Scaling in one and two dimensional NMR spectroscopy in liquids

SCALING IN ONE AND TWO DIMENSIONAL SPECTROSCOPY IN LIQUIDS R. V. HOSUR Chemical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India (Received 10 August 1989) CONTENTS 1. Introduction

Two Dimensional NMR Spectroscopy :: Theoretical Concepts

4. 5.

6.

I.

3.1. Scaling of NMR parameters 3.2. Average Hamiltonian theory 3.3. Product operator formalism for the density operator 3.3.1. Basis sets 3.3.2. Evolution of product operators 3.3.3. Effect of RF pulses 3.3.4. Observables One Dimensional Scaling Experiments 4.1. Off-resonance heteronuclear decoupling 4.2. Concertina effect Two Dimensional Scaling Experiments 5.1. 2D J-resolved spectroscopy 51.1. Homonuclear case 51.2. Heteronuclear case 5.2. 2D correlated spectroscopy 52.1. COSY spectra 5.2.I. I. Basic pulse sequence 5.2.1.2. Scaling modifications 5.2.1.3. Relaxation effects 5.2.2. 2D NOESY spectra 5.2.2.1. Basic pulse sequence 5.2.2.2. Scaling modifications 5.3. 2D Multiple quantum spectroscopy 5.3.1. Basic pulse sequence 5.3.2. Scaling modifications Applications of 2D Scaling Experiments 6.1. Observation of long range correlations 6.2. Measurement of coupling constants 6.3. Measurement of exchange rates Summary Acknowledgements References

: 4 4 as 8 9 10 11 II II 14 17 18 I8 IQ 21 22

22 25 34 36 :: 39 39 41 43 43 44 48 50

1. INTRODUCTION The concept of scaling in nuclear magnetic resonance (NMR) spectroscopy dates back to the continuous wave (CW) era. When high radiofrequency (RF) powers were used for observation of multiple quantum transitions (MQT), it was noticed that multiple quantum transition frequencies are scaled down by a factor equal to the difference in the total magnetic quantum numbers of the two concerned levels (say ‘a’, ‘b’) giving rise to the multiple quantum transition!‘-‘) this factor is called the coherence order pob. Thus the MQTs are actually observed in the same spectral region as the single quantum transitions (SQT). It was also observed that the multiple quantum line widths l/T, and the

2

R. V. HOSUR

spin lattice relaxation rates l/T, are also scaled by the same factor p; the line widths are scaled down while the spin lattice relaxation rates are enhanced. However, in these experiments, the experimenter has no control over the scaling factor p. which is fixed by the MQT being observed. The scaling of the parameters observed is thus not intentional, but rather a natural outcome of the experiment. In other words, it is not possible, for instance, to scale double quantum transitions by a factor 3, and vice versa. In double resonance experiments, spin decoupling in liquids(s-9) is a special case of scaling wherein the spin-spin coupling constants are completely eliminated in certain portions of the NMR spectrum. The decoupling is a selective phenomenon and only a few coupling constants can be eliminated at one time. The technique is extremely useful in identifying resonance frequencies of coupled spins and the simplifications of multiplet patterns achieved, due to removal of some coupling constants, allowing an estimation of the remaining spin-spin coupling constants. In heteronuclear decoupling experiments (e.g. “C-‘H), off-resonance decoupling leads to scaling of most 13C-‘H coupling constants in the “C NMR spectrum; the scaling factors for different “C multiplets are different depending upon the relative positions of the ‘H multiplets in the ‘H NMR spectrum with respect to the irradiation frequency. This technique is extremely useful for assignment of r3C spectra from the knowledge of ‘H assignments. It also results in an increase in the intensities of 13C multiplets because of the elimination of long range 13C-‘H coupling constants, which are usually buried in the linewidths and contribute to increased linewidths in the 13C multiplets. Scaling down of the one-bond “C-‘H coupling constants also results in substantial enhancement of the separation between the multiplets in the 13C NMR spectrum. The discovery of Fourier transform (FT) NMR in 1966(131added another dimension to the NMR methodology. As against the sequential excitation and detection of the individual resonance frequencies in the swept CW methods, in FTNMR, all the frequencies are excited simultaneously by the application of an intense RF pulse and the signal is detected as the transverse magnetisation created by the RF pulse relaxes back to the equilibrium state. Apart from the sensitivity enhancements achieved by this process, which is of immense value to the chemist community, a significant aspect of this development is that the excitation of the nuclear spin system and the detection of its response are separated in time. It was shown that this signal which is collected in the complete absence of any perturbation and hence is referred to as a free precession signal, or free induction decay (FID), is a Fourier pair of the conventional NMR spectrum.“” Another major advantage of FT NMR is that the Fourier transforms can be calculated numerically by use of computers which can also be used for discrete acquisition of the time domain data. This implies that the signal is sampled and digitized and the time between two data points is available for manipulation of the effective nuclear spin Hamiltonian. The fact that the FID and the NMR spectrum are Fourier pairs, a relation which is purely mathematical in nature, suggests the possibility of manipulation of the time domain data (FID) to achieve specific results in the frequency domain (NMR spectrum). Thus several window functions have been devised for multiplication of the FID data so that after Fourier transformation, the resonances have narrower lines (linewidth scaling), or a higher signal to noise ratio etc. (Is) These procedures are in a sense scaling procedures through data processing. The use of computers in NMR spectrometers for precise data manipulations results in significant improvements in the quality of NMR spectra. As a consequence of the separation of the excitation of the spin system and the detection of the response along the time axis of the FT NMR experiment, additional time periods began to be introduced between the two periods, and this led to the birth of the modern multiple-pulse experiments. The spin system can be prepared in any desired type of non-equilibrium state and the behaviour of these systems can be studied in great detail. The fact that during the intermediate periods no signal is detected opens up the possibility of a variety of manipulations of the nuclear spin Hamiltonian, scaling being one of these. Scaling of chemical shifts has been observed in several multiple pulses experiments designed to remove dipolar broadening from NMR spectra of solids.“‘j- ‘s) This fact led to the development of a pulse sequence for exclusive scaling of chemical shifts in liquid samples by arbitrary factors.ir9J The ideas were further used to eliminate the offset dependence of J-scaling factors in heteronuclear off-resonance decoupling experiments. (2o-22) Several reviews have been written to date covering various aspects of multiple pulse experiments in liquids and in solids.‘23-37’ The present review is restricted to scaling procedures in one and two dimensional NMR experiments on liquids.

Scaling in 1D and 2D NMR spectroscopy

3

The organisation of this review is as follows. Section 2 gives a brief account of 2D NMR spectroscopy. Since scaling is a technique which has to do with manipulations of the nuclear spin Hamiltonian, we describe in Section 3 the principles of scaling and the mathematical tools, namely; (a) average Hamiltonian theory(38*39’ which is best suited for the calculation of average effects of a series of perturbations of the nuclear spin system, and (b) the product operator formalism~40-42~ of the density operator description of the nuclear spin system. In the subsequent sections various scaling experiments and their applications will be described. Most of the illustrations in this review are spectra of deoxyribonucleic acid (DNA) segments.* 2. TWO DIMENSIONAL

NMR SPECTROSCOPY

Two dimensional NMR spectroscopy is by far the most significant of the multiple pulse experiments(43*‘4) and has led to results which could not have been obtained otherwise, the foremost of these being the detailed solution of structures of biological macromolecules such as proteins and nucleic acids. The principles and most modern developments in two dimensional NMR spectroscopy have been described elegantly in recent books and reviews. (*s-51) Applications to structure determination of biological macromolecules have also been described in recent books’sO-sL) and review articles.‘S3-76’ For the sake of completeness and because of its relevance to what follows later in the description of scaling we give below a brief description of 2D NMR techniques in general. The basic experimental scheme of 2D NMR spectroscopy is shown in Fig. 1. The time axis is divided, in general, into four time periods labelled as preparation, evolution, mixing and detection. The evolution (t,) and the detection (t2) periods are the two running time variables leading to the time domain signal S(t,, fz). As the name suggests, data is collected only during the period t,. which means several sets of data are collected by systematically incrementing the evolution period t,. Two dimensional Fourier transformation of the time domain signal yields the 2D NMR spectrum W‘, 4 The mixing period between the evolution and the detection periods is a crucial time segment which helps to establish correlations between the spins evolving in the t, and tl periods. Thus the effects of a perturbation occurring during t, can be detected indirectly by observing the modulation of the first data point in t2, as t, is incremented systematically during the course of the experiment. Therefore the evolution period may be called the ‘indirect detection period’ of the 2D experimental scheme. In most cases the mixing period is a constant time period, there are a few cases to be discussed latter, where the mixing period is varied synchronously with f, or randomly, so as to achieve specific results. Finally, the preparation period is designed to create the spin system in a suitable non-equilibrium state. For example, in a 2D double or zero quantum experiment, the preparation period is used to create double or zero quantum coherences which evolve with their characteristic frequencies in the subsequent evolution period.

DETECTION

FIG. 1. Segmentation of the time axis for multipulse experiments. Tie domain signal is collected during the ‘detection period’. The ‘evolution period’ generates the second time variable in two dimensional NMR experiments. The ‘preparation period’ is used to create the spin system in a suitable non equilibrium state and during the mixing period, transfer of magnetisation occurs between nuclear spins.

* Deoxyribonucleic acid (DNA) is a biological macromolecule built from repeating units of the type shown in Fig. 37. The chain consists of a sugar-phosphate backbone and to each sugar is attached a base. There are four types of bases namely cytosine (C), adenine (A), guanine (G) and thymine (T) occurring in DNA. Two such long chains are held together by H-bonds between bases on opposite strands. A pairs with T and G pairs with C in the double stranded structure.

4

R. V. HOSUR

Each of the above mentioned time periods is amenable to a variety of perturbations, the vast majority being by RF pulses or pulse trains. It is thus easy to guess the enormity of 2D NMR techniques one could generate, and during the last decade there has really been an explosion in this area with numerous applications in chemistry, biology and medicine. Two dimensional NMR techniques can be grouped into three broad categories, according to Ernst and coworkers.(45) (i) Techniques dealing with separation of different kinds of interactions between spins; these can be pooled under ‘2D resolved spectroscopy’. (ii) Techniques which reflect correlations between spins; these are pooled under ‘2D correlation spectroscopy’. (iii) Techniques which enable observation of multiple quantum transitions in a given spin system. This classification is by no means exclusive and techniques have been devised which make use of combinations of each of these principles. For example, double quantum filtered COSY makes use of principles of double quantum excitation in correlation spectroscopy. 3. THEORETICAL

CONCEPTS

3.1. Scaling of NMR Parameters The observables in an NMR spectrum are the resonance line frequencies and the line intensities. These have to be analysed to derive the relevant NMR parameters of the nuclear spin system, such as the chemical shift (6) the spin-spin coupling constant (J), the dipolar coupling constant (D) in solids and in liquid crystals, quadrupolar coupling constant (Q), spin-spin relaxation times (T,) related to the linewidths of the signals, etc. The analysis involves calculation of the various line frequencies and line intensities for the nuclear spin system starting from a knowledge of the nuclear spin Hamiltonian operative during the course of detection of the NMR signal. ‘77-80’ Thus, in an FT NMR experiment, the information content of the NMR spectrum is determined by the Hamiltonian operative during the free precession of the nuclear spins following the application of the RF excitation pulse. Of course, the intensities of the signals also depend upon whether the system was in an equilibrium state or not, prior to the application of the RF pulse. For example, selective saturation of the transitions before the application of a non-selective RF pulse leads to the so-called nuclear Overhauser effect (NOE),‘s’) which has been extremely useful in resonance assignment and structure determination of molecules in solution. For our further discussion in this section we will assume that the experiment is performed by starting from an equilibrium state of the spin system. More specifically, the populations of the various energy levels obey Boltzmann statistics and there are also no coherences between the spins in the different energy levels. Let .%’be the total Hamiltonian of the system operative during the course of detection of the signal, i.e. between every two sampling points of the FID, the spin system evolves identically under the influence of the Hamiltonian %‘. If .%’consists of the contributions Ho, H,, X, . . . etc. .#=.#o+~I+.W~+...

(1)

which represent the different interactions such as the Zeeman interaction with the external magnetic field, spin-spin interactions through the electrons, direct dipole-dipole interactions between spins, etc. then the NMR spectrum will reflect these NMR parameters. If by some means the Hamiltonian &’ is modified to W .W=&o+/Wr+5).@~+...

(2)

between every two sampling points, then the different interactions will be scaled by the factors x, A 7, etc. and will be reflected in the NMR spectrum. This is the general principle OTscaling and is shown schematically in Fig. 2.; z, 8, 7 . . . are the scaling factors for the different interactions. A variety of manipulations leading to scaling effects are possible and these are of two types as shown in Fig. 3. In one case, the spin system is subjected to a series of RF pulses of varying flip angles and phases and these affect the different interactions differently, effectively modifying the NMR parameters. In the

Scaling in ID and 2D NMR spectroscopy

5

time +

1

Scaling

FIG. 2. The generalised principle of scaling of NMR parameters. The Hamiltonian 3’ during each time interval between two successivetime domain data points is modified to a new Hamiltonian X’ which represents scaled NMR parameters.

second approach the interactions are separated in time by suitable strategies so that different components of the Hamiltonian are operative for different lengths of time. The first approach is used extensively in decoupling experiments by composite pulses@*-s*) while the second one is used in differential chemical shift and J-scaling experiments. The information content of an NMR spectrum obtained in the presence of a Hamiltonian JEPcan be obtained by calculating the evolution of the density operator u(t), which satisfies the equation b(t) = - i[Juo, u(r)]. If the Hamiltonian

Z is explicitly independent

(3)

of time, then the solution of (3) is given as

a(r) = e -

Mr

g(O)

eiJVt

(4)

The manipulations mentioned above can be described by an average Hamiltonian which will be identified with .P discussed above. For example for the situation in Fig. 3b, the solution in eqn. (4) is written

as u(t) = e - $X&r)

= e - i[oJy, +

+ M,(br)

+ ..

] ,,(o)

ei[J4,(ar)

+ H,(br)

bJP, + . . .]r ,,(o) ei[aJP, + bJP, +

_ e - iJY’r a(o) eiWr.

. . ]r

+

.

]

(3 (6)

(7)

Thus, the task of understanding the performance of a particular experiment involves calculation of an average Hamiltonian and the evolution of the density operator through the experiment under the influence of the average Hamiltonian. Both the average Hamiltonian theory and the density operator description of NMR have been elegantly described in recent books(45*4” and we will give brief accounts of these so that we can use the concepts later when describing the different scaling experiments. 3.2. Average Hamiltonian Theory If the Hamiltonian is being manipulated as a function of time in order to achieve specific results, then one has to deal with a time dependent Hamiltonian X(t) for calculation of the evolution of o(t)

R. V. HOSUR

6 (a)

(b)

k’=

o&,+ bZ1 +

Z’t= ito(

C&+

-----

-

,%!,(bt)+X,(ct)+----

FIG. 3.Two ways of achieving scaling of NMR parameters: (a) during each time interval (DW) a series of radio frequency pulses are applied in such a way that the spin system undergoes several transformations and returns, at the time of data point collection, to a state as though it were evolving under the influence of an average Hamiltonian .#I”’which has the various interactions scaled by different factors u, b. c, etc. (b) The different constituent interactions of the Hamiltonian are made to operate for different lengths of time during the dwell time (DW) of data collection. The time durations of the individual interactions may or may not be completely exclusive. i.e. during a part of cr(P”, evolution time), S’, evolution may also be occurring, etc. This depends on to what extent the different interactions can be separated. The approach ‘a’ has been most extensively used for removing dipolar coupling in solid state NMR spectra and in ID scaling experiments in liquids. The latter approach ‘b’ has been extensively used for 2D experiments in liquids.

in eqn. (1). However, if the time interval between two consecutive sampling points of the NMR experiment can be divided into several time intervals, TV,72, . . . rk during which the Hamiltonian can be assumed to be constant, .#(t) = H, for (T* + 72 + . . . rk- ,) < t < (7, + rz + . . . Tk)

(8)

the solution of eqn. (3) is then written as c(t) = v(r,) 40) WC)- l

(9)

with U(t,) = exp( - iH,r,) . . . exp( - iH,rl)

w

and t, = f

=k.

k=f t,

is called the cycle time or the period of the Hamiltonian.

(11)

7

Scaling in tD and 2D NMR spectroscopy U(t,) in eqn. (10) is a unitary transformation ation under an average Hamiltonian k

and can be expressed in terms of a single transform-

U(t,) = exp( - ikt,).

(12)

U(t,) can also be expressed by a Magnus expansion by explicitly expanding each of the exponentials in eqn. (10): U(r,)=

1 + (-i)[H,r,+ H2f2 + ...H,r,] (- i)' +T[(H:~:

+ H:r; + ...H.T:)

+Z(H,H,r,r, +H,H,r,r, + ...H,H,_lr,r,_,)]

+ 3(H,H:t,r:+ H:H,T:~, + . ...) + 6(H,H,H,r,r,r, + H,H,H2s,s,s,+ . ..)I.

(13)

Alternatively, it is also written as U(r,) = 1 + ( - i)(H,r, + . . . H,r,)+q(CH,r,, + [H,t,,H,r,]+

+

. a.)+

H,s,l + CH,s,.H,r,l

Ha?,,CH,r,vfhl]

;[H,*,,CH,r,, &?,I] +;[CH,r,,

+ [CW,, H,4, HA]

H,r,],H,:,] + . . .) + . . . .

(14)

From eqns (12) and (14) one obtains 2 I j@Jr + j@lJ + @2, + . . .

(1%

where

*co,

= ; [H,T~ c

j@(l) =

+ H,r,

+ . . . H,,TJ

(16)

geez*z, -

*

c

H

H,*,l + CH,r,, H,r,l + . . 4

*(*’ = $([H,*,.[Hz*,, H,4] + [C&r,, H,?,l, HA] c

+ fCH,Mf,r,,

H,s,l] + $CH,r,. H,r,l, i-b,] + . . .).

(17)

The terms $7(ot, k”), +(2). . . represent contributions of different orders to the average Hamiltonian k. If the individual Hamiltonians (H,) in different time intervals commute with each other, then one obtains the most simple expression for the average Hamiltonian: 2 = 2(O).

(18)

In this review, we will be dealing most often with this approximation for the average Hamiltonian. A general condition for the existence of a time independent average Hamiltonian for any sequence of aperiodic perturbations and free evolutions has been worked out. (*‘I Consider the general sequence of perturbations shown in Fig. 4. The total evolution period f is divided into several segments of durations a,7 separated by RF pulses Ri.

R. V. HOSUR

FIG. 4. A general&d scheme of aperiodic perturbations which can be treated by average Hamiltonian theory. Radio frequency pulsesR, are applied sequentially, separated by evolution periods q T,where 5 representsthe cycle time of the perturbation. The Hamiltonian during each time interval can be different.

5 =

[EOi =I]

i$,air;

(19)

During each segment, the Hamiltonian Hi is assumed to be constant. If u(0) is the density operator at the beginning of the evolution period, then the density operator a(r) at the end is given by U(T) = e -iHna,.rRZ,-iHza*fR, e-iH~ntr a(~) eiHlolr R; I eiH20*r,. . eiH,a.rs (20) This can be rearranged as u(~)=e-~~;~“~[e-~H:-,a”-,r..

. e-iH;o,rR,_,

. . . R, a(O) R;’

. . . R;?, ei*;-,~.-t~]

ei*Lv

(21)

with H;=R,-,R,_,...R~+,R,H,R;‘...R;:l.

(22)

Defining U’(O)= R,_,

. . . R,u(O) Ri’

. . . Rcl,

(23)

eqn. (21) can be written in a compact form as a(r) = e - iH;a,r_ . . e - iH;a,r u’(O) eiH;a, r. . . eiH;cl,r with

(24)

H:,=H,.

(25)

Equation (24) is similar to eqn. (9) in form but with the Hamiltonians and the initial density operator transformed by the various RF pulses. Therefore the average Hamiltonian can again be written as &+(r) = j@O) + *+)

+ ...

(26)

with j+‘) jpw

= - + [H;a,, ‘1

H;a,]

= t H; ai ill

+ [H;a,,

H;a,]

This suggests that a time independent average Hamiltonian Hamiltonians commute with each other.

Then one obtains,

(27) + [H;a,,

H; a2] + . . .} .

(28)

can be defined if all the transformed

[Hj, H;] = 0.

(29)

*r = k$‘O’.

(30)

Whenever such a suitable average Hamiltonian cannot be defined, time evolution operator will have to be calculated explicitly at every step of the pulse sequence.

of the density

3.3. Product Operator Formalism for the Density Operator 3.3.1. Basis Sets. Explicit calculation of the density operator evolution in a complex multipulse experiment can become a very tedious and cumbersome process if matrix notations are used for pulses

Scaling in 1D and 2D NMR spectroscopy

9

and density operators. Simplifying procedures have therefore been devised’*0-42*45’ whereby the calculations can be performed in symbolic notations and in a more convenient manner. The density operator is expressed as a sum of a set of basis operators which constitute a complete set: (31) where B, are the basis operators and b,(t) are the respective coefficients. The basis operators are constructed either from the Cartesian angular momentum operators IX, I,, I, etc. for individual spins, or from raising, lowering, polarisation operators, etc. These treatments are applicable to weakly coupled spin systems, but a few developments applicable to strong coupling situations have appeared in the literature.*9 In the case of Cartesian single spin operators as constituents of basis operators, the basis operators are defined as: E, = 2’4-1) fi (IkrF

(32)

k=l

where N = total number of spin-l/2 nuclei, k = index of nucleus, v = x, y or z, q = number of single spin operators in the product, a, = 1 for q nuclei and 0 for the remaining N - q nuclei. The product operators satisfy the relation Tr {B,B,} = a,., 2N-2.

(33)

For example, for a two spin-l/2 system the basis set consists of the following operators, where E is the identity operator, q=o q=l

In certain situations it is more convenient to use the raising and lowering operators to describe an experiment. In such a case, the basis set for a two spin-l/2 system may be constructed as: {B,} =;

E

q=o q=l

21: II,. 21: Ii, 21: I:

Using polarization

21; IL*, 21; I;,

21; I;

2I,,I,,,

2I,,I;.

2I,,Ii,

+,

(35)

operators I;, If, etc. {I3,) are constructed as B, = ki, Iy,. r

wherep,=

q=2

(36)

-_,z or!.

3.3.2. Evolution of Product Operators. The product operators transform within themselves on application of RF pulses or under free precession under the influence of a Hamiltonian. In liquids, the nuclear spin Hamiltonian for spin-l/2 nuclei is fairly simple and consists of a Zeeman term (S’“,) and

10

R. V. HOSUR

a spin-spin coupling term (*,I: Jf=~viliz+

z J,fiI,

i

(37)

icj

=*“,+I,

(38)

where i goes over all the spin-l/2 nuclei in the system. vi is the precessional frequency and J, is the coupling constant. In weak coupling situations, JIP, is approximated to contain only the f,I, products. Under these conditions evolution of each product operator under the influence of &’ can be split up into ‘evolution under .Wz’ and ‘evolution under .#‘,’ which can be evaluated separately and the sequence of their calculation is not important, since both the operators S’, and Jlpx commute. The following rules can be utilised for the calculation of individual evolution: I

(*‘) +I,, cos Q - I,, sin 4 Iry 41,.

I

(Jrz) + I,, cos C#J + I,, sin4 kx 6L,

(39)

where ~75 = 2~ vkr is the angle of rotation in physical space about the z-axis. (9,)

I kr.

COS dk,T

b i,,

+

21k,l,,

Sill

dk,

?

~J,F.~LL

I ky

(2,)

* tk,

COS dk,f

-

21,, f,,

Sin RJk,T

nJ,,r. X.I,,

(40)

(41) (42) where

a, j3, y and a’, /I’, y’ are cyclic permutations Similarly I: A

91,.

of x, y, z and $ = n.fklr. 1: exp( T iqb)

(43) (44) (45) (46)

with a = x, y or z. 3.3.3. E@ct CJJRF Pulses. Pulses are rotation operators in the physical space and thus their effects will be similar to evolution under chemical shift terms of the Hamiltonian. Thus I kz

I ky-

1:

es sL

l

I,, cos 9 + It, sin 9 I,, cos f#~- I,, sin I$

(47)

Scaling in ID and 2D NMR spectroscopy

I1

Here 4 refers to the flip angle of the pulse applied along the x-axis. For more details about these aspects we would refer the readers to the original literature’40-4r’ and books.‘45.47n50’ 3.3.4. Obseruables. The various product operators have been ascribed different physical meanings. The observable property of a particular operator depends on whether it has a non-vanishing trace with the I,( = z It,) or lY( = Elk,), since these traces refer to the magnetisations one can observe in an NMR experiment. Operators such as I,., I,,, I,,, I,, are referred to as in-phase magnetisations of the k-spin or the l-spin along the x- or y-axis. These have non-zero ‘trace’ with I, or I, operators and hence are observable operators. On the other hand, operators of the type Ik, I,=, Irr I,, have zero trace with I, or I, and are therefore not directly observable. These are termed as anti-phase k-magnetisation with respect to the I-spin along the x- or y-axis. However, it must be noted that these operators give rise to observable operators when they evolve under the influence of J-coupling Hamiltonian. Operators such as It, IIXrIt, I,,, etc., containing two transverse components are also not observable and it can be easily shown that they contribute to double quantum (DQ) and zero quantum (ZQ) coherences. The operator lk,l,, represents 2-spin z-order and is also not observable. In systems containing more than 2-spins, there will be operator products corresponding to higher quantum coherences and higher spin orders. In general, the observable operators are those which contain single transverse components in the products; the antiphase magnetisation operators such as 21t,li,, 411,1,,1,, lead to observable magnetisation after J-evolution. It is of course necessary that all the ‘active’ couplings are well resolved for each anti-phase operator to lead to observable magnetisation. Similar product operator descriptions exist for spin-l nuclei in the literature, but we do not intend to list those descriptions since we will be mostly concerned with spin-l/2 nuclei in the entire review; the largest volume of literature concerning scaling experiments today concerns protons which are spin-l/2 systems. 4. ONE DIMENSIONAL

SCALING

EXPERIMENTS

4.1. Off-Resonance Heteronuclear Decoupling The experimental scheme of heteronuclear off-resonance decoupling is shown in Fig. 5A; 13C and ‘H have been chosen for the purpose of illustration, and simple doublets are considered in each case. A continuous irradiation is applied in the ‘H region at an offset A from the ‘H doublet, while the 13C FID is being collected. In Fig. SB, the effect of changing the offset A, of the proton irradiation, on the i3C-‘H coupling constant is indicated schematically. It is observed that the heteronuclear coupling constant J(CH) is scaled differently depending upon the offset value. In real systems, containing several “C and ‘H multiplets, CW irradiation at any one frequency in the ‘H region will correspond to different offsets for different ‘H multiplets and accordingly the J-values in different “C multiplets will be scaled to different extents. The total Hamiltonian of the system consists of an unperturbed part (Ho) which would be operative in the absence of the double resonance irradiation and a contribution (.#,) from the CW irradiation applied in the ‘H spectrum. X0= *s + His + z,, (48) where S and I refer to *‘C and *H spins. .%‘s is the Zeeman part of S-spins, 2,s represents I-S interactions and &ii represents homonuclear I-I interactions. The S-S interactions are not included since the S-spin is considered to be the rare spin and these interactions are absent. If, for example, “C labelling is done on adjacent carbons, then these interactions will also have to be included. The perturbation term X, due to CW irradiation is given by:

where w2 is the irradiation frequency, ut is the resonance frequency of k”’ spin and B, is the amplitude of the irradiation field. The index k goes over all the I-multiplets in the spectrum. For sufficiently

R. V. HOSUR

1%

I

ACWI!WlON

A* 1.0

I I

A’ 2.0

FIG. 5. (A) Off-resonance decoupling in heteronuclear experiments (e.g. *HJ3C). During “C data acquisition, continuous wave (CW) irradiation is applied on the ‘H channel, at a particular frequency which may or may not correspond to a *H resonance line. A typical situation is illustrated in the box for a ‘H doublet of a IH-‘)C two spin system. A is the offsetof irradiation from the chemical shift of the proton.(B) The effectof varying offsetORthe “C doublet observed. As A decreases, the ‘HJ3C coupling constant J is progressively scaled down. A = 0 corresponds to full decoupling.

strong irradiation fields, the average Hamiltonian calculated as:‘4s1 .%“b”’= k,

2b”’ characterising the motion of the spins can be

+ 1 27rJ,, cos eh I&,

+ 3;;‘.

(50)

k.1

Ikz represents the z-magnetisation of k-spin along the new quantisation presence of the RF field B, . The angle ok is defined as: tan ok = yB,/(o,

- uk)

axis of the k-spin in the (51)

&”

represents the modified I-I interactions. Equation (50) tells us that the I-S interactions are scaled by a factor cos 0, which will be different for different I-spins. J:, = cos 8, Jt,

(52)

where Jh is the reduced kl coupling in the S-spectrum. While such scaling of J-coupling helps in resolving overlapping multiplets, the offset dependence of the scaling factor is a disadvantage. Measurement of the true coupling constant would require a cumbersome calculation of correction factors different for different spins. Therefore it is more desirable to have uniform scaling of the Js so that a single scaling factor would be applicable to all the S-spin multiplets. Such uniform J-scaling has been achieved 1*o-22’ by employing multipulse techniques instead of the continuous irradiation of the I-spins. These ideas were derived from procedures used for chemical shift scaling which are described in the following section.

Scaling in ID and 2D NMR spectroscopy

13

Two pulse sequences have been proposed for uniform scaling of J-values in heteronuclear experiments and these are shown schematically in Fig. 6. In scheme ‘u’ pairs of ‘H pulses with altemant phases and equal flip angle I (pulse width = zJ2) are applied separated by a free precession period (r,/2) and this repetitive cycle is synchronised with “C sampling. In scheme ‘6’ two pulses (width = 7J2) with flip angle a and opposite phases are applied in quick succession and this is followed by a free precession for a period rl. This repetitive cycle is again synchronised with the ‘% sampling. The pulse flip angle a and the periods T, and r2 have to be optimised to achieve best performance; a and tl are dependent on the decoupler power and r, is dependent on the decoupler offset from the resonance frequencies. The performance of these sequences can again be calculated using the average Hamiltonian theory. The Hamiltonian for a heteronuclear system consisting of n I spins (e.g. ‘H) and a single S spin is given by:

-@s + c (Wkfk, + 2~JkJkJZ) + -@II.

x0 =

(53)

k

In the presence of the two repetitive pulse sequences shown in Fig. 6a and b, the average Hamiltonians have been calculated and in both cases these have the form (up to 1st order):

+SF(f)C{wkfk,+2nJk,fk,S,) +.%,,

.j?,=.%,

(54)

k

where i = a, b and,

SF(4

=&rc

I12r: + 27:/a’)

a

(a 1

7:

3

c: ---

‘Ii

“C

a J”’

(I + cosa) + Z/rr,r,sin

-

+a

90” -*-.._

-a

_#

--a--

+a

+a

-a

--., --.

-__-

ct-

(b)

)'2? t

'H

+a -a

+a -a

+a -a

+a -a

90” “C

--x__ -1-

_*--’

__

--.-,

. .

-L-.__--

FIG. 6. Two multipulse sequences for uniform J-scaling in heteronuclear ‘HJ3C experiments. (a) During the acquisition time pulses with alternating phases and flip angle a, separated by free precession periods 5J2 are applied repetitively. If r,, is the pulse width, corresporiding to flip angle JI and if z is the dwell time of data acquisition, the condition T = 21, + zI is satisfied. (b) The pulses with alternating phases are applied in quick succession and each pair of such pulses is separated by a free precession period. Here also the condition z = Zr, + I, is satisfied (reproduced with permission from Ref. 20).

R. V. HOSUR

I4

1000

2000

0

PROTON RESONANCE

1000

2000”:

OFFSET

FIG. 7. Calculated J-scaling factors for a two spin-l/2 system as a function of the resonance olfset for the pulse sequences (a) and (b) of Fig. 6. The calculations have been performed by including higher order terms in the average Hamiltonian expansion. For comparison, the dependence of the scaling factor on offset for the conventional CW off-resonance decoupling is also shown. Proton resonance offset corresponding to scaling to 10% in the ofi resonance experiment is taken as zero for comparison. Scheme (b) seems to produce better uniformity in J-scaling and works over a wider range of offsets than scheme (a). Experimental data points are shown by open circles for scheme (b). The following parameters were used for scaling to IO?4 T = 278 ps, 5P = 112 /IS, ;E,.lZn = 7500 Hz (reproduced with permission from Ref. 20).

SF(b) = + I

J

5: + 25:/r’

(1 + cosa) + 2/x r,rs sin z] ‘I*.

that both sequences perform equally well and the scale factors do not depend explicitly on offset. However, this is because of the truncation of the average Hamiltonian to 1st order; in reality, higher order terms do contribute and these have explicit offset dependences. Figure 7 shows the exact computer simulation of scale factors as a function of the proton resonance offset for an AX system for the two pulse schemes mentioned above. For comparison, the scale factor for CW off-resonance irradiation is also indicated. It is seen that while both pulse schemes in Fig. 6 achieve better uniformity in scaling as compared to off-resonance decoupling. scheme b is much superior, and is insensitive to offsets over a wider range. Figure 8 shows an experimental demonstration of uniform J-scaling by pulse scheme b. Two quartets which are not discernible in the normal “C spectrum (bottom trace) are clearly resolved in the J-scaled spectrum (top trace). It is to be noted that in both schemes, the ‘H chemical shifts are also scaled by the same factor as are the heteronuclear coupling constants. It appears

4.2. Concertina Effect The ‘concertina effect’ refers to scaling of chemical shifts in liquids by arbitrary scale factors between 0 and 1. The basic ideas emerged from the experiments designed for removal of dipolar coupling from the NMR spectra of solids.tr6-‘st It was noticed in these experiments that the chemical shifts are scaled by a factor of (3)‘/*. This observation was used to devise pulse sequences to scale the chemical shifts in liquid samples by arbitrary factors, the motivation being to obtain spectra corresponding to various magnetic fields on the same spectrometer.

Scaling in ID and 2D NMR spectroscopy

15

I

40

20 PPM FROM TMS

FIG. 8. Scaled “C spectra of ethyl acetate using sequence (b) of Fig. 6. (I) No CW irradiation. (II) Scaling factor = 0.356 (gB,/tn = 6.66 kHz, T, = 118 ps. 27, = 160~s). (III) Scaling factor = 0.178 (‘~B,/2n = 6.66 kHz, 51 = 78 ps, 25, = 200 ps). The ‘H decoupler frequency was placed 350 Hz downfield from the CH, resonance. In (III) all the three multiplets are clearly resolved (from Ref. 20).

FIG. 9. Phase alternated pulse sequence for homonuclear chemical shift scaling in liquids, referred to as ‘chemical shift concertina*. r, is the cycle time of the sequence. Two pulses 0 and - 0 are interspersed with a free evolution period t,. Data is sampled at the end of every cycle (with permission from Ref. 19).

Figure 9 shows the basic pulse sequence designed for achieving chemical shift scaling in liquids. The experiment consists of a repetitive cycle of pulses separated by free precession periods;“” t, is the cycle time of the experiment and each t, consists of two pulses with alternant phases (applied along opposite axes, x, - x or y, - y, etc.) and two free precession periods of duration t,. The angle 0 is the flip angle of the pulse which is related the amplitude of RF field and t, is the width of the pulse. Data is sampled at the end of each cycle t,. The experiment has been analysed using average Hamiltonian theory!“) The rotating frame Hamiltonian for n homonuclear spins in the absence of any perturbation is

where Ao, is the precession frequency of spin i in the rotating frame. In the presence of a series of pulses as in Fig. 9, the average Hamiltonian can be calculated to different orders, and the first two

R. V.

16

HOSUR

terms of the expansion are: *‘“‘=T

AWi( 2/CC t t I cos

812

+

(zt,/@

sin

e/2])

X (COS d/2

-

Sin

012

fiy)

i-

1

Jjk IjI,

(58)

i
.ik’(l) = 1 (A@# (2t,)- I x [tf sin 0 + (2t,t,/O)] i

x (CGV

Ii,

I
- cos6) + (2t,t,/O) (0 - sin@] (f,I,,

- lj,I,,)

+ [ t.Zsin 0 + (2t,t,/O) (1 - cost?)] ( I, I,, - I, I,,)) .

(5%

It is easy to see from these equations that the chemical shifts (Awi) are scaled and the scale factors can again be calculated to different orders of approximation. The zero order scaling factor SF”’ is given by: SF(O) = 2/t, [t, cos O/2 + Z(r,/O) sin 6/Z].

NQ

If the first order correction of eqn. (59) is included, then the scale factor is given by: SF

=

1[

2t f(t, cos O/2 + f sin O/2) C

2

+

Aw.

tf sin0 +

i;’

(61)

C

While the zero order scale factor is independent of the chemical shifts or the offsets (Awi), the first order correction to the scale factor seems to depend on the offsets. Similarly the higher order corrections also depend upon the offsets themselves.

I 420.0

OFFSET

(Hz)

ALTERNATED

540.0

500.0

460.0 RESONANCE

PHASE

OFFSET

(Hz)

HIGH

RESOLUTION

EXPERIMENT

500.0

SPEtYRCIK

E23.0

SPECTRC:~;

FIG. 10. Experimental spectrum demonstrating chemical shift scaling in a high resolution NMR spectrum by the use of phase alternated pulse sequence of Fig. 9 (with permission from Ref. 19).

17

Scaling in 1D and 2D NMR spectroscopy PULSE

ANGLE

90’

0

4

8

I2 PULSE

135’

I6 WIDTH

20

180’

24

225.

26

32

(psccl

FIG. Il. Dependence of scale factor on the pulse width in the phase alternated pulse sequence. Experimental data points obtained with a water sample and cycle time of 175 ps are shown by open circles (from Ref. 19).

The scale factor is seen to depend critically on the experimental parameters 0, t w, t ,Vt C.in addition to the offsets. The results of average Hamiltonian theory up to 2nd order are faithfully reproduced when the offsets are less than 2/t,; when they are larger than this limit, differences have been found between the experimental and theoretical scale factors. In such situations, the classical approach of calculating the net rotation in the rotating frame of the classical magnetic dipole Seems to yield reliable results.~t9’ Figure 10 is a visual demonstration of chemical shift scaling achieved by the phase alternated experiment cycle of Fig. 9. Figure 11 shows the effect of pulse width t, on the scale factors. A good fit is observed between theoretical predictions (solid curve) and experimental data (discrete points).

5. TWO DIMENSIONAL

SCALING

EXPERIMENTS

The generalised experimental scheme of two dimensional (2D) NMR spectroscopy discussed before has effectively two detection periods: (i) the usual detection period (tz) during which time domain data is actually collected, and (ii) the evolution period (t,) during which data is not acquired but its information content is transferred to the detection period by the mixing process. This information appears along the o,-axis of the 2D spectrum obtained after 2D Fourier transformation of the time domain data. Therefore, obviously, the possibilities of scaling of NMR parameters are much more enhanced in 2D spectroscopy than in the 1D experiments. These add to the power of 2D spectroscopy and open up vistas for new applications. In addition to the manipulations described in connection with scaling in one dimensional experiments, other different scaling manipulations are possible during the evolution period of the 2D experiment without the constraint of data acquisition by the receiver of the spectrometer. The preparations and mixing periods are also amenable to such manipulations, but in order that these reflect as scaling in the 2D NMR spectrum, the two periods will have to be synchronously varied with t, in the course of the experiment. Data processing techniques have also been used for achieving scaling of NMR parameters. We will. discuss in the following sections scaling experiments in the three

R. V. Hosua

18

different types of 2D NMR experiments, namely, 2D resolution, 2D correlation (J-correlation and NOE correlation) and multiple quantum experiments. Although certain common features exist with regard to scaling during the evolution period in the latter two types of experiments, the spectral characteristics are different and therefore the optimisation of the experimental parameters such as scaling factors, number of t, experiments, etc., are different in the two cases. Therefore, while we describe the evolution period scaling schemes in detail in the context of 2D J-correlated spectroscopy we shall only discuss their optimisation in the context of multiple quantum spectroscopy and 2D NOE correlation spectroscopy. Scaling possibilities stemming from manipulations in the other periods will be dealt with explicitly in the NOE-correlated and multiple quantum experiments. 5.1. 2D J-resolved Spectroscopy 51.1. Homonuclear Case. This is an example where data manipulation procedures have been used to achieve scaling of spin-spin coupling constants (J). The experiment was designed primarily to separate the spin-spin and Zeeman interactions in liquids. The basic experimental scheme for homonuclear systems in the isotropic phase is shown in Fig. 12j9” The 180” pulse in the centre of the evolution period refocusses chemical shits (for weakly coupled spin systems) and thus the average Hamiltonian 2’ during t, consists essentially of spin-spin interactions. j+

= &zf,* =

c

Jijfifj.

(62)

i
In the t, period, the full Hamiltonian operative:

(X”)

with chemical shifts (X”,) and coupling constants is

.P=.@L+z,,. In the weak coupling situation the Hamiltonians

(63)

kc and .Wd commute:

w

[iP,.Jr”d] =o

and consequently there is no coherence transfer between spins, and the total number of peaks in the 2D spectrum is identical to that in the 1D spectrum. For two coupled spins I and S, (I = S = l/2), the density operator u(t, + t2), through the detection period t, can be calculated using the product operator formalism and is given by““’ cr(t, + tz) = - C,(t,)C,(t,)

G&2)&

(65) 180

90

+

!.L

‘2

FIG. 12. Experimental scheme of homonuclcar two dimensional J-resolved spectroscopy: t, is the evolution period and t2 is the detection period. The 180” pulse applied in middle of tI refocuseschemical shifts in weakly coupled systems.

Scaling in 1D and 2D NMR spectroscopy

19

J

8+J FIG. 13. A schematic 2D J-resolved spectrum resulting from the data collected by the scheme of Fig. 12. The multiplets lie on a line inclined at an angle of 45” to the o,-axis. which has J as well as 6 (chemical shift) information while the q-axis has only J information.

with the following definitions:

C,(t) = cos nJt; S,(r) = sin lrJt C,,(t) = cos 2x&t; Sza(t) = sin 2x&,t;

(66)

6, is the chemical shift of I-spin. It is clearly seen that only spin-spin couplings are present during t, , while the t2 period has both chemical shift and spin-spin coupling information. Equation (65) indicates that the spectrum will have mixed line shapes along both the frequency axes. This necessitates absolute value presentation of the spectrum. The appearance of the homonuclear 2D J-spectrum in the absolute value mode is shown schematically in Fig. 13. The multiplets belonging to a particular spin lie on an axis making an angle of 45” to the w,-axis of the spectrum. Projection of this spectrum onto the w,-axis generates the 1D spectrum, with the usual chemical shifts and coupling constants. However, if one takes projections of the absolute value spectrum on to axes which are at different angles to the w,-axis, the various multiplets will appear with J-values scaled. This is shown schematically in Fig. 14. The axis of projection o is defined with respect to the angle cpit makes with the w,-axis, and the projections on to this axis are taken as indicated by thick arrows in Fig. 14a. In Fig. 14b, the multiplet structures obtained for three different orientations of the axis are indicated and it is clear that arbitrary scaling of multiplets can be achieved by appropriate selection of the orientation (angle cp). In fact the scaling factor can be easily computed and is given by: SF=

1 -cotrp.

(67)

SF = O(cp= x/4) corresponds to a fully decoupled spectrum. Figure 15 shows an experimental demonstration for a 3-spin AMX system.‘9” 5.1.2. Heteronuclear Case. The heteronuclear J-resolved experiment can be performed in two different ways as shown in Fig. 16. The scheme ‘a’ is similar to the homonuclear case except that the heteronuclear coupling is removed during the 1, period by broadband decoupling. But this leads to a great improvement in the sense that peaks will have pure absorptive line shapes along both frequency axes of the 2D spectrum. This is easily seen by stepwise calculation of the density operator evolution and for a two spin-l/2 IS system, the final density operator for evolution of spin S is: a(l, + rt) = C,(r,)

CZdk)I,

-

c,(t,)s,d(t,)l,,

with C&)

= cos 2nb,t; S,,(t) = sin h6,t

C,(f) = cos nJt; S,(r) = sin dt.

(68)

R. V. HOWR

20

(a)

(b) WI

1

FIG. 14.

(a) Principle of scaling of homonuclear J-values by data manipulation of 2D J-resolved spectra. Projections can be taken on different axes making different angles (cp)to the q-axis. (b) Schematic appearance of spectra obtained by different projections. cp = 45’ corresponds to complete homonuclear decoupling. For

90 c cp c 180 Js are scaled up, while for 0 < cp c 90, the Js are scaled down. cp = 90 corresponds to the usual one dimensional spectrum.

Thus the signal detected as a function of t2 is amplitude by J-evolution in t, . Clearly the spectrum can be phased to pure absorption along both axes of the 2D spectrum (one can consider detection of y magnetisation alone during t2). The wz-axis of the spectrum has only chemical shifts and the w,-axis has only the IS coupling constant. Thus, in contrast to the homonuclear case, the peaks appear on an axis parallel to wi and not inclined at 45” to the w,-axis. As in the homonuclear case, J-scaled multiplets can be obtaind by taking projections on different axes making different angles to the q-axis. In scheme b of Fig. 16, broadband decoupling to I spins is applied during the t, period and part of the t, period (yt, ). In addition, since the refocussing 180 pulse is applied only on the S-spin, the heteronuclear J-evolution occurs effectively only for the period yt, rather than the entire t, period. The S-chemical shifts are refocussed as in the other experiment. The average Hamiltonian for the t, period of this experiment can be written as: (6%

This simple representation has the following assumptions: (i) SS interactions are not present, which is a good approximation in many instances, e.g. 13C-13C coupling interactions are usually not seen in natural abundance 13C spectra.

Scaling in

ID and 2D NMR spectroscopy

21

FIG. 15. Experimental J-scaled spectra as per the procedure of Fig. 14. The scale factors indicated on the right correspond to cp= 45”. 53.1”. 63.4” and 90” respectively from top to bottom (reproduced with permission from Ref. 91.

(ii) The I-spin Hamiltonian modified by the RF, %‘i does not influence S-spin evolution. This is valid only under special conditions.“” Ck,, a(O)1 = 0

(70)

where a(O) is the initial density operator before the application of I-perturbation in the ti period. If the experiment is started from an equilibrium state where a(O) consists of essentially I, and S, operators and if the II interactions are weak, eqn. (70) can become a valid condition. Returning to eqn. (69), it is observed that the heteronuclear couplings are scaled by a factor y and these scaled multiplets will appear along the w,-axis of the 2D J-spectrum. However, modifications of the scaling factors occur when the conditions described by eqn. (70) are not met. These also lead to intensity anomalies in the spectra. 5.2. 20 Correlated

Spectroscopy

Two dimensional correlated spectroscopy is by far the most important and widely used experiment in chemistry and biology for structure determination of small as well as large molecules in liquids.‘s1-76J In these experiments, the different correlations or interactions between the spins are

22

R. V. HOSUR

(a)

FIG. 16. (a) I-spin flip pulse scheme of heteronuclear 2D J-resolved spectroscopy. Broad band I-spin decoupling is applied during detection and relaxation delays. This results in a 2D spectrum which has exclusively J-information along w, and exclusively S chemical shifts along w2. (b) Gated decoupler method of heteronuclear (IS) 2D J-resolved spectroscopy which can be used for scaling of J-values. Broad band I-decoupling is applied for a fraction of the evolution period. The heteronuclear IS coupling will be scaled by a factor 7.

displayed on a frequency plane with the two frequency axes representing the chemical shifts of the correlated spins. For example, in heteronuclear IS correlation experiments, one frequency axis has exclusively S-spin frequencies while the other axis has exclusively I-spin frequencies. In homonuclear experiments, both axes have frequencies of the same spin. The different interactions are manifested in the form of ‘cross peaks’ which are characterised by different co-ordinates along the two frequency axes. In liquids, there are two types of interactions between spins which have found extensive applications and resulted in a variety of 2D correlated spectra. These are (i) indirect spin-spin interactions leading to J-correlated spectra, and (ii) direct dipole-dipole interactions leading to nuclear Overhauser effect (NOE) correlated spectra. The first type of experiments are classified under the acronym ‘COW spectra and the second type are grouped under ‘NOESY’ spectra. The general principles of these experiments have been described in detail iri several works(45-49) and, briefly, the COSY spectra are based on coherent transfer of magnetisation between the spins mediated by J-coupling, while the NOESY spectra are based on incoherent transfer of magnetisation between spins mediated by dipole-dipole interactions. Since the information contents of these spectra are clearly different, we will consider these two experiments separately for the discussion of scaling of NMR parameters. The motivation for scaling of NMR parameters in these two types of experiments has been manyfold; enhancement of intensities of cross peaks, enhancement of resolution in the spectra, observation of long range correlations ordinarily not seen in COSY spectra, etc. These aspects will be considered along with the individual experiments, since the scaling requirements are different for different purposes. Spectra 5.2.1.1. Basic pulse sequence. The basic experimental COSY scheme proposed originally by Jeenert4” and analysed in detail by Aue et al.‘U) is shown in Fig. 17A. It is worthwhile discussing the

5.2.1. COS Y

23

Scaling in 1D and 2D NMR spectroscopy

aa

dd

WI

--“k

/

/ dd

J

aa

i

t

“k

“I QJP

FIG. 17. (A) Basic pulse sequencefor homonuclear 2D J-correlated spectroscopy (COSY); p is the flip angle of the mixing pulse, I, and t, are the usual evolution and detection periods.f43*441(B) Schematic COSY spectrum of a weakly coupled 2-spin system. Cross peaks have anti-phase ( + - ) components while diagonal peaks have in-phase ( + + ) components. The cross peaks have absorptive character along both frequency axes while diagonal peaks have dispersive character along both frequency axes. The separation between the components is the coupling constant.

salient features of this experiment before one can go into the scaling modifications of the experiment. A simple analysis of the COSY experiment is obtained from the product operator formalism described in previous sections. Starting from the equilibrium density operator which corresponds to z-magnetisation, the time evolution of the density operator can be calculated through the evolution period t, of the experiment. For two weakly coupled homonuclear spins k and 1,the evolution of the magnetisation leads to the density operator (pulses applied along x-axis and relaxation during t, ignored). a(t,, 1, =

0) =

c- ~k,Ck(r‘Kk,(t*P,+ ~k,u~,r,,(t,)

- I,,C,(t,K,,(t,K,

+ 21k,IIzSk(t,)Skh, )s,c, + 21,,1,,C,(t,&(f, )c, + 2rk,I,,s,(t,)s,,(t,)C,L - 2rk,r,,sk(t,)s,,(t,)s,2 - 21k,f,,C,(f,)sk,(t,)S,

+ 1-

-

2~,,~,,S,(t,)Sk~(t,)S~C~]

'I~Cl(r,)Ck,(t, )s, + lI,S,(t, )Ck,(t,) - I,,C,(f,)ck,(t,)C,,

+

21kz1fz-%(t,

-

2l,l,~s,(t,)sk,(r,)s,2~

)skh,

)s,c,, + 21krII,C,(t,)Sk,(t,)Cp + -

21kyI,,C,(r,)Sk,(t,)SU

-

2l,,l,rs,(t,)s,,(t,)cf

21k,~l,S,(rI)Sk~(tI)SUCUl

(71)

24

R. V. HOSUR

with C,(r) = cos wLl, S,(t) = sin w,(t) C,(t) = cos w,t, S,(t) = sin ml(t) C,,(r) = cos ~JIJ, L(t) = sin nJL,f C, = cos fi, S, = sin p The terms in the first square bracket arise from evolution of k-spin while those in the second bracket originate from /-spin. Among the various terms in eqn. (7 I ). the /Lz, IL1II,, lk. I!,, fk, I,, and It, I,, terms do not contribute to observable magnetisation during r2 and hence need not be considered. The other terms evolve in r2 under the chemical shift and J coupling Hamiltonians to produce observable magnetisations. The terms containing It, or lIryevolve with characteristic frequencies ( + wt & aJkl) (rad.‘s) while terms containing I,, or I,, evolve with frequencies ( & co,f nJkl) (rads) in the t2 period. After two dimensional Fourier transformation of the data. the It, or fk, terms lead to peaks at [w2 = + (wa f rrJk,), Q, = + (tv, k- nJkI)] and [ w2 = + (w,, ? nJkl). co, = k (CJ, + ‘TJ,,)]. The former peaks are the diagonal peaks while the latter are the cross peaks. Similarly, time evolution of I magnetisation from equilibrium through t2 period leads to peaks at w, = f (0, + nJk, and II+ = + wI f nJ,, or UJ~= f (I)~f KJ,, in the 2D spectra. The spectrum will have mirror symmetry with respect to tr), = 0 axis. The appearance of the COSY spectrum on one side of co, = 0 axis, for two spins k and 1.is shown schematically in Fig. 178. It consists of two diagonal peaks and two cross peaks both of which have multiplet structures. The cross peaks have both positive ( + ) and negative ( - ) components while the diagonal peaks have only positive components. The separation between the components along either axis is equal to the coupling constant Jkl between the two spins. Another important feature which emerges from the detailed expressions is that the diagonal peaks have dispersive line shapes (dd) along both frequency axes while the cross peaks have absorptive character (aa) along both the axes. The COSY spectrum is extremely useful in identifying networks of coupled spins in complex molecules. Resonance positions of hundreds of protons in many biological macromolecules have thus been identified from the COSY spectra. However, one often encounters difficulties when cross peaks lie very close to the diagonal. The diagonal peaks are usually much larger because of coaddition of in-phase (+ +) components and also they have very long tails because of the dispersive line-shapes of these peaks. The cross peaks lying close to the diagonal thus get masked by the tails of the diagonal peaks. A remedy to this problem has been suggested which relies on the use of a double quantum filtering procedure to obtain COSY spectra. (93) In these (DQF COSY), the diagonal peaks also acquire largely absorptive line shapes and will have antiphase ( + -) components corresponding to the active coupling constant. (Note: for two spins and linear spin systems the peak shapes are purely absorptive and for more complex coupling networks, however, some admixture of lineshapes does occur.) In large complex spin systems the flip angle p of the second pulse in the COSY sequence plays an important role in determining the nature of the COSY spectrum.‘J5’ If fi is less than ~‘2, different components within a cross peak will have different intensities and this has been used to simplify cross peak patterns and measure the relative signs and magnitudes of coupling constants.“” For /I = rt;4. 50% of the components have about 17% intensities of the other components, contributing to substantial improvements in the resolution. However, selection of flip angles other than Z./Zleads to mixed line shapes for the diagonal peaks which is a disadvantage; the cross peaks would, however, still have pure phases. The existence of + and - signals in the cross peaks in COSY and DQF COSY spectra with /I = n/2, helps in proper recognition of the coupling constants involved in the cross peak. For example, in a weakly coupled four spin AMKX system with a coupling network as in Fig. 18, the A-M cross peak will have antiphase splitting due to J AMcoupling alone, the A-K cross peak will have antiphase splitting due to JAK, etc. These coupling constants are referred to as active couplings of respective peaks. All other couplings lead to in-phase splittings. Thus if all the components of a cross peak are well resolved, it is possible to measure all the individual coupling constants.

Scaling in 1D and 2D NMR spectroscopy A

JIAX)

X

J(KX)

J(AM)

M

25

K

J(MK)

FIG. 18. A four spin AMKX system with coupling patterns as indicated. Each spin is coupled to the other three spins.

The antiphase splitting due to active couplings also leads to a disadvantage, namely, cancellation of component intensities, if the multiplet resolutions are not adequate. In a 2D spectrum, the multiplet resolution depends on the experimental parameters such as ry (maximum value oft,) ty’ (acquisition time along tz), digital resolution, which is related to the disk storage space available, and finally of course, on the relative magnitudes of the participating coupling constants in the cross peak. While the disk storage space is a physical limitation, t I;,, is limited by the available instrument time. Thus, often the resolution along the w,-axis of the 2D spectrum is poorer than that along the w,-axis. It is often recommended”” that for optimum sensitivity in COSY spectra, one should use the minimum affordable resolution in the w, dimension. From the storage point of view, it is also important to use the minimum possible sampling rate, which implies the spectral width along W, should be minimally selected avoiding intolerable folding of the frequencies. 5.2.1.2. Scaling modifications. Several experimental pulse schemes have been proposed for the scaling of NMR parameters in 2D COSY-type spectra’96-‘02’ and the schemes for homonuciear experiments are shown in Fig. 19. Among these, A and B are the two basic schemes’96. 97) which were published independently and the other ones are off-shoots from the basic sequences. It is seen that although ail the sequences employ the same combination of 90, 180 and fl pulses for scaling purposes, they differ in the actual positioning and incrementation of the time periods during the individual experiments. The underlying principles of scaling in all these experiments are essentially similar; the chemical shift evolution and J-evolution are partly separated along the t,-axis. The experiments can be analysed for weakly coupled spin systems using the product operator formalism described earlier. In the following paragraphs, the individual schemes will be discussed separately and their merits and demerits pointed out explicitly. Scheme A Starting from the equilibrium z-magnetisation the density operator for two weakly coupled spins k and 1can be calculated and the observable part of the operator originating from k spin is given by (the l-spin evolution also leads to similar terms as seen explicitly in eqn. (71) for the COSY sequence). c,(t,, t, = 0) = [lrxSJt’)Str(t”)

- r,,c,(t’)c,,(t”)c#

+ 21~,II=C,(t’)S~,(t”)CB + 21,,r,,S,(t’)SLl(t”)CgL - 2uJ,(t’)Sdt”)S;

]

(73)

with t’ = ? - (2 - p)t,; t” = T - pt,.

(74)

Here again, relaxation effects during tI are ignored. For p = n!Z, eqn. (73) reduces to a simple form “*(f,,

ty = 0, B = n/2) = I&&“)&(f)

- 2r,,l,,S,,(t”)S,(t’).

(75)

The first term leads to diagonal peaks and the second term leads to cross peaks in the 2D spectrum. Both these terms generate w1 frequencies (rad/s) at [ + (2 - p)q + npJJ as against the frequencies ( + at + dH) in the usual COSY spectrum. This indicates that in the scheme A, the chemical shifts

26

R. V. HOSUR 90x

1% r-t,

A

90x

Bx (I-Ph,

180,

PX

0 I

1805

FIG. 19. Pulse sequences for w,-scaling in COSY spectra (A-D). Phase cycles on the individual pulses can be selected to eliminate artefacts such as quadrature images, effects of imperfections of 180” pulses, P Peaks, axial peaks, etc.‘96-99’ E and F are pulse sequences for scaling in double quantum filtered COSY.‘100*103~Phases of individual pulses and receiver are cycled so as to effect double quantum filtering and to eliminate the above artefacts. In F, 180’ is a selective pulse and 180g is a composite pulse.

will be scaled by a factor (2 - p) while the Js will be scaled by a factor p. Selecting p = 0 corresponds,

therefore, to a fully decoupled spectrum with the chemical shifts scaled by a factor 2. It is clear from the nature of the pulse sequence that the 180” pulse moves as the experiment progresses through different values oft, . The separation between the two 90” pulses is (r - pt,) which indicates that the two pulses move progressively closer as the t, value is systematically incremented. Thus the shape of the time domain signal (FID) as a function oft,, an increasing function oft,, is quite different from that in normal COSY data where it is a decreasing funtion of t,. The fact that the chemical shift scaling factor (2-p) and the J-scaling factor p are related to parameter p, implies that the two quantities cannot be scaled by arbitrary factors. However, depending upon the spectral condi-

27

Scaling in ID and 2D NMR spectroscopy

: URACIL

0

3:

1000

)I,

“v4

‘r 6.I’

!

“6

O

‘\ / ! !’

I

II

Hz

*I-

.

.*-

[I--_ I

.*

I-e

1I .. t

Ill --__ --__ --__

,-

2000

*-

*1

WI

304 WI -

1. .

FIG. 20. 2D J-scaled spectrum of a weakly coupled 2.spin system. The J-scaling effect is clearly seen in the expansion of a cross peak shown on the side. The spectrum has been recorded using scheme B of Fig. 19 with y = 4 and 01= 1 (reproduced with permission from Ref. 96).

tions, it is often possible to select a compromise p, to achieve particular results. A major disadvantage of the technique is that neither the cross peaks nor the diagonal peaks have pure line shapes along the o, axis of the spectrum. This results in significant loss of resolution in the spectrum. Scheme B

The density operator equation for this scheme can again be calculated for weakly coupled systems and for two spins k and I ignoring relaxation, the observable part of density operator for the situation /I = 42 is given as

It is again seen clearly that the chemical shifts appear scaled by the factor a while the Js are scaled by

the factor y. Both cross and diagonal peaks have pure phases as in COSY and the scaling factors a and y can be selected almost arbitrarily within the constraint that y has to be larger than a. The flip angle (/?) dependence of @,-scaled COSY spectra is similar to that of a conventional COSY spectrum. For fl it n/2 the cross peaks have pure phases and diagonals have mixed phases. If /? s z/4 the cross peak patterns are simplified resulting in better resolution. As against Scheme A, the FID along t, in this scheme is a decreasing function of t, and the usual processing parameters applicable in COSY are equally suited for this experiment as well. Figure 20 shows an experimental demonstration of the J-scaling effect with a 2-spin system giving rise to two diagonal and two cross peaks. All the four components of each cross peak are clearly visible which was not the case for a COSY spectrum recorded under identical conditions of data acquisition and processing. Figure 21 shows the appearance of a phase sensitive o,-scaled spectrum in a schematic fashion for a more complex network of coupled_spins with several possible cross peaks and diagonal peaks. The

R. V. HOSUR

28 wl- Scald

COSY

FIG. 21. Schematic q-scaled COSY spectrum of a weakly coupled multispin system obtainable with scheme B of Fig. 19.

here is to show the asymmetry in the spectrum with respect to the diagonal. The peak separations within and between the cross peaks are different on the two sides of the diagonal. This must be expected since scaling of NMR parameters is performed exclusively along the @,-axis of the spectrum. Each cross peak has several coupling constants and not all coupling constants present along one axis are present along the other axis. In fact the Js which are present along the w,-axis in a particular cross peak on one side of the diagonal are present along the w,-axis of the corresponding cross peak on the other side of the diagonal and vice versa. Thus, as a result of w,-scaling, the Js which are scaled on one side are not scaled on the other side and this leads to the asymmetry of the spectrum. For the same reason, by monitoring any particular cross peak between two given spins on both sides of the diagonal, one can essentially gain advantages of J-scaling along both frequency axes in any of the single peaks. Scaling down of chemical shifts may often lead to overlap of certain cross peaks. However, this does not necessarily constitute a loss of information, since the same two cross peaks will not overlap on both sides of the diagonal. On the other hand, scaling down of chemical shifts has certain advantages: (i) because of the reduced spectral width, the sampling rate along t, axis is reduced (dwell time increased), which implies that for a given number oft, increments, t? will be higher. This results in a better inherent resolution for the same amount of spectrometer time as compared to COSY; (ii) since the spectral width is smaller, less number of zeros would have to be filled to achieve a particular digital resolution in the spectrum. This also leads to advantages with regard to disk storage space, data processing time, etc. Figures 22 and 23 show, for illustration, the Hl’-HZ’, HZ” cross peak regions on the two sides of the diagonal of an o,-scaled COSY spectrum (with /? = n/4) of an oligonucleotide sample. The spectra clearly show the asymmetry discussed above, but all the relevant information can be derived by using both the spectra. In this spectrum the chemical shifts are scaled by a factor of 0.5 and the Js are not scaled. emphasis

29

Scaling in ID and 2D NMR spectroscopy

CGCGAGTTGTCGCG I.6

2.2

H2’.H2”

aa @06

*. B.

%

II0

@6

2.6

*ii 6.2

6.0

1

56

5.6

HI’ FIG. 22. Hl’-(H2’, HZ”) cross peak region of a 500 MHz phase sensitive w,-scaled COSY spectrum of an oligonucleotide. d-CGCGAGTTGTCGCG. at 25°C recorded with scheme B of Fig. 19 using a flip angle /? = 45”. This cross peak region occurs in the left half (above the diagonal) of the full COSY spectrum. The Hl’-HZ” peaks are labelled.

-1

I ZM

I

I

I

H2’,H2”

I

2.0

2-4

FIG. 23. Hl’-(HZ’, H2”) cross peak region in the lower half of the same q-scaled The labels identify HI’-HZ’ peaks.

COW spectrum as in Fig. 22.

Scheme C

This pulse sequence, referred to as COSS, (9*)is designed to scale up the chemical shifts while having the coupling constants unaltered. Once again, the observable part of the density operator in a system of two spins k and 1 is given by: a,@,, r, = 0, fi = 42) = Ikz’%(r, + ALW,) +

II,Ctdt,

+

A)S,W,)

-

2h,~fySkl(t, -

+

W,(~t,)

2Lh,Sn(~~ + AM=,).

(77)

The extended evolution period A which is a constant time period, appears as a phase factor and does not contribute to J-scaling. On the other hand, the chemical shift evolution goes for a longer period

R. V. HOSIJR

30

d- GAATTCCCGAATTC A [ COSY

J

22

0 P6

64

60

*6

‘5-l

60

56

H I’

wp(Ppm)

FIG. 24. Comparison of HI’-(HZ’. HZ”) cross peak regions of COSS and COSY spectra at 500 MHz of an oligonucleotidc d-GAATTCCCGAATTC, recorded under identical conditions. Both are a result of 512 t, experiments with 2048 r2 points each. Scale factor LXin COSS is 1.5 (reproduced from Ref. 98). (at,) than the ‘effective’ J-evolution period (t,) and thus the chemical shifts appear scaled in the 2D spectrum. In a sense, the situation is equivalent to recording a spectrum at higher magnetic field along the o,-axis in so far as the dispersion between the peaks is concerned. The sensitivity, however, is the same. The scaling factor ILcan be chosen fairly arbitrarily. The choice of this factor will be determined by the affordable value of A, since there is a certain loss of signal during this period from T, relaxation. The value of A should he at least equal to (I - 1)t y. Therefore, in practice, a suitable combination of z and tr’ has to he arrived at for maximum possible dispersion of signals, with sufficient intensities of the peaks. Figure 24 shows a comparison of COSS and COSY spectra of an oligonucleotide molecule recorded under identical conditions Jg8) The spectral dispersion in COSS is clearly superior and several cross peaks which overlap in the COSY spectrum are clearly resolved in the COSS spectra. The time period A which appears as a phase factor in eqn. (77) changes the phase characteristics of both cross and diagonal peaks in the spectra. In general it introduces mixed phases in both types of peaks and thus warrants an absolute value presentation of the spectra. Although, in general, this feature is a disadvantage since it results in loss of resolution, changing of phase characteristics can be used with an advantage in favourable circumstances. In eqn. (77) if A = 1/2J,.., then

-

~dkf(~l)%~~l)

-

&~kr(t$&tL)-

(78)

This indicates that the diagonal peaks acquire anti-phase character while the cross peaks acquire in-phase character. If the diagonal peaks are phased to absorptive line shapes, the cross peaks will have dispersive line shapes and vice versa. This feature is similar to the characteristics of the SUPER COSY technique.~‘oJ) Thus, if the o,-resolution in the spectrum is not adequate, which is usually the case because of instrument time limitations, the diagonal peak components tend to cancel their intensities while the cross peak components co-add. Thus, cross peaks lying close to the diagonal can be observed. Such situations are often encountered in spectra of oligonucleotides and an illustration of the utility of the COSS procedure is shown in Fig. 25. It is clearly seen that several diagonal peaks have very diminished intensity compared to COSY and many cross peaks very near the diagonal are easily discernible. In general any value of A introduces some in-phase character into the cross peaks

31

Scaling in ID and 2D NMR spectroscopy

3-o

--! wp( PPm)

w2mm)

FIG. 25. Comparison of H2’-H2” cross peak regions of COSS and COSY spectra of the same oligonucleotide as in Fig. 24. Scale factor a in COSS is 2.0. COSY spectrum is a result of 512 t, and 2048 tl points in the time domain data while COSS spectrum is a result of400 t, experiments with 2048 t2 points. Reduction of the diagonal is clearly seen in the COSS spectrum (from Ref. 99).

and some anti-phase character into the diagonal peaks. In low resolution spectra (multiplet components are not resolved), this partially compensates the loss of intensity due to decay of the signal, since the cross peak components now co-add rather than cancelling as in COSY spectrum. A consequence of scaling up of chemical shifts is that the increment between two successive t, experiments is reduced and hence a larger number of t, experiments will have to be performed to achieve a particular ty value, as compared to the COSY sequence. The merit of the technique however lies in that greater dispersion of the cross peaks can be obtained without having to have a large ,$” value. Since the J-values are not scaled, the widths of the peaks are largely dependent on the line widths which can be artificially reduced by using suitable apodization functions during data processing. This is acceptable for absolute value data presentation of the spectra. Scheme D

This is a simple variation of Scheme C (COSS) and can be used for scaling of both chemical shifts and coupling constants in a fairly arbitrary manner. This is abbreviated as S. COSY in the literature.“06’ The density operator equation for this scheme, corresponding to eqn. (77) of scheme C, is given by:

All the discussion relevant to Scheme C regarding the selection of scaling factors, A value, dispersion of peaks, etc. is applicable to Scheme D as well. An additional feature which is of significance is that Scheme D can be used in conjunction with Scheme C to obtain pure phase spectra and eliminate the disadvantages associated with mixed phase characteristics of the individual schemes themselves. This is achieved by adding COSS and S. COSY spectra recorded with identical A values and y = 1 for the S. COSY scheme. The density operator equation representing this situation is as follows (keeping only the observable terms and ignoring relaxation):

32

R. V. HOSUR

Simplifying eqn. (81) one obtains: bT =

21k,CkI(A)Ck,(t,)Sk(zt,) +

-

~~~.Ckd~)~kr(~,

41k,ff,Sk,(A)Ck,(r,)Sk(zt,)

)S#,)

-

4~dk,&,@)C&

)S,(zt,

1.

(82)

period A now as amplitude factor than as phase factor. diagonal peaks amplitude factor cos nJk,A that in cross peaks sin nJk,A. the time A can adjusted so to enhance cross peaks diminish the peaks. Further, both the and the peaks, the Ck,(tl)&.(xr,) [or leads to pure dispersive shapes. Absorptive shapes can obtained by phase correction. is easy see that of the spectra (COSS S. COSY): = UC

b,J

(83)

leads to line shapes both the of peaks the w,-axis the 2D This will introduce anti-phase in both peaks. The nature of components, in eqn. indicates that is an net transfer magnetisation from diagonal peaks the cross Such calculations also been out for containing more two weakly spins. It observed that general conclusions above are general valid pure phase, scaled (r l), spectra be obtained. success of pure phase scaled spectra discussed above on proper of the components on of the spectra. A examination of procedure reveals for any value oft, the total times A t, and - f, the COSS S. COSY respectively are and therefore relaxation of signal which to different affects the differently in two experiments. it becomes to take weighted addition the two to achieve cancellation of components. The factor (q) be easily by including explicitly in density operator (see Section and is given by: q=e -211/rz

(84)

Each free induction decay of S. COSY must be multiplied by a t,-dependent factor rt co-added to of COSS to the f, value.,‘06’ 26 demonstrates experimental procedure a 2-spin A 1D is carried to demonstrate ideas. If and L’ the relaxation corresponding to and S. respectively, then of the spectra as before leads the observable UT =

It, -

[(r-

+

21,,1k,[(L

~)~k,(~,)~k,(~) +

~)&,(~)~k,(~,)

+

(r

-

&%,(‘%(t,)] +

(L

-

‘%d~)&,(~,)]

skbt,) &b.t,).

(85)

Therefore, if two FIDs are collected one each with COSS and S. COSY sequences and with r, = 1/2Jkl and A = 3/2JkI and when they are coadded, with proper weighting factors so as to compensate for L # L’, there should be no signal. In Fig. 26, it is seen that ‘c’,which is the result of addition of ‘a’ and ‘b’ which correspond to COSS and S. COSY respectively, has substantially reduced intensity. This residual intensity reflects imperfect weighting, spectrometer instabilities and improper settings of A and t, values. Scheme E This scheme encompasses the advantages of double quantum filtering and scaling within one single experiment. The experiment requires a phase cycling procedure which is indicated in Table 1. In addition to the basic cycle for double quantum filtering, an additional phase cycling of the 180” pulse is needed to correct for imperfections in the 180’ pulse. The general aspects of scaling in this technique are similar to those in w,-scaled COSY except for the flip angle dependence of the cross peaks. In this scheme, if B # n/2, the cross peaks in general (for systems with larger than two coupled spins) will not have pure phases, in contrast to the p.ure phases obtainable with scheme B. Therefore, o,-scaled DQF

Scaling in 1D and 2D NMR spectroscopy

a

33

b

FIG. 26. One dimensional (ID) demonstration of the procedure of adding COSS and S. COSY spectra for recording pure phase shift scaled spectra. Boxes (a) and (b) contain ID spectra recorded using COSS and S. COSY pulse sequences respectively, of a three spin system, and (c) is a result of addition of the two. The experimental parameters are selected in such a way as to cause complete elimination of dispersive contributions on addition of the two spectra and box (c) should have no signal. The residual magnetisation is a result of imperfect settings of delays and pulses (from Ref. 106).

TABLE 1. Basic phase cycling for scheme E of Fig. 19 Receiver

A X

-x

X -x

X -x

X -x

-Y

-Y

-Y -Y -Y -Y -Y -Y -Y

-Y

X X Y Y -x -x

-Y -Y -x -x Y Y

X X

spectra have to be perforce recorded with /3 = x/2 for the detection pulse. Figure 27 shows the multiplet resolution enhancement achieved by w,-scaled DQF COSY as compared to the usual DQF COSY experiment.t99) COSY

Scheme F

This is a modification’*03) of the w,-scaled DQF COSY experiment to selectively scale certain coupling constants in a network of several coupled spins. A selective 180” pulse applied near the middle of the t, period to invert only a few of the spins refocusses all couplings to these spins from other spins which are not affected by the selective pulse. The non-selective 180” pulse in the middle of the second half oft, period is a composite pulse which performs better in refocussing chemical shifts and inhomogeneities. For a weakly coupled AMKX spin system of the type shown in Fig. 18, application of the selective 180” pulse to the A-spin, for instance, decouples the A-spin from every other spin. By suitably selecting the position of the central selective pulse, it is possible to eliminate different couplings from the q-axis of the spectrum. All the chemical shifts will be scaled by a factor of l/2 along the wr- axis of the 2D spectrum.

34

R. V. HOSUR A3-I’H

AT-I’M I

XIF- COSY

I

I

N,- scaled Z’OF-COSY

A7-2’n 2.6

” E d s I

A3-2’H2.6

A7- 2’;1-

3.0

A3-2% -

6.3

6.1 w,tp.p.m.)

63

61

FIG. 27. Comparison of resolutions in cross peaks in normal double quantum filtered and w,-scaled double quantum filtered COSY spectra of an oligonucleotide, at 500 MHz. The spectra were obtained under identical conditions with 568 t, points. Final digital resolutions after suitable data processing are 3.9 Hz,‘pt along wI and 1.9 Hz/pt along wj (from Ref. 100).

A critical aspect of the homonuclear scaling experiments discussed above is the accuracy of the 180 pulse. This pulse separates the evolution periods for the chemical shift and J-coupling parts of the Hamiltonian. If the pulse is not accurate it will lead to coherence transfer effects which result in stray additional diagonals in the correlated spectrum. Besides, chemical shift refocussing will not be perfect which affects the scaling factors and eventually folding of frequencies occurs along the o,-axis of the spectrum. Therefore it is important to measure the 180” pulses properly and use of composite pulses which compensate for RF inhomogeneities will be preferable. The general scaling techniques discussed in the above paragraphs can also be incorporated into other forms of correlated spectroscopy such as relayed COSY,“‘“‘) Z-COSY,““*’ TOCSY,‘log) SECSY,‘l’o) etc. A similar scaling technique has also been reported for the heteronuclear correlation experiment.“” 5.2.1.3. Relaxation eficts. In all the previous discussion of the scaling experiments, relaxation effects were not considered explicitly for the sake of simplicity of analysis of the experiments. However, relaxation does occur during the evolution period of each of the schemes and, intuitively, it is expected to change (scale) the linewidths along the w,-axis of the 2D spectrum. The density operator equation of motion in the presence of relaxation is:‘45’ b(C)= - ic*o,, g(t)1 - l-b(t) - a,)

(86)

where r is the relaxation superoperator, co is the equilibrium density operator. Under certain conditions such as neglect of non-secular terms in the relaxation matrix (Matrix representation of relaxation superoperator in the eigenbase of the Hamiltonian) and in the absence of degenerate

35

Scaling in ID and 2D NMR spectroscopy

transitions, eqn. (86) simplifies to: u.*,(r) = - iw,(t)

u,.(t) + R,.,,

u&t)

= ( - r&,,(r) + R,G,e)u,r(t)

(87)

where a, a’ refer to the eigen states of the Hamiltonian. u,(t) o,. = (w, - w,,). The solution of eqn. (87) can be written as: u,.(r) = u,(O) e - i+f

refers to the ax element of u(t),

e - rITzp..

(88)

Tzmme is the spin-spin relaxation time of a transition between x and z’ states. Thus, each element of the density operator oscillates with its characteristic frequency o,,, and decays in a manner determined by its characteristic relaxation time. In weakly coupled spin systems, considering the dipole-dipole interaction as the sole relaxation mechanism, one can assume identical relaxation times for all transitions belonging to a particular spin. In such a case it is possible to talk of relaxation of a particular spin rather than of a transition. Thus, the relaxation effects can then be incorporated into the product operator description of the various multipulse experiments. For example, one can say that the I*, operator which corresponds to y-magnetisation of &-spin should be multiplied by exp ( - t/Tzt) where Tzt is the spin-spin relaxation time of k-spin. As an illustration we explicitly show below the calculation of relaxation effects for pulse scheme A of Fig. 19. The fate of the k-spin magnetisation in a weakly coupled 2 spin system is traced below. I kz (equilibrium)

1 90, I ky

I I ky

evolution under isotropic 31” during (r-pt,)

1kxlIz

I

1kx

Ikybz

T, relaxation of k-magnetisation during (T-pt,)

I’kx

1 I kz

90, L

11,

Ii:

I kx

11,

I

Among these the observable elements are Ii, and Ii,Ir, , the latter leading to observable magnetisation after evohrtion during t, . I;, or Ii, represent the modifications to IkXor Ikr due to relaxation during the period (r - ptr). Since the 180” pulse in the experimental sequence refocusses the field inhomogeneities during a part of the evolution period, the individual transverse magnetisations decay for a part of the time with characteristic time constant Tf and for the rest of the time with the time constant T2. Thus: ri, = I,, e - CT- t2 - P)r*llTf e - 31 - P)rl/T* = I,, e - r/V e - r,/rIC - (2 -

PI +

31

-

~)fflrzl.

(89) (90)

R. V. HOSUR

36

For a conventional

COSY experiment: I;, (COSY) = Ik, e - f~irl

(91)

comparing the t ,dependent parts in eqns (90) and (91), it is clear that (Tf )- 1or the line width will also be scaled by a factor L in the scaling experiment of scheme A. L = - (2 - p) + 2(1 - p) Tf/T, If Tz = T, , i.e. if field inhomogeneities

.

(92)

are totally absent, then L=-p

(93)

which indicates that the line widths will be scaled by the same factor as the coupling constants (J) but with a negative sign. This amounts to resolution enhancement along the o,-axis of the spectrum. Calculations of the type described above are important to assess the performance and utility of a particular scaling experiment. The line width scaling factors for the pulse schemes described in Fig. 19 are listed in Table 2 along with the shift and J-scaling factors. It is observed that for the ideal case, Tf = T,, schemes A and D achieve resolution enhancement, while in schemes B, C and E and F the line width scaling factor is positive and equal to the J-scaling factor. Under these conditions J-scaling by the schemes B, C and E and F does not lead to specific advantages. However, in practice the condition Tf = T, is very rarely satisfied and J-scaling does become useful as will be demonstrated later. Exclusive shift scaling downward by the schemes B, D, E and F helps in increasing the multiplet resolution along the w,-axis, since higher ty” values can be achieved for any given number of t,-experiments as compared to the COSY experiment. However, this must be carefully performed with judicious choice of scaling factor so as to avoid loss of information due to overlap of cross peaks. Scaling up of shifts exclusively by scheme C or D helps in increasing the separation between the cross peaks along the o,-axis of the correlated spectrum. It is important to note here that the widths of the cross peaks along the o,-axis are not significantly altered, since the coupling constants are not scaled and the line widths also are not scaled. This is in spite of the fact that for a given number of t, experiments ty in COSS (scheme C) will be (X)-I times that in COSY, since a reduction in t1m*rcontributes more to the loss of resolution in a complex multiplet and not so much to the total width of the multiplet. A lower value of tI”g’ is also suited for better optimisation of sensitivity. 5.2.2. 20 NOESY Spectra. In contrast to the display of J-coupling correlations in COSY type spectra, the NOESY technique was developed to display the dipolar coupling correlations or chemical exchange correlations in molecules. (1f1.112) The former interaction is responsible for the nuclear Overhauser enhancement (NOE) and hence the name NOESY for the experiment. Again, before we discuss scaling in NOESY spectra, we will give a brief account of the NOESY experiment itself. 5.2.2.1 Basic pulse sequence. The basic experimental scheme for NOESY is shown in Fig. 28A. During the evolution period t the various spins are frequency labelled and at the end the magnetisation in the xy plane is converted into z-magnetisation, multiple quantum coherence. etc. by the application of the second RF pulse. From these only the z-magnetisation is retained by suitable phase cycling of the first two pulses. This situation constitutes the non-equilibrium state whose evolution is L ,

TAELE 2.

Scaling factors of NMR parameters for the pulse sequences in Fig. 19

Experimental sequence A B C D

E and F’ ’ In F, p = l/2.

Chemical shift (4

Coupling constant (1)

Line width (L)

2-P 1 I

- (2 - p) + 2(1 - P)T;/T, x+c;-z)T*‘T 21 2 x-(xI)T*‘T 21 2

If,

1 - (; + Xx) TT/T, 1 - p(l - G/T,)

Scaling in ID and 2D NMR spectroscopy

37

FIG. 28. (A) Pulse sequence for 2D NOESY experiment; T, is the mixing time during which magnetisation migrates from one spin to another. This time is synchronously varied with t, in the ACCORDION experiment. (B) Simulated signal shapes of diagonal and cross peaks along the w,-axis in an ACCORDION experiment. On the left the shapes of the FIDS are shown and on the right the frequency domain line shapes are drawn. The top signal corresponds to the diagonal and the bottom one lo the cross peak (reproduced with permission from

Ref. 128). monitored in the NOESY experiment. As the system returns to equilibrium during the next period T,,, called the mixing period, exchange of z-magnetisation occurs between the spins. This exchange can occur via chemical exchange or via cross relaxation or both depending upon the experimental system under investigation. At the end of the period T,, the partially relaxed z-magnetisation is converted into observable x, y magnetisation by the application of the third RF pulse and finally the signal is detected as a function oft, . Two dimensional Fourier transformation of the data leads to (i) diagonal peaks which represent residual z-magnetisation of each spin at the end of rrn period, (ii) cross peaks which represent transfer of magnetisation between spins either by cross relaxation or chemical exchange, and (iii) axial peaks which occur at w, = 0 and represent z-magnetisation which has recovered during t, period and thus has lost its frequency labels. The axial peaks can be eliminated by suitable phase cycling schemes. Considering a two site exchange process, A o B, the z-magnetisation at the beginning of T, period may be written in a simplistic form as MA,

= - MA0 cos o,t,

e-*1’T*

MS&J

= - MB,, cosq,fl

e-‘L/T2.

(94)

These t,-modulated z-magnetisations migrate from one site to another during I, from cross relaxation or chemical exchange, while spin-lattice relaxation attenuates the memory of frequency labelling. The time evolution of these magnetisations can be calculated from the rate equation:

-&AM&,,,) = LAM,&,,) m

with

AM&,) = M,k,,) - Me L=-R+K

(96)

R. V. HOSUR

38

where R is a matrix of auto and cross relaxation rates, K is the chemical exchange matrix containing exchange rates as its elements and M, is the equilibrium magnetisation. The solution of eqn. (95) is written as: M,(r,)

= M, + eLrmAM,(r,

= 0) .

(97)

More explicitly, this solution can be written .for the two site case as: MA,(T,)

=

MAt(Tm

= 0)

:[I + emzkrm]ewrm’l”

+ hfBz(r,

]e-h/Tl

+

MAr(~,

= 0) i [ 1 -

=

ewzkrm] e-‘m’r’

())_![I_ e-2k’~)e-‘m’T1

(98) (99)

where k is the rate constant for magnetisation transfer process. The first term in the eqns (98) and (99) leads to the diagonal peak and the second term leads to a cross peak after 2D Fourier transformation of the time domain data. The amplitudes of diagonal and cross peaks for the two site case can be written as: I,, (Or ISB)

=

uAA

(Or

oBB)

MAO

uw

IAB

=

aAB

(or

OBA)

MB0

(loo’)

(or

IBA)

with BAA

= Afro

= f [1 + e-2krm] e-rm’T1

UAB(~,) = UgA(T,) = f [* - eezkr=] e-rm’T1.

(101)

The diagonal peak intensity decreases monotonously with increasing r,,, while the cross peaks build up initially because of exchange process and then die off because of spin lattice relaxation. In systems containing more than two sites, say, k, I, m, n, , . . , analytical solution of eqn. (95) is difficult; however, if r, is extremely short, an approximate solution can be obtained by setting e“m=

1 +Lr,.

(102)

For diagonal and cross peak intensities, one then obtains: akk = 1 + Lkk&,

(103)

OkI =

ww

LkI

5,.

This is referred to as initial rate approximation. This procedure has been used extensively for estimation of cross relaxation rates in several biological macromolecules; the cross relaxation rates are then used to estimate interproton distances in these systems.(113-‘25) The structures of the peaks in the NOESY spectra depend upon whether the cross relaxing or exchanging spins are J-coupled to other spins in the molecule. In such an event, both the diagonal and cross peaks will have multiplet structure. However, all these components will have the same phases which is a consequence of net transfer of magnetisation between the interacting spins. Therefore, even in NOESY spectra where all cross peak components are not well resolved, the integrated intensity of a cross peak is entirely dependent on the cross relaxation efficiency which is in contrast to 2D COSY type spectra where cancellation of the positive and negative components in a cross peak is an important deterrent to sensitivity. In this context, it must be mentioned that sometimes, particularly at short mixing times, zero quantum artefacts may occur and alter the intensities of NOESY cross peaks. Methods have been described in the literature to eliminate these artefacts so that one can reliably estimate cross relaxation rates from the intensities of the cross peaks.t126*12’) X2.2.2. Scaling modifications. The experimental schemes (A-D) shown in Fig. 19 and discussed in the context of 2D COSY spectra can be incorporated in the NOESY experimental scheme as well. The portion of the experimental scheme up to the first two 90” pulses is replaced by those in Fig. 19, and

Scaling in ID and 2D NMR spectroscopy

39

the phase cycling schemes are also different for the selection of z-magnetisation. The criteria for the selection of scaling factors z and 7 is different in view of the different structures of the cross and diagonal peaks. Since in pure phase NOESY spectra, all the cross peak components have the same phases, one can select a 7 < 1, so that the width of a cross peak along w1 is reduced and the cross peak components overlap to a larger extent resulting in greater peak heights in the spectra. In this context the selection of the experimental scheme for scaling is important. Schemes derived out of A, C and D of Fig. 19 produce mixed phases and hence are not suitable. However, o,-decoupling by schemes A (p = 0) and D (7 = 0) can be used which not only removes all multiplicity along w1 but results also in greater separation between the peaks. But signal decay due to relaxation during the constant time period of the experiment may sometimes prove detrimental. If the scheme B of Fig. 19 is selected, pure phases can be obtained, but if 7 has to be less than 1, I, the shift scaling factor, also has to be less than 1 (note a c ‘J). This would imply a reduction in the separation between the cross peaks along the w,-axis. However, if the peaks are well separated, this scheme would be the ideal choice, since there is no loss in sensitivity and a particular ty value can be achieved with a smaller number of experiments as compared to NOESY without scaling. This will also save instrument time and disk storage space. Thus, better intensities of cross peaks are obtained in an w,-scaled NOESY experiment performed with z = 0.5, 7 = 0.6 and 256 t, increments than in a NOESY experiment with 512 t, increments; tyX will be identical in both cases. An important point to note in the optimisation of NOESY experiments is that multiplet resolution is not as much of a concern as the peak heights. A second type of scaling experiment is possible in the NOESY by carrying out manipulations during the mixing period of the experiment. A typical example of this has been reported and is called the ‘ACCORDION’ experiment.~‘28~129) H ere the mixing period is synchronously incremented with t, in the course of the experiment, (105) Ll =zt,. Substitution

of eqn. (105) in eqn. (98) for a two site exchange process leads to: Mar

= - MAocoso,t,

‘C

e-‘IITa j

1+

e-zk~~l]e-x’l/TI

The first term represents the diagonal peak while the second represents the cross peak arising out of magnetisation transfer from spin B to spin A. It can be seen from eqn. (106) that after 2D Fourier transformation of the data, the line widths as well as line shapes along the wI axis appear altered in all cross and diagonal peaks. Equation (106) can be rearranged to:

with

R, = l/T,;

R, = l/T,.

Thus, each diagonal peak will be a sum of two lorentzian absorption lines with line width factors R,[I + %R, T,] and R2%T2(R, + 2k) and each cross peak will be a difference of the same two absorption lines. Figure 28B shows simulations of cross and diagonal peak line shapes for a two site exchange process. It is interesting to note that one of the linewidth factors has the exchange rate k and therefore a linear combination of diagonal and cross peaks provides a method for estimating k from a single NOESY experiment (see below). 5.3. 20 Multiple Quantum Spectroscopy 5.3.1. Basic Pulse Sequence. The basic experimental scheme of 2D multiple quantum spectroscopy is shown in Fig. 29A.“30’ A pair of 90” pulses separated by a period 2r is used to create multiple

40

R. V. HOSUR

A-M-X

B

.

-

.-sm

P P - co+

0

. .O.‘\ .00

‘\

P co+B

r+sm

2

$ ;

=+s~rG-

2

.O

;

2

I

I‘ I’ .A.

,‘i m e

. . ,” .

.

t

. .

.*

0

s7x5-

D

2

=-sin -B cos5-B

. = + srPf

2

.D

0

.m.

cos -

\\

. * + 5,” -P 2

.m.,’

2

:

-

; co@-

2

P co& 4

s+-

0’ . ;2 -=- 2, 2 =+srP-

;

cos -

sm5- cos -

.C.

.m.

;

FIG. 29. (A) Pulse sequence for two dimensional multiple quantum spectroscopy; r is a constant time period used to treat multiple quantum coherences. (B) Schematic double quantum spectrum of a linear A-Xl-X spin system with JAx = 0. Filled and open symbols represent positive and negative signals. Different symbols are chosen CO indicate different intensities (from Ref. 130).

coherences (MQC) which then evolve during t, with characteristic frequencies. The third pulse converts the MQC into observable magnetisation which evolves in t, with characteristic single quantum frequencies. Figure 29B shows schematically the appearance of a double quantum spectrum of a linar three spin system A-M-X, with J,,x = 0. Excitation efficiency of MQC (or also referred to as multiple quantum transitions MQT) is critically dependent on the period r and in general in a system with widely ranging coupling constants, all MQTs will not be excited uniformly. Several pulse schemes have been proposed in this context,“30-134’ and we will see later that scaling happens to be one of the approaches. The peaks in 2D double or zero-quantum spectra can be classified into two categories. (1) Direct peaks which arise from the transfer of double or zero quantum coherences to single quantum coherences of the participating spins. (2) Remote peaks which arise from transfer of DQ/ZQ coherences to single quantum coherences of a passive spin which is coupled to both the participating spins. Thus, in Fig. 298, the peaks at (w, = o, $ wx. w1 = W, or ux) are direct peaks, and the peak at (w, = We + wx, w2 = uh() is a remote peak. In more complex spin systems many more peaks occur and the details of their multiplicity and phases have been discussed in the literature.“30’ It may be noted in Fig. 29B that the direct peaks have in-phase character along the w,-axis while the remote

quantum

41

Scaling in 1D and 2D NMR spectroscopy so

180

T I I I I BI

I

82

I ( I

83

B4

;-(,,2kt,( I

90

r

a tl I I 1

I I I Y I I 1I

P

I I

II j /-(A-n,

1-j I ytl

I I

!_I

I r(2b+2ktl$ I

I80 -

I I =I t I

yt~

-

’ k--t

I

I

I

I

I A-Q

)A

I

I

I

FIG. 30. Pulse sequencesfor scaling in 2D multiple quantum spectroscopy (from

Ref. 136).

peaks possess anti-phase characteristics. This is in general true of linear spin systems. An important problem associated with MQT spectra recorded with the scheme on Fig. 29A is of mixed line shapes (linear systems however have pure phases). Therefore, in complex spin systems it becomes necessary to present MQT spectra in the absolute value mode, resulting in lower resolution compared to phase sensitive spectra. Recently a remedy has been suggested to alleviate this problem.t*35) A Z-filter is used to convert MQT into 2-spin xx order which is then converted into observable magnetisation. 5.3.2. Scaling Modifications. Figure 30 shows some pulse sequences which have incorporated the basic ideas of scaling into the pulse sequence of Fig. 29A.(136) While schemes Bl and B2 reflect manipulations of the type discussed irl connection with 2D COSY spectra, B3 and B4 are aimed at achieving uniform excitation of multiple quantum transitions. All the four-pulse schemes achieve shift scaling of multiple quantum coherences by a factor r and the Js are scaled by a factor 7. In schemes B3 and B4 ‘accordion* type of excitation is incorporated. This typ of excitation increases the multiplicity of peaks and therefore the constant k has to be chosen small enough so as not to lose resolution significantly in the spectrum. Schemes Bi and B4 are useful for o,-decoupling (‘1= 0) which can be used to selectively enhance particular peaks in the spectrum. In all the schemes in Fig. 30 the scaling factors x and 7 can be chosen independently. Scaling down of the shifts (z < 1) allows reduction of the spectral widths along the @,-axis and thus t? will be higher compared to the normal multiple quantum spectrum for the same number of r, increments. The J-scaling factor 7 can also be chosen smaller than 1 so that the width of the peak along the o,-axis will be reduced. Both these factors lead to increased resolution along the o1 axis as compared to that in normal multiple quantum spectrum. Figure 31A shows a comparison of a section of a normal double quantum spectrum with an identical section of a or-scaled double quantum spectrum of a 3-spin system. The multiplicity of the peak at the double quantum frequency Ok + urx, which is a doublet with a separation of 15.4 Hz, can be seen clearly in the @,-scaled spectrum. The lower peak at the frequency cc,, + oh( has a smaller separation of the components (7.4 Hz) and hence is not resolved in both the spectra. Figure 31B shows the comparison of intensities of direct (D) and remote

R. V. HOSUR

42

a

b

1

FIG. 31. (A) Comparison of two direct peaks from a normal double quantum spectrum (b) and an or-scaled double quantum (a) spectrum of a three spin system (vinyl acetate). The top peak in the w,-scaled spectrum shows higher resolution than in the normal spectrum. In the lower field peak the components are not resolved because of smaller values of coupling constants involved. Shift scaling factor = 0.5 and both are a result of 256 t, and 2048 t2 data points collected under identical conditions. (B) UJ~cross sections at a particular double quantum frequency through normal and w,-scaled double quantum spectra of the same 3-spin system as in (A). In the o,-scaled DQ, shift scaling factor = 0.5 and J-scaling factor = 0.6. Enhancement of direct peak intensity in the o,-scaled spectrum may be noticed (from Ref. 136).

(R) peaks in the normal double quantum and w,-scaled (I = 0.5; y = 0.6) double quantum spectra. A horizontal cross section is taken at the co, frequency of wA + w, . It is seen that the direct peaks are stronger in the o,-scaled spectrum (bottom trace) than in the normal double quantum spectrum (top trace). This can be attributed to larger contribution of in-phase components to the direct peaks in the 2D spectrum.

Scaling in 1D and 2D NMR spectroscopy 6. APPLICATIONS

43

OF 2D SCALING EXPERIMENTS

We have seen in the previous sections that incorporation of scaling ideas into different forms of 2D spectroscopy leads to specific advantages with regard to intensities of peaks, resolution in spectra, dispersion of peaks, etc. These features enable some specific applications which we shall discuss in the following paragraphs. 6.1. Observation of Long Range Correlations Long range correlations provide useful assignment tools to the chemists and biologists in a variety of molecular systems. In conventional COSY spectra, J-correlations between spins which are separated by more than three bonds are usually not observed, exceptions are unsaturated conjugated systems where 4-5 bond correlations are indeed seen. The reasons for this are quite clear, and are related to the magnitude of the coupling constants, and resolution in the spectra. Often the resolution

i

6

Id

J- Seokd

COSY

WI

IPPY)

,

_‘1 6bi6Ci6

~6lPPw

&I- 'CMI'

6

r -1mw

6

FIG. 32. (A) Pulse scheme for q-scaling and w,-refocussing in COSY for observation of long range correlations. The refocussing delay r2 should be set to approximately l/U where J is the long range coupling constant. (B) (a) Experimental COSY spectrum showing long range (4 bond) correlations between base protons and sugar Hl’ protons observed in a dinucleotide s%C-2’. rs = 0.15 s, J-scaling factor = 6. (b) wt-cross section at ws = chemical shift of CH6 proton. The two doublets around 6 ppm correspond to CHS and CHI’ protons - (reproduced from Ref. 137).

R. V. HOSUR

54

along the o,-axis of 2D COSY spectra is quite poor because of limited instrument time, disk storage space, sensitivity, etc. Therefore, if the coupling constants are small, the positive and negative signals arising from these active couplings are not well resolved and as a result cancel their intensities. In such situations the J-scaling procedure whereby J-values can be scaled up by an order of magnitude has been found useful. Two approaches have been used in the literature. (i) J-scaling along w, and refocussing along w2 in COSY: while J-scaling increases the separation between the positive and negative components by an order of magnitude along w,, refocusing along wz changes the + - character into + + character in the cross peaks as in the SUPERCOSY experiment. Thus, cancellation of components along both frequency axes is avoided. Refocussing is achieved by giving a delay (approximately of the order of 1/2J) after the second n/2 pulse of the COSY sequence.““’ An alternative to this procedure of avoiding cancellation of + - components along the o2 axis would be to acquire a large number of data points along the t, axis so that a very high resolution can be obtained along the w,-axis. If a coupling constant of the order of 0.3-0.5 Hz has to be resolved, it may be necessary to acquire 32K (IK = 1024) or more data points along the t,-axis; this of course assumes that the line widths are smaller than the coupling constants. While this procedure saves n sensitivity to some extent, it is highly demanding in terms of disk space and data processing time. $ herefore, a compromise solution has often to be obtained by partly refocussing and partly increasing resolution along the w,-axis. Figure 32 shows an experimental demonstration of the above procedure”37’ on a dinucleotide molecule; 4-bond correlations can be seen between the base protons and the sugar Hl’-protons in the two units. An WI-cross section shown below clearly shows two doublets, which belong to CH,-CH, and CH,-CH’, couplings in the cytosine unit of the molecule. (ii) J-Scaling in spin echo correlated spectroscopy. Figure 33A shows the experimental scheme for J-scaling in spin echo correlated spectroscopy (SECSY), which is a variant of the COSY sequence, J’ lo) the /I pulse is applied in the middle rather than at the end of the evolution period. From the view-point of observation of long range correlations, the disposition of the cross peak components in a spin echo correlated spectrum (Fig. 33B) offers some advantages. It is evident that J-scaling along o,is adequate to avoid cancellation of intensities altogether; refocussing is not needed along w2 and also high resolution data collection is not warranted in the t,-domain. Figure 33C shows an experimental observation of long range correlations in a decapeptide recorded without employment of refocussing delays prior to data acquisition in the pulse sequence.“00’ A general disadvantage of the SECSY, however, is that peaks have mixed line shapes and absolute value presentation is necessary. But, for the purpose of observation of long range correlations in many peptides and nucleotides, this may not be a deterrent, since one is not hoping to resolve the components and measure the coupling constants here. Observation of long range correlation is primarily used as an assignment tool. 6.2. Measurement of Coupling Constants Coupling constants are important inputs to the structure determination of molecules in solution. Three bond coupling constants are related to torsion angles around the central bond by the so-called Karplus type relations: 3J = A COG 4 + B cos 4 + c

(log)

where 4 is the relevant torsion angle and A, B, C are empirical constants. Coupling constants can be measured by 2D J-resolved spectroscopy either by taking w, cross sections from a spectrum which has exclusive chemical shifts along o2 and exclusive Js along w1 or by taking projections on a suitable axis so that a well resolved 1D spectrum is obtained. In the latter case, the measured J-values will be scaled coupling constants and appropriate multiplications can be carried out to obtain the true values. The 2D J-resolved spectroscopic technique often suffers from problems of low sensitivity. Different portions of the spectrum have different ‘sensitivity’ which is a consequence of different multiplicities and different relaxation times of spins. Consequently all components of peaks may not be visible. For example, in J-resolved spectra of oligonucleotides, the sugar Hl’ multiplets are often easily discernible

Scaling in ID and 2D NMR spectroscopy

ye.

I

Wl

I_ .

0

.

45

. .Wa

0

. . 0

C

-l.O-

s t5 4

I I.0

-

2.0

-

I I

1

7.0

5.0

JO

FIG. 33. (A) Pulse sequence for J-scaling in spin echo correlated spectroscopy (SECSY). (B) Schematic J-scaled SECSY spectrum of a two spin system. Filled and open circles indicate positive and negative signals. (C) Experimental J-scaled SECSY spectrum of a decapeptide (Luteinising Hormone Releasing Hormone) showing long range (4 and 5 bond) correlations between aromatic protons and /I protons (also II protons) in the peptide. J-scaling factor = 9. No refocussingis done along 0s. Connectivities are shown for His residue. The peaks marked A and B belong to another aromatic residue in the molecule (from Ref. 101).

H2’, H2” and H3’ multiplets are very rarely completely resolved. In such a situation the measured sets of coupling constants (from the Hl’-region in the above example) from a given multiplet cannot be assigned uniquely to individual coupling constants. In this sense, 2D correlated spectroscopy offers certain advantages, since frequency information is present along both axes and analysis of cross peak patterns yields the appropriate coupling constants with stereospecific assignments. For instance, by the analysis of a Hl’-H2” cross peak in an oligonucleotide or NH-C’H cross peak in a peptide one can identify specifically the HI’-HZ” and HI’-HZ’ coupling constants in the former and NH-C’H, C*H-CbH in the latter. The COSY technique has relatively higher sensitivity as compared while, sugar

46

R. V.

HOSUR

to J-resolved spectroscopy and also pure phase spectra can be obtained in the former as against absolute value spectra in the latter. Extraction of coupling constants from COSY cross peaks requires of course that all the cross peak components be well resolved. As discussed in previous sections, the multiplet resolution in a cross peak or in a diagonal peak is determined by the CT and ty values of 2D data acquisition. It is relatively easy to increase ty and obtain good resolution along the o,-axis of the 2D spectrum. However, ty* is severely limited by the instrument time available. In such a situation, two alternative approaches can be adopted to resolve the components of the cross peaks along the o,-axis. (i) The J-values can be scaled up along the w,-axis so that separations between the cross peak components get enhanced. The scaling factor should be minimally chosen so as not to increase the width of the peak beyond what is necessary; this is to avoid overlap of peaks along the @,-axis as far as possible. This procedure is exemplified in Fig. 21. (ii) The chemical shifts may be scaled down without altering J-values. In this case resolution enhancement is achieved by virtue of an enhanced ty* value. Reduction of chemical shifts leads of course to reduction in spacing between the cross peaks and may

A 2.0

“I-IO

? h

R GIL? R

G9

6.0

G4 G2

2.8

5.8

5.6

HI’

l-10

H2”

Hl’ FIG. 34. (A) Pure phase q-decoupled COSY spectrum of an oligonucleotide d-CGCGAGTTGTCGCG recorded on a 500 MHz FT-NMR spectrometer at 2s”C. 400 f, experiments were performed with 2048 rs points each. The data was zero filled to achive 0.87 Hx/pt digital resolution along ws in the final spectrum. ry was 56 ms. (B) A wscross section through T10 cross peak in (A) showing the Hl’ multiplicity and the HI’-HZ’, HI’-H2” coupling constants (from Ref. 146).

Scaling in ID and 2D NMR spectroscopy

47

lead to overlaps; therefore the scaling factor has to be again optimally chosen. However, it must be remembered that in favourable cases, the information can be retrieved by monitoring peaks on both sides of the diagonal. A combination of both the above approaches has also been used in certain instances. Yet another scaling approach which is useful for estimation of the homonuclear coupling constant is w,-decoupled COSY. In Fig. 19, if p = 0 in scheme A or y = 0 in scheme D one obtains an otdecoupled pulse scheme. (14s*146)Note that under these conditions it is possible to obtain pure phase spectra. Cross peaks can be phased to pure absorption along both the o,- and w,-axis. In this experiment the entire multiplicity along the w,-axis is removed so that a very high separation between the cross peaks is obtained in the 2D spectrum. This, of course, does not lead to loss of information; in fact, in COSY spectra, there is a redundancy of information since the active coupling constant of a particular cross peak is present along both the frequency axes of the spectrum. Thus, from the w,-decoupled COSY spectrum, measurement of coupling constants always has to be performed along the w,-axis. Each cross peak will have the multiplicity of a single proton and hence can be analysed as in highly resolved ID spectra. Peaks on both sides of the diagonal will have to be used to measure all the coupling constants in a given molecule. Figure 34A shows the same portion of an w,-decoupled COSY spectrum of the same oligonucleotide as in Fig. 22. Comparison of these spectra clearly demonstrates the high dispersion between the peaks obtainable by o,-decoupling. A o+cross section through one of the peaks shows the positive and negative signals which can be used for coupling constant measurements (Fig 34B). Accurate values of the coupling constants can be obtained by simulation of antiphase multiplets. Coupling constants have also been estimated by selective J-scaling of certain coupling constants while eliminating certain other coupling constants from the spectra.“03’ Such a procedure becomes necessary when cross peaks in a COSY or DQF COSY spectrum have very complex multiplet structures consisting of several coupling constants. This is exemplified in Fig. 35 with the H3’-H4

Cl

n6

4.1

42

PPM

4’

F2 43

G4 _ _

tf’ G2

GIO

44

45 I

5.0

4.9

48

4.7

PPM F,

FIG. 35. H3’-H4’ cross peak region of a spectrum recorded using the pulse scheme F of Fig. 19.The o, (F,) axis has exclusively H3’-H4’ coupling constant, and the I values are ctTectivelyscaled by a factor of 2. The digital resolutions in the spectrum are 0.5Hz and MHz along o, (F,) and r.u2(FJ axes respectively (reproduced with permission from Ref. 103).

48

R. V. HOSUR

cross peaks of an oligonucleotide. (“‘I The spectrum has been recorded with the pulse scheme F of Fig. 19. All couplings to H3’ except the active coupling, J(H3’-H4’). are eliminated along the o,-axis and consequently the cross peak has a simple doublet structure along the o,-axis. In conventional COSY spectra, H3’-H4’ cross peak would have the H3’-P, H3’-HZ’. H3’-H4’ and H3’-H2” coupling constants along the H3’-axis and H3’-H4’, H4’-HS and HI-HS” couplings along the HI-axis of the cross peak. w,-scaling techniques in multiple quantum spectra are also useful for coupling constant determinations. Peaks in multiple quantum spectra have less complexity, since certain coupling constants do not

appear along the w,-axis.

6.3. Measurement of Exchange Rates

The exchange rates ‘k’ can be measured from the NOESY spectrum. of mixing time vs. cross peak intensity

In the so called ‘linear regime’ curve, (i.e. if /CT, Q l), eqn. (98) reduces to:

M,,,( 5,) = - MAOcos wAt, e -rl’Tz (1 - kr,)

e-r”‘iT1 - MB0 cos wBtle-‘iiT2 kr, e-rmirl. (109)

Thus the diagonal peak intensity ( IAA) and the cross peak intensity (I,,) at a given mixing time T,,,are proportional to the coefficients of the t, dependent 1st and 2nd terms respectively in eqn. (109): IAAz MAo( 1 - ks,) e- bdr, fABrMsoks,

eSrmirI.

(110)

FIG. 36. Linear combinations of w,-cross sections taken at two chemical shifts w2 = R, and wI = R, (connected by a cross peak at (cu,, WJ = (fi,, Q,)) from the ACCORDION spectrum of cisdecalin, at four different temperatures. The spectra on the left show the sums while the spectra on the right show the differences. Each peak results from addition (or subtraction) of a diagonal peak line shape and a cross peak line shape. The di&rence in the linewidths on the right and left side spectra can be used to measure exchange rates (reproduced with permission from Ref. 128).

Scaling in 1D and ZD NMR spectroscopy

H5”

OH

“(q-cNH2

Ci

0

c HS’

N --c

H3’ \

49

0

H2’ H’ ’

H4’ b

s 0 O\

P

H2”

/O

II

%l

FIG. 37. Molecular structure of a segment of DNA. Four types of bases exist and these are adenine (A), thymine (T), guanine (G) and cytosine (C). The figure shows the nomenclature of the atoms and the mode of covalent linkages (for G and C) in a long chain of such repeating units. Thick arrows identify observable J-coupling correlations.

For a homonuclear

symmetrical two site exchange MA0 = MB0 and then:

1 - kl

I AA

_

I AB

-kr,-

(111)

This provides a quantitative estimate of the exchange rates. Although the peak intensities at a single mixing time are used for estimation of k, experiments will have to be carried out at several mixing times to assess the validity of ‘initial rate’ or ‘linear regime’ approximation. An alternative to the above procedure wherein exchange rates can be obtained from a single experiment is to use the ACCORDION experiment discussed earlier. Here the mixing time is incremented synchronously with t, and the exchange rate then influences the line shapes and the line widths of cross and diagonal peaks. While the diagonal peak shape along o, will be an addition of two Lorentzian absorption signals, one being dependent on k, the cross peak shape will be a subtraction of the same two line shapes along the ot-axis. The sum of the diagonal and cross peak line shapes yields a narrow Lorentzian defined by:

S+(%‘)=

gfq& 1

(112) m

while the difference of the two lineshapes leads to a broader Lorentzian: S_(w,) =

2k+

R,

(2k+R,)'+o:Z

‘M,.

(113)

50

R. V. HOSUR

The difference in the line widths (in Hz) of these two line shapes is given by II [ - S+(Av,) + S- (Av,)] = 2k k

Or

=

;

[S_ (Av,) - S, (Av,)] .

(114)

Figure 36 shows an experimental demonstration of the procedure. Exchange rates have been measured at different temperatures to obtain estimates of activation energy of the system. 7. SUMMARY The principles of scaling in NMR spectroscopy have been described. We have tried to unify several isolated experiments under the umbrella of ‘scaling in NMR spectroscopy’. After a brief description of the mathematical tools, namely ‘average Hamiltonian theory’ and ‘product operator description’ of density operator treatments of nuclear spin systems, several one dimensional scaling experiments have been analysed. Extending to two dimensions, the versatility of scaling manipulations in the experimental scheme of 2D NMR spectroscopy has been demonstrated. Specific pulse sequences designed to scale the chemical shifts, coupling constants and line widths in three different categories of 2D NMR spectroscopy, namely J-resolved spectroscopy, correlated spectroscopy and multiple quantum spectroscopy, have been discussed. It has been shown that for optimisation of sensitivity and resolution in 2D spectra, the scaling requirements are different in the three different kinds of experiments. At every stage the merits and demerits of the scaling experiments have been discussed. Apart from the general&d applications in sensitivity and resolution enhancements, specific applications in observation of long range correlations, measurement of coupling constants and measurement of exchange (or cross relaxation) rates have been discussed. We believe that this general exposition should enable the NMR spectroscopists to select a suitable pulse scheme and optimise the scaling factors to derive specific results on the systems of interest. Acknowledgements-A significant portion of the work reported in this review has been carried out using the 500 MHz FT NMR National Facility suonorted by Deoartment of Science & Technologv. Government of India. and located at Tata Institute of Fundamental Research.‘Bombay (TIFR). I thank Profe&r’G. Govil at TIFR and Dr. H. T. Miles at National Institutes of Health, U.S.A. for some of the DNA samples. I am thankful to several authors and publishers who have given permission to reproduce their work in this review. I am thankful to Mr. A. Majumdar and Dr. U. Sasidhar for careful reading of the manuscript. Some of the spectra (Figs 22.23 and 34) were recorded in the laboratory of Professor Dinshaw J. Pate1 at Columbia University.

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