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Optimization of Carbo Molecular´Similarity Index Using Gradient Methods

ALAN J. MCMAHON and PAUL M. KING*Department of Chemistry, Birkbeck College, University of London, Gordon House, 29 Gordon Square,London WC1H 0PP, United Kingdom

Received 30 November 1995; accepted 26 April 1996

ABSTRACT

A steepest descent method for optimizing the Carbo molecular similarity index´was implemented and evaluated. Comparisons were made between thisprocedure and the extensively used simplex method. Several data sets wereconsidered, and in each case the gradient method showed a substantialimprovement in the time taken for the optimization to converge whilecomparable similarity values were obtained. In some cases, performanceenhancements of up to an order of magnitude were observed. Q 1997 by JohnWiley & Sons, Inc.

Introduction

olecular similarity methods provide a sim-M ple quantitative measure of how similartwo molecules are. These methods have foundwidespread use within the general area of rationaldrug design.1 In the modeling of unknown recep-tor sites molecular similarity is used to superim-pose molecules known to bind to a particular siteand to elicit specific responses, so that the shapeand charge distribution of the site can be deducedfrom its complimentarity to the aligned molecules.2

The superimposition of molecules based upon theirmolecular similarity also finds application in

* Author to whom correspondence should be addressed.E-mail: p.king@chem.bbk.ac.uk

Quantitative Structure Activity RelationshipŽ .QSAR studies, where attempts are made to corre-late observed chemical or biological response withcertain molecular properties and characteristics ofthe molecules.3 ] 6 The design and molecular mod-eling of bioisosteres and transition-state mimicsalso relies upon the quantitative nature of molecu-lar similarity to validate the results of such stud-ies. Molecular similarity is also used as a scoringfunction in the screening of data bases for poten-tial drug molecules having related behavior to aknown ‘‘lead’’ compound. The related concept ofmolecular ‘‘dissimilarity’’ is used to quantify howdifferent the enantiomeric forms of a chiral com-pound are. This can produce a chirality coefficientthat is often found to correlate well with the eudis-mic ratio of enantiomeric pairs. There are severalrecent reviews of molecular similarity to which theinterested reader is referred.7] 9

( )Journal of Computational Chemistry, Vol. 18, No. 1, 151]158 1997Q 1997 by John Wiley & Sons CCC 0192-8651 / 97 / 020151-08

MCMAHON AND KING

All of the applications cited above depend cru-cially on the superimposition of the two moleculesupon which a similarity calculation is being per-formed. Molecular similarity indices are often seento be very sensitive to the alignment of themolecules concerned. The simplest means of su-perimposition involves aligning the molecules, us-ing, for example, centers of mass, centers of charge,moments of inertia, electrostatic multipole mo-ments, least-squares fitting of selected atomic posi-tions, etc. However, there is no guarantee thatsuperimposition based on these relatively simplemeasures of geometric shape or charge distribu-tion will generate alignment of molecules in orderto produce optimal, or even realistic, similarityvalues. Given one molecule, A, and another, B,with position coordinates R and R , respectively,A Bthe similarity is a function of both the molecularconformation of each molecule and their relativeposition and orientation.

Similarity of A and B

Ž . Ž .s S s S R , R , R , Q , 1A B A B COM

where R is the displacement vector of theCOMcenters of mass and Q is the vector of three Eulerangles defining the relative orientation of themolecules. The function S will thus map out aA Bmultidimensional surface containing many station-ary points and, ideally, a single maximum thatrepresents the maximum similarity value corre-sponding, in a similarity sense, to optimal align-ment of the two molecules. The best way, there-fore, to align molecules such that they produce anoptimal similarity value is to search the similarityspace spanned by the conformations and relativeconfigurations of the two molecules. The simplestsearch procedures assume that the two moleculesare rigid, perhaps in independently energy mini-mized structures, so that the search only entailsexploring the 6-dimensional space of relative dis-placement and orientation. Earliest applications ofmolecular similarity adopted this approach.10, 11

Ideally, however, the ability of molecules to changeconformation should be included in the optimiza-tion process, and flexible-fitting methods were em-ployed.12 These methods essentially add the inter-nal degrees of freedom of each molecule to thesearch variables, and include a molecular mechan-ics penalty function to validate optimizationmoves.

To date, optimization of molecular similarityhas relied upon algorithms that use solely functionevaluations, primarily within the simplex ap-

proach.13 Methods aimed at improving the speedof similarity calculations, as outlined below, pro-vided the opportunity of using searching proce-dures that utilize gradient information. In this arti-cle we present the first application of searchingusing gradient-based methods.

Background

A quantitative measure of the molecular simi-larity between two molecules, A and B, was firstdefined by Carbo et al.14 as follows:´

Ž . Ž .r r r r drH A BŽ .R s , 2A B 1r2 1r2

2 2Ž . Ž .r r dr r r drH HA Bž / ž /Ž . Ž .where r r and r r are the electron densities ofA B

molecules A and B, respectively. However, this isnot a unique functional form for the index andneither is the choice of electron density the only, orindeed optimal, choice for the molecular propertyupon which to base the similarity measure. Analternative and commonly used index was intro-duced by Hodgkin and Richards10 that is moresensitive to the magnitudes of the electron densi-ties than the Carbo index.´

Most practical calculations of molecular similar-ity on large molecules and molecules of potentialpharmacological importance have not used theelectron density as the molecular property of inter-est, but rather the electrostatic potential or themolecular shape. This is because many applica-tions on large data bases of molecules androrlarge molecules require a classical rather thanquantum mechanical description of the electrostat-ics of the molecules in question. The classical elec-trostatic potential of a point charge distributionprovides a description of how the molecule will berecognized by others at medium- to long-rangedistances. In the discussion that follows, wedemonstrate the differentiation of the Carbo elec-´trostatic potential similarity index, and the use ofthe derivatives obtained to optimize the similarity.Our software also allows the differentiation of theHodgkin index and the use of shape, rather than

Želectrostatic potential for either of these Carbo or´.Hodgkin indices.

In the present work we consider the calculationof the Carbo similarity index using the electrostatic´potential of the molecules instead of the electron

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Ž . Ž .densities. This involves replacing r r of eq. 2 byŽ .V r . The electrostatic potential has the advantage

that it is straightforward to calculate classicallyusing atom-centered point charges:

N qiŽ . Ž .V r s , 3Ý < <r y R iis1

where q is the charge on atom i centered atiposition R and N is the number of atoms in theimolecule. Evaluation of the integrals required forthe evaluation of molecular similarity was origi-nally performed numerically on a grid. However,the use of a 2- or 3-Gaussian expansion for the 1rr

Ž .terms of eq. 3 allows the grid-based determina-tion of the electrostatic potential to be replaced byanalytic evaluations that make the calculation 2orders of magnitude faster. Using a Gaussian ex-pansion the electrostatic potential becomes15

NN Gauss2ya ŽryR .j iŽ . Ž .V r s q g e 4Ý Ýi j

is1 js1

where N is the number of Gaussians used inGaussthe expansion. All the integrals required for theevaluation of the molecular similarity can now bedetermined analytically using standard results.16

The optimization of the molecular similarity,i.e., obtaining the optimal superimposition of twomolecules, involves the use of an algorithm tosearch the space formed by the relative orientationof both molecules for the best similarity value.When numerical integration of the relevant inte-grals is utilized only optimization proceduresbased upon function evaluations or that involvethe numerical calculation of derivatives can beused. Traditionally the simplex algorithm hasfound widespread use. However, if use is made ofthe Gaussian expansion in the electrostatic poten-

w Ž .xtial eq. 4 , not only can the similarity index becalculated analytically but so too can the deriva-tives of the similarity index with respect to therelative orientation of the two molecules. It wasthis realization upon which the work described inthis article is primarily based. The availability ofrapidly calculated analytic derivatives enables awhole range of gradient-based search proceduresto be utilized.17 In this study we used the simplestof the gradient-based methods, that of steepestdescents.

If molecule B is considered to be stationarywhile molecule A is moving, the expression forthe derivative of the Carbo index can be written as´

follows:

­ R R ­ IA B A B A B Ž .s , 5A Až /ž / ž /I­ R ­ RA Bi i

where

N N NA B tA B 2A B Ž t <R yR < .k i j Ž .I s q q s e , 6Ý Ý ÝA B i j kž /

is1 js1 ks1

and

N NB t­ I A B 2A B A B A B Ž t <R yR < .k i js q q 2 s t R y R e .Ž .Ý Ýi j k k i jAž /­ R i js1 ks1

Ž .7

Here N s 3 for a 2-Gaussian expansion and N s 6t tfor a 3-Gaussian expansion. The various constants

Ž .in the expansion of eq. 6 are given in Table I. It isstraightforward to transform the atom-basedderivative of the similarity index expressed in eq.Ž .5 to forces acting upon the center of mass andtorques about a molecule-centered axis frame.These derivatives can then be used in a searchingalgorithm to optimize the superimposition of thetwo molecules. As outlined earlier, we can also usethis method for optimization of the Hodgkin indexand for the optimization of shape similarity.18

Computer Implementation

All molecules were built using the PIMMS19

molecular modeling package. The structures weresubsequently optimized by semiempirical quan-tum mechanical calculations using the MNDO20

Hamiltonian, while atom-centered partial chargeswere obtained by fitting to the electrostatic poten-tial calculated at this level of theory according tothe method of Besler et al.21 These calculationswere performed using the MOPAC22 program.

The program for molecular similarity calcula-tions was written in ANSI standard C on a SiliconGraphics Indigo-2 workstation. Prior to optimiza-tion of the similarity index the centers of mass ofthe lead and the analogue under considerationwere superimposed. The optimization of the simi-larity index proceeds by minimizing the negativeof the similarity and hence maximizing the simi-larity itself. This was performed using a basicsteepest descent algorithm. This is clearly not asophisticated algorithm but serves as an initial

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MCMAHON AND KING

TABLE I.Terms in Gaussian Expansion Evaluation of Molecular Similarity.

2-Gaussian Expansion 3-Gaussian Expansion

k s t s tk k k k

3/2 3/2p a p a1 12 21 g y g y1 1ž / ž /ž / ž /2a 2 2a 21 1

3/2 3/2p a a p a a1 2 1 2

2 2g g y 2g g y1 2 1 2ž / ž / ž / ž /a q a a q a a q a a q a1 2 1 2 1 2 1 23/2 3/2

p a p a a2 1 323 g y 2g g y2 1 3ž /ž / ž / ž /2a 2 a q a a q a2 1 3 1 33/2

p a224 g y2 ž /ž /2a 223/2

p a a2 35 2g g y2 3 ž / ž /a q a a q a2 3 2 3

3/2p a326 g y3 ž /ž /2a 23

benchmark for a gradient-based method withwhich to compare previously used simplex meth-ods, and also subsequent more powerful gradient-based searching methods. Optimizations were per-formed in the 6-dimensional space spanned by thedisplacement of the centers of mass and the threeangles specifying relative orientations. Bothmolecules were considered to be rigid throughoutthe optimization process. The steepest descentmethod is implemented by updating the positionof the moving molecule according to the formula

Ž .R s R q ag, 8new old

where g is the normalized rigid-body force actingon the moving molecule and a is a variable thatforms the step size in the search procedure. Theoptimization ends when either the similarity valueis greater than 0.99, all partial derivatives are

Ž y4 .within some value conv tol, typically 10 ofzero, or the maximum number of iterations areexceeded, i.e.,

FOR n s 1 TO MAX ITERATIONSBEGIN

calculate similaritycalculate gradients

Ž .IF similarity has improved

a s a)1.2

ELSE

a s ar2.0

Ž .IF similarity ) 0.99 OR function converged

BREAK

w Ž .xalter position of moving molecule eq. 8END

The performance of the gradient-based opti-mizations and the similarity values obtained wascompared against the frequently used simplexmethod. This was performed on two sets of datapreviously described in similarity works,6, 15 andone new set.23 The molecules used in this studyare shown in Figures 1]3.

Results

Table II gives the calculated similarity valuesfor the three series of molecules considered in thisstudy. The simplex average column represents av-eraged results from 100 simplex calculationsstarted with different random seeds. The simplexbest column lists the highest single similarity valueobtained from the 100 different runs. The steepestdescent column gives the results from a singleoptimization using the methods described here.

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FIGURE 1. Structures of hypoglycemic active lead fragment and ring analogues.

The results demonstrate that the use of the steep-est decent procedure for the optimization ofmolecular similarity does produce, in most cases,the same optimal alignment of the analogues asthe simplex method. This is clear from the correla-tion between the similarity measures obtained fromthe simplex best calculations and that from thesteepest descent procedure. The overall correlationcoefficient between these two measures is 0.93

Ž .with a root mean square rms deviation of 0.062.Only five of the molecules appeared to have lo-cated significantly different similarity maxima: hy-poglycemic fragment ring analogues 4 and 5, dihy-drofolate analogue 16, and serotonin analogues 22and 23. Removing these five molecules from the

analysis gives a correlation coefficient of 0.999 andan rms deviation of 0.009. The fact that differentlocal maxima were located in these five cases isseen from trajectories of nonoptimal simplex runsthat appeared to get stuck at the same position inmany cases. The location of local stationary pointsis a well known failing of gradient-based optimiza-tion methods and of the relatively primitive steep-est descent method in particular. Other gradi-ent-based methods, such as conjugate gradientapproaches, would suffer similar drawbacks, al-though performing a few runs with different ran-domized starting configurations might easily over-come this problem. However, what Table II clearlyillustrates is that for the vast majority of cases

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MCMAHON AND KING

FIGURE 2. Structures of dihydrofolate analogues.

virtually identical similarity maxima are obtainedfrom the best simplex result and that produced bya single steepest descent run.

Table III compares the time taken to optimizethe similarity using different methods. The sim-plex average column gives the average time takento converge for 100 different runs. The simplexbest column gives the time taken for the simplexrun that produced the best similarity value. Bothtimes are measured relative to the time takenusing the gradient-based approach. Typical runtimes for the gradient-based calculations were lessthan a second.

The results clearly show the improvement inspeed obtained using the steepest descent proce-dure. Compared to the simplex average results thesteepest descent method is approximately 11 timesfaster, while it is approximately 8 times faster than

the best simplex result. Thus we can conclude thatthe steepest descent method is approximately anorder of magnitude faster than the simplex ap-proach. There is only a single exception to this, thedihydrofolate analogue 21. This is due to the gra-dient method not converging, a failure of thesteepest descent algorithm that is likely to be im-proved using more sophisticated gradient-basedapproaches.

The time-consuming part of the similarity calcu-lation is the evaluation of the exponential terms of

Ž .eq. 4 . However, the additional calculation of gra-dients does not involve any further evaluations ofexponential functions, and hence there are virtu-ally no extra computational overheads. This, plusthe fewer number of similarity evaluations re-quired in any single run, leads to the enhancedperformance of the steepest descent optimization.

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´CARBO MOLECULAR SIMILARITY INDEX OPTIMIZATION

FIGURE 3. Structures of serotonin analogues.

Conclusion

A method for optimizing the Carbo and´Hodgkin molecular similarity indices using asteepest descent algorithm was described. Its per-formance for the Carbo index was compared with´the simplex optimization procedure. In the vastmajority of cases, the new method is considerablyfaster. Frequently both methods locate the samemaximum value.

The more rapid calculation of optimized molec-ular similarity will enable faster data base search-ing. As previously discussed,24 the initial superpo-sition of structures poses a problem for data basesearching software. Previous methods requirestructural features in the data base query to bealigned, resulting in an extra data base search. Ifour method of optimization was to be incorpo-rated in the search software, then we feel thisproblem could become redundant. Alternativelythe saving in time could be utilized by performingmore in-depth calculations on smaller sets ofmolecules, such as comparing more than onemolecular property or similarity index.

TABLE II.Electrostatic Potential Similarity Values ofCompounds Obtained Using DifferentOptimization Methods.

Simplex Simplex SteepestAverage Best Descent

Hypoglycemic ringfragment analogues

1 0.338 0.379 0.3752 0.473 0.493 0.4933 0.405 0.479 0.4394 0.515 0.635 0.4435 0.541 0.617 0.4876 0.636 0.636 0.6437 0.644 0.659 0.6588 0.827 0.827 0.8279 0.858 0.858 0.858

10 0.885 0.884 0.88411 0.896 0.896 0.896

Dihydrofolate analogues12 0.730 0.730 0.73013 0.648 0.744 0.74414 0.715 0.715 0.71415 0.653 0.706 0.70616 0.555 0.555 0.66417 0.727 0.731 0.73018 0.768 0.771 0.77119 0.485 0.493 0.49220 0.776 0.780 0.78021 0.905 0.905 0.905

Serotonin analogues22 0.493 0.540 0.36923 0.586 0.614 0.53724 0.567 0.597 0.59625 0.570 0.574 0.56326 0.544 0.544 0.54427 0.602 0.602 0.602

Current opinion suggests that shape may be auseful descriptor for the repulsive force betweenreceptor and ligand, with electrostatic potentialbeing the main contributor to the attractive forces.Comparison of both molecular properties may lead

Žto more accurate in terms of correlation with.experimentally determined data similarity calcu-

lations. Our gradient-based optimization methodswere implemented into shape similarity calcula-tions using Gaussian functions, and significantperformance enhancements were also observed inthis area.

Our program that performs this gradient-basedoptimization is currently command driven, al-though we are developing a user-friendly X-windows interface. In addition to the implemen-

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MCMAHON AND KING

TABLE III.Relative Times Taken for Optimization.

Simplex Simplex SteepestAverage Best Descent

Hypoglycemic ringfragment analogues

1 8 4 12 7 4 13 10 5 14 9 3 15 7 4 16 4 3 17 8 3 18 6 2 19 6 3 1

10 4 3 111 9 3 1

Dihydrofolate analogues12 14 13 113 28 12 114 9 6 115 34 18 116 4 2 117 14 12 118 6 5 119 15 10 120 3 3 121 1 0.5 1

Serotonin analogues22 4 4 123 23 25 124 10 23 125 21 18 126 22 16 127 13 7 1

tation of gradient-based optimization of shapesimilarity, it is our intention to incorporate othermeasures of similarity, as well as improved con-formational searching algorithms.

Acknowledgments

A. J. M. is supported by an EPSRC Quota stu-dentship. The work was also funded by a CollegeResearch Grant from Birkbeck College.

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