are clearly visible in both dimensions. As x = 20 was used, the true linewidths corresponding to multiexponential T1 behaviour are only l/20 of the apparent values (eq. (13)). For a quantitative determination the lineshape needs to be deconvoluted with the Lorentzian shape resulting from the exponential decay of the three-quantum coherence (eq. (13)).
7 November 1986
Several off-diagonal peaks are also visible in fig. 2. Most of these can be explained by coherence transfer to protons coupled to the multi-exponentially relaxing CH3 groups. During the period t1 the X3(S) threequantum coherence in an AX, system precesses under the influence of the coupling to the A nucleus with I= I/2. In terms of tensor operators
[6] this is
2nJt, I,S,
TS3,*3 -
T&3 cos(3nJtl)
- 21/2iTfOT&3
sin(3nJtl)
(14)
in addition to the Zeeman precession described by eq. (6). It is now possible to transfer the antiphase threequantum coherence of group spin S to a one-quantum coherence of spin I by the rf pulse 0’:
l’&,T&3 5
~T~,_,T&(5/S)1~2
sin4$.
(15)
In contrast to the third-rank tensor on the righthand side of eq. (7) with I = 3, this tensor product is observable as an anti-phase I-spin quartet. These signals are analogous to the “remote peaks” in conventional multiplequantum NMR [ 181. Their phase is orthogonal to that of the peaks on the skew diagonal and they have the following multiplet structure: -W
1
+-I -3
-1
t3
+3 -1 02. -3 +1 4
(16)
IOOHz
Fig. 3, Cross sections of selected peaks in the spectrum of fig. 2. For these plots the spectrum was re-transformed without window multiplication. In (a) the vertical cross section of the Met-52 singlet exhibits sidelobes, of a sign opposite to the central lobe, that are the obvious manifestations of multi-exponential longitudinal relaxation during 7. The horizontal cross section (b) is also flanked by such lobes, which however reflect multiexponential transverse relaxation. In (c) the antiphase doublet structure of the cross peak in fii. 2b in the WI dimension can be seen. The sidelobes are due to the longitudinal relaxation functionf(!i(T).
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CHEMICAL PHYSICS LETTERS
A practical example is shown in fig. 2b: The peak centered at o1 = 3S$, o2 = 52,, with OL,p indicating the 1yand methyl protons of alaninein BPTI, exhibits the multiplet structure predicted by eq. (16). The spectrum has been phase-corrected by n/2 in both dimensions for this enlarged plot. In the cross section along w1 in fig. 3 one can also see the sidelobes characteristic of longitudinal multi-exponential relaxation. Further examples of such remote peaks can be found in fig. 2a for Be-18 and Ile-19. It must be emphasized that although multiexponential longitudinal relaxation is necessary for these remote peaks to occur, multi-exponential transverse relaxation is not involved, in contrast to the skew-diagonal peaks in a MERCY spectrum.
7 November 1986
menting on an earlier version of the manuscript. A gift of Trasylol@ , (BPTI) obtained from Bayer AC, Vienna (Austria), is gratefully acknowledged.
References
111A.D. McLachlan, Proc. Roy. Sot. A280 (1964) 271. 121 P.S. Hubbard, J. Chem. Phys. 53 (1970) 985. [31 U. Piantini, O.W. Sbrensen and R.R. Ernst, J. Am. Chem. Sot. 104 (1982) 6800.
141 N. Mtiller, G. Bodenhausen, K. Wiithrich and R.R. Ernst, J. Magn. Reson. 65 (1985) 531.
[51 M. Rance and P.E. Wright, Chem. Phys. Letters 124 (1986) 572.
[61 N. Miiller, G. Bodenhausen and R.R. Ernst, to be published ,
[71 G. Jaccard, S. Wimperis and G. Bodenhausen, to be published.
4. Conclusion
[81 A.A. Ribeiro, R. King, C. Restive and 0. Jardetzky, J. Am. Chem. Sot. 102 (1980) 4040.
The new multiple-quantum 2D NMR method described in this Letter provides a means of monitoring multi-exponential relaxation. Skew-diagonal peaks exhibit lineshapes characteristic of multi-exponential T1 and T2 relaxation functions along the two frequency axes. The appearance of remote peaks in coupled spin systems requires only longitudinal relaxation to be non-exponential. Together with related methods [6,7] this may provide novel access to relaxation parameters.
[91 W.P. Aue, E. Bartholdi and R.R. Ernst, J. Chem. Phys. 64 (1976) 2229.
I101 P_L. Corio, Structure of high resolution NMR spectra (Academic Press, New York, 1966).
[Ill D.M. Brink and G.R. Satchler, Angular momentum (Oxford Univ. Press, Oxford, 1968).
[I21 G. Bodenhausen, H. Kogler and R.R. Ernst, J. Magn. Reson. 58 (1984) 370.
1131 N. Murali and A. Kumar, Chem. Phys. Letters 128 (1986) 58.
[I41 G. Bodenhausen and R.R. Ernst, J. Am. Chem. Sot. 104 (1982) 1304.
I151 M.H. Levitt and R. Freeman, J. Magn. Reson. 33 (1979) 473.
Acknowledgement
I161 S. Macura, Y. Huang, D. Suter and R.R. Ernst, J. Magn.
The author is indebted to G. Bodenhausen for ample access to the Bruker AM400 spectrometer at the University of Lausanne (Switzerland), for his continued interest and discussions concerning this work, and for a preprint of ref. [7] as well as to R.R. Ernst for com-
1171 W.J. Chazin, DP. Goldenberg, T.E. Creighton and
Reson. 43 (1981) 259. K. Wiithrich, European J. Biochem. 152 (1985) 429.
[181 L. Braunschweiler, G. Bodenhausen and R.R. Ernst, Mol. Phys. 48 (1983) 535.
223
Volume 131, number 3
OBSERVATION LONGITUDINAL Norbert Insiitut
OF MULTI-EXPONENTIAL AND TRANSVERSE RELAXATION
IN ‘IWO-DIMENSIONAL
7 November 1986
NMR
MijLLER
fiuChemie, Johannes Kepler Unrversrttit, A-4040 Linz, Austria
Received 21 August 1986
In a spin system that exhibits multi-exponential longitudinal relaxation, multiplequantum coherence may be created by a n/2 pulse acting on a state of partially recovered polarization. This, together with recently discovered violations of COherence-transfer selection rules through multiexponential transverse relaxation, allows one to design a new twodimensional NMR method. The shapes of the peaks situated on the skew diagonal of the spectrum are determined by longitudinal and transverse relaxation functions along the wt and wp axes, respectively.
1. Introduction Multi-exponential relaxation has long been known to occur in systems with equivalent spins [l] or quadrupolar nuclei [2]. Coherence-transfer selection rules [3] have recently been shown to break down because of unequal decay of degenerate coherences [4-61. This has stimulated efforts to apply two-dimensional (2D) multiplequantum NMR for the detection of multi-exponential relaxation [6,7]. Such an approach should be superior to one-dimensional (1D) NMR methods such as inversion recovery and spin echo techniques in cases where signal overlap prevents complete interpretation of 1D spectra, for example in spectra of large molecules. The correlation times that become accessible through the relaxation parameters may provide insight into molecular dynamics [6-81.
2. Theory The new pulse sequence shown in fig. 1 may be interpreted in two different ways, First, it may be regarded as a 2D correlation spectroscopy (COSY) pulse sequence [9], preceded by inversion and partial relaxation. In contrast to ordinary correlation spectroscopy, however, the experiment is designed to monitor the evolution of multiple-quantum coherence, which in this case is not created through other coher218
Fig. 1. (a) Pulse sequence and (b) coherence transfer pathways for multiexponential relaxation spectroscopy (MERCY). The phases rpand up’are cycled together with the receiver reference phase to select multiplequantum coherence of order p. The special case of three-quantum order selection is shown in (b). The variation of 7 is described by eq. (12).
ences but, as is seen below, from types of longitudinal spin order other than Zeeman order. Second, the sequence can be regarded as a modified inversion recovery experiment, where a multiple-quantum evolution time t1 and an observation pulse p’, that converts multiplequantum into onequantum coherences, have been added. The transformations occurring under the influence of this pulse sequence will be discussed with particular attention to X, spin systems and implications for AX, 0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CHEMICAL PHYSICS LETTERS
Volume 131, number 3
spin systems (neglecting cross-relaxation between the A and X spins for simplicity). The experiment is, however, applicable to any system that exhibits both longitudinal and transverse multi-exponential relaxation. (In the presence of resolved spin-spin couplings non-exponential decay of longitudinal magnetization is sufficient for certain peaks to occur, as will be seen below.) An extensive account on related methods applied to quadrupolar nuclei is given by Jaccard et al.
7 November 1986
j$‘(~ = 0) = 1, while the other relaxation functions consist of linear combinations of exponentials with amplitudes such that @(7 = 0) = 0 for I > 1 [4,6,7]. The tensor terms of second and third ranks (I > 1) are transferred into coherences of orders Ip I = 1, .... I by the radio-frequency pulse /3 of phase cpin fig. 1, while their ranks I are conserved. Using the elements dLo of the Wigner rotation matrices [ 1 l] * this transformation is described by [6]
171. The three magnetically equivalent protons of a methyl group can be described in a symmetry-adapted basis [IO], where the spin system is represented as a superposition of three group spins, two with S = l/2 and one with S = 312. In the absence of symmetry breaking cross-relaxation, only the latter need be considered if one focuses on multiplequantum coherences [6]. The transformations of the density matrix u will be discussed in terms of irreducible tensor operators T$ that have recently been found convenient for handling the effects of multi-exponential relaxation [6,7]. After the initial inversion pulse one can describe the displacement from equilibrium by u((7 = 0) - ueq = -20’122$,
(1)
which represents inverted Zeeman order. Under the influence of a relaxation operator IQ=O) the rank 1 of the tensor changes during the time interval 7 while the coherence order p is conserved. In the notation of refs. [6,7] this is
r(O)7
Go-
c
By a suitable cycle of the phase p the desired coherence level can be selected among the various orders p [ 121. In conventional inversion-recovery experiments Tf _1 is detected at this point in the pulse sequence. Ih a COSY experiment on the other hand, +l quantum coherences would be selected in the evolution period by the phase cycle. Murali and Kumar [13] have recently attributed spurious peaks in COSY spectra to multiple-quantum coherences created from incompletely recovered polarization by similar pathways. In the experiment discussed here multiplequantum coherence is allowed to precess during t 1. For methyl groups the relevant terms comprise only two- and three-quantum coherences. The latter can only be created from third-rank tensor terms, while twoquantum coherence is generated both from second- and third-rank tensors, as can be seen from the explicit forms of eq. (3) (neglecting terms with Ip I <2):
T$,$;tO,
0<1<3
where Tfo and Tfo represent different forms of longitudinal spin order and are analogous to quadrupolar and octupolar orders in I = 3/2 nuclei. The functions @(r) describe the build-up and decay of the tensor terins T&. In order to keep the discussion as general as possible - but excluding symmetry breaking crossrelaxation -we will not assume any particular relaxation mechanism but treat an arbitrary 4 X 4 relaxation matrix I’(o) with three different negative eigenvalues RI, R,, and Rg (one eigenvalue,Ro, is always zero). The functionsfif) will therefore generally be triexponential. In many cases bi-exponential behaviour is found to a good approximation [6,7]. The coefficients of all exponentials in #(T) have equal signs and
P GO
-
T&2
(3/8)1j2 si&,
P
@O -
Tf, * 2 (1 5/8)li2 sin20 cos fl T T&3
(5/16)112 sin3&
(5)
Selecting threequantum coherence during tl allows one to focus on third-rank tensor operators. However, while it is possible to select twoquantum coherences of second rank by setting 0 equal to n/2, there is no obvious way to select Tf k2 without interference from Tf f 2. Zeeman prec&sion changes neither the rank nor * The relevant elements in this context have been compiled in ref. [6]. 219
CHEMICALPHYSICS LETTERS
Volume I3 1, number 3
the coherence order of the tensors [6]. During tl the Hamiltonian simply leads to the transformation n&S1
Ti$-
T& exp(-ipaStL)
t other terms.
(6)
This precession labels the signals with the multiplequantum frequency pS$ in the tl domain. The last pulse /3’converts multiplequantum coherence (among others) into observable +l quantum coherence, i.e. in analogy to eq. (3)
P’ 1
T,f,2 T&ld!_ Q@‘) exp[-i(-1
-PM].
(7)
7 November 1986
The relaxation functionsf#(t2) describe the multiexponential behaviour of one-quantum coherences in an analogous manner to the way the functions fif)(r) in eq. (2) describe multi-exponential longitudinal relaxation. In a 2D NMR spectrum obtained via the pulse sequence of fig. 1 the above transformations give rise to peaks situated at the coordinates (wl = pas, a2 = as), i.e. on the skew diagonal of a multiple-quantum spectrum of order p. The existence of such a peak requires that both longitudinal and transverse relaxation be multi-exponential. Therefore the acronym MERCY (multi-exponential relaxation spectroscopy) is sugges-
A(Z = 2, *2 + -1) = sin p’ cos p’,
(8a)
ted, which may be extended to pQ MERCY to indicate the order Ip I of multiplequantum coherence selected. Note that this method applies primarily to singlets without scalar, dipolar or quadrupolar splittings, which were previously considered a prerequisite for the generation of multiplequantum coherence. In order to obtain quantitative information about the multi-exponential longitudinal relaxation functions #(r), the delay r can be varied synchronously with the incrementation of tL, in a similar manner to the accordion experiment [ 141:
@1=2,+2-t-l)=sin$,
(8b)
T=Tmax-Xll.
A(I = 3, +2 + -1) = (S/2)li2 sin /3’cos /I’,
(9a)
If the experiment is run with a constant delay 7, it can only serve to prove the existence of multiexponential longitudinal and transverse relaxation. If r is, however, varied according to eq. (12), the functions .#(r) influence the lineshape along ol. This lineshape function is a convolution of the Lorentzian shape imposed by the exponential decay of the multiplequanturn coherence of order p during tl (with time constant Tip)) and the multi-Lorentzian shape caused by multi-exponential behaviour during the delay 7. The functions@)(r) are scaled with respect to tl by the factor x, which serves to adjust for the different lifetimes of longitudinal spin order and multiplequantum coherence. In the w2 domain the lineshape is given by the Fourier transform of the relaxation functions@(t2) (eq. (1 l)), which are characteristic of multi-exponential transverse relaxation of degenerate onequantum coherences. The lineshape along w2 consists therefore of a superposition of at least two Lorentzians of different signs and widths, which correspond to the eigenvalues of the relaxation operator F(l) [6,7].
In order to obtain pure phase peakshapes, coherence transfer pathways from tp and -p quantum coherences must be selected simultaneously by the phase cycle [ 121, For the particular case of interest one obtains the following transfer coefficients A(Z, kp + - 1) andD(I, &p + -1) for the absorptive and dispersive components of the 2D peaks. By substituting the Wigner matrix elements into eq. (7) and using the procedure outlined in ref. [6], one obtains
D(I = 3, +2 --f -1) = (5/8)Lj2 sin 0 (3 cos2$ - l), (9b) A(1=3,+3+-1)=(15/16)1/2sin2/3’,
(lOa)
D(I=3,+3+-1)=-(15/16)1~2sin2~‘cos~’.
(lob)
From these equations it can be seen that if p’ = a/2 the two-quantum peaks obtained from second-rank tensor terms (eq. (8b)) will be of opposite phase to those stemming from third-rank tensors (eq. (9b)). A flip angle of n/2 also enables one to obtain pure phase peak shapes. The one-quantum tensor operator terms Tf_l (eq. (7)) precess with their characteristic Larmor fiequency during ~2. However, as their ranks exceed one, they are not observable unless conversion to first-rank terms is effected by multi-exponential transverse relaxation (described by the operator F(P=L)) as explained in ref. [6] : I-
q-1 220
(‘)tz
T;,-I~?(Q-
(11)
(12)
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CHEMICAL PHYSICS LETTERS
The two-dimensional time envelope of the free induction decay in such an experiment with threequanturn selection for an uncoupled S = 312 group spin is accordingly:
S(Q) t2) = fg(Tmax - xfl) exp(-tllT~3))fl13((t2).
7 November 1986
3. Experimental verification Multi-exponential transverse relaxation within the S = 3/2 subsystem in a system of three magnetically equivalent spins with I = l/2 can only occur outside the fast-motion limit 161. Therefore experimental evidence was sought by investigating a sample of a small protein (basic pancreatic trypsin inhibitor, BPTI, molecular weight a6000). Fig. 2 shows partial contour plots of a 3Q MERCY spectrum (i.e. three-quantum coherence is selected during fl). In order to suppress po-
(13)
Examples of theoretical lineshapes for the case of biexponential decay can be found in refs. [4,6,7].
r
1
1 ,,P 1 T3; )’
2.0
I18 U”II9119 IOHz
3.98 ppm
30Hz _---__
--Ii
-
$6
A48 4.0
w
A27 A40 A.58
I 9
T’l 1
5.0
A25 T54 6.0
7.0
t 2.4
2.2
a 2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
PPm
-9
Fig. 2. Partial contour plots of a 2D NMR spectrum of BPTI (36”C, 18 mM in *HaO, p*H = 4.7) obtained with the pulse sequence given in fig. la, with p = p’ = n/2 and X = 20 (eq. (12)). Threequantum coherence is selected during tl. The peaks on the skew diagonal (which appears at a slope of 1 because the plot scale ratio is 3: 1) in (a) stem from aII methyl groups in BPTI and are Iabelled accordingly. In (b) an enlarged off-diagonal multiplet stemming from coherence transfer between the methyl- and ar-protons of Ala-58 is shown. The signs of these multiplet components are shown in (c), where only the lowest contour levels have been drawn and painted black for positive signs. The arrows in (a) and (b) indicate positions of cross sections shown in fig. 3. The spectrum was recorded on a Bruker AM-400 spectrometer equipped with a digital phase shifter. 144 scans for each of 800 t 1 values were accumulated, using time-proportional phase incrementation in steps of 30” to separate positive and negative frequencies with the carrier in the center. The spectral width was 10 kHz in wi. For the contour plots the data were multiplied by Gauss-Lorentz window functions in both dimensions (LB1 = -20, GBl = 0.3, LB2 =. -10, GB2 = 0.2) before zero filling to 4K X 4K and twodimensional real Fourier transformation.
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tential artifacts that might obscure the weak peaks that are expected, several precautions were taken. In addition to the phase cycle for selection of three-quantum coherence, the phase of the inversion pulse (which actually was a (r/2), --(n& -(n/2), sandwich [ 151) was cycled in six steps of IT/~ while the receiver reference phase was kept constant to suppress any spurious coherences of orders lower than six during the interval r. This delay was also varied randomly within ~5% for each t1 value to suppress zero-quantum coherences 1161. All peaks centered on the skew diagonal of the spectrum in fig. 2a can be assigned to methyl groups of the amino acid residues alanine, threonine, valine , leucine , isoleucine and methionine. All methyl groups in BPTI give rise to such “skew diagonal peaks”. They are labelled according to their assignments as given by Chazin et al. [17]. An in-phase doublet structure along w2 is evident for the Ala, Thr, Val and Leu peaks. Cross sections of the Met-52 singlet along the two axes are shown in figs. 3a and 3b. No window functions have been applied to the raw data in this case to obtain undistorted peakshapes. Sidelobes with signs opposite to the central lobe, indicating bi-Lorentzian lineshape [4,6,7], are clearly visible in both dimensions. As x = 20 was used, the true linewidths corresponding to multiexponential T1 behaviour are only l/20 of the apparent values (eq. (13)). For a quantitative determination the lineshape needs to be deconvoluted with the Lorentzian shape resulting from the exponential decay of the three-quantum coherence (eq. (13)).
7 November 1986
Several off-diagonal peaks are also visible in fig. 2. Most of these can be explained by coherence transfer to protons coupled to the multi-exponentially relaxing CH3 groups. During the period t1 the X3(S) threequantum coherence in an AX, system precesses under the influence of the coupling to the A nucleus with I= I/2. In terms of tensor operators
[6] this is
2nJt, I,S,
TS3,*3 -
T&3 cos(3nJtl)
- 21/2iTfOT&3
sin(3nJtl)
(14)
in addition to the Zeeman precession described by eq. (6). It is now possible to transfer the antiphase threequantum coherence of group spin S to a one-quantum coherence of spin I by the rf pulse 0’:
l’&,T&3 5
~T~,_,T&(5/S)1~2
sin4$.
(15)
In contrast to the third-rank tensor on the righthand side of eq. (7) with I = 3, this tensor product is observable as an anti-phase I-spin quartet. These signals are analogous to the “remote peaks” in conventional multiplequantum NMR [ 181. Their phase is orthogonal to that of the peaks on the skew diagonal and they have the following multiplet structure: -W
1
+-I -3
-1
t3
+3 -1 02. -3 +1 4
(16)
IOOHz
Fig. 3, Cross sections of selected peaks in the spectrum of fig. 2. For these plots the spectrum was re-transformed without window multiplication. In (a) the vertical cross section of the Met-52 singlet exhibits sidelobes, of a sign opposite to the central lobe, that are the obvious manifestations of multi-exponential longitudinal relaxation during 7. The horizontal cross section (b) is also flanked by such lobes, which however reflect multiexponential transverse relaxation. In (c) the antiphase doublet structure of the cross peak in fii. 2b in the WI dimension can be seen. The sidelobes are due to the longitudinal relaxation functionf(!i(T).
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A practical example is shown in fig. 2b: The peak centered at o1 = 3S$, o2 = 52,, with OL,p indicating the 1yand methyl protons of alaninein BPTI, exhibits the multiplet structure predicted by eq. (16). The spectrum has been phase-corrected by n/2 in both dimensions for this enlarged plot. In the cross section along w1 in fig. 3 one can also see the sidelobes characteristic of longitudinal multi-exponential relaxation. Further examples of such remote peaks can be found in fig. 2a for Be-18 and Ile-19. It must be emphasized that although multiexponential longitudinal relaxation is necessary for these remote peaks to occur, multi-exponential transverse relaxation is not involved, in contrast to the skew-diagonal peaks in a MERCY spectrum.
7 November 1986
menting on an earlier version of the manuscript. A gift of Trasylol@ , (BPTI) obtained from Bayer AC, Vienna (Austria), is gratefully acknowledged.
References
111A.D. McLachlan, Proc. Roy. Sot. A280 (1964) 271. 121 P.S. Hubbard, J. Chem. Phys. 53 (1970) 985. [31 U. Piantini, O.W. Sbrensen and R.R. Ernst, J. Am. Chem. Sot. 104 (1982) 6800.
141 N. Mtiller, G. Bodenhausen, K. Wiithrich and R.R. Ernst, J. Magn. Reson. 65 (1985) 531.
[51 M. Rance and P.E. Wright, Chem. Phys. Letters 124 (1986) 572.
[61 N. Miiller, G. Bodenhausen and R.R. Ernst, to be published ,
[71 G. Jaccard, S. Wimperis and G. Bodenhausen, to be published.
4. Conclusion
[81 A.A. Ribeiro, R. King, C. Restive and 0. Jardetzky, J. Am. Chem. Sot. 102 (1980) 4040.
The new multiple-quantum 2D NMR method described in this Letter provides a means of monitoring multi-exponential relaxation. Skew-diagonal peaks exhibit lineshapes characteristic of multi-exponential T1 and T2 relaxation functions along the two frequency axes. The appearance of remote peaks in coupled spin systems requires only longitudinal relaxation to be non-exponential. Together with related methods [6,7] this may provide novel access to relaxation parameters.
[91 W.P. Aue, E. Bartholdi and R.R. Ernst, J. Chem. Phys. 64 (1976) 2229.
I101 P_L. Corio, Structure of high resolution NMR spectra (Academic Press, New York, 1966).
[Ill D.M. Brink and G.R. Satchler, Angular momentum (Oxford Univ. Press, Oxford, 1968).
[I21 G. Bodenhausen, H. Kogler and R.R. Ernst, J. Magn. Reson. 58 (1984) 370.
1131 N. Murali and A. Kumar, Chem. Phys. Letters 128 (1986) 58.
[I41 G. Bodenhausen and R.R. Ernst, J. Am. Chem. Sot. 104 (1982) 1304.
I151 M.H. Levitt and R. Freeman, J. Magn. Reson. 33 (1979) 473.
Acknowledgement
I161 S. Macura, Y. Huang, D. Suter and R.R. Ernst, J. Magn.
The author is indebted to G. Bodenhausen for ample access to the Bruker AM400 spectrometer at the University of Lausanne (Switzerland), for his continued interest and discussions concerning this work, and for a preprint of ref. [7] as well as to R.R. Ernst for com-
1171 W.J. Chazin, DP. Goldenberg, T.E. Creighton and
Reson. 43 (1981) 259. K. Wiithrich, European J. Biochem. 152 (1985) 429.
[181 L. Braunschweiler, G. Bodenhausen and R.R. Ernst, Mol. Phys. 48 (1983) 535.
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