Detection of topologies in homonuclear pseudo-3D NMR

JOURNAL

OF MAGNETIC

RESONANCE

95, 320-340 ( 199 1)

Detection of Topologies in Homonuclear Pseudo3D NMR B. N. GRAY AND L. R. BROWN Research School of Chemistry, The Australian National U,riversity, Canberra ACT 2601, Australia Received February 19, 199 1; revised May 20, 199 1 A general three-period homonuclear NMR experiment is analyzed in terms of the minimal scalar coupling topologies which can be detected. It is shown that the number of such minimal topologies is small, that specific topologies can be selected by appropriate choice of pathway types and quantum orders in the experiment, and that in experimental spectra each resonance can be ascribed to a particular minimal topology. The information content of 3D multiplequantum spectroscopy and of many common homonuclear 2D NMR experiments is analyzed in terms of whether a given minimal topology is observable and can be interpreted. The use of pseudo-3D multiple-quantum spectroscopy as a method for automated identification of homonuclear scalar coupling topologies is explored using the computer programs Predictor, Equivalent, and Comparator. o 199I Academic press, IK.

It is now widely accepted that multidimensional NMR spectroscopy is a powerful method for determining structure for quite complex molecules. In a general sense, there are three steps in such analyses: (i) data acquisition, (ii) mapping of NMR spectral parameters extracted from spectral data onto the covalent structure of a molecule, and (iii) calculation of three-dimensional structures using NMR parameters as constraints on possible conformations. All three of these areas are interdependent. For example, the use of NMR spectroscopy to determine 3D structures of biological macromolecules in solution depends critically on the use of constraints derived from cross relaxation between hydrogen atoms. This means that mapping of ‘H resonances to specific hydrogens in a molecule will undoubtedly remain a critical step for structural determinations. Indeed, in practice much of the effort necessary to produce 3D structures from NMR data is expended in the step of mapping resonances to covalent structure and a major potential gain in the efficiency of NMR structural analyses lies in the area of improved methods for performing such mapping. Recent developments in this area have included the development of data-acquisition methods based on isotope labeling, new data-acquisition methods that select for specific topological features of homonuclear scalar coupling networks ( 1)) three-dimensional NMR (2)) and attempts to automate the interpretation of homonuclear multidimensional NMR spectra. In the present paper we address the possibility of using a series of complementary homonuclear experiments which select for different scalar coupling topologies as a means for automated interpretation of homonuclear NMR experiments. There have been a number of attempts to automate the interpretation of homonuclear multidimensional NMR experiments (3--26). The strategy which has been most intensively pursued is high-resolution analysis of single-quantum correlation experiments. An important advantage of this approach is that a single type of exper0022-236419 1 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

320

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iment (COSY or a modified version thereof) contains complete information in the sense that both the number of spins and the topology of the scalar coupling network can be determined by examining the cross peaks and their multiplet fine structures. Unfortunately, this approach will probably prove to be inadequate for large molecules such as proteins because of resonance overlap and because multiplet fine structures are obscured by relatively large linewidths. In practice the analysis of protein spectra usually proceeds by comparison of a series of related spectra including COSY, TOCSY, and NOESY. The analysis of scalar coupling networks is very often based purely on the presence or absence of cross peaks over the series of different experiment types (19-26). The notion of using a complementary series of spectral types to unravel complex multidimensional NMR spectra of macromolecules raises the questions of what is the most useful set of complementary experiments, whether this set contains complete information in the sense noted above, what are the most efficient ways to acquire such data sets, how to compare information from different experiments, and what to do about experimental ambiguities such as overlap of peaks or missing peaks due to limited sensitivity. In the present paper, a method for the automated determination of homonuclear scalar coupling topologies is proposed and demonstrated. The method is based on analysis of the minimal coupling topologies which can be detected in homonuclear 3D spectroscopy, the use of interleaved pseudo-3D multiple-quantum spectroscopy to generate pseudo-3D spectra of quantum orders 2, 3, and 4 from a single data set, and computer programs for automated assignment of individual resonances to appropriate minimal coupling topologies. THEORY

Topological Description of Scalar Coupling Networks Scalar coupling networks can be described in terms of simple, connected, nondirected graphs by considering spins to be vertices and couplings to be edges in the graphs. Thus, for systems with three types of spins, there are two possible types of topologies (Fig. 1). For computational purposes such topologies can be described by adjacency

i

010

i

011

j

101

j

101

k

010

k

110

FIG. I. Graphs and adjacency matrices for the two possible types of connected topologies involving three spins.

322

GRAY

AND BROWN

matrices, A, where Aii = 0 and A, = Aji = 1 if spins Ii and Ij have a scalar coupling or A, = Aji = 0 if 1; and Ij do not have a (measurable) scalar coupling (Fig. 1). It should be noted that the adjacency matrices, like their associated graphs, can be changed by labeling permutations. For a given type of scalar coupling topology involving n spins, there are n ! labeling permutations. This is analogous to the fact that in an experimental NMR spectrum the same underlying spin topology may have many different representations in frequency space depending on the chemical shifts of the individual spins in the topology. For purposes of automated spectral analysis, the possibility of such permutations means that there are two distinct steps to be achieved. The first is to recognize a scalar coupling topology independent of the particular expression in frequency space. The second is to map the frequency space representation of the topology onto a chemical structure. In the present paper we are concerned solely with recognition of scalar coupling topologies. In the limit of weak coupling, any given multidimensional NMR experiment can be characterized in terms of the fundamental types of spin topologies for which the experiment will give a nonzero response function. In these terms, the process of determining the nature of a molecular spin topology can be thought of as probing the spin topology with one or more multidimensional NMR experiments, each of which is sensitive to certain types of topological features which may be contained within the complete spin topology. Determination of the complete spin topology requires logical identification of the complete topology from the topological features detected in one or more multidimensional NMR experiments. The General-Three-Period

Experiment

This experiment corresponds to the pulse sequence shown in Fig. 2, where (/3)+ represents an RF pulse of tip angle ,L3and phase I$ and where t, , t2, and t3 represent three different time periods. Various common 2D NMR experiments are special cases of this general experiment: (i) COSY with multiple-quantum filters ( t2 = 0), (ii) relayed COSY ( t2 = 7, a fixed delay), and (iii) 2D multiple-quantum spectroscopy ( tI = T). As will be seen below, the response functions for these experiments can be obtained from the general result in which t, , t2, and t3 are considered variable time periods. In the following it is assumed that the pulse sequence begins from thermal equilibrium magnetization and that detection occurs during t3. These assumptions mean that only single-quantum coherences are of interest during t, and t3. The spin which is active during t, will be denoted by Ii and the spin which is detected during t3 will

FIG.

2. A general 3D homonuclear NMR experiment.

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be denoted by 4. During t2, the coherences can be of any quantum order which is consistent with the scalar coupling topology which includes Ii and Ii. In general, a coherence-transfer pathway which begins with single-quantum coherence of spin Ii and ends with single-quantum coherence of spin Ij can be described by a graph where the spins shown at each level of the graph are those which are active during the respective time period (Fig. 3). On the basis of symmetries between the three time periods, six general types of pathways can be distinguished on the basis of the following criteria. ( 1) Type of excitation: (2) Type of detection: ( 3) Matching of pathway:

Direct-Ii is active in both t, and t2; Indirect-Ii is active in t , and passive in t2, Direct-Ii is active in both t2 and t3; Indirect-I, is passive in t2 and active in t3, Matched-Ii and Ij are the same spin ; Unmatched-Ii and Ij are different spins.

A pathway which involves direct excitation, direct detection, and a matched pathway is denoted by DDM and a pathway which involves indirect excitation, direct detection, and an unmatched pathway is denoted by IDU. Of the eight possible ways to classify pathways inherent in the above three criteria, the DIM and IDM pathways are physically impossible. The remaining six general types of pathways, together with illustrative

I-I-M:

t2

FIG. 3. Ciraphs showing the pathway types, the active spins in each time period, and the required edges for a general 3D homonuclear NMR experiment with four active spins during tz

324

GRAY

AND

BROWN

graphs, are shown in Fig. 3. It is clear that for a particular pathway to be possible, the active spin during t, (Ii or Ii) must be coupled to all active spins in t2 (other than itself) and the active spin in t3(Ii) must also be coupled to all active spins in t2(other than itself). The quantum order in t2 is not uniquely determined by the graph since combination lines may occur. However, as will be seen below, the type of pathway and the number of active spins during t2determines the nature of the scalar coupling topology which can be detected.

TopologicalSelectionRules Before detailed selection rules for the existence of the six general types of pathways are given, it may be helpful to give a qualitative description of the information content of the general three-period experiment. In the experiment of Fig. 2, after the excitation of Ii coherence by the first pulse, the remaining two pulses transfer coherences twice. The consequence of this is that, independent of the quantum order during t2,topological relationships amongst a group of weakly coupled spins can be investigated only over a span of at most two scalar couplings from spin Ii. In this sense, the selection rules given below for the existence of the different types of pathways can be regarded as describing correlations amongst the possible walks on the adjacency matrix which are of length 2 and which include both Ii and 1,. The selection rules for the general threeQUANTUM PATHWAY

0

1

2

D-D-M

-

0

@-@

D-D-U

-

-

w

D-I-U and I-D-U

-

0RDE:R

3

4

FIG. 4. The minimal topologies required for the existence Iof the six different possible types of pathway as a function of the number of active spins during t2 for the general 3D homonuclear NMR experiment.

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period experiment can then be summarized in terms of the minimal topology which is required for a given type of pathway to exist. These are shown in Fig. 4. The minimal topologies can be derived graphically from graphs of the type shown in Fig. 3; elimination of the redundant vertices and edges for each graph in Fig. 3 leads to the minimal topologies shown for four active spins in Fig. 4. A notable feature of the minimal topologies is that for any given type of pathway, the topologies observed for quantum order Q, are such that they are contained within a topology observed for quantum order Q2 if Q2 > Q, . Furthermore, the topologies for the IIM pathways are observable via the DDM pathway at one higher quantum order and the topologies for the DIU and IDU pathways are observable via the DDU pathway at one higher quantum order. This means that all possible minimal topologies in the general three-period experiment can in principle be observed via the DDM, DDU, and IIU pathways. For up to quantum order 5 in t2, there is a total of 13 distinct topologies involving two or more spins which are directly detectable in the general three-period experiment. A major advantage of formulating the selection rules in terms of the minimal topology required for existence of a pathway is that the spectrum which will be obtained from any arbitrary topology can then be expressed in terms of the numbers and types of minimal topologies which are contained in the overall topology. A precise definition of the selection rules and the possible multiple-quantum states in t2 for each of the six possible types of pathway can be given in terms of the adjacency matrix, A, appropriate for a given spin system (Table 1) . The specific pathways which are possible can be summarized in terms of the sets of frequencies, neglecting splittings due to scalar couplings, which are allowed in the three-dimensional space (0, , w2, w3). These are given in Table 2 in terms of the number and type of spins involved in

TABLE I Selection Rules for Existence of Pathways Pathway DDM IIM DDU DIU IDU IIU

Condition for existence” Wii a Q - 1 W, a Q A,= 1 A,WuaQ-2 A,,= I A,W,,> QA,,= 1 AilWi:jQ-

Wi, a

Q

Number of spinsb

; (321

Maximum quantum order 1 w,i+ 1

Q+l

Wii

2

2 w, + 2 1 w, + 1

1

(a-3) 2 Q + 1 (>3)

1

; + 1 (33)

w, + 1

Q+2

wtj

Q

I

a W = A* denotes a walk matrix for walks of length 2. Q denotes the number of spins which are active in tZ. Where more than one condition is given, for small numbers of active spins, pathways are possible in which only 1, (DDM) or Zi and 4 (DDU, DIU, IDU) are necessary for the pathways to exist. b This is the total number of different spins involved in the pathway.

326

GRAY

AND BROWN TABLE 2

Allowed Pathways Pathway DDM

Number of pathwaysb

Specific pathways”

1

[I~,~,,~jl

[Q?, 1 [Kyi, 1

[I, 34 + (Q - 1Vk,r,l IIM

[I, 3QIkpIII

Q

DDU

[Ii, I, + I,>r,l [I,

DIU

> I,

+

1,

+

1

(Q -

2)1k,

I,]

[ 1 Qw”2

1

[Ia1I, >r,l [I,,

I,

+

(Q -

l)Ik,

r,]

[r,

3 I/

+

(Q -

1 )Ikr

I,]

IIU a The spins which are active in [t,, tz, t,] are indicated. I, stands for any other spin which satisfies either Alk = 1 (matched pathway) or A,kAk, = 1 (unmatched pathway). b The number of pathways is given in terms of binomial coefficients for the number of ways in which the requisite spins other than I, and/or 4 can be chosen. Except for accidental degeneracies amongst the Ik spins, all of these pathways correspond to different frequen ties during tz .

the different pathways. Because different numbers of spins are involved in the various pathways, the pathways will also involve different phases and different antiphase states of the multiplet fine structure. The phase for each pathway is given in Table 3. Mixing of Selection Rules The selection rules embodied in Fig. 4 and Tables 1 and 2 are appropriate when the RF mixing pulses & , p3 = 90”. These rules can be altered in two ways. When all of the RF mixing pulses are 90” pulses, the minimal topologies shown in Fig. 4 for quantum order Q can also be observed in experiments involving quantum orders Q2, Q-4, . . . , due to the presence of combination lines in which Q spins are active during tZ, but the net quantum order is Q-2, Q-4, . . . . Whether these combination lines are likely to be observed depends on the particular experiment being performed. The second way in which the selection rules can be altered is by the use of pulses with p2 and/or & # 90”. In essence, /3 # 90” allows pulses to give transformations

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TABLE 3 Relative Phases for Different Pathways” Pathway DDM IIM DDU DIU IDU IIU

(i)*’

Ha

1 I

(i)Q-’ (i)Q-’

t )

WQ WQ

0) US

(i)Q-’

WQ (i)Q-’ WQ (i)Q-’

(da

’ ~5,denotes the phase during time period ti If t,isafixedperiod,&= 1.

of spin operators of the type I, + I,. Since either Ii or Ij is active in ti , Ij is active in t3, and Ik denotes all other spins active in t2, the transformations P2 Ii,

-

Ijz

-

P3

I.JZ

E+

Ikz

-

L

(unmatched

pathway),

I-JZ

(unmatched

pathway),

I.JZ

(matched pathway),

P2@3

are incompatible possibilities

Ikz

with the required active spins in each time period. This leaves the P3 Iiz

__*

P2

L

(unmatched

pathway),

(unmatched pathway), where I,, corresponds to a spin which is not directly involved in the minimal topology, i.e., is never an active spin, but is scalar coupled to at least one of Ii, Ii, and/or lk. These possibilities do not alter the set of (0, , w2, w3) frequencies which are obtained in the general three-period experiment. In other words, the minimal topologies given in Fig. 4 and Table 1 remain valid in the sense that the combinations of active spins given in Table 2 remain valid and no new resonances are seen. What does happen for ,f3# 90” is that the multiplet fine structure of the resonances is altered and this feature is the basis of experiments such as E.COSY (27). Ijz

-

1,JZ

Observation and Interpretation of Minimal Topologies Most analyses of multidimensional correlation NMR spectra proceed by identifying relationships amongst the frequencies at which a number of different resonances are

328

GRAY

AND

BROWN

observed. An alternative approach is to attempt to extract the maximum amount of information from each single resonance. In the present context this means trying to identify and assign minimal topologies from a single resonance. This requires that a given minimal topology must give a distinct resonance; i.e., each pathway type must give a unique and detectable combination of frequencies. Conversely, amongst the set of resonances observed in a particular experiment, it must be possible to identify the type of pathway which gives rise to each resonance. If chemical shifts are assigned to each spin in a minimal topology, then the spin combinations of Table 2 can be used to predict, in the limit of weak coupling, the ( (IJ 1, w2, w3) frequency combinations that are possible in the full three-period experiment. Barring accidental degeneracies in the chemical shifts of the different spins, all the frequency combinations are unique. This means that in a full 3D multiple-quantum experiment, each of the minimal topologies in Fig. 4 can in principle be observed, with the appropriately chosen quantum order, as a unique resonance in the three-dimensional frequency space. Matched and unmatched pathways are readily distinguished on the basis of the experimental singlequantum frequencies in t, and t3. Because of the different phase relationships involved in the various pathways (Table 3 ), it is then possible in principle to identify the type of minimal topology which gives rise to each resonance in the spectrum. For a single resonance, the only ambiguity is that, for pathways where multiple spins Ik are involved, only a linear combination of the chemical shifts of these spins is determinable. A set of 3D experiments with different quantum orders will require very long acquisition times, even if interleaving is used to record all desired quantum orders in t2 simultaneously. It is therefore of interest to also examine various 2D experiments to see whether similar information can be obtained. This can be expressed in terms of which topologies can be observed and interpreted in the various experiments. (a) COSY with multiple-quantumJilters. With t2 = 0, no evolution with respect to scalar couplings is possible during this period. This means that only the DDM, IIM, and DDU pathways contribute to the experimental spectrum. This reduces to eight the number of different types of minimal topologies of two or more spins which can be detected with up to five active spins during t2 (Fig. 4). In addition, within each minimal topology, chemical shifts can be observed only for Ii and Ii. This means that DDM and IIM pathways, which have different phases (Table 3 ) , can give superimposed diagonal resonances. In this case, the presence of a diagonal peak is only safely construed as indicating the simpler topology corresponding to the DDM pathway. A further disadvantage of COSY with multiple-quantum filters is that information is obtained only on the number, but not the chemical shifts, for the Ik spins. (b) Relayed COSY. This experiment is normally performed with Q = 1 during the fixed period t2 = 7. In this case, only two different minimal topologies involving at least two spins are observable (Fig. 4). If higher numbers of active spins are used during t2, responses from DDM and IIM pathways can give overlapped diagonal resonances and responses from DDU, DIU, IDU, and IIU pathways can give overlapped off-diagonal resonances. In these cases, the presence of a resonance is only safely interpretable as indicating the simpler topology from the DDM or DDU pathways. For topologies where DDU, IDU, and DIU pathways are not possible, i.e., where Ii and Ij are not scalar coupled, an IIU pathway will give an off-diagonal resonance with a pure phase. Because the DDM, DDU, and IIU pathways include all the possible

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329

minimal topologies, relayed COSY is sensitive to all 13 possible topologies contained in the full three-period experiment. Unfortunately, no information is obtained about the chemical shifts of the Ik spins. (c) 20 multiple-quantum spectroscopy. In this experiment t, = r and the experiment is sensitive to all the minimal topologies. Because of the loss of information on the frequency during tl , resonances from several minimal topologies can overlap. In particular, the minimal topologies for the DDU and IDU pathways contain the minimal topology for the DDM pathway and resonances from all three pathways have a common frequency during t2. Similarly, the DIU and IIU minimal topologies contain the IIM minimal topology and all three pathways have a common frequency during t2. In both cases, the overlap of the different pathways leads to mixed phases during the evolution period (Table 3 ) . In addition, the IIM and IIU pathways differ in phase by 180” during evolution, which can lead to partial cancellation. The consequence of these features is that the unmatched pathways cannot be reliably determined and only the presence of the DDM and IIM minimal topologies can be safely inferred from the presence or absence of resonances in 2D multiple-quantum spectroscopy. With up to five active spins during t2, these pathways include only five topologies of two or more spins. The advantage of this experiment is, of course, that information on the chemical shifts of the Ik spins is obtained. Because frequencies are measured in only two of the three time periods, all of the above 2D experiments are degenerate in the sense that multiple pathways, i.e., multiple minimal topologies, can contribute to the same experimental resonance. In this case, the information available in the 2D experiment is an assignable topology, which represents the simplest minimal topology amongst those minimal topologies which can contribute to a given experimental resonance. These assignable topologies are shown in Fig. 5. Pseudo-3D Multiple-Quantum

Spectroscopy

Each of the above experiments has serious drawbacks in terms of either experimental practicality or inadequate information. Recently it has been suggested that time periods t, and t2 (Fig. 2) be incremented in a concerted fashion to provide a single evolution period prior to detection in t3. This idea was applied with selection of double-quantum coherence during t2 to give so-called one-and-a-half-quantum spectroscopy (28). The same idea is applicable to higher quantum orders and we therefore prefer the name pseudo-3D multiple-quantum spectroscopy. A variant of this experiment, described in the following, provides a means to obtain most of the information on the minimal topologies which could be extracted from the full 3D multiple-quantum experiment. In the general three-period experiment, there are always four possible coherence-transfer pathways: +l + +Q --t -1, -1 +-Q+-l,+l+-Q+--1,and-l-,+Q+ -1. If concerted incrementation oft, and t2 is such that ( 1 - a) t, = at2, the first two of the above pathways lead to an effective frequency -+[ CYW 1 + ( 1 - LX)w2] and the latter two pathways to st[ - (~wi + ( 1 - (Y)w~]. With appropriate phase cycling, each of these pathways can be observed individually, and it is therefore possible to obtain two different types of phase-sensitive spectra in which the frequency in t, is either added to or subtracted from the frequency in t2. In general, the two types of spectra

330

GRAY quantum order

multiple quantum filtered COSY

1

relayed COSY

multiple quantum

-

i

3

j

i

i

G

j

c-a-o

77

0

4

AND BROWN

j

i

j

43

FIG. 5. Assignable topologies as a function of quantum order for 2D multiple-quantum 2D relayed COSY, and 2D multiple-quantum spectroscopy.

filtered COSY,

will differ in that resonances involving matched pathways will differ in chemical shift by 2awj and resonances involving unmatched pathways will differ in chemical shift by 2ao;. A possible complication in the pseudo-3D multiple-quantum experiment is the presence of combination lines in which Q active spins give net quantum orders of Q - 2, Q - 4, etc. These give a large number of additional resonances which also occur at unique frequencies. In general, the full pseudo..3D multiple-quantum spectrum is highly redundant because of the presence of combination lines and because the DIU, IDU, and IIM pathways repeat minimal topologies which are observable via the DDM, DDU, and IIU pathways. Clearly, it is desirable to alter the full experiment in such a way that much of the redundant information is suppressed. In principle this can be achieved by the use of p2, & # 90”. However, the use of mixing pulses with tip angles other than 90” leads to intensities that are unequal between pairs of pathways which are symmetric about Q = 0 during the evolution period. This would preclude phasesensitive transforms in which positive and negative frequencies are distinguished in t, . There are several possible ways to overcome this difficulty. The f 1 * + Q * - 1 pathways can be selected with equal intensity using p2 = 30” and & = 90”. With this choice, the intensities of the pathways are expected to be in the ratio DDM, DDU, DIU:IDU, IIM, IIU = 7:2. Combination lines involving quantum order Q - 2 would be reduced to $ of these ratios. Similarly, the kl + TQ * - 1 pathways can be selected with equal intensity using ,& = 210” and p3 = 90” with the same intensity

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ratios amongst the various pathway types. Alternatively, four experiments can be performed using the pairwise combinations of 182= (30”, 210”) and & = (30”, 210”) and one pathway selected for each combination of ( P2, &). For example, if the - 1 + -Q + - 1 pathway is selected from the combination (& = & = 30”), the relative intensities of the different pathway types will be in the ratio DDM, DDU, DIU:IDU, IIM, IIU = 49:4. In this case combination lines involving Q - 2 would be reduced to & of these ratios. An alternative method for reducing the intensity of the IDU and IIU pathways is to keep t2 relatively short, i.e., (Y = 1. This hinders in two ways the scalar coupling evolution during t2 which is necessary for the existence of these pathways (Fig. 3 ). First it reduces significantly the time available to resolve the necessary couplings, and second, the size of the couplings to be resolved in t2 is also scaled down (by a factor of 1 - cu). Although it seems at first that the DIU pathway should also be strongly suppressed by this method, this is not the case. This is because with & = 90”, the spin which is observed in t3, Ij, can only become antiphase during t2. However, using & # 90” allows Ij to become antiphase during t, and hence does not require any couplings to be resolved during t2. This DIU-type pathway would then be expected to be observed with almost the same sensitivity as the DDM and DDU pathways. By use of ,& = 30” and (Y = 1, the resulting experiment selects for the DDM, DDU, and DIU pathways and is similar to COSY with multiple-quantum filters except that each resonance is briefly labeled with the appropriate multiplequantum frequency during t2. RESULTS

Ally1 iodide ( 3-iodoprop- I-ene) has been used to test the pseudo-3D multiple-quantum experiment and methods for analyzing the spectra based on adjacency matrices. The pseudo-3D multiple-quantum spectrum of ally1 iodide was recorded using the pulse sequence 90”(~1)-at,-&(&)-( 1 - ~)t,-90”(~3)-t2(~rec), with the 1% step phase cycle 4, = - 4, = 0,90”,&=0,~3=n(2?r/9),n=Oto8.Theexperiment was run in an interleaved fashion to enable spectra of quantum orders 2, 3, and 4 to be generated from the one data set (29). Each of the 18 2D time-domain sets ( 320 X 2048) was stored separately during acquisition. They were then transferred to a VAX computer, where 2D NMR software developed in our laboratory (PROC2D/ ANALYS2D) was used to extract and transform the two-, three-, and four-quantum spectra. The resulting two-quantum pseudo-3D spectrum is shown in Fig. 6. The spectrum shows four distinct single-quantum frequencies, which suggests that the methylene hydrogens are degenerate. For purposes of the following discussion, the values of the coupling constants in ally1 iodide are Ji2 = 16.9 Hz, J13 = 9.9 Hz, J,4 = 7.9 Hz, Jz3 = 1.1 Hz, J14 = 1.1 Hz, and J34 = 0.4 Hz. The range of values for these couplings is noteworthy, as is the presence of a number of small couplings, including the very small J34 coupling. The presence of these small couplings might be expected to cause difficulties in an analysis which is based on the assumption of an adjacency matrix in which a coupling is either present or absent. This is discussed below. The analysis carried out for the pseudo-3D spectra of quantum orders 2, 3, and 4 is quite different from usual spectral interpretations. In particular, no attempt has been made to simulate an exact spectrum and no attempt was made to interpret all spectral responses. Instead, an attempt has been made to find a minimal topology

332

GRAY

AND BROWN

250 -

O-

!G = m

;; :: *

: :: I

Iyu +

II

; II

P

-250 -

-NO-

1

I 6.0

FIG. 6. Double-quantum

1 5.0

I 4.0

pseudo-3D NMR spectrum of ally1 iodide.

appropriate for each experimental peak which has sufficient signal/ noise and resolution to be easily observable in the experimental spectra. This approach is chosen to facilitate dealing with complex spectra where not all possible resonances may have adequate resolution or sensitivity to be observed experimentally. As will be seen below, the pseudo-3D experiment is highly redundant in that the number of potentially observable minimal topologies vastly exceeds the set of minimal topologies necessary to deduce the correct overall topology; i.e., it is not necessary to observe peaks corresponding to all possible minimal topologies to deduce the appropriate adjacency matrix. In practice, the pseudo-3D spectra have been analyzed by seeking an adjacency matrix which can account for all peaks which can be readily picked in the experimental spectra. A computer program (Predictor) has been written which, for a given adjacency matrix, transmitter offset, and set of single-quantum frequencies, uses the selection rules (Table 1) to generate a predicted list of all possible peaks for each of quantum orders 2,3, and 4. A second computer program (Comparator) is then used to compare the lists of predicted and experimental peaks, using a definable tolerance along each

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of the two frequency axes to identify matches between the two lists. The output of the comparison program consists of the following subsets: { P, and E} : The set of matches between a single predicted peak (Pi) and the experimental peaks (E) . Due to inadequate resolution, a given predicted peak may have matches to more than one experimental peak. { P and E,} : The set of matches between predicted peaks and each individual experimental peak (.I$). Due to inadequate resolution, there may be more then one predicted peak for a single experimental peak. { P, and J!?}: The set of predicted peaks for which there is no observed experimental peak. { p and E,} : The set of experimental peaks for which there is no predicted peak. This classification was chosen since it facilitates the checking of whether each predicted peak has any possible match with experimental peaks and, conversely, whether each experimental peak has any possible match with the predicted peaks. Although this feature will not be used in the present work, this classification also facilitates the identification of alternative interpretations for experimental peaks. From the first two of the above sets, a further set, {Pi and E,} , which consists of matches between single experimental peaks and single predicted peaks, is produced. Each match in this latter set identifies a frequency-labeled minimal topology. Figure 7 shows traces along wr through the w2 frequency of 6.06 1 ppm (Fig. 6) for the two-, three-, and four-quantum pseudo-3D spectra. In Fig. 7, several of the peaks have been marked with the type of pathway and the labeled minimal topology which gives rise to the peak. The derivation of these labels is described below. The identification of an adjacency matrix which can account for all picked experimental peaks proceeded from the assumption of a simple topology to the assumption of more complex topologies. The first level of analysis assumed that there are four spins in ally1 iodide, each of which corresponds to one of the four frequencies observed in the ID spectrum. The second level of analysis assumed that each of the four spin types could consist of multiple, magnetically equivalent spins. In the third level of analysis, the possibility was considered that these latter spins are isochronous rather than equivalent, e.g., that two spins with identical chemical shifts may have a coupling that is manifested in the experimental spectra. A total of 80 resolved, moderately to strongly intense peaks, i.e., 36 two-quantum, 33 three-quantum, and 11 four-quantum peaks, were picked in the corresponding spectra. An advantage of the present method is that highly accurate comparisons of chemical shifts between different peaks are not required and, for present purposes, it was adequate to pick each peak manually at the approximate center of gravity of the multiplet pattern. No attempt was made to identify the individual transitions in the cross peaks. The experimental spectra showed other weak peaks as well as regions containing overlapped peaks. No attempt was made to pick such peaks. The present experimental data were recorded using values of (Y = 0.9 and P2 = 35”. This means that only the direct excitation pathways can realistically be expected to be observed in the experimental spectra and, therefore, that only the DDM, DDU, and DIU pathways were used for initial identification of minimal topologies. At the first level of analysis, the four different spin types were assumed to be mutually pair-wise coupled

334

GRAY Cm D-D-M +

AND BROWN 00 D-D-U 4

sb 1 4 2 + D-I-U

A

-750

-500

-250

0 01

250

500

750

Hz

FIG. 7. Traces along w, through w2 = 6.061 ppm for the two-, three-, and four-quantum pseudo-3D NMR spectra of ally1 iodide (A-C, respectively). For some of the peaks, the minimal topologies corresponding to the experimental peaks are shown.

(topology A in Fig. 8 ) . By so choosing a complete graph, which can always be done for a graph with an arbitrary number of nodes (spins), there is no need to make any prior assumptions about the actual coupling in the spin topology. Table 4 shows the number of predicted pathways and predicted peaks for this topology. Because the resonances of ally1 iodide are all within a chemicalshift range of 2.2 ppm and because a relatively small value of ( 1 - CI) was used in .this experiment, only some of the predicted peaks are expected to be well resolved from each other (Pi . Pj in Table 4 ) . This leads to some accidental degeneracies in the spectra so that some peaks may therefore not be picked in the experimental spectra. A total of 57 experimental peaks

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NMR

FIG. 8. Topologies used in the analysis of the two-, three-, and four-quantum pseudo-3D spectra of ally1 iodide.

TABLE 4 Comparison of Predicted and Experimental Pseudo-3D Spectra for Ally1 Iodide Topology

Quantum order

Pathway type

Predicted pathways

P

PyP,

P,*Ek

(E,.P,).P,

A

2

DDM DDU DIU DDM DDU DIU DDM DDU DIU DDM DDU DIU DDM DDU DIU DDM DDU DIU DDM DDU DIU DDM DDU DIU DDM DDU DIU

12 12 24 12 24 12 4 12 0 18 18 42 24 42 30 14 30 6 20 20 60 30 60 60 20 60 20

12 12 24 12 24 12 4 12 0 12 12 24 15 24 18 10 18 6 13 12 21 18 30 24 13 30 9

8 8 16 6 12 6 4 12 0 8 8 16 9 12 12 6 13 3 9 8 19 8 13 I1 I 17 4

8 8 13 5 10 6 4 3 0 8 8 13 6 10 I 3 3 0 8 8 13 5 10 6 3 3 0

2 3 2 3 I 0 0 0 0 2 3 2 3 I 0 2 0

3 4 B

2 3 4

C

2 3 4

p.Ek

Note. P, total number of predicted peaks; P,* pj, the number of predicted peaks which are resolved from all other predicted peaks by b 15 Hz; P,* Ek, the number of matches between a single predicted peak and a single experimental peak using a tolerance of * 15 Hz along oi and wz; (EkeP,). Pj, the number of matches between a single predicted peak and a single experimental peak, but with another predicted peak within I5 Hz of the matched predicted peak; PmEk, the number of experimental peaks with no predicted peak within +15 Hz.

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could be matched with a unique predicted peak (Pi * Ek in Table 4). A further 17 experimental peaks matched well with a single predicted peak, but had a second predicted peak within 5 15 Hz of the predicted peak which best matched the experimental peak. There were also 6 experimental peaks for which there was no predicted peak. The minimal topologies corresponding to the 57 uniquely assigned peaks are shown in Fig. 9A. In many cases, a given minimal topology can be observed via multiple different resonances in the spectra; i.e., the number of minimal topologies is expected to be less than the number of experimental peaks assigned. Inspection of the minimal topologies in Fig. 9A shows numerous instances of all possible pairwise couplings among the four spin types; i.e., a fully coupled topology appears to be necessary. Of the total of 72 predicted resonances which were expected to be resolved in the experimental spectra, assignments were found for 54. An examination of the remaining 18 resolved, predicted resonances which were not observed experimentally revealed that all of the missing peaks involved pathways which required resolution of the very small 0.4 Hz coupling (&) in t3 and/or the 0.36 Hz (0.9 X 0.4 Hz) coupling in t,. On the other hand, the topologies shown in Fig. 9A arise from observation of 20 resonances for which the pathways concerned require resolution of

@@@@I~@-@@@@@ &&&&& &&&g&g @f&&g@ &h-&z& A +&+f+ -------------------_---------B&F+&

&&or&

FIG. 9. Labeled minimal topologies detected in pseudo-3D spectra of ally1 iodide using the topologies shown in Fig. 8. (A) Minimal topologies assigned using topology A. (B) Additional topologies assigned using topology B. (C) Additional topologies assigned using topology C. Because the predicted peaks were within 15 Hz, the involvement of spin 2 or spin 3 could not be differentiated for some of the topologies in (B) and (C).

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the JJ4 coupling. It appears that this small coupling is only sometimes manifested in the experimental spectra, probably as a consequence of the detailed nature of the multiplet splittings in a given resonance. Indirectly this provides evidence that Jj4 is probably small. Similar, though less pronounced, behavior is observed for some of the other small couplings. While a fully coupled topology appears to be a necessary condition for interpreting the experimental spectra, a four-spin topology is not sufficient since there were still six resonances which could not be assigned to any of the six pathway types, including combination lines (Table 4). The program Equivalent was therefore used to determine what combination of specified single-quantum frequencies could lead to an observed multiple-quantum frequency under the assumption that any given single-quantum frequency may correspond to up to three equivalent spins. For four of the six unassigned peaks, solutions which corresponded to spin 4 consisting of two equivalent spins were found by this program. A second level of analysis of the experimental data was therefore carried out with topology B shown in Fig. 8. In addition to the previously identified topologies, three new topologies, which involve two equivalent spins of type 4, were identified (Fig. 9B). These accounted for four of the six experimental peaks without possible predicted peaks in the original analysis. The remaining two experimental peaks for which there was no possible predicted peak could be accounted for by assuming that the two spins of type 4 were isochronous (topology C in Fig. 8). With this assumption, two further minimal topologies were identified (Fig. 9C). The minimal topologies shown in Fig. 9 have been identified on the basis of assignment to direct excitation pathways. To check that indirect excitation pathways have been adequately suppressed in the experimental spectra, Predictor was used to predict peaks from indirect excitation pathways which would be well resolved in the experimental spectra. Of 40 such peaks, very weak intensities, observable only in traces and not in contours of the 2D data, were found for 5 peaks. The lack of peaks for indirect excitation pathways provides confirmation for the minimal topologies of Fig. 9. These minimal topologies may then be regarded as necessary conditions which must be included in the overall topology. The necessary features of the overall topology can be determined by assembly of the minimal topologies, which in the present case leads to topology C. It should be noted that not all possible experimental peaks inherent in topology C have been observed. For quantum orders 2, 3, and 4, a total of 176 peaks is predicted for this system. Of these predicted peaks, 90 involve the small Jx4 coupling constant. Furthermore only 96 of the 176 predicted peaks are expected to be well resolved in the spectra and 49 of these resolved peaks involve J34. In contrast, 56 experimental peaks could be matched with unique predicted peaks and a further 24 experimental peaks had only 2 possible predicted peaks, 1 of which was a much better match to the experimental peak. Only 4 peaks were observed which required the presence of two spins of type 4, but only 12 peaks were predicted which were resolved and did not involve J34. Similarly, only 2 peaks were observed which suggested that the two spins of type 4 were isochronous, but only 6 peaks were predicted which were resolved and did not involve J 34. Considering that some of the other couplings are also quite small, it therefore appears that, in addition to accounting for all experimental peaks which were picked, topology C gives a reasonable agreement with the predicted spectrum.

338

GRAY

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If desired, final confirmation that the spectra correspond to topology C could be obtained by quantitative simulation and fitting of the NMR spectra. This would be necessary to determine the values of scalar couplings and of relaxation parameters. Simulation would also be helpful in deciding whether the two spins of type 4 are isochronous witb a scalar coupling between them or are strictly magnetically equivalent with the observed experimental peaks arising from relaxation violations of simple selection rules ( 30). This distinction cannot be made simply from the presence of the observed experimental peaks. In the present case, the simulation used to extract coupling constants for ally1 iodide indicates that the two spins of type 4 are indeed equivalent. Whether such simulation and fitting, which may be computationally expensive, is desirable and/or necessary will depend on the goal of the NMR experiments. In introducing the idea of minimal topologies and the reassembly of minimal topologies to give overall topologies, our goal has been to lind simple methods for extracting from NMR spectra enough information on spin topologies to allow mapping of these topologies to molecular structures. In the present case, it has been shown that an analysis based purely on the assumption of an adjacency matrix can be used to delineate the nature of the spin topology. In particular, the adjacency matrix analysis was able to proceed from a highly simplified first assumption of a connected topology involving four spins to evidence of mutual coupling amongst all four spin types and to a correct identification of five spins, two of which were at least equivalent, and possibly isochronous. It has been possible to deduce this spin topology because it contains a set of minimal topologies which is vastly redundant in terms of the information needed to reconstruct the spin topology; i.e., it is not necessary to observe all predicted peaks to deduce the correct topology. By the same token, the use of an analysis based on assignment of minimal topologies to experimental peaks followed by reassembly of the minimal topologies allows the determination of the information content which is available from prominent features of the experimental spectra. This can be particularly useful in analyses of more complex spectra where, at least in the initial stages of analysis, there may be ambiguity in the correct interpretation. This will be described in more detail elsewhere. CONCLUSION

When using multidimensional NMR spectra to analyze scalar coupling networks, there are two possible extreme strategies. At one extreme is the possibility of running a single, simple experiment such as COSY and attempting to extract all information on the spin topology from a high-resolution analysis of this spectrum. This requires the elucidation of the relationships between different cross peaks and an analysis of the multiplet patterns for each cross peak. As noted above, this strategy becomes increasingly difficult to apply as the size and complexity of the molecule increases. At the other extreme would be the possibility of having a series of experiments, each of which selects for a particular scalar coupling topology. Provided that a sufficiently selective set of experiments could be devised, a positive response with a given experiment would be incontrovertible evidence for the entire topology. Unfortunately, the number of different topologies increases rapidly with the number of spins, e.g., there are 2 1 distinct topologies for five spins, 112 for six spins, and 853 for seven spins, and

TOPOLOGIES

IN

HOMONEUCLEAR

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NMR

339

it therefore seems unlikely that the extreme version of this approach can be made practical. The recognition that a general three-period experiment selects for a relatively small number of minimal topologies provides a third, middle course. If the relatively small number of topologies to which this class of experiments is sensitive can be individually identified in experimental spectra and if this set of minimal topologies is adequate to identify all possible complete topologies, then complete topologies could be determined on the basis of the number and type of minimal topologies which they contain. This leads to a strategy for spectral interpretation in which any multidimensional NMR experiment is regarded as defining a set of minimal topologies for which the experiment can give observable spectral responses. Conversely, any observed peak must correspond to one of the minimal topologies appropriate for the given experiment. Similarly, any molecular spin topology can be decomposed into the appropriate minimal topologies which are detectable in a given type of experiment. Conversely, a molecular spin topology can be reconstructed by assembling minimal topologies. Furthermore, in many cases it will not be necessary to observe all possible minimal topologies in order to have sufficient information to deduce the appropriate molecular adjacency matrix. As a consequence of these features, graphs provide a common language for relating molecular topologies to the results of different types of NMR experiments. Adoption of a graphically based analysis has the additional advantage that a wide variety of computational methods which are already available for dealing with graphs can be applied to the analysis of NMR spectra. Indeed, as will be described in more detail elsewhere ( 31) , the present version of Predictor makes use of graphical algorithms to rapidly generate combinations of active spins and the associated frequencies in t, , tz, and t3 for a full homonuclear 3D experiment. This means that by choice of the appropriate frequencies, it is equally easy for Predictor to generate appropriate spectral predictions for homonuclear 3D NMR, pseudo-3D NMR, COSY with or without multiple-quantum filters, relayed COSY, and 2D multiple-quantum spectroscopy. A similar analysis of molecular topologies could also be based on these other types of 2D experiments as long as due care is given to the inherent degeneracies (Figs. 4 and 5). The necessary predictions can be made very rapidly; e.g., prediction of a full 2D multiple-quantum spectra of a small protein involving 10 12 pathways and 56 1 distinct resonances has been obtained in five seconds on a Sun-4 computer. This allows Predictor to be used in interactive analysis of complex spectra and/or in further computerbased reasoning programs. Furthermore, having adopted graphical methods, heteronuclear NMR experiments can be readily dealt with by provision for colored edges and colored nodes and this is currently under development. It is obvious that such a graphical analysis is not a quantitative method in that no quantitative interpretation of NMR spectra is sought. In particular, no attempt is made to account for the values of scalar couplings or of relaxation parameters, which would be necessary for exact computation and/or fitting of the NMR spectra. As a consequence, there are two types of spectral characteristics which are not explicitly included in Predictor. The first is strong coupling, which will certainly have to be dealt with by other methods. The second is relaxation-induced violations of simple selection rules. However, as shown in the case of ally1 iodide, Predictor is capable of recognizing the possibility of such violations by treating them as instances of isochronous spins;

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i.e., it is possible to find graphs which will account for spectral peaks which arise from relaxation-induced violations of simple selection rules. This allows the interpretation of such graphs to be deferred to the stage of relating the graph to molecular structure. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

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