Advances in Molecular Similarity, Volume 2

ADVANCES IN MOLECULAR SIMILARITY

Volume 2 • 1998

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ADVANCES IN MOLECULAR SIMILARITY Editors:

R A M O N CARBO-DORCA Institute of Computational Chemistry University of G iron a Giron a, Spain P A U L G . MEZEY Departments of Chemistry and Mathematics and Statistics University of Saskatchewan Saskatoon, Canada

VOLUME 2

•

1998

uCii) JAI PRESS INC. Stamford, Connecticut

London, England

Copyright © 1998 byJAI PRESS INC 100 Prospect Street Stamford, Connecticut 06904-0811 JAI PRESS LTD. 38 Tavistock Street Covent Garden London WC2E 7PB England All rights reserved. No part of this publication may be reproduced, stored on a retrieval system, or transmitted in any form, or by any means, electronic, mechanical, photocopying, filming, recording, or otherwise without prior permission in writing from the publisher. ISBN: 0-7623-0258-5 Manufactured in the United States of America

CONTENTS

LIST OF CONTRIBUTORS

vii

PREFACE

xi

Q U A N T U M SIMILARITY Ramon Carbo-Dorca, Liuis Amat, Emili Besalu, and Miquel Lobato

1

FUZZY SETS A N D BOOLEAN TAGGED SETS; VECTOR SEMISPACES A N D CONVEX SETS; Q U A N T U M SIMILARITY MEASURES A N D ASA DENSITY FUNCTIONS; DIAGONAL VECTOR SPACES AND Q U A N T U M CHEMISTRY Ramon Carbo-Dorca

43

PATTERN RECOGNITION TECHNIQUES IN MOLECULAR SIMILARITY W. Graham Richards and Daniel D. Robinson

73

TOPOLOGY A N D THE Q U A N T U M CHEMICAL SHAPE CONCEPT Paul G. Mezey

79

STRUCTURAL SIMILARITY ANALYSIS BASED O N TOPOLOGICAL FRAGMENT SPECTRA Yoshimasa Takahashi, Hiroaki Ohoka, and Yuichi Ishiyama

93

ANALYSIS OF THE TRANSFERABILITY OF SIMILARITY CALCULATIONS FROM SUBSTRUCTURES TO COMPLEX COMPOUNDS Guido Sello and Manuela Termini

105

vi

CONTENTS

SIMILARITY IN ORGANIC SYNTHESIS DESIGN: COMPARING THE SYNTHESES OF DIFFERENT COMPOUNDS GuidoSello

137

BROWSABLE STRUCTURE-ACTIVITY DATASETS Mark Johnson

153

CHARACTERIZATION OF THE MOLECULAR SIMILARITY OF CHEMICALS USING TOPOLOGICAL INVARIANTS Subhash C. Basak, Brian D. Cute, and Gregory D. Grunwald

171

OPTIMIZING HYBRID DENSITY FUNCTIONALS BY MEANS OF QUANTUM MOLECULAR SIMILARITY TECHNIQUES Miquel Sola, Marta Fores, and Miquel Duran ATOMIC SIMILARITY THROUGH A NEURAL NETWORK: SELF-ASSOCIATIVE PERIODIC TABLE OF ELEMENTS Jose Fayos

187

205

COMPARISON OF QUANTUM SIMILARITY MEASURES DERIVED FROM ONE-ELECTRON, INTRACULE, AND EXTRACULE DENSITIES Xavier Fradera, Miquel Duran, and Jordi Mestres

215

THE COMPLEMENTARITY PRINCIPLE AND ITS USES IN MOLECULAR SIMILARITY AND RELATED ASPECTS Jerry Ray Dias

245

CORRELATIONS AND APPLICATIONS OF THE CIRCUMSCRIBING/EXCISED INTERNAL STRUCTURE CONCEPT Jerry Ray Dias

259

LEAST-SQUARES AND NEURAL-NETWORK FORECASTING FROM CRITICAL DATA: DIATOMIC MOLECULAR fe AND TRIATOMIC AHa AND IP Jason Wohlers, W. Blake Laing, Ray Hefferlin, and W. Bradford Davis INDEX

265 289

LIST OF CONTRIBUTORS

Liufs Amat

Institute of Computational Chemistry University of Girona Girona, Spain

Subhash C. Basak

Natural Resources Research Institute University of Minnesota Duluth, Minnesota

Em Hi Besalu

Institute of Computational Chemistry University of Girona Girona, Spain

Ramon

Carbo-Dorca

Institute of Computational Chemistry University of Girona Girona, Spain

W. Bradford Davis

Department of Chemistry Southern Adventist University Colegedale, Tennessee

Jerry Ray Dias

Department of Chemistry University of Missouri Kansas City, Missouri

Miquel

Duran

Institute of Computational Chemistry University of Girona Girona, Spain

Jose Fayos

Departamento de Cristalografia Instituto Rocasolano, CSIC Madrid, Spain

Marta Fores

Institute of Computational Chemistry University of Girona Girona, Spain

LIST OF CONTRIBUTORS

VIII

Xavier Fradera

Institute of Computational Chemistry University of Girona Girona, Spain

Gregory D.

Grunwald

Natural Resources Research Institute University of Minnesota Duluth, Minnesota

Brian D. Gate

Natural Resources Research Institute University of Minnesota Duluth, Minnesota

Ray Hefferlin

Department of Chemistry Southern Adventist University Collegedale, Tennessee

Yuichi Ishiyama

Department of Knowledge-Based Information Engineering Toyohashi University of Technology Toyohashi, Japan

Mark Johnson

Pharmacia & Upjohn Kalamazoo, Michigan

W. Blake Laing

Department of Chemistry Southern Adventist University Collegedale, Tennessee

Miquel

Institute of Computational Chemistry

Lobato

University of Girona Girona, Spain Jordi Mestres

Institute of Computational Chemistry University of Girona Girona, Spain

Paul G. Mezey

Departments of Chemistry and Mathematics and Statistics University of Saskatchewan Saskatoon, Canada

List of Contributors Hiroaki

Ohoka

Department of Knowledge-Based Information Engineering Toyohashi University of Technology Toyohashi, Japan

W. Graham Richards

New Chemistry Laboratory Oxford University Oxford, England

Daniel D. Robinson

New Chemistry Laboratory Oxford University Oxford, England

Guido Sello

Dipartimento dei Chimico Organica e Industriale Universita'degli Studi de Milano Milano, Italy

Miquel Sola

Institute of Computational Chemistry University of Girona Girona, Spain

Yoshimasa Takahashi

Department of Knowledge-Based Information Engineering Toyohashi University of Technology Toyohashi, Japan

Manuel a Termini

Dipartimento dei Chimico Organica e Industriale Universita'degli Studi de Milano Milano, Italy

Jason Wohlers

Department of Chemistry Southern Adventist University Collegedale, Tennessee

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PREFACE

This new volume of the book series on Advances in Molecular Similarity is devoted to a selection of topics and problems presented at the Third Girona Symposium on Molecular Similarity, University of Girona, Girona, Spain, May 30-31,1997, held in conjunction with the Seventh International Conference on Mathematical Chemistry, University of Girona, Girona, Spain, May 26-29, 1997, both organized by Professor Ramon Carbo-Dorca, Director, Institute of Computational Chemistry, University of Girona, Girona, Spain. These two international scientific meetings were sponsored by several sources. Special thanks are due for the financial support provided by the following institutions and agencies: • • • • • • •

Institute of Computational Chemistry, University of Girona, Girona, Spain University of Girona, Girona, Spain Ministerio de Educacion y Cultura Fundacio Catalana per a la Recerca Generalitat de Catalunya Ajuntament de Girona Diputacio de Girona

The coverage of the two conferences provided a detailed cross section of the current advances in the rapidly expanding field of molecular similarity research, with a strong emphasis on both the fundamentals—quantum similarity measures.

xli

PREFACE

molecular shape analysis, molecular topology, and structural invariants—and the applications in such important experimental and industrial fields as pharmaceutical drug design, toxicological risk assessment, and molecular engineering for nanotechnology. This volume offers some of the highlights of the advances presented at the two conferences. In their presentations, our authors have made a remarkable effort to emphasize the underlying connections between the fundamentals and applications. Molecular similarity research is a dynamic field where the rapid transfer of ideas and methodologies from the theoretical, quantum chemical, and mathematical chemistry disciplines to efficient algorithms and computer programs used in industrially important applications is especially evident. These applications often serve as motivating factors toward new advances in the fundamental and theoretical fields, and the combination of intellectual challenge and practical utility provides mutual advantages to theoreticians and experimentalists. It is the aim of the Editors to present our readers with an overview of the current methodologies of molecular similarity studies, and to point out new challenges, unsolved problems, and areas where important new advances can be expected. We are convinced that this volume will serve our readers well, and it will represent a valuable, special source of information in their studies of chemistry, where molecular similarity continues to play a central role. Ramon Carbo-Dorca Paul G. Mezey Series Editors

QUANTUM SIMILARITY

Ramon Carbo-Dorca, Liuis Amat, Emili Besalu, and Miquel Lobato

I. Introduction 2 II. Quantum Similarity Measures 4 III. The Nature of Approximate First-Order Density Functions: Atomic Shell Approximations 5 A. Density Functions 6 B. ASA Coefficient Constraints 7 C. Quadratic Error Function 8 D. ASA Coefficient Optimization Using Elementary Jacobi Rotations 9 E. Alternative Approximate Expression of Density Functions: Complete ASA 12 F. Approximate Expectation Values 14 IV. Molecular Representations 14 A. MQSM Surfaces, Molecular Superposition, and Density Transformations . 15 B. Density Maps and Overlap-Like Measures 17 C. Discrete Matrix Representations 18 V Manipulation of Similarity Measures: Similarity Indices 21 A. C-Class Similarity Indices 22 B. D-Class Dissimilarity Indices 23

Advances in Molecular Similarity, Volume 2, pages 1-42. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 1

2

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

C. Generalized QSI D. Transformations between QSI E. A Discussion on Discrete Representation Indices VI. The Origin of QSAR and Related Problems A. The Success of QSAR B. Convex Sets and QSPR C. MQSM and Molecular Topology D. MQSM Topological Indices VII. Similarity Over Energy Surfaces A. Boltzmann Distributions and Boltzmann Similarity Measures B. General Distributions and Similarity Measures VIII. Conclusions Acknowledgments References

23 24 25 27 27 29 31 33 38 38 39 40 40 41

I. INTRODUCTION As is well known, the first rough description of quantum similarity was made in a naive paper by Carbo et al. in 1980.^ In this work was discussed the initial basic concepts, related to molecular similarity measures, but seen from a quantummechanical point of view. Since then, a large amount of research has been performed on the specific subject of quantum similarity. Several laboratories and individuals have developed the seminal ideas, and in the current literature have been published many papers^ as well as book chapters'^ and monographs'^ are available. Even a specialized series^ has been devoted to the study of the broader concept of molecular similarity. Theoretical settings of the Quantum Similarity framework have been studied in several aspects by various authors.^ A general discussion of the theory enveloping quantum similarity has also been constructed in our laboratory.^ Among these last contributions must be noted the description of the so-called Mendeleyev postulates,^ a set of several points of view trying to govern the ideas constructing quantum similarity background. Different concepts have emerged from the research experience of the present decade in the field of quantum similarity. Among the most relevant, in our opinion, are the following: 1. Quantum similarity measures (QSM) are a natural vehicle to obtain a discrete representation^^ of density matrix elements, with particular emphasis on the representation of first-order density functions. 2. Results on quantum similarity constitute a new way to practically connect all fields of chemistry^ with quantum theory. This new relationship has to be mostly considered based on purely geometrical grounds, as a consequence of the application of quantum-mechanical postulates.

Quantum Similarity

3

3. Computation of QSM can be achieved in a fast, approximate although highly accurate framework, based on density function fitting to a spherically symmetric atomic basis set,^^ by means of the so-called atomic shell approximation. 4. Molecular QSM (MQSM) surfaces can be easily computed,^^ generalizing the density surface analysis and producing alternative ways to observe pictorially the molecular shapes. 5. Molecular superposition^^ may be accomplished in a very efficient way, using the fact that any MQSM, being a definite positive function of the molecular relative positions, can be easily maximized. 6. Use can be made of the discrete, n-dimensional, description of the density functions,^^ achieved by means of QSM, performed on a known quantum object set. Zermelo's theorem [14] can be invoked to consider a possible order, which can be induced within a given quantum object set, opening a natural way to construct particular p^noJ/c tables over the set. 7. Quantum similarity indices are to be considered as a set of parameters strictly dependent on QSM, which can be arbitrarily described in numerous ways.^^ 8. The success and scientific foundations of the well-known QSAR or QSPR procedures^'^'^ may be easily deduced from considerations attached to the molecular quantum similarity framework. 9. Molecular topological parameters, with identical structure as the classically defined ones,^^ but bearing an important amount of three-dimensional information, may also be deduced using simple molecular quantum similarity ideas. 10. Extensions of quantum similarity^^ to chemically interesting functions, other than those belonging to the density matrix element family, can be envisaged without further effort. As some discussions of the main subjects listed above can be found dispersed in the literature, our aim here is to furnish a coherent and comprehensive presentation of all these quantum similarity related topics. To achieve this goal, we will first make a simple presentation concerning the nature of similarity measures. An analysis will follow, dealing with first-order density functions, which will open the way to introduce the reader to atomic shell approximations and to the procedures to compute them. Next, the framework of the discrete molecular representation connected with quantum similarity, and a short presentation of the features of quantum similarity indices will be developed. A triad of discussions, closing this contribution, will follow, consisting of the following points: 1. The origin of QSAR and related problems, studied from the point of view of the inherent discretization, associated with QSM.

4

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

2. Connection of QSM with molecular topology. Manipulation of quantum similarity matrix elements to compute topological indices. 3. A discussion on the possible ways of computing similarity measures using energy surfaces or other quantum functions.

II. QUANTUM SIMILARITY MEASURES QSM, although defined within the scope of an arbitrary number of density matrix elements, bearing arbitrary orders,^ when studied from the computational, practical, point of view, are based essentially on first-order density functions. This rule will be followed throughout this paper. A QSM involving two quantum objects (QO) {A, B}, described by the density functions {p^, p^}, may be defined by means of the integral ^AB(^)

= J J PAiriMr,, r^)p,ir^)dr,dr^

(1)

where the presence of the operator symbol Q(rj, r^) corresponds to a general possible selection of any positive definite linear operator form employable within Eq. 1. The most usual operator's choice has been a Dirac's delta^^ function 6(rj - r^), which defines a QSM related to the spin-spin contact^^ correction, a term forming part of the Breit Hamiltonian.^^ Taking into account the properties of the Dirac's delta function when used as an operator, Eq. 1 can be written using a more simplified integral form: M5

= |p^(r)Pg(rVr

(2)

which constitutes the oldest definition of a QSM.^ Due to the involved parts and computational structure, Eq. (2) has been customarily called an overlap-like QSM. In both previous QSM definitions, the possible integrals involving the same density function, z^, are referred to as quantum self-similarity measures (QS-SM). Other operators may be used in the defined measure integral 1, as Coulomb or gravitational operators^^ have been, but a particular operator choice turns out to be a matter of integral complexity, computational advantages, system description, and problem environment. Among feasible operator selections, as the previously mentioned ones, there can be present a density matrix element by itself, corresponding to another system C, like Pc(ri» ^2^ Then, a very interesting QSM could be constructed: ZAB;C

= 11

PA(^I)PC(^V

r2)PB(r2)dr^dr^

^^^

constituting one of the several possible forms associated with the so-called tripledensity QSM.^^ This integral form, as presented in Eq. 3, opens the way to outline

Quantum Similarity

5

the formal structure of multiple-density QSM. These measures may be constructed, for instance, as the integral of the product of the density function set D = {pi(r)}, attached in turn to the elements of some chosen quantum object set (QOS); that is, choosing the simplest formal notation within the integrand functions:

One can see, in this manner, how it is possible to define QSM, in the best-suited form, to study any problem related to the computational manipulation of QO. A large variety of positive definite operators can be used to allow a tailor-made description of QO by means of QSM. An obvious way becomes apparent when, once a QOS S = {Sj} is chosen, the QSM between a set element and the rest are computed. For example, using definifion 1, every element of S can be associated to all of the others belonging to the same QOS, including itself. This may produce a column vector, whose dimension will be attached to the set cardinality. The elements of this vector can be obtained computing each QSM between the chosen QO density function and the rest.

III. THE NATURE OF APPROXIMATE FIRST-ORDER DENSITY FUNCTIONS: ATOMIC SHELL APPROXIMATIONS Over a long period, our laboratory has been interested in the elementary Jacobi rotations (EJR) technique.'^^ EJR constitutes a body of straightforward procedures to obtain ^-dimensional vector norm-conserving variation. In this field, a large theoretical and computational contribution dealing with quantum electronic energy direct optimization^^ has been developed over time. Some work has also been performed on the many aspects of the Jacobi diagonalization algorithm,^"* proposing a new parallelizable procedure, which constitutes a practical computational scheme, able to deal with large matrices and producing a chosen subset of eigenvalues and eigenvectors. On the other hand, within the purpose of a general search of optimal QSM algorithms, the subject of electronic density function fitting has been considered in a preliminary paper,^^ by using a superposition of atomic spherical shells. Afterwards, a conceptual and practical refinement of the previous formalism, the so-called atomic shell approximation (ASA),^^ has also been extensively studied. This section will deal, in a broad manner, following the path of the accumulated experience on both EJR and ASA directions, with the proposal of using EJR transformations to solve the problem, associated with the constrained fitting of electronic density, using ASA-type functions. A brief description of the main ideas that will be employed seems necessary.

6

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO A. Density Functions

From a theoretical point of view, accurate MQSM may be obtained using ab initio molecular electronic density expressions constructed within the common LCAO approach, corresponding to the expression

where {D^J are the charge-bond order matrix elements, and {x^} represent the atomic orbital (AO) basis set functions. Using this approximation, four center integrals over the AO basis set have to be computed in MQSM calculations, like the ones attached to Eq. 2. For instance, taking expression 5 and the equivalent one, corresponding to the second system density function LCAO expression, the following quadratic form in terms of the charge and bond order matrices is found:

\i,veA X,GEB

^

where the four index symbol hypermatrix elements {^y ^} are the four center overlap-like integrals over the corresponding AOs on both systems: ^ f ^ } = jx:(r)X»xf(r)xr(rMr

^'^

Besides these computational difficulties, MQSM evaluation has in addition the problem of the measure maximization.^^ The function defined by the integral z^^ depends on the relative position of the implied molecular systems A and B, and the best similarity matching between them could only be defined when the maximal integral value is reached. To speed up QSM integral maximization, computational algorithms have been designed to calculate high-accuracy approximate density functions. ^^'^^ QSM is an ideal mathematical construction, logically placed inside the theoretical and conceptual structure of quantum mechanics, which may be used whenever it is necessary to compare two or more density functions. In fact, this may be seen from a more general point of view, associating the density functions appearing in Eq. 1 with square summable definite positive functions. From this mathematical perspective, a similarity integral as Z^Q can thus be interpreted as SL positive valued weighted scalar product. On the other hand, use of ASA-like density functions can be traced up to the initial papers on QSM,^ where a CNDO-like^^ approach was invoked to deal with the computational evaluation problem of the quantum similarity integrals, z^^, for molecules. From this initial viewpoint, the concept of approximating a given density function has evolved to consider ASA as a superposition of spherical nS-type, STO or GTO, functions. Recently, another similar approach, but circum-

Quantum Similarity

7

scribed to the expression of core-electron density,^^ has been published, although no constraint conditions have apparently been used in this case, like the complete ones stressed in this paper, established in the following section. The density function in ASA form may be written in terms of an atomic function set superposition {oj,

pfV) = EcT,(r)

(8)

aeA

where the sum runs over all of the atoms {a}, present in a given molecule A. At the same time, the atomic function set {oj, may be constructed using another function set chosen in such a way as to describe atomic shells {5.}, using another sum: ^a(^) = Z ^i(^)

(9)

and the sum in Eq. 9 is performed over all of the atomic shells of atom a, belonging to molecule A. Finally, the spherical function set [s-], describing some sort of atomic shells, can be defined, in a very easy manner, as follows:

kei

where now the sum is carried out over all of a chosen positive definite function set {cp^}, belonging to the atomic iih shell. The set of coefficients {c^}, is sought to be positive in all cases, so as to keep, in general, positive definite the probability density distribution structure of the atomic shells, {5.}, and thus transferring this characteristic to the approximated density function p^^"^ too. The above ASA partition is equivalent to writing Eq. 8, in a more compact notation, as a linear combination of a definite positive function set {G-}: P n r ) = SvvA(r)

(^1>

ieA

where the sum is performed over all of the basis function set {9.}, and the set of positive coefficients [w-] must be determined. One must insist on the necessary positive definition of the usual density distributions, which becomes translated in the ASA approach as the approximate density functions, being defined in turn with the form and properties of n-dimensional simplexes}^ B. ASA Coefficient Constraints

The most interesting case of function fitting in the realm of MQSM is constituted by the ASA approximation of first-order density functions, but other high-level

8

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

density function forms may be studied as well. At any level, both the exact and the ASA density functions may be supposed normalized to one particle, by dividing the function by the appropriate particle number combinatorial coefficient. Also, in Eq. 11, considering the involved basis functions normalized in the usual sense: Je.(r)t/r=l,ViG A

(^2)

then, necessarily the set of ASA coefficients {w.}, besides the imposed positive definite condition: >v.>0, ViG A

(13)

must fulfill the additional constraint: (14)

Although the second condition may be easily taken into account, employing a Lagrange multiplier technique,^^ the first one as expressed in Eq. 13 cannot be so easily introduced^^^ into the computation process. It will be shown that both conditions can be kept throughout the optimization procedure by using adequate algorithmic tools. C. Quadratic Error Function

A significant MQSM computational simplification, while preserving measure integral accuracy, is achieved in the so-called promolecular approximation, where the total molecular electronic density function is written as a sum of individual atomic electronic densities:

pfV) = EpfV)

(i^>

aeA

Every atomic density function p^^"^ is built up with the same formalism as in Eq. 11, replacing 0- by squared nS-type functions pT\r) = J^w,S,{rf

(16)

Using this approximation, overlap-like QSM between two atoms can be expressed by ^ab = Yj ^i S "^j^ij ie a

je b

(17)

Quantum Similarity

9

where the elements {z-}, which can be collected into some positive definite matrix Z, are defined by the integral over the nS-type functions: Zy-j

5,(r)2s/r)2«fr

(18)

For a given atom a, the ASA coefficients {w.}, can be calculated minimizing the quadratic error integral function between the atomic ab initio and ASA electronic density functions. A particular form of the quadratic error function is easily written as

e*'* = ||Pa(«-)-pfV)f^«ije a

ie a

pi,v6 a

where z^a corresponds to the ab initio QS-SM of atom a, z.a-iPairfdr

(20)

which is computed in turn within the LCAO approximation, replacing electronic density p^(r) by the equivalent form in Eq. 5. Equation 19 may be rewritten in matrix form as £(2) = ^^^ + w7';2w-2A^W

(21)

where the elements of the vector A = {A-} are given by the integral ^. = S ^ . v | W ' x ; ( r ) X v ( r ) ^ r

(22)

and w is the normalized column vector (w^w =1) containing the ASA coefficients. With the corresponding modifications within every implied integral, the quadratic error integral 19 may be rewritten using a definite positive weight operator. In fact, the quadratic error integral can be considered from the point of view of QSM as a self-similarity integral involving the exact and ASA density functions difference. Thus, form 19 is nothing than an overlap-like definition, as the one appearing in Eq. 2. While choosing a positive definite operator, a measure form like the general one provided by integral 1 will be present as an alternative quadratic error. D. ASA Coefficient Optimization Using Elementary Jacobi Rotations The set of positive coefficients collected as a vector w = {w-} can be defined as the square modules of some auxiliary vector components x = {x-}, which will be called the generating vector:

10

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

>v, = k,p; v/

(23)

and in this way condition 13 is fulfilled. Moreover, due to the fact that the Z matrix is definite positive, a unitary matrix U can be found such as U-'ZU = D

(24)

where D is a diagonal matrix with positive real elements. The first step in the procedure consists in diagonalizing the matrix Z, so as to obtain their eigenvalues and eigenvectors. Then, the initial coefficients {x-} can be made equal to the most suitable normalized eigenvector of the matrix Z, and consequently the required constraints specified in Eq. 14 are automatically fulfilled. Starting from this generating vector, and applying orthogonal EJR, the constraints will hold along the optimization process. ASA coefficients are obtained by an optimization procedure, which minimizes integral 19. Substituting every ASA coefficient w. by Ix-P, and only considering the case of real generating vector coefficients, the quadratic error integral function can be rewritten as (25) ij£ a

ie a

EJR are easy tools to obtain unitary or orthogonal transformations usable over vectors or matrices. The origin of such transformation matrices can be found in the 1846 paper of Jacobi.^^ Being orthogonal, EJR may also be viewed as rotation matrices over real (or complex)-valued n-dimensional spaces. Applied to a given ^-dimensional vector, an EJR, which will be written here as Jpq{o), will transform the vector components p and q only, keeping invariant the rest of them. The EJR transformation on the generating vector chosen components is defined by the equations

^q^^p^^^q

(26)

where c and s are the cosine and sine of the EJR angle a. The norm of the transformed vector remains invariant with respect to the initial one. Over the generating vector coefficients in Eq. 25 it is easy to apply the EJR represented by Eq. 26, and then the variation of e^^^ with respect to the active pair of elements {p,q} may be expressed as

+ 25x^1 ^^,, + 25.^ E^?^„ i*p,q

i*p,q

Quantum Similarity

11

-2A„5xl-2A8xl p

p

q

(27) q

To compute 8e^^^ in Eq. 27 it is necessary to evaluate the second- diwd fourth-order variation of the elements x and x . The second-order ones are easily obtained:

5-^ = (^ - ^9) = s Vp - 4 ) + 2cs V , ] = A

(28)

The fourth-order variation terms are obtained in turn from the second order ones, giving

8(^;^) = ( ^ ^ ^ - ^ ^ ^ ) = ( ^ - ^ ) A - A 2

(29)

Further development of the 5x1 and 5x^ expressions, as well as the one associated with the crossed term 5(xV) in Eq. 29 provides the dependence of the quadratic error from the EJR sine {s} and cosine {c}. Substituting the expressions dx^, 5x^, 8x^, 5^^, and 6(A^;C^) into Eq. 27 and collecting terms, one finally arrives at a quartic polynomial on the rotation sine: 5e^2^ = EQ/ + £:i3cs^ + i ^ o / + E^cs

(30)

where the parameters {Ejj} are described as follows: ^04 = i^pp + ^,, - 2Zp,)[(A^ - 4 ) ' - 4 ^ ^ ] ^13 = (Zpp + z^^ - 2zp^)ixl - xl)x^^ ^02 = 4(z^^ + z^^ - 2z^X^^ - 2 ( ^ - ^ ) G

and G=

Tj'^(Zpi'-^qi)^4^pp-'^q^qq-(4-'''lK-^^^

i^p,q

The optimal sine can be chosen with the gradient condition dde^^^/ds = 0, —T— = 4EQ/ + i^i3(-^s^ + 3cs^) + 2EQ2^ + E^^(-ts + c) = -c[(£i3s2 + E,y

- 2 ( 2 £ o / + E,^)t - (3E,,s^ + ^jj)]

(31)

12

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

= -C{T/

- IT^t -T^) = 0

(32)

where s/c = t and dc/ds = -t. The best Jacobi rotation angle is found solving the quadratic polynomial equation in the EJR tangent {?}, appearing in expression 32. The optimization is conducted through an iterative procedure, until the global variation of Jacobi rotation angles or the quadratic error integral function becomes negligible. A Newton procedure has also been used to optimize the exponents of the fitted nS-type functions appearing in Eq. 16. This search algorithm is available because the analytic gradient and the Hessian matrix of the quadratic error can be made easily available. A program called GATOMIC^^ has been codified to compute fitted atomic shells using nS-GTO or nS-STO functions. The initial basis set exponents have been systematically taken from an even-tempered^^ geometric sequence. A coefficient optimization is sought, followed by a Newton search, which is used over this initial exponent set so as to obtain ameliorated values. Next, a new coefficient optimization using EJR is performed to obtain the most accurate fitted positive coefficients, until convergence is reached. Details and computational examples will be given elsewhere. E. Alternative Approximate Expression of Density Functions: Complete ASA

Although losing the elegant simplicity of the ASA approach, there appears to be an alternative very natural way to express the first order density function, using the same generating vector concept as before, within the ASA approach, as discussed in previous sections. Suppose known a spherical basis set made as in ASA environment of nS-type functions [S-]. Then, the first-order density function may be approximated by a function like

where {;c.}, are the elements of the generating vector x. This approach will be called the complete atomic shell approximation (CASA). The quadratic error function, using Eq. 33, can be written in a similar manner as in Eq. 25:

i,j,kM a

ije a

where the hypermatrix elements {z-.^J are overlap-like similarity integrals involving four different 5-type spherical basis functions:

Quantum Similarity

13

Zyu = ! S^(^)SM)S,{r)Siir)dr

(35)

while the matrix B = [B-j] corresponds to an integral between two AO and two S-type functions: ^ij = E ^.v I S.(r)Sjir)x;(r)Ur)dr

(^6)

EJR can be used in the same way as in the ASA approach to optimize the new CASA quadratic error, adapting different variation terms for the generating vector coefficients. But the structure of the CASA, as in Eq. 34, permits an alternative proposal, which leads to elegant matrix formalism, identical to the one used in the monoconfigurational SCF^^ computational structure. Here the new approximate expression of the density function leads to the normalization condition, which must fulfill the CASA generating vector coefficients: J pCASA(r)jr = ^ x^xj J S,(r)5/r)Jr i,jea

=

^XiXjSy=l

(37)

i,J^a

That is, in matrix form x^Sx = 1, provided that the matrix S = {5.} collects the metric matrix elements of the S-type basis function set. Thus, the problem is reduced to minimizing Eq. 34, submitted to constraint 37. A Lagrange multiplier technique can produce a more elegant result here than EJR. The Euler equations of the constrained optimization problem are written easily in terms of a generalized secular equation^^: GX = YSX

(38)

where y is half of the necessary Lagrange multiplier, and the matrix structure G depends on the generating vector coefficients and the involved integrals already defined:

Klea

The CASA generating vector coefficient computational procedure should be made iterative, and in the process, the eigenvalue y has to be chosen as the one with minimal module. This is so, due to the fact that eigenvalues in Eq. 38 can be shown

14

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

to be the same as the scalar product between the CASA density and the difference between this approximate density function and the exact one. This approach is under study in our laboratory, and the practical results will be published elsewhere. F. Approximate Expectation Values

The ultimate use of ASA or CASA fittings to an exact density function lies in the possible fast, but accurate, computation of QSM integrals. ASA or CASA approaches become somehow essential owing to the need to compute huge amounts of integrals, so as to obtain the optimal measure values, when using quantum similarity to superpose two molecular structures.^^ See Section IV.A for more details. But another possible application of ASA-type functions may be found in the approximate calculation of expectation values of other operators than those employed in QSM. Numerical experiments show that ASA fitted density functions perform quite well when self-similarity values are computed, but fail when it is time to compute expectation values like kinetic energy, <^>, which at any approximation level must be estimated using

<^> = - ? S E ^6- j5,(r)V'5/r)^r

(40)

If sound expectation values have to be evaluated, then optimal quadratic error functions shall be optimized in the way discussed above, but adding the expectation value errors, computed under ASA, with respect to the ab initio ones. The overall self-similarity values possess a greater error than the computed ones with the previous optimization technique, but several expectation values can be perfectly adjusted in the process. Numerical details will be provided elsewhere.

IV. MOLECULAR REPRESENTATIONS In light of the previous discussion, many ways can be envisaged so as to have an appropriate, discrete or continuous, QO representation. Starting from the typical quantum-mechanical density function continuous description, using the QSM adequately, one can obtain new functions, which may produce additional information on the molecular shape and environment, in the same way as other density function manipulations do, like the well-known electrostatic potential. In this section these possible additional functions will be discussed along with the related problem of molecular superposition, which has been studied in our laboratory. A description of the QO discrete representation and possible manipulations will end this section.

Quantum Similarity

15

A. MQSM Surfaces, Molecular Superposition, and Density Transformations MQSM Surfaces Suppose one is dealing with the QSM involving two molecular structures {A,B} with attached density functions {p^jP^}. The MQSM integral of type 2, and more sophisticated ones too, should be written taking into account the relative coordinate positions of both molecular frames {X^,X^}, where vectors X collect all of the atomic coordinates of both molecules respectively. Supposing the molecular internal atomic position degrees of freedom constant, MQSM integral 2 can now be written in a slightly different form: z^(T;R) = J p^ir;X^)Pgir;{T,R}[Xg\)dr

(41)

where M = {T, R} are the three translation and rotation vector elements related to both molecular frames. Here, it is supposed that molecule A coordinates are kept invariant and the atomic structure B is translated and rotated with respect to the A molecular coordinate system. The net result is such that the integral ZJ^Q will depend on these six parameters collected on the vector M, and being such a six-variable function, it will expand a seven-dimensional surface. Keeping some of the translation-rotation parameters constant, the MQSM could be transformed into a similar function, which can be depicted as the density or the electrostatic potential ones. However, in a very particular situation no such kind of restrictions need to be considered: This case corresponds to the MQSM integral involving a molecule A and an atom B. Certainly, in this situation the rotation angles are irrelevant, and only the translation vector survives in M; thus, Z^Q ( T ) will behave as a function of three variables, which depict the atom B position in space, and as such could be represented directly as plain density or electrostatic potential functions are. Some examples can be found in Ref. [11]. Integral 41 dependencies on the translation-rotation parameters M, as mentioned above, produce a very interesting problem, which has been considered and taken into account since the first time a MQSM was computed, ^'^^ although until recently has not been efficiently solved. ^^ MQSM, being positive definite functions, could have a maximal positive value with respect to the variation of the M = {T,R} parameter vector pairs. This feature can be explained in such a way as that the integral associated with the measure Z^Q reaches a maximal value, when the moving structure matches, in a natural way, the fixed molecular frame. That is, one can search for an MQSM, which superposes maximally, according to this particular measure integral choice, both sets of involved atomic coordinates.

16

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

Molecular Superposition

Molecular superposition has been a problem, whose solution has many implications in chemical process knowledge, and certainly carries a great relevancy in pharmacological studies. Although many solutions have been described in the literature, none as far as we know has been based on a coherent and natural theoretical basis. From the molecular superposition point of view, the optimal overlap-like similarity matrix, containing the MQSM between the elements of a QOS, can be formally computed using the following integral definition: z^" = max J p^(r,X^)p,(rM[X,])dr

(42)

The problem has been solved recently, and a set of algorithms described. ^^ A varied set of examples proved that MQSM maximal values can be easily reached. Using an ASA approach or even simpler similarity integral forms the process can be extended to proteins. Thus, maximal MQSM constitute the most appropriate theoretical structure to obtain molecular superposition and matching. When constructing the similarity matrix Z ={z^}, whose elements are made by the maximally matching molecular structure pairs, the added problem now is such that, in general, one cannot suppose Z to be a positive definite matrix any longer. Usually, the similarity matrix bears a positive definite structure. Whenever every column or row of the matrix corresponds to an MQSM calculation using the same Density Function, with the molecular coordinates remaining unchanged by a transformation of type M, the positive definition is present. The same remark holds, even when choosing a positive definite weight operator in the MQSM integral, because then, the Similarity Matrix, Z, also acquires a metric structure. A final remark must be proposed with respect to this metric property assigned to similarity matrices. This is so provided that the molecules in the QOS are chosen essentially different, that is, described by linearly independent density function descriptors. Matching can be extended to sets of molecular structures as optical isomers, molecular excited states, and conformational forms of a given molecular structure. Density Transformations

From Eq. 41 QSM expressions can be seen as a way to obtain density integral transforms (DIT). Indeed, in the above integral, one can so look at the integrand density function pair, as to consider the product of densities as a properly defined function and a transform kernel, respectively. The MQSM, z^^, can be considered as a transform of p^ employing the transform kernel p^. This situation can be easily generalized, defining a DIT of a known density function p^ as the integral A^(R) = r(p^) = J K{K T)p^{T)dT

(43)

Quantum Similarity

17

where A^(R, r) is the transform kernel, an operator that produces another function by performing the integration DIT A^, which can be used in turn as a new representation of the QO, attached to the former density function p^. The formal scheme in the DIT definition can be extended owing to the fact that the density function can still be regarded to depend on another position vector set, RQ, as occurs under the universally admitted Born-Oppenheimer approximation framework. This dependence, which can also be attached to the kernel, produces as a result a DIT dependence of this vector too: A^(R, R„) = r(pJ = J K(R, r, R,)p^(r, R^)dr

(44)

Usual calculations consider the dependence of all of the involved functions on the coordinate vector RQ as implicit, and in this way it is not explicitly written in the developed formulas. In any case, if DIT are used over a density function set P = {p^}, obtained over a QOS S = {sj}, the corresponding elements of the DIT set D = {Aj} can be considered as sound representations of the set S elements as the former density function set can be. In any case, the following relationships may be established: VSjE S A VpjE P-^SjSf^Aj

(45)

B. Density Maps and Overlap-Like Measures

An electronic density distribution map may be connected to a similarity measure in the following way. Suppose known a density function for some molecular structure M: p^(r), say. In the same manner, suppose known the density function of a given atom a, centered at some space position R, and symbolically expressed as p^(r, R). The overlap-like QSM W R ) = IpM(r)Pa(r.R¥r

(46)

corresponds to an MQSM map of molecule M with respect to atom a, placed at the 3D space position R. Let us suppose that the atomic density p^(r, R) can be safely associated with a Dirac's delta function A^^ 5(r - R), with A^^ being the nuclear charge. Then, one will obtain ^M«(R) = ^A J PM('-)8(r- R)dr= N^p^iR)

(47)

That is, both MQSM and density function map become a unified concept. MQSM maps have as a limit the density map.

18

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO C. Discrete Matrix Representations

From the discussion of the QSM problems presented so far, similarity measures of this kind can be considered as the way to obtain a discrete, numerical, matrix representation of a given molecular structure, which will depend on the rest of the molecular structures taken into account. The following points illustrate the general background of the discrete matrix representation of a molecule. The matrix representation of a molecule, with respect to a given molecular set, can be associated in turn with a set of numerical values forming a finite-dimensional vector, representing the studied molecule, as mentioned in Section II. If the studied molecular set contains m molecules, the original ©o-dimensional molecular representation by means of the density matrix or the corresponding DIT elements, associated at the same time with every molecule belonging to a QOS can be considered projected into an m-dimensional space. Of course, when one is dealing with hypermatrices, the m-dimensionality can be achieved by using a dimension reduction, as will be described below. However, in this case, this m-dimensional projection is not compulsive and one can freely work into higher dimensional spaces. Several discrete representations may be obtained, since the matrix of the QO coordinates may come from the QSM values collected into matrices of any kind. A practical way to codify the QOS into a matrix may be done as follows. Suppose a given QOS, 5, and an attached set D collecting the chosen density functions of every QO in a one-to-one correspondence: V5G5^3pGD=>5<->p

(48)

Also, all possible QSM, involving QOS elements, may be considered defined in the tensorial product space D(8)D, and finally collected into a similarity matrix: Z=[zjj{Q)}^{Zu]

(49)

The matrix Z contains, in this manner, information about the relationships between coupled elements of the QOS. The columns of matrix Z Z = (Zi,Z2, . . . , Z ; , . . . , z j

(50)

can be interpreted as the matrix representation of every element in D, in the vector space spanned by the QO density functions, which, in turn, act as a basis set. In this way, a finite-dimensional set of vectors Z represent the density funcdons D, by means of the correspondence VpG D - > 3 z e Z=>p<-^z

(51)

so, from these consideradons, the following connecdon between QO and matrix columns can be obtained too:

Quantum Similarity

19

V J G 5->3zG Z=>5<->z

(52)

In the case of a molecular set study, this point of view leads to the concept of point-molecule, that is, any column ZjE Z of the similarity matrix. The collection of all point-molecules in the matrix Z is known as a molecular point-cloud. Molecular point-clouds can also be constructed by means of the eigenvectors of matrix Z. This is possible because the set of QSM [Sjj] can be considered a set of scalar products between density function pairs. In this manner, the QSM matrix Z is somehow a kind of Gram matrix, constructed using the elements of the density set D. The Z matrix column eigenvector coordinates are to be considered, in any case, a set of point-molecules possessing a canonical behavior. They are normalized vectors orthogonal to the rest of the molecular point-cloud elements. They constitute, in this manner, some sort of uniform coordinate system, which may be defined for any QOS, provided that a similarity matrix is known and their eigenvectors computed. An even more interesting uniform coordinate set of this sort may be formed, taking into account the Z matrix eigenvectors, collected as row vectors, they can be considered a dual space representation of the former molecular point-cloud. When the weight operator Q in the QSM is chosen as another density function or as a product of them belonging to the elements of D, as in the triple-density QSM defined in Eq. 3, then the similarity matrix Z can be considered a hypermatrix. As a consequence a synmietric matrix representation of every object in S is obtained. In general, when a multiple QSM, consisting of a product of p density functions, is chosen, a (/7-l)-dimensional hypermatrix can be attached to every object in set S. The procedures outlined so far may be modified in the following way. Until now it has been supposed here that all elements of a given molecular set have been used to represent the same molecular elements, producing, in this manner, square dimensional numerical collections of the active molecular structures. But it is not necessary to proceed in this way, in order to obtain discrete molecular representations. A given molecule or molecules can be compared with a given molecular set, which may serve as a basis set for the discrete representation. Suppose that T is a molecular set acting as a pattern structure, to it a one-to-one correspondence with a density function set P is known. Then, the following relationships may be obtained, using the similarity matrix U derived from the tensorial product space D®P or from some high-order direct product similarity measures: V5; 6 5 -> 3u; G U =» 5; <-> u^

(53)

and the column matrix elements {M^} G U^ are obtained using the similarity measures between the associated densities to Sj and all of the density functions of the reference molecular set T: p ^ G P; V^.

20

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

From here, it is easy to see how this algorithm can be taken either as a way to obtain a molecular representation by means of a rectangular array or as a procedure leading to the square discrete representation described previously. When the present numerical process is followed, it may be seen as acting as a tool to augment the discrete molecular matrix representation dimension. To understand this last comment, let us define by means of a direct sum a new matrix set V = Z©U such that \/sje S^3\je

V=>5^<->v^ = z^eu^

(54)

From the diverse points of view discussed above, one can see that QSM, obtained considering some or all QOS elements, may lead to a projection of any given molecular element representation. This projection goes from the oo-dimensional molecular space density set, D, into a finite-dimensional space, whose dimensions may be of diverse, arbitrary, finite magnitude, depending on the chosen conventional rules used, when gathering the MQSM into a matrix form. It is possible to generalize the previous definitions: as can be deduced from the above discussion, every system set S = {sj} can be represented by a hypermatrix whose elements are taken as the MQSM: {Z,. ,

,.(£2)} = {Z[^,.,^,.,...,i,.](Q,R)}

(55)

where a multiple QSM made of n density elements attached to the corresponding QOS elements {s-,s^,..., s^ ] has been used. A possible explicit dependency of an operator Q and a coordina?e vector R is expressed on the left-hand side of the equation. To obtain a computationally manageable form of the similarity hypermatrix, it is possible to reduce the n-dimensional hypermatrix information into a matrix. A, dependent on the weight operator Q, say, involving all available pairs of molecules contained in S: A(Q) = {Ajj(Q)}

(56)

This can be reached by means of the following hypermatrix reduced product definition:

A,/Q) = X

(i = l,ml,l)Z/(i)Z/i)

(57)

which possesses a structure similar to a scalar product. There a nested summation symbol^"^ (NSS) formulation has been used and the terms Z/i) are written as a shorthand notation of the hypermatrix elements: Zj(i) = Zr ''l''2'-Vl

(Q)

(58)

Quantum Similarity

21

Most frequent representation cases can now be envisaged into this new formulation: 1. When n = 2, one is dealing with the {Zj-} matrix elements, as those defined in the previous paragraph. Then, (59) 1=1

2.

where the NSS appearing in Eq. 57 has been reduced to a unique summation symbol. This kind of reduction is a scalar product between the rows (or columns) of the matrix Z. If n = 3, then every molecular structure is represented by a square matrix, but even so one can also find a method to compact the data generated by triple-density QSM hypermatrix elements {Z^-}; for example, it is only necessary to use the following straightforward rule: m m

(60)

1=1 ; = l

At this stage, we can describe the general methodology to follow in a QSM study. When a family of QO is defined, one constructs the attached hypermatrix representation of every set element, once a given kind of similarity measure has been chosen. In general, a matrix or some hypermatrix, which is reduced to manageable dimensions, can always be directly obtained, following any one of the contractions previously mentioned. Afterwards, it is convenient to analyze somehow the matrix or hypermatrix structure. The final conclusions can be used to suggest relationships between the QOS elements.

V. MANIPULATION OF SIMILARITY MEASURES: SIMILARITY INDICES As has been previously discussed, once the set of QO to study is formed and the operator related to the QSM definition chosen, the resultant value of the QSM itself, related to the QOS, is unique. However, the similarity matrix elements, obtained as discussed in the previous sections, can be transformed or combined so as to obtain auxiliary terms of a new kind, here named quantum similarity indices (QSI). A vast quantity of possible QSM manipulations leading to a consequent great variety of QSI definitions exists. The most common, which can be considered as the standard ones, arise from the manipulation of QSM between two molecules as those defined in Eq. 1 or, more generally, by means of some reduction like that described in Eq. 57.

22

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

In this manner, the QSM values can always be referred to as the elements of the similarity matrix: Z = {Zjj(Q)}. For reasons that will be obvious through the following lines, two kinds of similarity indices may be described and collected into two well-defined classes. These classes are related to the most elementary similarity-dissimilarity Indices defined so far in molecular similarity studies or related fields, as pattern recognition theory. The nature of this classification of the similarity indices will be developed next. A. GCIass Similarity Indices A well-defined similarity index constitutes the leading member of this index class. It is nothing more than an index formal expression belonging to the correlation-like class. In fact, the mathematical interpretation of such an index is that of the generalized concept of the cosine of the angle subtended by two vectors, weighted by a chosen positive definite operator and defined in a suitable oo-dimensional functional space, containing the density matrix elements. The cosine-like similarity index between two molecules / and J may be constructed as z,/Q)

(61)

This C-class QSI, for any pair of compared systems, can have any value belonging to the [0,1] interval. A 0 figure corresponds to a total dissimilarity, whereas a value of 1 indicates complete similarity of the two compared objects. These values depend on the similarity matrix elements, associated with both molecules. The two cited extreme situations have a geometrical meaning too, corresponding to a couple of orthogonal or collinear density matrix elements, respectively. Some authors refer to this index as the Carbo similarity index. Indices of this kind are not unique to quantum similarity but actually have become extended to other scientific areas. Take for example crystallography, where recently the cosine-like form 61 has been described.^^ Cosine-Like Indices and Multiple QSM

How must one proceed to extend the concept outlined before, when three or more systems are to be explored? It seems that the natural way to extend the cosine-like index, for three molecular structures simultaneously, can be expressed as ^IJK

~ ^IJK\^III^JJJ^KKK)

^

^

Thus, when I = J = K, and someone tries to evaluate these general ideas, the plain C^y^ index goes to 1. So, defining the Mh order QSM as the integral

Quantum Similarity

23

.nW = J

dv

(63)

V J where I = (Ij, I 2 , . . . , I^y) is a set of indices associated with A^ QO; then an Mh order QSI may be defined as f N

\

-l/N

c
(64)

k=i

z^^^ (4I) is the Mh order QS-SM, associated with the ^ QO: (65)

B. D-Class Dissimilarity Indices A distance similarity index, forming a distance-hke class, can also be attached to the construction of a dissimilarity index. The mathematical interpretation of this alternative manipulation of the QSM matrix elements is such that it represents a distance, defined again in some suitable 00-dimensional space, between two density matrix elements. The dissimilarity index may be defined as some kind of generalized Euclidean distance. The simplest way to obtain this index is via the classical formula D,j(Q) = ZjjiQ) + Zjj(Q)-2z^j(Q)

(66)

where the dissimilarity index values can be found, in this case, lying in the interval [0, +00]. The lowest value corresponds this time to complete similarity between compared objects, whereas the higher the value of the index, the more dissimilar the objects are. It must be mentioned here that QSI provides SL fuzzy relationship to the connections between the elements of QOS. See Ref. 36 for more details. This relationship can be extended to triple-density QSM involving three or more quantum objects. In the present discussion, all possible descriptions of similarity indices, computed by means of QSM manipulations, will necessarily belong to one or another of these two classes. A generalization of the initial QSI definition couple follows. C. Generalized QSI Both previous QSI definifions, as they appear in Eqs. 61 and 66, are associated with the oldest QSI described in this context: our originally described cosine and Euclidean distance indices, respecfively.^ But other choices can be considered. Once one of the QSI entities has been computed, the other can be obtained by means

24

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

of a new transformation over the first results. There are many alternatives for the generalized definitions of QSI.^^ Some assorted choices inside the two index classes are given here. D-Class Generalized QSI

A generalized Euclidean distance-like SI may be described by using the following form, depending on two parameters: D,j{k,x,Q) = I (zjjiQ) + Zjj{Q)) -xzjjiQ),

X e [OM

^^^^

which for /: = jc = 2 reduces to a Euclidean distance index, as in Eq. 66. Another D-class index Sjj can be defined from the cosine-like one of Eq. 61 in the following way:

jjmm

(68)

using the variation range of the cosine-like index and the definition of entropy. C'Class Generalized QSI: The Girona Index Other C-class QSI have been defined in the literature but they are not as easy to interpret as those defined initially here. For example, the Hodgkin-Richards^^ or Tanimoto indices can be cast into a generalized formula: ZjXQ)

(69)

where the D-class index in the denominator corresponds to the general definition associated with expression 67. Also, when k = 2, with ;c = 0 the Hodgkin-Richards index is reproduced and in the same manner, when ;c = 1 the Tanimoto index is obtained. Thus, the Girona index contains many possibilities and suggests that the contained indices may be related to the initial cosine-like form. D. Transformations between QSI

From the previous discussion on the many possible definitions of QSI, various relationships between the described indices can be presented, which permit transformation of C-class into D-class indices and vice versa. Only two examples of each transformation family are given.

Quantum Similarity

25

D-Class into C-Class

Knowing a set of D-class indices, it is easy to obtain a new set of C-class ones using one of the following recipes:

1. ^^^Cjj=l-iSinh{Djj). 2. ( % , = (l+(D,,)V/^;/7>0. C-Class into D-Class

Defining the scale factor K, the following relationships may also be used to transform C-class into D-class indices: 1. ^^%j = K^TCOs(Cjjy

2. ^'^Dj, =

K(Cj,r\l-Cj,).

In this way, a set of one class of indices can be transformed into the complementary class without problems. This situation allows great freedom at the moment of using QSI sets to obtain information, derived from the molecular point-cloud sets, which can be correlated with the characteristic properties of the QOS. E. A Discussion on Discrete Representation Indices

When analyzing the Carbo index construction, as written in Eq. 61, one can see easily the origin of the expression. This source appears when writing the scalar product of two density functions {p^, p^}, as is usual, and connecting such a product with the functions QSM, as shown below: ^AB = 9APB = 9A 9 B cos(a^) = Z^Z^B cos(a^)

(70)

in terms of the two function modules and the cosine of the angle subtended by both functions in ©o-dimensional space. Equation 70 furnishes the elements of Eq. 61, and as a consequence the Carbo index is nothing more than the cosine of the angle between the pair of involved density functions. An interesting situation appears when the discretization ideas, involved in the QSM representation, are used in such a simplified QOS, formed by a pair of objects only. In this case, even in a very general representation, using the techniques outlined in Section IV, the reduced similarity matrix Z could be brought to a simple (2x2) symmetric matrix form, whose diagonal elements can be associated to the squared function modules {z^, z^^}, and the off-diagonal element is naturally attached to the scalar product {z^}. In this naive context, the discrete representation of both functions has to be forcibly two-dimensional, with the representation vectors defined by the similarity matrix columns as

26

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO ^AB

PA^^A

=

(71)

^BB

^'AB

V ""J

Then, the vector pair {z^, z^} may be employed too, to compute QSI of the kind discussed in this section. For instance, the Carbo index can be expressed in this two-dimensional context as T

KsR =

T

r

= cosCa,)

(72)

Obviously enough, it is hard to imagine that the resulting angles in expressions 61 and 72 will be the same. However, some relationships could be obtained, developing the scalar products between the column vectors in Eq. 72 above, which on some straightforward manipulation can be written as ^2 _

1

_DetZ

^ ^ AA5

.__.

^A^B

In this equation, the ratio x ^ behaves like a D-class similarity index. The obtained ratio, x ^ , between the determinant of the (2x2) similarity matrix Z, DetlZI, and the scalar product of the two constituent columns, could also be employed in the study of QOS possessing a larger than two cardinality. Thus, Eq. 73 may be considered the definition of a new possible similarity index. The interesting dependence between K^ and C^^ has been given in detail in previous papers^^^'^^ and the final relationship between the two related angles {CC2, cc^}, can be written in a very simple way in terms of the respective tangents «^z, = t g ( a 2 ) t g - V j

C^^)

and the ratio co^ is nothing but half of a C-class index defined formerly by Hodgkin and Richards, as an alternative to the Carbo index,^^ which can be written as ZAB ^AB

(75)

=

^AA "^ ^BB

As can be deduced from the discussion and results of this section, the image of the nature and significance of QSM and QSI appear more clearly drawn. QSM can be defined univocally, once the density functions for a given QOS and a suitable weighting operator are chosen. QSI instead, being simple manipulations of QSM, can be redefined in many different ways, and are far from being uniquely defined. Even so, this late discussion on the relationships, between the subtended angles of the density functions and their discrete projections seems to indicate that the Carbo index can be considered as a general background reference to all possible definitions of QSI.

Quantum Similarity

VI.

27

THE O R I G I N OF QSAR A N D RELATED PROBLEMS

The usual problem in chemistry, since the nineteenth century, has been to obtain some kind of relationship between the chemical properties or the biological activity of molecular structures, and some set of parameters, easily available from theoretical considerations about the nature of the involved molecules. These empirical relationships have been called quantitative structure-property or -activity relationships (QSPR or QSAR). Curiously enough, although being employed for more than a century, no theoretical general reasoning has yet been adduced to explain scientifically, not just statistically, the success of QSPR. Recently, with a purely theoretical paper,^^ our laboratory has begun the publication of some contributions^^ to prove, by means of the QSM body developed above, that this kind of theoretical background can be useful to explain the origin of the QSPR achievement in chemical problems. Furthermore, the sources of classical QSAR procedures may be attached to a primary discrete representation of molecular structures, obtained by means of the approximate estimation of projections 78 and 79, defined below, implicit in the calculation of MQSM. This section contains a theoretical discussion of the problem. First, the connection between QSPR computational background structure and the present MQSM analysis will be addressed, and the fundamental relationship will be shown to be again the n-dimensional discrete representation of a molecule. Second, the relationship between convex sets and QSPR will be examined. Third, the link between QSM and molecular topology will be shown, this time using QS-SM. Finally, we will introduce a new collection of molecular topological indices,^^ based on MQSM considerations, and bearing somehow the molecular three-dimensional information. A. The Success of QSAR

Even if the expectation value, , of a given quantum operator Q needs to be stated as in expression 40, it can certainly be formally expressed by means of the following integral: <(0> = JQ(r)p(ryr

C^^)

using any kind of appropriate density function: p(r). The previous discussion on the possible matrix representation of density functions can be resumed by means of the associations defined in Eq. 53 or 54. From there one can safely suppose the existence of an already constructed, ad hoc, n-dimensional vector space O^. The space will contain all of the matrix representations of the operators, like the density functions, associated with the QOS oo-dimensional representation.

28

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

The QOS representation could be associated in turn with some kind of subset P, belonging to some sort of ©o-dimensional vector space O^, where also all operators depending on the same particle coordinate vector, r, can be supposed to belong. That is, Q(r)GO^Ap(r)GPcO^

(77)

Then, once admitting the partnership proposed in Eq. 77, Eq. 76 can be viewed as some sort of scalar product between two elements of the O^ space. Precisely, it is the proposed interpretation above that differential operators must possess expectation values, defined in the way quantum-mechanical kinetic energy is, just to preserve the symmetrical structure of the scalar products between the involved operators. Projection of the O^ space vectors into some finite-dimensional one may be considered the consequence of some linear transformation T\ O^ -^ O^. Then, not only can one consider the density functions projected into the space O^, but one may consider a general situation, which can be described by means of the following relationship: 3 7 : O^ -> 0 „ A VQ(r) G O^ - ^ 3w G O,

(78)

Then, this possibility will produce a similar relationship between scalar products in both spaces, that is, VQ, p G O^ : Q p = J Q(r)p(r)Jr=» T{Q) = w A r(p) = z - ^ f2 p = w^z

(79)

and one can consider that the expectation value of the operator Q can be expressed in the discrete n-dimensional framework by the scalar product of its own projection and that of the density function, respectively. Thus, -w^z

(80)

When considering the QOS 5 = {5^} associated with the density representation set D = {p^} and the attached QO n-dimensional representation Z = {z^}, one can express the expectation value of Q with respect to each QOS element by the expression ; = w \ : V 5 ; G 5

(81)

QSPR produces a linear equation, which may be written by means of a scalar product like 7i = w^p

(82)

Quantum Similarity

29

where n stands for some molecular property, w is an n-dimensional vector, whose coefficients must be determined, and p is a molecular structure vector, whose elements are attached by some computational relationship to the described molecule. Their elements may be called molecular descriptors. The problem is defined by the knowledge of a set of property values {71^} and a set of molecular descriptor vectors {p^}. Equation 82, in this situation, permits setting up a linear system, which, on application of the well-known least-squares technique, can produce the coefficient vector, w, as a result. In the MQSM framework, everything looks the same, as one can see when inspecting Eq. 81; the molecular descriptor vectors can be safely taken by the set Z= {Zj}, formed by the MQSM in the way discussed in Section IV. The differences between MQSM techniques and the usual empirical QSPR procedures can be explained with the following two considerations: 1. In the MQSM framework the theoretical molecular descriptors can be attached to nonempirical values, whose origin can be traced to quantummechanical principles. The procedures to obtain them are well established for any molecular structure: The conditions and restrictions for the calculation of the parameter set are those related to MQSM integral evaluation. 2. No one has yet attached in the QSPR framework any background significance to the coefficient vector values, w, except on rare occasions.^^ Contrarily, from the reasoning given at the beginning of this section, the vector w always has a possible meaning, produced by the plausible formal connection between Eqs. 80 and 82. w can be considered the discrete representation of some unknown operator, related to the studied molecular property 71, which, in turn, can be seen, from the quantum-mechanical side associated with the MQSM point of view, as an expectation value of some molecular observable. Thus, owing to these considerations, perhaps the QSPR success over the past 150 years, as a body of empirical procedures to obtain molecular structure and molecular property relationships, may be explained in the ground of MQSM theory. Classical molecular descriptor values, which can be collected into some ^-dimensional vector, p, being a discrete representation of a given molecular structure, can be considered in whole as an empirical attempt to construct an approximate theoretical MQSM vector, z. B. Convex Sets and QSPR

A very interesting and far-reaching question may be related to the construction of composite definite positive operators to be used in QSM calculations. Suppose a set of definite positive operators W= {co (rj, r2)} and construct with them the operator

30

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

Q{r,.r^) =

'Za^(^^{r,,r^)

(83)

p

Employ a set of real valued weights a= {a } submitted to the restrictions 0
(84)

p

Then, the new operator Q is a definite positive one too. Using the composite operator, a QSM involving a density function pair {p^, p^} can be defined in the usual way:

p

p

In this way, a possible similarity matrix, obtained with the elements z^g{^), can also be written as a superposition of similarity matrix elements such as Z(Q) = X ^p^%)

^^^^

p

Owing to the constraints, affecting the superposition weights, it is easy to see that the similarity matrix Z(Q) is a convex set, defined in the matrix vector space of the appropriate dimension. As a consequence of the convex set structure of the similarity matrix defined previously, one can also transfer this mathematical feature to the discrete representation of QOS. Indeed, using the /th column of the similarity matrix z- e Z(Q) and calling z j e Z(co ) the same column obtained from the similarity matrix, associated with the pih operator, it can be easily seen that every discrete representation is also a convex set: (87) p p\i p

Using the general form of a convex set discrete representation of any QO, that is dropping the QOS identification index, for simplicity,

z = y az JLI

P P

p

and, thus, preparing a usual QSPR of the sort, over the convex set structure

(88)

Quantum Similarity

31

ZJ

P

P

P

then the least-squares quadratic error function will depend on two sets of parameters: {a, w}. The parameter set w corresponds to the conventional QSPR leastsquares coefficients, obtained by means of the matrix algorithm w = Z(Qr^7i

(90)

while the convex set coefficients a shall be computed concurrently, although using a procedure exactly the same as in the ASA case, as discussed in Section III.D. However, this computational situation may provide the possibility to accurately describe the QOS and at the same time tune up this QSPR description by means of the convex set coefficients. C. MQSM and Molecular Topology To the MQSM discrete molecular representation can also be added another empirical connection, this time with molecular topology. Topological matrices (TM), known since the introduction of the Hiickel method,^^ can be associated with the numerical representations of molecular graphs, and thus, following the line of thought of Section VI.A, TM can be considered in several aspects as a procedure to attain the discretization of the molecular information. This fact is obvious when TM are used as the background information for the computation of molecular descriptors known as topological indices (TI).^^ Then, MQSM has to be connected somehow with TM, and in this way with the possibility to construct TI from MQSM taken as molecular descriptors, as discussed below. To obtain a first-level relationship between TM and MQSM, the explicit expression of the molecular quantum self-similarity measure (MQS-SM) can be used. From an ASA approach for a given molecule, as written in Eq. 16, the MQS-SM can be described by

^AA = J pA(^fdr= X X ^.^/(, = ^'"Zw

(91)

ieA jeA

where Z is the metric of the spherical basis set {l5.(r)F}(/ e A). To establish a better pictorial association between z^ and molecular TM, the MQS-SM integral in the equation above may be interpreted as being computed using the density matrix off-diagonal elements instead of the diagonal ones, as has been customarily done in the previous discussion. This means that one can write Eq. 91, in the following form, producing a Cioslowski-like^^ MQS-SM, 6 ^ : ^AA = l \

PAC^P ^2)PA(^V h)d^idv^

32

R. CARB6-DORCA, L. AMAT, E. BESALO, and M. LOBATO

ieA

jeA

ieA

jeA

^"^"^ ieA

w.WjSl = y^^S^^^yy

(92)

jeA

this result being equivalent to Eq. 91, and where S^^^ = {I5..P} is the matrix whose elements are the squared modules of the s-type function {5.(r)} metric matrix elements. Matrix S^^^ may be considered as a good approximation of the QSM metric matrix Z. So the approximate relationships, which can be found from this point on, can apply to any similarity metric matrix over the ASA s-type basis set. The principal characteristic of the metric matrix S is that some values (5.. = 1) when squared remain almost unaltered, while others (5.. = 0) become negligible. The elements o = 15..P =1 are to be associated with index pairs (ij) corresponding to neighbor molecular centers, while negligible elements will forcibly correspond to distant centered s-type functions. When only one s-type function is chosen for every molecular center, then it is immediate, assuming the previous considerations, to make the association: S^^^ = I + aX

(93)

where I is the unit matrix, a a constant value almost unity, and T the molecular TM obtained by means of QSM considerations. Using this approach, Eq. 92, or what is the same, Eq. 91 too, can be rewritten as ^A4 ~ ^^^ "^ aw^Tw

(94)

in this way it can be easily seen that MQS-SM are directly related to molecular topology. As 0 ^ can be considered the expectation value of the density function considered as an operator, then in this context the expectation value of any observable, expressed as the discrete n-dimensional image vector u, say, may be obtained as ^ = u^w + au^Tw

(95)

so molecular properties can be determined related to the TM. As the discrete observable vectors may be decomposed as a linear combination of the density vector coordinates w and an orthogonal complement vector v, say u = aw + Pv

(96)

Quantum Similarity

33

Then, it is also noteworthy that observable expectation values may be related to QMS-SM, because substituting Eq. 96 into expression 95, the following approximation is obtained: <^>A == cce^ + Yv^Tw = a e ^ + Yco^

(97)

where y = Pa. Now, after inspecting Eq. 97 and using self-similarity, 9 ^ , like an available variable as the first term suggests, then 0)^ and thus <^>^ can be developed in a power series such as <^>A-S^P^

(98)

In the above series {a } are appropriate coefficients, depending on the operator and the nature of the MQS-SM. This possible feature is currently being investigated in our laboratory. Preliminary interesting relationships between self-similarity and conformational energy have already been published."^^ Equation 98 also suggests that nonlinear QSPR may be found in some particular cases. The successive powers of the MQS-SM could be substituted by multiple QS-SM if necessary. For example, using the equivalence

9^M = ^S = Jp.('-r'^r

(99)

the integral definition can be changed according to computational conveniences and the availability of the density matrix elements. D. MQSM Topological Indices

Due to the connection between MQSM and molecular topology as evidenced in the previous section, it can be assumed that the ASA metric matrices, involved in the computation of self-similarity measures, may act as good substitutes of TM. Two interesting points must be stressed with respect to this association: 1. There could be several metric matrices accessible, which can be used for the purpose of constructing a quantum similarity TM (QSTM). Depending on the weighting operator or the density matrix element chosen to compute the metric MQSM framework there will appear to be a somewhat different matrix and thus another kind of QSTM will be available. 2. Any chosen QSTM will bear in a natural way information on the three-dimensional structure of the molecule represented in such a manner. The drawback of this approach lies in the necessary use of molecular atomic coordinates to compute the QSTM metric elements, in front of classical TM, where only a molecular graph is needed.

34

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

Once admitting the possibility of using QSTM to compute molecular quantum topological indices (MQTI), they can then be computed in the same way as is customary in the TI calculations. Topology and Molecular Similarity Matrix Elements The traditional description of a molecule, by means of hydrogen-suppressed graphs, has been widely used in QSPR studies, giving good results when correlating numerical quantities derived from them with respect to physicochemical and biological properties."*^ The classical topological representation of a molecule by a graph can be coded by means of its TM, T, whose elements are 1 if the associated atoms can be considered connected (bonded) and 0 otherwise, as previously mentioned. Thus, only links between bonded atoms are taken into account when constructing the classical TM. This matrix is used in the definition of TI, which are taken afterwards as molecular descriptors, susceptible to being correlated with properties. The TI most widely applied and successful in QSAR studies are the Wiener and Randi6 approaches on TI definitions. Also, the subset of generalized connectivity indices as defined by Kier and Hall^^ has proved to be extremely successful. However, it is often found that the three-dimensional molecular structure can be of key importance for describing some of the molecular properties and in this manner forcing more accurate QSAR results. This may be so, because part of the spatial information of the molecule can be lost when a topological matrix is used. To include this information when defining molecular descriptors, Randic has introduced a geometrical distance matrix representation as an ersatz input data set^^ related to this question. Among the TI that can be defined and may be relevant in this part of the present study are the well-known Wiener path number, Randic, Schultz, and Balaban indices, Harary number, and generalized connectivity indices '"x^. All of them are widely discussed in Ref. 44. To compute TI, it is just necessary to define some auxiliary matrices as shown in Table 1. The TI definition and the relationship with the matrix elements, shown in Table 1 is summarized in Table 2.

Table 1, Definition of Matrices Used in TI Calculations^ Matrix

Elements Definition

^ in^ n)

Classical topological matrix: J _ j1 if atoms / and j are bonded u [0 otherwise

D (n X n) V in)

Djj'. topological length of shortest path from atom / to atom j Vf. sum of entries in the /th row or column of matrix T

Note: ^n is the number of atoms.

Quantum Similarity

35

Table 2. Definition of Several Assorted Topological Indices Index

Deiinuton

Wiener path number

Randi6 index

n

n

n

n

^"Z S "V^'^y y>'

Schultz index (molecular topological index)

Harary number

n

M7/=5;[V(T+D)]y n

n

^-

y>'

Balaban index

n

H+1 f

n

^ (D),
ji: number of cycles ne: number of edges {P\\ sum of distances from vertex / Generalized connectivity indices Order: m, type: f

"m m f l

s=l

/=1

n^: number of connected subgraphs of type t

To obtain the new QSTI, associated with the presentframework,a set of matrices related to the one used in QSM is evaluated for each molecule and introduced as the background computation of the indices. These new indices do not derive from the classical TM elements, but they can arise from the use of the new TM set associated with QSM, the QSTM—using the QSTM elements afterwards, and dealing with the same computational structure as in classical TI evaluation over TM. To compute MQTI, it is necessary to know some molecular geometry data and use a simple set of atomic basis functions, sufficient to describe in a very schematic way the molecular density. The geometry can be obtained by means of either ah initio or semiempirical methodology or by any other reliable procedure or source. Once the molecular geometry is known, the procedure associates a basis function with each atom. The active basis set can be defined using a normalized Is GTO

36

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

function {^^(r - R^)} for atom a, with a scale factor ^^. In practice, these functions can be constructed transforming single-zeta STO exponents, derived from atomic SCF energy optimization,"^^ to GTO function exponents by minimizing the integral quadratic error between both function types. But other approaches may be imagined and are currently being tested in our laboratory. The similarity matrix used in computation of MQTI, which can be used to substitute the TM and the topological distance matrix as used in classical index evaluation, has to be defined as previously discussed. It must be noted that, while MQSM concepts are usually defined between molecular pairs, within the MQTI framework every matrix is defined using only a unique molecular structure, as in self-similarity calculations, according to the analysis carried out before. In the following definitions, a and b are atoms belonging to molecule A. Integrals are computed between two normalized Is GTO functions corresponding to atoms a and b with centers R^ and R^, respectively. The most relevant QSTM elements are computed as follows: Overlap-like S(Q): Each element is computed taking into account the structure of the metric matrix with a positive definite operator, as the general integral:

The use of such an integral form may become justified once one is aware of the simplified structure, which is associated with the present discussion, where only one Is GTO function is ascribed to a given atom. Because spherical GTO can be taken as probability density functions directly, Eq. 100 corresponds to the next expression below: Coulomb-like E(Q): Each element £^^ is defined as the two-electron Coulomb repulsion integral:

Both integrals above, leading to the TM elements, can be described in a unique form, using a new more general expression: Gab = \ \ So (ri - R / i 2 ( r p T^g,{T^ - R,rdr,dr^

(102)

where p = 1 orp = 2 produces the pair of previous integrals respectively. Moreover, if /? = 1 and Q = 6(ri - r2), one has an overlap integral and the corresponding overlap-like similarity integral is associated with the same operator but taking p = 2. A Coulomb repulsion matrix element is obtained with /? = 1 or 2 but Q = IFJ - r2l - 1, while a gravitational matrix element will be computed using the operator Q = Ir. - r,!"^.

Quantum Similarity

37

Exchange-like 0(Q): Besides the kind of definitions described in the general context of the last equation above, each element of the topological matrix O^ can be defined as the integral involving an exchange-like structure:

where depending on the operator's choice, that is, when taken in the same order as before, the exchange-like measures as Cioslowski overlap-like, exchange repulsion, and gravitational exchange integrals are reproduced. Also, some transformations can be applied to the above-defined matrix elements, to avoid the presence of large values. The matrix transforms can be chosen to constrain all of its values within the interval between zero and one. Diagonal elements are defined as zeros following the custom in the classical TM. The matrix transforms of electron repulsion, gravitational, and overlap QSTM lead to three new matrices, where each transformed matrix C(A) is defined from the original matrix A elements, using a Carbo index construction rule: CiAJ-

Kb

(104)

^a^. bb

Thus, several, immediate QSTM can be defined in this way: topological, overlap, Cioslowski-like overlap, electron repulsion, electron exchange, gravitational, gravitational exchange, and also the matrix transforms of these matrices can contribute quite generally to the TM construction. Finally, MQTI can be obtained from the same classical formulas used in the TI computation, as shown in Table 2, but using any of the QSTM elements instead. Practical Computation and Use of MQTI In a practical computation, all MQTI obtained from the SM or their transforms, as defined above, are to be evaluated. The generalized connectivity indices are computed from order m = 0 up to 6th order. This gives quite a large set of indices, which can be collected as molecular descriptors in an appropriate matrix. This matrix is manipulated in the same manner as can be done with the MQSM matrix Z. The practical MQTI selection works as follows: The best individual index is chosen; then, all possible pairs are generated and the best correlation is saved. Later on, three index subsets are obtained and the best is saved, and so on. In this context, best means a multilinear regression model achieving higher cross-validated r^ (Q^), which is the parameter that determines the model's prediction accuracy."^^ The whole procedure is performed applying a nested summation symbol^"* parallelizable algorithm. The algorithm was coded into a Fortran 90 program: NESTED-MLR, developed in our laboratory."*^ Results will be published elsewhere.

38

R. CARBO-DORCA, L. AMAT, E. BESALU, and M. LOBATO

VII. SIMILARITY OVER ENERGY SURFACES Until now the principal characteristic of the functions attached to QOS elements was on their positive definition, being associated to normalizable probability distributions. But in quantum chemistry, under Born-Oppenheimer approximation, there are several function types, mainly associated with electronic energy variation, which are not positive definite, but nondefinite functions. Indeed, electronic energy surfaces (EES)^^ may have either positive or negative zones: The typical most available and widespread such function is the molecular electrostatic potential (MEP), first defined by Bonnacorsi et al.'^^ Although the electronic part of the MEP has been inserted in the MQSM theoretical structure,"^^ the whole MEP function is not definite. MEP functions even present evident discontinuities, making similarity measure integrals divergent, in the same way as the simple Coulomb potential function integral in the interval [0, +00] diverges. Thus, comparisons of nondefinite functions of quantum theoretical origin are not yet included in a, mathematically speaking, sound manner within the QSM structure. This section will try to propose a possible way to carry out this task. The starting point of the possible QSM treatment over such functions is agreement that only positive definite probability functions can be safely and generally used to compute measure integrals. Once this circumstance is accepted, then the problem is to design a procedure to transform such functions into adequate measurable functions. Fortunately, the answer is already here in the form of statistical mechanics probability distributions. This simple statement will be the starting point of the present discussion. A. Boltzmann Distributions and Boltzmann Similarity Measures

Statistical mechanics probability distributions, such as the Boltzmann Distribution^^ (BD), or partition functions, are ideal constructions to relate EES with a probability distribution capable of being integrated to form some kind of similarity measure integral. The procedure will demand here compulsively to transform the original function into a definite positive BD. Once obtained, the new function is the procedure will be the same as in the usual QSM methodology. Suppose that a QOS, 5 = {5'^}, is known and that a set of EES are associated with each object in 5: £ = {efR)], where the coordinate vector, R, defines the variables on which every surface depends. The limitation here is such that the vector R must be uniformly and coherentiy defined for every function in the set E, in the same way it was when discussing QSM over density functions. It is possible to find a transformation such as a new set of functions B = {P/R)}, corresponding to a BD of every function in E, that is, \/sjeS--^

3efR)

SEA

0(^/R))

= P/R)

EB

(105)

Quantum Similarity

39

The transformation O leading to the BD can be defined as in the usual theoretical definition: 0(^/R)) = (3/R) = exp(-^/R)(/:rr^)

(106)

where k is the Boltzmann constant and T the temperature in K. The BD must be normalizable in the sense that the integral over the volume V, containing the variation of R, converges:

and 0(R) is a suitable operator to force, if necessary, the integral to converge. Then, the BD set 5 = {P^}, can be normalized using the above integral, that is, Pf(R) = 'a7^P/R)

(108)

the original or the new normalized BD set B may, then, be used to compute QSM in the same way as in the usual electronic density distribution framework. That is, a Boltzmann similarity measure (BSM) involving two QO, {A, fi}, corresponds to the integral \x^^(Q) = J P^(R)Q(R)P^(R)^

(109)

just as in Eq. 1, Q(R) is a weighting operator and the densities used here are the associated BD (P^, P^}. Thus, everything said in the realm of electronic density distributions can be repeated here. A paper dealing with this kind of problem has just been published. ^^ Within it detailed examples are given, and will not be repeated here. That discussion has led to the definition, besides the pure QSM and BSM, of mixed quantum and Boltzmann similarity measures, where one of the probability density distributions appearing in the measure integral could be of quantum and the other one of Boltzmann origin. B. General Distributions and Similarity Measures

Other transformations can be envisaged in the light of the BD form shown in Eq. 106, so as to obtain positive definite functional forms to be compared. An immediate one could be imagined as some sort of Gaussian distribution (GD) definition, although the interesting statistical mechanics connection of the previous section will be in this case lost. That is, the BD transformation definition may be changed by the alternative GD expression: r(^.(R)) = Y/(R) = cxp(-e^R)(kT)-'

^)

d 10)

40

R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

where all of the previous definitions hold, but the Boltzmann exponential has been transformed into a GD. The set of GD, G = {Y/}, attached in this way to the QOS, 5, can be used in the usual way to produce any kind of similarity measure. The discussion above carried out on the possible QSM methodological extensions, using, for example, BD or GD, to obtain other plausible molecular similarity measures, can be imagined even more extensive. Some general aspects of the question were discussed early on,^^ and in Section IV.A, with respect to the fact that, prior to the computation of any QSM, the density matrix elements may be, in turn, transformed. In this manner, the similarity measure will contain these density transforms, instead of the raw density distributions. Allan and Cooper employed a related procedure when they used a well-known Fourier transform in so as to work the QSM over the momentum space representation.^^ The situation can be finally generalized, because the construction of a general transformation rule, over any quantum distribution function descriptor (QDFD) set A = {X/R)}, having the positive definite structure that has been associated with such functions, can be formally written as the integral transform: cp/r) = J 0(R, rmX^R))dR

d ^D

where (A,/R)) can be any previous suitable function transformation and Q(R, r) a convenient operator, the above integral transform produces over the original set A a new QDFD set F = {cp.(r)}, which can be used afterwards in the calculation of QSM integrals.

VIII. CONCLUSIONS It can be agreed at this stage of the MQSM question that a vast and adapted to almost chemical problem set of similarity measures can be computed for any chemical system. So, a complete collection of tools to measure any chemical similarity aspect is set. The description possibilities of these procedures, being of quantum mechanical origin, and so generally defined, encompass the territory of chemistry and can even be thought to extend to other areas dealing with Quantum Objects, as atomic nuclei for instance, where density distributions can also be easily defined.

ACKNOWLEDGMENTS This work has been partially sponsored by project SAP 96-0158 of the CICYT L. A. is a fellow of Ministerio de Educacion y Cultura and M. L. benefits from a University of Girona grant. The authors are grateful for lively discussions with Dr. P. Constans, which enlightened many aspects of this work.

Quantum Similarity

41 REFERENCES

1. Carbo, R.; Amau, M.; Leyda, L. Int. J. Quantum Chem. 1980, 77, 1185-1189. 2. See, for example: (a) Carbo, R.; Calabuig, B. Comput. Phys. Commun. 1989, 55, 117-126. (b) Carbo, R.; Calabuig, B. J. Chem. Inf. Comput. Sci. 1992, 52, 600-606. (c) Carb6, R.; Calabuig, B. J. Mol. Struct. (Theochem) 1992,254,517-531. (d) Carb6, R.; Calabuig B. In Computational Chemistry: Structure, Interactions and Reactivity. Vol. A; Fraga, S., Ed.; Elsevier: Amsterdam, 1992, pp. 300-324. 3. Carb6, R.; Calabuig, B. In Concepts and Applications of Molecular Similarity, Johnson, M.A.; Maggiora, G., Ed.; Wiley: New York, 1990, pp. 147-171. 4. Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches; Carbo, R. Ed.; Kluwer: Dordrecht, 1995. 5. Advances in Molecular Similarity, Carbo-Dorca, R.; Mezey, RG. Eds. JAI Press: Greenwich, CT, 1996, Vol. 1. 6. See, for example: (a) Concepts and Applications of Molecular Similarity, Johnson, M.A.; Maggiora, G., Eds.; Wiley: New York, 1990. (b) Molecular Similarity. Sen, K., Eds.; Topics in Current Chemistry, Vols. 173 & 174; Springer Verlag: Berlin, 1995. 7. Carbo, R.; Besalii, E. In Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches; Carbo, R. Ed.; Kluwer: Dordrecht, 1995, pp.3-30. 8. See, for example: (a) Carb6, R.; Besalu, E.; Amat, L.; Fradera, X. J. MatkChem. 1995, 75, 237-246. (b) Carb6, R.; Besalu, E. Afinidad 1996, 53, 77-79. 9. See, for example: (a) Sola, M.; Mestres, J.; Carbo, R.; Duran, M. J. Am. Chem. Soc. 1994,116, 5909-5915. (b) Sola, M.; Mestres, J.; Carbo, R.; Duran, M. J. Chem. Inf Comput. Sci. 1994,34, 1047-1053. (c) Mestres, J.; Sola, M.; Carbo, R.; Luque, F J.; Orozco, M. J. Phys. Chem. 1996, 100,606-610. (d) Fradera, X.; Amat, L.; Besalu, E.; Carbo-Dorca, R. Quant. Struct.-Act. Relat. 1997,16, 25-32. 10. See, for example: (a) Constans, R; Carbo, R. J. Chem. Inf Comput. Sci. 1995, 35, 1046-1053. (b) Constans, R; Amat, L.; Fradera, X.; Carbo-Dorca, R. \n Advances in Molecular Similarity; Carbo-Dorca, R.; Mezey, R G., Eds.; JAI Press: Greenwich CT, 1996, Vol. 1, pp. 187-211. 11. See, for example: (a) Carbo-Dorca, R.; Besalu, E.; Amat, L.; Fradera, X. In Carbo-Dorca, R.; Mezey, P. G., Eds.; Advances in Molecular Similarity; JAI Press: Greenwich, CT, 1996, Vol. 1, pp. 1-42. (b) Amat, L.; Fradera, X.; Carbo R. Scientia Gerundensis 1996, 22, 97-107. 12. See, for example: (a) Constans, P.; Amat, L.; Carbo-Dorca, R. J. Comput. Chem. 1997, 18, 826-846. (b) Amat, L.; Carbo, R.; Constans, P Sci. Gerundensis 1996, 22, 109-121. 13. See, for example: (a) Carbo, R.; Calabuig, B. Int. J. Quantum Chem. 1992, 42, 1681-1693. (b) Carbo, R.; Calabuig, B. Int. J. Quantum Chem. 1992,42, 1695-1709. 14. Bourbaki, N. Theorie des Ensembles. Elements de Mathematique; Hermann: Paris, 1979. Carbo, R.; Besalu, E.; Amat, L.; Fradera, X. J. Math. Chem. 1996,19, 47-56. 16. See for example: Kier, L.B.; Hall, L.H. Molecular Connectivity and Drug Research; Academic Press: New York, 1976. 17. Carbo-Dorca, R.; Besalu, E. J. Math Chem. in press. See also IQC Technical Report IT-IQC-011996. 18. Arsenin, V. Y Basic Equations and Special Functions of Mathematical Physics; Iliffe Books: London, 1968. " 19. See, for a computational formula over GTO's: Matsuoka, O. Int. J. Quantum Chem. 1973, 7, 365-381. 20. Bethe, H. A.; Salpeter, E. E. Quantum Mechanics of One- and Two-Electron Systems; SpringerVerlag: Berlin, 1957. 21. See, for example: Carbo, R.; Besalu, E.; Calabuig, B.; Vera, L. Adv. Quantum Chem. 1994,25, 253-313. 22. Carbo, R.; Calabuig, B.; Besalu, E.; Martinez, A. Mol. Eng. 1992, 2, 43-64.

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R. CARB6-DORCA, L. AMAT, E. BESALU, and M. LOBATO

23. See, for example: (a) Carb6, R.; Domingo, L.; Peris, J. J. Adv. Quantum Chem. 1982, 75, 215-265. (b) Carbo, R.; Mir6, J.; Domingo, L.; Novoa, J. J. Adv. Quantum Chem. 1989, 20, 375-441. (c) Carbo, R.; Domingo, L.; Peris, J.J.; Novoa, J. J. J. Mol. Struct. 1983, 95, 15-33. (d) Carb6, R.; Calabuig, B. Comput. Phys. Commun. 1989, 52, 345-354. 24. Carb(5, R.; Molino, L.; Calabuig, B. J. Comput. Chem. 1992,13,155-159. 25. Mestres, J.; Sola, M.; Duran, M.; Carbo, R. J. Comput. Chem. 1994, 75 ,1113-1120. 26. Pople, J.A.; Beveridge, D.L. Approximate Molecular Orbital Theory; McGraw-Hill: New York, 1970. 27. Cioslowsky, J.; Piskorz, P J. Chem. Phys. 1997,106, 3607-3612. 28. See for a detailed analysis of EES: Mezey, P.G. Potential Energy Hypersurfaces. Studies in Physical and Theoretical Chemistry, Vol. 53; Elsevier: Amsterdam, 1987. 29. Jacobi, C. G. J. J. Peine Angew. Math. 1846, 30, 51-94. 30. GATOMIC Program, Amat, L.; Carb6-Dorca, R. Institute of Computational Chemistry, University of Girona, 1997. 31. See for example: Schmidt, M. W.; Ruedenberg, K. J. Chem. Phys. 1979, 71, 3951-3962. 32. See for example: Carbo, R.; Riera, J. M. A General SCF Theory. Lecture Notes in Chemistry,Vol. 5; Springer-Veriag: Beriin, 1978. 33. See for example: Wilkinson, J. H.; Reinsch, C. Linear Algebra; Springer- Veriag: BerUn, 1971. 34. See, for example: (a) Carb6, R.; Besalu, R. J. Math. Chem. 1993,13, 331-342. (b) Besalu, E.; Carbd R. In Strategies and Applications in Quantum Chemistry: from Astrophysics to Molecular Engineering; Defranceschi, M.; Ellinger, Y, Eds.; Kluwer: Dordrecht, 1996, pp. 229-248. (c) Besalu, E.; Carb6, R. / Math. Chem. 1995, 78, 37-72. 35. Lunin, V. Y; Lunina, N. L. Acta Crystallogr 1996, A52, 365-368. 36. See for example: (a) Carbo, R.; Domingo, L. Int. J. Quantum. Chem. 1987, 23, 517-545. (b) Carbo, R. Sci. Gerundensis 1987,13, 177-184. 37. Hodgkin, E. E.; Richards, W. G. Int. J. Quantum Chem. 1987,14,105-110. 38. See for example: Carb6, R.; Martin, M.; Pons, V. Afinidad 1977, 34, 348-353. 39. Coulson, C. A.; Streitwieser, A., Jr. Dictionary of n-electron Calculations. Pergamon Press: Oxford, 1965. 40. Cioslowski, J.; Fleishmann, E. D. J. Am. Chem. Soc. 1991, 775, 64-67. 41. Oliva, J. M.; Carbo-Dorca, R.; Mestres, J. In Advances in Molecular Similarity; Carbo-Dorca, R.; Mezey, P G., Eds.; JAI Press: Greenwich, CT, 1996, Vol. 1, pp. 135-165. 42. See, for a review: Balaban, A. T. J. Chem. Inf Comput. Sci. 1995, 35, 339-350. 43. Randic, M.; Jerman-Blazic, B.; Trinajstic, N. Comput. Chem. 1990,14, 237-246. 44. Mihalic, Z.; Trinajstic, N. J. Chem. Educ. 1992, 69, 701-712. 45. Clementi, E.; Raimondi, D. L. J. Chem. Phys. 1963,38, 2686-2689. 46. Montgomery, D. C ; Peck, E. A. Introduction to Linear Regression Analysis. Wiley: New York 1992. 47. NESTED-MLR Program. Lobato, M.; Besalu, E. Institute of Computational Chemistry, University of Girona, 1996. 48. Bonnacorsi, R.; Scrocco, E.; Tomasi, J. J. Chem. Phys. 1970, 52, 5270. 49. See for example: (a) Carb6, R.; Sufie, E.; Lapena, F; Perez, J. J. Biol. Phys. 1986,14, 21-28. (b) Carb6, R.; Lapena, F.; Sufie, E. Afinidad 1986,45, 483-485. 50. See for example: Eyring, H.; Henderson, D.; Stover, B. J.; Eyring, E. M. Statistical Mechanics and Dynamics; Wiley: New York, 1964. 51. See, for a recent review: Allan, N. L.; Cooper, D.L. Top. Curr Chem. 1995,775,85-111.

FUZZY SETS AND BOOLEAN TAGGED SETS; VECTOR SEMISPACES AND CONVEX SETS; QUANTUM SIMILARITY MEASURES AND ASA DENSITY FUNCTIONS; DIAGONAL VECTOR SPACES AND QUANTUM CHEMISTRY

Ramon Carbo-Dorca

Abstract I. Introduction II. From Fuzzy Sets to Boolean Tagged Sets and Beyond A. Preliminary Definitions B. Tagged Classes C. Operations over Boolean Tagged Sets D. Boolean Tagged Vector Spaces E. Applications

Advances in Molecular Similarity, Volume 2, pages 43-72. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 43

44 44 46 46 47 48 48 49

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F. Extensions Tagged Sets, Convex Sets, and QSM A. QSM B. Convex Sets C. ASA D. Convex Operators E. Finely Tuned QSAR IV. On the Statistical Interpretation of Density Functions: Diagonal Vector Spaces and Related Problems A. The Nature of Discrete QO Representations B. The Structure of the Generating «-DimensionalVS: DVS C. Expression of the Density Functions and Other Problems V. Conclusions Acknowledgments References

III.

50 51 51 56 57 61 63 65 66 67 68 70 70 71

ABSTRACT Fuzzy set structure is analyzed from the point of view of a new definition: Boolean tagged sets, which are constructed as a straightforward generalization of the fuzzy set conception, but fully adapted to computational purposes. Boolean tagged sets are simply structured as sets whose members are tagged by a binary string, attached in turn to the vertices of a unit hypercube. Boolean tagged sets are designed to be useful in any field of applied mathematics, but they also can be seen as obviously prepared for chemical applications, associated with molecular information gathering and manipulation. Concepts and definitions of tagged and convex sets are applied afterwards, with the aim to discover a general mathemafical pattern enveloping the quantum similarity measures framework. As a consequence, several aspects of the quantum similarity theoretical structure become beautifully related to a mathemafical construction, adopting the form of some interwoven essential formalism, connecting quantum theory, molecular similarity, convexity, and tagged sets. Finally, following a statistical interpretation analysis, as a discrete probability distribution, of the atomic shell approximation linear coefficient set, a new description of quantum object sets is given, using a discrete representation, based on diagonal vector spaces. Previous definitions, involving essentially tagged sets and vector semispaces, are used.

I. INTRODUCTION The development of quantum similarity (QS) theoretical structure in recent times, as well as the applications, extensions, and prospective scenarios have been studied collectively in various books, published before the present volume.^'^

Fuzzy Sets and Boolean Tagged Sets

45

From the initial naive QS formulation, where first-order density functions were approximated by a CNDO-like expression,^ up to the actual atomic shell approximation (ASA) formalism,^^ there has not been any published attempt to produce a comprehensive study of the mathematical background attached to this field. Even more lacking appears to be a discussion, as complete as possible, of the possible influential consequences of the QS formalism, into discrete quantum-mechanical procedures, so heavily related to usual quantum chemistry computational practice. This work, which conglomerates a set of three separate studies,^^'^^'^^ shall be considered a primitive sketch to make up for the absence of a basic mathematical formalism in QS. The present study has three main sections, as briefly described below. In Section II an alternative definition of fuzzy sets will be given. This can be understood as a primitive resource, which can be employed to open the way to describe, in Section III, the needed mathematical formalism adapted to the purposes of quantum similarity measures (QSM). Section IV will analyze some consequences that the previous developments can produce, within the realm of discrete quantum chemistry. Then, this paper will be organized as follows: 1. Section II will be devoted to the definition and mathematical settings of an alternative, almost trivial, definition of the concept of fuzzy set. Boolean tagged sets will be introduced to the reader, as a natural way to describe molecular structures, by means of an appropriate mathematical formulation. To construct and discuss this kind of tagged set structures. Section II will be organized as follows. First, some definitions will be given, then operations on tagged sets and tagged vector spaces will be described, and finally some applications, sketching the main formalism provided later on, will be presented. 2. In Section III, the role that could be played by tagged set definition and convex sets together, will be studied within the framework of QSM and along the related techniques and algorithms. Therefore, this section will be structured as follows. First, the appropriate definitions allowing construction of the general theoretical background of QSM theory will be given. Then, some necessary notions associated with tagged set extensions and convex sets will be provided. The connection between both subjects will be discussed next. Finally, an application example involving quantitative structure-activity or -property relationships (QSAR, QSPR) will be presented. 3. Section IV provides an analysis of the nature of the linear coefficients appearing in some expressions of the first-order density function (DF), as in the ASA formalism. This preliminary discussion will reveal that the ASA coefficients can be associated with a discrete probability distribution. Finally, the need for diagonal vector spaces (DVS) will appear as a natural choice of the theory. This will permit obtaining in the discrete quantum theory framework the same formalism as in the continuous case.

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RAMON CARB6-DORCA

11. FROM FUZZY SETS TO BOOLEAN TAGGED SETS AND BEYOND Since Zadeh's original definition of the fuzzy set concept/ there has been a growing body of literature dealing with the theory and applications of this interesting generalization. Fuzzy sets are perfect tools to deal with the current everyday life situations, where no sharp cut between logical, {true, false}, sentence contents is present.^ The aim of this section is to describe an alternative general way of dealing with the same problems as fuzzy set theory does, while presenting an immediate connection with actual information structure and manipulation. The fact is that, owing to the current technological situation, information is gathered and manipulated within a pattern, which can be reduced to the binary equivalent of the logical contents, {1,0}, previously mentioned. Due to this situation, electromechanical computational devices deal with sequences of information units, bit strings, which in turn can easily be seen as the vertex set of unit n-dimensional cubes or hypercubes, whose origin can be arbitrarily placed at the vertex made of an appropriately dimensional {0} bit string. More than this, when one considers any kind of information structure, it can be realized, with a simple imagination effort, how this information can be transported into some vertex of the unit n-dimensional cube. Then, as a set of obvious examples, one can describe, using the vertex bit string of some appropriate dimension unit hypercube: the CPU contents of a given computer in a given time slice, or the information structure of a hard disk, or the contents of a stable CD-ROM, and so on. A not so typical example may correspond to the books of a library translated into binary code. Using a completely similar situation, the set of all molecules reported within Chemical Abstracts, with the known information on them, translated to binary code and ordered forming a bit string, constitutes another obvious example of a set, with the appropriate characteristics as those that will be described here. Another interesting example of a Boolean tagged set form, connected to a particular molecular discrete description, may be briefly discussed. Consider a given set of molecular structures. Take it as a background set. Construct the attached topological matrix to every molecule in the background set and order it as a column vector. When finished with this task, homogenize the dimension of these column topological vectors by, for instance, adding the appropriate number of zeroes. Take the homogenized column topological vectors as the tag set. A Boolean tagged set has been defined, in this manner, over the molecular set. A. Preliminary Definitions

Definition 1. Hypercube. A unit-length n-dimensional cube or hypercube is a set of 2" vertices, whose elements can be constructed with the binary string vectors, associated with the integer sequence: {0, 1, 2 , . . . , 2""^}. Two vertices should be

Fuzzy Sets and Boolean Tagged Sets

47

underlined from the abovementioned sequence, namely, 0 = (0, 0, 0, . . . , 0), the zero vertex, where the origin of coordinates can be supposed to be, and 1 = (1, 1, 1, . . . , 1), the unit vertex. Let us use K„ to note such n-dimensional regular polyhedron, n Hypercubes of this sort can be constructed iteratively as K^^^ <— K^. Using the algorithm: Vv, € A:„: V'^ = V, e (0)AV^,, = V, © (1) =* {y'^, v^,,} € K„,,

(D

The algorithm above amounts to the same as considering a direct sum construction: A^^^i =K^® Ky This also allows admitting the possible existence of more general possible constructs as K^_^_^ = K^® K^ and even of still more generic forms like ^yv = ©/^n(/)WithAr = Z.n(0. Definition 2. Boolean Tagged Set. Suppose any set, 5 = f^}, the background set, and a hypercube, AT^. A Boolean tagged set, B„, is constructed by means of the ordered pairs: B„{S) = SxK„ = {zeB,\z = [s,Y,];seSAy,eK„}

(2)

The order of a Boolean tagged set will indicate the dimension, n, of the underlying hypercube K„. So as to ease the notation, the equivalent symbols B„ = B„(S) will be used whenever no confusion could be generated. A classical set can be associated with a Boolean tagged set of unit order: 5,. D B. Tagged Classes It is apparent that the Boolean tag, associated with every element of the background set S, placed in the second moiety of the ordered pairs when constructing Boolean tagged sets, can be understood as a membership tag class. A trivial example is given by the unit order Boolean tagged set, where the following interpretation holds: VzG ^1 -> {z = [s, 1]ASE S] V {Z = [S,0]AS€ S). The integer value, k, of a given Boolean tag, v^, could be defined by the symbol x(vj^) = k, so one will have, for example, x(0) = 0 and x(l) = 2'* - 1. In general, now, and following the same path as in fuzzy set theory, when observing any Boolean tagged set, B^, only those background elements, bearing the unit vertex Boolean tag, can be considered true members of the background set 5. Likewise, only those background elements, bearing the zero vertex Boolean tag, can be considered true nonmembers of 5. The rest of the background elements may be interpreted as the marginal elements of B^. Definition 3. Tagged Class. A tagged class C,^ of a Boolean tagged set B^ can be defined by means of the following condition: x(v,) = VAT = [t, vj e

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RAMON CARB6-DORCA

Cj^czB^.A tagged class will be attached to a subset 7^ of the background set 5, such that Vr G 7^ c 5 -^ Q = T; X {v^}. D From the definition of tagged classes, the following relationships hold:

k

U 7 ; =5Ar^nr^ = 0

(3)

k

That is, \/s£ 5: 3 v^ G K^-^z = [s, v j G C^ c B^. Thus, Boolean tagged sets may be defined in such a way that every element is a member of some disjoint subset, associated with a given precisely well chosen hypercube vertex, also representing an integer value in the range ke {0, 1, 2 , . . . , 2" - 1}. A nondegenerate Boolean tagged set has the available 2" classes with one or nil attached elements of the background set. On the other hand, degenerate Boolean tagged sets would give to the observer the meaning of lack of information on the tag moiety, whereas the presence of void tagged classes could signal that possible candidates of the background set remain unknown. C. Operations over Boolean Tagged Sets Any operation defined over the elements of a Boolean tagged set, B^, has to be decomposed into two distinct operations, involving the background and the tag set parts separately. That is, a general situation may be depicted in the following way: Va,Z7G ^ „ : a * ^ - ^ [ a , ^ i ] * [ p , v ] - > [ a o p , i L i * v ] G 5^

(4)

At the same time, transformations of the elements of B^ can be defined in three possible ways, according to how they affect the Boolean tagged set elements. There may thus be transformations of the background part only, or the tag part could be the one affected, or both parts may be changed. Background, tag, or total transformations may be defined accordingly. D. Boolean Tagged Vector Spaces A Boolean tagged vector space is simply a Boolean tagged set whose background set is a vector space. The structure of vector space is preserved providing that appropriate operations could be defined in the tag moiety. To the vector sum of the background part, there must be defined a corresponding composition on the tag set part. That is, if W^ corresponds to such a Boolean tagged vector space, then V a, Z? G W^: a + Z? = [a, jx] + [(3, v] = [a + p, |i • v]

G

W„

(5)

Fuzzy Sets and Boolean Tagged Sets

49

In this way, choosing the tag set part operation as •=A, for example, then true members are preserved as such in the final sum result, and when summed up to true nonmembers the result is a true nonmember. Sums performed over a tagged class of marginal set members preserve the tagged class association of the result. Within this trend, marginal members not of the same tagged class, when summed, may yield true nonmembers. Other associations from the tag set part point of view may be defined, using diverse Boolean operators. When taking into account the product of a vector by a scalar, it is sufficient to consider this operation as a background set transformation, leaving the tag part invariant. As a consequence, linear combinations of Boolean tagged vector spaces are perfectly defined in this way. Metric background vector spaces may merit some attention, because scalar products or norms will produce a scalar result in the background part. Thus, in the tag part some sound transformation must be defined too, providing the projection of the whole Boolean tagged space into a boolean tagged unit order set with a scalar background set. Then, metric Boolean tagged vector spaces are easily defined. Distances can also be defined quite effortlessly over Boolean tagged vector spaces. The background part is immediate and can be obtained using some common form, as in the corresponding space and in the tag part a Minkowski formula may be used, which amounts to the same as counting the number of noncoincident bits in the pair of implied tags. The biggest distance number obtained in any usual way will be the one involving the extreme vectors 0 and 1. E. Applications

Obvious applications may be envisaged in the domain of chemistry, as well as in other areas, like Boolean tagged logic, which do not directly concern this contribution. The background set S may be taken, from a chemical optics, as a set whose elements are made of molecular structures. The hypercube tag elements can be identified as a set of chosen molecular descriptors, transformed into a corresponding set of bit strings. In doing so, the molecular classes may be easily compared and ordered. Proceeding in this manner, it can be straightforwardly deduced how the Boolean tagged sets are perhaps to be considered in the backyard of all of the attempts and procedures to classify molecules. Molecular classification and ordering can be achieved by means of studies, essentially based on molecular properties of any origin as molecular descriptors. The molecular magnitudes, obtained through theoretical considerations or experimental measures, being rational numbers in the worst case, can be easily transformed into bit strings. A molecular set can be associated with Boolean tag descriptors, and transformed in this way into a Boolean tagged set. The atomic periodic table is nothing but an early nice example of this situation. Another trivial example of such molecular tagged sets can be constructed

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RAMON CARB6-DORCA

using as a tag set part the molecular coordinates of every background set molecular structure. The coordinates are fixed at some conformation and can be understood as a set of positive definite rational numbers, describing a three-dimensional molecular structure. Positive definition of the coordinates can always be achieved by means of an appropriate translation of the origin. The realm of QSM^, a field where the author has been active in recent years, provides us with a perfect example of the possible construction of a Boolean tagged molecular set. It is well known that molecular QSM (MQSM)^'"^ can produce a discrete representation of molecular structures within a given molecular set, the so-called molecular point-cloud^ of the set. Each molecule, by using this procedure and described in the molecular point-cloud, is represented by a vector, SLpoint-molecule, obtained by purely computational means. For this reason, the QSM molecular discrete description can be easily transformed into a bit string. A molecular point-cloud is, from this point of view, a representative example of a potential Boolean tagged set. It is not an accident that in early examples of molecular point-cloud representations in the form of graphs, n-dimensional hypercubes^ of the appropriate number of vertices were successfully used. F. Extensions

There are multiple extensions of the previous description, but they will lose the inherent direct Boolean computational flavor, which was the main objective of the previous Boolean tagged set definitions. However, this drawback can be compensated by the construction of possible linking of tagged sets with QSM. Suppose, again, a set 5, and a set D, made of positive definite (PD) functions of some variable(s): D = {5(x) I Vx G C„ -> 5(x) eR"). The general Boolean tagged set structure may be transformed into a function tagged set. This can be easily done, just by defining the ordered pairs B^(S) = Sx D. Then, to every element of S can exist an attached function with some additional properties, not only a bit string, but possessing a continuous functional structure, that is, BJS) = SxD={ze

5 ^ | z = [5,5(x)];5G S A 5 ( X ) 6 D]

It is immediately seen that fuzzy sets are nothing but a very particular case of functional tagged sets. It is much more interesting, though, to recognize that functional tagged sets contain as particular cases quantum object sets (QOS).^ For this purpose it is only necessary to associate the background set S with a collection of microscopic systems in a given well-defined state and, according to quantummechanical postulates, one can consider D as the set of associated DF. This possibility will be studied in the next section. An even more general tagged set structure may be easily envisaged. Up to now a unique tag set has been used throughout our description of a tagged set. But nothing opposes the use of a broader definition, where the tag set part may be

Fuzzy Sets and Boolean Tagged Sets

51

formed by composite aggregates of two or more collections of appropriate tags. The earlier tag set part example, made of molecular coordinates, can be interpreted in this way.

III. TAGGED SETS, CONVEX SETS, AND QSM QSM first^, and tagged sets as defined above and convex sets (CS)^ in the second term, constitute interestingly interconnected fields. The relationship may be found in the associated characteristic definitions and mathemafical structures, which have been developed over time in the realm of QSM.^^ In the recent literature, the structure of such QSM theory has always been associated with integrals involving the description of quantum objects (QO). By a QO must be understood a microscopic system that possesses all of its information in an associated PD function. Such functional descriptive power lies in the assignation to the attached PD function of some statistical probability distribution formalism, the so-called density function}^ which in turn is nothing but a result of the system wave function squared module manipulation. Such initial function information, in the form of wave functions, is obtained as a solution of the system's Schrodinger equation, in the well-known quantum mechanics scenario. The whole conceptual structure may be cast as a quantum-mechanical postulate. Also, besides the PD DF dependence of any QSM, these can be easily attached to PD operators too, as well as to the functional spaces where PD DF belongs. This apparent PD mathematical background makes of QSM theory a good candidate to be related, somehow, to the structure and definitions of convex sets, because the particular structure of this kind of set deals, preferentially, with PD linear combinations of vector space elements. This collection of problems will be discussed in the present section. A. QSM QSM have been defined in several previous papers (e.g., see Refs. 5-8), as have the basic concepts associated with the theoretical body, involving the ideas underlying the basic integral structure of QSM. To keep these primary ideas present at the time to deepen in the QSM theory, several definitions, which depend on the previously presented ones, are given below. Definitions

Tagged Sets and QO. A parallel definition to the one associated with Boolean tagged sets, defined in Section II and the subject of a recent report^^ as an alternative to fuzzy sets, will be employed to construct QOS. The term quantum object^^ will be used from this point on as a synonym for any microscopic system composed of a numerable set of particles, associated with a

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RAMON CARB6-DORCA

probability DR Within the current quantum-mechanical structure the QO DF acts as a PD descriptor, which possesses all of the information contained in the system. Definition 4. Quantum Object. Any microscopic-sized system can be called a QO, if it can be constructed with a numerable set of particles, possessing all of the extractable information contained within a probability distribution: the system's DF. The DF is a PD function of random variables made by the system's particle positions or momenta. When dealing with nonstationary system states, time should be added as an additional random variable. Due to this attachment between QO and DF, one can refer to the DF as a QO continuous descriptor. D Definition 5, Density Function Tagged Set. The Boolean tagged set definition can be easily generalized, considering various possible tag set extensions, as presented in Section II.F. Suppose now known a set of microscopic systems, 5, and let us associate it with a background set. Suppose also a set made of PD DF, P, and let us call it the tag set. According to quantum mechanics, to every element of S there can be found a one-to-one correspondence with a DF of P. The situation may be cast into a new set, 2L density function tagged set, T, defined in the following way: r = 5 x P = { T l V 5 e 5 , 3 p 6 P - ^ T = (j,p)}-

(6)

Within this definition a QOS may be associated with a density function tagged set, r, and a QO can be considered in consequence as an element of T. D This structure may be considered as the simplest case of a possible general tagged set form, which for the moment is not relevant here, where Definition 5 will be used throughout. In fact, in this possible general definition, QOS may be considered tagged sets, with the tag set part, formed not only employing a unique function, but the collection of all possible DF of the system's quantum states. The tag, p, could have, from this point of view, necessarily a vector structure, whose elements will be the state DF. An alternative way can be associated with the tagged set formed with the elements of the microscopic system background set and one quantum state density as the tag part: In this case the tagged set elements will possess the same background part and diverse tags. QSM. Let us define primarily a QSM as a composition involving two QO, constructed with the rule: ya,bETA{a

= (s^,pJ',b = (s^,p,)}:

Z,, (Q) ^ = J J p,(ri)Q(ri, v^)p,(r^)dr,dr^

(7)

Fuzzy Sets and Boolean Tagged Sets

53

where ^ is a PD operator, whose dependence on the variable set {r^ r2} must be coherent with the ones associated with the tag functions of the involved QO. The case of computing a QSM when both involved systems are the same, a = b, produces a quantum self-similarity measure (QS-SM). Definition 6, Convex Conditions. By convex conditions K„(c) held over an n-dimensional vector c, it is understood that the elements have the following properties: K^(c) = {c = {c,} G K(RO: c, > 0; V/ A ^

c, = 1} D

^^^

Definition 7. Positive Definite Operators. A PD operator, Q > 0, acting over a vector space, "U, such that Q'.'U-^'U, can be defined in the usual way as follows:

Suppose that convex conditions K„(yv) hold over a coefficient vector w. Suppose known a set of PD operators W={Q„}, the convex linear combination: ^ = E^a^aA/^„(w)

(10)

produces a new PD operator. DF can be considered PD operators too, then this property also applies to this collection of functions, a Due to the PD nature of all of the involved elements in the QSM, the values of the measure integrals, described in Eq. 7, are always positive. Thus, a QSM, as previously defined, can be considered an operation transforming the ordered pairs of tagged set elements into the set of positive real numbers: Z:(Tx T) -^ R"*". Integral defined scalar products with weights, associated with the PD operators Q, are good candidates to be connected to QSM too. The most popular operator, used to date in QSM integral expressions, corresponds to Dirac's delta function 5(^1 - r^^, and in this case, integral 7 transforms into a so-called overlap-like QSM. A third element of the QOS can be used instead, producing several possible forms of triple QSM.^^ Multiple products of DF substituting the operator will yield a multiple QSM. Similarity Matrices and Discrete Representations ofQO

In any case, given a QOS, the computation of QSM involving, at least, element pairs produces a new set, which has been discussed in the literature in many ways.^"^ In the simplest situation, every QO can be connected with the rest of the QOS elements, including itself.

54

RAMON CARB6-DORCA

Definition 8. Similarity Matrix. When all of the QOS elements in pairs are involved in the QSM calculation and ordered, this gives as a result a symmetric matrix: The similarity matrix (SM), Z. The dimension of the SM will depend on the cardinality of the QOS: If #(7) = n, then Dim(Z) = (n x n). The SM can be considered a row hypermatrix whose elements are n-dimensional column vectors, collecting all of the matrix elements associated with a given QO. That is, Z = (z„ z , , . . . ,z„) = {z,} c C„(R^) A iS:„(z,); V,

d D

As already mentioned, the SM elements are computed within some kind of scalar product formalism as in Eq. 7. Because the DF tag part, if the background set elements are chosen to be very different, can be considered a linearly independent function set, the collection of QSM may be easily seen as forming a metric matrix, computed over the DF tag set, and thus the SM, Z, may be considered a PD matrix too. This is so whenever the QSM are calculated with DF bearing the same orientation in the particle coordinates space, for all matrix elements, a An interesting feature, derived from the PD nature of the DF tag set and from the resulting QSM, corresponds to the elements of such vectors and matrices: the column vectors, {z.}, elements are positive too, as represented schematically in Eq. 11. Given a QOS, constructed as a function tagged set, as the one previously defined for QO, and provided the QSM among the elements of the QOS, as defined in Eq. 7, Eq. 11 introduces the possibility of constructing another tagged set kind. This can be performed employing the following definition, which has a parallel structure to Definition 5: 0 = 5 x Z = { e | V 5 G 5 , 3 z G Z - ^ e = (5,z)}

(12)

where the symbol for the tag set, Z, has been chosen the same as the one used for the SM. The new vector tagged set is another possible representation of the QOS. The vector tagged set constructed as in Eq. 12 is immediately connectable to concepts defined early in the QSM context. ^° Indeed, a QOS in the form of a vector tagged set has been called, when molecular QOS were studied, a molecular point-cloud, and the elements contained in it point-molecules. Definition 9, Point-Clouds. Vector tagged sets or QO point-clouds can be derivedft-omthe original function tagged sets, T, by projecting the PD DF tagged set, T=SxP, into a PD vector tagged set, 9 = 5 x 2 : ^ (7) = e => VT G T: (P (X) = T(is, p)) = (s, !P(p)) = (^, z) = e G 0

D

Fuzzy Sets and Boolean Tagged Sets

55

The new projected QOS corresponds to a discrete QO description, where every system from the background set is no longer tagged by a continuous DF, but by an n-dimensional PD vector. As such, it can be easily transformed into a Boolean tagged set, according to comments in previous work,^^ given also in Section lI.E and based on the evidence of the usual computational practice. Thus, computations are made within the field of rational numbers, and as such the QSM values involved in the elements of the tag set part of any QO point-cloud can be associated with this numerical field. Rational numbers are expressed within the usual computational structure in the form of bit strings. Already in this numerical form, they can enter as the elements of the tag part of vector tagged sets, transforming them into the equivalent Boolean tagged sets. The sets P and the projection Z=!P(P) constitute very peculiar ensembles. In fact, they are properly defined by elements, which have values only in the PD set of real numbers. Both correspond to sets whose elements belong to some vector space, with operations defined in R"*". The vectorial addition cannot possess a complete group structure, but a semigroup one instead,^^ lacking reciprocal elements. Everything else can be considered preserved. One can refer to structures of this kind as vector semispaces. Definition 10, Vector Semispace. A vector space defined over the positive real field, R"^, only and provided with the usual operations of addition and product by a scalar. In a vector semispace (VSS) the additive Abelian group is substituted by a semigroup}^ As a consequence, no vector reciprocal elements can be present in a VSS, and the linear combination coefficients will be made by positive real numbers only. A two-dimensional image of this kind of VSS may be given by the vectors lying over the positive {4-JC, + J } quadrant of the real plane. D The following definition will serve to summarize all of the concepts already discussed. Definition 11. Discrete QO Description. Suppose known a QOS Q = S x P. This means that \/ se S, in the QOS background set part, there exists a DF, p G P, in the tag set part, which is a continuous descriptor of s. A discrete representation of the involved QO may be obtained choosing a PD operator, Q, and defining the QSM^^: Q = SXP: Vp,, p^eP=>z^=

jPa^p^dV

(13)

The SM, Z = {Za,b}y ^s described in Definition 8, which collects all of the integrals between the DF pairs of the QOS, can be considered as a row hyper vector formed by a set of column vectors Z = {Zj, Zj, . . . , z„}. Every column vector z^, say, is formed by the QSM involving the QO, s^, DF, p^, with all of the QO elements in S, including itself. Being the involved operator PD and considering that the QOS is

56

RAMON CARB6-DORCA

made of essentially different QO, the matrix Z could be considered PD. The whole matrix may belong to some n-dimensional VSS, Z„(R^). Then, starting from here, a new TS, 0 , can be constructed with the same background set part as the former QOS Q, but with the tag set part formed by the columns of Z, that is, 0 = S x Z. This corresponds to obtaining a PD operator-dependent discrete representation of the QO. One can name the column vectors z^ G Z as discrete descriptors of the QO. D B. Convex Sets CS can be referred to as VSS for our purposes, although there are huge libraries full of interesting problems related to this concept, which can be easily found in the literature. ^^ A CS in the present discussion will be considered as a collection of linear combinations of vectors in a VSS, thus having positive coefficients, which fulfill an extra constraint such that the coefficient addition yields the unity. That is, a convex set is a subset !I(^ofa, VSS !H, which is defined by means of

[3a = {a.} c R^: X = 5 i ttjCj A ^ ttj = 1] <= K^ia)

(14)

A curious question can be formulated now about how CS, as defined above, can be considered made using vectors of a general vector space. Whenever a set of complex numbers are defined such as to form a normalized column vector: {Y,.} c C A g = (Y,, Y2. • • •. Y„)^e K(Q A g'g = ^ IViP = 1 i

A[if: a= a,. = |Y,.p e r ; Vi} ^ ^ «i = H ' ^ ^«(«)

^^^^

i

Then, CS may be constructed from an n-dimensional vector space over the complex field, ^JjC), associating normalized vectors of this general space to the positive coefficients summing unity of Eq. 14. The squared modules of the general vector elements are the needed positive coefficients of the CS. Such a vector space, as l^^(C), may be called a generating vector space of the CS,!}(, By extension, any vector g G ^J,C) will be called the generating vector of the attached CS element. A generating rule can be defined formally as 3ge'^',(QAg^g=l=» ^(g->f)^

(16)

This bears, no doubt, a discrete quantum-mechanical flavor. Quantum-mechanical DF are computed as manipulated squared modules of some complex-valued wave

Fuzzy Sets and Boolean Tagged Sets

57

function, which can act as a generating oo-dimensional vector. The generating vector space of DF is the Hilbert space whose elements are the quantum system's wave functions. This aspect of the question will be studied in detail below. It must be remarked now that DF constitute some kind of PD functions, admitting only positive coefficients for linear combination purposes. Moreover, the volume subtended by any DF shall be finite and constantly equal to some function of the particle number. As a consequence, DF can be, without loss of generalization, considered normalized to unity: possessing a unit volume. Thus, DF could be considered the elements of a CS defined in oo-dimensional space. Indeed, suppose a set of DF, possessing coherent particle coordinates {p,(r)} c P, such as V/:Jp.(r)Jr=l

(1'^)

and construct a new DF, -©(r), using a set of PD coefficients {c.} c R^: (18)

Thus, forcing new linear combinations of DF to have constant unit volumes is the same as embedding the DF set into a CS structure. It will now be an easy task to construct the following: Definition 12, Convex Set. A convex set is made of vectors as elements, whose linear combinations and elements altogether fulfill convex conditions. The set of linear combinations of a subset belonging to a VSS, whose coefficients and generating vectors are constrained within the unit sphere, is a convex set. D C. ASA Recently, various papers have been devoted to the problem of constructing properly defined first-order DF, by using spherical function basis set superpositions.^^ The so-called ASA has been discussed in various molecular and atomic environments as well as practically studied through several methodological algorithms. According to the previous generalization and analysis, the ASA becomes nothing but a consequence of the nature of the DF and of the vector semispaces where they belong. ASA within a CS Environment

Suppose a PD basis set of totally symmetric functions S = {a-}, which can be built up from a normalized function set O = {cp.}. This can be obtained using simply the equivalence a,. = l(p-P; V/. Taking into account the normalization conditions of the functions belonging to the set O, the set S acquires automatically the property of unit volume, mentioned earlier, when dealing with the CS structure of DF:

58

RAMON CARB6-DORCA

V,;J|9,(r)pdr=Ja,.(ryr=l

d^)

Suppose a DF, p(r), with an appropriate coherent variable set, r, as the one associated with the basis function set Z. The ASA approach consists in expressing the DF as a CS element generated from the basis set E: Vp(r) ^ J3c = [c.] c IT A ^ c , = l | ^ K„(c):p{r) = ^ ^ A

(20)

The coefficients {c.} of Eq. 20 are to be estimated using a constrained leastsquares technique. Recently/^ it has been shown how elementary Jacobi rotations (EJR)^^ can be employed efficiently to deal with problems of this kind. For atoms, the ASA procedure produces first-order DF with almost negligible quadratic error integral measures. Also investigated was whether SLpromolecular approach can be constructed as an accurate ersatz for molecular first-order DF. The promolecular formalism states that the first-order DF tag part for any molecule can be approximated by a simple sum of atomic ASA functions, computed by means of a constrained least-squares procedure as already mentioned. Results have been accurate enough for QSM computational purposes.^^ More refined approaches can use a completely equivalent formalism as in Eq. 20, employing atomic ASA optimal functions as the components of the basis set E. ASA in Atoms

First-order atomic DF can be approximated by means of the so-called ASA function. ^^ These approximated functions for atoms can be constructed in the following fashion. Suppose known a set of ns-type AO {5'-(a,,r)}, an atomic ASA function can be written as pA(^) = Yc,\s,(r)f'AK„(c=[c,})

(21)

where the coefficient vector c = {c.} has to be fitted to any given "ab initio " atomic DF, previously computed by any meaningful procedure, using a constrained leastsquares technique, so as to obtain the solution while keeping the convex conditions KJjc). Constrained fitting within the convex restrictions associated with Eq. 21 over the coefficient set can be easily accomplished using an EJR scheme. ^^ The main idea, before applying the EJR technique to fit the atomic ASA function to an exact DF, appears when gathering the coefficients into the column vector c, and associating with it a new column vector of the same dimension, the generating vector, X, in such a way that the following generating rule ! ^ (x -^ c) holds: iRJix ^ c) = {3x e 'J/„(Q A x+x = 1 => c = {c,. = |;c,.p} -^ ^^(c)

(22)

Fuzzy Sets and Boolean Tagged Sets

59

Thus, EJR can be applied to transform the generating vector elements {JC.}, and then, indirectly, those of the ASA coefficient vector are varied, while preserving the convex conditions ^„(c). ASA Structure in Molecules With the above considerations known, it is easy to think that, by performing a previous computational task, a set of fitted atomic DF, A = {p^}, can be gathered, having the ASA form as in Eq. 21. The set A can be taken, being strictly formed by convex linear combinations of PD functions, as a PD operator set. Thus, A can be used as a generating set of new ASA-type DF. A molecular DF, p^, for example, can be approximated by a linear combination of A elements, with a coefficient vector w = {w^}, fulfilling the convex conditions ^„(w): pA/('-) = E'^AP^(«--r4)A^„(w)

(23)

A

where {r^} are the molecular atomic coordinates, on which the ASA DF are centered. According to the properties of the PD operators, described in Definition 7, p ^ is also a PD function. This allows the possibility of fitting the coefficient vector, w, to the molecular DF, in the same fashion as described for the atomic case. This also means that some generating vector u can be defined too, fulfilling a similar set of conditions as shown in Eq. 22, provided x <— u and c <- w, that is defining a generating rule like !l{Ju —> w). A promolecular approach to p ^ may correspond to choosing the ASA-type coefficient vector w with all of its components equal, that is w = n~4, and 1 = (1, 1 , . . . , 1)^. Another possibility is to associate with each atomic density a coefficient w^=Z^Ar\ where Z^ is the atomic nuclear charge and A^ = Z^Z^ is the total molecular number of electrons. A CNDO^ approximation, as was employed in early stages of QSM calculations, will correspond to using, as the vector w elements, normalized Mulliken gross atomic populations. Considerations around MO Theory Another example of this first-order DF convex form may be found within MO theory. Indeed, if an orthonormalizedMO set is at our disposal—{cp.}—then it is well known that the expression of the DF may be obtained as PM = E'0,.|cpf

(24)

where the set co = {(oj can be constructed to fulfill the usual discrete convex conditions ^„(a)). In this case the elements of vector co are interpreted as the set of MO occupation numbers, which in the usual SCF monoconfigurational theory are

60

RAMON CARB6-DORCA

just integers, 2 or 1. In the present framework they must be trivially transformed in such a way as to fulfill the convex conditions. In other computational environments, as in MC SCF or in natural orbital algorithms, the MO occupation can be made of real numbers, so the convex conditions may become more naturally defined. Equation 24 has the same structure as the previous ASA described expressions. In closed shell systems, studied in a monoconfigurational way, the vector (O just corresponds to a simple promolecular approach with co = n'^l, with n being half the number of electrons. The dependence of the MO basis set from an AO basis, within the well-known LCAO MO approach, has not been exploited in the ASA approaches. This possibility will be studied in forthcoming work. Here it can be said that the LCAO form of MO might be related to the DF generating VS. An interesting point to be noted is the possibility of transforming SCF theory and electronic energy expressions into a normalized form, so as to use a first-order DF with the appropriate ^„(co) conditions. It is a trivial matter to use Eq. 24 with the adequate convex conditions in monoconfigurational energy expressions, and the effect of this point of view in the Fock operator definition will be immediately grasped. Keeping convex conditions over the DF will scale monoconfigurational electronic energies by a factor that corresponds to the inverse of the squared number of electrons: G^ = N^^. This means that 1. Repulsion integrals need not be scaled at all. 2. One-electron Hamiltonian must be scaled in full by a factor equal to the inverse of the number of electrons: a = N~^ . Taking these simple rules into account, one can see that any SCF computation under this convex scaling of the DF will translate the process of any atomic or molecular system into some computational algorithm, with such a structure that it will become independent of the number of system particles. Thus, in this scaled framework the SCF iterative structure could be studied with respect to convergence, extrapolation, and so on, for all of the systems on an almost equal footing. The only variable aspects of the SCF process will derive from the basis set size and the nature, as well as the three-dimensional space position, of the atomic centers. Another interesting aspect of this scaling process corresponds to the numerical size of electronic energies, which will become constrained within a shorter interval than before scaling. For example, absolute values of atomic SCF electronic energies, from H to U atoms, will belong to the approximate interval {0.5; 4.}. Resizing them to the usual values needs only a scaling by a factor N^. The Continuous ASA Case

Equation 23 can be made much more general if it can be studied as embedded in a continuous environment. Provided some PD DF basis set is known beforehand—

Fuzzy Sets and Boolean Tagged Sets

61

{p(t, r)}—while considering that the DF basis elements could depend on some continuous parameter set t, then the continuous construction of a PD DF, p^^Cr), can be set up using the integral form p„(r) = J(o(t)p(t, f)dt A K„ (co(t))

(25)

where co(t) is a function of the parameter set t, fulfillling the continuous convex conditions ^^((o(t)), which in this case shall be written as ^oo(co(t)) = {Vt: co(t) G R+ A Jco (t)dt = 1}

(^^)

At the same time can be defined a generating function, Y(t), contained in some Hilbert space, which fulfills an equivalent generating rule as the one shown in Eq. 22 in the discrete case: !^(Y(t)^(0(t))^

3Y(t)e5/JC)AjY(t)*Y(tMt=l

^(o(t) = |Y(t)p^^J{o(t))

(27)

In this way, it is easy to see how a discrete ASA framework can be extended into an oo-dimensional environment. Equation 27 has interesting features, which provide the generating function, Y(t), with a nearby kinship to a QO wave function. Also the coefficient function, co(t), closely resembles a DF itself, through the corresponding equivalent definition. On the other hand, Eq. 25 also can be considered as an integral transform^ ^ of the coefficient function co(t), with the DF p(t, r) acting as a transform kernel. Thus, somehow, the continuous DF linear combination 25 can also be considered a scalar product between two functions, having an equivalent DF nature. A convolution transformation may also be considered as a particular, but nonetheless appealing, situation, instead of a typical integral transform. However, this possibility can lead us far from the present discussion objectives, and will perhaps be considered elsewhere. D. Convex Operators In the previous analysis of the ASA approach, the CS nature of the DF has been fruitful in designing an efficient approximate algorithm, so as to obtain reasonably accurate approaches to first-order DF. From there, and from the relationship between CS and DF, as stated in Eq. 18, one can try to go beyond the functional CS, and, using the PD operator nature of DF,^^ extend the previous ideas to PD sets of operators. In such an extension, the QSM themselves will be affected, being based on PD operator weighted integrals. The present discussion, from this point on, will be devoted to this extension of the CS framework and, finally, as a corollary it will

62

RAMON CARB6-DORCA

attempt to determine if there is some possible application of the resulting formalism. Convex Linear Combinations ofPD Operators Suppose a set of PD operators, Q = {co^^}. They fulfill the following relationship, when studied over the coherently defined DF VSS, P: Q = {co|co:P -> R^} A Vco 6 ^ ; Vp G P ^ Jco(r)p(ryrG R"

(^^)

Nothing opposes considering the following situation: Q c P => V(0, p: Jco(r)p(ryr= e R""

(^^)

where the application of the PD operator set over the PD DF set can be interpreted as a noncommutative scalar product, defined over the VSS, where both PD operators and DF sets belong. Moreover, the scalar product in Eq. 27 can be regarded according to the usual quantum-mechanical interpretation as the expectation value, , of the system observable, represented by the particular operator co, in terms of the QO particular state DF tag part, p. Because it is possible to consider the operator set, Q, as forming part of the VSS, P, the situation applied on DF as stated in Eq. 18 can be used over the elements of the operator set. In such a manner, if a set of coefficients w = {w^} exists, and the following constraints, similar to Eqs. 14 and 20, are set on it, then a linear combination of the operator set Q will yield a PD operator, y, such as ^»^{W={WJCR^AX>V„=1}=^ a

Y = X H-aCO^: e R* a

(30)

a

The second constraint has been introduced so as to obtain a pattern comparable to the CS structure of the VSS P and transfer it to Q, but it is not strictly necessary to keep this unit coefficient sum, if not needed. The most interesting thing is the obvious result, according to Definition 7, that PD operators can yield, in a CS environment, new PD operators. Tuned QSM^ SM^ and QO Descriptors The previous conservation of the PD property on linear combinations of PD operators in a CS environment can be employed in the evaluation of new kinds of QSM, by constructing a new breed of PD operator weights. The y-type operators appearing in Eq. 30 can be tuned up, while maintaining the identity of the operator set, just by changing the values of the CS coefficient set, w, conserving the initial

Fuzzy Sets and Boolean Tagged Sets

63

chosen constraints. A QSM, following the definition provided in Eq. 7, can be built up, under these circumstances, as:

The resulting tuned up SM elements, Z^^(Y), produce another obvious result for the SM set, {Z(cOj^)}, associated with every operator in Q. With each SM attached to a PD operator, every such SM can be considered as some discrete matrix representation of the associated operator in the corresponding basis set of the involved DF. These matrices, as already mentioned, can be considered as PD matrices. Thus, Eq. 31 can be written in whole matrix form as Ziy) = ^wj^(0j

(32)

Being the resultant matrix PD, because if in the SM set, 0 = {Z((0(j)}, all of the SM elements are PD, then the following property will hold: Vx e C„ A VZ(a)„) € e-> x^Z(coJx e R+=> x'-Z(Y)x = ^ w„x+Z(co^x 6 R^; if V„: w^ e R+

(33)

a

These results demonstrate that a finely tuned set of QO descriptors can be obtained in this way. This is so because Eq. 32 holds for the SM columns too, in such a way as z,(Y) e Z(Y) A z,((o„) e Z({o„) -> Z,(Y) = S ^a^ii^^a)

^^"^^

a

Thus, all of the findings and definitions up to this point can be summarized as follows. A QOS is chosen in form of a DF tagged set. A PD set of suitable operators is used, as a set of weights, in the evaluation of QSM between QO. A set of SM is thus computed for each operator. A CS with suitable coefficients is chosen to combine the elements of the SM set. The resultant SM columns are convex descriptors of the corresponding QO, and provide a discrete vector tagged set representation of the QOS. E. Finely Tuned QSAR

If an immediate application of all of the previous development has to be chosen, quantitative structure-activity or -property relationships (QSAR or QSPR) constitute a good candidate field. In our laboratory, the basic theory connecting QSM

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RAMON CARB6-DORCA

and QSAR or QSPR was developed^^ some time ago and various practical applications have been reported^^ more recently. It has been deduced that molecular properties have to be, in some manner, related to the discrete representation of molecular descriptors furnished by the columns of SM, constructed in turn from QSM over the molecular QO. As a consequence of Eqs. 29 and 34, a given property value, 7i, for a particular molecular QO, described in turn by a discrete descriptor, Z(Y), can be related by means of 7i = u'^z(Y)

(35)

where the vector u corresponds to an unknown discrete representation of some operator over the same PD DF basis set, used to construct the convex discrete molecular descriptor Z(Y).^^ The usual procedure is to use a least-squares algorithm so that, knowing the pairs {71, Z(Y)} for a molecular QOS, the values of u can be obtained. Taking into account the tuned construction of the vectors Z(Y), it can be easily seen that the vector u will depend on the tuning parameter set, Y- We use the least-squares solution of the problem u = (Z(Y)^Z(Y))-^Z(Y)^p

(36)

where the vector p = {71^} contains the values of the property for each molecular QO, and Z(Y) is the SM of the QOS computed according to Eq. 32. Equation 36, however, has been written taking into account the possibility that the SM may no longer be square symmetric, but rectangular. This will constitute the more general case, where instead of a unique tagged set, two QOS with different cardinalities, m and n, are used to compute the QSM. The resultant SM will be of dimension (m x n). From this previous definition, one can easily deduce that the vector u will depend on the tuning coefficients w. Opfimization of the tuning set coefficients w can be done at the same time as the classical least-squares problem is solved, keeping in mind the associated CS constraints, which the tuning set w bears. A parallel nonlinear constrained opfimization on a quadratic function of the w elements will appear. The interesting feature here is that CS constraints can be studied in the same way as these are kept in the optimal ASA problem. Thus, to the usual least-squares problem, involving the operator associated vector, u, there will appear another least-squares equation, which starts defining the residual vector: A = p-Z(Y)u = p - ^ w „ Z ( c o J u = p - ^ w „ v „ = p - V w

(37)

where the previous definitions of the involved matrices have been employed. Also, the matrix V collects the vector set {v^ = Z(cOj^)u}, and the vector w = {w^}, contains the coefficients of the tuning set W. The residual vector 37 is obviously dependent on the classical least-squares solution u in Eq. 36. From the inspection

Fuzzy Sets and Boolean Tagged Sets

65

of the residual vector A, it is easy to see that the quadratic error will depend on a generalized quadratic function with a variable set formed by the new unknown vector w. This least-squares problem has to be solved under the constraints associated with the PD nature of the vector w. A CS constraint structure may be very convenient in normalizing the problem form. Thus, the quadratic function and the constraints may be written, using a compact matrix form, as follows: e^^^ = %-2qV

+ w^Qw A K^(W)

(38)

where the following simplifications have been used: X = p^pAq = V^pAQ = V^V

(39)

It is important to define the appropriate SM set, {Z(co^^)}, so that the matrix V = {v^^}, possesses its elements linearly independent, to obtain a PD matrix Q. This is equivalent to saying that the SM set shall provide images of the least-squares solution u, which must be linearly independent. The solution of the second optimization problem for the vector w could be sought using a generating vector, for example x, which will substitute the w elements in the following convex constraints A^„(w) and the generating !^ (x -> w) rule. Expression 38 will transform into a quartic function in terms of the components of vector X. Optimization under the unit norm of the generating vector x'*"x = 1 may be obtained by means of EJR, as in the ASA case.^^ The whole optimization process shall be made in an iterative manner: 1. Using a starting approximate tuning vector w obtain u, solving Eq. 36. 2. Knowing u, compute a new w, minimizing function 38. 3. Go to step 1 while the vector pair {u,w} remains inconsistent with respect to the previous iteration. Changing the number and nature of the SM composite Z(Y) will obviously produce different results, but within a given choice these can be coherently tuned up. This can add extraordinary possibilities to QSAR procedures.

IV. ON THE STATISTICAL INTERPRETATION OF DENSITY FUNCTIONS: DIAGONAL VECTOR SPACES AND RELATED PROBLEMS One of the main contributions of the present paper is the definition of n-dimensional diagonal vector spaces (DVS). The objective in introducing DVS will be to find some discrete vector representation so that it can consistently fit some usual properties of oo-dimensional Hilbert spaces,^"^ containing the relevant functions, which are subsequently employed to describe QO, in accordance with the von Neumann^"^ point of view. Thus, the main concern here will be to obtain, in a natural

66

RAMON CARB6-DORCA

way, the CS structure of approximate DF within DVS, in the same natural manner as the DF is obtained from the squared module of the QO system wave function. A. The Nature of Discrete Q O Representations

Let us now suppose a QOS, Q, constructed in the usual way as a TS, that is, Q = S X P. Let us also suppose that the elements of the tag set part are ASA-type DF, built as in Eqs. 21 or 23. Accepting this scenario is the same as considering that a QO is described under some finite PD functional basis set 0 = {l(p.(r)l^} with coordinates: CO = {O)-} fulfilling the convex conditions ^„(co) and belonging to a given Ai-dimensional VSS, such as (O G Wj^(R'^). The TS constructed as Q„ = S X {O c W^(R^)} corresponds to a QOS which has as tag set part a Subset of some VSS of finite dimensions. A discrete representation of QO can thus be reached in this way, besides the one discussed in Section IILA. Considering the nature of the continuous AS A-type transform 25, it can be seen that in the discrete case, the PD basis set O and the coefficient vector O) shall bear some equivalent structure. As the convex conditions A'^(co) hold for the coefficient vector, it is easy to interpret this feature in such a way that the elements of the coefficient vector CO constitute a discrete probability distribution. For example, in a promolecular approach as well as in an MO monoconfigurational closed shell structure, CO can appear as a homogeneous discrete probability distribution. Thus, the coefficient vector, co, bears the equivalent statistical features of a DF in discrete n-dimensional spaces. It is not strange that there exists a generating vector, Y G '^(C), producing the PD ca elements by application of the generating rule !^(Y -^ ca). The structure of the rule is not one that can be attached to a linear transformadon, but has to bear a nonlinear form. This nonlinear relationship between generating vector and coefficient vector appears nonnatural from an algebraic point of view. This is more obvious when examined in the continuous situafion, as discussed previously. The image appears even more conspicuous when observing the nature of the DF from the quantummechanical side, because any DF has to be considered as a squared module of the QO wave function, acting thus as a generating vector. The problem can be stated transparently using a simple, well-known mathematical device, associated with the most basic aspects of quantum mechanics. Suppose a QO wave function is known for some system state ^ (r) G i^(C). The corresponding DF is simply computed as p(r) = I 4^(r)p. The interesting fact is that the DF thus defined may belong either to the Hilbert space direct product 9{{C) (8) i^(C), when considered as an operator, or to some functional VSS HCR"*"), when considered as a PD real-valued function. In this sense, the generating rule can be applied here, and immediately written as %J^ -^ p). However, it can be interpreted in the following way, using the practically unmodified structure of Eq. 27:

Fuzzy Sets and Boolean Tagged Sets

^(^-^P)^

3^(r) G itf (C) A ||^(r)p^r= 1

67

(40)

This continuous generating rule must imply a closer relationship between the vectors involved in the discrete case. There, the generating rule !l(Jy —> co) means that, while the normalization part for y possesses a simple algorithm to be computed, 1 = Y'^Y, it is no such simple operation in the second part of the generating rule. That is, when one must attach the O) coefficient vector elements to the squared modules of the generating vector y, the algorithm is not naturally isomorphic to the one in Eq. 40. Insisting on the problem: there is a lack of simple, naturally obtained, isomorphic operation in the second part of the discrete generating rule, as stated in Eq. 22, when compared with the continuous case as defined in Eqs. 27 or 40. A possible solution of this interesting situation will be discussed next. B. The Structure of the Generating n-Dimensional VS: DVS The generating rules 22 and 40 are a shorthand notation of some nonlinear transformation involving the generating VS, '^'^(C), and the final VSS containing the coefficient vectors, 'H^^(R^). The lack of a simple natural operation, producing the results, implicitiy stated in the generating rule in the discrete case, can be circumvented using the following scheme. Suppose such an isomorphic pair of n-dimensional VS, which will be named ^„(C) and !FJR^) Both can be good substitutes for the original "UjiQ and ^„(R^) VS described above, respectively. A sound isomorphism of column or row VS is constituted by DVS, whose elements possess the structure of diagonal matrices. Let us consider that the isomorphic ^n(C) and !FJJR^) VS elements are chosen as diagonal matrices. This element choice has not been arbitrary, because matrix multiplication is closed in DVS, that is, matrix products of diagonal matrices yield new diagonal matrices. Moreover, diagonal matrix products are commutative. Considering only the diagonal part of the matrix elements, and discarding the off-diagonal elements, the DVS possess the same dimension as their isomorphic column-row vector counterparts. Then, it is easy to see that using this simple isomorphic device both the discrete and continuous generating rules acquire the same formal structure. Indeed, the discrete rule in Eq. 22 will be rewritten within any DVS framework as

^(D^A)^

3DG^,(C)A
(41)

A = D*D = Diag(|rf,.p) A (A) = 1 ^ K„iA) The symbol (D) = Z,- dj is, in this case, equivalent to the trace of the corresponding diagonal matrix, but it can be used as a simple symbol to mean the sum of all matrix

68

RAMON CARBO-DORCA

elements, and so was defined and used for the same purposes before?^ The convex condition has to be slightly modified to take into account the new DVS element structure: K^(A) =\A = Diag(7i.) G jT^Cr): n. > 0; V/ A (A) = ^n. = 1

(42)

thus, working with DVS instead of conventional VS and VSS, the coefficients in discrete DF description possess the same structural properties as the DF themselves. The generating DVS elements, D e ^n(^)' ^^^ ^^ ^^® ssuno manner as do the QO wave functions. And the resultant coefficient diagonal matrix, A e !FJJR^), satisfying the convex conditions A'^(A), can be written as a squared module of the former diagonal matrix. This can be done using a discrete form of the generating rule i^(D —> A), similar to the wave function-DF generating rule ^Q¥-^p):A = D'^D = DD"" = IDP = Diag(ld.P), as described in Eq. 41. The DVS of ^n(C) type may be considered normed spaces, with one of the possible norms defined as the trace of the squared matrix module. As a consequence, the DVSS ^n(R^) elements are constructed in such a way that their trace is always normalizable, and thus easily made unit. A diagonal TS, (D^, can be derived in the usual way by using a given background set part, 5, and a DVSS convex subset, HC as the tag set part, that is, !D„ = 5 x{!^ c ir„(R^)}. The question, now, may not need to be: why do the n-dimensional DVS fulfill in a natural way the same conditions as oo-dimensional functional VS? But it could be much better stated as follows: which kind of consequences, if any, will this situation have in the development of a discrete quantum chemistry framework? The next section will try to describe some of the possible features of this DVS structure. C. Expression of the Density Functions and Other Problems

It has been shown that the best discrete representation of the DF, having an ASA-like form, as in Eqs. 21, 23, and 24, is better described as a diagonal matrix, instead of a vector, as is usually done. Then, the scalar-like expression of the ASA DF type could be redefined in terms of the natural operations presented in the discussion of the preceding section. To obtain a coherent view of all of the possible redefinitions, which can be found as a consequence of the adoption of the DVS representation, some preliminary considerations will be made next. As the formal structure of ASA generating rules is better represented from the point of view of diagonal matrices rather than vectors, both the generating and coefficient vectors are thus transformed into elements of some DVS. The ASA forms discussed in Section III.C, besides the coefficient vector, are associated with a PD function set, which is in turn connected to the squared module of another function set, further belonging to another structure which can be termed a generat-

Fuzzy Sets and Boolean Tagged Sets

69

ing function VS. This situation can be managed in the same way as in the preceding discussion. Suppose that a function basis set is known: 0 = {(p.}. Nothing opposes the situation in which the set O can always, without loss of generality, be arranged into a diagonal matrix structure, and considered constructed as O = Diag((pj, (p2,..., (p„) G F(C). Then, it is obvious that when the following diagonal matrix product is made: p = 0*0 = Diag(l(Pil^, I(p2p , . . . , IcP;,!^)} G PCR"^), it will always produce a new diagonal matrix, whose elements belong to a special function VSS made of PD functions, that is, made of function squared modules. Thus, taking into account the definition of the diagonal product of the initial basis set, one can consider that the above result produces an entirely new PD basis set: P = {l(p,P}. Also, having defined the generating and coefficient VS, one can construct the following hybrid diagonal matrix: VD = Diag(J.) G ^(C) A VO = Diag((p.) G F(C) => ^ = DO = Diag(J.(p,) eJiOQg

(C) xF(C)

(43)

Once mixed structures of this kind are constructed, the ASA-like DF could be simply built by computing traces of squared modules of the diagonal structures, ^ , as defined in Eq. 43. That is,

p==;^K.(p,f=SKI > , f = S m I

/■

(^^

/

The formalism is now clear on how to construct the necessary generating elements and the road is open to obtain, in a very natural way, the structure of ASA-like DF. The most interesting feature of the whole procedure, perhaps, will consist in finding out how closely the deducible formal rules, based on discrete DVS, are equivalent to the formalism based on continuous quantum mechanics. But it seems that nothing opposes this possibility. In fact, it only remains to express the formal problem as to how an expectation value (Q) of some observable, associated with an operator Q, can be computed within a DVS formalism. A possible way could be:

{a} = JQpdV=JQ{^*^)dV= XKP J^kP = S (oJQp,dv = S COJ(p;Q(p,^/y=/(T'Q^yy /

(^^s)

i

The last linear combination of integrals is suited to differential operators, and can be naturally obtained when considering the operator £2 as a scalar matrix QI.

70

RAMON CARBO-DORCA

V. CONCLUSIONS A general framework, where quantum objects can be described in a systematic way, has been constructed. The concept of density function tagged set encompasses an early generalization that is proposed as a sound substitution of fuzzy set definitions, to describe molecular structures, namely, the Boolean tagged sets. At the same time, the definition of quantum-mechanical density functions has been used to put in evidence its essential positive definite nature. This fundamental property of density functions, often forgotten in the current literature, has also been used to connect quantum similarity measures, a simple concept, which compares two or more quantum objects, with the spaces containing positive definite operators. Vector semispaces and, more conventional, convex set algebra have been put into the context of the computation of approximate density functions, as in the ASA framework. This kind of computational algorithmic experience has been extended to positive definite operators and their matrix representation, the similarity matrices, from the point of view of quantum similarity measures. Positive definite operators can be used to construct a convex set of new positive definite operators and consequently their matrix representations remain positive definite. In this way, a new window is opened to obtain discrete,finelytuned, molecular descriptors in the form of positive definite vectors belonging to n-dimensional vector semispaces. The utility of the presented theoretical results in the context of quantitative structure-activity relationships is but one of the vast prospective application fields. The quantum chemical, statistically coherent, significance of the expansion coefficients, satisfying convex conditions, in ASA-like DF forms, which can be considered as a discrete probability distribution has been shown. Moreover, there is apparently no problem in using the fitted atomic densities to obtain expectation values of quantum chemical operators. The formalism, based on DVS and TS, becomes in this manner a fruitful tool, where one can fundament further work.

ACKNOWLEDGMENTS This work was partiallyfinancedby CICYT Research Project SAP 96-0158. Professors J. Karwowski and P. G. Mezey are thanked for lively debates on the subject of fuzzy and tagged sets, and Dr. E. Besalu for constructive criticism and advice. The author warmly thanks Mr. LI. Amat for stimulating conversations on ASA and QSAR, which led to various Fortran 90 implementations, as well as for patiently performing preliminary calculation tests on some tuned QSAR problems. Enlightening, informal discussions with Dr. J. Mestres have been carried out in previous stages of this work.

Fuzzy Sets and Boolean Tagged Sets

71

REFERENCES 1. Zadeh, L. A. Inf. Control 1965 S, 338. 2. Trillas, E.; Alsina, C ; Temcabras, J. M. Introduccion a la Logica Difusa\ Ariel Matematica: Barcelona, 1995. 3. Carbd, R., Ed. Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches; Kluwer: Dordrecht, 1995. 4. Carbo-Dorca, R.; Mezey, R G., Eds. Advances in Molecular Similarity, Vol. 1; JAI Press: Greenwich, CT, 1996. 5. Carb6, R.; Calabuig, B.; Vera, L.; Besalu, E. Adv. Quantum Chem. 1994, 25, 253-313. 6. Carb6, R.; Calabuig, B. Int. J. Quantum Chem. 1992, 42, 1681-1709. 7. Carbo, R.; Besalu, E. In Molecular Similarity and Reactivity: From Quantum Chemistry to Phenomenological Approaches', Carb6, R., Ed.; Kluwen Dordrecht, 1995, pp. 3-30. 8. Carbo, R.; Amau, M.; Leyda, L. Int. J. Quantum Chem. 1980,17, 1185-1189. 9. Stoer, J.; Witzgall, C. Die Grundlehren der matematischen Wissenschaften in Einzeldarstellungen. Vol. 163; Springer-Verlag: Berlin, 1970. 10. See for example: (a) Carb6, R.; Calabuig, B. Comput. Phys. Commun. 1989, 55, 117-126. (b) Carbo, R.; Calabuig, B. J. Mol. Struct. (Theochem) 1992,254, 517-531. (c) Carbo, R.; Calabuig B. In Computational Chemistry: Structure, Interactions and Reactivity; Fraga, S., Ed.; Elsevier: Amsterdam, 1992, Vol. A, pp. 300-324. 11. See for example: (a) Lowdin, R O. Phys. Rev 1955,97,1474-1489. (b) McWeeny, R. Rev. Mod. Phys. 1960, 32, 335-369. 12. Carbo-Dorca, R. Fuzzy Sets and Boolean tagged sets; Technical Report IT-IQC-5-97, see also: J. Math. Chem. 1997, 22, 143-147. 13. Carb6, R.; Calabuig, B.; Besalu, E.; Martinez, A. Mol. Eng. 1992, 2, 43-64. 14. Carbo, R.; Calabuig, B. J. Chem. Inf. Comput. Sci. 1992, 32, 600-606. 15. Encyclopaedia of Mathematics; Reidel-Kluwer: Dordrecht, 1987. 16. See for example: (a) Constans, R; Carbo, R. J. Chem. Inf Comput. Sci. 1995, 35, 1046-1053. (b) Constans, R; Amat, L.; Fradera, X.; Carbo-Dorca, R. In Advances in Molecular Similarity; Carb6-Dorca, R.; Mezey, R G., Eds.; JAI Press: Greenwich, CT, 1996, Vol. 1, pp. 187-211. (c) Amat, L.; Carb6, R.; Constans, R Sci. Gerundensis 1996, 22, 109-121. (d) Amat, L.; CarboDorca, R. QSM and Expectation Values under ASA: First Order Density Fitting Using EJR; Technical Report IT-IQC-2-97, see also: J. Comp. Chem. 1997,18, 2023-2039. 17. Jacobi, C. G J J. Peine Angew. Math. 1846, 30, 51-94. 18. Constans, R; Amat, L.; Carbo-Dorca, R. / Comput. Chem. 1997,18, 826-846. 19. Carbo, R.; Besalu, E.; Amat, L.; Fradera, X. J. Math. Chem. 1995,18, 237-246. 20. See for example: (a) Fradera, X.; Amat, L.; Besalu, E.; Carb6-Dorca, R. Quant. Struct.-Act. Relat. 1997, 16, 25-32. (b) Lobato, M.; Amat, L.; Besalu, E.; Carbo-Dorca, R. Estudi QSAR d'una familia de Quinolones; Technical Report IT-IQC-4-97. (c) Lobato, M.; Amat, L.; Besalu, E.; Carb6-Dorca, R. Structure-Activity Relationship of a Steroid Family using QSM and Topological QS Indices; Technical Report IT-IQC-8-97, see also: Quant. Struct-Act. Relat. 1997,16, 465-472. 21. Carbo, R.; Besalu, E. In Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches; Carb6, R., Ed.; Kluwer: Dordrecht, 1995, pp. 3-30. 22. Carb6-Dorca, R. Tagged Sets, Convex Sets and Quantum Similarity Measures; Technical Report IT-IQC-9-97, see also: J. Math. Chem. 1998, 23, 353-364. 23. Berberian, S. K. Introduccion al Espacio de Hilbert; Editorial Teide: Barcelona, 1970. 24. Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, 1955. 25. Encyclopaedia of Mathematics; Reidel-Kluwer: Dordrecht, 1987, Vol. 8, p. 249.

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26. See for example: (a) Encyclopaedia of Mathematics', Reidel-Kluwer: Dordrecht, 1987, Vol. 5, p. 126. (b) Zemanian, A. H. Generalized Integral Transformations; Dover Publications: New York, 1987. 27. Carb6, R; Besalu, E. J. Math. Chem. 1995,18, 37-72. 28. Carbd-Dorca, R. On the Statistical Interpretation of Density Functions: ASA, Convex Sets, Discrete Quantum Chemical Molecular Representations, Diagonal Vector Spaces and Related Problems; Technical Report IT-IQC-10-97, see also: J. Math. Chem. 1998, 23, 365-375.

PATTERN RECOGNITION TECHNIQUES IN MOLECULAR SIMILARITY

W. Graham Richards and Daniel D. Robinson

I. II. III. IV.

Abstract Introduction Two-Dimensional Representations Alignment Conclusion Acknowledgment References

73 74 74 74 76 76 76

ABSTRACT The speedup in molecular similarity calculations needed to cope with libraries of tens of thousands of compounds is achievable if we adopt techniques from pattern recognition and start with two-dimensional representations derived by nonlinear mapping of the three-dimensional distance matrices. Here we describe the use of invariant moments in this respect.

Advances in Molecular Similarity, Volume 2, pages 73-77. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 73

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W. GRAHAM RICHARDS and DANIEL D. ROBINSON

I. INTRODUCTION Molecular similarity has proved to be a useful tool since its introduction by Carbo et al.^ and later extensions to the use of molecular electrostatic potential^ and shape-^ as descriptors. In particular, the use of similarity matrices where every member of a series of compounds is compared with a lead compound or even with every other member of the series has been a powerful technique both for quantitative structureactivity studies'^ and as a measure of diversity. This utility was readily apparent when dealing with series of some tens of compounds. Now that combinatorial chemistry is providing actual libraries of tens of thousands of molecules and virtual libraries containing millions of compounds, new problems present themselves. These are difficulties of speed of calculation. If we are to align molecules optimally and then compute similarity, we need to gain an increase in speed of several orders of magnitude. In two-dimensional problems, such as optical character recognition, speeds that are appropriate can be achieved. Here we show how such pattern recognition approaches can be applied to molecular similarity if we can represent three-dimensional structures in two dimensions.

II. TWO-DIMENSIONAL REPRESENTATIONS The three-dimensional structure of a molecule may be represented by a distance matrix. We have shown^'^ that nonlinear mapping permits one to produce two-dimensional representations that retain the majority of the distance information from three dimensions. These figures are suitable material for pattern recognition.

III. ALIGNMENT There are two aspects to alignment of two-dimensional figures: putting the center of the figures at the same point (translational invariance) and rotating to achieve maximal similarity (rotational invariance). Borrowing the technique from pattern recognition, Hu's method^ of invariant moments may achieve both translational and rotational alignment. The method is based on a statistical analysis of the distribution and values of the property p to be compared. This distribution takes the form of the following equation: m

== J J yyp(;c, y)dxdy

for continuous data

or

% . =. -T^ T^ yVp(-^' y)

for discrete data

Pattern Recognition Techniques

75

Now from a uniqueness theorem due to Papoulis,^ provided that p(x, >^) is continuous, and has nonzero values only in a finite part of the x, y plane, moments of all orders exist and the sequence of moments m are uniquely determined by p(x, y). Conversely, it can be shown that the infinite sequence of m uniquely determines Inherent in the equations for generating the moments is the assumption that the property p(jc, y) is centered on the calculation coordinates. This may not be the case. However, let us define two parameters as follows: x=-

^0,0

y-

^0,1 ^0,0

These two parameters clearly give the center of the property p(x, y) in the calculation coordinate system. These have been shown to be consistently placed for all reasonably comparable systems. Then let us define the central moment |Lt„^ by |i

= J J (jc - Icfiy - 3^)^p(JC, y)dxdy

for continuous data

l^>,9 z> a =^^X La X (^ ~ ^y^y ~ >^)^P(-^' >^)

for discrete data

or

These central moments have the desired invariance to translafion. We have seen how aligning a structure along its principal axes enables us to remove any unwanted rotation. In the case of the moments Hu utilizes this property of the principal axes to rotate p(x, y) so that the property's distribufion is aligned along the calculation axes. He does this by forming a variety of combinations of the three second-order central moments and the four third-order central moments. In his paper Hu explains these combinafions, which are able to disfinguish between structures that exhibit mirror and rotational symmetry: ^ 0 = 1^0 + ^^02

^ 2 = ^Kl^so - ^1^12)'" + (3^^2i - |L^03)'' Tl3 = Vl(|ll3o + |Lli2)^ + (fi21 + ^^03)^'

./V

(M30 - 3fi,2)(H30 + ^^l2)[(^l3o + ^\T)^ - 3(Mai + ^03)^! + (3^21 - ^lo3)(^l2l + ^lo3)[3(^l3o+^\^^ - ^^i\ + ^03)^]

76

W. GRAHAM RICHARDS and DANIEL D. ROBINSON

^6

(3^12 - M(l^21 + |L^03)[3(|Ll3o + [i^^)^ - (^21 + ^03)^] These seven invariant central moments are all that is required to gain a high degree of sensitivity in pattern recognition. Indeed, in Hu's original paper he used only the first two invariant central moments to implement a crude character recognition system which by all accounts worked remarkably well. Utilizing the invariant central moments is fairly straightforward. All we have to do is to project our molecular property onto a grid and run through the calculations detailed above. This gives us a seven-dimensional vector X]^ 20 which represents the distribution of p^ 2Z)(^' y)- ^ ^ molecule to be compared can also be subjected to the same treatment, remembering that we no longer have to bother aligning A and B to get the correct answer. This yields a second seven-element vector r|^ 20The similarity, or more properly in this case the distance between the two molecules, is then given by the Euclidean distance of these two vectors in seven-dimensional space:

Clearly the above equation can be calculated in a fraction of the time required for the Carb6 or Hodgkin indices.

IV. CONCLUSION Using pattern recognition techniques of this type fuels the hope that we might be able to scan a whole library of compounds for examples that are similar to a chosen set of leads or other sources of ideas. At the same time we retain the essentials of the three-dimensional structure which are so important in the binding between a small molecule and its target receptor.

ACKNOWLEDGMENT This work was carried out in part pursuant to a contract with the National Foundation of Cancer Research.

REFERENCES 1. Carb6, R.; Leyda, L.; Amau, M. Int. J, Quantum Chem. 1980, 77,1185. 2. Hodgkin, E. E.; Richards, W. G. Int. J. Quantum Chem. Quantum Biol. Symp. 1987,14,105. 3. Meyer, A. M.; Richards, W. G. J. Comput. AidedMol Des. 1991,5,426.

Pattern Recognition Techniques 4. 5. 6. 7. 8.

77

Good A .C; So, S.-S.; Richards, W. G. J. Med. Chem. 1993, 36, 433. Barlow T. W.; Richards, W. G. I Mol Graph. 1995,13, 373. Robinson, D. D.; Barlow, T. W.; Richards, W.G. J. Chem. Inf. Comput. Set. 1997, 37, 943. Hu, M. K. IRE Trans. Inf. Theory 1962, 179. Papoulis, A. Probability, Random Variables and Stochastic Processes; McGraw-Hill: New York, 1965.

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TOPOLOGY AND THE QUANTUM CHEMICAL SHAPE CONCEPT

Paul G. Mezey

I. Introduction II. Topological Resolution and Molecular Shape III. Molecular Similarity Measures Based on Topological Resolution of the Shape of Electron Density IV. Summary Acknowledgment References

79 81 86 90 91 91

I. INTRODUCTION The concept of molecular shape is not based on direct observation. The size of most molecules is too small for visual examination; moreover, the wavelength range of visible light is not suitable to provide a detailed enough resolution for molecules. This fact has influenced in a fundamental way the special evolution of the molecular shape concept, since, in the absence of direct observation, the shape of molecules is usually perceived to be that of the molecular models used for their representation. Naturally, the early, somewhat simplistic molecular models, such as the "ball and

Advances in Molecular Similarity, Volume 2, pages 79-92. Copyright © 1998 by JAI Press Inc. Allrightsof reproduction in any form reserved. ISBN: 0-7623-0258-5

79

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PAUL G. MEZEY

stick" or fused sphere "space-filling" models, could reflect only some, highly simplified aspects of molecular shape, yet the shapes of these models have been treated by many chemists as if they were the actual shapes of the molecules. It is remarkable that, even today, one of the primary tools for conveying molecular shape information is the "ball and stick"-type stereodiagram used by many chemists. The more realistic, fuzzy, three-dimensional electron density cloud models, already easily calculable by quantum chemistry methods, only recently have started to become an appreciated tool for molecular shape representation. ^"^^ Molecular electron densities can be represented by three-dimensional density functions p(^, r), where ^ is a specified nuclear configuration and r is the three-dimensional position variable. With the introduction of the additive fuzzy density fragmentation (AFDF) methods,^^'^^ including the numerical MEDLA (molecular electron density loge assembler) technique^^""*^ and the more advanced analytical ADMA (adjustable density matrix assembler) method,"^^""^^ ab initio quality electron densities p(^, r) can be calculated, virtually for any molecule of chemical and biochemical importance, even for macromolecules, such as proteins."^^"^^ The shapes and similarities of these p{K, r) electron density clouds can be analyzed in great detail, and additional properties can be studied, such as the forces acting on the various nuclei in macromolecules,"*^ leading to conformational changes and changes of folding patterns of long, polymeric chain molecules. Electron density clouds are rather fuzzy objects, where any, detailed enough graphical representation of the peripheral, low electron density range is likely to hide from view the higher density range closer to the nuclei. This fact, and the fact that it is difficult to construct any macroscopic model that properly reflects the fuzzy nature of molecules have apparently contributed to the popularity of the simpler, and easily visuahzable "ball and stick" and fused sphere "space-filling" models, where molecules appear analogous to macroscopic, classical-mechanistic constructions. Whereas for classical-mechanical objects geometry is the appropriate tool of shape description, for fuzzy, quantum-mechanical molecules, geometrical methods are no longer efficient; by contrast, topology appears as an ideal tool of shape description. Topology allows for flexibility in a very natural way, and it also has the capacity to describe quantum-mechanical uncertainty and the associated fuzziness in a natural and intuitively transparent way. During the past decade, topology—specifically, algebraic topology—has been advocated as a powerful framework for molecular shape description that is compatible with the fundamental, quantum-mechanical nature of molecules. Among the relevant results, the introduction of the molecular shape group methods (SGM)^"^^ has led both to novel theoretical interpretation of molecular shape properties, as well as to various applications in molecular similarity and complementarity analysis in systematic pharmaceutical drug discovery approaches and in toxicological risk assessment.^^'"*^"^^

Topology and the Quantum Chemical Shape Concept

81

In this chapter, some of the more recent advances in the systematization of topological methods of molecular similarity analysis, involving the concept of topological resolution of molecular electron densities p(^, r) and their contour surfaces, will be reviewed.

II. TOPOLOGICAL RESOLUTION AND MOLECULAR SHAPE Before discussing the precise formulation of topological resolution and its applications to molecular similarity analysis, I shall briefly review the motivation for the approach and some of the relevant topological concepts. In the study of molecular similarity, structural features of molecules are the most commonly used properties that are analyzed and compared. Depending on the level of detail required, static shape features of molecules can be studied at various levels of resolution; these levels of resolution can be used as a tool for the introduction of various similarity measures.^^ Evidently, if three objects A, B, and C are indistinguishable at some low level of resolution, but at some higher level of resolution A is distinguishable from B and C, but B and C are still indistinguishable, then B and C are more similar to each other than to object A. Evidently, the level of resolution required to distinguish objects can be used as a measure of similarity. Since resolutions in the geometrical sense are easily characterized by numbers, this approach, the resolution-based similarity measures (RBSM) approach, provides numerical similarity measures.^^ Originally, the RBSM approach was formulated in terms of geometrical resolution,^^ for example, by placing objects on a regular, rectangular grid and using the occupied cells of the grid for comparisons. The finer the grid, the finer the resolution, and the grid size required served as a numerical measure of similarity. In this contribution I shall discuss some topological generalization of the idea of using resolution to measure the degree of similarity. The switch to topological resolution is the simplest if one focuses on the static shape features of molecules. However, static shape features provide a biased representation of the molecule, and in the evaluation of molecular similarity, it is also of importance to use information on chemical reactions and conformational changes, involving molecular interactions. Such interactions are typically determined by local molecular shape properties. In the long-range interactions during the initial stages of a chemical reaction, typically the large-scale features of local molecular moieties are dominant. However, as the reaction progresses, which usually involves a close approach of one molecule by another, details of local shape features become increasingly more important. Consequently, the level of resolution required for the analysis of shape features relevant in different stages of molecular interactions does change in the course of the reaction or conformational change induced by one molecule in another. Since the local shape features of both the static and the dynamic repre-

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PAUL G. MEZEY

sentations of molecules are characterized by their topological properties, it is natural to invoke the concept of topological resolution in molecular similarity analysis. In the following paragraphs some of the fundamental concepts of the relevant branches of point set topology are reviewed with special focus on topological resolution, followed by a description of a topological realization of RBSM. Consider a set X, and a family T of subsets T^ of X, X D 7^, where the members T^ of family T fulfill the following three conditions: (i)

X, 0 G T

(1)

that is, the original set X and the empty set 0 are included in the family T, furthermore,

(ii)

ur^eT

(2)

a

for any number of sets T^ in the family T, and

(iii)

T^r^TaeT

(3)

for any two sets T^, T^ e T. If properties (i)-(iii) are fulfilled, then the family T is called a topology on set X, and the members T^ of family T are called the T-open sets of set X. The pair (X, T) is called a topological space. The structure of set X provided with a topology T, and various functions defined on X can be studied in terms of the T-open sets of X. Note that nowhere in the above discussion was it assumed that set X has a geometrical structure, and there is no need even for a distance function for introducing a topology on a set X. Nevertheless, properties (i)-(iii) are precisely the most fundamental properties of open sets in a metric space, for example, in a Euclidean space provided with the ordinary, Pythagorean distance function. In a metric space, open sets are defined in terms of distance: A set Y is open within a metric space if for any point y of y one can find a ball of some nonzero radius, centered on the point y, such that the entire ball still falls within Y. Invoking balls with some nonzero radius involves the distance function of the metric space, since the ball is the collection of all points with distance from y less than the specified distance chosen as radius. What provides topology with a remarkable versatility is the fact that many of the properties of open sets, and the implied properties for continuous functions defined in terms of these open sets, are fully operational without any reference to distance. This provides a well-controlled flexibility, which is the special hallmark of topology. Furthermore, as evident from requirements (i)-(iii), on any given set X one can introduce many, different topologies. This provides a whole range of possible degrees of "flexibility," an important concern in the chemistry of actual, quantum-

Topology and the Quantum Chemical Shape Concept

83

mechanical, nonrigid molecules. Since there are many different ways a topology T can be chosen, in the study of the topological properties of any object X (for example, a molecule X), one must specify the actual topology T used. The following relations are crucial for the introduction of the concept of topological resolution. Consider a set X, and assume that for two topologies Tj and T2 on X the following relation holds: Every Tj-open subset of X is also a T2-open set. This implies that Tj is a subfamily of T2, T2DT1

(4)

If relation 4 holds, then topology Tj is said to be coarser (or weaker) than topology T2, or one can say that topology T2 is finer (or stronger) than topology Tj. Two topologies on a set X do not always relate to one another in such a clear-cut fashion; in fact, a relation such as 4 above is rather special. Two topologies are regarded incomparable if neither is finer than the other. The finer-coarser relation between topologies on a given set X gives only a partial ordering of the set of all topologies on the set X. Exploiting this partial order, the interrelations among topologies on a given set X can be studied using lattice theory. The detailed analysis of a topology, as well as the construction of new topologies on a given set X, can be carried out using the concepts of base and subbase of a topology T. Consider a subfamily B of family T: T DB

(5)

This subfamily Bis a, base for topology T on X if and only if every T-open set G e T is a union of some sets in B. Consider a subfamily S of family T: T D5

(6)

This subfamily 5 is a subbase for topology T on X if and only if finite intersections of elements of S form a base for T. The special role of subbases is illustrated by the fact that they can be used to define topologies. In such cases we refer to subfamily 5 as a defining subbase. By choosing a family of subsets of X as a subbase 5, and generating a base B by the above recipe (of generating allfiniteintersections), one can indeed generate a family T that fulfills all three conditions (i)-(iii). Note that a special finite intersection, the empty intersection of subsets of space X, is the full space X, consequently, X is automatically included in the base B generated by this recipe, and hence X is also included in the family T. The empty union of sets from the base B is the empty set 0 , hence the empty set 0 is automatically a member of the family T generated by this recipe. The subbase-base approach provides a very versatile method for generating topologies. Also note that, if for two generating subbases 5j and 5*2 the relation

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PAUL G. MEZEY

52=)5i

(7)

holds, then the corresponding topologies are necessarily comparable, and topology T2 isfinerthan topology Tj, T2DT1

(8)

Consider a set X and a family T of topologies T- on X, where these topologies T- are fully ordered by thefiner-cruderrelation: T={Ti,T2,...T,,...}

(9)

T,,,DT,

(10)

where

for every index / for which T-^j is included in the family T of topologies. We shall also use the notation (X,T.,i)D(X,T.)

(11)

to express the same fact in terms of topological spaces, if the specification of the underlying space X is important. If relation 11 holds, we say that the topological space {X, T-^j) is of higher topological resolution than topological space (X,T.). Topological resolution is defined in terms of thefiner-cruderrelations. In comparison 11, the former topological space, (X, T-^^), provides a more detailed topological description of the underlying space X than the topological space (X, T.). In particular, afiinction/thatmaps X onto itself may be T.-continuous but not T-^j-continuous (note that a function is continuous if and only if the inverse image of every open set is open, where openness is interpreted within the actual topologies used). A topological description with a higher level of topological resolution provides more information than one at a lower level of topological resolution. We consider the three-dimensional, fuzzy electron densities of molecules embedded in the ordinary 3D Euclidean space E^. The shape analysis of these electron densities, leading to the determination of the algebraic-topological shape groups, has been reviewed extensively,^^'^"^ and only a brief sununary will be given here. For each nuclear arrangement K, the molecular electron density function p(^, r) can be represented by an infinite family of molecular isodensity contour (MIDCO) surfaces G(K, a), where the density threshold a can take values from the [0, ©o) interval. Each MIDCO is defined as a set G{K,a) = {r:p{K,r) = a]

(12)

that is a surface with a specific shape for each nuclear configuration K and each value of the electron density threshold a along the MIDCO.

Topology and the Quantum Chemical Shape Concept

85

The local shape of each MIDCO G(Ky a) is tested against a range of reference curvatures b. For each reference curvature value b, the points r along each MIDCO G(K, a) are classified according to the local curvatures, as being a point where the contour surface is either 1. Locally convex relative to b (r belonging to a domain of type D2(b)), 2. Locally of the saddle type relative to reference curvature b (r belonging to a domain of type D^(b)), or 3. Locally concave relative to b (r belonging to a domain of type DQ(b)). For each MIDCO G(K, a), the domains D (b) for various curvature types jii with reference to a curvature value b generate a pattern P(K, a, b) of domains on the MIDCO G{K, a). These patterns P{K, a, b) can be analyzed by topological methods, leading to a description of the interrelations among the domains within each topologically distinct pattern P{K, a, b). Whereas no actual construction of new objects is needed, it is useful to picture the process of focusing on a given curvature type within the actual pattern P(K, a, b) by assuming that the corresponding domain is excised from the MIDCO surface G{K, a). For each MIDCO G(K, a), the domains DJb) of a specified curvature type \i with reference to a curvature value b are removed from the MIDCO, leading to a truncated object G {K, a, b) with a certain set of holes. The algebraic-topological homology groups of the truncated MIDCO G^{K, a, b) are denoted by H^JJC, a, b), and are, by definition, the shape groups of the molecule. There are three families of these groups, the zero-, one-, and two-dimensional shape groups, where in the notation H^(K, a, b) the dimension k, the truncation type \x, the nuclear configuration K, the density threshold a, and the reference curvature b are all specified. Note that some of these specifications are often omitted if they are evident from the context. The shape groups HHK, a, b) are invariant within small intervals of the threshold values a for electron densities, also within small intervals of reference curvature b, and also for some small molecular deformations changing the nuclear arrangement K. Note that, for a molecule of A^ nuclei {N > 3), the family of accessible nuclear configurations K form a subset of the nuclear configuration space M, where M is a metric space of 3N-6 dimensions. The local invariances of the shape groups H^(K, a, b) within the parameter plane {a, b) spanned by the density and curvature thresholds a and fc, and within domains of the nuclear configuration space M imply that there are only a finite number of shape groups for each molecule.^^'^'* Usually a separate shape group analysis is performed for each specified nuclear configuration K of interest. The finite number of shape groups HHK, a, b) within the parameter plane (a, b) can be characterized by their ranks, called Betti numbers, providing a numerical shape code for each conformation K of each molecule. In some sense, the topological shape group approach represents a reduction of the information content of a three-dimensional continuum of a fuzzy electron density

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cloud p(^, r) to a set of discrete Betti numbers, in a process that retains the essential shape information about molecules. The shape codes provide a concise representation of shape information. These shape codes can be compared numerically, in a process that is much simpler than the direct comparisons of molecular electron densities. Note that in direct density comparisons the mutual orientation of the molecules must be optimized in the initial step of shape comparison. No such optimum superposition is needed when evaluating the similarity of molecules based on their shape codes. These direct, numerical comparisons of the topological shape codes are used to compute numerical shape similarity measures and complementarity measures for the fuzzy electron density clouds of molecules. The shape group approach provides a detailed shape description and shape comparison. In most instances, such a detailed shape description is required to detect and interpret the shape features of the electron density p(^, r) relevant to a given chemical problem. However, in some cases, one does not need a complete shape analysis of the electron density p(^, r), and the focus can be shifted from the details to some of the more prominent shape features. Furthermore, for large molecules, the large amount of detail obtained in a complete shape group analysis of the entire electron density p(^, r) may render the computational task and the interpretation of the results cumbersome. In such cases, it is warranted to use alternative shape characterization methods where the level of detail studied can be appropriately modified. A natural condition that can be used to control the amount of detail is the level of resolution. In the next section, a new set of molecular similarity measures will be discussed, based on the concept of topological resolution of fuzzy electron density clouds p(^, r).

III. MOLECULAR SIMILARITY MEASURES BASED ON TOPOLOGICAL RESOLUTION OF THE SHAPE OF ELECTRON DENSITY Different ranges of the electron density threshold parameter a and of the reference curvature b provide a natural approach to shifting the emphasis between the local details and the large-scale features of the shapes of molecular electron densities p(^, r). For example, the MIDCO surfaces G{K, a) corresponding to low-electrondensity thresholds a usually exhibit less detail than the high-density contours G{K, a) running closer to the atomic nuclei in the molecule. However, for our present analysis, the role of range selection of the reference curvature parameter h is more important than the choice of density threshold a. When considering a reference curvature b of high negative value, one finds that at most points r along a MIDCO G{K, a), even for MIDCOs with high values of the electron density threshold a, both of the local canonical curvatures of the surface (that is, both eigenvalues of the local Hessian matrix expressing the local curvatures at point r) are greater than the reference curvature b. Consequently, most if not all

Topology and the Quantum Chemical Shape Concept

87

points r of the MIDCO G(K, a) belong to a curvature domain of type DQ(^) of relative concavity with reference to the curvature b. This, in turn implies that a truncation of type fx = 2 eliminates only a few domains or no domain at all from the MIDCO G(K, a). For example, if the local canonical curvatures at all points r of the MIDCO G{K, a) are greater than the reference curvature b, then no truncation occurs, resulting in the coincidence G2(K,a,b) = G(K,a)

(13)

and in the associated trivial group as shape group H\(K, a, b), Hl(K,a,b) = {0}

(14)

If the value of the reference curvature is gradually increased, eventually, more and more local canonical curvatures fall below the value b, and more domains become subject to elimination from the MIDCO G(K, a), resulting in topologically distinct objects, G2(K,a,b)i^G(K,a)

(15)

as well as in nontrivial groups as shape groups H\{K, a, b). In fact, more detail of the shape properties of the MIDCO G{K, a) of the molecular electron density function p(^, r) becomes accessible. It is possible to consider all of the topological changes of the truncated objects G2{K, a, b) as the value of the reference curvature b is increased, and this approach has merits if the goal is to detect the actual threshold values for b where the topological change occurs. However, it is computationally simpler if one focuses on a finite series of selected b values: b,,b2,...b,,.,.,b^

(16)

where fe, <£>,,,

(17)

and determines the truncated MEDCO G2(K, a, b) for each b- value. This leads to the series of truncated MIDCOs G^iK, a, b,\ G^iK, a, b^),...,

G^iK, a, b),...,

G^iK. a, bj

(18)

and, if the focus is not restricted to the curvature domains type D (b) of type ^l = 2, to the associated pattern series P(K, a, ^i), P(K, a, b^),...,

P(K, a, Z..),..., P(K, a, bj

(19)

Note that all of these patterns P{K, a,fc.)are generated on the same MIDCO G(K,a). Consider now the combined pattern

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PAUL G. MEZEY

P{K,a,b,...b;)

(20)

obtained by superimposing all patterns on G{K, a) involving the subsequence up to and including index /: P(K, a, b,l P(K, a, b^\ . . . , P(K a, b)

(21)

The set of DJ^b^ domains within each pattern P{K, a, b.) can be regarded as a defining subbase S{K, a, Z?.) for a particular topology T(K, a, b) on the MIDCO G(K,al S(K,a,b) = {D^(b;)}

(22)

each generating a formal topological space iG{K,alT{K,a,b))

(23)

The superposition of patterns on the same MIDCO G(K, a) generates finite intersections of domains belonging to various reference curvatures b-. Consequently, the subbase S\K, a,by.. b-) taken as the set of all such intersections of domains D^(bf) for / = 1, 2,. . . , / generates a topology T' that is the same as the topology T(K, a,by .. b^ obtained with a subbase S{K, a,b^... b^ defined as the union of subbases S{K, a, b^, S(K, a, b^,..., S{K, a, b-): S(K, a, by .. b) = u^i^,. S{K, a, bj)

(24)

S\K, a, by .. b) = S{K, a, by .. b)

(25)

where

hence T = T{K,a,by,.b)

(26)

Note that for the defining subbases S{K, a,by .. b^ of these topologies T(K, a,by ., bj) the relation S{K,a,by.,b.^,)^S{K,a,by..b)

(27)

must hold for any choice of index /, 1 < / < m - 1, as a consequence of the definition of subbase S(K, a,by.. fo.) as the union (24) of all of the individual subbases S(K, a, bj) of indices k up to and including the index /. Consequently, the corresponding topologies T(Ky a,by .. b) are also fully ordered, T(K, a, by .. Z7j D T{K a, by .. ^ i ) ^ • • • D T(A:, a, by .. Z?.) D . . . D T(A:, a, b^)

(28)

Topology and the Quantum Chemical Shape Concept

89

This complete ordering implies that these topologies T(K, a,by.. b.) provide a monotonia series suitable for a systematic adaptation of the techniques of topological resolution for the shape characterization of the MIDCO G(K, a), using gradually increasing topological resolution as index / sweeps over the interval [1, m]. Based on these topologies T(K, a,by .. b-), an RBSM, or more precisely, a topological RBSM (in short, TRBSM), can be constructed for molecules, as follows. Take two molecules, Mj and M2, of nuclear configurations K^ and K2, respectively, and consider two respective density thresholds a^ and a2> and the associated two MIDCOs G(K^, a^) and G(K2, a^. Generate the series of topologies T(A'p flj, Z7j... ^.) and T{K2, ^2, b^ .. ./?•), respectively, for the range 1 < / < m of indices /. We say that the two MIDCOs G{K^, a^ and G(^2' ^2) ^^^^ equivalent shapes at the level / of topological resolution if and only if there is a one to one and onto correspondence between the two defining subbases S{K^,a^,by..b^ and 5(A^2' ^2' ^r • • ^/) ^^^^ ^^^^ preserves the curvature index \k assignment of each element of the subbase for each sublevel k,\
eq[^i...Z7.] Gi^K^.a^)

(29)

where the relevant reference curvatures [by .. b-] are specified, or in short, by G(Kya^) eq GiK^.a^)

(30)

if the choice of reference curvatures and the resulting topologies used are evident from the context. Note that, if the first reference curvature b^ is chosen as a large enough negative value, then G(K,,a,)

cq[b^] GiK^.a^)

(31)

for practically any two molecules M^ and M2, of any two nuclear configurations ATj and K2, as long as both density thresholds are small enough so that both MIDCOs G{Ky a^ and G(A^2' ^2) ^^^ topological spheres, since then both patterns P{Ky a^ b^) and P(^2' ^2' ^1) contain only a single domain D^ib^). That is, a common starting point exists for any two molecules Mj and M2 at a rather crude level of topological resolution where their shapes appear equivalent, and this is not affected by how different the two molecules and their two nuclear configurations K^ and K2 actually are. However, by gradually increasing the resolution of the topologies, the shape differences gradually manifest themselves, and at some, possibly high positive value of the reference curvature b-, even slight conformational deviations of the same molecule are necessarily detected by the method, resulting in a nonequivalence of the shapes of the two

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PAUL G. MEZEY

MIDCOs at the corresponding level / of topological resolution, where this nonequivalence is denoted by G(K^,a^) noneq[foi... fc.] GiK^a^)

(32)

In general, a higher index / for equivalence implies a finer (stronger) topology, that is, a higher level of topological resolution. Equivalence at a higher level of topological resolution implies a higher level of similarity. For example, if for three molecules Mj, M2, and M3 the maximum indices of equivalence for the various pairings are /12 for G(A'j, flj) and G(K2, a^ /i3 for G{K^, aj) and G{Ky a^) 1*23 for G(i^2' ^2) ^^^ G{K^, ^3) where *23

(33)

then the similarity of the shapes of MIDCOs is the highest for the pair G(K2, a^ and G{K^, a^, and the lowest for the pair G{K^, a^ and G(^2» ^2)The similarity measure based on the above implementation of the concept of topological resolution is a tool that allows one to focus on the essential shape features of molecular electron densities in a systematic manner. By gradually increasing the topological resolution, finer and finer details of shape are detected by the method, and for each type of chemical problem, it is possible to select the level of resolution that reflects the relevant size, shape, and detail requirements. The quantity m - i^^ ^^Y serve as a dissimilarity measure, although m - i^2 is not a metric on the abstract space of all shapes, since for afixedselection of the sequence of reference curvatures, two actual shapes of slight geometrical differences may exhibit topological equivalence, unless the highest selected reference curvature value is sufficiently increased.

IV. SUMMARY After a brief review of the fundamental topological concepts of molecular shape analysis and topological similarity methods, a novel alternative method of similarity evaluation is described, based on the concept of topological resolution. The motivation and justification of the approach are presented in the context of the quantumchemical shape properties of fuzzy molecular electron density distributions. An implementation of the topological resolution method is described, utilizing the local convexity shape domains already available from the earlier approach of SGM.

Topology and the Quantum Chemical Shape Concept

ACKNOWLEDGMENT The original research reported in this account was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Mezey, P. G. Int. J. Quantum Chem. Quant. Biol. Symp. 1986,12, 113. Mezey, P. G. /. Comput. Chem. 1987, 8, 462. Mezey, P G. Int. J. Quantum Chem. Quant. Biol. Symp. 1987,14, 127. Mezey, P G. J. Math. Chem. 1988, 2, 325. Mezey, P G. In Concepts and Applications of Molecular Similarity; Johnson, M. A.; Maggiora, G. M., Eds.; Wiley: New York, 1990. Arteca, G.A.; Mezey, PG. Int. J. Quantum Chem. Symp. 1990, 24, 1. Mezey, P. G. In Reviews in Computational Chemistry; Lipkowitz, K. B.; Boyd, D. B., Eds.; VCH Publishers: New York, 1990. Mezey, P G. J. Math. Chem. 1991, 7, 39. Mezey P G. J. Math. Chem. 1992,11, 27. Mezey, P G. J. Chem. Inf. Comput. Sci. 1992, 32, 650. Luo, X.; Arteca, G. A.; Mezey, PG. Int. J. Quantum Chem. 1992,42, 459. Mezey, P G. J. Math. Chem. 1993,12, 365. Mezey, P. G. Shape in Chemistry: An Introduction to Molecular Shape and Topology, VCH Publishers: New York, 1993. Mezey, P G. J. Chem. Inf Comput. Sci. 1994, 34, 244. Mezey, P G. Int. J. Quantum Chem. 1994, 51, 255. Mezey, P G. Can. J. Chem. 1994, 72, 928. (Special issue dedicated to Professor J. C. Polanyi) Mezey, P. G. In Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches, Carbo, R., Ed.; Kluwer: Dordrecht, 1995. Mezey, P. G. In Molecular Similarity in Drug Design; Dean, P. M., Ed.; Chapman & HallBlackie: Glasgow, 1995. Walker, P D.; Mezey, PG. J. Comput. Chem. 1995,16, 1238. Walker, P D.; Maggiora, G. M.; Johnson, M. A.; Petke, J.D.; Mezey, PG. J. Chem. Inf Comput. Sci. 1995, 35, 568. Walker, P D.; Mezey, P G.; Maggiora, G. M.; Johnson, M. A.; Petke, J. D. J. Comput. Chem. 1995,16, 1474. Mezey, P. G. In Topics in Current Chemistry, Vol. 173, Molecular Similarity; Sen, K., Ed.; Springer-Verlag: Beriin, 1995. Mezey P G. In Advances in Molecular Similarity, Vol. 1, 1996, pp. 89-120. Zimpel, Z.; Mezey, P. G. Int. J. Quantum Chem. 1996, 59, 379. Mezey P G. J. Chem. Inf Comput. Sci. 1996, 36, 1076. Heal, G. A.; Walker, P D.; Ramek, M.; Mezey, P G. Can. J. Chem. 1996, 74,1660. Klein, D. J .; Mezey, P G. Theor Chim. Acta 1996, 94,177. Mezey, P G.; Zimpel, Z.; Warburton, P; Walker, P D.; Irvine, D. G.; Dixon, D. G.; Greenberg, B. J. Chem. Inf Comput. Sci. 1996, 36, 602. Mezey, P G. Int. J. Quantum Chem. 1997, 63, 105. Zimpel, Z.; Mezey, P .G. Int. J. Quantum Chem. 1997, 64, 669. Mezey, P G. Int. Rev. Phys. Chem. 1997,16, 361. Mezey, P. G. In Fuzzy Logic in Chemistry; Rouvray, D.H., Ed.; Academic Press: San Diego, 1997, pp. 139-223.

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33. Mezey, P.G. In From Chemical Topology to Three-Dimensional Geometry; Balaban, A. T., Ed.; Plenum Press: New York, 1997, pp. 25-42. 34. Mezey, P. G. In Conceptual Trends in Quantum Chemistry, Vol. 3; Calais, J.-L.; Kryachko, E. S., Eds.; Kluwer: Dordrecht, 1997, pp. 519-550. 35. Harary, E; Mezey, P G. Int. J. Quantum Chem. 1997, (52, 353. 36. Mezey, P. G. In Advances in Quantum Chemistry; Lowdin, P.-O.; Sabin, J. R.; Zemer, M. C , Eds.; Academic Press: San Diego, 1996. 37. Mezey, P G. Structural Chem. 1995, 6, 261. 38. Walker, P D.; Mezey, P G. Program MEDIA 93 (Mathematical Chemistry Research Unit, University of Saskatchewan, Saskatoon, Canada, 1993). 39. Walker, P D.; Mezey, P G. J. Am. Chem. Soc. 1993,115,12423. 40. Walker, P D.; Mezey, P G. J. Am. Chem. Soc. 1994,116,12022. 41. Walker, P D.; Mezey, P G. Can. J. Chem. 1994, 72, 2531. 42. Walker, P D.; Mezey, P G. J. Math. Chem. 1995,17, 203. 43. Mezey, P. G. Program ADMA 95 (Mathematical Chemistry Research Unit, University of Saskatchewan, Saskatoon, Canada, 1995). 44. Mezey, P G. J. Math. Chem. 1995,18, 141. 45. Mezey, P. G. In Computational Chemistry: Reviews and Current Trends; Leszczynski, J., Ed.; World Scientific Publishers: Singapore, 1996. 46. Mezey, P G. Int. J. Quantum Chem. 1997, 63, 39. 47. Mezey, P G. Pharm. News 1997,4, 29. 48. Mezey, P G.; Walker, PD. Drug Discovery Today (Elsevier Trend Journal) 1997, 2, 6. 49. Mezey, P G. In Soil Chemistry and Ecosystem Health; Huang, P M., Ed.; SSSA: Pittsburgh, PA, 1998, pp. 21-43. 50. Mezey, P. G. In Pauling's Legacy: Modem Theory of Chemical Bonding; Maksic, Z.; OrvilleThomas, W. J., Eds.; Elsevier: Amsterdam, 1998, in press. 51. Mezey, P. G. In Pauling's Legacy: Modem Theory of Chemical Bonding; Maksic, Z.; OrvilleThomas, W. J., Eds.; Elsevier: Amsterdam, 1998, in press. 52. Mezey, P G.; Zimpel, Z.; Warburton, P; Walker, P D.; Irvine, D. G.; Huang, X.-D.; Dixon, D. G.; Greenberg, B. M. Environ. Toxicol. Chem. 1998,17,1207. 53. Mezey, P. G. In Bolyai Society Mathematical Studies, 1998, in press. 54. Mezey, P G.; Fukui, K., Arimoto, S.; Taylor, K. Int. J. Quantum Chem. 1998, 66, 99. 55. Mezey, P. G. In Chemical Topology: Introduction and Fundamentals', Rouvray, D.; Bonchev, D., Eds.; Gordon & Breach: New York, 1998, in press. 56. Mezey, P. G. In Encyclopedia of Computational Chemistry; Schleyer, P. v. R.; AUinger, N. L.; Clark, T.; Gasteiger, J.; KoUman, P A.; Schaefer III, H. P.; Schreiner, P R., Eds.; Wiley: New York, 1998, Vol. 4, p. 2582. 57. Mezey, P. G. In Advances in Molecular Structure Research, Vol. 4,1998,4,115.

STRUCTURAL SIMILARITY ANALYSIS BASED ON TOPOLOGICAL FRAGMENT SPECTRA

YoshimasaTakahashi, Hiroaki Ohoka, and Yuichi Ishiyama

Abstract I. Introduction II. Methods A. Topological Fragment Spectrum B. Quantitative Evaluation of Structural Similarity Based on TFS III. Results and Discussion A. Structural Similarity Analysis of 42 Psychotropic Agents B. Subspectrum Use C. Application to Similar Structure Search in Chemical Database IV. Concluding Remarks Acknowledgment References

Advances in Molecular Similarity, Volume 2, pages 93-104. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 93

94 94 95 95 97 98 98 98 100 103 104 104

94

YOSHIMASA TAKAHASHI, HIROAKI OHOKA, and YUICHI ISHIYAMA

ABSTRACT This paper discusses an approach to structural similarity analysis of chemical compounds using the topological fragment spectrum. The latter is based on the enumeration and numerical characterization of all possible substructures from a chemical structure. Here, the subspectra based on structural fragments having up to a specified size (number of edges) were also used to describe the topological profile of a chemical structure. The performance of the approach was examined in a structural similarity analysis of 42 psychotropic drugs. The appHcation of this method to similar structure searching in a chemical database is also discussed.

I. INTRODUCTION The use of molecular similarity methods, especially structural similarity, is under active development in the area of drug design, for the selection of analogues for chemicals and for the estimation of molecular properties. ^'^ The rationale is that structurally similar compounds are likely to possess similar molecular properties and similar biological activities. There are two different approaches for structural similarity analysis. One approach focuses on the identification of common and similar parts between the molecules that are to be compared. There have been several important attempts to determine structural fragments that are common among a group of chemical compounds.^"^ This is also closely related with the maximal common substructure (MCS) problem.^'^^ In these studies, for a group of compounds that show the same or similar biological activities although they are different in chemical structure, it is assumed that the activity in question should be attributed to a particular molecular feature that they all have in common. ^^ The other approach focuses on the degree of similarity between structures. This is a problem for a quantitative evaluation of the similarity of their chemical structures. Substructural analysis described by Cramer et al.*"^ and Adamson and Bush^^ gives us a basis for approaching this problem. It is based on the correlation of activities and properties of chemical compounds with the occurrence of particular partial structures that are embedded within their structures. Adamson and Bush^^ also reported the quantification of the similarity between chemical structures on the basis of their substructural analysis. In their work, 39 local anesthetics were classified numerically by calculating similarity or dissimilarity coefficients between pairs of the structures and applying cluster analysis to the obtained results. In using such an approach, Willett and Winterman^^'^^ examined the effectiveness of various measures for the evaluation of molecular similarity for the clustering of chemical compounds. These approaches are based on the generation of substructural descriptors from chemical structures and their use in quantitative evaluation by any similarity measure. Graph theoretical analysis has also been applied to the ordering of chemical structures and followed by correlative analysis of molecular properties.*^'^^ A

TFS-Based Structural Similarity Analysis

95

topological index is a numerical value representing the topological nature of molecular structure. The structural similarity or dissimilarity has been examined using some kinds of topological indices to provide the ordering of molecules."^^'^^ However, in many cases, substructural descriptors are more often used to define intermolecular similarity.^^"^'^ In general, topological/2D substructure descriptors are extremely useful for large-scale data because most chemical databases contain only connection tables without three-dimensional information. Recently, Brown and Martin^^ have evaluated a variety of substructure-based clustering methods for use in compound selection. The results suggest that 2D descriptors and hierarchical clustering methods yield better results than 3D descriptors and other methods. However, quantification of such structural similarity sfill depends on the chosen set of substructures defined as the descriptors in advance. Nevertheless, the concept of structural similarity is quite important for the further intelligent use of computers in the chemical field. In that case, an approach is required to process structural informafion in a more flexible way so as to allow somehow the automatic evaluation of the more ambiguous global structural similarity; in other words, a method to examine the similarity of structures when they are regarded as whole enfifies. Here we describe an alternafive approach using the exhaustive fragmentation profile of a chemical structure for the quantification of structural similarity or diversity of chemical compounds.

II. METHODS A. Topological Fragment Spectrum

Topological fragment spectrum (TFS) has been proposed by the authors^^ as a tool for the description of the topological structure profile of a molecule. The approach is based on enumeration of all possible substructures from a chemical structure and numerical characterization of them. The procedure involves two major steps: (1) enumeration of all possible substructures from each chemical structure and (2) numerical characterization of the substructures. Both are described in the following section. Enumeration and Characterization of the Substructures

For a given structure represented as a chemical graph (hydrogen-suppressed graph), all of the possible substructures embedded in it are enumerated. Here, all of the substructures of the parent structure (the original chemical graph) are taken into account in the following processes even if they are isomorphic. Subsequently, all of the individual substructures are characterized with a numerical quantity. To perform this characterization we have used two methods in the present study as follows: (1) the overall sum of degrees of the nodes composing each subgraph and (2) the overall sum of the mass numbers of the atoms (atomic groups) corresponding

YOSHIMASATAKAHASHI, HIROAKI OHOKA, and YUICHI ISHIYAMA

96

to the nodes of the subgraph. With the first method the chemical structure is represented by a simple graph where the hydrogen atoms are omitted, the characterization of the structure thus depending only on the topology of the structural skeleton. An illustrative scheme of the procedure is shown in Figure 1. For the second method, attached hydrogen atoms are taken into account as augmented

S(e)=0

O S(e)=4

Occurrances

(b)

o 1 2

3

4

5

6

7

8

Sum of the degrees of nodes

(c)

X = (3,1,2,2,2,3,3,1)

Figure 1. An illustrative scheme forthe generation of a topological fragment spectrum (TFS). (a) Enumeration of all possible subgraphs from a chemical graph of 2-methylbutane. (b) TFS characterized by the sum of degrees of nodes in the subgraphs, (c) Vector representation of the TFS.

TFS-Based Structural Similarity Analysis

97

atoms and are represented by weighting correspondingly their respective nodes in the graph. Description of the Structural Profile by a Spectral Representation

We define as "topological fragment spectrum" (TFS or fragment spectrum in what follows) the histogram resulting from displaying the frequency distribution of a set of individually characterized substructures (structural fragments) according to the value of their characterization index. The fragment spectrum generated according to this manner is a representation of the topological structural profile of a molecule. The fragment spectra of promazine as characterized by two methods mentioned above are shown in Figure 2. B. Quantitative Evaluation of Structural Similarity Based on TFS

Obviously, the fragment spectrum obtained by these methods can be described as a kind of multidimensional pattern vector. Consequently, using this pattern representation of a spectrum it is possible to apply various quantitative measures

2000

IBOO

1000

"[ IT

BOO

c

M.MMlllll 10

20

1- "\"

1 30

40

(a) Fragment spectrum characterized by the sum of degrees of nodes in fragments.

Q^''\

S

N-/

]

,

1 0

1J [

ijUiJiXLIiMiil,ili, 11 .1 100

200

(b) Fragment spectrum characterized by the sum of atomic mass numbers in fragments. Figure 2. Promazine fragment spectra generated by different characterization methods.

98

YOSHIMASA TAKAHASHI, HIROAKI OHOKA, and YUICHI ISHIYAMA

for the evaluation of similarity. In the present work, the following measure was used for evaluating the similarity: D(X,,X^

S(X,Xp =

(1)

l — ^ max

where X- and X- are pattern vectors representing the fragment spectra of the iih and jth molecule, respectively. Z)(X., X) is the Euclidean distance between patterns Xand Xj. D^^ is the maximum distance among the set of molecules under analysis. The different dimensionalities of the spectra to be compared are adjusted as follows: If X. = (x.j, x.^,...,

x.^) and Xj = (Xj^, Xj^,..., Xj^, Xj^^^^y . . . , Xj^) (q
then X. is redefined as X. = (x-j, .x.2,..., x.^, x^^^^^y . . . , x.^) h e r e , A:,(^+I) = ■^/(^+2) =" • • • ~ ^/p = ^

This approach was fully computerized and used in the following structural similarity analysis and similar structure searching in a chemical database.

III. RESULTS AND DISCUSSION A. Structural Similarity Analysis of 42 Psychotropic Agents

The approach described above was applied to the structural similarity analysis for 42 psychotropic agents. The sum of the atomic mass numbers was employed in the characterization of substructures. The evaluation of their structural similarity was carried out with Eq. 1. The result obtained was summarized in a spanning tree of 42 psychotropic agents in the nearest-neighbor manner. The spanning tree is shown in Figure 3. Each link in the spanning tree shows the most similar pair of structures. It is difficult to conclude whether the resulting intermolecular similarity relationship is reasonable or not. On the basis of our intuitive analysis, it seems that the present approach yields a successful result. It is valuable to note that promazine (structure 7 in Figure 3) and its analogues were not clustered in a clear or rational manner. The first five nearest neighbors of promazine were compounds 27,31,19, 29, and 9 in this order. It is considered that there are too many "zero" elements to adjust for the different dimensionality between the fragment spectra that were being compared producing an unfavorable result. B. Subspectrum Use

In the previous section, structural similarity analysis was carried out for 42 psychotropic agents using their full fragment spectra. However, the computational time required for the exhaustive enumeration of all possible substructures from a chemical structure is often very large especially for molecules that contain highly fused rings. In addition, a large difference in the dimensionality between the

TFS-Based Structural Similarity

Analysis

99

_a Q _ 2 . _ \ o "9 6" °to °^o •o

s

0

°-o«-~

X

X)

P

(S-K -

T\

o 11

^ ^"x/ \

'X}

[^ J

N^

X)

L y

tV

V"0

X)

A

r 16

I OX)

^

39 1

&X)

^ \

Figure 3. A spanning tree of 42 psychotropic agents based on their structural similarities. The analysis was carried out with the full fragment spectra in which every fragment Is characterized by the fragmental weight.

fragment spectra to be compared may lead to an unexpected result. To avoid these problems, an alternative approach based on the use of subspectrum may be employed for such a similarity analysis, in which each spectrum can be described with structural fragments up to a specified size in the number of edges (bonds). Figure 4 shows an example TFS subspectrum for promazine. In this way, the similarity analysis was carried out for the same data set used above. The result of the structural similarity analysis was also summarized as the spanning tree shown in Figure 5, which seems to show more reasonable interrelationships for their structural similarity. In fact, for this case the first five nearest neighbors of pro-

100

YOSHIMASATAKAHASHI, HIROAKI OHOKA, and YUICHI ISHIYAMA

Full fragment spectrum of promazine

■li,.i....llli

id.^

Subspectrum for the set of fragments with S(e)^5. Figure 4. TFS full spectrum of promazine and its subspectrum derived from the small fragments. The spectrum is exemplified for fragments having 5(e) ^ 5.

mazine (7) were compounds 31, 32, 30, 38, and 33 in this order. Figure 6 shows their subspectra and that of promazine. C. Application to Similar Structure Search in Chemical Database

Obviously the TFS can be applicable to similar structure searches in chemical databases. To test the applicability of the TFS approach to this problem, we prepared a structure database consisting of 1600 perfume and flavor chemicals taken from Arctander's compilation.^^ The database contains a variety of molecules (e.g.,

TFS'Based Structural Similarity Analysis

101

■o

•o o

6 31

\ 7

^'r

»

ir-LJT

^

--v-A^o

"O

s-Q

s-Q

j-Q

,db

Figure 5. Result of similarity analysis for 42 psychotropic agents using TFS subspectra.

alcohols, terpenes, steroids) that are structurally diverse. The TFS subspectral approach was employed for the similar structure search. All of the spectra were characterized for fragments having five or less bonds within them. These spectra of the molecules in the database were generated and stored, in advance, as an index. A result of the similar structure search is shown in Figure 7, in which decamethylene oxalate was used as a query structure. Figure 7 shows the first ten most similar compounds to the query structure, the compounds being ranked according to their similarities. It is quite difficult to evaluate the result for the similar structure search, because the similarity evaluated depends on an individual's interests, experiences, and so on. Furthermore, it often depends on one's psychological situation. Nonetheless, we still believe that this result shows the structural similarities of these compounds very well. In conclusion, these results suggest that our approach succeeds in the evaluation of structural similarity or structural diversity of organic compounds in place of the

102

YOSHIMASA TAKAHASHI, HIROAKI OHOKA, and YUICHI ISHIYAMA

lOf

I

Ill.ll..

Il.lll.l.lllllllll.l

ii

mill

llllil,,!..

lllUJ

,A,jm

llUUJ

k

lliJI

jiw^

I

iiiiii,,

iiiiiiii....iiii,iiii„,i

Figure 6. The first five nearest neighbors of promazine (in the order shown) and their TFS subspectra from fragments having S(e) ^ 5.

TFS-Based Structural Similarity

103

Analysis Query structure

M -0

0-

Decamethylene oxalate

-0

0-

10

Figure 7. Result of similar structure search in a structure database of flavor and perfume chemicals.

human eye and brain. The current approach is also applicable to similar structure search on a chemical compound database.

IV. CONCLUDING REMARKS TFS can be viewed as a useful tool to describe topological structural profiles of a molecule. The work presented here apparently suggests that it can be applicable to structural similarity analysis in a quantitative manner. TFS is also useful for similar structure search in chemical databases. The advantages are that a molecule's TFS can be measured by a PC from an ordinary connection table, and it does not require any kind of a priori substructural definition like a fragment dictionary to generate it. In the present work, we used fragment weight for the characterization of structural fragments that are enumerated. It allows us to numerically describe the structural profile of a molecule. This approach can also be easily extended to use other characterization schemes for the fragments.

104

YOSHIMASA TAKAHASHI, HIROAKI OHOKA, and YUICHI ISHIYAMA

ACKNOWLEDGMENT This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan.

REFERENCES 1. Johnson, M. A.; Maggiora, G. M., Eds. Concepts and Applications of Molecular Similarity; Wiley: New York, 1990. 2. Sen, K., Ed. Molecular Similarity I, Top. Cum Chem. 1995, 173. Sen, K., Ed. Molecular Similarity II, Top. Cum Chem. 1995,174. 3. Carhart, R. E.; Smith, D. H. J. Chem. Inf. Comput.Sci. 1985, 25, 64. 4. Bawden, D. In Concepts and Applications of Molecular Similarity; Johnson, M. A.; Maggiora, G. M., Eds.; Wiley: New York, 1990, p. 65. 5. Judson, P. N. J. Chem. Inf Comput. Sci. 1992, 32, 657. 6. Barnard, J. M; Down, G. M. J. Chem. Inf Comput.Sci. 1992, 52, 644. 7. Down, G. M; Willett, P. Rev. Comput. Chem. 1995, 7, 1. 8. Lipkus, A. H. J. Chem. Inf Comput. Sci. 1997, 37, 92. 9. Varkony, T. H.; Shiloach, Y; Smith, D. H. / Chem. Inf Comput. Sci. 1979, 79,104. 10. Klopman, G. J. Am. Chem. Soc. 1984,106, 7315. 11. Takahashi, Y; Sukekawa, M.; Sasaki, S. J. Chem. Inf Comput. Sci. 1992,32, 639. 12. Bayada, D. M.; Johnson, P. In Molecular Similarity and Reactivity; Carb5, R., Ed.; Kluwer: Dordrecht, 1995, p. 243. 13. Takahashi, Y Top. Cum Chem. 1995,174, 105. 14. Cramer, R. D., Ill; Redl, G; Berkoff, C. E. J. Med Chem.. 1974,17, 533. 15. Adamson, G. W; Bush, J. A. Nature 1974, 248, 408; J. Chem. Inf Comput. Sci. 1975, 75, 215. 16. Adamson, G W; Bush, J. A. J. Chem. Inf Comput. Sci. 1975, 75, 55. 17. Willett, R; Winterman, V. Quant. Struct.-Act. Relat. 1986, 5, 18. 18. Willett, P. Similarity and Clustering Technique in Chemical Information System; Research Studies Press: Letchworth, England, 1987. 19. Kier, L. B.; Hall, L. H. Molecular Connectivity in Chemistry and Drug Research; Academic Press: New York, 1976. 20. Randic, M. / Math. Chem. 1991,17,155. 21. Randic, M. J. Chem. Inf Comput. Sci. 1992, 32, 686. 22. Downs, G. M.; Gill, G. S.; Willett, P; Walsh, P SAR QSAR Environ. Res. 1995, 3, 253. 23. Matrin, E. J.; Blaney, J. M.; Aiani, M. A.; Spellmeyer, D. C ; Wong, A. K.; Moos, W. H. J. Med. Chem.l99S,38,U3\. 24. Karcher, W; Karabunarliev, S. J. Chem. Inf Comput. Sci. 1996.36, 672. 25. Kearsley, S. K.; Sallamack, S.; Huder, E. M.; Andose, J. D.; Mosley, R. T; Sheridan, R. P J. Chem. Inf Comput. Sci. 1996, 36, 118. 26. Shememskis, N. E.; Weininger, D.; Blankley, C. J.; Yang, J. J.; Humblet, C. J. Chem. Inf Comput. Sci. 1996,36, 862. 27. Sheridan, R. P; Miller, M. D.; Underwood, D. J.; Kearsley, S. K. J. Chem. Inf Comput. Sci. 1996,56,128. 28. Brown, R. D.; Martin, Y C. / Chem. Inf Comput. Sci. 1996, 36, 572. 29. Takahasi, Y; Ishiyama, Y In Molecular Similarity and Reactivity; Carbo, R., Ed.; Kluwer: Dordrecht, 1995, p. 311. 30. Arctander, S. Perfume and Flavor Chemicals I, II1969.

ANALYSIS OF THE TRANSFERABILITY OF SIMILARITY CALCULATIONS FROM SUBSTRUCTURES TO COMPLEX COMPOUNDS

Guido Sello and Manuela Termini

Abstract I. Introduction 11. Methodology III. Results and Discussion A. ED Variations: Changing Structure B. ED Variations: Changing Conformation C. ED Variations: Changing System D. Similarity Index: 2D and 3D E. Similarity Index: SP Influence F. Similarity Index: Juxtapositioned Pairs G. Similarity Index: Triplets IV Calculations, Transferability, Similarity Measures and Indices V Conclusion Acknowledgments References and Notes Supplementary Material Advances in Molecular Similarity, Volume 2, pages 105-136. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 105

106 106 108 112 112 118 120 125 126 128 129 130 131 132 132 133

106

GUIDO SELLO and MANUELA TERMINI

ABSTRACT The possibility of transferring calculation results from simple, easy-to-handle structures to complex compounds containing many atoms that can be dissected into pieces is attractive but can produce unwanted side effects. It is therefore necessary to check thoroughly the validity of the transfer. The problem becomes more intriguing in the field of similarity evaluation where an elusive concept must be transformed into a quantitative measure. The results obtained from a series of calculations are reported and discussed in this perspective and the validity of the introduced approximation is emphasized. The level of the simplification that can be used with confidence is consequently derived. The calculations are made using an empirical methodology for evaluating molecular electronic distribution and, as a consequence, molecular electronic energy.

I. INTRODUCTION The possibility of transferring the calculations done on simple substructures to complex compounds has been studied and used in many fields of molecular modeling. It is sufficient to cite the parametrization of molecular mechanics or semiempirical methodologies, most of the time relying on calculations made on simple compounds at the ab initio level. More recently, the transfer of fragment values has been employed also in drug design^ using structural characteristics as bond lengths or atomic residual charges. The acceptability of those transfers has always been ensured by comparison of "true" and approximated calculations. Thus, it is common practice to "accept" a transfer if the results obtained at different levels of approximation are in agreement or if the property calculated by the model is correctly represented. Similarity between organic compounds has been used in the past as a purely qualitative concept^ whose importance was seldom clearly identified. Thus, it was very common to speak about "similar reactivity" or "similar geometry" without attaching any particular meaning to it. In the late 1980s a different and more powerful view on similarity appeared and is now rapidly developing. The application fields are various and spread over data management as well as quantitative biological activity prediction.^ This last area of similarity application can concern the specific case of QSAR measures as well as the wider and more general subject of drug design. It is not difficult to understand why similarity has been introduced in this field. In general we can identify two different methodological approaches to the problem of drug design, in accordance with the information available on it: 1. Those approaches where the structure of the receptor is known; in this case the goal is the identification of compounds that can fit in the best way with the receptor (limited utility of similarity).

Transferability of Similarity Calculations

107

2. Those where, on the contrary, the structure of the receptor is unknown; in this case the only possibility is to look for compounds having functionalities that can provide a specific activity. This second case can suggest the joining of similarity with computational techniques introducing a new methodological approach."* The quantitative measure of similarity brings to light the problem of its transferability. From a qualitative point of view it is instinctive to combine the similarity of small pieces so as to assess the similarity of the whole. This is obvious in reactivity as well as in biological activity. The combination is possible unless the perturbations introduced by the added piece are so important as to change the characteristics of the entire construction. Then all of the general difficulties related to the possibility of transferring values must be considered and a way of validation found. In the case of similarity measures the validation is very difficult by experimental proofs because similarity cannot be directly measured and the validity of its connection to a property is always questionable. Therefore, the most reliable approach must use calculation comparisons. In recent years we became interested in the usage of the similarity concept,^ mainly following our ongoing interest in chemical reactivity.^ Our approach considers the possibility of defining the similarity between substructures using their electronic description, more exactly, the perturbation induced by each atom or atomic group on the electronic energy of the substructure. Thus, we developed an original method^ that can quickly calculate electronic energy differences. The method is based on a previously described approach to atomic charge calculation^ and on the well-known relation existing between chemical potential and electronic energy.^ The energy differences can then be used to evaluate the similarity between substructures. We applied this approach to different substructure classes: from simple functional groups to complete molecules^. In our first analysis we examined topological (2D) molecular representations only; but the extension of our method for electronic distribution calculation to 3D representations^^ has stimulated subsequent work on the subject. First we repeated the calculation onto 3D substructure representations and compared the results; then we calculated the similarity to get a feeling as to the response of the approach to the introduced change. ^^ After having verified the reliability of the calculation, we went back to the analysis of one particular class of compounds that we have studied previously: the nucleic bases. The genetic information contained in nucleic acids is formally based on the juxtaposition of the sequence of base triplets furnishing the correct composition of the genes and of the corresponding proteins. This means that the very long chains of bases present in genes can be subdivided into sequences of triplets. As there are only four fundamental bases their combination can give only 64 different triplets. The consequence is a very high level of repetition of similar substructures. Therefore, we examined the possibility of transferring the calculations done on

108

GUIDO SELLO and MANUELA TERMINI

simple substructures to complex compounds using those substructures. To verify this possibility we analyzed the influence of different factors onto the results. As a final result we expected to get sufficient data supporting our hypothesis and therefore an easy way to simulate complex compounds without resorting to cumbersome calculations.

II. METHODOLOGY To be better understood we will briefly describe our calculation methodology. We developed an empirical method for calculating atomic charges. It is based on Gordy's approach^^ to the calculation of atomic electronegativities, i.e., on the formula X = 0 . 3 1 x ( n + l ) / r + 0.50 where n is the number of valence electrons; as modified by Pritchard^^:

where a and b are constants depending on the element row and Z^^ is calculated from Slater. We extended this formula to nonconstant Zs and rs, obtaining X = axZ^^^i)/r(i) + b where Z and r refer to a hypothetical perturbed state, Z^^ is naturally related to the effective (also fractional) electron number, and r is connected to the same factor by Pauling's equation^"*: r(0 = rxZ,ff/Z,ff(0 where Z' is equal to Z for complete electron shielding. Another Pauling equation correlates bond ionic percent to % difference: % = 100 X (1 - exp(-l/4 X Ax^)) and, as a consequence, (2 = ^x(l-exp(-l/4xA%^)) The equation can be extended to multiple interactions giving G T = 2:,G(0

At X equalization we reach a stable state, yielding the atomic charges. Given the correlation between electronic energy (e.e.), % (or chemical potential |Li), and hardness (r|) Ti = [diJJdn]z= [d\&.&.)/dn\

Transferability of Similarity Calculations

109

and considering the equation for % calculation as a continuous function (at least in the valence shell) we can derive the corresponding formulas for hardness and e.e. by the following mathematical manipulations: Z^ff = A^ - (aN^ + bN^ + c(A^3 - 1))

\x = -X = -k^x Z^^/r + ^2 = -^1 X [N-(aN^ + bN2 + c(N^ - l))]/r + ^2 = -k^ x[N- (aN^ + bN2 + c(N^ - 1))] x [TV - (aN^ + bN^ + cN^WiHj = k^x[N-

x r^) + k^

(aN^ + bN2 + c(N^ - 1))] x [A^ - (aN^ + bN2 + cN^)] - k^

and Ti = [dvi/dn\ = [k^ X {N^ x (A^3 - \))/N^ - k^]^ = ^3 x {2N^ - 1) e.e. = J |Lu//i = J \xdN^ = \k^ X [A^- (aN^+bN2-^c(N^ - 1))] x [A^- (aA^^ + Z7A^2 + ^^3)] "^^2 = ^3 X (A + 5 + O - /:2 X A^3 + A: where A = {N'^ + aN-2NN^-2bNN2'^N\

+2bN^N2'-aN^ +b^NJ

-abN2)xN^

B = 0.5x {-2aN + 2aN^ + 2abN2 - a^) x A^^

C = a^^xNl The formulas obtained permit the calculation of the atomic hardness and of the e.e.; concerning the latter quantity the corresponding molecular energy is simply the sum of the atomic contributions. Each molecule, each perturbation of a molecule, each transformation of a molecule, has a different energy because the different electron distribution influences the occupancy of Slater's shells and, consequently, the energy calculation. Nevertheless, the direct comparison of the molecular electronic energies does not give information on molecular similarity; in fact, while, for example, the energy of two molecules of linear alkanes (C4 <-> C^) constantly increases thus allowing their comparison, there is no guarantee that the energy of different molecules is different. On the contrary, a much more powerful source of information is furnished by the analysis of the variations of the electronic energy, mainly if it is compared on many points. It is the use of these variations that allowed us to introduce the already described new definition of functional groups. The immediate next step is

110

GUIDO SELLO and MANUELA TERMINI

the application of the just introduced concept to the analysis of the similarity between substructures. In fact, an interesting aspect of our approach is represented by the calculation mechanism. This is based on a very simple idea: If we think that the electronic energy of a molecule is strictly related to the energy of its atoms as results from their reciprocal interactions, we can affirm that the presence/absence of an atom at a specific position determines an energetic variation which can be thought of as the quantification of the difference between that molecule and a hypothetical molecule where the atom is isolated (annihilation principle). Therefore, if we consider two identical molecules and we work on atoms in complete correspondence, the energy variation will be identical; on the contrary, if we consider two different molecules and we work on similar atoms, or if we consider two identical molecules and we work on different atoms, the calculated energy differences will be different and they will represent a measure of the "importance" of the atom in the molecule. Finally, concerning the nucleic bases, we introduced a similarity index calculated as follows:

/=i:,w, where A^. is the ED calculated for atom /, obtaining values that are representative of the stabilization level given by the base pairing. To get the most information we sequentially analyzed structures of increasing complexity, beginning from a neutral isolated base in 2D and ending with a charged triplet in 3D. In our approach the efficiency of the base pairing depends on the similarity gain that results after pairing in comparison to the isolated compounds; we also postulated that a model for the pairing mechanism can be represented by the starting compounds with a charge on the atoms involved in the pairing mechanism (Figure 1). Passing from 2D to 3D representations we needed to be completely confident about the transferability of the calculated values, and therefore thoroughly examined the perturbation carried by each possible change in the analysis methodology. The first change concerns the differences related to the transformation of the 2D into the 3D approach. Those effects were initially examined on neutral and charged isolated bases considering the changes both of the energy differences and of the similarity indices. The 3D structures are built using a standard model builder^^ without geometry optimization because of the complete rigidity of the compounds. A methyl group takes the place of the substituent on the nitrogen atom usually carrying the sugar-phosphate residue. As a second step we considered the same bases but with the correct substitution (having interest in the examination of RNA bases, the sugar is a ribose residue) to verify its influence on the calculation. The structures are considered both without

Transferability of Similarity Calculations

111

U -tfi H-HN

H-HN

Figure 1, Base-base pairing and its model simulation.

and with geometry optimization.^^ The phosphate residue is always in the correct physiological ionic form. In our model we simulated the interaction between bases by charging the atoms involved in the coordination. To ensure the reliability of such approximation we also built some base pairs, and then calculated their similarity and compared the results with those from the charged isolated bases. The calculations were made in 2D and in 3D. In this regard a further problem can occur: the geometry of the nitrogen atom in the NH2 group. In fact, we found that the model builder uses, as a standard, a planar NH2 group and that such geometry is also usually reported in

H /

\

H

Figure 2. Standard nucleic base chain conformation. The triplet shown is GUG.

112

GUIDO SELLO and MANUELA TERMINI

biological papers. As we would have liked to eliminate all doubts concerning the correctness of this assumption, we also calculated our energy differences using tetrahedral nitrogens. The last but very important problem in the transfer of results is represented by the effects that vicinal bases can produce in a long base chain. More precisely, we would like to look at the interactions between bases in triplet chains particularly in the case of 3D calculations. Therefore, all 64 possible triplets were built and calculations made on both neutral and partially (i.e., with charged central bases) charged bases. We adopted the standard conformations of the builder^^ (Figure 2).

III. RESULTS AND DISCUSSION In Figure 3 the bases used for the analysis are shown together with the numbering of the examined atoms. For the sake of clarity we will compare ED variations and similarity index variations separately. A. ED Variations: Changing Structure The resulting presentation and discussion will be articulated as shown in Scheme 1; each branch of the 2D tree is compared with the corresponding branch of the 3D

*01

5

*0

1 R

NH2*

I R NH2*

I R *0

R

R OH*

*HNA^N

*0

-N^N--^^

*HN-^^

I X > 11 R OCK,

I I>

k

R

*0

I X> R

ID R

Figure 3. Bases used in the calculations with atom numbering. Starred atoms are charged when simulating pairings. The bases shown are (from left top to right bottom): thymine (T), uracil (U), cytosine (C), adenine (A), guanine (G), adenine tautomer (A ), thymine tautomer (T ), inosine (I), methyl-guanine (M), xanthine (X).

Transferability of Similarity Calculations

113 Nitrogen geometry

± SP residue NBn+SPNitrogen planar

± Charges NBn-f-Sugar&Phosphate

NBn+SPNitrogen tetrahedral NBneutral NBn-SPNitrogen planar NBn-Sugar&Phosphate

/

NBn-SPNitrogen tetrahedral

NB(2D or 3D) NBc+SPNitrogen planar NBc+Sugar&Phosphate

\

NBc+SPNitrogen tetrahedral NBcharged NBc-SPNitrogen planar NBc-Sugar&Phosphate NBc-SPNitrogen tetrahedral

Scheme T.

tree. The trees branch first on the charge status. Then the sugar-phosphate (SP) residue is introduced as a second element of complexity. Finally the geometry of the nitrogen atoms of the amine functions is considered. The structure consequently presents eight final branches for each tree. The discussion will follow each branch upward. Only three of the standard bases (A, G, C) possess amine functions that can be configured tetrahedrically (Table 1) or planarly (Tables 2, 3). In all cases we can observe a complete agreement between planar and tetrahedral nitrogens when the atoms are charged. On the other hand, neutral nitrogen atoms show a constant decrease (-0.05) of the calculated ED values passing from planar to tetrahedral geometry. ^^ The presence of the SP residue on the separate bases does not influence the ED values too much (Tables 2, 3). In fact, the only visible change concerns a small difference (±0.1) of the ED for amine nitrogen atoms when charged. This effect is common to 2D and 3D cases. The difference is, however, negligible where the ED calculation is concerned. Going upward we can compare neutral and charged bases. The ED variations are very small or zero for all multiple bonded atoms without differences due to the different situations (presence/absence of the SP residue, 2D or 3D tree). On the contrary, all single bonded atoms (both N and O) are heavily influenced by the presence of the charge. But the observed variations are scarcely sensitive to the

GUIDO SELLO and MANUELA TERMINI

114

Table 1, Values of Energy Differences of Compounds with Tetrahedral Nitrogen Compound^ Calc. type^ Adenine Guanine Cytosine Calc. type*^ Adenine Guanine Cytosine Calc. type*^ Adenine Guanine Cytosine Calc. type^ Adenine Guanine Cytosine Calc. type^ Adenine Guanine Cytosine Calc. type^ Adenine Guanine Cytosine Calc. type^ Adenine Guanine Cytosine Calc. type*^ Adenine Guanine Cytosine

Atom 1^

Atom^

0.451 2.872 0.452

3.692 2.886 2.820

1.851 2.845 2.843 1.852 1.851

3.701 2.878 2.876 2.847 2.850

0.471 2.846 0.468

3.692 2.894 2.823

1.840 2.810 2.811 1.871 1.871

3.704 2.890 2.891 2.849 2.851

0.453 2.873 0.455

3.690 2.887 2.822

1.854 2.850 2.850 1.854 1.854

3.712 2.880 2.879 2.846 2.850

0.443 2.848 0.468

3.691 2.896 2.823

1.791 2.804 2.804 1.810 1.811

3.713 2.894 2.894 2.852 2.854

Atom J(b

Atom 4^

2D, -SP, Nt, neutral 3.790 3.743 2.824 0.280 2.841 2.885 2D, -SP, Nt, (zharged1 3.752 3.809 1.492 2.825 2.824 1.493 2.874 2.866 2.864 2.882 3D, -SP, Nt, neutral 3.780 3.743 2.821 0.339 2.838 2.906 3D, -SP, Nt,
Atom^

0.455 2.898

1.854 0.455 2.891 2.911

0.480 2.808

1.835 0.477 2.774 2.817

0.454 2.899

1.851 0.451 2.891 2.910

0.482 2.792

1.838 0.480 2.749 2.802

Notes: ^Compound names refer to Figure 3. ^Atom numbering refers to Figure 3. *The specific structure and the calculation type are indicated in the column and refer to each whole block. 2D and 3D indicates the dimensionality; ± SP indicates the presence/absence of the sugarphosphate residue; std indicates the standard conformation of the SP residue; Nt indicates the tetrahedral structure of the amine nitrogen; neutral/charged indicates that calculation are made on neutral/charged compound.

Transferability of Similarity Calculations

115

Table 2. Values of Energy Differences of Neutral Compounds Compound ^

Atom 1^

Atom^

Atom 3^

Atom 4^

Calc. type*^ Adenine Guanine

0.503 2.872

3.691 2.885

Thymine

2.882

Uracil Cytosine Adenine**^

0.279 0.280

2.881 2.882

0.505

2.882 2.880 2.821

2.843

2.953

2.819

0.250

2.885 2.807

Thymine*^ Me-guanine

0.775 0.204

2.780

2.875

2.882

3.829

3.768

0.280

2.803

0.278

2.920

2.880

2D, -SP, Np, neutral 3.740 3.789 2.824 0.280

Atom^

0.506 2.902 2.903 2.898 2.899 0.504

Inosine

2.872

3.646 2.887

Xanthine

2.882

2.881

3.691

3.782

Guanine

0.527 2.847

2.895

0.340

2.823

0.532

Thymine

2.762

2.913

0.370

2.909

Uracil

2.825

0.373

2.910

2.786 2.787

Cytosine Adenine*^

0.522

2.900 2.824

2.839 0.260

2.906 2.804

Thymine*^

0.823 0.202

Calc. type^ Adenine

2.940

3D, -SP, Np, neutral

2.965

2.816 2.788

3.740

2.808

2.863

2.903

2.809

3.643

3.800

0.529

0.895

Xanthine

0.846 2.857

0.326 0.366

3.766 2.802 2.932

2.883

Calc. type^ Adenine Guanine Thymine Uracil Cytosine

0.502 2.872 2.882 2.880 0.505

3.692 2.886 2.882 2.880 2.823

3.790 0.279 0.280 0.280 2.844

2.823 2.880 2.880 2.884

2.953 0.774

2.820

0.249

2.808

2.778

2.875

2.882

3.646 2.887

3.830

3.769

Inosine

0.203 2.872

0.280

2.802

Xanthine

2.885

2.884

0.279

2.898

Me-guanine Inosine

Adenine*^ Thymine**^ Me-guanine

2.891

2D, +SPstd, Np, neutral 3.741 0.508 2.902 2.902 2.899 2.900 0.502 2.917

{continued)

GUIDO SELLO and MANUELA TERMINI

116

Table 2, Continued Atom 1 ^

Atom 2^

Calc. type^ Adenine Guanine

0.495

3.692

Thymine Uracil Cytosine

2.830

2.898

0.499

Adenine*^

2.963

2.823 2.814

Thymine*^

0.810

Me-guanine Inosine

0.213 2.847

Xanthine

2.859

Compound^

Atom 3^

Atom 4 ^

Atom 5^

2.848

3D, +SPstd, Np, neutral 3.741 3.780 2.822 0.342 2.896

0.538

2.761

2.911

0.369 0.372

2.917

2.772

2.917

2.773

2.840

2.791

0.264

2.910 2.804

2.794

2.872

2.909

2.794

3.646

3.797

3.766

0.533

2.896

0.326

2.802

2.893

0.365

2.919

2.844

Notes: ^Compound names refer to Figure 3. ^Atom numbering refers to Figure 3. ^ h e specific structure and the calculation type are indicated in the column and refer to each whole block. 2 D and 3D indicate the dimensionality; ± SP indicates the presence/absence of the sugar-phosphate residue; std indicates the standard conformation of the SP residue; Np indicates the planar structure of the amine nitrogen; neutral indicates that calculations are made on neutral compounds. ^Starred bases refer to the tautomeric forms of the corresponding standard bases.

Table 3. Values of Energy Differences of Charged Compounds Compound^ Calc. type^ Adenine Guanine Thymine Uracil Cytosine Adenine* Thymine* Me-guanine Inosine Xanthine

At. 1^

At. 2^

1.861 2.843 2.845 2.861 2.881 2.889 2.880 1.861 1.862 3.003 1.630 1.631 0.196 2.844 2.858 2.878

3.699 2.876 2.877 2.870 2.882 2.867 2.882 2.848 2.850 2.853 2.811 2.816 3.661 2.878 2.869 2.881

At. 3^

At. 4^

2D, -SP, Np, charged 3.806 3.749 1.492 2.825 1.492 2.826 1.608 2.881 1.609 2.869 1.608 2.881 1.607 2.869 2.869 2.875 2.864 2.881 1.525 2.808 2.867 2.903 2.902 2.877 3.857 3.778 1.494 2.804 1.488 2.920 1.488 2.916

At. 5^

1.863 0.505 2.901 2.885 2.900 2.885 2.892 2.910 2.889 2.909 1.860 2.936 2.929

Notes ^

3 charges 2 charges shift shift 3 charges 2 charges 3 charges 2 charges

shift {continued)

Transferability of Similarity Calculations

117

Table 3, Continued Compound ^ Calc. type^ Adenine Guanine Thymine Uracil Cytosine Adenine* Thymine* Me-guanine Inosine Xanthine

Calc. type^ Adenine Guanine Thymine Uracil Cytosine Adenine* Thymine* Me-guanine Inosine Xanthine Calc. type"^ Adenine Guanine Thymine Uracil Cytosine

At. / ^

At:P

1.787 2.808 2.811 2.700 2.755 2.780 2.819 1.884 1.882 3.001 1.675 1.674 0.194 2.811 2.826 2.855

3.699 2.889 2.891 2.918 2.912 2.896 2.900 2.849 2.854 2.851 2.823 2.822 3.662 2.892 2.883 2.891

3D,-SP, Np, charged 3.771 3.748 1.387 2.821 2.822 1.366 1.441 2.912 1.441 2.914 1.435 2.911 1.437 2.914 2.904 2.858 2.857 2.902 2.807 1.484 2.901 2.885 2.897 2.881 3.776 3.815 2.803 1.385 2.932 1.311 2.936 1.311

1.861 2.850 2.850 2.860 2.880 2.859 2.879 1.861 1.853 3.005 1.631 0.120 2.849 2.863 2.882

3.712 2.880 2.879 2.868 2.882 2.866 2.881 2.848 2.850 2.852 2.814 3.671 2.879 2.872 2.881

2D, +SPstd, Np, charged 3.821 3.751 1.614 2.825 1.612 2.823 2.880 1.609 2.866 1.608 2.880 1.608 2.868 1.608 2.874 2.870 2.864 2.880 2.806 1.637 2.877 2.903 3.779 3.869 2.802 1.614 1.607 2.898 1.609 2.886

2.918 2.903

1.790 2.804 2.804 2.697 2.755 2.787 2.824 1.785 1.785

3.713 2.894 2.894 2.920 2.913 2.896 2.900 2.852 2.854

3D, +SPstd, Np, charged 3.752 3.802 1.497 2.820 2.819 1.491 2.919 1.440 1.442 2.923 2.916 1.436 2.927 1.439 2.918 2.855 2.908 2.855

1.840 0.482 2.767 2.707 2.767 2.711 2.748 2.801

At 3^

At. 4^

At. 5^

Notes ^

1.812 0.530 2.780 2.729 2.780 2.733 2.774 2.818

3 charges 2 charges

2.775 2.817 1.844 2.881 2.857

1.861 0.456 2.902 2.886 2.902 2.886 2.891 2.909 2.911 1.861

shift shift 3 charges 2 charges 3 charges 2 charges

shift

3 charges 2 charges shift shift 3 charges 2 charges 3 charges

shift

3 charges 2 charges shift shift 3 charges 2 charges

GUIDO SELLO and MANUELA TERMINI

118

Tables, Continued Compound ^ Adenine* Thymine* Me-guanine Inosine Xanthine

Notes:

At. 1^ 3.000 1.710 0.210 2.802 2.820 2.851

At 2^ 2.848 2.830 3.671 2.899 2.888 2.892

At 3^ 1.599 2.898 3.810 1.511 1.432 1.435

At 4^ 2.805 2.904 3.778 2.853 2.919 2.922

At 5^ 2.807 1.829 2.834 2.795

Notes'^ 3 charges

shift

^Compound names refer to Figure 3. '^Atom numbering refers to Figure 3. Special notes are reported in this column indicating either the number of charges on the compound, or the need of shifting the base so as to have coordination therefore requiring a different position for the charges. ^The specific structure and the calculation type are indicated in the column and refer to each whole block. 2D and 3D indicate the dimensionality; ±SP indicates the presence/absence of the sugar-phosphate residue; std indicates the standard conformation of the SP residue; Np indicates the planar structure of the amine nitrogen; charged indicates that calculations are made on charged compounds.

particular case, always varying by 0.1 or 0.2 at most. The results are easily understandable considering the limited influence of the through space interactions. The last comparison concerns the differences existing between the 2D and the 3D calculations. Here it is again possible to observe an overall similarity between the ED values (Tables 2, 3). It is clear that the 3D calculations show, as a constant characteristic, a greater variability (e.g., the values for the €26^ carbonyl in thymine and uracil become differentiated). This effect is expected because more interactions are considered in 3D calculations. In conclusion, we can affirm that the ED values are consistently invariable and insensitive to the diverse changes introduced. However, it is fundamental to confine the comparisons to a specific calculation area (2D or 3D) without mixing the results. This is an obvious precaution to limit the chance of errors. B. ED Variations: Changing Conformation

In this section we discuss the variations of the ED values caused by changes in conformations. Two aspects will be considered: the relative position of the SP residue, and the effects of small changes of the overall conformation. If we minimize the rotational position of the SP residue with respect to the base, the standard nucleotidic conformation is lost (we use the a-helix conformation as built by the model builder). As a consequence, the through space influence of the SP residue onto the base atoms could change. In fact, some small differences are found (e.g., neutral amine A^s), but the ED values remain more or less constant (Table 4). We also made some calculations on two bases (guanine and thymine) in different conformations chosen between the minima of the potential energy surfaces

Transferability of Similarity Calculations

119

Table 4, Valuesof Energy Differences of Compounds with Minimized SP Residue Compound^ Calc. type*^ Adenine Guanine Thynnine Uracil Cytosine Adenine*^ Thymine*^ Me-Guanine Inosine Xanthine Calc. type^ Adenine Guanine Thymine Uracil Cytosine Calc. type^ Adenine Guanine Thymine Uracil Cytosine Adenine*^ Thymine*^ Me-Guanine Inosine Xanthine Calc. type^ Adenine Guanine Thymine Uracil Cytosine

Atom 1^

Atom 2^

0.503 2.872 2.882 2.880 0.505 2.952 0.774 0.204 2.872 2.883

3.691 2.886 2.881 2.880 2.823 2.819 2.780 3.646 2.887 2.882

1.861 2.847 2.849 2.860 2.858 1.862 1.861

3.713 2.877 2.878 2.868 2.867 2.848 2.852

0.495 2.835 2.768 2.830 0.481 2.970 0.816 0.200 2.846 2.857

3.692 2.898 2.914 2.901 2.824 2.818 2.789 3.643 2.897 2.892

1.794 2.873 2.786 2.706 2.785 1.889 1.890

3.713 2.896 2.897 2.920 2.898 2.854 2.855

Atom 3^

Atom 4^

2D , +SPmin, Np, neutral 3.788 3.740 0.279 2.823 0.280 2.881 0.280 2.880 2.844 2.887 0.250 2.806 2.881 2.875 3.767 3.828 2.802 0.280 0.277 2.921 ,2D, +SPmin, Np, charged 3.821 3.750 2.824 1.615 1.612 2.825 1.609 2.880 1.609 2.880 2.874 2.868 2.865 2.883 3D,, +SPmin, Np, neutral 3.780 3.740 0.339 2.821 0.370 2.918 2.921 0.368 2.841 2.916 0.262 2.805 2.869 2.912 3.800 3.766 0.327 2.799 0.365 2.937 ;3D, +SPmin, Np, charged 3.800 3.749 1.497 2.819 1.491 2.820 1.444 2.919 2.921 1.439 2.921 2.863 2.859 2.911

Atom^

0.506 2.902 2.902 2.900 2.899 0.504 2.941

1.864 0.506 2.900 2.902 2.891 2.911

0.496 2.754 2.760 2.760 2.761 0.526 2.864

1.812 0.496 2.747 2.739 2.709 2.769

Notes: ^Compound names refer to Figure 3. ^Atom numbering refers to Figure 3. ^ h e specific structure and the calculation type are indicated in the column and refer to each whole block. 2 D and 3 D indicates the dimensionality; + SP indicates the presence of the sugar-phosphate residue; min indicates the minimized conformation of the SP residue; Np indicates the planar structure of the amine nitrogen; neutral/charged indicates that calculation are made on neutral/charged compound. *^Starred bases refer to the tautomeric forms of the corresponding standard bases.

GUIDO SELLO and MANUELA TERMINI

120

Table 5, Values of Energy Differences of Guanine and Thymine Calculated In Different Conformational Minima Compound ^ Calc. type*^ Guanine

Thymine

Atom 1 ^

Atom 2^

2.848 2.861 2.842 2.848 2.843 2.837 2.844 2.774 2.766 2.780 2.770 2.766 2.773

2.896 2.892 2.899 2.896 2.897 2.898 2.895 2.912 2.910 2.913 2.909 2.911 2.909

Atom 3 ^ 3D, +SP, Np, neutral 0.340 0.325 0.336 0.340 0.338 0.337 0.317 0.369 0.367 0.369 0.371 0.358 0.371

Atom 4 ^

imi 2.811 2.822 2.822 2.818 2.819 2.809 2.917 2.918 2.917 2.909 2.913 2.908

Atom 5^ 0.535 0.564 0.537 0.535 0.541 0.538 0.513 2.780 2.765 2.782 2.811 2.777 2.797

Notes: ^Compound names refer to Figure 3. ^Atom numbering refers to Figure 3. T h e specific structure and the calculation type are indicated in the column and refer to each whole block. 3D indicates the dimensionality; SP indicates the presence of the sugar-phosphate residue; Np indicates the planar structure of the amine nitrogen; neutral indicates that calculations are made on neutral compounds.

obtained by rotation of two dihedral angles (between the base and the SP residue and between the sugar and the phosphate residue). The results are shown in Table 5 where we can observe an overall stability of the values therefore making consistent the assumption that the EDs would be constant for the conformational minima. It is obvious that the conformational changes can affect the ED values, but very small variations cannot change too much the weights of the atoms. C. ED Variations: Changing System

Last but not least of the analyzed perturbations come two massive changes of the reference system. The first concerns the pairing simulation by juxtaposition of the base pair without charging the atoms. The second analyzes the behavior of the base triplets. Positioning the bases at the standard pairing distance (as built by the model builder) it is possible to calculate EDs where the presence of the second base influences the result (Table 6). Of course, the comparison of those values with the EDs of the separate neutral bases shows small differences because the through space interaction is weak, but not zero. Therefore, extending the comparison to three

Transferability of Similarity Calculations

121

Table 6. Values of Energy Differences of Base Pairs Calculated in 3D without SP Residue, with Planar Nitrogen, and with Neutral Bases Pair^

At. 1^

At. 2^

At. 3^

At. 4^

Inosine

2.791

2.909

0.334

2.800

Adenine

0.610

3.682

3.772

3.740

Inosine

2.801

2.903

0.403

2.797

At. 5^

Notes^

Index^

0.121 2.740

shift

0.017

2.740

shift

0.030

2.913

2.789

shift

0.020

2.914

2.803

Uracil

2.835

2.896

0.468

2.915

Inosine

2.795

2.904

0.403

2.796

Thymine

2.771

2.907

0.471

2.915

Inosine

2.811

2.904

0.377

2.795

Xanthine

2.868

2.893

0.456

Xanthine

2.795

2.910

0.387

Adenine

0.592

3.683

3.782

3.735

Xanthine

2.786

2.907

0.435

2.903

2.836

Cytosine

0.595

2.810

2.835

2.906

2.794

Xanthine

2.819

2.900

0.459

2.910

2.827

0.106

Uracil

2.831

2.903

0.464

2.911

2.757

Xanthine

2.818

2.900

0.457

2.910

2.827

Thymine

2.765

2.914

0.461

2.911

Xanthine

2.866

2.891

0.457

2.913

0.071 shift

0.006

2.757

shift

0.009

2.784

shift

Guanine

2.809

2.902

0.384

2.813

0.504

Xanthine

2.819

2.900

0.427

2.913

2.856

Xanthine

2.867

2.890

0.462

2.908

2.803

Xanthine

2.833

2.916

0.413

2.925

2.800

Me-guanine

0.224

3.649

3.814

3.758

0.624

0.146

Adenine

0.578

3.683

3.775

3.741

Guanine

2.791

2.909

0.341

2.820

0.538

0.099

Adenine

0.572

3.683

3.781

3.737

Uracil

2.765

2.914

0.405

2.919

2.761

0.073

Adenine

0.582

3.685

3.781

3.736

Thymine

2.705

2.923

0.401

2.918

2.759

0.080

Adenine

0.598

3.689

3.773

3.740

Me-guanine

0.250

3.658

3.804

3.764

Adenine

0.556

3.689

3.765

3.739

Adenine*

2.934

2.814

0.297

2.803

Guanine

2.778

2.910

0.392

2.814

Cytosine

0.588

2.812

2.833

2.921

2.736

Guanine

2.808

2.901

0.421

2.823

0.499

0.565

0.021 shift

shift

0.008

0.023 0.006

0.576

Uracil

2.829

2.902

0.461

2.914

2.742

Guanine

2.807

2.901

0.419

2.822

0.498

Thymine

2.764

2.913

0.458

2.914

2.741

Guanine

2.805

2.906

0.413

2.813

0.572

Thymine*

0.863

2.775

2.864

2.917

2.744

Cytosine

0.608

2.812

2.831

2.917

2.762

Inosine

2.796

2.910

0.388

2.974

2 charges

0.077

shift

0.100

shift

0.004 0.064 0.066 {continued)

122

GUIDO SELLO and MANUELA TERMINI Tables. Continued

Pair^

At. 1^

At. 2^

At. 3^

At. 4^

At. 5^

Cytosine Uracil Cytosine Me-guanine Cytosine Adenine* Thymine Cytosine Thymine Uracil Thymine Me-guanine Thymine Thymine* Thymine Adenine* Uracil Thymine Uracil Me-guanine Uracil Adenine* Uracil Thymine*

0.605 2.716 0.565 0.245 0.509 2.954 2.704 0.603 2.766 2.795 2.736 0.229 2.739 0.885 2.753 2.977 2.819 2.756 2.796 0.226 2.813 2.978 2.780 0.887

2.814 2.921 2.820 3.658 2.820 2.813 2.922 2.815 2.914 2.903 2.916 3.650 2.916 2.782 2.917 2.807 2.907 2.909 2.910 3.648 2.908 2.807 2.907 2.780

2.834 0.421 2.829 3.806 2.823 0.329 0.420 2.835 0.449 0.422 0.429 3.815 0.443 2.860 0.398 0.378 0.445 0.421 0.432 3.815 0.402 0.380 0.443 2.861

2.904 2.908 2.916 3.765 2.920 2.816 2.909 2.905 2.911 2.914 2.919 3.758 2.906 2.903 2.924 2.819 2.912 2.916 2.920 3.759 2.924 2.819 2.907 2.902

2.801 2.788 2.775 0.559 2.729 0.588 2.788 2.801 2.760 2.784 2.730 0.596 2.798 2.797 2.718 0.500 2.759 2.784 2.729 0.600 2.719

Nofes ^

Index "^ 0.140

shift

0.003 0.086 0.086 0.034 0.079 0.009

shift 0.169 shift

0.001 0.085

shift 0.172

2.796 2.798

0.017

Notes: ^Compound names refer to Figure 3. ''Atom numbering refers to Figure 3. ^Special notes are reported in this column indicating either the number of charges on the compound, or the need of shifting the base so as to have coordination therefore requiring a different position for the charges. '^The similarity indices are obtained using ED values deriving from these pairs in the place of the charged bases used in the previous tables.

elements (neutral separate base, neutral base in the pair, charged separate base) we can observe a consistent trend, where the charged base shows the largest variations. The uniform behavior of the bases submitted to diverse perturbations (in the first case by the presence of the partner base; in the second case opportunely charged) can guarantee the validity of our model of pairing. The last part of the present section concerns the calculations made on base triplets. These calculations are quite time demanding (~2 h of cpu time, corresponding to -6 h of real time, on an IBM-RISC/6000, 520, medium loaded) and therefore we limited our interest to standard bases (A, C, G, U), building all 64 possible

Transferability of Similarity

Calculations

123

combinations, in 3D only.^^ It is interesting to control two different aspects: the sensitivity of the method to the molecular neighborhood, and the possibility to save time using isolated bases as models of base chains. The triplets have been realized using the model builder in our hands and, as a consequence, they present the standard substitution pattern with a 3' OH terminal and a 5' P terminal (Figure 4). This apparently small difference is nevertheless significant; in fact, both adenine and cytosine in position 3' OH show evident changes (Table 7) of the ED of the nitrogen atoms, particularly when in the central position of the triplet either guanine or uracil (a carbonyl group is positioned exacdy below the nitrogen atom) is present. When positioned in the central or in the 5' P positions, all of the bases show only small variations in the ED values. More exactly we can observe a general decrease in the ED values with corresponding differences of < -0.08 for 5' P bases, < -0.13 for central bases, and < -0.23 for 3' OH bases (the values are obtained comparing the EDs of the atoms of the corresponding isolated bases with those of the triplets and represent the greatest calculated differences). These differences are not meaningless and, in the case of analysis of short base chains, they should be considered; but in the case where all of the bases are well inside a long chain they can be compared with 5' P bases in the triplets. To verify the base behavior in the pairing we calculated some examples with the central base charged (Table 7) and one triplet example with all of the bases charged (Table 8). The comparison with the corresponding isolated charged bases shows a variation always positive and of less than 0.04 and 0.06, respectively. This result is clear if we consider that the perturbation introduced by the charge is great enough

A C T U G G

C U I A I X X X X X A U T A U G M A*

CI

C A*

T r

U M

U T*

Figure 4, Comparison of similarity indices calculated using standard base pairing and simulated base pairing. The dashed line represents standard values.

124

GUIDO SELLO and MANUELA TERMINI

Table 7. Values of Energy Differences of Base Triplets Calculated in 3D with SP Residue and Planar Nitrogen Atom^

ACU^

AUA^

CAG^

CGC^

CUC^

1 2 3 4

0.383

0.381

0.288

0.421

0.428

3.690

3.694

2.818

2.815

2.815

3.793

3.797

2.842

2.847

2.854

3.742

3.743

2.917

2.920

2.912

2.769

2.766

2.795 2.716

5 1 2 3 4

2.763

2.721

0.460

2.737

2.909

2.917

3.683

2.916

2.919

0.330

0.329

3.792

0.339

0.339

3.732

2.803

2.924

0.517

2.721

2.785

0.448

0.461

2.914

2.815

2.814

3.795

0.330

2.843

2.839

3.733

2.813

2.918

2.918

0.491

2.746

2.734

2.808

2.924

5

0.492

2.718

1 2 3 4

2.813

0.445

2.909

3.687

0.347 2.923

5

2.734

Atom^

GAC^

CGG^

GUG^

UCA^

UUU^ 2.770

1 2 3 4

2.762

2.785

2.795

2.746

2.916

2.912

2.911

2.917

2.907

0.320

0.327

0.330

0.359

0.372

2.820

2.818

2.820

2.920

2.912

5

0.501

0.507

0.506

2.773

2.779

1 2 3 4

0.489

2.786

2.768

0.445

2.769

3.680

2.903

2.906

2.805

2.901

3.799

0.344

0.352

2.838

0.364

3.731

2.806

2.920

2.919

2.915

0.493

2.723

2.728

2.732

2.790

2.818

2.825

0.437

2.810

2.913

2.907

2.906

3.692

2.908

0.328

0.342

0.337

3.788

0.361

3.734

5 1 2 3 4

2.813

2.810

2.809

5

0.486

0.491

0.502

2.918

Atonrf

AGW

AUA""

CAC

CGC

cue

0.393 3.688 3.793 3.740

0.389 3.692 3.796 3.742

0.289

0.432

0.428

2.817

2.812

2.814

2.844

2.847

2.854

2.917

2.919

2.912

2.772

2.765

2.796

2.720

{continued)

Transferability of Similarity Calculations

125

Table 7, Continued Atonff

ACt/

AUA""

CAC

CGC

cue

1 2 3 4

2.683 2.922 1.486 2.808 0.492

2.629 2.729 1.465 2.920 2.711

1.837 3.707 3.811 3.745

2.656 2.930 1.487 2.808 0.516

2.714 2.921 1.467 2.938 2.631

0.454 3.684 3.795 3.731

5

2.814 2.907 0.351 2.921 2.735

2.782 2.914 0.330 2.814 0.492

0.455 2.813 2.844 2.917 2.747

0.461 2.814 2.840 2.921 2.735

Atom^

GAC

GGC^

GUC

UCA"^

UULF

1 2 3 4

2.762 2.915 0.320 2.819 0.502

2.786 2.911 0.329 2.817 0.507

2.796 2.910 0.323 2.818 0.491

2.746 2.916 0.359 2.920 2.774

2.775 2.912 0.373 2.915 2.779

2

1.839 3.703

2.710 2.917

2.688 2.917

1.844 2.836

2.771 2.906

Atom^

GAC'

GGG^

GUa

UCA""

UUU^

3 4

3.820 2.743

1.468 2.806 0.492

1.442 2.918 2.717

2.845 2.916 2.735

1.438 2.936 2.648

2.787 2.912 0.329 2.812 0.488

2.820 2.905 0.345 2.809 0.492

2.826 2.904 0.340 2.807 0.504

0.438 3.691 3.786 3.733

2.810 2.910 0.359 2.924 2.726

5 1 2 3 4

5 1

5 1 2 3 4 5

Notes: ^Atom numbering refers to Figure 3. ^Triplets are indicated by the first letter of the names of the corresponding bases. Neutral triplets. H'riplets are indicated by the first letter of the names of the corresponding bases. Triplets have only the central base charged.

to make the charged centers less sensitive to their neighborhood. Even charging all of the bases in the triplet is not sufficient to introduce large changes in the calculated EDs.i^ D. Similarity Index: 2D and 3D The calculation of our similarity indices is a combination of the ED values, thus it changes as much as they change. As a consequence, we expect an overall

GUIDO SELLO and MANUELA TERMINI

126

Table 8. Values of Energy Differences of Base Triplets Calculated in 3D with SP Residue and Planar Nitrogen Atom^

Cb

C^

A^

Atonr^

/\^

C"

A""

1 2 3 4

1.835 2.849 2.866 2.910 2.804

1.842 2.839 2.832 2.918 2.727

1.853 3.711 3.805 3.756

1 2 3 4 5

0.385 3.701 3.793 3.741

0.413 2.820 2.836 2.919 2.744

0.254 3.703 3.790 3.747

5 Notes:

^Atom numbering refers to Figure 3. ''Bases are indicated by the first letter of their names. All three bases are charged. '^Bases are indicated by the first letter of their names. All of the bases are neutral. The 5' P adenine is rotated by 90° with respect to the usual conformation.

agreement with ED variations. It is worth repeating that the importance of the indices' reliability is entirely dependent on their use, i.e., on the property they are representing. In the present case the goal would be the possibility of ordering base-base pairing by the use of the corresponding similarity indices. The property (similarity of pairing) is difficult to measure; therefore, it is difficult to have an experimental validation of the model. We could be satisfied by a qualitative agreement among the ordering given by different index approximation, or, better, by a consistent stability of the ordering inside each type of calculation (e.g., 2D or 3D). In Table 9 the indices calculated using structures with different characteristics in 2D and 3D approaches are shown. By comparing the two halves of the table it is possible to note an overall agreement between the two approaches with increasing index values from left to right. Besides that, we can locate six different groups in both halves: AT, AU, TC, UC; TG, UG; TU, UT; AC; AG; CG. The possibility of locating groups in all calculations (particularly in the 3D approach) is interesting because it represents a signal of a transferable influence of the main effects. E. Similarity Index: SP Influence

Going beyond standard bases we analyzed the indices where mutant or tautomeric bases are also involved. This analysis has been done only in 3D with the aim of studying the influence of the presence of the SP residue. From Table 10, we see that the SP residue changes the index values, as expected, but it is also possible to locate a more articulated grouping of the bases: A, M, C; U, T, X; I, G; A*; T*. In fact, with few exceptions, where one member of the group pairs with a base we can also find the other members. This behavior is consistently presented by both of the calculation classes (with or without SP). In addition, pairing among members of the first group always gives high indices, while in the second group omopairing

Transferability of Similarity

127

Calculations

Table 9. Values of Similarity Indices Calculated Using Neutral and Charged Isolated Bases -SP

+SPmin

+SPstd

Np^

Np^

Np^

Adenine-thymine

0.067

0.013

0.084

0.126

Adenine-uracil

0.068

0.013

0.085

0.128

Cytosine-guanine

0.194

0.068

0.058

0.113

Adenine-guanine

0.190

0.081

0.080

0.124

Thymine-guanine

0.143

0.018

0.018

0.019

shift

Uracil-guanine

0.140

0.019

0.017

0.018

shift

Thymine-cytosine

0.071

0.070

0.061

0.114

0.062

0.116

Paif^

-hSPstd N^

Notef

2D

Uracil-cytosine

0.073

0.071

Thymine-uracil

0.018

0.021

Uracil-thymine

0.022

0.020

Adenine-cytosine

0.044

0.013

2 charges

shift shift inversion

0.033 3D

Adenine-thymine

0.241

0.306

0.313

0.362

Adenine-uracil

0.233

0.293

0.298

0.346

Cytosine-guanine

0.388

0.324

0.194

0.248

Adenine-guanine

0.260

0.215

0.213

0.259

Thymine-guanine

0.075

0.036

0.049

0.043

shift

Uracil-guanine

0.065

0.039

0.049

0.050

shift

Thymine-cytosine

0.370

0.414

0.294

0.351 0.335

Uracil-cytosine

0.361

0.401

0.279

Thymine-uracil

0.046

0.052

0.002

Uracil-thymine

0.084

0.042

0.014

Adenine-cytosine

0.128

0.109

0.019

2 charges

shift 0.011

shift inversion

Notes: ^Pair names refer to Figure 3. '^he specific structure and the calculation type are indicated in the column header. ± SP indicates the presence/absence of the sugar-phosphate residue; std/min indicates the standard/minimized conformation of the SP residue; Np/Nt indicates the planar/tetrahedral structure of the amine nitrogen. '^ Special notes are reported in this column indicating either the number of charges on the pair, or the need of shifting/inverting the base so as to have coordination therefore requiring a different position for the charges. If the bases need to be shifted, the first base is submitted to the shift.

is always represented by low indices. Finally, there are the special cases of A* and T*. A* is the only base that uses an NH imine group as proton acceptor; T*, on the other hand, presents an OH enolic group as proton donor. As a consequence, the A* pairings are always in the upper part of the table and the T* pairings in the lower part.^^ In conclusion, the indices can be considered transferable at least for what concerns the grouping of similar bases.

128

GUIDO SELLO and MANUELA TERMINI

Table 10, Ordered Values of Similarity Indices Calculated Using Neutral and Charged Isolated Bases^ in 3D with Planar Amine Nitrogen Without SP residue

xc 0.464 XA 0.336 IT* 0.154 XU 0.096 AA* 0.010

T*M 0.460 UM 0.320 UT* 0.147 UT 0.084 IT 0.009

XM T*A* CT CG CU 0.411 0.391 0.388 0.361 0.370 TM UA* AT AG TA* 0.250 0.241 0.316 0.260 0.247 TT* GT* CA* IX AC 0.121 0.104 0.139 0.128 0.118 CM XG TU TC UG 0.071 0.074 0.065 0.048 0.046 XX TT UU 0.007 0.005 0.008 With SP residue in standard conformation

CI 0.355 AU 0.233 AM 0.100 XT* 0.044

XA* 0.341 lA 0.227 XT 0.100 lU 0.013

T*A* 0.446 CT 0.294 lA 0.177 CA* 0.069 AM 0.007

T*M 0.383 XA 0.289 CI 0.160 AA* 0.056 XX 0.007

UA* 0.368 XM 0.289 lU 0.135 TG 0.049 CM 0.006

TM 0.302 CG^ 0.196 XG 0.085 UU 0.017

AU 0.298 GT* 0.179 TT* 0.080 UT 0.014

TA* 0.365 CU 0.279 IT 0.132 UG 0.049 TU 0.002

XA* 0.352 XC 0.272 IX 0.119 XU 0.023 TT 0.001

AT 0.313 IT* 0.213 XT* 0.101 XT 0.020

UM 0.305 AG 0.213 UT* 0.095 AC 0.019

Notes: ^A = adenine; C = cytosine; G = guanine; T = thymine; U = uracil; A* = adenine tautomer; T* = thymine tautomer; M = methylguanine; I = inosine; X = xanthine. ^ h e CG index calculated on three points is 1.155 and a better comparison could be with a weighted percentage of this value (e.g., 6 7 % = 0.77).

F. Similarity Index: Juxtapositioned Pairs

The index calculation of the pairings simulated by juxtapositioning the bases gives values that are smaller than those calculated by our model (Table 6). This is due to the small changes of the EDs that are induced only by the through space interactions between hydrogen donors and acceptors. However, the indices group the pairings in dependence on the base similarity in a fashion comparable to the above-reported effects. The groups are not so clearly formed for two principal reasons: (1) the small values obtained from differences of large numbers inherently contain greater errors; (2) the juxtapositioning of the base for nonstandard pairings has been forced by hands, particularly where purine-purine or pyrimidinepyrimidine interactions are involved.

Transferability of Similarity Calculations

129

G. Similarity Index: Triplets The last case of index calculation represents a further modification of the pairing model. Here the pairing concerns triplet pairs where only the central base has been charged for the simulation. Following are some aspects of the results (Table 11). 1. In all of the examples the index values increase with respect to the isolated bases (last line in the table). This effect was expected because the greater the number of interactions, the greater the weight of each atom, being the weight representative of the atom's importance. 2. On the contrary, the smaller the index value of the isolated base, the greater the variance of the triplet indices. This was expected for the same reasons that make the least important atoms more sensitive to the perturbations. 3. Even if the indices show differences in dependence on the triplet composition, they are each similar enough to the others to be equally usable. For the sake of clarity, we will briefly summarize the work done and the results obtained. To verify the possibility of transferring the similarity calculations from small substructures to complex compounds, we analyzed the changes introduced by adding new features to the starting simple model. Therefore, we added complexity elements to the model step by step, analyzing the perturbations caused by each new element. The first and most important modification was the passage from topological to 3D calculations. Here the changes were important and, as a consequence, we should work either in 2D or in 3D without mixing results. Further, only the 3D values can distinguish different situations caused by through space interac-

Table 11, Values of Similarity Indices of Base Triplets Calculated in 3D with SP Residue and Planar Nitrogen A-U^ Paii^ CAG-GUG GAG-GUG CAG-AUA GAG-AUA CAG-UUU GAG-UUU CAG-GUG GAG-CUC

Notes:

A-G^

C-C^

C~U^

Index

Paii^

Index

Paii^

Index

Paii^

Index

0.386 0.361 0.351 0.326 0.320 0.295 0.270

CAG-GGG CAG-CGC CAG-AGU GAG-GGG GAG-CGC GAG-AGU

0.387 0.368 0.359 0.323 0.304 0.295

GGG-UCA CGC-UCA AGU-UCA

0.398 0.379 0.370

UCA-GUG UCA-AUA UCA-UUU UCA-CUC

0.396 0.361 0.330 0.280

0.245

^Bases are identified by the first letter of their names, "^he central bases of each triplet are used in pairing.

GUIDO SELLO and MANUELA TERMINI

130

NHz

^ HO-P-O-CHi O ' H

O

NH2

OH

HO-P-O-CHfe |i

O

'

H

H\| O

H^

/H OH

<J^X

HO-P-O-CH,

o

H HN OH

H" /H OH

Figure 5. An example triplet (ACG) used in the calculations.

tions. In this view, all of the other changes (presence/absence of SP substitution, planar or tetrahedral nitrogen atoms, isolated bases or triplets) introduced visible differences in the calculations. As a matter of fact, we should consider each case differently so as to obtain the correct results; but, in the aim of studying similarity, we can choose a standard approximation [SP substitution in standard RNA conformation (see Tables 2 and 3), planar nitrogen atoms (see Tables 2 and 3), isolated bases] and use it with confidence. In fact, the increase of the calculation accuracy does not justify the simultaneous increase in computer time, being that the quantitative calculation of similarity is still difficult to validate.^^

IV. CALCULATIONS, TRANSFERABILITY, SIMILARITY MEASURES AND INDICES In the Introduction we briefly mentioned the problems connected with the use of an elusive concept like similarity and it was clear that even the idea of quantifying this concept can lead to erroneous conclusions and/or methodological approaches. In this respect, if we add to the mere calculations the aim of transferring results between structures and substructures so as to get similarity measures or indices we could find ourselves in quicksand. In fact, we are adding the approximations ingrained into calculations to those from the transfer of values from simple to complex structures with the objective of getting a quantitative measure of something, like similarity, that is even difficult to define. Therefore, we would like to reconsider the work we have presented. The initial idea was to develop a methodology for measuring the similarity between structures; this measure would have been based on the calculation of a well-defined physical molecular characteristic (electronic energy difference). Supposing the calculation of the characteristic is exact (this is never true but the

Transferability of Similarity Calculations

131

4U0 2

,♦

'A

350 -

300 i

i

\\ *\

''/

v/

50 -

AU

-trl«

/

\\

100 -

std

h

\\

150 -

\\ *

^ '

200 -

\\

f'

\\

250 -

0 -

p ,''

\\

\ H

1

1

1

CG

UG

AG

— 1 — CU

1 UU

Figure 6, Comparison of similarity Indices calculated using triplets and Isolated (dashed line) bases.

approximation is clear), it could be possible to compare the values obtained for different structures. The passage from these values to a similarity measure is not immediate but it is easy. The difficulties appear as soon as we would like to attach a meaning to the similarity measure, for example affirming that structures that are similar in our system will have similar chemical reactivity. Most of the time this step is realized using brute force. It is, therefore, clear that the transfer of calculations from simple to complex structures will be problematic if, and only if, it will change the connection between the numerical value and the similarity meaning attached to it. At this point the question is: did we demonstrate that the value transfer does not invalidate the connection between calculations and similarity measures? The answer is positive. It must be clear that the qualitative sense of similarity is maintained throughout the transfer; it is also clear that the ordering of the structures is more or less constant.^^ We think that, at the present state of the development of the project, it would be possible to transfer ED values with confidence and index values with sufficient reliability at least for what concerns the grouping of the base pairings.

V. CONCLUSION The use of a modified version of the algorithm for the calculation of the similarity between substructures introduces the problem of the transferability of results from

132

GUIDO SELLO and MANUELA TERMINI

simple substructures to complex compounds. The obtained results clearly show that, at an optimal level of approximation, the operation can be done with enough confidence. The level of approximation that is necessary to get reliable results is represented by the calculations made on isolated bases with SP substitution using 3D representation and planar aminic nitrogen atoms.

ACKNOWLEDGMENTS Partial financial support by the Consiglio Nazionale delle Ricerche, and by the Ministero deirUniversita e della Ricerca Scientifica e Tecnologica, is gratefully acknowledged.

REFERENCES AND NOTES 1. (a) Chan P. L.; Dean P. M.; / Comput.-AidedMol Des. 1992,6,385-396. (b) Chan, P L.; Dean P M.; J. Comput-AidedMol. Des. 1992, 6,407-426. 2. Rouvray D. H. In Concepts and Applications ofMolecular Similarity; Johnson, M. A.; Maggiora, G. M., Eds.; Wiley-Interscience: New York, 1990, pp. 15-42. 3. As recent general references see: Concepts and Applications of Molecular Similarity] Johnson, M. A.; Maggiora, G. M., Eds.; Wiley-Interscience: New York, 1990. Molecular Similarity and Reactivity: From Quantum Chemical toPhenomenologicalApproaches; Carbd, R., Ed.; Kluwer: Dordrecht, 1995. 4. (a) Good, A. C ; So, S.; Richards, W. G. J. Med Chem. 1993,5(5,433-438. (b) Richards, W. G. PureAppl. Chem. 1994, 66,1589-1596. 5. Leoni, B.; Sello, G. In Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches; Carb6, R., Ed.; Kluwer: Dordrecht, 1995, pp. 267-289. 6. Sello, G, J. Chem. Inf. Comput. Sci. 1992, 52,1\?>-1\1, and references cited therein. 7. Baumer, L.; Sello, G. /. Chem. Inf. Comput. Sci. 1992, 32, 125-130. 8. (a) Baumer, L.; Sala, G.; Sello, G. Tetrahedron Comput. Method 1989, 2, 37-46. (b) Baumer, L.; Sala, G.; Sello, G. ibid 1989, 2, 93-103. (c) Baumer, L.; Sala, G.; Sello, G. ibid 1989, 2, 105-118. 9. \i = (dE/dN)z. 10. Sello, G. Theochem 1995, 340, 15-28. 11. Sello, G.; Termini, M. Unpublished results. 12. Gordy, W.; Thomas, W. J .0. J. Chem. Phys. 1956,24, 439-444. 13. Pritchard, H .O.; Skinner, H. A. Chem. Rev. 1955,55,745-786. 14. Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, 1960. 15. Molecular Advanced Design', Aquitaine Systemes: Paris, Version 2,1990. The standard conformation used by the builder is a-helix, which is the most representative for mRNA and tRNA. 16. A fast search for rotational minima has been performed using a Montecarlo-Metropolis algorithm. 17. Also 2D calculations feel the effect of the geometry because the bond lengths are different. 18. The full set of the calculated EDs is available as supplementary material. 19. To demonstrate the sensitivity of the method to the atom position in the space we considered a last case concerning a triplet (ACA) where one base (the 5' P) has been rotated by 90° (Table 8). Both the triplet and the base to rotate were chosen arbitrarily. The variations of the ED values are quite large as expected. 20. GT* is a three-point pairing whose index is quite lower than the comparable GC index.

Transferability of Similarity Calculations

133

21. In similarity evaluation the possibility of having very accurate calculations could have a dangerous side effect. In fact, the more precise the model is, the less extensible is its applicability. This situation is not surprising because similarity can be seen as a fuzzy property. 22. The sensitivity of the measuring method does not exclude the possibility that, in some cases, the result will depend on the approximation level used.

SUPPLEMENTARY MATERIAL (Values calculated for the complete set of 64 base triplets.)

Values of Energy Differences of Base Triplets Calculated in 3D with SP Residue and Planar Nitrogen Atom^

AAA^

AAC^

AAG^

AAU^

ACA^

ACC^

ACG^

ACU^

1 2 3 4 5

0.261 3.694 3.789 3.744

0.260 3.693 3.789 3.744

0.262 3.694 3.789 3.740

0.260 3.693 3.788 3.743

0.270 3.696 3.793 3.744

0.268 3.693 3.792 3.742

0.270 3.695 3.795 3.743

0.269 3.693 3.792 3.740

1 2 3 4 5

0.416 3.687 3.788 3.735

0.419 3.688 3.791 3.734

0.457 3.683 3.792 3.733

0.448 3.684 3.794 3.731

0.413 2.810 2.836 2.919 2.732

0.420 2.810 2.841 2.917 2.751

0.457 2.804 2.840 2.919 2.730

0.460 2.803 2.844 2.915 2.752

1 2 3 4 5

0.437 3.690 3.785 3.735

0.421 2.817 2.836 2.900 2.741

2.787 2.912 0.329 2.814 0.487

2.785 2.914 0.336 2.924 2.727

0.418 3.690 3.785 3.734

0.413 2.818 2.828 2.922 2.728

2.776 2.916 0.321 2.812 0.494

2.765 2.918 0.338 2.926 2.713

Notes: ^Atom numbering refers to Figure 3. H-ipiets are indicated by the first letter of the names of the corresponding bases. Neutral triplets.

Atom^

AGA^

AGC^

AGG^

AGU^

AUA^

AUC^

AUG^

AUU^

1 2 3 4 5

0.374 3.692 3.793 3.742

0.377 3.691 3.794 3.742

0.379 3.691 3.794 3.740

0.383 3.690 3.793 3.742

0.381 3.694 3.797 3.743

0.381 3.693 3.797 3.743

0.387 3.692 3.798 3.741

0.386 3.690 3.796 3.739

1 2 3 4 5

2.734 2.915 0.322 2.812 0.481

2.745 2.915 0.328 2.811 0.491

2.755 2.909 0.330 2.810 0.485

2.763 2.909 0.330 2.808 0.492

2.721 2.917 0.329 2.924 2.718

2.727 2.917 0.329 2.924 2.718

2.744 2.910 0.340 2.922 2.716

2.749 2.910 0.339 2.919 2.738

1 2 3 4 5

0.468 3.689 3.793 3.734

0.444 2.814 2.843 2.919 2.749

2.818 2.906 0.341 2.811 0.494

2.813 2.909 0.347 2.923 2.734

0.445 3.687 3.795 3.733

0.441 2.812 2.838 2.919 2.739

2.817 2.907 0.335 2.808 0.501

2.805 2.908 0.353 2.922 2.725

134

GUIDO SELLO and MANUELA TERMINI

Atom^

CAA^

CAC^

CAG^

CAiP

CCA^

CCC^

CCG^

CCU^

1 2 3 4 5

0.288 2.818 2.842 2.917 2.770

0.287 2.818 2.842 2.918 2.770

0.288 2.818 2.842 2.917 2.769

0.288 2.816 2.844 2.918 2.770

0.295 2.819 2.848 2.915 2.791

0.297 2.818 2.850 2.914 2.794

0.395 2.816 2.851 2.914 2.792

0.297 2.818 2.850 2.913 2.794

1 2 3 4 5

0.420 3.687 3.787 3.734

0.423 3.687 3.790 3.733

0.460 3.683 3.792 3.732

0.452 3.683 3.794 3.730

0.415 2.811 2.829 2.922 2.718

0.421 2.811 2.834 2.920 2.736

0.461 2.804 2.833 2.922 2.715

0.461 2.805 2.839 2.919 2.739

1 2 3 4 5

0.442 3.690 3.785 3.735

0.428 2.818 2.837 2.920 2.739

2.785 2.914 0.330 2.813 0.491

2.782 2.915 0.336 2.923 2.725

0.436 3.690 3.786 3.734

0.434 2.819 2.829 2.923 2.724

2.782 2.915 0.331 2.812 0.511

2.769 2.919 0.344 2.927 2.710

Atom^

CGA^

CGC^

CGG^

CGLP

CUA^

CUC^

CUG^

CUU^

1 2 3 4 5

0.419 2.815 2.846 2.919 2.765

0.421 2.815 2.847 2.920 2.766

0.423 2.814 2.847 2.918 2.765

0.427 2.810 2.848 2.920 2.766

0.430 2.815 2.852 2.914 2.794

0.428 2.815 2.854 2.912 2.795

0.431 2.813 2.854 2.913 2.795

0.430 2.814 2.854 2.910 2.796

1 2 3 4 5

2.729 2.915 0.333 2.810 0.506

2.737 2.916 0.339 2.803 0.517

2.750 2.910 0.341 2.807 0.511

2.754 2.910 0.341 2.806 0.518

2.706 2.918 0.337 2.926 2.704

2.716 2.919 0.339 2.924 2.721

2.726 2.912 0.347 2.925 2.702

2.738 2.912 0.345 2.922 2.724

/\fO/T7^

CGA^

CGC^

CGG^

CGU^

CUA^

CUC^

CUG^

CUU^

1 2 3 4 5

0.469 3.689 3.793 3.733

0.448 2.815 2.843 2.918 2.746

2.816 2.907 0.342 2.810 0.497

2.809 2.910 0.347 2.922 2.731

0.463 3.688 3.795 3.732

0.461 2.814 2.839 2.918 2.734

2.822 2.907 0.342 2.808 0.520

2.809 2.909 0.357 2.923 2.721

/Atom^

GAA^

GAC^

GAG^

GAU^

GCA^

GCC^

GCG^

GCU^

1 2 3 4 5

2.760 2.917 0.319 2.822 0.499

2.758 2.917 0.319 2.822 0.502

2.762 2.916 0.320 2.820 0.501

2.760 2.916 0.321 2.821 0.503

2.774 2.918 0.322 2.823 0.487

2.772 2.914 0.332 2.822 0.505

2.779 2.916 0.323 2.821 0.487

2.775 2.914 0.332 2.819 0.507

1 2 3 4 5

0.448 3.684 3.795 3.733

0.451 3.685 3.798 3.732

0.489 3.680 3.799 3.731

0.481 3.681 3.802 3.730

0.436 2.806 2.841 2.918 2.739

0.445 2.805 2.846 2.916 2.758

0.480 2.800 2.846 2.918 2.737

0.484 2.800 2.850 2.914 2.760

Transferability of Similarity Calculations

135

1 2 3 4 5

0.437 3.691 3.786 3.734

0.424 2.816 2.836 2.917 2.744

2.790 2.913 0.328 2.813 0.486

2.788 2.913 0.338 2.921 2.729

0.416 3.693 3.789 3.767

0.415 2.819 2.830 2.924 2.731

2.781 2.918 0.321 2.813 0.495

2.770 2.918 0.337 2.928 2.717

Atom^

CGA^

GGC^

GGG^

GGU^

GUA^

GUC^

GUG^

GUU^

1 2 3 4 5

2.782 2.912 0.325 2.820 0.504

2.776 2.912 0.322 2.820 0.504

2.785 2.912 0.327 2.818 0.507

2.780 2.910 0.323 2.820 0.506

2.791 2.913 0.320 2.821 0.490

2.789 2.901 0.330 2.820 0.506

2.795 2.911 0.330 2.820 0.506

2.793 2.908 0.332 2.816 0.510

1 2 3 4 5

2.762 2.909 0.336 2.809 0.488

2.711 2.908 0.342 2.808 0.498

2.786 2.903 0.344 2.806 0.493

2.791 2.902 0.344 2.805 0.500

2.744 2.912 0.340 2.923 2.726

2.747 2.912 0.344 2.920 2.743

2.768 2.906 0.352 2.920 2.723

2.770 2.904 0.351 2.917 2.745

1 2 3 4 5

. 0.467 3.689 3.794 3.733

0.443 2.810 2.842 2.916 2.748

2.818 2.907 0.342 2.810 0.491

2.818 2.905 0.349 2.920 2.736

0.449 3.687 3.800 3.734

0.443 2.813 2.840 2.920 2.742

2.825 2.906 0.337 2.809 0.502

2.807 2.909 0.354 2.924 2.727

Atom^

UAA^

UAC^

UAG^

UAU^

UCA^

UCC^

UCG^

UCU^

1 2 3 4 5

2.731 2.917 0.359 2.922 2.752

2.731 2.917 0.356 2.923 2.752

2.734 2.916 0.355 2.922 2.753

2.735 2.916 0.356 2.923 2.754

2.746 2.917 0.359 2.920 2.773

2.750 2.917 0.367 2.919 2.775

2.748 2.916 0.360 2.920 2.775

2.752 2.917 0.367 2.919 2.776

Atom^

UAA^

UAC^

UAG^

UAU^

UCA^

UCC^

UCG^

UCU^

1 2 3 4 5

0.451 3.682 3.796 3.731

0.454 3.683 3.799 3.730

0.491 3.679 3.801 3.730

0.483 3.679 3.803 3.728

0.445 2.805 2.838 2.919 2.728

0.452 2.805 2.843 2.916 2.748

0.491 2.799 2.842 2.919 2.726

0.491 2.799 2.847 2.915 2.750

1 2 3 4 5

0.443 3.688 3.787 3.734

0.429 2.814 2.838 2.918 2.741

2.792 2.911 0.330 2.814 0.493

2.788 2.913 0.337 2.922 2.726

0.437 3.692 3.788 3.734

0.434 2.818 2.830 2.922 2.727

2.786 2.916 0.332 2.812 0.511

2.771 2.918 0.344 2.926 2.713

Afom^

UGA^

UGC^

UGG^

UGU^

UUA^

UUC^

UUG^

UUU^

1 2 3 4 5

2.755 2.914 0.368 2.925 2.749

2.751 2.913 0.357 2.945 2.776

2.758 2.912 0.365 2.923 2.749

2.756 2.910 0.363 2.925 2.751

2.767 2.913 0.364 2.920 2.775

2.769 2.913 0.374 2.914 2.774

2.770 2.910 0.365 2.918 2.777

2.770 2.907 0.372 2.912 2.779

136

GUIDO SELLO and MANUELA TERMINI

1 2 3 4 5

2.766 2.908 0.344 2.806 0.512

2.774 2.907 0.350 2.805 0.523

2.787 2.902 0.352 2.803 0.516

2.793 2.902 0.353 2.802 0.524

2.739 2.911 0.352 2.922 2.715

2.747 2.908 0.358 2.917 2.730

2.760 2.905 0.363 2.921 2.713

2.769 2.901 0.364 2.915 2.732

1 2 3 4 5

0.472 3.686 3.795 3.733

0.452 2.812 2.844 2.917 2.748

2.822 2.904 0.343 2.811 0.500

2.816 2.906 0.349 2.921 2.733

0.467 3.688 3.796 3.731

0.465 2.813 2.840 2.915 2.734

2.826 2.906 0.345 2.807 0.521

2.810 2.908 0.361 2.918 2.720

SIMILARITY IN ORGANIC SYNTHESIS DESIGN: COMPARING THE SYNTHESES OF DIFFERENT COMPOUNDS

Guido Sello

I. II. III. IV. V.

Abstract Introduction Similarity Measures Comparison Methodology Results and Discussion Conclusion Acknowledgments References

137 138 139 140 142 150 150 150

ABSTRACT The possibility of using similarity concepts to compare syntheses of different compounds is examined and discussed. New similarity measures and indexes are described; they analyze both the strategic and tactical aspects suggesting a systematic

Advances in Molecular Similarity, Volume 2, pages 137-151. Copyright © 1998 by JAI Press Inc. Allrightsof reproduction in any form reserved. ISBN: 0-7623-0258-5 137

138

GUIDOSELLO

approach to the problem. The examples reported are used to help the reader understand the principles and methods introduced. Discussion of the results illustrates the advantages that similarity can bring into synthesis planning and emphasizes the real applicability of the realized procedures.

I. INTRODUCTION Organic synthesis planning is one of the most creative and difficult tasks that can be faced by chemists. It needs the assistance of many human intellectual abilities that all contribute to the attainment of the final result: an efficient and often elegant synthesis. In this respect, we can predict a very high productive application of similarity, as is demonstrated by several literature references.^"^ However, the explicit use of similarity in thisfieldis scarce and incomplete. Very few examples^"^^ exist that attempt to introduce similarity in the design of synthesis and most of them just mention its possible use without making an accurate analysis of its importance and contribution. But, when examining many of the best known syntheses, the impression of its presence is immediate. It is often possible to note the intelligent use of well-known synthetic steps in the planning of the synthesis of new and diverse compounds. Recently we became involved in a project fully dedicated to the introduction of similarity concepts into synthesis design.^"* We could thus elaborate on some preliminary ideas that helped the development of a rough initial system based on similarity measures. After some further modifications the system was applied to different evaluation phases and its contribution was certain. However, the greatest part of the studies^ was devoted to the application of similarity to the synthetic design of a single target with the aim of selecting the best path among many possibilities, of locating alternative steps, and of predicting the most different solutions. Now we are interested in the application of the analysis to the comparison of the syntheses of different compounds. It is evident that this problem is more difficult to solve and that it could even be difficult to assess the quality of the solution. Our previous experience has shown that even structurally similar compounds (e.g., the same compound) can be synthesized by very different routes where comparisons can be problematic. In addition, while the comparison of structures is possible by diverse methods, the comparison of transformations is often done by using substructure changes, neglecting reagents. As a consequence, it becomes hard to compare different molecules where substructures might not be similar at all. For example, the reduction of a carbon-oxygen double bond is keyed by a different substructure with respect to the reduction of a carbon-carbon double bond, despite the evident similarity from the viewpoint of synthesis planning. Herein I will address the complex problem of the comparison of the syntheses of different compounds, developing new ideas and calculation methods at the same time. The synthetic routes, used as examples, are taken directiy from the literature^^

Similarity in Organic Synthesis Design

139

and are not the best possible routes, but they can serve to assess the utility of similarity as a tool in synthesis design.

II. SIMILARITY MEASURES To compare synthetic routes we need both structure and reaction descriptors. In fact, I will develop a system that can consider the strategic aspect of the synthesis and its tactical realization. It is generally accepted that strategy is mainly a structure problem because the strategic approach to the synthesis of a target cannot be conditioned by the state-of-the-art development of transformation methodologies. On the contrary, tactics are concerned with the application of the strategic principles to the current target and depend on the transform management. During the development of a synthetic plan it is possible to conceive a system that, using similarity, can enhance strategic and tactical efficiency. However, when comparing existing synthetic routes we are forced to use similarity measures only to weigh the alternative options. We selected two structure and one reaction descriptors. In synthesis design it is important to consider two aspects of the similarity between structures: The first is a classical substructures comparison between educts and products (substructure similarity measure, SSM); the second aims at measuring the effectiveness of the synthetic step (globularity similarity measure, GSM). Every synthetic chemist instinctively feels that a good synthetic step must correlate two compounds that partially share structural features, but at the same time are as different as possible. In principle the best synthetic step transforms an educt into a product that is the most diverse where the change was predicted, but that maintains every other part of the structure unchanged. In other words, we can say that in a good educt-to-product passage all of the building blocks are conserved and all of the reacting blocks are affected. Our two structure descriptors aim indeed at measuring the realization of this goal. On the contrary, the comparison between transformations is best represented by a single descriptor that can measure the efficiency of a transformation in association with its group. The use of the reaction classification scheme developed by us should sustain the system by defining a global transform similarity measure (GTSM). SSM is directly derived from our previous work in the field of substructure similarity. ^^ We defined a substructure similarity index as SS\ = Nx{A + B)l{AxB)

(1)

where A^ is the number of similar atoms, and A and B are the numbers of atoms in molecules A and B, respectively. This index is well suited for comparing structures. Nevertheless, we slightly modified its definition to take into account two problems: the first concerning the different weight that the similarity of a connected and an unconnected substructure

140

GUIDOSELLO

must have; the second changing the limits of the index that now ranges between zero and one. The calculation is thus obtained as SSM = VSSFf

SF. = 2xN/(A-^B)

(2)

where A^. is the number of similar atoms in fragment /, and A and B are the numbers of significant atoms of molecules A and B, respectively. It follows that 0 < SF- < 1, equal to 0 if A^- is equal to 0, and equal to 1 if A^. is equal to A and A equal to B, and consequently 0 < SSM < 1. Globularity is a measure of the structure complexity and of its distribution on the molecule. ^'^ It is calculated by G = MAXD/COMPTOT

(3)

where MAXD is the greatest of the smallest distances of atom pairs measured as atom complexity, and COMPyQj is the molecular complexity measured as the sum of all atom complexities; from this descriptor we derive the similarity measure as G S M = AGAB

W

where AG is the difference between globularity of molecules A and B, A being the educt and B the product. From our reaction classification scheme we chose one descriptor that is used to measure the similarity between transformations. It is based on the calculated chemical potential ^^ of changing atoms and is obtained as ii = dE/dn = -IC,Z,^,^/(ZZXJ

+ k^

(5)

where Z^^ = Z-G = Z-[N^ + 0.85A^2 + 0.35(A^3 - 1)] Z^j^, = Z - (A^i + 0.S5N2 + 0.35A^3) Z is the atomic nuclear charge, a is Slater's core screening factor, N^ is the number of inner-shell electrons, N2 is the number of medium-shell electrons, and A^3 is the number of outer-shell electrons. Using this descriptor we can obtain the corresponding similarity measure as GTSM = ^ A ^ .

(6)

where A|i. is the difference of chemical potential of atom i in the product and the educt.

III. COMPARISON METHODOLOGY The definition of the similarity measure is necessary for the realization of a system that can compare synthetic routes; nevertheless, it is also necessary to conceive a

Similarity in Organic Synthesis Design

141

methodology that can guarantee the correct use of similarity measures. The methodology must be clear and stable; but, in addition, it is worth remembering that we are comparing quite different objects using an approach that must conserve its flexibility and large scope. Consequently, I am not going to calculate similarity indexes by just combining the corresponding similarity measures, but instead will develop a procedure where the similarity measures are used following a precise scheme. The final result will still be a number but its value will be highly dependent on the current comparison. In other words, the same synthetic step either can contribute to the calculation of the similarity index with reference to one synthesis, or can be neglected when considering a different synthesis. We can define three similarity indexes, one for SSM, one for GSM, and one for GTSM, that are calculated as reported below. SSI (substructure similarity index) is obtained by taking the geometric mean of the similarity percentages of the SSMs of all of the synthetic steps that have an SSM similar to the corresponding partner to an extent greater than or equal to 80%. For example, consider step 1 of routes A and A', and define SSM of A equal to 0.75 and SSM of A' equal to 0.82; because their ratio is equal to 0.91, step 1 contributes to SSI. On the contrary, if step 2 shows an SSM of A equal to 0.70 and an SSM of A' equal to 0.50, because their ratio is equal to 0.71, step 2 does not contribute to SSI. The rationale is that a synthetic step of two different syntheses can be considered sufficiently similar if the level of similarity of the educts and the products is at least 80%, neglecting the absolute value of the SSM. It is clear that this rationale is valid only when comparing educts and products, i.e., when we can be confident that the compounds we are considering cannot be too dissimilar. Calculation of the index is as follows: SSI = (n„ (SSM/SSM^)^/'^

V SSM/SSMj > 0.80

(7)

GSI (globularity similarity index) is based on the use of the descriptor globularity to evaluate the strategic efficiency of a synthetic step. As shown above, G is a measure of the molecular complexity and has a lower value for more complex structures; therefore, it should increase going down from target to precursors. Nevertheless, in real syntheses, G can either increase or decrease and GSM can thus be positive or negative. The corresponding index must consider this situation and we again chose to add together only GSM that show similar trends, i.e., GSI is the geometric mean of the ratio only of GSMs that have the same sign (positive or negative): GSI = (n„ (GSM. / GSM^.))^'''' V GSM/GSM^. > 0

(8)

The rationale, in this case, is that we can compare, and then add their contribution to the similarity, only those changes in globularity that have the same strategic effect, i.e., either increase or decrease the molecular complexity. Those synthetic steps that have opposite strategic meaning cannot be compared. The importance of

142

GUIDOSELLO

the contribution is measured by the similarity between the complexity changes; thus, even very small complexity variations that are strategically meaningless, can contribute to the overall strategic similarity between syntheses. GTSI (global transform similarity index) is naturally connected to the similarity between transformations, i.e., between reactions transforming educts into products. Its calculation uses the measure of the global changes in chemical potential of all of the atoms. Also in this case we must reckon when two reactions in two different syntheses can be compared. We have already reported^^ a system for reaction classification based on two descriptors, electronic energy and chemical potential, that allows the hierarchical subdivision of reactions into ordered sets. Thus, we are in the position of using that classification scheme in our procedure. However, the classification scheme is too articulated and, for the sake of synthesis comparison, we decided to use only two levels of the hierarchy. GTSI is the geometric mean of the ratio of the GTSMs of the reactions that belong to the same energy class; i.e., we consider comparable only two reactions that are both additions, or eliminations, or substitutions. GTSI is calculated as follows: GTSI = (n^ (GTSM./GTSM^)^''"

iff class(/?.) = class(/?p

(9)

In this case it is clear that we can use for similarity evaluation only those reactions that have similar reactivity bases. The GTSI values vary much more than their structure counterparts and their mere comparison can be misleading. As a consequence we decided to maintain a trace of the number of participating reactions (PRN) to the calculation and to use both GTSI and PRN as indexes of synthesis similarity.

IV. RESULTS AND DISCUSSION The results presented in the following concern two molecular sets: the first composed of four molecules of the same class (four prostaglandin derivatives), the second composed of the first set augmented by two different compounds, Sirenin and Methoxatin, chosen because the size and the length of their synthetic routes are of the same extent (Figure 1). The syntheses have been selected from the same literature source, thus their description is sufficiently uniform. However, not all of the synthetic steps have been explicitly considered and in the course of the discussion I will point out the differences that can appear just because of different descriptions. Before beginning the discussion I will repeat some general guidelines. The analysis is always carried out in the synthetic sense; thus, when I speak about, for example, step 3,1 mean the third step from the starting material. The values of the similarity measures (SSM, GSM, GTSM) are always obtained by comparing two intermediates of the same synthetic route (e.g., PGl-5 with PGl-6); on the contrary, the values of the similarity indexes (SSI, GSI, GTSI) are calculated by comparing similarity measures of corresponding steps in different syntheses.

143

Similarity in Organic Synthesis Design NHCHO PGE1

0" > r O

"N" "COOH METHOXATIN

Figure 1, Set of examined compounds.

Finally, it must be clear that, while the similarity measures represent the values of the descriptors and consequently have constant values, the similarity indexes are strictly dependent on the calculation procedure and can be changed very easily. In the first set we have four prostaglandin derivatives: PGl and PG2 are the same intermediate in two syntheses of PGEl; PG3 and PG4 are also intermediates in the syntheses of PGEl, but they are different from PGl and PG2. The four synthetic routes are sketched in Figures 2, 3, 4, and 5. The syntheses of PG3 and PG4 are very similar differing only in the last step. Nevertheless, I have freely chosen to describe the two routes in two different ways at the second and third steps because we would like to verify the response of the system. Values of SSM, GSM, and GTSM are reported in Table 1. From Table 2 we can observe that the SSI factors for PG3 and PG4 are over 80% similar in the last three steps only, in agreement with the different weight that the ethereal chain has in the two structures; as a consequence the SSI is the smallest in the PG series. Looking at the GSI, on the contrary, it is immediately obvious that the two syntheses are strategically very similar; all six steps show homogeneous variations and the final result is very clear. In conclusion, the two syntheses are similar for the strategy concerned, but it must be clear that the use of big protective groups in small molecules can influence the overall yield. The transformations are obviously very similar, being in agreement for all but one of the steps. The GTSM values are influenced by the diverse reagents used only in step 5 and in step 3, where we deliberately used water for the hydrolysis of PG3, and NaOH for that of PG4. Comparing PGl and PG2, which are exactly the same target, we can note that different synthetic routes of the same molecule are not certainly similar. All of the

GUIDO SELLO

144 NO,

NO, As^(CH2)eCN

(CH2)5CN

/

•

<

.^-TN^^..

Base ^

OCH^OCH^CHO

CHO

y^Iy^'*^

PG1-7

\CW^fM

o

o

(CH20H)2

(CH2),CN _PIS_^

^nAm PG1-2

O

Figure 2. First synthesis of PGE2. (ChyXN

^^^^^

(CH2)6CN

A J U ^^

OH

NHCHO'oH

PG2-8

PG2-7 NHCHO

NHCHO

^A^(CH2}6CN

^A^(CH,),CN

° V/ PG2-5 NHCHO

NHCHO

Js^{CH,),CN

.(CH2)5CN H2SO4.

OAc

OH°W°

PG2^

NHCHO

^A/(CH2),CN

-{CH2}sCN

OAc PG2-2

^^

O

PG2-1

^

F/g«/re J. Second synthesis of PGE2.

Similarity in Organic Synthesis Design

145

/tic. Mo^ ^ ° 3 ^

£l

2)MCPBA

^^

\

PG3-5

PG3-7 CO,H

J

1)NaOH

HjO

PG3-4

PG3-3

ri

2) (Bu)3SnH

2)Cr03

OAc

\^0

OAc

PG3-1

PG3-2

Figure 4, Synthesis of one precursor of PGE2.

indexes have PRNs equal to 2 or 3, demonstrating the low similarity between the two syntheses. On the other hand, PGl shares a good SSI with PG3 and PG2 shares a medium SSI with PG4. GSI and GTSI (Tables 3 and 4) are always very scarce; this permits us to affirm that the syntheses of PGl, PG2, and PG3 (or PG4) are different enough to represent synthetic alternatives. More interesting and more pertinent to our discussion is the analysis of the second molecule set because it can reveal the power of the similarity analysis applied to the syntheses of different

OBn

^

6 'X- "^ -^ "^ PG4-5

PG4-6

CO.H

J

1) NaOH

Wa

..ri

HjO

2)C02

OH

OBn

OH PG4-3 O

O

1)pB2CI ►

2) (Bu)3SnH

OpBz

V-OBn PG4-2

OpBz

V-OH PG4.1

Figure 5. Synthesis of another precursor of PGE2.

146

GUIDO SELLO Table 1, Values of Substructure Similarity Measure, Globularity Similarity Measure, and General Transform Similarity Measure Step

Compound

/ +2

Methoxatin Sirenin

0.55 0.74

PCI

2+3

3+4

4+5

5 +6

6 +7

0.68

0.88 0.82 0.72

0.68 0.81

0.71

0.67

0.61

0.93

0.63

0.75

0.58 0.50

0.49

0.56

0.38 0.52

PG2

0.88 0.71

0.71

0.72

0.89

0.40 0.57

PG3

0.77

0.60

0.74

0.64

0.40

0.38

PG4

0.85

0.71

0.82

0.86

0.91

0.65

Methoxatin

26

Sirenin

-46

PGl PG2

-32

PG3

60 14

PG4 Methoxatin Sirenin PGl PG2 PG3 PG4

12

7 +8

-85 -14

39

-52

115

-54

-8

10

-35

-10

56

73 25

-24

0

-26

-37

-86

-7

5 -67

-15

22

-27 -27

2

-46 -23

12

-19

El-10.07 A l 8.93 A l 13.02 A 1 8.64 IC 1 0.08 A 1 5.09 E 1 -4.38 E 1 -5.41 El-6.11 IC 1 0.89 IC i 1.49 IC 1 1.71

ICI-0.10 ICI-1.73 IC 1 0.56 A 15.33 A 1 2.96 A l 3.39

10 21

-66

IC 1 -0.30 IC 1 2.74 A l 6.19 IC 1 -3.91 A 1 3.99 A l 5.23 ICI 1.66 A 1 6.55 A 1 4.86 ICI 1.32 A 1 4.07 IC 1 -0.54 A 1 5.75 E 1-10.84 A l 4.27 IC 1 0.48 ICI-0.99 A 18.16 IC 1 -3.34 IC 1 -2.45 A 1 8.30

compounds. Both Methoxatin and Sirenin have structures that are clearly dissimilar from prostaglandins. We have only taken care of the general complexity and of the length of the synthetic routes. These two characteristics are unnecessary to make a comparison, but it is obvious that they make it easier to understand the result. In the case of comparisons between syntheses of different lengths, we need to

Table 2. Values of Substructure Similarity Index^ Sirenin Methoxatin Sirenin PGl PG2 PG3

PCI

PG2

PG3

2/0.80/0.89 3/0.67/0.88 2/0.78/0.88 3/0.69/0.88 4/0.51/0.85 5/0.69/0.93 2/0.86/0.93 3/0.72/0.89 5/0.73/0.94 3/0.76/0.91

PG4 3/0.87/0.93 3/0.82/0.94 3/0.74/0.90 4/0.72/0.92 3/0.70/0.89

Note: ^First value is Participating Reaction Number; second value is the product of SSMs; third value is the geometric mean.

Similarity in Organic Synthesis Design

147

Table 3, Values of Globularlty Similarity Index^ Sirenin Methoxatin

2 / 0.03 / 0.1 7

Sirenin

PCI 1 / 0.69 / 0.69

PG2

PG3

PG4

3 / 0.005 / 0.1 7 6 / 0.005 / 0.41 6 / 0.002 / 0.35

4/0.046/0.46 1/0.88/0.88

2/0.05/0.22

2/0.09/0.30

PG1

3/0.017/0.26 1/0.55/0.55

1/0.56/0.56

PG2

3/0.04/0.34

PG3

3/0.06/0.39 6/0.02/0.52

Note: ^First value is Participating Reaction Number; second value is the product of GSMs; third value is the geometric mean.

introduce a preliminary step that consists of the selection of the synthetic subroute of the longer synthesis. This function can be effected in two ways: either selecting the subroute containing the precursors of a size similar to the smaller compound, or selecting, after comparison, the subroute most similar to the shorter synthesis. In any case the analytical procedure is always the same, even if the result changes. Proceeding in the study of our molecule set we get the following results. It is obvious that the comparison could not suggest much similarity between compounds and syntheses that are so different, but the curiosity of determining what can be obtained by this approach is nevertheless sparkling. Sirenin. Its synthesis is sketched in Figure 6 and shows five steps where the structure remains acyclic; in step 6 two cycles are formed giving rise to the final architecture. A similar trend is present in the synthesis of PGl (cyclization at step 6) and is reflected by both SSI and GSI (PRNs equal to 4). Looking at the result of each step as measured by SSM and GSM, we can observe that (1) step 2 is a condensation of PGl and not of Sirenin, thus the two measures are in disagreement; (2) step 4 shows a bigger change for Sirenin with respect to PGl, and this is again reflected by the similarity measures; (3) steps 5 and 7, where the differences are less clear, show disagreement in only one measure. All of the remaining steps are

Table 4. Values of General Transform Similarity Index^ Sirenin Methoxatin Sirenin PGl PG2 PG3

PCI

4/0.004/0.25 2/0.09/0.30 3/0.12/0.49

PG2

PG3

2/0.24/0.49 4/0.06/0.49 2/0.10/0.32 2/0.07/0.26 2 / 0.81 / 0.90 0 / 0.00 / 0.00 3/0.19/0.57

PGA 3/0.03/0.31 2/0.54/0.73 1 / 0.07 / 0.07 2/0.10/0.32 5/0.06/0.57 (4/0.43/0.81)

Note: ^First value is Participating Reaction Number; second value is the product of GTSMs; third value is the geometric mean.

148

GUIDO SELLO 1)BuLi, TMS 2) BuLi

Br

3) AgNOs 4) NaCN

CHjO

S8

Ni(C0)4 ^

1

^

^

1)CH2N^

S r;^co,H

^

^^""^^

OH S6

S5

C03CH3

1)NH2NH2 2) Mn02

( >s

COXH, S3

C03CH3

1)Se02 ^ 2) LiAIH4

Figure 6. Synthesis of Sirenin.

similar for the strategy concerned. Also, PG2 shows an interesting similarity with Sirenin in SSM, despite the fact that PG2 cyclization is at the third step as evidenced by GSI; this result demonstrates that the similarity in the changed or maintained substructures can exist even for compounds that are quite dissimilar. The similarity between reactions is a different matter. PGl has three steps that are classified similarly to those of Sirenin (step 3 is an addition; step 5 is a substitution; step 6 is an addition) and only step 3 has a similarity value that is of the same magnitude. The most similar synthesis is, however, that of Methoxatin with four of six steps falling in the same class (step 1 of Methoxatin and step 2 of Sirenin are additions; step 3 of Methoxatin and step 4 of Sirenin are eliminations; step 4 of Methoxatin and step 5 of Sirenin are substitutions; step 5 of Methoxatin and step 6 of Sirenin are additions) with the two addition steps showing similar values. Methoxatin. Methoxatin is a poly cyclic aromatic compound; the synthesis of one of its precursors is sketched in Figure 7 and is compared with all other members of the set. Despite the clear diversity of this molecule with respect to the others, it is still possible to have some hints of the similarity of its synthetic route. SSI is always

Similarity in Organic Synthesis Design

149

1) HCOOAc. HCO2H

1) NaNOo. 0.3N HCI =

2)R02. H j . EtOH. 65

►

2) KOH, CH3OH/H2O, 0 'C NHCHO OCH3

^

o

M6

HN

Y

6,"

Acetone/H20. A

NHCHO

NHCHO

OCH3 M5

CP2CH3 ,/ - N H H O

CO2CH3

CH2CI2

NHj

s<:^^CO,CH, cOjCH,

CO,CH, C02CH3

Figure 7, Synthesis of Methoxatln.

relatively important (PRNs equal to 2 or 3) because the Methoxatin synthesis has much more structural change at each step than any other compound in the set. Somewhat surprisingly, GSI behaves differently. Both PG3 and PG4 show exactly the same trend as Methoxatin and the GSI values are very near those between PG3 and PG4 themselves, regularly alternating increases and decreases of globularity. In the case of Methoxatin the alternation is evidently justified by the corresponding alternation of condensation-cyclization steps. GTSI also shows some interesting similarity in the PG3 and Sirenin syntheses (PRNs equal to 4 of six compared steps for both). The addition and elimination steps contribute the most to the similarity, as expected. In concluding the analysis of this second molecule set, we can affirm that even the syntheses of quite different compounds can be compared providing interesting suggestions on similar reactivity or strategy weight of each route.

150

GUIDOSELLO

Some final remarks are necessary. Similarity has shown its potential use also in the analysis of synthetic routes; it is clearly possible to compare the syntheses of very different compounds; the results are often easy to understand. Concerning the values of SSM and GSM, we think their meaning can be immediately grasped, i.e., a great value of SSM means a largely conserved structure, a positive change of GSM means a molecule with a more distributed complexity. In the same way we can interpret SSI and GSI values, i.e., two synthetic steps can be considered similar if their SSM share the same level of substructure change or their GSM share the trend in complexity distribution. GTSM and GTSI are less easily bound to the reactivity behavior. First, we had to define a classification scheme to compare transformations; second, the calculated GTSMs are highly dependent on the reaction class. In particular, additions and eliminations generally have more similar values and, when it is the case, their comparison gives greater contributions. On the contrary, substitutions have, by definition, small values that change size and even sign and their comparison has a lesser influence. This fact must be kept in mind when comparing the values of GTSI, because the same number of similar transformations can give different index values if we are considering additions-eliminations or substitutions. Finally, it is important to remember that the PRNs must always be considered because two synthetic routes that share many steps are more similar than two routes that share few steps with higher values.

V. CONCLUSION The importance of similarity measures has been demonstrated in the field of synthesis planning. Using a limited amount of descriptors it is possible to effectively compare the syntheses of different compounds and to obtain, as a final result, the level of their relative efficiency. In addition, by defining a reference synthesis it could be possible to order synthetic routes, thus suggesting a preferential order of application. It is clear that the "best" synthesis does not exist, but it is likely that any reference synthesis can be used for the goal of determining the most appealing between a limited set of synthetic routes.

ACKNOWLEDGMENTS Partial financial support by the Consiglio Nazionale delle Ricerche, and by the Ministero deirUniversita e della Ricerca Scientifica e Tecnologica, is gratefully acknowledged. The author thanks the Organizing Committee of the 7th International Conference on Mathematical Chemistry for the invitation to present part of this work.

REFERENCES 1. Johnson, A. P.; Marshall, C; Judson, P. N. Reel. Trav. Chim. Pays-Bas 1992, 111, 310.

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2. (a) Hendrickson, J. B. Anal. Chim. Acta 1990,235,103. (b) Hendrickson, J. B. Reel Trav. Chim. Pays-Bos 1992,111,323. 3. Gordeeva, E. V.; Lushnikov, D. E.; Zefirov, N. S. Tetrahedron 1992, 48, 3789. 4. (a) Barone, R.; Arbelot, M.; Chanon, M. Tetrahedron Comput. Method 1988, 7, 3. (b) Azario, P.; Arbelot, M.; Baldy, A.; Meyer, R.; Barone, R.; Chanon, M. New J. Chem. 1990,14, 951. 5. (a) Long, A. K.; Kappos, J. C ; Rubinstein, S. D.; Walker, G. E. / Chem. Inf. Comput. Sci. 1994, 34, 922. (b) Long, A. K.; Kappos, J. C. J. Chem. Inf. Comput. Sci. 1994, 34, 915. (c) Corey, E. J.; Long, A. K.; Lotto, G. I.; Rubinstein, S. D. Reel. Trav. Chim. Pays-Bas 1992, HI, 304. 6. Johnson, A. R; Marshall, C ; Judson, R N. Reel Trav. Chim. Pays-Bas 1992, HI, 310. 7. Gasteiger, J.; Hondelmann, U.; Rose, R; Witzenbichler, W. J. Chem. Soc. Perkin Trans. 2 1995, 193. 8. Nakayama, T. J. J. Chem. Inf. Comput. Sci 1995, 35, 885. 9. Gelemter, H.; Rose, J. R.; Chen, C. / Chem. Inf Comput Sci 1990, 30, 492. 10. Hamm, R; Jauffret, R; Kaufmann, G. Reel Trav. Chim. Pays-Bas 1992, 111, 317. 11. Fontain, E.; / Chem. Inf Comput. Sel 1992, 32, 748. 12. (a) Gasteiger, J.; Ihlenfeldt, W. D.; Rose, R R. Reel Trav. Chim. Pays-Bas 1992, 111, 270. (b) Ihlenfeldt. W.; Gasteiger, J. Angew. Chem. Int. Ed Engl 1995, 34, 2613. (c) Gasteiger, J.; Ihlenfeldt, W. D.; Rose, J. D. J. Chem. Inf Comput. Sel 1992, 32, 700. 13. Wochner, M.; Brandt, J.; v.Scholly-Pfab, A.; Ugi, I. Chimia 1988, 42, 111. 14. Sello, G.; Termini, M. Tetrahedron 1997, 53, 3729. 15. Corey, E. J.; Cheng, X. The Logic of Chemical Synthesis; Wiley: New York, 1989, pp. 141,165, 251,253,255,258. 16. Sello, G.; Termini, M. In Advances in Molecular Similarity; Carbo, R.; Mezey, P., Eds.; JAI Press: Greenwich, CT, 1996, Vol. 1, p. 213. 17. Baumer, L.; Campagnari, I.; Sala, G.; Sello, G. Reel Trav Chim. Pays-Bas 1992, 111, 297. 18. Sello, G. Theoehem 1995, 340, 15. 19. Sello, G.; Termini, M. Tetrahedron 1997, 53, 14085.

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BROWSABLE STRUCTURE-ACTIVITY DATASETS

Mark Johnson

I. II. III. IV. V. VI. VII.

Abstract Introduction The Problem of the Merchandiser A First Look at a Structure-Activity Dataset Molecular Equivalence Numbers as Primary Structural Browsing Variables Level Sets as Primary Structural Browsing Variables Similarity-Based Projections as Primary Browsing Variables Summary and Conclusions References

153 154 155 156 . 160 164 167 169 169

ABSTRACT Ever-larger structure-activity datasets are motivating the need for ways of rapidly finding and validating interesting structure-activity patterns. Patterns among chemical and biological-activity variables are easily viewed and browsed with currently available graphical methods. Substructure and similarity searching provide an analogous capability offindinginteresting compounds represented in a structural database, but these techniques are slow and unsystematic. Structural browsing variables provide

Advances in Molecular Similarity, Volume 2, pages 153-170. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 153

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MARK JOHNSON

a complementary and systematic browsing alternative. The features that make these variables useful for structure browsing are discussed and illustrated using the perillartine sweetener dataset of Acton and Stone.

I. INTRODUCTION In the pharmaceutical industry, the race to discover new chemical entities for treating disease is dictating larger project teams that encompass more extensive and diverse synthetic efforts directed at increasingly complicated activity spectra. These changes are now being fueled by rapid technological advances in high-throughput screening/ combinatorial chemistry,^ and genomics."^ Chemists will increasingly be confronted by ever-larger structure-activity datasets in terms of both numbers of compounds and numbers of activities. This raises the browsing question: What is required of an effective method for efficiently gaining an intuitive and reasonably comprehensive view of the basic factors determining the structure-activity relationships buried in these datasets? The time-honored way of accessing structural information in chemical structure database is substructure searching. It serves well if you have a clear hypothesis or idea of what types of structures you should be examining. The notion of browsing arises when structure-activity data must be organized and examined in a manner that facilitates the formulation of hypotheses and ideas and the isolation of structural effects that those hypotheses must explain. Similarity searching has been proposed as a browsing tool."* Each similarity search forms a group of those structures having a specified similarity with a query structure. By performing a sequence of similarity searches in which the next query structure is a "hit" from a preceding search, one has the sense of "wandering about in structure space." However, similarity searching restricts one's attention to those compounds that can be "reached" from the original query structure by always taking steps no larger than the similarity search radius. These reachability sets, also called level sets, can be confined to quite small regions of structure space when the compounds are highly clustered. As the reachability sets grow in size, a second difficulty is encountered. Many structural regions in the reachability set will never be visited because of the unsystematic paths one is likely to take in a "similarity walk" through structure space. In this study, I address the browsing problem by stepping back to the very first steps of a structure-activity analysis. Here one is formulating ideas concerning the important substructures and critical structural comparisons that shape our intuitive understanding of the structure-activity relationship. Section 2 begins by analogically looking at the general nature of the problem raised by the browsing question. The concepts of primary and secondary browsing variables are motivated. Section 3 presents some data from an analysis of some oxime sweeteners and illustrates the use of activity variables as browsing variables.

Browsable Structure-Activity Datasets

155

Section 4 defines the concept of a molecular equivalence number (MEQN) and gives an example of an MEQN that works very effectively as a primary browsing variable. It also argues by example that many of the common quantitative chemical descriptors are not likely to be primary browsing variables, but may work well as secondary browsing variables. Section 5 shows that indexed level sets can be very effective primary browsing variables. Finally, Section 6 examines a particular combination of two one-dimensional multidimensional scaling projections that work well for browsing.

II. THE PROBLEM OF THE MERCHANDISER The manager of a large department store encounters a problem analogous to that raised by the browsing question: How can the merchandise be structured so that a first-time shopper can effectively find what he or she wants in the store? A purely random assortment of the merchandise may suffice if one has only a dozen articles or so, but as the number increases, a one-dimensional arrangement is almost invariably employed, usually decomposed into linear or curvilinear segments defined by the aisles of the store. The shopper can systematically browse the items in the store because of this basic linear structure. Systematic browsing also requires that the items be meaningfully organized along the aisles. A purely random order is never used. Neither are size or color used even though they are critical attributes of merchandise. Rather the items are functionally grouped into men's wear, lady's wear, hardware, appliances, and so forth. These groups must then be indexed to positions along the aisles. A variety of indexings would work; it is only important that one be implemented. The formation of these groups and the selection of an indexing defines what I will term SL primary browsing variable. Within these primary groupings, the articles are arranged by other attributes to influence or facilitate the shopper. For example, if the group is men's suits, they might be further ordered by size. Because color and size operate at this second ordering level, I will refer to them as secondary browsing variables. Mathematically, a variable is a mapping of a set of objects to the real line. If these objects are molecular structures, that variable is often called a chemical descriptor. As a variable, any chemical descriptor effectively orders groups of structures along the real line, the entity that corresponds to the aisles in our analogy. If one thinks of a group as those molecular structures mapped to a particular number or interval of numbers, it becomes apparent that any chemical descriptor is a basis for defining groups of structures. But are these groups interesting, and how do we tell? Our merchandising analogy provides a clue. If the merchandiser is clever, he or she will form groups that facilitate the comparisons a shopper wishes to make in purchasing an item. In buying a suit, you will be wanting to make comparisons between suits of your size, and you can be reasonably assured of finding the suits

156

MARK JOHNSON

that interest you in close proximity to each other. You will not find those suits dispersed throughout the store. With that clue in mind, we seek a basis for forming groups of structures so that one can make interesting structural comparisons within a group.

III. A FIRST LOOK AT A STRUCTURE-ACTIVITY DATASET Acton and Stone^ present a number of analogues of perillartine, an oxime sweetener. The taste potency relative to sucrose along with percent sweetness and water solubility were reported for each analogue. These values are given in Table 1. Acton and Stone wanted a compound that would elicit a strong, sweet taste. As such, they required high values for percent sweetness, potency, and solubility. However, they found that the more potent compounds tended to be the more insoluble compounds. This trade-off between potency and solubility is given in Figure 1. The log of the potency plus the log of the solubility is an approximate measure of deliverable taste strength which will be called delivered potency. It is apparent from Figure 1 that aldoxime, compound 44, represented the best combination of percent sweetness and delivered potency among the compounds in the study. Consequently, our analysis will focus on these two activity variables. A simple listing of the structures found in Ref. 5 is not given here because the function of such a listing becomes largely archival as the number of structures gets large. This study emphasizes the role that systematic structure browsing plays in searching out the critical structures that shape one's perception of a structureactivity relationship. Consequently, only a few selected subsets of structures are presented as might be constructed by various browsing techniques. This results in some structures appearing in more than one figure. Such reporting is inefficient from an archival viewpoint, but helps to "explain" why such structures come to play such critical roles in our intuitive understanding of a structure-activity relationship. The development of this intuitive understanding is the principal function of structure-activity browsing. Activity as a function of structure has often been likened to a mountainous surface where the height corresponds to activity and the latitudinal and longitudinal coordinates correspond to locations in structural space. When first venturing into a large structure-activity dataset, it is natural to want to look at those structures with the most desired activity spectrum, i.e., the "highest" points of the activity landscape that have been visited in a lead optimization effort. The first row of Figure 2 lists the first five structures with the highest delivered potency. The diversity of structures gives one great freedom in formulating possible hypotheses as to the types of structural features associated with good delivered potency. We might seek hypotheses related to the minimal number of those structural features. This leads to compound 3. Alternatively, we might seek hypotheses related to the largest number of features consistent with good delivered potency.

Browsable Structure-Activity Datasets

157

Table 1, Structure-Activity Table. Biological and physico-chemical data from [5]. Num

LogSol

LogPot

1 -3.3 3 -0.22

2.57 0.9

4 -0.82 6 -2.7

1.6 1.98

7 -2.3 8 -3.7

1 2.24

9 -3.7 10 -2.52

3.06 1.7

11 -1.96 12 -2.7

1.08

13 -1.57 14 -1.92

%Swt

DelPot

60 -0.73 15 0.68 55 0.78 17 -0.72 0 -1.3 14 -1.46 50 -0.64 65 -0.82

ShrbSiz Rnglso

BdSetO

BdDel2

6 5

1 2

1 14

1 2

6 12

2 1

15 17

12

1

5 5 4

1 1

18 2

BdDelRPI

BdDelMPI

-1.09 4.81

1.44 -4.56

2

4.81

-3.8

12

-1.09

7.79

13 14

-1.09

7.79 -1.5 0.17

-1.09

2

1

3

19

3

-1.09 -4.91

1 1

20

15

-1.09

21 22

16 17

-1.09

1.43 2.11

-1.09

4

23 24

18

-1.09

5.69

19

-1.09

25

20

-1.09

6.93 5.61

4.05

6

0.9

23 -0.88 55 -0.8 4 -0.67

15 -1.8

0.6 1.04

6 -1.32 4 -0.76

10 11

16

0.3

1.48

5

1.78

10

17 -1.1

1.74

48

0.64

3

4

26

21

2.73

-5.42

18 -2.22

1.38

6

4

3

22

2.73

-5.73

19 -1.3 20 -1.96

0.9 1.2

39 -0.84 42 -0.4

5

1

-1.09

-0.09

6

1

3 4

4 4

-1.09

-0.12

21 -1.1 22 -0.15

0.48 0.18

45 -0.76 22 -0.62

3 3

5 6

5 5

23 24

1 0.88

-3.83 -4.01

23 -1.1

0.3 1

3

6

5

25

0.88

5

6 7

3

26

0.88

27 28 6 4

27 1 1

0.77 -1.09 -1.09 -1.09 -1.09 -1.09 -1.09 -0.42

24 -1.64 25 -1.26 26 -1.77 27 -3.22 28 -1.52 29 -2.52 32 -1.66 33 -2.05 34 -1.74 35 -3.22 37 -2.4 38 -0.85

1.9

0

0.03

1.95 2.18

9 10 3 4

2.13

65 -0.27

7

0.6

2 -0.25

1.3 1.74 2.4 1.72 1.9 0.9 1.7

39 -1.6

1.23

4 -0.37

2.15 2.61

0 -0.85 0 -0.79

42 -1.82

2.3 2.7

45 -2.4

8

2 -0.8 0 -0.64 0 0.04 40 -0.03 16 -0.82 53 0.2 50 -0.62 0 -0.76 0 -0.35 50 0.21 22 -1.04

40 -3 41 -3.4 43 -3.22 44 -1.7

8

70

0.48

2.35

78 -0.52 90 0.65

2.48

92

0.08

3 3 4

1 1 1 1

6 7

-4.06 -3.04 -3.91 -1.01 0.14

8

7 8 29 9

3 3 5 5 6

8

10

6

-0.42

8

12

7

-0.42

9

9

5

30

4.81

-3.93

11 7 4

2

3 30

31

4.81

-0.96

2 1

4.81

-4.42 -1.26

3 4

10

-1.09 -0.7

2 1

6

2.18 3.3 4.87 6.16 -2.16 -0.7 3

8 8 9

-0.7

6

10 10

9 10 11

-0.7

-1.01 1.14

7

10

12

9

-0.7

242

-2.38

{continued)

MARK JOHNSON

158 Table 1. Continued Num

LogSol

LogPot

46 -3.4 47 -2.22

2.51

48 -2.52 49 -1.7 50 -2.4

1.45 2.15

-1.92

51 Note:

%Swt

DelPot

34 -0.89 12 -0.7

1.52

ShrbSiz Rnglso

BdDel2

BdDelRPI

10

1

32

4

33

2.73

1 10

3 7 11

34 10

-1.09 -0.7

10

12

10

-0.7

-1.24 -4.22 -3.24

11

1

35

-3.92

-1.83

2 -1.07

1.96

0 0.45 0 -0.44

6 7

2.51

0

3

-0.7

BdDelMPI

6 6 7

0.59

BdSetO

1.25 -5.47

Num corresponds to the Acton and Stone ordering [5]; LogSol and LogPot are the logs of the solubility and the potency; %Swt is percent sweetness; DelPot = LogSol+LogPot; ShrbSiz is the number of atoms outside the ring system; Rnglso is an arbitrary indexing of the ring systems; BdSetO (BdDel2) are the 0-level (2-level) sets based on the bond-set (edge-deletion) distance; BdDelRPI (BdDelMPI) are 1 -dimensional multidimensional-scaling projections of the edge-deletion pairwise distances between the ring systems (chemical graphs).

This leads to compound 16. As yet, we have no basis with which to reject the hypothesis that any structure "in between" these two extremes would have good delivered potency. All experimental bases for excluding hypotheses have been filtered out by our singular focus on compounds with good delivered potency.

Sweetness

«ooooOO

o (A D)

eg

— I —

0.5

1.0

1.5

2.0

— I —

I

2.5

3.0

Log potency

Figure 1. Joint plot of potency, solubility, and sweetness showing compound 44, the selected sweetener aldoxime.

Browsable Structure-Activity

Datasets

159

The second row of Figure 2 lists the first five structures with the highest percent sweetness. Four of the five structures are close analogues of each other and possess a common ring system. On the one hand, this greatly reduces the diversity of structures from which we might spawn hypotheses. On the other hand, these close analogues reveal a number of small structural changes at the para position that do not change percent sweetness. Thus, we have located part of a "sweetness plane" in this particular response region of structure space. Because compounds 42, 43, and 45 are not found in the first row of Figure 2, it might be suspected that we have identified a "delivered-potency cliff." We shall see that other browsing variables are more specifically designed for systematically locating those cliffs. Again, by construction, this set of structures excludes from our consideration those critical changes that do reduce percent sweetness, i.e., the "sweetness cliffs." To get a better feel for the activity planes and cliffs of our activity landscape, we would like to know more about the structural changes that have been tried and the effects of those changes. By suitably narrowing our focus, we can easily collect together all of the compounds in a small region of structure space. Substructure searching and similarity searching are common methods of "drilling down" and focusing on smaller regions of structure space. For example, one could list all of the 1,4-cyclohexadiene derivatives or perform a similarity search with compound 44 as the query structure. Both represent hypotheses that restrict one's attention, and by suitably narrowing one's focus, one eventually reaches the point where all

First five compounds ordered by delivered potency

U First five compounds ordered by percent sweetness Figure 2. Structures with the best five delivered potency values and the best five sweetness values.

160

MARK JOHNSON

of the structures within a substructure search can be examined. This process of drilling down takes us out of the browsing mode which concerns us here. It is natural to return to the browsing mode when we seek the next interesting region of structure space.

IV. MOLECULAR EQUIVALENCE NUMBERS AS PRIMARY STRUCTURAL BROWSING VARIABLES A subregion of structure space becomes interesting if it contains some of the peaks, planes, and cliffs of the structure-activity surface that stand out from the vast regions of inactive space. The preceding section used delivered potency and percent sweetness as browsing variables. By concentrating on the most desired values for these variables, we located the delivered potency and sweetness peaks and, in doing so, found a sweetness plane. These two browsing variables made no use of structure. The remaining part of this study will focus on browsing variables that take structure into account. We began with a simple count, called ShrbSiz in Table 1, of the number of atoms outside of the ring system. Figure 3 maps ShrbSiz, the number of atoms that lie outside the ring system, against our two activity variables. The values of ShrbSiz have been jittered by the addition of a small random number so as to distinguish compounds with common tied values. No global trends are apparent in Figure 3

I O

6
Number of atoms outside of theringsystem Figure 3. Joint plot of jittered ShrbSiz, delivered potency, and sweetness.

Browsable Structure-Activity Datasets

161

except that the better values for these two variables tended to be restricted to the lower values of ShrbSiz. The absence of interesting global trends does not rule out the possibility that by examining compounds with particular values of ShrbSiz we might find some interesting structural comparisons as we did in the bottom half of Figure 2. This possibility is ruled out by considering Figure 4, which contains those structures with a ShrbSiz of 6. One can find a few interesting structural comparisons, such as compounds 44 and 49. But even with this small set of compounds, such comparisons are difficult to locate because of a lack of a dominaang and obvious commonality among the structures. ShrbSiz is an example of SL globally quantitative chemical descriptor. A difference in ShrbSiz for two compounds always means that the two compounds differ in the number of atoms that lie outside of the ring system; it is only a matter of scale. Babaev and Hefferlin^ provide interesting arguments for using the number, A^, of atoms and the number, Z^, of valence electrons in a molecule as globally quantitative chemical descriptors for organizing molecular structures when one is looking for molecular properties that reflect the fundamental periodicities found in atomic structure. However, in the regions of (iV,Z^)-space where most drugs are found, it has yet to be demonstrated that these two descriptors will prove effective as primary

6^° 47

o 28

4

Figure 4. Compounds with a ShrbSiz of 6.

49

MARK JOHNSON

162

browsing variables. This does not mean that quantitative properties are not useful in a browsing context. In fact, ShrbSiz will be seen to be very effective as a secondary browsing variable for viewing and comparing structures within a group defined by another primary browsing variable. Opposite the quantitative chemical descriptors are the nominal chemical descriptors. Here we only have to give a structural reason why two compounds have identical values for the descriptor. There is no requirement that compounds given different descriptor values have anything in common or are similar to each other in some regard. In other words, a difference in the value of a nominal descriptor has no meaning unless this difference is 0. A nominal chemical descriptor can be defined by simply specifying a manner in which two compounds are to be viewed as equivalent. For example, two molecules may have the same molecular formula, the same chemical graph, or the same ring system. Each of these equivalence relationships partitions every collection of compounds into disjoint classes so that each compound falls into one and only one class. An MEQN is an arbitrary indexing of the classes of an equivalence relation that is defined over all compounds. Rnglso in Table 1 groups the oximes according to a common ring system. (Acyclic structures are equivalent to methane in this equivalence relation, and the

B

Q

d

in

d

Ring equivalence number

Figure 5. Joint plot of jittered Rnglso, delivered potency, and sweetness.

Browsable Structure-Activity Datasets

163

number of atoms in the ring system is defined to be 1.) By replacing ShrbSiz in Figure 3 with Rnglso, we obtain Figure 5. Because of its construction, we cannot expect to see global structure-activity trends between Rnglso and activity. However, by systematically examining the structural classes, we see that the most desired compounds have a Rnglso value of 10. These structures are listed in Figure 6. The structures in Figure 6 share an obvious commonality, namely, their ring system. Consequently, one can expect that almost any pairwise comparison would prove interesting, especially if one takes into account how medicinal chemists use functional groups to explore a structure-activity relationship. When one notes the disparities in the sweetness of compounds in this set, one knows that a region of structure-activity cliffs has been isolated. However, these cliffs are still to be specified structurally. This is easily done visually using structure-activity maps, one of which for the dataset can be found in Ref. 7. However, structure-activity maps are slow to construct relative to the rate at which large structure-activity datasets must be browsed. Consequently, there is a need for a simpler method of visually isolating the cliffs and planes. The structures in Figure 6 have been ordered by ShrbSiz. Using this ordering together with Figure 5, we see that adjacent compounds 42 and 43 are part of a sweetness plane and a delivered-potency cliff. Further on, adjacent compounds 49 and 44 are part of a sweetness cliff and a delivered-potency plane, and adjacent compounds 50 and 45 are again part of a sweetness cliff and possibly a deliveredpotency cliff. If these observations are not to remain anecdotal curiosities, possibly explicable in terms of experimental variation or inexplicable because of the complexities of the underlying mechanisms, we must group them in terms of commonalities and

'

r

'

°\

42

49

Figure 6. Compounds with a Rnglso of 10.

MARK JOHNSON

164

trends. For example, the sweetness cliffs defined by compounds 49 and 44 and compounds 50 and 45 are both associated with meta versus para substitution. As we begin to piece together these cliffs and planes, an intuitive understanding of a structure-activity relationship emerges that must be taken into account by any quantitative description of that relationship.

V. LEVEL SETS AS PRIMARY STRUCTURAL BROWSING VARIABLES Molecular similarity measures are playing an increasingly important role in drug discovery. ^"^ ^ A variety of algorithms exist for clustering a collection of compounds once the pairwise similarities or distances between the compounds have been calculated.^^ In hierarchical clustering, one can speak of the clusters associated with a given level, v, of similarity. ^^ These clusters partition the collection of structures into disjoint sets of compounds, the level sets at level v. Unlike the equivalence classes used in the construction of an MEQN, the membership of these sets changes as the collection of compounds changes. By adding another compound to the collection it is possible for two level sets to merge into one even though the level value, V, does not change. Indexing the level sets of a hierarchical clustering of structures gives rise to another class of interesting browsing variables. Table 1 includes two level-set

Sweetness

••••••# •

4 4 «

•

(D Q. •

•

28

O

42

49*

34 45

• 22

d

•

• •20

36

37 4 3 »

21 .23

• ^

6

•

50*

29 • 32

41 27

35»

• 48

4

•

8

10

Level sets Figure 7. Joint plot of jittered BdSetO, delivered potency, and sweetness.

12

Browsable Structure-Activity Datasets

165

browsing variables: BdSetO and BdDel2. BdSetO is an arbitrary indexing of the single-linkage level sets for the bond set distance at level 0. (The bond set distance between molecules A and B is the total number of bonds in both molecules minus the number of bonds they have in common when ignoring connectivity.) At a level of 0, two compounds are in the same level set if and only if they are indistinguishable by the bond set distance. BdDel2 is an arbitrary indexing of the level sets of the bond deletion distance at level 2. (The bond deletion distance between molecules A and B is the minimum number of bonds that must be deleted from either A or B that results in a common substructure. This conmion substructure need not be connected.) By letting BdSetO be the primary browsing variable we obtain Figure 7. Looking at compounds 44 and 49, we know we have located a sweetness cliff and a delivered potency plane. Turning to Figure 8, which contains the structures of a number of level sets in Figure 7, we see that the sweetness cliff is defined by differences associated with the meta Sind para positions. This cliff was noted in the preceding section when we confined our attention to a single ring system. Level sets often allow comparisons across ring systems. For example, a comparison of compounds 42 and 34 suggests that a 1,4-cyclohexadiene analogue will have better sweetness and potency than the 1,3-cyclohexadiene analogue. This "substructure" hypothesis is not only confirmed by the comparison of compounds 45 and 37, but suggests that compound 45 may be approaching an optimal structure because the same structural

/ ^ 23 o

'-><^<»

Figure 8. Structures of annotated points in Figure 7.

MARK JOHNSON

166

f?

s & ?
>

in

d o o

■<5 Q

Level sets Figure 9, Joint plot of jittered BdDel2, delivered potency, and sweetness.

change had a much more dramatic effect, an argument very much in spirit with the Pfeiffer rule.^^'^"* Unfortunately, the 1,3-cyclohexadiene analogue of compound 44 was not isolated.^

°v/

Figure 10, Structures of annotated points In Figure 9.

Browsable Structure-Activity Datasets

167

By letting BdDel2 be the primary browsing variable, we obtain Figure 9. Using a different similarity measure gives rise to slightly different groupings, but the nature of the arguments remains unchanged. Compounds 44 and 45 define a sweetness plane and a modest delivered-potency cliff. A similar statement can be made for compounds 49 and 50, except now the delivered-potency cliff is more pronounced and neither compound is sweet. Looking at the corresponding structures in Figure 10, we see the negative effect on sweetness is associated with substitution at the meta position and the negative effect on delivered potency is associated with a methyl addition alpha to the ring.

Vl. SIMILARITY-BASED PROJECTIONS AS PRIMARY BROWSING VARIABLES Given the pairwise similarities or distances between the compounds, multidimensional scaling methods can array the compounds in R^ so that their pairwise distances in /?" optimally correlate with their pairwise similarities or distances in structure space. By letting n = 1, we obtain another potential browsing variable. The variables BdDelRPl and BdDelMPl are two such projections. BdDelMPl represents a projection of the distance matrix computed over all molecular structures using the bond deletion distance. To form BdDelMPl, the multidimensional scaling function cmdscale of Splus^^ was run on the pairwise bond deletion distances calculated on the hydrogen-reduced chemical graphs of the molecules using the R^ projection option. BdDelMPl is the x-axis component of that projection. BdDelRPl was computed in a similar manner except the pairwise distances were based on the chemical graphs of the ring systems. BdDelMPl may be thought of as a one-dimensional molecular projection and BdDelRPl as a one-dimensional ring-system projection. Recall that Rnglso was an arbitrary indexing of the ring systems. The x-axis of Figure 5 represents only one of the many indexings that would order the compounds along a line while preserving the integrity of the ring system groups. However, one might be interested in arranging these ring systems along the line in a manner that puts similar ring systems close to one another. In our department store analogy, this would be analogous to arranging the item groups so that men's shirts are close to men's suits. The ring projection, BdDelRPl, can be viewed as replacing the arbitrary indexing in Rnglso with a more relevant ordering without destroying the makeup of the ring system groupings. Figure 11 plots the molecular and ring projections, BdDelRPl and BdDelMPl, against each other. Percent sweetness is expressed as the size of the plotted point and delivered potency as the type character used for the plotted point. The best sweetener would be a large, solid disk. Compound 14 is represented by a solid but small disk. Compound 44, the selected sweetener, is represented by a large circle indicating a better than 90% sweet taste and better than average delivered potency.

MARK JOHNSON

168 8 >♦ 7 « 15

Delivered potency

■ 33 * 14

Sweetness

+X5KAO®« D • • • • • • •

•

X *

37

+ 0

K 2*

• 1

1

Q

2 ^ 3 '♦^ 22

1

1

1

Ring projection

F/gure / / . Joint plot of jittered BdDelRPI, BdDelMPI, delivered potency, and sweetness.

The sweet compounds, indicated by the large plotted characters, cluster in the middle of the figure. The five labeled compounds in this region form the upper row of Figure 12. This is a highly organized set of structures with an easily recognized commonality consisting of a six-membered ring substituted with analogously sized substituents in thtpara position.

"v><„

Figure 12, Structures of annotated points in Figure 11.

Browsable Structure-Activity Datasets

169

Other regions of this plot also organize structures for meaningful visual comparison. The two other sets of annotated points at the top and bottom of the plot illustrate two of these regions. These corresponding compounds are not sweet as judged by the small characters associated with the plotted points. The corresponding structures are given in the second and third rows of Figure 12. The second row of structures share the same ring system as compound 28 in the center of the first row, but have a different group of substituents in ihopara position. Interestingly, the acid and salt, compounds 6 and 7, are separated in Figure 11 from the remaining three compounds in the second row.

VII. SUMMARY AND CONCLUSIONS This study has compared a number of chemical descriptors with regard to their potential utility in a single context involving roughly 50 compounds. In this limited context, the browsing variables have accomplished the purpose for which they were designed: to group structures in meaningful ways that facilitate the construction of an intuitive understanding of the activity cliffs and planes associated with the structure-activity surface. Clearly no single browsing variable is going to solve all needs. This raises the question as to what combinations of browsing variables are needed to most quickly bring one to an intuitive understanding of the structure-activity relationship. In this regard, the department store analogy begins to break down. As was pointed out to me by Doug Klein at the Girona Conference on Mathematical Chemistry and Molecular Similarity, the store manager changes his or her arrangement of the store items at great cost. With computers, we can rapidly reorganize the structures to suit a variety of browsing needs. The ability to define collections of browsing variables creates the need for flexible and powerful software tools for dynamically exploring visual presentations of tabled data. As the number of points grows and the number of variables increases, it becomes increasingly important to have access to a dynamic data visualization package for putting together various visual combinations of browsing variables and activity variables. For this study, Spotfire^^ was used for dynamic data visualization, and Splus^^ was used to construct the printed figures.

REFERENCES 1. Hams, A. L. P/i^rm. A^^w5 1995, 2, 26. 2. Czamik, A. W.; EUman, J. A., Eds.; Combinatorial Chemistry. Special issue of Ace. Chem. Res. 1996, 29. 3. Beeley, L. J.; Duckworth, D. M. Drug Discovery Today 1996, 7, 474. 4. Bawden, D. In: Concepts and Applications of Molecular Similarity, Johnson, M. A.; Maggiora, G. M., Eds.; Wiley-Interscience: New York, 1989, pp. 65-76. 5. Acton, E. M.; Stone, H. Science 1976, 2, 584.

170

MARK JOHNSON

6. Babaev, E. V.; Hefferlin, R. In Concepts in Chemistry: A Contemporary Challenge; Rouvray, D. H., Ed.; Wiley-Interscience: New York, 1997, pp. 41-100. 7. Johnson, M. A. J. Biopharm. Stat. 1993, 3, 203. 8. Willett, R Similarity and Clustering in Chemical Information Systems', Research Studies Press: Letchworth, 1987. 9. Johnson, M. A.; Maggiora, G. M., Eds.; Concepts and Applications of Molecular Similarity; Wiley-Interscience: New York, 1989. 10. Dean, R M., Ed., Molecular Similarity in Drug Design; Blackie Academic & Professional: London, 1995. 11. Carbd, R., Ed.; Molecular Similarity and Reactivity: From Quantum Chemistry to Phenomenological Approaches; Kluwer: Dordrecht, 1995. 12. Jain, A. K.; Dubes, R. C. Algorithms for Clustering Data; Prentice-Hall: Englewood Cliffs, NJ, 1988, p. 66. 13. Pfeiffer, C. C. Science 1956, 7, 29. 14. Lehmann, P A. F Quant. Struct.-Act. Relat. 1987, (5, 57. 15. Splus is a commercial statistics package marketed by StatSci, a division of MathSoft, Seattle, Washington, http://www.mathsoft.com. 16. Spotfire is a commercial data visualization package marketed by Spotfire AB, Goteberg, Sweden, http://www.spotfire.com.

CHARACTERIZATION OF THE MOLECULAR SIMILARITY OF CHEMICALS USING TOPOLOGICAL INVARIANTS

Subhash C. Basak, Brian D. Cute, and Gregory D. Grunwald

Abstract I. Introduction IL Methods A. Database B. Calculation of Indices C. Classification of the Indices D. Statistical Methods and Computation of Similarity III. Results A. Principal Component Analysis B. Analogue Selection C. /C-Nearest-Neighbor Property Estimation IV. Discussion Acknowledgments References Advances in Molecular Similarity, Volume 2, pages 171-185. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 171

172 172 173 173 173 178 178 179 179 183 183 183 184 184

172

SUBHASH C. BASAK, BRIAN D. CUTE, and GREGORY D. GRUNWALD

ABSTRACT Three similarity spaces were used in the selection of analogues and i^T-nearestneighbor (KNN)-based estimation of normal boiling points for a diverse set of 2926 chemicals. The similarity spaces consisted of principal components derived from (1) 40 topostructural indices, (2) 61 topochemical parameters, and (3) the full set of 101 topostructural and topochemical indices. The three methods selected sets of analogues with a substantial number of structurally analogous molecules. For the KNN method of property estimation, the similarity space that used the full set of indices was superior to either of the subsets (topostructural or topochemical). For all three methods, K = 6-10 gave the best estimated values for boiling point.

I. INTRODUCTION Interest in quantifying the similarity of molecules using computational methods has increased. ^"^ In particular, a recent trend in the characterization of similarity/ dissimilarity of chemicals makes use of graph invariants. Molecular structures can be represented by planar graphs, G = [V,E], where the nonempty set V represents the set of atoms and the set E generally represents covalent bonds.^ These graphs can be used to adequately represent the pattern of connectedness of atoms within a molecule. Graph invariants, values derived from planar graphs, are graph theoretic properties which are identical for isomorphic graphs. A numerical graph invariant or topological index maps a chemical structure into the set of real numbers. Various graph invariants have been used in ordering and partial ordering of sets of molecules. ^'"^"^ Various topological indices (TIs) and principal components (PCs) derived from TIs have been used in quantifying the similarity/dissimilarity of molecules and in the similarity-based estimation of physical and toxicological properties."^'^'^^^^ Such TIs include those derived from simple planar graphs which contain adjacency and distance information for vertices. These TIs could be considered topostructural indices. Other TIs, which are derived from weighted chemical graphs, could be regarded as topochemical indices because they contain explicit information regarding the chemical nature of the atoms (vertices) and bonds (edges) in the molecular structure, in addition to quantifying the adjacency and distance relationships within the graph. Our earlier studies made use of a combination of topostructural and topochemical indices to select analogues of chemicals and estimate properties of molecules in large and diverse databases using the A'-nearest-neighbor (KNN) method. In this paper we have carried out a comparative analysis of similarity-based analogue selection and KNN-based estimation of normal boiling point using: (1) a set of 40 topostructural indices, (2) a group of 61 topochemical indices, and (3) the combined set of 101 indices.

Molecular Similarity Using Topological Invariants

1 73

II. METHODS A. Database

The normal boiling point database consisted of 2926 compounds taken from the U.S. EPA ASTER^^ system. The data comprised a set for which chemical structures and normal boiling values were available, and for which it was possible to compute all 101 TIs. B. Calculation of Indices

The TIs calculated for this study are listed in Table 1 and include Wiener number/^ molecular connectivity indices as calculated by Randic^^ and Kier and Hall,^^ frequency of path lengths of varying size, information theoretic-indices defined on distance matrices of graphs using the methods of Bonchev and Trinajstic^^ as well as those of Raychaudhury et al.,^^ parameters defined on the neighborhood complexity of vertices in hydrogen-filled molecular graphs,^"^"^^ and Balaban's J indices."^^"^^ The majority of the TIs were calculated using POLLY 2.3.-^^ The J indices were calculated using software developed by the authors. The Wiener index (W), the first topological index reported in the chemical literature,^^ may be calculated from the distance matrix D(G) of a hydrogensuppressed chemical graph G as the sum of the entries in the upper triangular distance submatrix. The distance matrix D(G) of a nondirected graph G with n vertices is a symmetric nxn matrix (J.), where d.. is equal to the distance between vertices v. and v- in G. Each diagonal element d-- of D(G) is zero. We give below the distance matrix D(G^) of the unlabeled hydrogen-suppressed graph Gj of n-propanol (Figure 1): (1) (2) (3) (4) 1 D(G,) = l 4

0 1 2 3 1 0 1 2 2 1 0 1 3 2 1 0

W is calculated as

W=l/2'^d,j = 'Zh.g,

(1)

where g^ is the number of unordered pairs of vertices whose distance is h. Thus, for D(Gy), Wheis a value often. RandiC's connectivity index,^^ and higher order connectivity path, cluster, pathcluster, and chain types of simple, bond and valence connectivity parameters were

174

SUBHASH C. BASAK, BRIAN D. GUTE, and GREGORY D. GRUNWALD

Table 1. Symbols, Definitions, and Classifications of Topological Parameters Topostructural 1^

Information index for the magnitudes of distances between all possible pairs of vertices of a graph

1^ W

Mean information index for the magnitude of distance Wiener index = half-sum of the off-diagonal elements of the distance matrix of a graph

P

Degree complexity

H^

Graph vertex complexity

H^

Graph distance complexity

7C

Information content of the distance matrix partitioned by frequency of occurrences of distance h Order of neighborhood when / Q reaches its maximum value for the hydrogen-

O

filled graph M^i

A Zagreb group parameter = sum of square of degree over all vertices

M2

A Zagreb group parameter = sum of cross-product of degrees over all neighboring (connected) vertices

^X

Path connectivity index of order h = 0-6

^X

Cluster connectivity index of order h = 3-6

^XpQ

Path-cluster connectivity index of order h = 4 - 6

^Xch

Chain connectivity index of order h = 3-6

P^

Number of paths of length /i = 0-10

J

Balaban's J index based on distance

'ORB

Topochemical Information content or complexity of the hydrogen-suppressed graph at its maximum neighborhood of vertices

IC^

Mean information content or complexity of a graph based on the rth ( r = 0-6) order neighborhood of vertices in a hydrogen-filled graph

SIC^

Structural information content for Ath ( r = 0-6) order neighborhood of vertices in a hydrogen-filled graph

CICr

Complementary information content for rth (r = 0-6) order neighborhood of vertices in a hydrogen-filled graph

^A^

Bond path connectivity index of order h = 0-6

^Ac

Bond cluster connectivity index of order h = 3-6

"A^h

Bond chain connectivity index of order h = 3-6

^A^c

^0"<^ path-cluster connectivity index of order h = 4 - 6

^A^

Valence path connectivity index of order h = 0-6

^XQ

Valence cluster connectivity index of order h = 3-6

'^Ach

Valence chain connectivity index of order h = 3-6

^XpQ

Valence path-cluster connectivity index of order h = 4 - 6

f

Balaban's J index based on bond types

/

Balaban's J index based on relative electronegativities

J

Balaban's J index based on relative covalent radii

Molecular Similarity Using Topological Invariants (1)

(2)

(3)

175 (4)

G, Figure 1, The unlabeled hydrogen-suppressed graph (Gi) of n-propanol.

calculated using the method of Kier and Hall.^^ The generalized form of the simple path connectivity index is as follows:

paths

where v., v.,.. . ,v^^j are the degrees of the vertices in the path of length h. The path length parameters (P^), number of paths of length /z (/i = 0, 1, . . . , 10) in the hydrogen-suppressed graph, are calculated using standard algorithms. Information-theoretic TIs are calculated by the application of information theory on chemical graphs. An appropriate set A of n elements is derived from a molecular graph G depending on certain structural characteristics. On the basis of an equivalence relation defined on A, the set A is partitioned into disjoint subsets A • of order n. (/ = 1, 2 , . . . , /z; S-n- = n). A probability distribution is then assigned to the set of equivalence classes: Aj, A2, . . . , A^ PvP2^'--^Ph

where/?. = n/n is the probability that a randomly selected element of A will occur in the ith subset. The mean information content of an element of A is defined by Shannon's relation:^ ^ h

The logarithm is taken at base 2 for measuring the information content in bits. The total information content of the set A is then n x IC. To account for the chemical nature of vertices as well as their bonding pattern, Sarkar et al.^^ calculated the information content of chemical graphs on the basis of an equivalence relation where two atoms of the same element are considered equivalent if they possess an identical first-order topological neighborhood. Since properties of atoms or reaction centers are often modulated by stereoelectronic characteristics of distant neighbors, i.e., neighbors of neighbors, it was deemed

176

SUBHASH C. BASAK, BRIAN D. CUTE, and GREGORY D. GRUNWALD

essential to extend this approach to account for higher order neighbors of vertices. This can be accomplished by defining open spheres for all vertices of a chemical graph. If r is any nonnegative real number and v is a vertex of the graph G, then the open sphere ^(v, r) is defined as the set consisting of all vertices v- in G such that ^(v,v•) < r. Therefore, 5(v,0) = (]), 5(v, r) = v for 0 < r < 1, and 5(v, r) is the set consisting of V and all vertices v- of G situated at unit distance from v, if 1 < r < 2. One can construct such open spheres for higher integral values of r. For a particular value of r, the collection of all such open spheres 5(v,r), where v runs over the whole vertex set V, forms a neighborhood system of the vertices of G. A suitably defined equivalence relation can then partition V into disjoint subsets consisting of vertices that are topologically equivalent for rth-order neighborhood. Such an approach has been developed and the information-theoretic indices calculated based on this idea are called indices of neighborhood symmetry.^^ In this method, chemicals are symbolized by weighted linear graphs. Two vertices UQ and VQ of a molecular graph are said to be equivalent with respect to r-th-order neighborhood if and only if corresponding to each path WQ, Wj,.. ., M^ of length r, there is a distinct path VQ, Vj, . . . , v^ of the same length such that the paths have similar edge weights, and both UQ and VQ are connected to the same number and type of atoms up to the rth-order bonded neighbors. The detailed equivalence relation has been described in earlier studies.^^'^^ Once partitioning of the vertex set for a particular order of neighborhood is completed, IC^ is calculated by Eq. 2. Basak et al. defined another informationtheoretic measure, structural information content (SIC^), which is calculated as SlC^^IC/log^n

(4)

where IC^ is calculated from Eq. 2 and n is the total number of vertices of the graph.^"^ Another information-theoretic invariant, complementary information content (CIC^), is defined as C/C, = log2n-/C,

(5)

CICj, represents the difference between maximum possible complexity of a graph (where each vertex belongs to a separate equivalence class) and the realized topological information of a chemical species as defined by IC^}^ In Figure 2, the calculation of IC2, SIC2, and C/C2 is demonstrated for the labeled hydrogen-filled graph (G2) of n-propanol. The information-theoretic index on graph distance, I^, is calculated from the distance matrix D(G) of a chemical graph G as foUows:^^ (6)

l'^=W\og2W'-'Zg,'h\og2h

Molecular Similarity Using Topological

Invariants

H2

\77

Hg

I y^ I

G2: n-propanol

H3

I

H7

H5

Second order neighbors: I

m

n

^1

^^2

I . I 0

•

C

'^3

H4

1 .

1

C

1

H/I\ H/I\

C

" 0 ^

IV

1 1

c

c

H Q C

H Q C

•

r

H5

1

ih ih ih c

" 0 ^

V

VI

^0.

^c.

Vffl

vn H I S'

J\\

H

HV|

H i C " H

/W

JK

H H

Subsets:

I

V

VI

vn

vni

(OO

(Ci)

(C2)

{C3)

IV

V

VI

vn

vm

3/12

1/12

1/12

1/12

1/12

n

m

(H2-H3)

(H4-H5)

(He-He)

I

n

ffl

1/12

2/12

2/12

(Hi)

IV

Probability:

IC2 = 5*1/12*1092 12 +2*2/12*1092 12/2 + 3/12*log2 12/3 = 2.855 bits SIC2 = ICi/log2 12 = 0.796 bits CIC2 = 1092 12 - IC2 = 0.730 bits

Figure 2, Calculation of the indices IC2, SIC2, and CIC2 for the hydrogen-filled, labeled graph (C2) of n-propanol.

178

SUBHASH C. BASAK, BRIAN D. GUTE, and GREGORY D. GRUNWALD

The mean information index, ^ , is found by dividing the information index I^ by W. The information-theoretic parameters defined on the distance matrix, H^ and //^, were calculated by the method of Raychaudhury et al.^^ Balaban defined a series of indices based on distance sums within the distance matrix for a chemical graph which he designated as J indices.^^"^^ These indices are highly discriminating with low degeneracy. Unlike W, the J indices have a range of values that is independent of molecular size. The general form of the J index calculation is as follows: / = ^(ji + l)-' ^ ( v / ' ^ 2

(7)

ij, edges

where the cyclomatic number |LI (or number of rings in the graph) is\x = q-n-\-l with q edges and n vertices, and 5- is the sum of the distances of atom / to all other atoms and s- is the sum of the distances of atom; to all other atoms.^'^ Variants were proposed by Balaban for incorporating information on bond type, relative electronegativities, and relative covalent radii.^^'-^^ C. Classification of the Indices

The set of 101 TIs was partitioned into two distinct subsets: topostructural indices and topochemical indices. Topostructural indices encode information about the adjacency and distances of atoms (vertices) in molecular structures (graphs) irrespective of atom type or factors such as hybridization states and number of core/ valence electrons in individual atoms. Topochemical indices quantify information regarding specific chemical properties of the atoms comprising a molecule as well as the topology (connectivity of atoms). Topochemical indices are derived from weighted molecular graphs where each vertex (atom) is properly weighted with selected chemical/physical properties. These subsets are shown in Table 1. D. Statistical Methods and Computation of Similarity Data Reduction

Initially, all TIs were transformed by the natural logarithm of the index plus one. This was done since the scale of some TIs may be several orders of magnitude greater than other TIs. A principal component analysis (PGA) was used on the transformed indices to minimize intercorrelation of indices. The PCA analysis was accomplished using the SAS procedure PRINCOMP.^'* The PCA produces linear combinations of the TIs, called principal components (PCs) which are derived from the correlation matrix. The first PC has the largest variance, or eigenvalue, of the linear combination of TIs. Each subsequent PC explains the maximal index variance orthogonal to the previous PCs, eliminating any redundancies that could occur within the set

Molecular Similarity Using Topological Invariants

1 79

of TIs. The maximum number of PCs generated is equal to the number of TIs available. For the purposes of this study, only PCs with eigenvalues greater than one were retained. A more detailed explanation of this approach has been provided in a previous study by Basak et al."^ These PCs were subsequently used in determining similarity scores as described below. Similarity Measures Intermolecular similarity was measured by the Euclidean distance (ED) within an n-dimensional space. This /x-dimensional space consisted of orthogonal variables (PCs) derived from the TIs as described above. ED between molecules / and j is defined as 1/2

EDy-

1(D,,-Djf

(8)

lc=l

where n is the number of dimensions or PCs retained from the PCA. Z).^ and DM are the data values of the kth dimension for chemicals / and 7, respectively. K'Nearest'Neighbor Selection and Property Estimation Following the quantification of intermolecular similarity of the 2926 chemicals, the A'-nearest neighbors {K= 1-10,15,20,25) were determined on the basis of ED. This procedure can be used to select structural analogues (neighbors) of a probe compound or the neighbors can be used in property estimation. In estimating the normal boiling point of the probe compound, the mean observed normal boiling point of the A'-nearest neighbors was used as the estimate and the standard error {s) of the estimate was used to assess the efficacy of the set of indices.

III. RESULTS A. Principal Component Analysis From the PCA of the 40 topostructural indices, seven PCs with eigenvalues greater than one were retained. These seven PCs explained, cumulatively, 90.8% of the total variance within the TI data. Table 2 lists the eigenvalues of the seven PCs, the proportion of variance explained by each PC, the cumulative variance explained, and the three TIs most correlated with each individual PC. The PCA of the 61 topochemical indices resulted in the selection often PCs, all having eigenvalues greater than one. The ten PCs explain a total of 92.1% of the variance within the TI data. Table 3 presents a summary of the information regarding these ten PCs.

Table 2. Summary of Principal Component Analysis of 40 Topostructural Indices for 2926 Chemicals

PC 1 2 3 4 5 6 7

Eigenvalue

Proportion of Explained Variance

Cumulative Explained Variance

28.2

46.2

46.2

11.0

18.0

64.3

Top Three Correlated Indices

5.9

9.6

73.9

4.1

6.7

80.6

Pi,Po. >^ 4y 5V 6y '^PC/ '^PC/ ^PC 3 y 5 y 4y ^O ^O ^PC L '>^Ch. 'Xc

2.8

4.6

85.2

XcU' '^Ch/ '^Ch

1.9

3.1

88.3

1.5

2.4

90.8

XcUf XQ\^f '^Ch Xn PlO/ P9

Table 3. Summary of Principal Component Analysis of 61 Topochemical Indices for 2926 Chemicals

PC 1 2 3 4 5 6 7 8 9 10

Table 4.

PC

Eigenvalue

Proportion of Explained Variance

Cumulative Explained Variance

Top Three Correlated Indices

20.4

33.5

33.5

^A^, 2;^, ^A*'

10.8

17.8

51.2

5/C4, S/C3, S/C5

8.1

13.3

64.6

3A^, ^A^, ^A^C

6.1

9.9

74.5

3.0

5.0

79.5

2.4

3.9

83.4

1.7

2.8

86.2

1.4

2.2

88.4

-^h/ ' ^ h / -^h 3yb 3yv 4yb '^Ch/ ^Chf -^Ch ICQ, SICQ, / Q 6 vb 5 vb Sy/ ^0 ^C' '^C ^A^, 2;^, ^A^

1.2

2.0

90.4

1.1

1.8

92.1

^A^, ^A^, ^A^ 4yb 4yv 6yv C' C' '^PC

Summary of Principal Component Analysis of 101 Topological Indices for 2926 Chemicals

Eigenvalue

Proportion of Explained Variance

Cumulative Explained Variance

Top Three Correlated Indices

1

42.6

41.6

41.6

2 3 4 5

13.3 11.4 8.9 5.1

13.0 11.1 8.7 5.0

54.7 65.8 74.5 79.6

Pi, PQ/^'^

6

3.7

3.6

83.2

ICQ, SICQ, SIC^

7

2.6

2.6

85.8

^A^, ^A^ ^A^

8

2.0

1.9

87.7

9

1.7

1.7

89.4

"^X^, ICQ, SICQ

10 11

1.4 1.1

1.4 1.1

90.8 91.9

^A^/A^c/^'^ ICyf^JCo

12

1.0

1.0

92.8

PS.PW^PQ

"^A^o % o ^-^c SICs, SICe, CICe ^^ck^'Xcu^'X^u y/^ch/'^'^h

^'^/^'^/'^h

Molecular Similarity Using Topological Invariants

181

Twelve PCs were retained from the PC A of the full set of 101 TIs. Each of these PCs had an eigenvalue greater than one and, cumulatively, they explained 92.8% of the variance within the full set of TIs. These PCs are summarized in Table 4.

Probe: 3-methyl-4-chlorophefX)l

CH3

Structural:

OL O a OL O ^CHa

CI

OH

(1) 0.00

Chemical:

CI

CI

(2) 0.00

(3) 0.01

dLtt'O ^Cl

^ ^

^CHg

(1) 0.01

All:

NHj

(1) 0.01

"Y

CI

NHj

(2) 0.02

NH2

(3) 0.02

NH2

(4) 0.01

on

T^

^ ^

6H

(4) 0.02

(5) 0.01

^T^

^CHg

CI

(5) 0.03

a,a.oco CI

OH

(2) 0.02

(3) 0.02

CI

(4) 0.03

(5) 0.03

Figure 3. The five analogues selected for the probe 3-methyl-4-chlorophenol using three molecular similarity spaces: topostructural, topochemical, and all indices. The numbers under the structures indicate the ranking of the analogues and the Euclidean distance to the probe.

SUBHASH C. BASAK, BRIAN D. CUTE, and GREGORY D. GRUNWALD

182

Table 5. Comparison of the Three Sets of TIs and Their Derivative PCs for Prediction of Normal Boiling Point (°C) Using K-Nearest-Neighbors {n = 2926) Indices

K

r

s

Topostructural Topochemical Topostructural + topochemical

10 6 8

0.881 0.883 0.896

39.0 38.6 36.6

0.92 0.90 -

0.88

«

0.86

a> o ^ 0.84 Topostructural indices Topochemical indices All Indices

0.82 H 0.80

T

5

10

15

20

25

30

Number of neighbors (K)

50 Topostructural indices Topochemical indices All indices

48 46 44 42 40 38 36 34 10

15

20

25

30

Number of neighbors (K) Figure 4. Pattern of (top) correlation (r) and (bottom) standard error (s) of the estimates according to the /C-nearest-neighbor selection for 2926 normal boiling points using three molecular similarity spaces.

Molecular Similarity Using Topological Invariants

183

B. Analogue Selection

Figure 3 shows an example of analogue selection using PCs to derive a Euclidean distance space. The first five analogues (neighbors) for the probe compound, 3-methyl-4-chlorophenol, are presented for each of the three similarity spaces. The analogues selected by the topostructural model show a repetition of the same skeletal structure, ignoring substituents, throughout the first five analogues. In the topochemical model and the full set model some variability in the skeletal structure arises (chemical analogues 2 and 5, full set analogue 4). Also of interest is the repetition of chemicals between the sets of analogues. While the ordering varies between the methods, the topostructural and topochemical models select two identical structures, the topostructural and the full set have three analogues in common, and the topochemical and full set select four of the same analogues. 2-Chloro-5-methylphenol appears in all three sets, while there are only three unique compounds (topostructural analogues 4 and 5, topochemical analogue 5). C. fC-Nearest-Neighbor Property Estimation

Figure 4 presents the correlation (r) and the standard error (s) of the prediction of the normal boiling points for the 2926 chemicals for the three groups of indices over the full range of i^ values examined (K= 1-10,15, 20, 25). Table 5 shows the best normal boiling point model for each set of indices. The best boiling point estimates for all three sets were for K in the range of 6 to 10. The full set of indices gave the best result, although there was only a small difference between models.

IV. DISCUSSION The purpose of this paper was to study the relative effectiveness of three similarity spaces derived from graph invariants in the selection of structural analogues and in the KNN-based estimation of properties. The similarity spaces were created using a PCA of calculated graph invariants. Tables 2-4 summarize the results of the PCA of the three sets of indices. The first PC is always correlated with indices that quantify molecular size. In the case of the topostructural indices, the second PC is most correlated with branching indices. In the case of PCs derived from either topochemical or the full set of topostructural and topochemical parameters, the first PC was strongly correlated with molecular size, while the second PC was highly associated with the molecular complexity indices. These results are in line with our earlier studies on different sets of chemicals."^'^'^^'^^'-^^ All three spaces were used in the selection of five analogues of a particular structure (Figure 3). Perusal of the three sets of structures shows that there is a substantial degree of similarity among the three groups of five chemicals selected. It is interesting to note that all five nearest neighbors of the probe selected by the topostructural method had isomorphic skeletal graphs when hydrogen atoms are

184

SUBHASH C. BASAK, BRIAN D. GUTE, and GREGORY D. GRUNWALD

suppressed. For the two similarity spaces created by topochemical indices alone and the combined set of topostructural and topochemical indices, four of the five selected neighbors are common (Figure 3) although the ordering of the molecules is different. This shows that these two similarity methods are not intrinsically very different. Our earlier results showed that analogues selected by similarity methods derived from experimental physical properties, atom pairs, and TIs select very similar sets of analogues.^^ In the case of KNN-based estimation of boiling points of chemicals from their analogues, K was varied from 1 to 25. The best estimated value was obtained in the range of ^ = 6-10. This is in line with our earlier studies with different properties."-'2 In conclusion, the three similarity spaces derived in this paper have reasonable power for selecting analogous molecules from a very diverse database of chemicals. The KNN-based estimation shows that selected analogues can be used for the estimation of boiling points of diverse chemicals if more accurate methods are not available.

ACKNOWLEDGMENTS This is contribution number 161 from the Center for Water and the Environment of the Natural Resources Research Institute. Research reported herein was supported in part by grants F49620-94-1-0401 and F49620-96-1-0330 from the United States Air Force, a grant from Exxon Corporation, and the Structure-Activity Relationship Consortium (SARCON) of the Natural Resources Research Institute of the University of Minnesota.

REFERENCES 1. Johnson, M. A.; Maggiora, G. M. Eds. Concepts and Applications of Molecular Similarity; Wiley: New York, 1990. 2. Carbd, R.; Leyda, L.; Amau, M. Int. J. Quantum Chem. 1980, 77,1185. 3. Bowen-Jenkins, P. E.; Cooper, D. L.; Richards, G. J. Phys. Chem. 1985, 59, 2195. 4. Basak, S. C ; Magnuson, V. R.; Niemi, G. J.; Regal, R. R. Discrete Appl. Math. 1988, 79,17. 5. Basak, S. C ; Bertelsen, S.; Grunwald, G. J. Chem. Inf. Comput. Sci. 1994, 34, 270. 6. Rum, G.; Hemdon, W C. J. Am. Chem. Soc. 1991,113,9055. 7. Willett, P.; Winterman, V. Quant. Struct.-Act. Relat. 1986, 5, 18. 8. Wilkins, C. L.; RandiC, M. Theor. Chim. Acta 1980,58, 45. 9. Trinajstie, N. Chemical Graph Theory Vols. I c& 77; CRC Press: Boca Raton, PL, 1983. 10. Basak, S. C ; Grunwald, G. D. Math. Model. Sci. Comput., in press. 11. Basak, S. C ; Grunwald, G. D. SAR QSAR Environ. Res. 1994, 2, 289. 12. Basak, S. C ; Grunwald, G. D. New J. Chem. 1995, 79, 231. 13. Basak, S. C ; Grunwald, G. D. / Chem. Inf Comput. Sci. 1995,35, 366. 14. Basak, S. C ; Grunwald, G. D. SAR QSAR Environ. Res. 1995,3, 265. 15. Basak, S. C ; Grunwald, G. D. Chemosphere 1995,31, 2529. 16. Basak, S. C ; Gute, B. D.; Grunwald, G. D. Croat. Chim. Acta 1996, 69,1159. 17. Lajiness, M. S. In: Computational Chemical Graph Theory, Rouvray, D. H., Ed. Nova Science Publishers: New York, 1990, p. 300.

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18. Russom, C. L. Assessment Tools for the Evaluation of Risk (Aster) v. 7.0; U.S. Environmental Protection Agency, 1992. 19. Wiener, H. J. Am. Chem. Soc. 1947, 69, 17. 20. Randie, M. J. Am. Chem. Soc. 1975, 97, 6609. 21. Kier, L. B.; Hall, L. H. Molecular Connectivity in Structure-Activity Analysis; Research Studies Press: Hertfordshire, U.K., 1986. 22. Bonchev, D.; TrinajstiC, N. J. Chem. Phys. 1977, 67, 4517. 23. Raychaudhury, C ; Ray, S. K.; Ghosh, J. J.; Roy, A. B.; Basak, S. C. J. Comput. Chem. 1984, 5, 581. 24. Basak, S. C; Roy, A. B.; Ghosh, J. J. In Proceedings of the Second International Conference on Mathematical Modelling; Avula, X. J. R; Bellman, R.; Luke, Y. L.; Rigler, A. K., Eds.; University of Missouri-Rolla, 1980, p. 851. 25. Basak, S. C ; Magnuson, V. R. Arzneim.-Forsch. Drug Res. 1983, 33, 501. 26. Roy, A. B.; Basak, S. C ; Harriss, D. K.; Magnuson, V. R. In Mathematical Modelling in Science and Technology; Avula, X. J. R.; Kalman, R. E.; Liapis, A. I.; Rodin, E. Y, Eds.; Pergamon Press: New York, 1984, p. 745. 27. Balaban, A. T. Chem. Phys. Lett. 1982, 89, 399. 28. Balaban, A. T. Pure andAppl. Chem. 1983, 55, 199. 29. Balaban, A. T. Math. Chem. (MATCH) 1985, 21, 115. 30. Basak, S. C ; Harriss, D. K.; Magnuson, V. R. POLLY v. 2.3 (copyright University of Minnesota), 1988. 31. Shannon, C. E. Bell Syst. Tech. J. 1948, 27, 379. 32. Sarkar, R.; Roy, A. B.; Sarkar, R. K. Math. Biosci. 1978,39, 299. 33. Magnuson, V. R.; Harriss, D. K.; Basak, S. C. In Studies in Physical and Theoretical Chemistry; King, R. B., Ed.; Elsevier: Amsterdam, 1983, p. 178. 34. SAS Institute Inc. In SAS/STAT User's Guide, Release 6.03 Edition; SAS Institute Inc.: Gary, NC, 1988, p. 751. 35. Basak, S. C ; Niemi, G. J.; Veith, G. D. J. Math. Chem. 1991, 7, 243. 36. Basak, S. C.; Magnuson, V. R.; Niemi, G. J.; Regal, R. R.; Veith, G. D. Math. Model 1987, 8, 300.

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OPTIMIZING HYBRID DENSITY FUNCTIONALS BY MEANS OF QUANTUM MOLECULAR SIMILARITY TECHNIQUES

Miquel Sola, Marta Fores, and Miquel Duran

Abstract I. Introduction II. Methodology III. Results and Discussion A. The CO System B. The N2 System C. The LiF System IV. Conclusions Acknowledgment References and Notes

188 188 190 192 192 199 200 201 201 201

Advances in Molecular Similarity, Volume 2, pages 187-203. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 187

188

MIQUEL SOLA, MARTA FORES, and MIQUEL DURAN

ABSTRACT The ao, a^, and ac semiempirical parameters of the original three-parameter method of Becke have been optimized by minimizing the difference between the density deUvered by this method and the singles and doubles quadratic configuration interaction (QCISD) generalized density using quantum molecular similarity measures. The optimization is performed employing the relaxed geometry at each set of ao, ax, and ac parameters. This method has been applied to a series of small molecules (N2, CO, and LiF) that have experimentally known properties and molecular bonds of diverse degrees of ionicity and covalency. Results show that, at least in these diatomic molecules, it is possible to obtain a set of parameters that reproduces almost exactly the electron density obtained from the QCISD methodology. Especially interesting are the values obtained for the ao parameter, which reflect how much exact exchange should be included in the description of a particular system.

I. INTRODUCTION The density-functional theory (DFT)^ of electronic structure has seen significant advances since its original formulation by Hohenberg, Kohn, and Sham.^ The use of density gradients in exchange and correlation corrections to the so-called local spin-density approximation (LSDA) has largely improved the computed molecular properties like geometries, vibrational frequencies, dipole moments, and particularly the molecular bond energies.^'^"^ Despite the unquestionable success of DFT in many fields, there is still a need to improve the current DFT schemes. For instance, the difficulties that DFT has in describing weak interactions like hydrogen bonds, charge transfer, and van der Waals complexes are well established,^ and they will be surmounted only with the advent of more precise exchange-correlation potentials. In this sense, the analysis of the performance and the improvement of different exchange-correlation functional is a subject of great current interest. So far, two approaches have been followed. On one hand, some authors have investigated nonlocal schemes of the generalized gradient approximation (GGA) which include the Laplacian of the electron density and the kinetic energy density.^ On the other hand, different hybrid schemes^"^^ have been proposed in which the Hartree-Fock (HF) exact treatment of exchange is incorporated to some extent into the available functionals. These hybrid DFT methods, which are based on the adiabatic connection formula, ^^ attempt to improve the exchange part of the exchange-correlation functional. In fact, it has been shown that errors in molecular descriptions arise mainly from the treatment of exchange, which is the dominant part of the exchange-correlation energy.^^'^'^"^ So, it is commonly argued that a partial inclusion of the exact exchange must improve the overall accuracy of the exchange-correlation functional. Indeed, results show that these hybrid methods

Optimizing Hybrid Density Functionals

189

yield energetic and structural results of an accuracy comparable to those obtained by methods that are much more demanding computationally.^^ Probably the most popular hybrid scheme for the exchange-correlation functional is Becke's three-parameter method, which was originally formulated as^

Here the E^^^^ term corresponds to the HF exchange energy based on Kohn-Sham orbitals, while E]^^ is the uniform electron gas exchange-correlation energy, AE^^ is Becke's 1988 gradient correction for exchange,^^ and AE^^^ is Perdew and Wang's gradient correction for correlation.^^ Commonly, this procedure is referred to as the B3PW91 method. The coefficients a^, a^, and a^ were determined by Becke fitting 56 atomization energies, 42 ionization potentials, and 8 proton affinities. The values obtained, which are in some sense semiempirical, were a^ = 0.20, a^ = 0.72, and a^ = 0.81. It is worth noting that, in this fitting, single point energy calculations were performed at experimental geometries. Furthermore, the exact exchange and the gradient corrections were added in the evaluation of energies in a non-self-consistent fashion using converged LSDA densities.^^ Probably, the so-called B3LYP functional is even more popular than the B3PW91 method. In the Gaussian-94^^ implementation the expression used is similar to Eq. 1 with slight differences: ^xc = Ex'""^ + %(^r'

- ^ ^ ^ ^ ) + ^.^T

+ E^"" + a.iAE"^ - E^^^) (2)

As the Lee-Yang-Parr (LYP) functional^^ already contains a local part and a gradient correction, one has to remove the local part to obtain a coherent implementation. This can be done in an approximate way by subtracting £^^^ to AE^^. The method has been normally used with the same three parameters derived originally for the B3PW91 functional. Other common hybrid schemes are those based on Becke's half-and-half (BHandH)^ linear interpolation of the adiabatic connection integral, which takes the values of a^ = 0.5 and a^ = a^ = Oin Eq. 1. If one has a high-quality electron density for a particular system, the DPT functionals, and in particular the three parameters of the B3LYP method, can be optimized so that one can obtain an electron density in better agreement with the reference density for the system considered. Ideally, the reference density should be taken from experimental results. In practice, however, it is very difficult to obtain a reliable electron density from a direct experimental measurement. In this case, an electron density obtained from a high-level ab initio method, such as the singles and doubles quadratic configuration interaction (QCISD),^^ can be used as the reference density. The aim of this work is to optimize the a^, a^, and a^ semiempirical parameters of the B3LYP method by minimizing the differences between the density furnished by the B3LYP method and the QCISD generalized density^^ which is taken as the reference density. This minimization is performed by maximizing the quantum

190

MIQUEL SOLA, MARTA FORES, and MIQUEL DURAN

molecular similarity measure (QMSM) between the B3LYP and the QCISD densities. The expression used to compute the QMSM between two first-order electron density functions {p^, Pj] is given by^^ ZjjiS) = J J p/r^) e(ri, r^) p/r^) dr, dv^

(3)

0(rp r^) being a positive definite operator depending on two-electron coordinates. Overlap-like QMSM are obtained when the ©(r^, r^) operator is chosen as the Dirac delta function 5(rj - r^. Use of the operator l/rj2 or l/r\2 gives rise to Coulomblike QMSM and gravitational-like QMSM, respectively.^^ In the particular case that P/ = Py» 01^^ g^ts Zji, which is the so-called self-QMSM.^^ In previous works,^^ it has been shown that comparison between densities through use of QMSM can provide detailed information on the similarities and differences between various methods. Such comparative studies should increase our understanding of the behavior of the different DFT schemes, and they should also provide hints toward improved treatments. In this work, we show how QMSM can be used to improve standard hybrid methodologies.

II. METHODOLOGY Let us define the function difference: ^(r) = pB3LYp(r) - pQCISD(r)

(4)

If Becke's a^, a^, and a^ parameters are allowed to change, the function difference P(r) will depend on these three parameters. Then, one can obtain the set of parameters that yield the B3LYP density closest to the QCISD density by minimizing the quadratic error integral:

D = lp\r)dr

(5)

which is obtained as a function of a^, a^, and a^. The Gaussian-94^^ program has been used to perform singles and doubles quadratic configuration interaction (QCISD)^^ and DF (B3PW91, B3LYP,^ BHandHLYP, and BHandrf) calculations. In all of these DF calculations, the nonlocal energy functional is incorporated into the optimization of both electronic and nuclear degrees of freedom self-consistently. B3LYP and B3PW91 calculations with (2Q, a^, and a^ coefficients different from the standard coefficients provided by Becke,^ have been performed with Gaussian-94 using internal options.^^ To minimize basis set effects, which may produce relevant QMSM differences,^^ the 6-311++G** basis set^^ has been used throughout. All calculations have been done within the restricted Hartree-Fock formalism except ionization potentials, which have been calculated with the unrestricted Hartree-Fock method.

Optimizing

Hybrid Density

191

Functionals

QMSM have been obtained from the Gaussian-94 electron densities using the Messem program developed in our group.^^ For QCISD calculations, generalized densities^^ have been used. Likewise, the DF electron densities have been calculated from self-consistently converged Kohn-Sham orbitals. All QMSM are Coulomblike, i.e., ©(FJ, r2) = l/rj2 in Eq. 3, except self-QMSM, which are overlap-like. Gradients of the D function in Eq. 5 with respect to the a^, a^, and a^ parameters have been computed numerically, and the optimization of this function has been performed using the quasi-Newton Davidon-Fletcher-Powell (DFP) algorithm.^^

INITIALIZE a„. a„ and ac

OBTAIN D(ao,a,.a.)

COMPUTE 5D/5ak ak = ak + Ei ak = ak - ei i = i +1

T

OBTAIN D(ao.a..ac)

_ilfl_

a. a., and a^ L

Y«

XConvergedV_Jifl

Scheme 1.

DFP OPTIMIZATION J NEW SET OF ao. a,, a.

192

MIQUEL SOLA, MARTA FORES, and MIQUEL DURAN

Scheme 1 depicts the program structure chart of the present algorithm. In this chart, shadow rectangles indicate calculations that use the Gaussian-94 program for geometry optimization and obtention of the electron density. Thus, starting from the initial set of a^, a^, and a^ parameters given by Becke, the QCISD and B3LYP electron densities are computed allowing complete electronic and nuclear relaxation of the system. With these two densities and using the Messem program one can obtain the value for the D function through Eq. 5. After that, the gradients of the D function with respect to the a^, a^, and a^ coefficients are computed numerically using the central difference approximation, i.e.,: dP

Z)(a^ + S ) - D ( a ^ - 5 )

daj^

25

(6)

with 5 = 10"^. Finally, and after checking the convergence of the process, a DFP optimization step is performed. We have considered that the optimization has been converged when the norm of the gradient vector was below 2 x 10"^ au. Throughout this contribution, the final converged a^, a^, and a^ parameters will be referred to as the QCISD-density-optimized parameters. Bader topological analyses^^ and maps of electron density differences were carried out with the Electra program developed in our group.^^

III. RESULTS AND DISCUSSION To test the methodology reported above, a study of the CO, N2, and LiF systems has been carried out. We have selected these three molecules because, from the point of view of Bader's atoms-in-molecules theory,^^ they exhibit different bonding nature: LiF is a typical case of closed-shell ionic interaction; N2 is an appropriate example of a molecule with a shared interaction; and CO is a well-known case of an intermediate interaction. Analyses of B3LYP electron densities in these molecules can put forward the behavior of the a^, a^, and a^ parameters in these different types of bonding. A. The CO System

Before starting the process of optimization, it is interesting to represent the quadratic error integral as a function of the a^, a^, and a^ parameters to obtain information about possible multiple minima on this surface. Figures 1-3 plot the quadratic error integral as a function of two variable Becke's parameters, while the remaining parameter is kept frozen. For all three surfaces we have computed 100 points changing each variable parameter from 0.1 to 1 by 0.1 unit each time. During the calculation of these surfaces, the geometry of the CO molecule has been kept frozen at the QCISD-optimized geometry.

Optimizing Hybrid Density Functionals

193

a,=0.81

0.014 <

--

0.012-1

3

0.010 i

3^ c 0.008 -1 o '^ o c 3

0.006 j 0.004'

H-

0.002'

o

0.000 1.2

0.0

0.0

Figure 1, Plot of the quadratic error integral (in au) as a function of 1 - ao and ax computed at the QCISD/6-311++G** optimized geometry. In this graph the ac has been kept frozen at 0.81.

Figure 1 depicts the quadratic error integral as a function of the a^ and a^ parameters. The a^ coefficient has been kept frozen at 0.81, as suggested by Becke in the B3PW91 method.^ As can be seen, the quadratic error integral is minimized along the ca. I -a^ = a^ line. This is not surprising, and indicates that the presence of the E^^^^ term reduces the need for the nonlocal correction AE^^ to approximately the same extent. For the same reason, Becke in his original work^ pointed out that a^ should be lower than 1. The value of a^ is especially interesting because it reflects how much exact exchange should be included in the description of a particular system. What is more striking is the fact that the value of the a^ coefficient does not have much influence as long as one takes a^=l -a^. In fact, along the \ - a^ = a^ line, the quadratic error integral is almost constant, reaching a minimum for ca. a^ = 0.5. Interestingly, a recent simplification sets a^=l -a^, and a^ = 1, with a^ = 0.l6oTa^ = 0.28,^^'^"^ and thus a unique parameter is needed to define this new hybrid DFT method. Further, in a recent paper, Perdew et al.^^ have provided a qualitative physical explanation to this relationship and have indicated that the a^ parameter must have a value of ca. 0.25. Our result reinforces the existence of this simple correlation between a^ and a^ parameters.

MIQUEL SOLA, MARTA FORES, and MIQUEL DURAN

194

a,=0,72

0.008]

6

0.0071

^ c o o c

0.0061 0.0051 0.0041

3

0.0031

o

0.0021 0,00V 0.000

1-a„ Figure 2. Plot of the quadratic error integral (in au) as a function of 1 - ao and BQ computed at the QCISD/6-311++G** optimized geometry. In this graph the ax has been kept frozen at 0.72.

Figure 2 illustrates the quadratic error integral as a function of the a^ and a^ coefficients; the a^ has been kept fixed at 0.72. Interestingly, this surface has a small dependence on the a^ coefficient. The minimum is found approximately along the line \-a^ = a^ (0.72) value. Finally, in Figure 3 the quadratic error integral is represented as a function of a^ and a^, for the a^ value of 0.2. Again, the surface has a small dependence on the a^ parameter, and presents a minimum near the a^=l-a^ (0.8) value. Analysis of the shape of these plots shows that in all cases there is only a minimum on the surface, and so multiple minima problems are not expected. However, the presence of flat regions on the surface around the minima may complicate the process of optimization. The QCISD-density-optimized Becke parameters obtained for the CO molecule are 0.099, 0.857, and 0.935 for the a^, a^, and a^ parameters, respectively. As expected, the a^ value is smaller than 1 and follows approximately the I -CIQ relationship. The results obtained for some experimental quantities and for the different hybrid schemes analyzed are presented in Table 1.

Optimizing Hybrid Density Functionals

195

a, = 0.20

0.008] ^ 0.0071

6

o.ooej

'^'^

0.0051

CO

c o 'ip o c 3 M— o

0.0041 0.0031 0.0021 0.0011 0.000

Figures, Plot ofthe quadratic error integral (in au) as a function of ax and ac computed at the QCISD/6-311++G** optimized geometry. In this graph the ao has been kept frozen at 0.20.

Comparison between results obtained from the different hybrid methodologies studied and those yielded by QCISD and experiment, reveals that the largest differences appear in the two functionals that make use of Becke's half-and-half exchange functional.^ This is especially true in the case of the CO dipole moment, which has the wrong direction when computed with the BHandH procedure. However, it is well known that most DFT functionals have success in computing the sign of the dipole moments with values close to zero, like CO and NQ,^'^^^'^^ and in fact the rest ofthe functionals analyzed give the correct sign for the CO dipole moment. The two B3LYP schemes analyzed here (the so-called B3LYPBE which uses the standard three parameters of Becke and the so-called B3LYPNP which employs the QCISD-density-optimized parameters) give values quite close to the QCISD and experimental results. The proportion of exact exchange in the QCISD-optimized parameters B3LYP method is lower (9.9%) than in the B3LYP standard method (20%). The B3LYP QCISD-density optimized parameters method reproduces the QCISD geometry. Also, there is a significant improvement from B3LYPBE to

MIQUEL SOLA, MARTA FORES, and MIQUEL DURAN

196

Table 1. CO Experimental and B3LYP,' BHandHLYP, BHandH, and QCISD 6-311++G** Optimized Bond Lengths,^ Harmonic Frequencies,^ Dipole Moments,^ Ionization Potentials,^ Overlap-like Self-QMSM,^ and Density^ and Laplacian of the Electron Density^ at the Bond Critical Point BHandH liC-O) v(C-O)

\^ IP Zii

P(rc) V2p(r,) Notes:

1.111 2354.2 -0.019 14.00 112.34402 0.5136 0.7889

BHandHLYP BSLYPBE^ B3LYPNP^ 1.114 2325.7 0.027 14.18 113.39620 0.5085 0.7755

1.127 2210.3 0.072 14.20 113.23945

1.133 2162.1 0.104 14.04 113.27957

0.4930 0.5499

0.4860 0.4659

QCISD

Exp.^

1.133 2177.2 0.090 13.71 113.41322

1.128 2170 0.112 14.05

0.4776 0.7854

— — —

^B3LYPBE refers to the standard B3LYP procedure, while B3LYPNP indicates the B3LYP method with the a^, a^, and a^ QCISD-density-optimized parameters. b|nA. ^All frequencies are harmonic and are reported in c m " \ ^In debyes. ^In eV. ^In au. sprom Ref. 3a.

B3LYPNP as far as harmonic frequency, dipole moment, and ionization potential are concerned. In particular, the improvement in the ionization potential was unexpected because Becke's original three parameters were optimized so as to reproduce experimental ionization potentials among other energy-related properties.^ As a whole one may conclude that the density afforded by the QCISD-optimized parameters is closer to the QCISD density than the B3LYPBE density. The good pattern of the B3LYPNP density is also demonstrated by the value of the density at the bond critical point,^^ which is the closest to the QCISD one. Indeed, Figure 4 reveals the small differences between the B3LYPNP and QCISD densities. The maximum difference is 5.781 au (2.18% of error) and is located at the oxygen nucleus. There is a slight reduction of density at nuclei when going from QCISD to B3LYPNP, which is transferred to the bonding region and around nuclei. As already reported,^^^'^^ the LSDA electron density is too diffuse in the region near the nuclei. Nonlocal corrections pull the density to the regions close to the nuclei, although, as can be seen in Figure 4, only partially correct the LSDA deficiency. This B3LYPNP density reduction at nuclei when compared with the QCISD density is also reflected by the value of the self-QMSM measure of Table 1. As pointed out earlier,^^ quantum molecular overlap self-similarity measures are good indexes to estimate concentration of electronic charge in molecules: the larger the self-QMSM, the more concentrated the charge density. Values of self-QMSM in Table 1 already indicate that the QCISD density is slightly more concentrated than the B3LYPNP

Optimizing

Hybrid Density

197

Functionals

- a O O - 8 . 8 0 - 5 . 6 0 - 4 . 4 0 - 3 . 2 0 - 2 . 0 0 - 0 . 8 0 0.40 1.60 2.80 4.00 5.20 piiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiH

6.40

7.60

7.60

I 7.60

6.40

6.40

5.20

5.20

4.00

4.00

2.80

2.80

1.60

1.60

0.40

0.40

-0.80

-0.80

-2.00

-2.00

-3.20

-3.20

-4.40

-4.40

-5.60

-5.60

-6.80

-6.80

-8.00 - a O O - 6 . 8 0 - 5 . 6 0 - 4 . 4 0 - 3 . 2 0 - 2 . 0 0 - 0 . 8 0 0.40

-8.00 1.60

2.80

4.00

5.20

6.40

7.60

Figure 4, Plot of the 6-311++G** electron density difference comparing the density obtained from the QCISD methodology with that computed at the B3LYPNP level, for the CO molecule at their optimized geometry. In this map the carbon nucleus is on top. The minimum contour is 1 x 10""* au and increases to 2, 4, 8, 20, 40, 80, . . . X 10~ au. Dashed lines correspond to negative values, that is, points where QCISD density is larger.

density. Further, the B3LYPNP Laplacian of the density at the bond critical point is the worst one when compared with the QCISD result. Remarkably, most DFT schemes provide strikingly too low values for the Laplacian of the electron density at the bond critical point of the CO molecule as compared with the QCISD values.^^^ However, because the CO electron density distribution is quite complex, one cannot infer from this result that B3LYP Laplacian values are defective. Finally, as Becke's three parameters were originally optimized together with the PW91 nonlocal correlation correction (B3PW91),^ we have investigated how the B3LYP QCISD-density-optimized parameters are modified when the nonlocal correlation correction is changed from A£^^^ - E^"^^ to A£^^^^ The minimization of the ^Q, a^, and a^ parameters with the B3PW91 functional has yielded the values of 0.708, 0.853, and 0.964, respectively. The differences with the parameters

Table 2, H^ Experimental and B3LYP/ BHandHLYP, BHandH, and QCISD 6-311++G** Optimized Bond Lengths,^ Harmonic Frequencies,*^ Ionization Potentials/ Overlap-like Self-QMSM,^ and Density^ and Laplacian of the Electron Density^ at the Bond Critical Point BHandH KN-N) y(N-N) IP Z// P(rc) V2p(r,) Notes:

BHandHLYP B3LYPBE^ BSLYPNP^

QCISD

1.080 1.082 1.095 1.104 1.104 2582.4 2441.4 2607.6 2353.7 2387.4 16.03 16.15 15.88 15.68 15.37 104.39067 105.41418 105.25520 104.96788 105.40628 0.7119 0.7090 0.6827 0.6657 0.6649 -2.8776 -2.8544 -2.5843 -2.4135 -2.4425

Exp! 1.098 2360 15.58

— — —

^B3LYPBE refers to the standard B3LYP procedure, while B3LYPNP indicates the B3LYP method with the a^, a^, and a^ QCISD-density-optimized parameters. b|nA. *^All frequencies are harmonic and are reported in cm"^ '^In eV. ^In au. ^From Ref. 3a.

- 8 L 0 0 - 6 . 8 0 - 5 . 6 0 - 4 . 4 0 - 3 . 2 0 - 2 . 0 0 - 0 . 8 0 0.40

1.60

2.80

4.00

5.20

6.40

7.80

7.60

6.40

6.40

5.20

5.20

4.00

4.00

2.80

2.80

1.80

1.60

0.40

0.40

-0.80

-0.80

-2.00

-2.00

-3JZ0

-3.20

-4.40

-4.40

-5.60

-5.60

-6.80

-6.80

II""'"" "'" iiMiiMriimimimimn mii iimiii iiinmiimminiininiiii -8.00 -8.00 - & 0 0 - 6 . 8 0 - 5 . 6 0 - 4 . 4 0 - 3 . 2 0 - 2 . 0 0 - 0 . 8 0 0.40 1.60 2.80 4.00 5.20 6.40 7.60

Figure 5. Plot of the 6-311++G** electron density difference comparing the density obtained from the QCISD methodology with that computed at the B3LYPNP level, for the N2 molecule at their optimized geometry. The minimum contour is 1 x 10'"* au and increases to 2,4,8,20,40,80,... x 10"^ au. Dashed lines correspond to negative values, that is, points where QCISD density Is larger. 198

Optimizing Hybrid Density Functionals

199

obtained using the B3LYP functional are quite small. As expected, the largest difference corresponds to the a^ parameter, which has changed by 0.029 unit. The small change in the parameters observed when going from B3LYP to B3PW91 justifies the use of Becke's B3PW91 optimized parameters in B3LYP calculations. B. The N2 System

For the N2 molecule we have obtained a^ = 0.021, a^ = 0.761, and a^ = 0.704. The results calculated for some experimental quantities and for the different hybrid schemes analyzed are shown in Table 2. Here too, the largest errors are provided by methods that use Becke's half-andhalf exchange. The B3LYP method with standard parameters yields excellent numbers. However, the B3LYP method with the QCISD-optimized parameters provides a significantly better harmonic frequency and ionization potential. Again, the values of the density and the Laplacian of the density in the bond critical point show that the B3LYPNP method yields a density closer to the QCISD density than does B3LYPBE. The good pattern of the B3LYPNP density can also be seen in Figure 5. The largest difference is now 0.337 au (0.20% of error) and is located at the nitrogen nuclei. All B3LYPNP computed values are closer than B3LYPBE values to QCISD results, except the self-QMSM, although in this case this is probably related to the shorter N-N bond length when computed with the

Table 3. LiF Experimental and B3LYP,' BHandHLYP, BHandH, and QCISD 6-311++G** Optimized Bond Lengths,^ Harmonic Frequencies,'^ Dipole Moments/ Ionization Potentials/ Overlap-like Self-QMSM/ and Density^ and Laplacian of the Electron Density^ at the Bond Critical Point BHandH KLi-F)

1.553

y(Li-F)

952.8 6.339

L |A

IP

10.68

Zu

BHandHLYP 1.568 925.4 6.431 11.74

B3LYPBE^

B3LYPNP^

1.582

1.595

903.9

880.6 6.317

6.355 12.23

1.595 886.8 6.544 11.55

122.01616

122.99495

Plz-c)

0.0765

0.0721

0.0697

0.0670

0.0663

V2(rc)

0.7654

0.7259

0.6785

0.6415

0.6550

Notes:

122.87479

12.48

QCISD

123.11443

123.02987

fxp.8 1.564

914 6.33

— — — —

^BSLYPBE refers to the standard B3LYP procedure, while B3LYPNP indicates the B3LYP method with the a^, a^, and a^ QCISD-density-optimized parameters. b|nA. '^All frequencies are harmonic and are reported in cm~\ ^\n debyes. ^In eV. ^In au. ^From Ref. 3a.

MIQUEL SOLA, MARTA FORES, and MIQUEL DURAN

200

- 8 . 0 0 - 6 . 8 0 - 5 . 8 0 - 4 . 4 0 - 3 . 2 0 - 2 . 0 0 - 0 . 8 0 0.40

1.60

2.80

4.00

5.20

6.40

7.60

7.60

6.40

I 6.40

5.20

5.20

4.00

4.00

2.80

2.80

1.60

1.60

0.40

0.40

-0.80

-0.80

-2.00

-2.00

-3.20 I

-3.20

-4.40 I

-4.40

-6.80 I"'" miiiimm iiiiiiimiiiniiiiii itiiiiiimiimiiiiiiiiiiiiiiiiiniiiiiiiiiimimiiiig _. illllllllllllllllllill - & 0 0 - 6 . 8 0 - 5 . 6 0 - 4 . 4 0 - 3 . 2 0 - 2 . 0 0 - 0 . 8 0 0.40 1.60 2.80 4.00 5JZ0 6.40 7.60

Figure 6. Plot of the 6-311++G** electron density difference comparing the density obtained from the QCISD methodology with that computed at the B3LYPNP level, for the LiF molecule at their optimized geometry. In this map the fluorine nucleus is on top. The minimum contour is 1 x 10""* au and increases to 2, 4, 8, 20, 40, 80, . . .

B3LYPBE methodology. A shorter N-N bond length produces an increase in the concentration of electron density. As for the CO case, the B3LYPNP ionization potential is closer to QCISD and experimental ionization potential than the B3LYPBE value. C. The LiF System

The QCISD-density-optimized Becke parameters obtained for the LiF system are a^ = 0.029, a^ = 1.003, and a^ = 0.730. The small value ofa^ allows a^ to take a value slightly larger than 1, but which also follows approximately the I - CL^ relationship. Table 3 presents the results obtained for the different hybrid schemes analyzed. Experimental results are also included. For this molecule, all methods perform similarly when compared with the available experimental results. The most interesting fact again is the important

Optimizing Hybrid Density Functional

201

similarity between the B3LYPNP and QCISD densities, which can also be appreciated in Figure 6. Both the electron density and the Laplacian of the density, as well as the self-QMSM, provided by the B3LYPNP method are the closest ones to the QCISD values.

IV. CONCLUSIONS In this work, we have analyzed a methodology that allows one to obtain a set of a^, a^, and a^ parameters that reproduces properly any density of reference. In particular, the QCISD density has been chosen as the reference density because it yields reliable geometries and energetic parameters, although in principle any density could be used. In the three systems studied, the proportion of exact exchange is found to be smaller than 20%, which is the percentage used in the standard B3PW91 method, although we expect that in more correlated systems or during bond breaking processes this percentage may be larger. For both CO and LiF, the relationship a^ = 1 - AQ is essentially preserved, while for N2 the a^ value is somewhat smaller than 1 - <3Q. The QCISD-optimized value of the a^ parameter ranges from 0.935 in CO to 0.704 in N2. Quadratic integral error surfaces for the CO molecule have provided evidence as to the fact that this parameter (a^) does not play a leading role in defining hybrid schemes. Despite the conclusions summarized above, the examples studied in this contribution do not provide consistent suggestions and clear conclusions about which are the best QCISD-density-optimized Becke parameters. Thus, to draw general conclusions that can be applied to most molecular systems, we will have to analyze a larger set of molecules. Furthermore, we think that the present algorithm would provide more interesting results when applied to the description of chemical species with strong correlation or which are not well described by the current functionals, like weak interacting complexes and excited states. In this case, the application of this methodology can offer the most convenient parameters. Research in this direction is currently under way in our laboratory.

ACKNOWLEDGMENT This work has been funded through Spanish DGICYT Project No. PB95-0762.

REFERENCES AND NOTES 1. (a) Parr, R. G.; Yang, W. Density-Functional Theory ofAtoms and Molecules; Oxford University Press: London, 1989. (b) Ziegler, T. Chem. Rev. 1991, 91, 651. (c) Ziegler, T. Can. J. Chem. 1995, 75, 743. 2. (a) Hohenberg, P; Kohn, W. Phys. Rev. B 1964,136, 864. (b) Kohn, W.; Sham, L. J. Phys. Rev. A 1965,140,1133.

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3. (a) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. (b) Murray, C. W.; Laming, G. J.; Handy, N. C ; Amos, R. D. Chem. Phys. Lett. 1992, 799, 551. 4. Andzlem, J.; Wimmer, E. Physica B 1991,172, 307. 5. (a) Stanton, R. V.; Merz, K. M., Jr., J. Chem. Phys. 1994, 100, 434. (b) van Leeuwen, R.; Baerends, E. J. Int. J. Quantum Chem. 1994,52,711. (c) Becke, A. D. J. Chem. Phys. 1986,84, ASIA, (d) Andzlem, J.; Wimmer, E. / Chem. Phys. 1992, 96, 1280. (e) Schmiedekamp, A. M.; Topol, I.A.; Burt, S. K.; Razafmjanahary, H.; Chermitte, H.; Pfaltzgraff, T.; Michejda, C. J. J. Comp. Chem. 1994,15, 875. 6. (a) Ruiz, E.; Salahub, D. R.; Vera, A. J. Am. Chem. Soc. 1995,117, 1141. (b) Kim, K.; Jordan, K. D. J. Chem. Phys. 1994, 98, 10089. (c) Del Bene, J. E.; Person, W. B.; Szczepaniak, K. J. Phys. Chem. 1995, 99, 10705. (d) Pudzianowski, A. /. Phys. Chem. 1996,100, 4781. 7. Proynov, E. I.; Vela, A.; Salahub, D. R. Chem. Phys. Lett. 1994, 230, 419. 8. Becke, A. D. J. Chem. Phys. 1993, 98,1372. 9. Becke, A. D. J. Chem. Phys. 1993, 98, 5648. 10. Becke, A. D. J. Chem. Phys. 1996,104, 1040. 11. Gorling, A.; Levy, M. J. Chem. Phys. 1997,106, 2675. 12. Gritsenko, O. V; van Leeuwen, R.; Baerends, E. J. Int. J. Quantum Chem. 1996, 30, 163. 13. (a) Harris, J.; Jones, R. O. J. Phys. F1974, 4, 1170. (b) Gunnarsson, O.; Lunqvist, B. I. Phys. Rev. B1976,46,6671. (c) Langreth, D. C ; Perdew, J. P Phys. Rev. B. 1977, 75,2884. (d) Harris, J. Phys. Rev. A 1984, 29, 1648. 14. Gunnarsson, O.; Jones, R. O. Phys. Rev. B 1985, 31, 7588. 15. (a) Bauschlicher, C. W., Jr.; Partridge, H. Chem. Phys. Lett. 1995, 240, 533. (b) Wang, J.; Eriksson, L. A.; Johnson, B. G.; Boyd, R. J. / Phys. Chem. 1996, 100, 5274. (c) Torrent, M.; Duran, M.; Sola, M. J. Mol. Struct. (Theochem) 1996,362,163. (d) Barone, V; Adamo, C. Chem. Phys. Lett. 1994, 224, A2>1. (e) Barone, V. Chem. Phys. Lett. 1994, 226, 392. (f) Bauschlicher, C. W, Jr., Chem. Phys. Lett. 1995, 246, 40. (g) Hack, M. D.; Maclagan, R. G. A. R.; Scuseria, G. E. J. Chem. Phys. 1996,104,6628. (h) Jursic, B. S. J. Chem. Soc. Perkin Trans. II1996,697. (i) Jursic, B. S. J. Chem. Soc. Perkin Trans. II1997, 637. (j) De Proft, P.; Geerlings, P J. Chem. Phys. 1997,106, 3270. 16. Becke, A. D. Phys. Rev. A 1988, 38, 3098. 17. Perdew,J. P;Chevary,J. A.;Vosko,S. H.;Jackson,K. A.;Pederson,M. R.;Singh,D. J.;Fiolhais, C. Phys. Rev. B 1992,46, 6671. 18. Adamo, C.; Lejl, F. J. Chem. Phys. 1995,103, 10605. 19. Frisch, M. J.; Trucks, G. W; Schlegel, H. B.; Gill, P M. W; Johnson, B. G.; Robb, M. A.; Cheesman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V; Foresman, J. B.; Peng, C. Y.; Ayala, P Y; Chen, W; Wong, M. W; Andres, J. L.; Replogle, E. S.; Gomerts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94 Revision A.l, Gaussian, Inc., Pittsburgh, PA, 1995. 20. (a) Lee, C ; Yang, W; Parr, R. G. Phys. Rev. B 1988,37, 785. (b) Miehlich, B.; Savin, A.; StoU, H.; Preuss, H. Chem. Phys. Lett. 1989, 757, 200. 21. Pople, J. A.; Head-Gordon, M.; Raghavachari, K. J. Chem. Phys. 1987, 87, 5968. 22. (a) Handy, N. C ; Schaefer, H. F, III. J. Chem. Phys. 1984, 81, 5031. (b) Wiberg, K. B.; Hadad, C. M.; LePage, T. J.; Breneman, C. M.; Frisch, M. J. J. Phys. Chem. 1992, 96, 671. 23. (a) Carbd, R.; Amau, M.; Leyda, L. Int. J. Quantum Chem. 1980, 17, 1185. (b) Carbo, R.; Calabuig, B. Int. J. Quantum Chem. 1992, 42, 1681. (c) Carbo, R.; Calabuig, B. Comp. Phys. Commun. 1989,55, 117. (d) Carbo, R.; Calabuig, B. J. Mol. Struct. (Theochem) 1992,254, 517. (e) Carb6, R.; Calabuig, B. Int. J. Quantum Chem. 1992,42, 1695. (f) Carbo, R.; Calabuig, B.; Vera, L.; Besalu, E. Adv. Quantum Chem. 1994, 25, 253. 24. Besalu, E.; Carbo, R.; Mestres, J.; Sola, M. Molecular Similarity. Topics in Current Chemistry Series, Springer-Verlag: BerUn, 1995, Vol. 173, pp. 31-62.

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25. Sola, M.; Mestres, J.; Oliva, J. M.; Duran, M.; Carb6, R. Int. J. Quantum Chem. 1996, 55, 361. 26. (a) Solk, M.; Mestres, J.; Carbo, R.; Duran, M. J. Chem. Phys. 1996,104, 606. (b) Torrent, M.; Duran, M.; Sola, M. Adv. Mol. Sim. 1996,1,167. 27. In this case, the route must contain the following internal options: #B3LYP IOP(5/45 = Pi P2) IOP(5/46 = P3 P4) IOP(5/47 = P5 P^). The values Pj = 1000, Pj = 0200, P3 = 0720, P4 = 0800, P5 = 1000, and Pg = 0810 reproduce the B3LYP results. The formula for the ^BSLYP energy is E^,^yp = P2E^^+ pMBk'''' + P3^'')+P6Er''+ Psi^h^^- E^^ where Pi - Pj\^M^, for / = 1 to 6. Thus, Becke's parameters AQ, A^* and a^ are p^oiX- pi -p^, pi -p^y and P5, respectively. The p^ parameter has been fixed at 1, while p^ has been 1 in all B3LYP calculations and zero in all other cases. For instance, BHandH results are obtained for/?! = 1, P2 = 0.5, P2 = 0, /?4 = 0.5, and p^=p^ = 0. 28. Carb6, R.; Domingo, L. Int. J. Quantum Chem. 1987, 32, 517. 29. (a) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72,650. (b) Clark, T; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P v. R. J. Comp. Chem. 1983, 4, 294. (c) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J. Chem. Phys. 1984, 80, 3265. 30. (a) Mestres, J.; Sola, M.; Besalu, E.; Duran, M.; Carbo, R. Messem, Girona, CAT, 1993. (b) Besalu, E.; Carbo, R.; Duran, M.; Mestres, J.; Sola, M. Modem Techniques in Computational Chemistry: METTEC-95; Clementi, E.; Corongiu, G., Eds., STEP, Cagliari, 1995, pp. 491-508. 31. (a) Davidon, W. Variable Metric Methodfor Minimization; Argonne National Laboratory Report ANL-5990, Argonne, IL, 1959. (b) Hetcher, R.; Powell, M. J. D. Comput. J. 1963, 6, 163. 32. (a) Bader, R. F W. Ace. Chem. Res. 1985, 18, 9. (b) Bader, R. F W. Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford, 1990. 33. Mestres, J. Electra, Girona, CAT, 1994. 34. Ernzerhof, M.; Perdew, J. P.; Burke, K. Density Functional Theory; Nalewajski, R., Ed.; Springer: Berlin, 1996. 35. Perdew, J. P; Ernzerhof, M.; Burke, K. J. Chem. Phys. 1996,105, 9982. 36. (a) Aarnst, M.; Stufkens, D. J.; Sola, M.; Baerends, E. J. Organometallics 1997,16, 13. 37. (a) Wang, J.; Shi, Z.; Boyd, R. J.; Gonzalez, C. A. J. Phys. Chem. 1994, 98, 6988. (b) Wang, J.; Eriksson, L. A.; Johnson, B. G.; Boyd, R.J. J. Phys. Chem. 1996,100, 5274.

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ATOMIC SIMILARITY THROUGH A NEURAL NETWORK: SELF-ASSOCIATIVE PERIODIC TABLE OF ELEMENTS

Jose Fayos

I. 11. III. IV. V.

Abstract 205 Introduction 206 Architecture and Function of a Neural Network for the Periodic Table . . . . 207 Self-Association of Elements and Properties 209 Prediction of Properties for Elements 211 Conclusions 212 Acknowledgment 213 References 213

ABSTRACT Atomic similarity analysis by a neural network produces self-association of chemical elements into three main groups preserving the conventional periodic table structure. Seventy-two elements and seven solid-state properties of each are taken as neurons to build an lAC (ftferative activation and competition between neurons)-type neural

Advances in Molecular Similarity, Volume 2, pages 205-213. Copyright © 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 0-7623-0258-5 205

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JOSE FAYOS

network. To create associations, the range of values for each property is divided into about seven intervals or categories, and the net is genetically determined by 446 excitatory or activating contacts between the elements and their known property categories. The neural competition consists of inhibitory contacts within elements and within categories of the same property. Besides the simpler database retrieval functions, like that of finding elements sharing one property, this artificial net reproduces more intelligent (Mendeleyev-like) properties of human memory. Thus, starting from an all-inactive neuron pattern, when a neuron is activated by an external input, multiple parallel activation and inhibition processes expand through the net, converging to a pattern of activated neurons, which are revealed to be self-associated. If such self-associations are produced by activating successively, one by one, the above elements and categories, the following results ensue: (1) the 72 elements are approximately grouped into only three main famiUes, each sharing close properties; (2) although no assumption about the atomic structure is made, these families pack byfillingthree different areas in a conventional periodic table, showing that the electronic configuration of the atoms is implicit in the solid-state properties; (3) one of the atomic families contains halogen and rare-gas atoms together with alkaline and alkaline-earth atoms, which gives rise to a closed cylindrical periodic table; and (4) the net is able to predict unknown properties for elements, with an average error less than one category.

I. INTRODUCTION The parallel processes between neurons of an artificial neural network allow neuron self-associations, without including any explicit rule. We use this advantage to discover hidden associations between chemical elements when they share close properties. The IAC net proposed by McClelland and Rumelhart^ is applied to create associations between 72 elements (those of thefirstsix periods of the periodic table excluding lanthanides) and seven solid-state properties; namely, density, crystal structure, atomic diameter, cohesion energy, ionization energy, melting point, and compressibility. The observed values of these properties were taken from tables in Chapters 1 and 3 of Kittel's Introduction to Solid State Physics? To generate such associations, there must be elements sharing some properties, thus the range of the observed values for each property was divided into 7 or 8 intervals or categories, trying to homogenize the intervals by their length and by the number of values included. Table 1 shows the average of the values taken in each interval, which are included in the name of the resulting 49 categories. Exception is made for crystal structures (STR) as they are already categorical, where only the 5 more common are considered as categories for the net. Besides, some other ambiguous or uncommon properties were also excluded. Thus, of the 12x1 = 504 possible element-category connections, we considered 446.

Atomic Similarity througti a Neural Network

207

Table 1. The Property Layers: Density (RHO), Crystal Structure (STR), Atomic Diameter (DIA), Cohesion Energy (COH), Ionization Energy (ION), Melting Point (MEL), and Compressibility (COM)^ RHO: R1.1. R2.7, R4.6. R6.1. R7.4, R9.3, R12.4, R20.4 (0.1 -22.6 g/cm') 12 10 7 7 5 9 8 7 STR: HCP. BCC. FCC, DIAMOND. RHOMBOHEDRAL 17 13 16 4 5 DIA: D1.5. D2.2. D2.6,D2.9. D3.2. D3.7. D4.6 (1.4-5.2 A) 2 4 18 14 9 9 6 COH: C0.1, CI .0, CI .8. C3.0, C4.2. C5.7, C7.6 (0.1 - 8.9 eV/at) 5 9 12 14 11 8 10 ION: 14.1.15.7,17.5. 19.6.111.7.113.7.116.6.123.1 (3.9-24.6 eV/at) 3 10 30 16 3 5 2 2 MEL: M90. M390. M740. M1170. M1720, M2260. M3160 (20-3700 K) 8 15 8 9 9 10 7 COM: CM0.4. CM1.2. CM2.4. CM3.8, CM6.1. CM10.5, CM42.2 (0.2-56x10" m^/N) 21 11 11 5 2 4 4 COL: lAIIA, IIIBIVB. VBVIBVIIB. VIII, IBIIBIIIA, IVAVA, VIAVIIA, VIIIA 11 6 9 9 11 10 10 6 Note: ^For each property, the names of the categories (including the average of the numerical values) follow and, below, the number of elements associated. In parentheses are shown the total range of values for each property. The last row corresponds to categories of the chemical group (COL) included to predict properties of elements.

II. ARCHITECTURE AND FUNCTION OF A NEURAL NETWORK FOR THE PERIODIC TABLE Figure 1 represents a fragment of our neural network. A total of 193 neurons are distributed among nine layers: one central hidden layer containing 72 hidden elements (X*) and eight external layers. The hidden layer is included to increase competitiveness between elements; of the external layers, one is for the 72 elements and the other seven are property layers, the latter including the 49 categories. Both elements and hidden elements and categories are interconnected neurons which process the incoming signals in the same way. The neural contacts or synapses work in two senses. Inside each layer there are inhibitory contacts (not represented in Figure 1) between every two neurons, a total of 10,524, allowing full neural competition intralayer. Between layers the contacts are excitatory, and each hidden element connects with its element and with its known categories, giving 72 + 446 = 518 excitatory contacts, which constitute the genetic base of knowledge of the net.

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JOSE FAYOS

R1.1 R2.7 ... R20.4

HCP BCC ... RHOM

CM0.4 CM1.2 ... CM42.2

H* He* Li* Be* B* (3*( D* F* Ne*

Cs* Ba* .... .... Po* At* Rn*

H He

Cs Ba

Li

Be

B C O F

Ne

.... Po At

Rn

Figure 1, Schematic architecture of the lAC neural network for the elements of the periodic table and their properties. Only some layers, some elements or categories, and some connections have been represented for clarity.

The external neurons can also be activated by a foreign input. Each neuron processes the input stimuli arriving either from foreign or from connected neurons, changes its own activation a by A<3, and send its new activation (only if it is positive) back to the connected neurons. Thus, in the lAC neural process,^ a target neuron can be excited by the outputs of connected neurons by ^ = oLexc-outputs or it can be inhibited by neurons of the same layer by / = iLinh-outputs, where a and y are overall filter parameters. The net neuron input is net = e-i + strength xforinput, where the last term is the strength of the foreign input if it exists. Finally, Afl = {a^^^ - a)net -{a- rest)decay if net > 0, or A(3 = (a - a^Jnet (a - rest)decay if net < 0, which allows that a^^