Fuzzy Structure Generation: A New Efficient Tool for Computer-Aided Structure
Elucidation (CASE)
Mikhail E. Elyashberg,§ Kirill A. Blinov,§ Sergey G. Molodtsov,‡ Antony J. Williams,*,† and
Gary E. Martin|
Advanced Chemistry Development, Moscow Department, 6 Akademik Bakulev Street, Moscow 117513,
Russian Federation, Advanced Chemistry Development, Inc., 110 Yonge Street, 14th Floor, Toronto, Ontario,
Canada M5C 1T4, Novosibirsk Institute of Organic Chemistry, Siberian Branch of Russian Academy of
Science, Lavrentiev Avenue 9, Novosibirsk 630090, Russian Federation, and Schering-Plough Corporation,
Pharmaceutical Sciences, Rapid Structure Characterization Laboratory, Summit, New Jersey 07901
Received November 21, 2006
Contemporary Computer-Aided Structure Elucidation (CASE) systems are heavily based on the utilization
of 2D NMR spectra. The utilization of HMBC/GHMBC and COSY/GCOSY correlations generally assumes
that these correlations result from 2-3JCH and 2-3JHH spin-spin couplings, respectively, and consequently
these values are used as the default setting in these systems. Our previous studies1,2 have shown that about
half of the problems studied actually contain some correlations of 4-6 bonds, so-called “nonstandard”
correlations. In such cases the initial 2D NMR data are contradictory, and the correct solution is therefore
not directly attainable. Unfortunately nonstandard correlations and the number of intervening bonds usually
cannot be identified experimentally. In this work we suggest a new approach that we term Fuzzy Structure
Generation. This allows the solution of structural problems whose 2D NMR data contain an unknown number
of nonstandard correlations having different and unknown lengths. Suggested methods for the application
of Fuzzy Structure Generation are described, and their application is illustrated by a series of real-world
examples. We conclude that Fuzzy Structure Generation is efficient, and there is no real alternative at present
in terms of a universal practical method for the structure elucidation of organic molecules from 2D NMR
data.
1. INTRODUCTION
In our previous reports1,2 we have already discussed the
problems that arise when an expert system is used for
molecular structure elucidation from 2D NMR data. As a
rule, a combination of GHMQC/GHSQC, GHMBC, and
COSY spectra make up an experimental data set with the
GHMBC and COSY correlations assumed to correspond to
2-3JCH and 2-3JHH coupling constants, respectively. In our
previous work1,3,4 we defined such correlations as “standard”
correlations. For n>3JHH/CH couplings we label these correla-
tions nonstandard. The origin and nature of nonstandard
correlations (NSCs) are discussed elsewhere in the literature.5
Unfortunately, there are no reliable and routine experimental
methods so far that unambiguously distinguish between
standard and nonstandard correlations though there have been
significant efforts to address the differentiation of 2JCH from
3JCH long-range heteronuclear couplings (note that although
the couplings are listed here as nJCH correlations, this
statement can also, albeit it less frequently, apply to nJNH
correlations as well).5
It is generally believed by others in this field of study that
correlations of nonstandard length are observed fairly
rarely.6-8 Results obtained in our work2 contradict this
opinion. Previously we investigated the solutions of more
than 250 problems whereby the expert system Structure
Elucidator (StrucEluc)3,4 was applied to the structural
identification of complex natural products from 2D NMR
and MS spectra. The studies indicated that almost half of
the problems (45%) contained nonstandard correlations in
the 2D NMR data. Nonstandard long-range heteronuclear
correlations are even more common when some of the
accordion-optimized long-range heteronuclear shift correla-
tion experiments are employed.3 These additional HMBC
connectivities can give additional constraints and simplify
the process of solving the problem. However, at the same
time new NSCs can appear so a method to address these
NSCs is necessary. This is the benefit of the fuzzy generation
approach discussed in this paper. If the spectroscopist can
indicate the connectivities associated with the NSCs, then
this information will of course be useful. Meanwhile, expert
systems are usually optimized to structure elucidation as-
suming a set of correlations of common (standard) length.
The presence of nonstandard correlations within one or more
2D NMR data sets generally produces a result that is
inconsistent with the real structure. For instance, the observa-
tion of a COSY correlation between hydrogen atoms attached
to two carbon atoms C-1 and C-2 suggests that these atoms
are connected by a carbon-carbon bond. In the case when
a given correlation is of nonstandard length, the distance
between the C-1 and C-2 atoms is actually of two or more
bonds in the real structure. Prior to actually establishing the
* Corresponding author e-mail: tony@acdlabs.com.
† Advanced Chemistry Development, Inc.
‡ Siberian Branch of Russian Academy of Science.
§ Advanced Chemistry Development, Moscow Department.
| Schering-Plough Corporation, Pharmaceutical Sciences.
1053J. Chem. Inf. Model. 2007, 47, 1053-1066
10.1021/ci600528g CCC: $37.00 © 2007 American Chemical Society
Published on Web 03/27/2007

structure of the unknown molecule, the presence or absence
of nonstandard correlations as well their number and real
lengths remains unknown. Therefore, the problem adds up
to molecular structure elucidation from spectrum-structural
information that is not only fuzzy by nature (2-3JCH in
HMBC) but can also be both contradictory and uncertain
(i.e., the number of nonstandard connectivities and their
lengths are unknown).
In a previous report1 we suggested approaches for solving
problems in the presence of correlations of nonstandard
lengths. The program first attempts to find whether non-
standard correlations are present in the spectral data. The
method is based on the logical analysis of a full set of
connectivities including HMBC and COSY data, and most
frequently the algorithm is capable of detecting skeletal atoms
that are involved in connectivities of a nonstandard length.
If nonstandard connectivities are associated with particular
atoms, then the program automatically lengthens all con-
nectivities emanating from such atoms by one bond and
attempts to generate structures. It is important to emphasize
that not all connectivities are lengthened, only the ones
identified by the algorithm as being nonstandard. As a result
the program is capable of generating structural solutions
consistent with the data within a reasonable period of time
even if there are a large number of nonstandard correlations.
In those cases when correlations are present in the 2D
NMR data with nJ where n > 4, this method, unfortunately,
does not work. The augmentation of the path between two
intervening nuclei by one bond obviously cannot lead to the
generation of a correct structure. Moreover, even in those
cases when n e 4 the algorithm gives no guarantee that all
nonstandard correlations will be found and corrected due to
a lack of constraints that are to be logically analyzed. For
example, the greater the number of carbon atoms with
accurately defined properties (the type of hybridization and
different heteroatom neighborhoods) and/or the higher the
total number of available 2D NMR connectivities, the higher
the probability of successfully performing logical analysis
to arrive at the correct structure. In contrast, severely proton-
deficient molecules can be among the most challenging.
Obviously, the problem becomes more computationally
complicated as the complexity of a molecule increases. Our
experience has shown that the number of nonstandard
correlations contained within the 2D NMR data associated
with a molecule, m, can be rather largesup to about 20
correlations. At the same time the augmentation of standard
correlation lengths, a, could be 1-3. As an example of such
situations several structures taken from literature9-12 are used
to demonstrate those examples with a large number of
nonstandard correlations including 5J and 6J coupling
constants (see Figure 1).
To overcome the described shortcomings a computational
approach is suggested that we have defined as Fuzzy
Structure Generation (FSG).1 During the process of fuzzy
generation the number of nonstandard connectivities is
restricted to a parameter m, and their lengths can be
augmented by a number of bonds equaling a. A strategy of
determining the values of the parameters m and a has not
previously been elaborated. The goal of our work was
therefore to develop a methodology that would identify both
the number and length of the nonstandard connectivities to
facilitate the structure elucidation of unknown organic
molecules from 2D NMR data. This goal is achieved by
fuzzy structure generation on the basis of m and a parameters
whose actual values become known during the process of
problem solving. As a result of our studies a solution to the
problem posed was determined, and the efficiency of fuzzy
structure generation was examined using a series of real-
world examples. We have shown that the procedure of
determining the correct parameters associated with fuzzy
structure generation can be conducted as a result of series
of iterations controlled by the user, and, in principle, it is
amenable to automation.
2. SOLVING PROBLEMS IN THE PRESENCE OF
NONSTANDARD CORRELATIONS
2.1. Modes of Fuzzy Structure Generation. Prior to
describing the strategy of fuzzy structure generation some
specific concepts will be described. As stated previously13 a
solution to the problem is Valid if the resulting structural
file contains the correct structure. Otherwise, the solution is
considered inValid. The number of structures included in the
output file is denoted as k. According to the methodology
common for StrucEluc, the solution quality is evaluated on
the basis of NMR spectral prediction in order to allow
identification of the most probable structure. As will be
shown, a comparison of structural output files obtained with
the aid of fuzzy structure generation at different stages of
the structure elucidation plays an important role in identifying
the validity of the final answer. Therefore we will briefly
explain the main features of the approach used since details
and numerous examples have been reported previously in
our articles.1,3,4,14
During the first step 13C NMR spectra are predicted for
all generated structures using an incremental method, our
so-called “Fast” method,3,4,15 and dF values, the average
deviation of an experimental 13C NMR spectrum versus
predicted chemical shifts, are calculated. Recently the
incremental prediction algorithms were revamped (an article
is presently in preparation reporting the associated advances).
The most recent iteration of the algorithm predicts 5000-
7000 shifts per second with an average chemical shift
Figure 1. An illustration of a number of structures containing
multiple nonstandard correlations. The nonstandard COSY cor-
relations are shown as blue arrows, and the GHMBC correlations
are shown by green arrows. In the legends for the structures m is
the total number of nonstandard correlations, and a is the value of
correlation lengthening allowed during the process of Fuzzy
Generation (see below).
1054 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 ELYASHBERG ET AL.

deviation of 1.8 ppm. In parallel a fast algorithm for 13C
NMR prediction using artificial neural networks was also
implemented in StrucEluc. Since both algorithms have
comparable speeds and accuracy, both were employed during
one specific run for initial spectrum prediction. After
duplicate removal (among the duplicates, the structures with
the smallest deviations are retained4), the structures are
ranked by the dF value and sorted in ascending order. The
smallest dF value indicates the best match between the
experimental and calculated spectra, and this structure will
be the first in the output list. Usually the fast calculation of
13C NMR spectra and their subsequent ranking places the
correct structure (if it exists in the output file) as the first or
the second in the list. Only in very rare instances is the
correct structure listed below fifth place. During the next
stage so-called “accurate” 13C NMR spectra are calculated
for the first 20-50 structures of the ranked file. These
predictions are performed using the database of 175 000
structures with the corresponding assigned 13C and 1H NMR
spectra.15 The description of each nuclear environment is
defined using the HOSE code approach16 (Hierarchical
Ordering of Spherical Environments). The average deviation
values between the experimental and calculated values (dA)
are found, and the structures are again rank-ordered. For
additional control over the correct choice of the output
structure, the accurate proton chemical shifts can be predicted
and displayed together with the corresponding deviation
value dH (the mentioned15 database is used). The same can
also be done for 15N chemical shift predictions when the
investigator has access to direct or long-range 1H-15N
heteronuclear chemical shift correlation data. For proton
NMR prediction, the predicted proton-proton couplings can
be enhanced by three-dimensional optimization of the
structure. An additional complex match factor d“ ) dA +
10âdH is also calculated. The complex match factor reflects
how well the structure matches to both the 13C and 1H NMR
spectra. Recalling that the 13C NMR shift range is about
0-200 ppm, while for 1H shifts the range is about 0-15
ppm we ensured an approximate equal weight to the two
deviations, d(13C) and d(1H), by multiplying dH by 10 and
defining the complex match factor as d“ ) dA + 10âdH. The
position of the correct structure in the file determines its rank
depending on the type of ranking parameter, i.e., dA, dF, dH,
or d“ correspondingly. The rates of the correct structure in
the ranked file are denoted as rA, rF, rH, and r“. If the correct
structure assumes primary position in the list ranked by dA
values, then rA ) 1, and the corresponding deviation is
denoted as dA(1). As a rule, the final structural ranking is
carried out according to the increasing values of the most
significant dA or d“ parameters, while the magnitudes of the
dF and dH parameters serve as additional aids for estimating
the reliability of the correct structure selection.
The first ranked structure is considered as the most
probable one. If the deviation di(1) calculated for the first
ranked structure is less than a threshold Di which depends
on the precision of spectrum prediction, then the solution is
classified as acceptable. Otherwise the solution is deemed
to be unacceptable. Experience has shown that acceptable
solutions are most frequently valid though exceptions can
occur.
Numerous computational experiments have allowed us to
conclude that if the program detects the presence of
nonstandard correlations but fails to resolve contradictions
in the 2D NMR data using algorithms,1 then fuzzy structure
generation should be used to solve the problem. Moreover,
it is quite probable that structure elucidation from 2D NMR
data on the basis of fuzzy structure generation can be
considered as a general CASE strategy because it is almost
independent of the presence or absence of nonstandard
correlations in the 2D NMR data.
Fuzzy structure generation can easily be controlled by
parameters that make up a set of options. The two main
parameters are m, the number of nonstandard connectivities,
and a, the number of bonds by which some connectivity
lengths should be augmented. Unfortunately, 2D NMR
spectral data cannot deliver definitive information regarding
the values of these variables and, as a matter of fact, both of
them can be determined only during the process of structure
elucidation. We have concluded that in many cases the risk
of choosing an erroneous value for a can be avoided, and
the solution of a problem can be considerably simplified if
the lengthening of the m connectivities is replaced by their
deleting. When set in the options the program can ignore by
deleting connectivity responses that have to be augmented
(by convention, the parameter a is set to a value of 16 in
these cases). Such an approach can be successful in those
cases when the number of 2D NMR connectivities is in some
sense optimal. In this sense we mean that the total number
of connectivities (structural constraints), N, must be large
enough to facilitate a description of the chemical structure.
In many instances, there are sufficient numbers of correla-
tions in the ensemble of 2D NMR data acquired to essentially
overdetermine the structuresin other words there is redun-
dancy in some of the connectivity information. It can then
be expected that deletion of m of the connectivities will not
dramatically influence either the generation time or the size
of the output file. On the other hand, the number of
combinations of N connectivities taken m at a time can be
very large. This can dramatically impede problem solving
to a point that it is not feasible to solve the problem. Indeed,
some workers have commented that some of the accordion-
optimized long-range heteronuclear shift correlation experi-
ments actually provide too many long-range correlations of
the type nJXH where n g 4.
If the number of connectivities, N, is small, then further
decreasing N by m in a connectivity combination can lead
to an excessive decrease in the number of structural
constraints required for solving the problem. In such a case
the problem may be difficult to solve because the 2D NMR
data structural constraints will only reduce the total number
of possible isomers very slightly.
Independent of the use of augmentation or removal of
connectivities, the crucial point in application of fuzzy
structure generation is the number of connectivity combina-
tions that should be checked during structure generation. For
instance, if N ) 60 and m ) 5, then the number of
connectivity combinations, nmath )CN
m
, is equal to 5.5
million. Any attempt at structure generation has to be
performed using each of these combinations. It is necessary
to perform generation of structures from each of the CN
m
data sets and obtain the output file as a unification of all of
the intermediate results. Even though the StrucEluc structure
generator is fast, the productivity is certainly insufficient in
A NEW EFFICIENT TOOL FOR CASE J. Chem. Inf. Model., Vol. 47, No. 3, 2007 1055

terms of coping with a combinatorial problem as outlined
here.
To overcome this difficulty the system is delivered with
an algorithm capable of reducing the number of combinations
without the risk of losing the correct solution. The first step
is to reduce the total number of connectivities N down to
N0, where N0 is the number of connectivities used to form
the connectivity combinations. The data are preprocessed
according to the following rules: (1) ambiguous connec-
tivities are excluded from consideration. Ambiguous con-
nectivities are those that appear due to accidental degeneracy
of chemical shifts associated with two or more nonequivalent
atoms; (2) if two connectivities C-1 to C-2 and C-2 to C-1
are present, then only one of them is included in a data set.
One of the two equivalent correlations is redundant and
corresponds to overdetermination of the data needed for
solution of the structure.
The second and most important step is based on the results
of logical analysis of the initial 2D NMR data. If connectivity
sets containing nonstandard connectivities are identified (see
details in ref 1), then groups of these connectivities are
utilized to produce connectivity combinations. As a conse-
quence connectivities that are suspected to be nonstandard
are included in all resulting combinations and the initial
number of combinations reduces (as will be shown later this
number can be reduced by many factors). In addition, the
algorithm is capable of immediately detecting combinations
ofconnectivitiesfromwhichstructuregenerationis impossibles
a connectivity combination of this kind still contains at least
one nonstandard connectivity. These combinations are skipped
during the structure generation process. As a result fuzzy
structure generation can be performed in a reasonable time
even in those cases when nmath is very large. If the MCD
checking process fails to detect nonstandard correlations in
the 2D NMR data (according to our studies the probability
of failure is about 10%), then the program is forced to try
all CN
m
connectivity combinations. This can drastically
increase the time to solve the problem, and the described
approach is inefficient. In these cases User Fragments and
Found Fragments3-4 can frequently be helpful. The ability
of the program to calculate and display the real number of
connectivity combinations to be validated during fuzzy
structure generation allows the user to approximately evaluate
the complexity of a given task even at the first stage of the
structure elucidation process.
When option parameters are combined in a different way,
it is possible to initiate the following modes of fuzzy structure
generation:
Mode 1. Structures are generated such that the number of
correlations that are extended is specified (m)m0) and
connectivity augmentation is also assigned (a)a0). In this
case for a GHMBC correlation having a length of 1-2
skeletal bonds both the lower and upper length limits are
updated and the connectivity length is extended to 3 bonds.
Mode 2. Structure generation is performed using the
following options: it is assumed that the number of
extendable (or ignored) connectivities cannot exceed
mmax, (m)1,2,...,mmax), while a is equal to a0. The mmax value
is defined as the maximum allowed number of nonstan-
dard correlations in the 2D NMR data. Typically the mmax
value is set equal to 20 thereby covering a wide range of
nonstandard connectivities (see Figure 1). The program
initially performs structure generation with a value of m )
1. If the attempt is unsuccessful, then the m value is
automatically incremented by 1, and a new run is made with
m ) 2 and so on. An iteration is declared unsuccessful if
either no structure is stored after structure generation and
spectral filtration or if an unacceptable solution was found.
When m reaches the mg value, then the program considers
the 2D NMR data to be consistent, and then fuzzy structure
generation is initiated with m ) mg. The program stops after
completing structure generation with m ) mg if the output
structural file is not empty and if an acceptable solution is
provided.
Mode 3. The number of connectivities m is allowed to
vary between mmin and mmax values (mminememmax), while
the fixed number of bonds a0 is set. The minimum number
mmin is usually derived as a result of checking the 2D NMR
data for consistency. The program stops when similar
conditions as described for mode 2 are achieved.
Mode 4. This mode is a generalization of mode 3 where
the interval for m value variation is defined by the condition
mmin e m e mmax at mmin ) 0. The peculiarity of this mode
is that it is a “generalized” mode of structure generation and
can be initiated with m ) 0. In this mode, the program starts
by checking the hypothesis that nonstandard correlations are
absent in a given 2D NMR data set. If the data set does not
contain nonstandard connectivities, then the program com-
pletes the process of structure generation, and the further
solution of the problem is carried out as described
previously.3-4,14 If an attempt with m ) 0 proves to be
unsuccessful, then the program automatically performs fuzzy
structure generation starting with m ) 1, a )16 and continues
problem solving in the manner described earlier for mode
3. The merit of such an approach is that no assumption
regarding the a value is necessary.
Mode 5. This mode is initiated if it is necessary to perform
fuzzy structure generation iteratively covering all values of
m starting from mmim to mmax without exclusion. For example,
if structure generation is successful at m ) mg, then the
program automatically switches to m ) mg+ 1 and so on
until it reaches m ) mmax. The structures generated at each
step are added to those generated during the previous step.
This mode is useful to check the solution for stability to make
sure that the best structures found at steps m ) mg and m )
mg + 1 or higher are equivalent.
Mode 6. This mode resembles mode 5, but the function
of this mode is to generate all structures for which the number
of nonstandard connectivities is less or equal to m at the
given value of a. The corresponding options are denoted as
{mem0, a)a0}. The number of connectivity combinations
from which fuzzy structure generation is performed depends
only on the N0 and m values. In contrast to the “step-by-
step” modes some combinations of the connectivities are
united by this approach, and this in principle can speed up
the calculations. When this procedure is performed, only the
maximal lengths of GHMBC connectivities (i.e., two skeletal
bond lengths) are enlarged. For example, consider a GHMBC
connectivity between C-1 and C-2 atoms whose “standard”
length is varied from 1 to 2 skeletal bonds. In this mode the
updated connectivity length varies from 1 to 3 skeletal bonds.
It is important to note that the number of nonisomorphic
structures generated in this mode is equal to the total number
1056 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 ELYASHBERG ET AL.

of nonisomorphic structures generated during all steps of
mode 5. However, the total time necessary for completion
of fuzzy structure generation can be significantly different
between these modes.
Mode 6 with the parameter a set equal to 16 can be
considered as the most comprehensive mode since in
principle it will solve a problem in which the 2D NMR data
contain an unknown number of nonstandard connectivities
of an unknown length. Experience has shown that depending
on the complexity of the problem the m value is typically
equal to 5, 10, or 15. If the problem is successfully solved
with a given set of options, then the real m and a values are
simply determined visually from the resulting structure which
is displayed along with all COSY and GHMBC connectivi-
ties. In other cases these parameters can be estimated only
by the trial method.
All modes described above are summarized in Table 1.
In addition to the approaches mentioned for controlling fuzzy
structure generation there is also a possibility to exclude the
COSY data from the process of fuzzy structure generation
as a user option. In some cases, especially those when the
COSY data contain many nonstandard correlations requiring
a > 1 while the GHMBC data are rich enough, the exclusion
of the COSY data both simplifies and accelerates the solution
of the problem.
2.2. The Strategy of Fuzzy Structure Generation
Application. The possibility of employing several different
modes of fuzzy structure generation provides a very flexible
analytical tool. However, the diversity of modes available
is also a source of complexity since the user has to choose
the optimal mode when solving a specific problem. Before
starting the calculations it is unclear which mode will lead
to a solution in a reasonable time. An attempt was made to
answer the question of whether there is a general strategy
of structure elucidation using fuzzy structure generation that
works best. A set of more than 100 problems was selected
where either the GHMBC or COSY spectra or both contained
a total of 1-18 nonstandard connectivities corresponding to
a range of coupling constants nJHH,CH where n ) 4-6. The
structures under investigation were all natural products, and
the number of skeletal atoms in the molecules varied between
15 and 75 skeletal atoms. The experimental data were
obtained from articles published mainly in the Journal of
Natural Products or from collaborations with various
laboratories.
For each problem the NMR spectral data were entered into
the program and graphically represented as MCDs (Molec-
ular Connectivity Diagrams, see refs 3 and 4). The procedure
for checking the 2D NMR data for contradictions1 was then
applied to every problem. If the presence of nonstandard
connectivities was revealed, then the program displayed the
minimum number of nonstandard connectivities and made
an attempt to automatically resolve the contradictions as
described in our work.1 In successful cases the updated
MCDs were displayed with modified connectivities marked
by specific color.
As a result of these studies all problems were classified
into three sets as follows: (1) 53 problems were identified
where NSCs were detected and the initial MCDs were
updated; (2) 34 problems were identified where the program
revealed presence of NSCs but failed to update the MCDs;
and (3) 13 problems were identified where the program failed
to detect NSCs.
This classification describes all conceivable results of
checking the MCDs. Depending on the results of checking
the MCD, various modes or combinations of modes can lead
to solution of the problem. Attempts to solve each problem
were made using different fuzzy structure generation modes
to investigate possible approaches. The problems for which
valid solutions could not be found during the first attempt
were eventually solved after utilizing different fuzzy genera-
tion options. Logical data preprocessing frequently allowed
significant reduction of the number of connectivity combina-
tions to be tested during the fuzzy structure generation. Figure
2 shows the ratio F describing the number nreal of tested
connectivity combinations to the theoretically calculated
number of combinations, nmath ) CN0
m for the entire problem
set. Figure 3 examines in greater detail these combinations.
The figures demonstrate that the theoretical number of
combinations can be hundreds of billions, but the real
Table 1. Main Features of Fuzzy Structure Generation in Different Modes
generation
mode
m, number of extendable or
ignored correlations a, connectivity length augmentation
peculiarities of the fuzzy structure
generation process
mode 1 M ) m0 a ) a0 Generation is stopped after checking
all possible combinations of
extendable correlations.
mode 2 m ) 1 f mmax, m ) 1, 2,... a ) a0 Stop after completing structure generation with
mmax, mmax is defined m ) mg if the output structural file is not
by the user empty and if an acceptable solution
is provided.
mode 3 m ) mmin f mmax, mmin e m e a ) a0 the same conditions as for mode 2
mmax, mmin and mmax are
defined by the user
mode 4 m ) mmin f mmax, mmin ) 0, Start with strict generation at m0 ) 0, The same conditions as for mode 3. No
mmax is defined a ) 0. If unsuccessfully, assumption about a value
by the user continue with mmin ) 1, a ) 16 is necessary.
mode 5 m ) mminmmin f mmax, mmin e a ) a0 or a ) 16 at user discretion Perform fuzzy structure generation iteratively
m e mmax, mmin and covering all values of m starting from
mmax are defined mmin to mmax without exclusion
by the user
mode 6 Generate all structures for which the a ) a0 or a ) 16 at user discretion Allows a problem to be solved in which the 2D
number of NSCs is less or NMR data contain an unknown number
equal to m, of nonstandard connectivities of
{m e m0, a ) a0} an unknown length.
A NEW EFFICIENT TOOL FOR CASE J. Chem. Inf. Model., Vol. 47, No. 3, 2007 1057

numbers reduce down to manageable dimensions. For
instance, in 20 problems the theoretical number dropped by
104-106 times, but the real numbers of combinations still
remained rather large. Nevertheless, the speed of the structure
generator algorithm was fast enough to solve almost all
problems.
Fuzzy structure generation did however fail for the
elucidation of structure S4 (C32H50NO2) in Figure 1. The 2D
NMR data contain 18 nonstandard connectivities (12 GH-
MBC and 6 COSY nonstandard connectivities; 5 connec-
tivities are of the type 5J). The theoretical number nmath of
connectivity combinations is equal to 43  1012 for this
case. The difficulty could be circumvented by using the
Fragment Mode, but no large appropriate fragment was found
in the database during the 13C NMR search. The application
of a large User Fragment led to an extremely large set of
MCDs with each containing the User Fragment with different
distributions of the carbon chemical shifts2. As a result these
two combinatorial “explosions” hampered problem solving.
The solution of such computationally difficult problems will
hopefully be eased by further development of the algorithm
providing fragment “implementation” in MCDs. Work in this
direction is presently in progress.
As a result of our studies, general traits were identified
that could help to find appropriate ways to solve a problem.
These strategies, as applied to the three problem subsets
mentioned above, are described in the following sections.
NSCs Were Identified and the MCD Was Updated.
Assuming that the MCD updating process was performed
correctly (with the lengths of all NSCs increased), then strict
structure generation is performed. If an acceptable solution
is obtained, then it should be checked for suitability. Fuzzy
structure generation with the options {m)mmin20; stop at
m)mg, a)16} is started from the initial MCD, not the
updated MCD. The previously found solution will be
confirmed if the first ranked structures for both strict and
fuzzy solutions coincide. When an inequality dstA(1)>dfuzA-
(1, mg) is observed (dstA(1)sthe deviation calculated for the
first ranked structure of the solution found by strict structure
generation, dfuzA(1, mg)sthe same found by fuzzy structure
generation at m)mg), then it is concluded that not all NSCs
were lengthened and fuzzy structure generation should be
repeated with mg + 1 and so on until the minimum value of
dfuzA(1, mg+V) and a valid solution is achieved at m ) mg +
V. The corresponding structure is then considered as the most
probable.
An unacceptable solution can be obtained as a result of
strict structure generation from the updated MCD, i.e., a
solution will be found for either dstA(1) >DA, where DA is a
threshold value or an empty structural file is obtained (k)0).
In both cases the program is automatically switched to the
mode where {m)mmin20, stop at m)mg, a)16}. Depending
on the mg values and the complexity of the problem (the
size of mreal and the calculation time) evaluated during the
first stages of solving the problem the user can initiate fuzzy
structure generation with the options {mem0, a)16}, m0 )
5, 10, or 15 to obtain the most reliable solution.
NSCs Were Identified but the MCD Failed To Be Updated.
If the program identified NSCs but failed to update the MCD,
then fuzzy structure generation is one manner by which to
solve such a problem. Since the software application only
displays the minimum number of NSCs while their associated
lengths remain unknown, the solution should be used in
Common Mode4 of FSG with the options {m)mmin-20, stop
at m)mg, a)16}. The real numbers of the connectivity
combinations, nreal, are displayed as well as the number of
combinations for a given m ) mg, and the predicted time
for structure generation allow the user to easily evaluate the
complexity of the problem and suggested time for execution.
If mode 6 can be applied as on the time estimates then it
should be used.
NSCs Were Not Detected. If nonstandard connectivities
were not revealed by checking the MCDs, then there are
two ways to interpret this result: either the 2D NMR data
are free of nonstandard connectivities or the NSCs are
present, but the program failed to detect them. Both of these
situations are covered by fuzzy structure generation with the
options {m)020, stop at m)mg, a)16}. If NSCs are
indeed absent from the 2D NMNR data, then structure
generation is performed with m ) 0 with a nonzero output
file and the values of deviation allow the user to determine
whether the solution determined is acceptable. Obtaining
deviation values that exceed the threshold DA or deriving
an empty output file after spectral filtering both serve as hints
to the presence of latent nonstandard connectivities.
Figure 2. The ratio of numbers of real connectivity combinations
to the numbers of theoretically possible combinations for the
problems solved using fuzzy structure generation. The program
failed to reduce the number of combinations mainly in those cases
when nonstandard connectivities were not detected during checking
of the MCD.
Figure 3. A plot of the logarithms of the theoretical (red) and
real (blue) numbers of connectivity combinations.
1058 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 ELYASHBERG ET AL.

When NSCs are not detected by the logical data analysis,
then the number of connectivity combinations that must be
tested during fuzzy structure generation cannot be reduced,
and it is equal to CN0
m
, m ) 1, 2, 3, ... at each mth step of the
fuzzy structure generation process. This situation can cause
significant difficulties due to an unmanageable number of
connectivity combinations needing to be processed; as
discussed previously, both Found and User Fragments can
assist in this situation.
It is difficult to describe the myriad of nuances associated
with fuzzy structure generation since these depend on each
2D NMR data set associated with a given problem. A series
of examples illustrating the strategies leading to valid
solutions with the minimum number of user assumptions will
be presented. Example problems were chosen where auto-
matic updating of the MCD to resolve contradictions was
inefficient. The structures are shown in Figure 1 where there
are a large number of NSCs used as examples.
2.3. Problem Solution in the Common Mode. Example
1. In the analysis of cleospinol A9 with molecular formula
C20H32O2 (1), the 2D NMR data are comprised of 21 COSY
and 55 HMBC correlations. These data were used to evaluate
the possibility of solving a problem in those cases when a
large number of nonstandard correlations were present. In
this case the 2D NMR data contained the following
combination of NSCS: 3 HMBC[2a(1), 1a(3)] + 12COSY-
[8a(1), 3a(2), 1a(3)] ) 15. This nomenclature describes the
fact that there are 3 HMBC nonstandard correlations, two
of which must be lengthened by 1 bond and one by 3 bonds;
the information about the 12 COSY correlations is interpreted
analogously. The total number of NSCs is 15. In this article
such expressions are used to provide short and unambiguous
designations of the numbers and lengths of the NSCs
contained within the 2D NMR data.
The COSY connectivities are represented below on the
structure by blue double-headed arrows, while the HMBC
correlations are defined by green unidirectional arrows from
the proton to the carbon to which it is long-range coupled.
The COSY, HMQC, and HMBC spectral data associated
with the compound were fed to the program, and the MCD
was generated. A check of the MCD was accompanied by
the automatic removal of contradictions. The software
program displayed a message declaring that the contradic-
tions had been detected and resolved, while the minimum
number of NSCs was estimated to be equal to 7. Unfortu-
nately strict structure generation from the automatically
edited MCD resulted in an empty output file. This result was
interpreted as evidence of the presence of either undetected
additional nonstandard correlations or those whose lengths
must be augmented by more than one bond.
There are two possible trajectories from this point to solve
the problem. Since in general there is no information about
the number of NSCs and their lengths, these values can be
determined using a trial and error method. If it turns out
that a ) 1, then there is a chance to find a solution in a
short time. The second approach is more general and allows
the user to ignore the problem of determining the maximal
a value. The cost, however, may be longer structure
generation times and a consequently larger output file. Both
approaches are described in detail below as applied to this
specific example.
With the first approach fuzzy structure generation was
initiated from the initial (not updated) MCD using mode 3
with the options {m)7-20, stop at m)mg, a)1}. The
program started the generation process automatically with
mg ) 14 (m)7-13 were immediately rejected), and the
process was aborted by the operator at m ) 16 with a zero
result. An attempt to repeat fuzzy generation with a ) 2
again gave an empty result file.
The possibility that one or more of the nonstandard
connectivities needed to be augmented by 3 bonds was
assumed. When the options {mmax)20, stop at m)mg, a)3}
were set, then the program automatically started with mg )
10 and completed the fuzzy generation process in 9 m 15 s
(Pentium IV, 2.8 MHz) with m ) 14. Three molecules were
generated and two were stored in the results file after the
removal of duplicate structures. The highest ranked structure
coincided with the actual structure, 1. The second structure
gave a dA(2) value with ¢2-1 ) dA(2) - dA(1) ) 1.5 ppm. A
solution was found at m ) 14 and not at m ) 15 since the
COSY and HMBC connectivities between C-5 and C-9
carbons are coincidental. The program displays the final
structure where all connectivities and their associated lengths
can be visualized.
The second more universal and systematic approach was
applied, and fuzzy structure generation was initiated assum-
ing only that the number of nonstandard connectivities is
not more than 15 (mode 6), i.e. options {me15, a)16} were
set. In this case 18 281 379 connectivity combinations from
40 225 345 056 theoretically possible combinations were
used for structure generation. The following result was
obtained: 769 structures were generated, 430 structures were
stored after spectral filtering, 245 structures remained after
removing duplicates (this is denoted as k)769f430f245),
a generation time of tg ) 29 min 9 s, and the correct structure
was ranked first by all methods of spectrum prediction, rall
) 1.
The program therefore identified the correct solution even
when 15 nonstandard connectivities existed in the 2D NMR
data and especially in the presence of HMBC and COSY
connectivities representing both 6JCH and 6JHH correlations.
Note that only 10-4 of the theoretically possible connectiv-
ity combinations were processed. The real number of
processed connectivity combinations nreal is more than 18
million. Nevertheless, the high-speed structure generator
present in the Structure Elucidator program completed the
process in a reasonable time.
Example 2. The 2D NMR data associated with a natural
product with molecular formula C20H28O isolated and identi-
fied by Mensah et al.10 contains 14 NSCS: 2HMBC-
[2a(1)]+12COSY[4a(1), 6a(2), 1a(3)] ) 13. The presence
of a minimum of 6 NSCs was detected, and the program
A NEW EFFICIENT TOOL FOR CASE J. Chem. Inf. Model., Vol. 47, No. 3, 2007 1059

displayed a message indicating that the MCD was success-
fully updated. Strict structure generation of the updated MCD
led to an empty output file indicating that fuzzy structure
generation must be used. Experience shows that the maxi-
mum value of a usually does not exceed 3. A solution to
this problem can be obtained with the options {m)6-15,
stop at m)mg, a)3}. The program initiated generation at a
value of mg ) 11 and stopped when 4 structures were
generated. The large deviations calculated for the best
structure suggested that the solution was invalid (see Table
2). The results of subsequent structure generations with mg
) 11, 12, and 13 are presented in the Table 2. All deviations
calculated for the best structure achieved minimum values
for the solution obtained at a value of mg ) 13: the best
structure coincided with structure 2. Carbon atom assign-
ments were identical to those suggested by the authors.10
The practical approach was applied with the options
{me15, a)16}. The following results were obtained: k )
66 538 f 38 407 f 12 070, tg ) 2 h 41 m, nmath  63â109,
nreal ) 48 525 735, F ) 7.6â10-4, rA ) r“ ) 1. It is worth
noting that the 13C NMR spectra of 38407 structures were
calculated in 2 min 23 s for preliminary ranking using the
incremental method.
Example 3. Natural product 3
was isolated and identified by Wellington et al.11 using a
combination of HMBC and COSY spectra. There are 8 NSCs
in the 2D NMR data: 5HMBC[2a(3), 3a(1)] +3COSY-
[3a(1)] ) 8. As a result of checking the MCD it was updated,
and a further analysis showed that the minimum number of
NSCs mmin was 4. An attempt to perform strict structure
generation was unsuccessful, and the program produced an
empty output file. The problem could be solved using “step-
by-step” methods analogous to those described in the
previous example, but we present here only the result of the
practical approach.
For modeling a situation where nothing is known about
the number and lengths of NSCs except that m g 4 the
following fuzzy structure generation options to ensure
minimization of the risk of losing the correct solution were
set: {me10, a)16}. The result is defined by k ) 114 638
f 68 668 f 23 213; tg ) 1 h 52 m; nreal ) 52 427 715,
nmath  0.6â109; F ) 0.08; rall ) 1. Using mode 6 the program
performed an exhaustive search of all possibilities to generate
structures (see section 2.1), and the output file is large. The
13C NMR spectrum prediction for about 70 000 structures
using the incremental method took about 5 min. The correct
structure was distinguished by examining the values of the
deviations calculated using all methods within the StrucEluc
system.
Example 4. Kirsch and coauthors17 reported the isolation
and structure elucidation of a natural product with a
molecular formula of C25H38O2 (4) using 2D NMR data
containing 23 COSY and 41 HMBC correlations. As
illustrated by the arrows the following set of NSCs are
observed in structure 4: 2HMBC[2a(2)]+7COSY[4a(1),
3a(2)] ) 9.
The following steps allowed identification of a correct
solution to this problem:
(1) When the MCD was checked, nonstandard correlations
were detected, but the program declared that the contradic-
tions in the 2D NMR data could not be resolved. The
minimum number of NSCs had a value of 6.
(2) Fuzzy structure generation was performed with the
options {m)610, stop at m)mg, a)1} but resulted in an
empty output file.
(3) Fuzzy generation with the options {m)6-10, stop at
m)mg, a)2} provided a valid solution in less than 7 min.
The solution was obtained with m ) 9 since m ) 1-8
resulted in empty output files.
When the more general approach was applied with options
{m)620, stop at m)mg, a)16}, a single and correct
structure was found in 2 min. With this 370 950 connectivity
combinations of 2.5â109 theoretically possible ones com-
binations (F10-4) were processed in this time.
The validity of the solution was checked by a time-
consuming process using fuzzy structure generation with the
options {me10, a)16} to give the following result: k )
18 f 17 f 15, tg ) 28 min 35 s, nreal ) 4 830 600, nmath 
10â109, F ) 4.7â10-4, rall ) 1. The first ranked structure and
associated atom assignment coincided with that determined
by the authors.
Example 5. Computational difficulties associated with
structure generation rise with an increase in the complexity
of the molecule even in those cases when the 2D NMR data
contain correlations of standard lengths only. The difficulties
become especially serious if the nonstandard correlations
exist in COSY and HMBC spectra. The possibility of solving
problems using fuzzy structure generation for a large
molecule with a large number of NSCs in 2D NMR data is
illustrated in the following example.
Feller et al.18 reported the isolation and structure deter-
mination of a new terpenoid 5 with a molecular formula of
C43H66O10 and therefore containing 53 skeletal atoms:
The 2D NMR data were composed of 53 COSY and 94
HMBC correlations including 10 nonstandard connectivities
1060 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 ELYASHBERG ET AL.

as shown on the representation of structure 5 and enumerated
as follows: 5HMBC[5a(1)] + 5COSY[4a(1), 1a(2)] ) 10.
When the MCD was created, four carbon atoms, whose
chemical shifts are marked in red, were not involved in any
correlations. The presence of such “free” atoms introduces
an additional obstacle to solving a problem and generally
leads to an increase in structure generation time.
MCD checking accompanied by automated resolution of
contradictions in the 2D NMR data produced a program
message declaring that the minimum number of nonstandard
connectivities was 7. The MCD was then updated by the
program to resolve the contradictions. Structure generation
from the updated MCD gave k ) 56 f 28 and tg ) 21 s.
Structure 5′ was distinguished as the best one.
The deviation value of dA(1) ) 2.74 ppm is typical for a
correctly recognized structure2 so there was no reason to
reject the first ranked structure. The visualization of the
nonstandard correlations in structure 5′ displays the presence
of 6 nonstandard COSY connectivities, including the con-
nectivity 19.8 to 40.7 corresponding to a 5JHH coupling
constant. This connectivity was lengthened by two bonds
following analysis of 2D data. Five of the six NSCs exist in
structure 5, but lengthening the connectivity 46.4-32.6 was
a mistake and leads to an incorrect structure similar to the
genuine one. To obtain a reliable solution fuzzy structure
generation was performed as outlined below.
Fuzzy structure generation was performed using the
options {m)7-15, stop at m)mg, a)16}. A nonempty
structural file was generated with mg ) 9 (nmath328â109,
nreal28â106, F)8â10-5) due to the accidental degeneracy
of two COSY and HMBC nonstandard correlations 40.7-
19.8 (otherwise only m)10 would be successful). The
following results were obtained: k ) 36 f 21 f 11, tg )
7 h 50 min, r(all) ) 1, dA(1) ) 2.62 ppm, and the best
structure coincided with that deduced by the authors.18 The
difference between the dA deviations for the correct (5) and
incorrect (5′) structures is only 0.12 ppm. The calculated
similarity coefficient was equal to 0.98 for these structures.
The structures differ only by the permutation of the carbons
at 47.9 and 46.4 ppm which resulted in the transformation
of part of the correct structure into its mirror reflection.
The solution was fairly time-consuming due to a large
number of connectivity combinations (28â106) as well as the
presence of four carbon atoms with no connectivities
demonstrated to other atoms. The example provides evidence
that the approach is efficient even in a situation when the
analyzed molecule is large and the number of NSCs is big
enough including correlations corresponding to a > 1.
The examples presented in this section demonstrate the
high efficiency of the procedure suggested for logical analysis
of 2D NMR data. The application of this procedure reduces
the total number of connectivity combinations by about 104-
106 times, and this allows the program to complete the fuzzy
structure generation within a reasonable time. The examples
given lead to the conclusion that the capability of fuzzy
structure generation in the StrucEluc system enables complex
tasks to be successfully resolved even in those cases when
the number of nonstandard correlations is large (10-15), and
when correlation lengths exceed the default values of 2-3
bonds by 2 or even 3 additional bonds.
2.4. Problem Solution in the Fragment Mode. As shown
in our previous publications,3-4,14 the use of fragments stored
in a content database (the so-called Found Fragments selected
as a result of a database search using 13C NMR spectra as
inputs) as well as User Defined Fragments allows problems
to be solved even when Common Mode structure generation
is time-consuming and cannot be completed in a reasonable
time. In this work peculiarities of the fragment approach have
been investigated as a tool for solving problems in the
presence of nonstandard connectivities.
As shown previously3 the need to apply fragments to solve
a problem arises when there is a deficit of hydrogen atoms
in the molecular formula or when the number of connec-
tivities in the 2D NMR data is simply not ample enough to
produce a set of efficient structural constraints. One might
assume that the larger the fragment involved in forming the
MCD then the quicker the time to arrive to a solution. In
reality for a large fragment there is a huge number of
permutations for different assignments of the experimental
chemical 13C shifts to the fragment carbon atoms, and this
can lead to an extremely large number of MCDs for
processing during structure generation. When fuzzy structure
generation is applied to the set of MCDs containing frag-
ments, the mg value can be different for each MCD, and the
Table 2. Results of Four Subsequent Steps of Fuzzy Structure Generation
m N0 nmath (approximately) nreal F (nreal/nmath) results
11 41 3.1â109 17 629 5.6â10-6 k ) 4 f 4, tg ) 1 m 35 s, dA(1) ) 6.22,
12 41 7.9â109 240 001 3â10-5 k ) 25 f 19, tg ) 2 m 40 s, dA(1) ) 5.15,
13 41 17.6â109 1 601 574 1.2â10-4 k ) 218 f 176 f 110, tg) 12 m, dA(1) ) 2.13
A NEW EFFICIENT TOOL FOR CASE J. Chem. Inf. Model., Vol. 47, No. 3, 2007 1061

mmin value estimated during checking of the MCD generated
in the Common Mode has low predictability. The specificity
of fuzzy structure generation from a set of MCDs containing
fragments will be demonstrated by examining a series of
examples.
Example 1. Mdee and co-workers19 isolated and identified
using 2D NMR data (9 COSY and 44 HMBC correlations)
a new bichalcone of molecular formula C30H22O8.
The COSY spectrum contained a set of nonstandard
correlations 3COSY[1a(1), 1a(2), 1a(3)] as illustrated by the
blue arrows shown in structure 6. Checking the MCD and
using automatic contradiction removal detected a minimum
number of 2 NSCs and resulted in updating of the initial
MCD. No structures could be generated from the resulting
MCD however. An attempt to perform fuzzy structure
generation from the initial MCD with the options {m)210,
stop at m)mg, a)16} was also unsuccessful: even though
only 213 connectivity combinations were processed with mg
) 2 and a ) 16 the fuzzy generation process turned out to
be very time-consuming and tens of hours were predicted
for the value tg. A search of the Fragment Database using
the 13C NMR spectrum as an input was performed, and 1138
fragments were selected. The first ranked fragment, 7,
showed good coincidence of its 13C subspectrum with the
chemical shifts of the unknown compound (spectrum com-
parison was easily performed due to a visual display
capability for spectrum representation4). The program created
four MCDs from this fragment.
Fuzzy structure generation was initiated {m)2-15, stop
at m)mg, a)16} and was completed with the following
result: k ) 46, tg ) 6 min 30 s, rall ) 1, i.e., the correct
structure was unambiguously identified. Analysis of the
conditions applied for fuzzy structure generation was per-
formed from each MCD which delivered a result and gave
In spite of the different conditions 156 structures were
generated from each MCD and resulted in 46 nonisomorphic
structures. It is evident that the fuzzy structure generation
process was not performed in the most rational way when
applied to each MCD, and further work is necessary to
optimize this procedure.
Example 2. Kehraus et al.20 reported the separation and
identification of a new natural product geometricin A,
C39H63N2O12P (8).
The reported 2D NMR data included 11 COSY and 99
HMBC correlations. Two NSCs characterized with a ) 1
are present in HMBC data set and are represented by green
arrows. The program detected the presence of NSCs and
determined mmin to be equal to 2 but failed to resolve the
contradictions. An attempt to apply fuzzy structure generation
to the MCD showed that the problem would be too time-
consuming without the introduction of some key fragments.
The following user fragments were introduced to assist in
solution of the problem.
Kehraus et al.20 reported that the 13C NMR chemical shifts
for carbons C-49, C-50, and C-53 together with a singlet
proton resonance at ä 7.76 characterized an oxazole ring.
With the chemical shifts calculated for fragment 9 the
program produced 18 MCDs containing both user fragments.
With mmin ) 2 fuzzy structure generation was performed
with {me3, a)16} to produce the following result: k )
6180 f 4900 f 2450; tg) 3 min 30 s, nmath ) 57 154, nreal
) 1056, F ) 0.02, rall ) 1. In reality a result was obtained
only from one of the 18 MCDs since the other MCDs
contained fragment 9 in a format whereby the carbon atom
assignment did not correspond to that specific to the target
structure. 13C NMR spectral prediction for the 4900 structures
via the incremental method took only 27 s. As a result of
prediction the correct structure was ranked first in the output
file.
2.6. Is There an Alternative to Fuzzy Structure Gen-
eration? To the best of our knowledge the question regarding
to what extent the lengthening of all 2D NMR correlations
can act as a method for contradiction resolution in 2D NMR
data has never been investigated. In this study an attempt to
identify the quantitative characteristics describing how
structure generation time increases and the amount of
structural information obtained decreases if all correlations
MCD #1: mg ) 10, nreal )
6060, nmath  20â10
6
, F ) 3â10-4
MCD #2: mg ) 6, nreal )
96, nmath  0.3â10
6
, F ) 3.2â10-4
MCD #3: mg ) 6, nreal )
3319, nmath  0.5â10
6
, F ) 7â10-3
MCD #4: mg ) 2, nreal ) 27, nmath ) 351, F ) 8â10
-2
1062 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 ELYASHBERG ET AL.

belonging in the 2-4J range have been made. In previous
work13 it was suggested that in principle the amount of
structural information obtained as a result of the application
of an expert system can be measured in the manner described
below.
Assume that the entire number of possible isomers N
corresponding to the molecular formula of an unknown
compound is fixed and nothing is known about the structure
to be analyzed. Let pi be the probability that the ith (1 e i
e N) isomer is the genuine structure. Before the task can be
solved all isomers must be equally probable and pi ) 1/N.
According to Shannon21 the initial entropy E0 characterizing
the solution can then be calculated from the equation E0 )
log2N.
If the task results in an output file containing k structures
and the solution contains the genuine structure, then the
entropy of the correct solution, Ec, can be calculated as Ec
) log2k. The amount of structural information obtained as a
result of solving the problem can be expressed as follows:
Obviously, if k ) 1 (the structure of the unknown
compound is unambiguously and correctly elucidated), then
the general amount of structural information I0 obtained as
a result of solving the problem will be equal to I0 ) log2N.
The value í(%), the portion of the full structural information
obtained at any given stage of the task, can be expressed as
follows:
Our computational experiments have demonstrated that it
is possible to generate a full set of isomers corresponding to
a given molecular formula for small molecules containing
no more than 17 skeletal atoms2 using the GENM22-24
generator present in StrucEluc. It takes tens or even hundreds
of hours when a PC Pentium IV, 2.8 GHz is used. For our
work examining structural problems via 2D NMR data only
problems with less than 20 skeletal atoms were selected. The
experimental data were borrowed from publications.
In Table 3 the structures, their molecular formulas, and
the total number of structural isomers are given. The table
shows that all selected structures can be considered as small
structures. For each problem the two sets of solution results
obtained with two different options enabled are listed. In
the first case all correlations were assumed to be 2-3J by
default. In the second case the option settings allowed the
connectivity length to vary between 2 and 4. Also listed is
the number of structures k contained in the output file and
the generation times tg(2-3) and tg(2-4) and their ratio ô )
tg(2-4)/tg(2-3). The ô value demonstrates a slowing down of
the generation process if generation is performed using
default correlation lengths corresponding to 2-4J. The í
coefficient estimates the loss of structural information that
takes place in this case.
Analysis of the table shows that even in the case of small
molecules the output file size increases considerably when
the 2-4J couplings are set as default. Under those conditions
the portion of extracted structural information drops from
95-100% to 60-70%. At the same time the generation time
increases by many times to hundreds or even tens of
thousands times greater. Problem 6 is the most distinctive
example where the size of the output file increased from 2
to 3036 structures, while the generation time increased by
6.5 million times!
For problems 2 and 5 in Table 3, the 2D NMR data
contained nonstandard correlations, and both tasks were
solved automatically using fuzzy generation with the output
file containing only one structure at the conclusion of the
run. The solutions to these problems found by simply
lengthening all correlations to 2-4J resulted in an increase
in the number of structures up to 25 (problem 2) and up to
1211 (problem 5) structures. The computing time grew by
25 times and 11 900 times, respectively.
Table 3 lists some examples that allow the examination
of the dependence of the results on the default option settings.
However, the most typical problems of the same structure
size are presented in Table 4. In example 134 8 out of 28
HMBC correlations were of nonstandard length. The task
was solved using the fuzzy generation mode and resulted in
a single correct structure with a generation time of 41 min
35 s. Strict generation with the coupling constant values 2-4J
set by default gave k ) 4 and tg ) 24 min. This example
shows that if a molecule is small and the number of
nonstandard correlations 2-4J is large, then while the longer
correlations (5J, 6J) are absent in 2D NMR data the
application of fuzzy generation and the lengthening of all
correlations both give comparable results in a case when the
number of skeletal atoms is less than 20.
The problem solution time and the number of generated
structures increase dramatically when the size of the molecule
increases, and it is evident that the default setting of 2-4J
for the correlation length can only assist when the number
of skeletal atoms is around 20.
The solutions for problems 2 and 335,36 with 2-3J coupling
constants set as the default were identified in several seconds.
However, when the coupling constants were set to 2-4J as
default, the program was aborted by the operator in about
15 h. At the same time the number of generated structures
was around 105, and it was impossible to predict the time
left in order for structure generation to be completed.
Investigation of the results of problem 437 also provided
interesting results. The correct solution was determined by
two approaches: using the method of automatic correction
of the MCD (k)1, tg )0.009 s) and using fuzzy structure
generation with the options {mmax)10, stop at m)mg, a )2}
and resulted with k ) 1, tg ) 0.031 s. Structure generation
with all correlations set to 2-4J by default gave 61 structures
in about 6 min. Rank ordering of the structural file according
to the deviation values gave a structure with the deviations
dA(1) ) dF(1) ) 8.6 ppm and dH(1) ) 0.66 ppm placed at
the top of the ranked file. According to previously defined
criteria dA(1) should be less than 5.5 ppm for the correct
structure, and under this constraint the solution is likely
incorrect. This implies that the generation process should
be repeated with the coupling constants set to 2-5J as the
default.
It could be expected that the number of structures would
be large so the structure generation process was executed in
a mode whereby the structures were not written to disk. The
program was later aborted by the operator after 6 min since
the number of resultant structures had already reached half
a million.
I0 ) E0 - Ec ) log2N - log2k ) log2(N/k)
í ) (Ic/I0)â100 ) (1 - log2k/log2N)â100
A NEW EFFICIENT TOOL FOR CASE J. Chem. Inf. Model., Vol. 47, No. 3, 2007 1063

The last example shows that even in the case of a molecule
being fairly small, in this case with n < 20, and even when
the both COSY and HMBC spectra contain a large number
of standard correlations, the process of increasing the
intervals allowed for the correlation lengths offered no
solution. In such a situation, the most effective way to solve
the problem appears to be fuzzy structure generation.
We assume that the results obtained in this work strongly
support the guideline that the lengthening of all correlations
should be rejected as a general method of solving problems
arising from the presence of nonstandard correlations in 2D
NMR data. This conclusion is even more obvious if we take
into account the distribution of problems solved using
StrucEluc.2 The distribution shows that molecules with less
than 20 skeletal atoms are rarely found among natural
products.
4. CONCLUSIONS
This study has shown that fuzzy structure generation
should be considered as the most general method for structure
generation in expert systems based on 2D NMR spectra. It
allows the process of structure elucidation to be initiated
under conditions whereby the user has no idea about the
presence or absence of NSCs or details regarding their real
lengths. This is attained by setting appropriate generator
options that provide a varying number of NSCs in the range
of 0 up to mmax. Experience has shown that the value of mmax
Table 3. Dependence of the Amount of Structural Information
Extracted from 2D NMR Data Based on the Nature of the Coupling
Constant nJ Value Set as the Default during the Structure
Generation Processa
a Calculations were performed with a PC Pentium IV, 2.8 GHZ with
1 Gbyte of RAM. The abbreviation NSC represents Non-Standard
Connectivity.
Table 4. Dependence of Structural Information Extracted Based on
the J Value Range Utilized for a Series of Fairly “Large-Sized”
Molecules
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is usually set equal to 10, 15, or 20 depending on the
minimum number of NSCs detected by the program during
the logical analysis of 2D NMR data. The real number of
NSCs and their lengths can be determined by two approaches.
The first approach allows identification of these variables
along with the structure of the unknown using a trial method.
In so doing, both variables, m (true number of NSCs) and a
(augmentations of connectivity lengths), are varied.
The second approach assumes only that the m variable
does not exceed some mmax value, while no information about
a value is necessary. This approach is advisable since it is
fully automated and more universal. Its shortcoming is that
fuzzy structure generation can in some cases be more time-
consuming and in addition the output file can be larger. This
can be of little impact since calculation time can be relatively
inexpensive in the present era of computational cost and
structure generation can be performed in background mode
anyways. At the very least computations can be left overnight
as is common for 2D NMR data acquisition. We suppose
that tg ) 5-15 h is still acceptable because the separation
and identification of a natural product by traditional methods
usually takes weeks and even months. To elucidate the
structure of a new compound without any danger associated
with the presence of an unknown number of NSCs of
unknown lengths is attractive enough to afford time-
consuming structure generation.
As a result of dramatically increasing the speed of 13C
NMR spectrum prediction (5000-7000 shifts/s) a large
output file no longer hampers fast candidate structure spectral
prediction that is necessary for the optimal elimination of
isomorphic structures and file ranking in order to select the
most probable structure. For instance, the 13C spectrum
prediction and average deviation calculation for 4900 isomers
generated from the molecular formula C39H63N2O12P (see
structure 8) took only 27 s. Fuzzy structure generation can
be concluded to be an appropriate analytical tool for
application to the structure elucidation of organic molecules
using 2D NMR spectra.
In each individual case a strategy for problem solving is
chosen using an estimation of the problem complexity. This
is possible on the basis of computing the number of
connectivity combinations which can be processed during
fuzzy structure generation. Preliminary logical analysis of
the 2D NMR data allows the reduction of the calculated
number of combinations by 103-105 times. In spite of a large
enough number of remaining combinations to be processed
(thousands and millions) the speed of the structure generator
in StrucEluc is such that it copes with a task in a reasonable
time.
The strategy of fuzzy generation has been illustrated in
this work using a series of real-world examples including
molecules whose 2D NMR spectra contained up to 15 NSCs
with lengths varying between four and six bonds. When a
data set lacked 2D NMR correlations but was accompanied
by the presence of NSCs, then the problem can be solved
using the fragments found in a Fragment Databases using a
13C NMR search and employing user proposed substructures.
However, the solution of such problems can be difficult since
a large number of MCDs can be created from the fragments.
To circumvent this difficulty in the future an algorithm for
fragment “implementation” in a MCD as well as fuzzy
structure generation from fragment-containing MCDs need
to be improved. This work is underway.
In the literature examples are described where lengthening
all correlations by one bond circumvented the problem of
nonstandard correlations in 2D NMR data. This possibility
has been investigated here and has shown that even the
structures of small molecules with less than 20 carbon atoms
in general cannot be elucidated using this approach. The
lengthening of connectivities by one bond is evidently in
vain when the NSCs are characterized by 5-6J coupling
constants present in the 2D NMR data.
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