Indian Journal of Chemi stry Vol. 42A, June 2003, pp. 1315-1329

Reviews

From molecular graphs to drugs. A review on the use of topological indices in drug design and discovery

Ernesto Estradal.· , Grace Patlewicz l & Eugenio Uriarte2

ISafety and Environmental Assurance Centre (SEAC), Unilever COlwOlth, Colworth House, Sharnbrook, Bedford MK44 ILQ, UK

"Faculty of Pharmacy , Department of Organic Chemistry, University of Santiago de Compostela, 15706 Santiago de Compostela, Spain

Received 28 January 2003

A review on the use of topological indi ces (Tis) in the process of drug design and development is presented on the basis of the most recent advances in thi s field . The first part of thi s work is focussed on introducing the definitions of terms used in drug design and di scovery, graph-theoretical chemistry as we ll as topological indices. The second part of the review dis­cusses the use of topological indices in the lead discovery and optimization processes, including simjlarity/dissimi larity studies, rational combinatori al library design, QSAR, the issue of 2D versus 3D QSAR, as well as the use of novel che­mometric techniques, such as artificial neural networks, genetic algorithms and parti al least squares in combination with Tis. Recent advances on the definition of TIs accounting for three dimensional molecular features (topographic indices) and

molecular chirality are also presented.

Introduction The main paradigm of medicinal chemistry is that

biological activity, as well as physical, physicochemi­cal and chemical properties of organic compounds is inherent on molecular structureI.2. Crum-Brown and Fraser published the first quantitative structure­activity relationship) in 1868 on the basis of this prin­ciple. Despite many advances in the field of theoreti­cal drug design, this principle still serves as a useful starting point for guiding the discovery of new lead compounds4

.6

.

Presently the number of known organic and inor­ganic compounds exceeds 20 million (http://www.cas.org/cgi-biniregreporLpl) . Asignifi­cant proportion of these compounds (> I ,000,000) are available from different chemical suppliers . Most pharmaceutical companies have in-house compound collections, which represent a very valuable source for lead discovery and/or optimization. With the ex­plosion of high throughput screening (HTS), large compound libruries can now be screened with ever increasing biological specificity . As a result of these screens, leads are selected and synthesized for further development through pre-clinical and toxicity testing. However, as this development increases so does the cost of lead failures.

The critical step in drug discovery remains the identification and optimization of lead compounds in

a rapid and cost effective way. Computational tech­niques have advanced rapidly over the past decade and accordingly have played a major role in the de­velopment of a number of drugs now on the market or going through clinical trials . These computational techniques are based on the use of a "virtual" world of computer-generated hypotheses, which are tested for practicality. In silico studies minimize the need for resource intensive synthesis and bioassays. It has been stated that the "Early use of computational approaches followed by high throughput experimental determina­tions may become the norm as best practices are adopted in order to improve the chances of successful preclinical and clinical development" 7.

Developing structure-activity relationships for drug compounds using computational or theoretical meth­ods, relies on appropriate representations of molecular structure. This review will focus on the role of topo­logical indices (TIs) from a drug design and discovery perspective. The traditional application of TIs in QSAR will not be covered in this review. Readers are encouraged to consult previous compilations, which can provide an introduction to these types of descrip­tors and examples of their applications. Here we will firstly introduce a number of basic concepts and defi­nitions from a variety of areas, including drug design and discovery, graph-theory and topological indices. These definitions are presented alphabetically. Sec-

1316 INDIAN J CHEM, SEC A, JUNE 2003

ondly we will discuss the role of TIs in drug di scov­ery, highlighting some of the most recent advances in lead discovery strategies, such as drug design, virtual screening, combinatorial library design, similarity­di ssimilarity database searching and lead optimiza­tion.

Definitions of drug design terms Artificial neural networks (ANNs): Algorithms

simulating the functioning of human neurons which may be used for pattern recognition problems e.g. to establish quantitative structure-acti vi ty relationships 8.

Bioisostere: A compound resulting from the ex­change of an atom or group of atoms with another, broadly similar, atom or group of atoms . The bioi so­steric replacement may be physicochemical or topo­logically based 9.

Combinatorial chemistry: The systematic and re­petitive, covalent connection of a set of different "building blocks" of varying structures to each other to yield a large array of diverse molecular entities 10.

Combinatorial synthesis: The process to prepare large sets of organic compounds by combining sets of building blocks9

.

Combinatorial library: The set of compounds pre­pared by combinatorial synthesis9

.

Comparative Molecular Field Analysis (CoMFA): The 3D-QSAR method that uses statistical correla­tion techniques for the analysis of the quantitative relationship between the biological activi ty of a set of 'compounds with a specified alignment, and their three-dimensional electronic and steric properties8

.

Computer-aided drug discovery (CADD): This in­volves all computer-assisted techniques used to dis­cover, design and optimize compounds with desired

d . 8 II structure an propertIes ' .

Discriminant analysis: This is a statistical tech­nique to find a set of descriptors which can be used to detect and rationalize separation between activity c1asses8

.

Genetic algorithm (GA): This is an optimization algorithm based on the mechanisms of Darwinian evolution which uses random mutation, crossover and selection procedures to breed better models or solu­tions from an originally random starting population or sample8

.12

.

High-throughput screening (HTS): The use of ro­botics to screen thousands to millions of compounds in an automated form l3

.15

Lead compound: Any chemical compound that dis­plays the biological activity of interest. It is not a drug but the first approach to search it.

Lead discovery: The process of identifying active new chemical entities, which by subsequent modifi­cation may be transformed into a clinically useful drug9

. It includes the search of compounds in nature (plants, animals , soils, etc.), the investigation of side effects of existing, and so forth I6

,17 .

Lead optimization: The synthetic modification of a biologically active compound, to fulfill all stereoelec­tronic, physicochemical, pharmacokinetic and toxi­cologic requirements for clinical efficacl.

Mass screening: The most tradi tional method used in drug discovery comprising large scale screening of chemicals in a battery of biological assays .

Molecular descriptors: Terms that characterize a specific aspect of a molecule8

. These are numbers containing structural information derived from struc­tural representations of molecules under study.

Molecular similarity/dissimilarity: This is a num­ber to express structural relatedness between pairs of molecules8

.

Multivariate linear regression (MLR): The use of statistical methods for modeling a set of dependen t variables in terms of combinations of predictors8

.

Orthogonalization of molecular descriptors: This is an approach in which molecular descriptors are trans­formed so they no longer mutually correlate. Both the non-orthogonal descriptors and the derived orthogonal descriptors contain the same information. This results in the same stati stical parameters of the QSAR mod­els 18-2 1. However, the coefficient of the QSAR model based on orthogonal descriptors is stable to the inclu­sion of novel descriptors , thus permitting interpret­abil ity of the regression terms and evaluation of the role of individual descriptors to the QSAR model.

Partial least squares (PLS): A robust multivariate generalized regression method using projections to summarize multitudes of potentially collinear vari­ables8

.22

.

Pharmacophore: The ensemble of steric and elec­tronic features that is necessary to ensure the optimal supramolecular interactions with a specific biological target structure and to trigger (or to block) its biologi­cal response9

.

Quantitative structure-actlVlty relationships (QSAR): These are mathematical relationships linking chemical structure and pharmacological activity in a quantitative manner for series of compoundsB

•

ESTRADA et al. : FROM MOLECULAR GRAPHS TO DRUGS 1317

Structural database: A large collection of molecu­lar structures in an electronic format.

Topological indices (TIs): Numerical values asso­ciated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activitl. They are derived from a topological representation of mole­cules and can be considered as structure-explicit de­scriptors in contrast with those structure cryptic de­scriptors, such as quantum chemical ones or structure­implicit descriptors such as hydrophobicity and elec­tronic constants23

.

TOpological Sub-structural MOlecular DEscrip­tors (TOSS-MODE): These descriptors are the spec­tral moments of the bond matrix of a molecular graph weighted to account for hydrophobic, steric and elec­tronic molecular effects and can be expressed as lin­ear combinations of molecular fragments. They also permit the computation of contributions of any part of a molecule to the pharmacological property studied.

Virtual screening: The use of computational tech­niques to select a reduced number of potentially ac­tive compounds from structural databases. The main objective of this approach is to discriminate potent candidate molecules from inactive or less potent molecules 7. 24.25.

Definitions of molecular topological terms Ad}acency matrix A: This is a square symmetric ma­

trix of order n whose elements aij are ones or zeros if the corresponding vertices i and} are adjacent or not.

Bond matrix E: This is a square symmetric matrix of order m whose elements eij are ones or zeros if the corresponding bonds i and} are adjacent or not26

.

Complete graph, K,,: A graph with any two vertices adjacent. The number of edges in KII is n(n-l )12.

Distance matrix D: This is a square and symmetric matrix of order n whose elements dij correspond to the topological distances between atoms i and}.

Edge or bond degree, 8(ek): This is the number of bonds adjacent to k . Two bonds are adjacent if they are incident to the same atom. The following relation is maintained, 8(ek) = 8(Vi) + 8(v) - 2, if Vi and Vj are incident to ek (ref.26).

Graph theoretical invariant: Any mathematical expression based on the elements of a graph that does not depend on numbering of graph elements is an in­variant. A topological index (TI)27 is the numerical result of any graph invariant. For instance, adjacency or distance matrices are not graph invariants as they

depend on the numbering of graph vertices. However, a simple invariant can be the order of the adjacency matrix, which is the number of vertices in the graph but which does not depend on the graph elements numberingt.

Graph: A mathematical object represented as G = (V, E), where V is the set of vertices and E is the set of edges. The number of vertices in the graph is desig­nated by n and the number of edges by m.

Hydrogen-depleted graph: A molecular graph in which hydrogen atoms are not considered.

Molecular graph: A topological representation of a molecule consisting of a collection of points repre­senting the atoms in the molecule and a set of lines representing the covalent bonds28

-3o

. These points are named vertices and the lines are named edges in the graph theory language.

Path: A sequence of vertices ViO, Vi i, Vi2, ... ,Vil of a graph, such that Vij_1 and vij are adjacent} = 1, 2, .. . ,1. The length of this path is equal to I.

Spectral moments of a matrix: These are the traces, i.e. the sum of the main diagonal, of the different powers of the corresponding matrix.

Subgraph of a graph G: A graph G*= (v" , E*) where v" is a subset of Vand E* is a subset of E.

Topological distance between atoms i and}: This is the shortest path between both atoms.

Topological distance: dij is the length of the short­est path between vertices i and j.

Topological representation: A representation of an object providing information only about the number of elements composing it and their connectivities.

Vertex degree or valency of atom i, 8(Vi): The number of bonds incident in i.

Weighted graph: A graph of the form G = (V, E, <p, V), where V is a set containing all the chemical sym­bols of the elements and <p is a subjective mapping of the elements of V onto the set V. It may be considered as a more realistic representation of the topology of a molecule because it identifies the different atoms in the molecule by labeling them with their chemical symbols.

Definitions of topological indices First generation TIs: These are integers based on

integer graph properties, such as topological dis­tances31

• The most representative indices of this class

t In a more exact mathematical way the graph invariant can be taken here as the trace of the Oth power of the adjacency matrix . The trace being the sum of the diagonal elements of such matrix

1318 INDIAN 1 CHEM, SEC A. JUNE 2003

are Wiener index32 W, Hosoya index33 Z and the cen­tric indices of Balaban34 Band C. The only one that has been used in drug discovery research is the Wie­ner index.

Wiener index: This is the half-sum of all entries in the distance matrix32

,

. . . (1)

Zagreb indices: These were developed around 1975 on the basis of atom valencies35

:

II

MI = ~)8(VJ]2 (2) i=1

(3)

Second generation TIs : These are real numbers based o n integer graph properties . Most of the TIs used in drug di scovery today are of this class. Some examples are defined below:

Molecular connectivity indices:These indices are based on a graph-theoretical invariant introduced by Randi6 25 years ago in order to compute a "branch­ing" index for alkanes36

.39

:

. . . (4)

where the sum is carried out over all adjacent atoms in the molecule, i.e. over all bonds.

Valence molecular connectivity: This is an exten­sion of the molecular connectivity indices of Randi6 carried out by Kier and Hall to account for heteroatom differentiation as well as for different subgraphs in the molecule and is expressed36

.38 as :

.. . (5)

where 8 v (Vi) is the valence degree of the atom 1,

which is equal to the count of valence electrons in the atoms minus the number of hydrogen bonded to it.

The remaining molecular connectivity indices are defined as follows 36

.37

:

. . . (6)

where ty means the type of the fragment for which the connecti vity index is defined . For instance, connec-

t1 vlty indices of path, clusters, path-clusters and chains have been defined.

Balaban J index: A modification of the molecular connectivity index introduced in 1982 by Balaban as an average distance-based connectivity index4o:

(7)

where D(v;) is the sum of all the topological distances related to atom i. That is, the sum of all the entries of the row or column of the matrix D corresponding to atom i.

Charge indices: This was introduced by Galvez et al in order to complement the structural information contained in molecular connectivity indices4 1

:

Gk = f fICT,j ~(k , dij) and i t = Gkl(n - I ) ;=1 j

.. . (8)

where CT;j = l71ij - 117 ji' where III are the elements of

the M matrix defined as: M = AxD*. The matrix D* is the matrix of the inverse square distances, in which their diagonal entries are assigned as 0 and 8 is Kronecker' s del tao

Bond connectivity indices: This is introduced by Estrada and calculated by using the same graph­theoretical invariant of Randi6 carrying out the sum by bond degrees instead of atom valencies4

2-46 :

- 1/ 2 p o

E = I [ok }5(e J ... (9)

s= l s

where the sum is can'ied out over all pairs of adjacent bonds, P2.

Kappa indices of molecular shape and flexibility: This was introduced by Kier4

7-49 in the 1980' s as the basis for devi sing a relative index of shape given by the re lationship of the number of path of length l in the molecule i, 'Pi, to some reference values based on molecules with a given number of atoms, 12, in which the values of 'P are maximum and minimum, 'PlIIlIX and 'Plllill • The first order shape attribute, IK, is given by the follow ing expression49

:

. .. (10)

The second and third order kappa indices are de­fined as follows49

:

2K=(1l-1Xn-2? /(2 ~'t .. . (11)

ESTRADA el al.: FROM MOLECULAR GRAPHS TO DRUGS 1319

3/(i = (n - 1Xn- 3?/(3 PJ whennisodd (12)

3/(i = (n - 3Xn -2f 1(3 ~ ') when fl is even (13)

In order to account for the variation in size contri­bution to shape from different atoms the radius of atom X relati ve to the covalent radius of a carbon Sp 3

hybrid atom is considered. The specific correction in computing IK is made by modifying the count of at­oms, fl, with a modifier, a, calculated as:

. .. (14)

where a represents a decrement or increment of n for a non-carbon Sp3 element X. The modified kappa shape indices are given by49:

l/(a=(n+aXn+a -l? /(I~+a) (15)

2/(a =(n+a-lXn+a-2f / (2 Pi +a') (16)

3Ka = (/1 +a -lX/1 +a -3? /(3 p; +a) fl is odd (17)

3/(a = (1l+a- 3Xn+a -2?/(3 ~+ a) niseven (18)

Electrotopological state (E-state) indices: Intro­duced by Kier and Hall in the 1990' s, these are based on graph invariants for each atom in the moleculeso-s4. These indices can be considered as local molecular descriptors. The E-state index for an atom in a mole­cule represents the electron accessibility of that atom. It is a combination of electron richness or deficiency together with topological accessibility. The E-state index for atom I in a molecule is defined as:

Si = Ii + "EMij j

.. . (19)

The sum is over all other atoms j within the molecular graph. The term for the perturbation of atom i by atom j is defined as:

. .. (20)

The intrinsic values are defined as :

... (21)

where N is the principal quantum number for the va­lence electrons.

Third generation TIs: These are real numbers based on real-number local properties of the molecu­lar graph. These indices are of recent introductionss-57,

have very low degeneracy and offer the possibility of a wide selection. Other third generation TIs are based on information theory applied to terms of distance sums or on newly introduced non-symmetrical matri­cesS8-60

• This paper will not discuss these kinds of TIs as there have been few applications of such descrip­tors in drug discovery research.

Topological Indices in Drug Design/Discovery

One of the first examples of successful design of novel compounds using TIs was described by the Upjohn and Pharmacy team in 1993. They were able to design a new class of heteroarylpiperazine com­pounds with activity versus HIV retrotransferase61

using Basak's method which was based on the appli­cation of a series of principal components derived from a pool of 90 TIs62-66. Another successful design, which included the discovery of new active com­pounds, was reported by Grassy et al. These workers designed immunosuppressive peptides by using Tls67 ,

h k h . d' 46-49 I I suc as appa s ape In Ices , mo ecu ar connec-tivity36,37, Balaban index39, Wiener index32

, E-state sum50-54, as well as other descriptors for the number of atom types including ellipsoid volume, molecular volume, molar refractivity , lipophilicity and molecu­lar mass . The peptides under analysis were derived from the heavy chain of HLA class I, which modulate immune responses in vitro and in vivo. One of them, derived from HLA-B2702, prolonged skin and heart allograft survival in mice. Using this strategy, these workers were able to design and synthesize a peptide that displayed an immunosuppressive activity ap­proximately 100 times higher than the lead compound tested in an heterotropic mouse heart allograft model.

A systematic approach for drug design and discov­ery has been applied from the mid 1990' s by the Gal­vez group at the University of Valencia in Spain. They developed a strategy for drug design based on the combined use of molecular connectivit/6,37 and charge indices40 together with other in-house mo­lecular descriptors accounting for molecular shape68. The strategy involved selecting a large data set of drugs containing active as well as inactive com­pounds, e.g. active meaning analgesic drugs and inac­tive non-analgesic compounds. A vector containing connectivity, charge and shape indices represented each compound. A classification model was found using linear discriminant analysis (LDA) for the com­pounds in a training set. This was then tested for pre­dictability by using an external prediction set. The

1320 INDIAN] CHEM, SEC A, JUNE 2003

classification model was used to search for 'active' compounds in a large database of structures which were then tested for activity. In this way the model could evaluate compounds not already synthesized.

In applying this approach, Galvez et at. has been able to discover novel activities for known com­pounds or design and synthesize new ones69.85. The most significant results obtained by this group include the discovery of compounds with analgesic, hypogly­cemjc, cytostatic, antibacterial , ~-blockers, broncho­dilator, antifungal, antiparasitic and anti histaminic activities among others.

The most recent applications of this approach have been reported for the prediction of antifungal activity of a diverse set of compounds86 in addition to dis­criminating "drug-like" from "non-drug-like" com­pounds87. Indeed the last study concluded that it was possible to achieve a pattern of general pharmacologi­cal activity based on molecular topological indices. The prediction of antihistaminic activity using multi­linear regression and linear discriminant analysis were also reported by this group88. The topological pattern of activity obtained allowed a reliable prediction of antihistaminic activity in drugs frequently used for other therapeutic purposes. Based on the results, the proposed pattern was apparently only valid for drugs which interacted with histamine through competitive inhibition with HI receptors . The prediction of some pharmacological properties of hypoglycemic drugs89

and antibacterial discriminant functions90 has also been reported using the same approach. In the last report physicochemical parameters, such as heat of formation, HOMO, LUMO, dipole moment, polari­zability, hyperpolarizability, PM3 generated IR vibra­tional frequencies, etc., were calculated together with TIs, such as connectivity and topological charge indi­ces.

Another recent strategy using topological descrip­tors for drug design and QSAR is the so-called TOPS­MODE approach. The TOPS-MODE approach has been successful in the identification of drug candi­dates in structural databases. Several compounds were identified as sedativelhypnotic by searching the Merck index database using a TOPS-MODE dis­criminant model91 . This model was also able to iden­tify more than 80% of Ras farnesyl transferase (FTase) inhibitors in a simulated virtual screening experiment92 for anti-cancer compounds. In another experiment TOPS-MODE was able to identify more than 88% of anti-convulsant lead compounds in a data

set using the discriminant model previously ob­tained93

. More recently, this approach has been used for discriminating nucleosides with anti-HlY activity from those being inactive or having low activitl4

.

The main advance of this approach is that it enables a calculation of structural contributions to the pharma­cological activity studied. Through calculating these fragment contributions, TOPS-MODE has been suc­cessfully applied to the identification of possible pharmacophores and bioisosteres. For instance, the role of the piperazinyl fragment in sedativelhypnotic compounds was analyzed with this approach. The role of di-substitution of nitrogen atoms and the possibili­ties of making isosteric substitutions of this ring with acyclic fragments was determined91. A series of active and inactive fragments were also identified in the case of anticancer compounds showing good agreement with active ones previously reported through using a completely different approach92. A 20 pharmaco­phore and a possible isosteric substi tution were identi­fied by studying the anticonvulsant activity of a large data set of drugs93 . The superposition of the 3D structures of such a fragment in a series of structurally different molecules revealed a good degree of map­ping between this 20 fragment and a possible 3D pharmacophore93. More recently, the TOPS-MODE approach has allowed the elucidation of a series of structural rules for the development of anti-HIY nu­cleoside compounds. These rules include, the role of different substituents at different positions of the py­rimidyl ring of nucleosides as well as the role of dif­ferent types of rings in substituting the frequently used furanosyl ring.

Another area of application for topological indices is in similarity/dissimilarity studies. There is recent interest in this area on account of advances in the synthesis and screening of combinatorial libraries as well as lead identification searching from large struc­tural databases95.96. Consequently, topological simi­larity has been used both in mining structural data­bases for new analogues based on a lead structure and for improving the efficacy of lead identification screening programs by using HTS. These types of similarity/dissimilarity studies have also been impor­tant in combinatorial synthesis programs for design­ing diverse or biased libraries for synthesis. In fact, molecular diversity is intimately linked to topological similarity/dissimilarity. The principle of similar­ity/dissimilarity studies is that, given a compound with a desired biological activity, compounds that are

ESTRADA el al.: FROM MOLECULAR GRAPHS TO DRUGS 132 1

"topologically similar" are likely to have sImI­lar/diss imilar activity. Consequently, the main issue is in identifying these lead compounds from a large pool of chemical structures contained in a database. The application of si milarity/di ss imil arity searching pro­cedures provides a solution95.IOO.

There are two approaches to similarity studies of chemical structures. According to the direct­comparison method the similarity between two mole­cules is calculated after superposi tion of the structures of both molecules 101. These methods are computer­intensive and with some exceptions, are not typically applied in database searching. The other approach is based on the use of molecular descriptors that are used to represent the structures 102. The simi larity of a probe molecu le and each database entry is calculated by comparing, (using an appropriate measurement), the li st of descri ptors in the probe with the list for the entry. There are many ways of encoding the structural information in the form of molecular descriptors. However, the aim of descriptor selection is to reduce the infinite number of potential descriptors to those most appropriate for a given application . Therefore, the use of TI s for similarity/di ss imilarity studies have been viewed as the ultimate form of data reduction, since each topological index represents a single num­ber which characterizes the structure as a whole, with the expectation that similar molecular structures will have similar values of such an index .

One of the TIs similarity/dissimilarity approaches was introduced by Basak et al. 62.66. These workers used a subset of 3,692 structures from the 25,000 contained in the U.S. Environmental Protection Agency 's database TSCA and computed more than 90 TIs. Ten principal components were selected using principal component analysis. These principal com­ponents explained more than 92% of the variance and were used as descriptors in a si milarity search of the whole database. The results obtained by these authors for the top-fi ve hits for a selection of randomly cho­sen queries were as good as those expected from the traditional structural-fragment similarity search . The distinction of functional groups, however, was poorer than expected from the second more traditional method .

Another approach to similarity searching in struc­tural databases is based on the use of atom-type elec­trotopological state indices53.54.103. An atom-type E­state value is defined for each type of atom in a mole­cule lD4

, e.g. , -Cl, -OH, -CHz-, etc. Each atom is classi-

fied according to its valence state, number of bonded hydrogens and aromaticity . For an atom type E-state index , the E-state values are summed or averaged for all atoms of the same type in the molecule. These E­state values, summed or averaged for an atom type, may be thought of as numerical components of a hy­perspace. Thus a 26 atom-type vector was built for each of the 23,310 structures of a database created by Hall and Kier54 (21,842 structures from the Pomona MedChem data set + 663 hydrocarbon structures + 515 polyaromatic hydrocarbons + 290 highly branched alkanes). The following general observa­tions were obtained by applying this approach to the 23,310 structures in the database: (i) nearest neigh­bors tend to have the same or very similar skeletal structure; (ii) nearest neighbours tend to have the same set of heteroatoms; (iii) nearest neighbours' tend to have the same number of atoms; and (iv) there is a pattern in the computed similarity distances (very similar molecules up to 0.10 units; similar molecules having one more (less) carbon atom 0.25-0.30 units; isoskeletal molecules with different heteroatoms lie at greater distances.

The application of thi s approach for searching si milar compounds to drug molecules shows great promise as evidenced by analogues of mefloquin, me­peridine, niridazole, and minoxidil.

Another very active area of research in applying TIs to drug discovery is related to the area of combi­natorial chemistry. The main objective of creating combinatorial libraries is to provide collections of compounds suitable for drug discovery and optimiza­tion lO

.I05. An important question that should be under­

stood in studying combinatorial synthesis is the im­portance of designing libraries with great molecular diversity. The main objective of selecting a diverse subset of compounds from a much larger population is to identify a subset that best represents the full range of chemical diversity of the larger popu­lation and in doing so saving time and money spent in either synthesising or screening "redundant" compoundsI06-108. Another important aspect of combi­natorial library design is that such a general method should work with virtually any potential chemical building block ; thus easily calculated molecular de­scri ptors are preferred I 08.

Tis have been excellent candidates for analyzing molecular diversity for designing combinatorial li­braries. Martin et al. showed that the discovery effi­ciency and productivity in combinatorial synthesis

1322 INDIAN J Cl-lEM, SEC A. JUNE 2003

can be further improved by using an experimental design to maximize molecular diversity for a g iven Iibrary ltJ9. They described new methods to quantify molecular diversity using TIs in combination with other descriptors characterizing lipophilicity, chemi­cal functionality, and specific binding features l09.

Another interesting approach to rational design of targeted combinatorial library design with Tis was in­troduced by Tropsha et al. In 1998 and is calledll0-112

Focus-2D. Thi s approach is based on randomly as­sembling the building blocks that are used in a com­binatorial synthesis to produce a vi:tual library. The individual compounds in such a library are then repre­sented by means of TI~ and the molecular similarities between them are evaluated by modified pairwise Euclidean distances in c'. multidimensional TIs space. Stochastic algorithms, such as, simulat.::d annealing, genetic algorithms and neural network methods, are used to search the potentially large structural space of virtual libraries in order to identify compounds similar to lead molecules. The building blocks that can be used in combinarorial sy nthesis of chemical libraries are identified by frequency analysis of building blocks of selected virtual compounds . The compounds ob­tained by combinatorial sy nthesis will thu s have great similarities to the lead molecules. This method was able to identify five building blocks wi th frequencies higher than random using morphine as a probe. The results show that Focus 2D provided results better than random with 87% confidencello-11 2. Ten building blocks were selected by this method when rnet­enkephalin was used as a probe with better than nm­dom prediction. at 95% confidence. Twelve building blocks were selected when a composite probe of met­enkephalin and morphine was used. The results indi­cated that Focus 2D performed better than random with >99% confidence.

In contrast, Linusson et al. have used TIs in a st rat­egy for the statistical molecul ar design of building blocks for combinatorial chemistry I 13. These authors found that it was of critical importance to investigate the building blocks space carefully and to select an appropriate number of blocks to result in an adequate diversity.

In spite of recent progress in the application of TIs to different areas of drug discovery such as simi lar­ity/dissimilarity studies or combinatorial library de­sign, their application to lead optimization continues to be the most active area of research. Consequently, many of the applications of TIs in biological research

have been dedi ca ted to QSAR studies. This trend is influenced by hi storical facts. In fact, the number of QSAR studies using TIs showed a dramatic increase after the introduction of the molecular connectivity indices in the 1970's. The explosion of QSAR using TIs is probably due to the fact that novel descliptors do account for a great deal of structural information and because the computation of such descriptors is now readily accessib le using a number of software products. Katritzky et al. 11 4 conducted a study which analyzed five sets of molecular descriptors for the development of quantitative structure-property rela­tionships (QSPRs) and QSARs. TIs were found to account for a significant proportion ()f the variance ciescribing both physicochemical and biological prop­erties. For the development of QSARs. TI s were best combined with additional parameters such as hydro­phobicity or quantum chemical ones, possibly due to !h~ influence of transport acros biological mem­branes 11 4. Several good reviews36,37.lls-11 7 have sum-

marized many of the applications of TIs in QSAR in the 1970' s and 1980's. Several QSAR applications of the E-state index have been compiled in an excellent book by Kier and Ha1l 54 . In light of th is we will only highlight the most significant advances of TIs in QSAR thar have been reported in the recent literature.

There have been two main sources of criticism re­garding the application of TIs in QSAR. The first is the non-inclusion of three-dimensional features of molecules into their definition , which avoids the de­scription of some important structural factors that could be directly related to the biological activity. The second is related to the " physical interpretability" of such mol ecular descriptors. The firs t of these difficul ­ties will be discussed below, the second one will be addressed later in the manuscript.

A new era of QSAR research was born following the introduction of Comparative Molecular Field Analysis (CoMFA)II B. Reasonably simpli stic, high degree of autonomy, good visual ization and clear physicochemical interpretation of the descriptors used (steric and electrostatic) made CoMFA one of the most popular methods for QSAR studi es l1 9. The num­ber of applications of this approach to QSAR studies registered an explosion in the 1990' s with more than 383 papers published from 1993 to 1996120. In spite of the great number of successful applications of CoMFA in QSAR, several problems have persis ted with thi s method. In two different papers by Tropsha et al. 121

,122 it was shown that the results of a conven-

ESTRADA et al.: FROM MOLECULAR GRAPHS TO DRUGS . 1323

tional CoMFA could often be non-reproducible due to the sometime strong dependence of the CoMFA cross-validated correlation coefficient, l , with the orientation of rigidly aligned molecules on a user's terminal. These alignment problems are typical for 3D QSAR methods and despite the proposal of possible solutions, the unambiguous 3D alignment of structur­ally diverse molecules still remains a difficult task.

A novel approach to 2D QSAR was proposed by Tropsha ef al. by combining the simplicity of TIs with the powerful statistical tools employed in 3D QSAR, namely PLS 123. These authors also incorporated GA to optimize the resulting PLS regress ion equations by removing irrelevant Tis ill a fully automated ap­proach. The use of this elegant 2D QSAR approach showed that the combinat ion of a GA-PLS method with TIs gave comparable or better results than CoMFA. This novel approach had the advantage of lacking 3D-alignment problems characteristic of 3D QSAR. Due to its simplicity, efficacy, and higher de­gree of automation this method provides a powerful alternative to 3D QSAR. More recently , this approach was used to model dopamine DI antagonists and com­pare outcomes obtained by a CoMFA method l24

. The work showed that this method was highly competitive for QSAR stud ies, lacked the problems of alignment and eliminated the possibility of inappropriate con­formation selection.

Despite this real advance in QSAR, namely, the use of TIs in combination with powerful statistical tech­niques , non-consideration of 3D structural features remains a problem. An important step forward in the development of graph-based molecular descriptors has resulted in the definition of topographic descrip­tors. This field of research will not be discussed in detail here. However, some of the key features of th~se descri ptors will be highlighted here. These kinds of molecular descriptors are based on molecular graphs with appropriate weights (weighted graphs) to account for 3D molecular features. Pioneering work in this area was performed by Milan Randic1 25. 126 at the end of the ! 980' s. Randic proposed the use of topographic di stance matrices, firstly based on a graph embedded into a hexagonal lattice and then into a 3D diamond lattice. Further work by Bogdanov ef a/. m proposed a 3D distance matrix to compute a 3D Wiener index. The different direction in the search of topographic indices was followed by Estrada who proposed the use of quar-tum chemical parameters as vertex and edge weights for building topographic ad-

. . 179-111 ] 1993 h' h d Jacency matrIces - ' . ntIs aut or propose the definition of topographic connectivity indices based on bond order weighted molecular graphs 129. Furthermore, this type of weighted graphs was used to defi ne a topographic edge adjacency matrix (3D bond matrix) as well as a 3D bond connectivity index 130. ]n a parallel approach Estrada introduced molecular graphs with vertices weighted by electron charge den­sity on atoms to define a 3D adjacency matrix and a

f h· .. d . 131 I set 0 topograp 1C connectivIty escnptors . n more recent years different authors have defined other types f· h' d . 112-135 o topograp 1C escnptors ' .

There have not been many applications of these de­scriptors to drug discovery and design . Recently topographic (3D) molecular connectivity indices based on molecular graphs weighted with quantum chemical parameters were used in QSPR and QSAR studies. These descriptors were compared to 2D con­nectivity indices (vertex and edge ones) and to quan­tum chemical descriptors in modelling partition coef­ficient (log P) and antibacterial activity of 2-furylethylene derivatives. ]n describing Jog P, the 3D connectivity indices produced a significant improve­ment (more than 29%) in the predictive capacity of the model compared to those derived with topological and quantum chemical descriptors. The best linear di scriminant model for classifying antibacterial activ­ity of these compounds was also obtained with the use of 3D connectivity indices. The global percentage of good classification obtained with 3D and 2D connec­tivity as well as quantum chemical descriptors was 94.1, 91.2 and 88.2, respectively . In general, all these models predicted the antibacterial activity of a set of 9 new 2-furylethylene derivatives correctly. The best resu lt was obtained with 3D connectivity indices which correctly classified 100% of these compounds versus 88.9% obtained with 2D connectivity or quantum chemical descriptors l36

.

A specific class of these descriptors, namely, the chirality topological indices (CTIs) are briefly consid­ered here. The main purpose of developing these de­scriptors is to be able to account for chiral molecules, which are well known to have an important role in medicinal chemistry. Until now there have been few of these descriptors reported in the literature but a serious effort in this direction has been recognized by researchers. Among the CTIs published in the litera­ture we can mention those derived by Pyka for descrihing enantiomers in thin layer chromato­graphy m-139. Some of these indices have been ration-

1324 INDIAN 1 CHEM, SEC A, JUNE 2003

alized from the mathematical viewpoint of Gutman and Pyka l40, in particular the optical topological index which distinguishes between enantiomers and the stereoisomeric topological index which distinguishes between stereoisomers. The relationships between these indices and the Wiener index have been estab­lished. Schultz et al.141 have also modified a series of Tis in order to introduce information on chirality of stereocentres in molecules.

The first attempt of usi ng CTls in describing bio­logical activity was carried out by De Juli an-OrtIs et al. 142 in developing descriptors that could differentiate between the pharmacological activ ity of pairs of en­antiomers. The ICso values of inhibitors of dopamine O2 receptor and the (J receptor of a group of 3-hydroxy phenyl piperidines were studied with this ap­proach.

The most recent approach to CTls was that intro­duced by Golbraik, Bonchev and Tropsha (G BT) 143. In the GBT approach several series of CTIs were introduced on the basis of known Tis such as connectivity indices, Zagreb group indices, charge indices, and others. They make use of a term called the chirality correction, which is added to the vertex degrees of asymmetric atoms in a molecular graph. A QSAR study was performed for a set of ecdysteroids with a high proportion of chiral and enantiomeric compounds. The results obtained using the GBT ap­proach compared favorably with those derived from CoMFA. In all these examples it is sti ll important to carry out further studies in order to demonstrate the advantages and disadvantages of using these molecu­lar descriptors in drug di scovery; however, the pre­liminary results are encouraging.

One of the difficulties arising from applying Tis in drug desi gn and discovery is due to the fact that many TIs are interrelated to some extent. Consequently, their inclusion in a QSAR equation does not meet the necessary requirement of variable independence. To solve this problem Randi6 proposed a procedure of orthogonalization of molecular descriptors that has been applied with much success to QSPR and QSAR studies 18.21.

This approach has been used to predict the ability of 19 flavone derivatives to inhibit cAMP phosphodi­esterase l44. The best QSAR model with orthogonal­ized descriptors showed the following statistical pa­rameters: R = 0.9365, s = 6.99, and F = 35.70. How­ever, a significant improvement was observed with three ordered orthogonalized descriptors from which

the following statistical parameters resulted 144: R = 0.9841, s = 3.54, and F = 153.38. Similar im­provements have been observed in other QSAR studiesI 4S.149. These results indicate that the use of this approach not only overcame the main drawback of using Tis in QSAR, the great intercorrelation between Tis, but it also is able to improve the quality (both statistical and interpretation) of the QSAR model s. Advances in orthogonalization strategies of TIs have been reported in recent years by several groupsI49.ISO.

In order to improve the quality of TIs used in drug discovery programs, several strategies have been pro­posed and applied. One of them is the construction of linear combinations of connectivity indices (LCC!), composite connectivity variables, which are found to be very usefu l in the development of QSAR models of various molecular properties. This procedure of linear combination and semi-empirical TIs introduced by Pogliani is based on a minimal and expanded set of connectivity indicesI SO-15S. This strategy has been fo­cused on the study of physicochemical properties of biochemicals, such as amino acids and DNA bases, as well as the study of such properties of other organic compounds. According to Pogliani some of the mai n features of LCCI are1SO-15S : (i) The combination with the maximal number of molecular connectivity indi­ces does not always show the best quality; (i i) succes­sive inclusion of the next good index to the best com­bination of indices does not always produce the next best combination of indices; and (iii) even highly in­tercorrelated indices can contribute to the best combi­nation of indices.

Another approach to improve the quality of TIs has been introduced by Randi6 et al.1 56- 164 and this has been named the "variable molecular descriptors". This approach has been proposed as an alternative route for the characterization of heteroatoms in molecules. It consists of replacing the zeros along the main diago­nal of the adjacency matrix by variables x, y, Z, ... for each type of (hetero )atom and then uses the same i n­variants for calculating the TIs. In using this approach the authors have been able to modify known TIs, such

. . 156- 161 · I d h as connectivity ones to mc u e eteroatoms as well as improve the quality of simple TIs in QSPR studiesI62.164. This approach opens up new possibili­ties in the development of TIs applied in the drug dis­covery process, as the number and quality of the indi­ces derived may be increased comparatively to the traditional ways of obtaining such descriptors. A gen­eralization of the "variable molecular descriptors"

ESTRADA el al.: FROM MOLECULAR GRAPHS TO DRUGS 1325

approach which enables the calculation of several known TIs using the same graph invariant has been recently introduced by Estradal6s . According to this generalized algorithm, Wiener index, Zagreb group indices, Balaban J index, Randie X index and some others are particular cases of an infinite set of TIs.

Another direction for improving the quality of the models in which TIs are used is based on the applica­tion of more powerful statistical techniques as in other areas of drug design and discovery. One of these powerful techniques now widely used in QSAR re­search is the artificial neural network. The theory and general practice of ANN applications is already well documented as well as its application to the solution of different chemical problems I66-168. The applications of ANN to TIs QSAR have shown important ad­vances in recent years. In a recent paper Huuskonen et af proposed a method for predicting the aqueous solu­bility of drug compounds by using TIs and ANN I69. In this work a total of 101 connectivity, shape, and atom­type E-state indices were calculated together with three indicator variables for aromaticity, number of hydrogen bond donors and number of hydrogen bond acceptors, respectively . The topological indices used

included' 2X 3X 6 X" 7XI' 9" 10 " 3 • . , c ' ell ' ell ' XcII' Xcii' K a as

well as the E-state for 22 atom types. The data set was composed of 211 drugs and related compounds for which the solubility was expressed as log units vary­ing from 0.55 to -5.60. The statistical data for the training set of 160 compounds showed a reasonable degree of accuracy: R2= 0.90 and s = 0.46. The pre­dictive power of such model was also acceptable

having R~,.ed = 0.86 and s = 0.53 for the 51 com­

pounds in the prediction set l69 .

The atom-type electrotopological state indices were also applied to predict the partition coefficient of a data set of 345 drug compounds or related complex chemical structures by using multilinear regression analysis (MRA) and ANN. Both MRA and ANN pro­vided reliable log P estimations, but the application of the ANN provided better prediction ability for train­ing and test series using the same series of parame­ters 170

Back-propagation ANNs were also used to model the structure-activity relationship of a large data set of 103 capsaicin analogues using TIs including molecu­lar connectivity and charge indices, as well as con­nection table theory, simple atomic decomposition of molecules into atom type and novel physicochemical

descriptors l71. The ANN QSAR model showed a high level of correlation between experimental and pre­dicted data. After optimization the ANN predicted the ECso of lO I capsaicin analogues, classifying correctly 97% of the 60 active compounds and 83% of the 41 inactive ones. Other applications of ANNs for the classification of drugs as active/inactive by using TIs have also been reported. The antimicrobial activity of a heterogeneous group of compounds was discrimi­nated from a series of other inactive drugs using con­nectivity indices and linear discriminant analysis (LOA) as well as ANNs l72. Although both methods were shown to be adequate in differentiating between active and inactive compounds, the ANN was more appropriate since it had a success of 98% in the pre­diction set versus 92% obtained by LOA. More re­cently, Tomas-Vert et af. used ANNs to discriminate antibacterial activity in a data set of 249 active mole­cules, (belonging to different groups of antibiotic and chemotherapeutic drugs) from a set of 415 inactive drugs (belonging to different thera­peutic groups but which did not have antibacterial activity) 173. A set of 62 topological descriptors were used including a series of indicator variables ac­counting for the number of different atom types in the molecule and a series of sums of distances from a given heteroatom. However, the last set of descriptors (from 51 to 62 in the original paper) that are designed as "summatory of distance x" really correspond to the extended Wiener indices as defined and applied to QSPR studies by Estrada et al. 174

. The results ob­tained by these authors show a good classification of 93.6% of actives and 95.9% of inactives in an external

f I I 169 II . test group 0 mo ecu es . An exce ent review on the use of molecular graphs descriptors in neural net­works models has been reported by Ivanciuc 175

•

Another recent example of the application of pow­erful statistical tools to TIs in QSAR is the use of PLS

to generate quantitative models . In a QSAR study based on the use of E-state indices for describing the activity of flavone HIV -\ integrase inhibitors Buo­lamwini et af. fOllnd highly predictive models 176. The correlation coefficients of R2 = 0.98 (3 PLS compo­nents) and R2= 0.99 (5 PLS components) and the cor­responding cross-validated coefficients of 0.5\ and 0.73 demonstrated the possibilities of a PLS approach to TIs QSAR. According to this report, specific areas of the flavone framework are important to the predic­tion of HIV integrase inhibitory activity . Similar re-

1326 INDIAN J CHEM, SEC A, JUNE 2003

gions of activity were found using CoMFA on the

same data set again showing the robustness of results obtained by E-state indices and PLS.

finally, we wish to make several comments re­garding the physical meaning of Ti s as a limitation . All TIs has been introduced in an ad hoc way based on graph-theoretical representations of molecules. It is clear that a graph picture of a molecule is an oversi 111 -

plification of the proper nature of the molecular structure. However, any sort of representation of the molecular structure, or even of any !Jhysical reality , is a si mpli fication of such reality. If this representation is able to describe the properties related to it well, it must be because this model represents a "good" de­scription of such reality . Tn this way this representa­tion of the physical world immediately takes on a physical meaning. There are several good examples of such "oversimplifications" in physical sciences. For instance, the use of the "liquid-drop" model for repre­senting very heavy atoms by Bohr and Wheeler 177

allows the prediction of di s integration products of uranium considering that the usual radioactive phe­nomena would be analogous to the evaporation of a molecule from a liquid-drop (!). It is important to say that the value of the theory resides in its predictabil­ity. At present , there is no doubt that the use of TIs in drug design and discovery shows this desired feature namely that the models are predictive.

There have been some attempts in relating mo­lecular connectivity indices to quantum chemical con­cepts as reported by Zefirov l78 and Galvez ' 79

, as well as on the basis of intermolecul ar encounters by Kier and HaII1 80.1 81. More recently , Estrada l82 identified

molecular connectivity indices as components of mo­lecular accessibi lity. The first and second order con­nectivity indices represented molecular accessibility areas and volumes, respectively, while higher order indices represented magnitudes in higher dimensional spaces. In identifyir.g accessibility perimeters the atom degrees were identified as a measure of the ac­cessible perimeter of the cOITespondi ng atom. The Randi6 and connectivity indices are identified as the two components of the molecular accessib ility area. The accessibil ity perimeter is computed here from the van der Waals and covalent radii of the atoms and the interpenetration angle between the van der Waals cir­cumferences of bonded atoms. The description of the accessi biJity area in terms of the first order Randic and connectivity indices accounts for the success of these descriptors in correlating different physico-

chemical and biological properties si nce they are a measure of the extension of intermolecular interac­tions. A theoretical justification for the selection of the exponent in the Randic invariant is provided by the relation between the valence degree and the acces­sibility perimeters calculated in this work. This war};: opens the doors to interpreting other T Is in terms of physical principles currently used in chemistry.

Conclusions The actual application of topological indices in

drug design and discovery covers a large part of the field including lead discovery and lead optimization. These field s have been mainly covered for this current review and include virtual screen ing, drug design, combinatorial library design , QSAR, structure· pharmacokinetics, structure-toxicity re lationships, and so forth.

We have dedicated some sections of thi s review to what we considered were important areas of TI devel­opment in drug design and discovery. These areas include the use of more sophisticated stat istical tech­niq ues for developing TI QSARs, such as GA, PLS, ANNs, etc. Such techniques wi ll provide researchers with more appropriate tools for developing robust models using these descriptors. W also considered how the application of topographic and chirality indi­ces in drug design and discovery represent an impor­tant step forward. We believe that the study of the structural nature of TIs and their derived models will be of great significance in continuing this area of re­search active over the next few years.

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