A Randomized Kinematics-BasedApproach to Pharmacophore-ConstrainedConformational Search andDatabase Screening

STEVEN M. LAVALLE,1 PAUL W. FINN,2,∗ LYDIA E. KAVRAKI,3

JEAN-CLAUDE LATOMBE4

1Department of Computer Science, 204 Atanasoff Hall, Iowa State University, Ames, Iowa 500112Pfizer Central Research, Sandwich, Kent, United Kingdom3Department of Computer Science, Rice University, Houston, Texas4Computer Science Department, Stanford University, Stanford, California

Received 14 June 1999; accepted 28 January 2000

ABSTRACT: Computational tools have greatly expedited the pharmaceuticaldrug design process in recent years. One common task in this process is thesearch of a large library for small molecules that can achieve both a low-energyconformation and a prescribed pharmacophore. The pharmacophore expressesconstraints on the 3D structure of the molecule by specifying relative atompositions that should be maintained to increase the likelihood that the moleculewill bind with the receptor site. This article presents a pharmacophore-baseddatabase screening system that has been designed, implemented, and tested on amolecular database. The primary focus of this article is on a simple, randomizedconformational search technique that attempts to simultaneously reduce energyand maintain pharmacophore constraints. This enables the identification ofmolecules in a database that are likely to dock with a given protein, which canserve as a powerful aid in the search for better drug candidates. c© 2000 JohnWiley & Sons, Inc. J Comput Chem 21: 731–747, 2000

Correspondence to: Steven M. LaValle; e-mail: lavalle@cs.iastate.edu∗Current address: Prolifix Ltd., 91 Milton Park, Abingdon,

Oxfordshire, UKContract/grant sponsor: NSF CAREER Award (to S.L.); con-

tract/grant number: IRI-9875304Contract/grant sponsor: NSF CAREER Award (to L.K.); con-

tract/grant number: IRI-970228, SA1728-21122NM, and ATP003604-0120-1999

Contract/grant sponsor: ARO MURI (to J.-C.L.); contract/grant number: DAAH04-96-1-007

Journal of Computational Chemistry, Vol. 21, No. 9, 731–747 (2000)c© 2000 John Wiley & Sons, Inc.

LAVALLE ET AL.

Keywords: drug design; conformational search; pharmacophore; protein–liganddocking; database screening

Introduction

T he development of a pharmaceutical drug is along, incremental process, typically requiring

years of research and experimentation. The goal isto find a relatively small molecule (ligand), typicallycomprising a few dozen atoms, that docks with areceptor cavity in a specific protein. Protein–liganddocking can stimulate or inhibit some biologicalactivity, ultimately leading to the desired pharma-cological effect. The problem of finding suitableligands is complicated due to both energy consider-ations and the flexibility of the ligand. In addition tosatisfying structural considerations, factors such assynthetic accessibility, drug pharmacology, and tox-icology greatly complicate and lengthen the searchfor the most effective drug molecules.

In the search for a new drug, a pharmacophoreoften serves as a template for the desired ligand.The pharmacophore is expressed as a set of fea-tures that an effective ligand should possess and aset of spatial constraints among the features. Thefeatures can be specific atoms, centers of benzenerings, positive or negative charges, hydrophobic orhydrophilic centers, hydrogen bond donors or ac-ceptors, and others. The spatial arrangement of thefeatures represents the relative 3D placements ofthese features in the docked conformation of the lig-and. The pharmacophore encapsulates a prevailingassumption in drug design that ligand binding isdue primarily to the interaction of some features ofthe ligand to “complementary” features of the re-ceptor. The interacting features are included in thepharmacophore, and are key for searching for newdrugs. The rest of the ligand atoms merely provide ascaffold for holding the pharmacophore features intheir spatial positions.

THE PROBLEM

This article deals with the following problem.Given a pharmacophore and a database of flexibleligands, identify those ligands that can achieve alow-energy spatial conformation that matches someof their features to the features of the pharma-cophore. We will say that the selected ligands can“satisfy” the pharmacophore.

We view the problem primarily as that of obtain-ing a constrained conformation of a known kine-

matic structure (the ligand). The 3D positions of cer-tain parts of the structure are predetermined by thepharmacophore model. We compute the “folding”of the rest of the structure in a way that it preservesall the pharmacophore matches while respecting allstructural constraints (i.e., bond lengths), all kine-matic constraints (i.e., torsional degrees of freedomand their allowed values), and all energy constraints(i.e., the energy of the ligand should be below athreshold).

We model the kinematics of the ligand using tech-niques common in robotics.1, 2 Although powerfulanalytical techniques exist for searching the solutionspaces of similar structures in robotics,3 such tech-niques are impractical for handling high degreesof freedom. Our ligands have many torsional de-grees of freedom; thus, we focus on randomizedsolutions to the conformational search problem. Theneed for efficiency has also motivated randomizedsearch techniques in robotics.4, 5 We use our experi-ence with these methods to develop a randomizedconformational search technique, which simultane-ously reduces energy and maintains the pharma-cophore constraints.

In pharmaceutical chemistry, a variety of phar-macophore models and representations have beenconsidered, and most of the current research ad-dresses the development, evaluation, and selectionof these models and representations. For example,as opposed to the 3D feature-matching model de-scribed so far, other possibilities include matchingand agglomerating molecular fragments, and alsodefining enrichment factors that assess the similar-ity of entire known inhibitors. Instead of dealingwith these issues, our work focuses primarily onthe development of computational techniques thatimprove our understanding of geometric issues in-volved in drug design. For this reason, we chose todevelop and demonstrate our methods for a partic-ular pharmacophore model and representation. Webelieve that the techniques from this article can beadapted to other kinds of pharmacophores; how-ever, such issues are beyond the scope of this articlebecause they generally depend on the particularpharmaceutical application.

SIGNIFICANCE OF OUR WORK

Our work is expected to improve the ability toscreen pharmaceutical databases for ligands that

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can satisfy the pharmacophore model by deliver-ing computational techniques that reason about thecomplicated geometric and kinematic constraintsthat arise from the matching problem. The tech-niques can be applied (in combination with otherfilters) to 3D structural databases, such as the Cam-bridge Structural Database, the Brookhaven ProteinDatabank, or proprietary pharmaceutical databases.As methods improve for screening databases, greatamounts of effort and expense can be spared byexploiting the results of past experimental efforts.6

In several cases, many properties of the ligandshave been systematically documented in the data-base. In other cases, the search may reveal ligandswith diverse chemical compositions that can stillsatisfy the same pharmacophore. The comparativeanalysis and modification of such ligands can leadto better drug candidates. In the area of combi-natorial chemistry, our techniques could be em-ployed to assist the design of databases by screen-ing enumerated structures (if pharmacophores areknown).

The identification of a pharmacophore is a chal-lenging and speculative task, and is beyond thescope of this article. Pharmacophore models areconstructed for cases in which the 3D structureof the receptor is known and in which it is not.Several three-dimensional quantitative structure–activity relationships, or 3D QSAR7 theories havebeen developed to capture some of the underly-ing chemical activity in a pharmacophore. If the 3Dstructure of the docking site is known, the phar-macophore is based on the properties of atomson or close to the surface of the receptor and theligands that are known to dock in that site. Of-ten the 3D structure of the receptor can not beobtained using techniques such as X-ray crystallog-raphy or NMR. In that case, the only informationavailable is a set of molecules that interact withthe specific receptor, and hence, exhibit the phar-macophore in their docked conformations. Eachof these ligands, however, has many torsional de-grees of freedom, making the identification of thepharmacophore an extremely difficult task. Com-putational techniques that automatically constructa pharmacophore from a set of molecules havebeen developed.7 – 10 Both in the case of knownand unknown receptor structure, database screen-ing techniques, such as the one presented in thisarticle, are powerful tools in the drug developmentprocess.

RELATED WORK

For over two decades, the drug degign processhas been aided by database-searching techniquesthat perform matching based on 3D structure. A re-cent survey of database-searching techniques thatconsider flexible molecules appears in ref. 11. Theearliest systems performed static matching of sin-gle conformations to queries.12 There has beenmuch interest in recent years to consider confor-mational flexibility in performing matches, whichwas first considered in ref. 13. There are gener-ally two types of approaches to database-searchingthat consider conformational flexibility: similarity-based methods and pharamcophore-based meth-ods. Many similarity-based approaches use super-position, such as the FlexS system, which computesa scoring function, and candidates are ranked andselected on the basis of enrichment factors.14

Pharmacophore-based methods represent themost common searching approach, and a contin-uing source of many computational challenges.15

In addition to conformational flexibility, pharma-cophores allow flexibility in the query through thespecification of geometric features that only par-tially characterize the active site.16 A wide vari-ety of conformational search techniques have beendeveloped and used with these methods,17 includ-ing systematic search,18 distance geometry,19, 20 ran-domized search,21 – 25 and genetic algorithms.26, 27

Several approaches to pharmacophore-basedsearch consider conformational flexibility while thequery is processed (as opposed to storing multi-ple conformations of each molecule).28, 29 For exam-ple, the “Directed Tweak Method”28, 30 minimizesa pseudoenergy function that combines the energyof the molecule and the sum of the squares of thedeviations of the distances found in the molecularstructure to the distances expressed in the phar-macophore query. Unfortunately, the pseudoenergyfunction contains a large number of local min-ima, and conformations having high energy arefrequently returned.17 Our work differs from previ-ous work in the sense that it rigorously treats thekinematics of the molecule while guiding the mole-cule into low energy conformations.

Problem Formulation

THE MOLECULE MODEL

A molecule is characterized by a pair (A, B), inwhich A represents a collection of atoms, and B

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LAVALLE ET AL.

FIGURE 1. The molecule is considered as a tree that is rooted at the anchor atom. Some of the bonds are consideredrotatable. For each rotatable bond, the structure below the black dot rotates about the bond’s main axis.

represents a collection of bonds between pairs ofatoms. An underlying graph can be considered forthe molecule, in which vertices represent atoms andedges represent bonds. Thus, usual graph-theoreticconcepts such as connectedness, paths, trees, andcycles can be applied to molecules. It will be conve-nient to choose one atom, aanch ∈ A, as the anchorfor the molecule (or the root of the correspond-ing graph). Figure 1 shows an example. We assumethat the underlying graph structure is a tree (i.e.,no flexible rings). We represent rigid rings by con-sidering the entire ring as a special “atom” that isattached by a “bond” to the rest of the molecule.Our assumption about rings remains valid for alarge set of molecules that are of interest in drugdesign. Our general approach could be extendedto cyclic molecules by exploiting computational al-gebra techniques that obtain kinematic solutions tocyclic chains,31 by using techniques for large cyclicchains as in ref. 32, or by breaking bonds and addingdummy constraints.

Information used for kinematic and energy com-putations is associated with each of the atoms andbonds. Each atom carries standard information,such as its van der Waals radius. Three pieces ofinformation are associated with each bond, bi ∈ B:(i) the bond length, li; (ii) the bond angle, αi, is theangle between bi and the previous bond, in the di-rection toward aanch; (iii) the set of possible torsionangles, θi ⊆ [0, 2π), which represents the ability ofthe bond to rotate about its own axis. The part ofthe molecule that is attached to b in the directionaway from the anchor will also undergo rotation

about this axis. If θi must remain constant, the bondis fixed; otherwise, it is considered rotatable. In mostmolecular studies,15, 19 bond lengths and bond an-gles are considered fixed, while torsions are allowedto vary. We follow this assumption in our work.From now on we represent the conformation of theligand as m-dimensional vector of torsion angles θ ,in which each component of θ corresponds to a ro-tatable bond.

THE PHARMACOPHORE MODEL

A pharmacophore is defined in terms of a finiteset of features. Typically, there are between threeand six features. In our system, a feature corre-sponds to an atom in A; however, it is straightfor-ward to extend our implementation to incorporateother features, such as the center of a rigid ring.In general, the kinematic models, given in (1)–(4),can be extended to include “dummy atoms” bydefining artificial bonds and atoms. These exten-sions are needed for some common pharmacophoremodels (e.g., for modeling hydrogen bond donorsand acceptors). The result is an extended molecule,which can be handled using the same techniquesas for the original molecule. The kinematic equa-tions can be applied to the extended molecule, andour search methods will work on the extended fea-ture set. Conformations for the extended moleculewill lead directly to conformations of the originalmolecule. The conformation of the original mole-cule would be used for energy computations, andthe conformation of the extended molecule would

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be used for kinematics computations. In the remain-der of this article we forego the consideration of anextended molecule, and assume that every featurecorresponds to an atom in the original molecule;however, our general approach can be extended toconsider additional types of features by replacingthe original molecule with an extended molecule.

The pharmacophore model also includes con-straints on the relative positions between features.Suppose for convenience that one of the featuresis designated as aanch, which lies at the origin of aglobal xyz coordinate system. For each remainingfeature, the corresponding atom is constrained tolie near a specified position in this new coordinatesystem. Any set of torsion angles that is chosen forthe molecule must place each of the feature atomswithin a small neighborhood of its prescribed po-sition. Another coordinate frame is attached to themolecule at aanch. The molecule coordinate frameand the feature coordinate frame can become mis-aligned by rotations; this will be handled shortly.For the purpose of assigning bond angles for bondsthat are attached to aanch, assume that aanch is at-tached to a fictitious “bond” that connects (0, 0,−1)and (0, 0, 0).

THE KINEMATIC MODEL

Molecular kinematics give the positions of all ofthe atoms of the ligands in terms of the torsion an-gles. The bond lengths, bond angles, and torsionangles can be conveniently used as parameters inthe Denavit–Hartenburg representation for spatialkinematic chains.1, 2 This representation is useful fordetermining the appropriate rigid-body transforma-tion to apply to any link in a series of attached links.

For the molecule, suppose that a local coordinateframe is attached at the beginning of each link (oratom center). If a bond bi follows a bond bi−1 in thechain, then the coordinate frame of bi is related tothat of bi−1 by the homogeneous (both rotation andtranslation are performed) transformation

Ti =

cθi −sθi 0 0

sθicαi−1 cθicαi−1 −sαi−1 −lisαi−1

sθisαi−1 cθisαi−1 cαi−1 licαi−1

0 0 0 1

, (1)

in which θi represents the torsion angle, cθi is cos θi

and sθi is sin θi. Figure 2 depicts the quantities thatappear in (1). The fictitious bond is used to define α0.If bi is not rotatable, then θi is a constant; otherwise,θi is a conformation parameter, included in θ .

The position of any atom in the molecule can bedetermined by chaining matrices of the form (1). Forexample, suppose bi, bi−1, . . . , b1 represents the se-quence of bonds in the path from a particular atom,a ∈ A, to aanch. The xyz position of a is given by

xyz1

= T1T2 . . .Ti

0001

. (2)

Before the expression of the kinematics is com-plete, there is one additional transformation thatmust be defined because we assume, without lossof generality, that aanch is a feature. The definitionof the features requires that the position of aanch isat the origin, but it does not impose any constraintson the orientation of aanch. Thus, it is possible that acoordinate frame attached to the molecule could berotated with respect to the global coordinate framefor the pharmacophore feature positions. It is, there-

FIGURE 2. The assignment of α, l, and θ parameters along the kinematic chain.

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FIGURE 3. Four features are shown for a hypothetical molecule. The coordinate frame is attached to one of thefeatures, which is designated as aanch. The anchor atom, aanch, must be allowed to rotate to preserve the originalfreedom of the molecule.

fore, necessary to allow the frame attached to themolecule at aanch to achieve any orientation with re-spect to the global frame (see Fig. 3).

The space of the 3D rotations can be parame-terized using Euler angles, γ ,φ,ψ , for which 0 ≤γ < π , 0 ≤ φ < 2π , and 0 ≤ ψ < 2π .1 Theparameterized rotation matrix can be placed intohomogeneous form by extending one row and col-umn to obtain

TR(γ ,φ,ψ)

=

cφsγ cψ −cφcγ sψ − sφcψ cφsγ 0

sφcγ cψ + cφsψ −sφcγ sψ + cφcψ sφsγ 0−sγ cψ sγ sψ cγ 0

0 0 0 1

.

(3)

Taking into account the anchor orientation, theposition of a particular atom, a ∈ A, at the end ofa path, bi, bi−1, . . . , b1, to aanch is given by

xyz1

= TR(γ ,φ,ψ)T1T2 . . .Ti

0001

. (4)

THE KINEMATIC ERROR FUNCTION

Using the kinematic expressions, an error func-tion can be specified to express how closely the

pharmacophore constraints are satisfied. The trans-formation (4) is expressed in terms of several con-stants and variables. The constants are the bondangles, αi, bond lengths, li, and torsion angles fornonrotatable bonds. The variables are γ , φ, ψ , andthe torsion angles for rotatable bonds which aregiven in θ .

Suppose that there are N features in the pharma-cophore model. Let Gk denote the prescribed posi-tion for the kth feature, for each k ∈ {0, 1, . . . , N− 1}.Let G0 = (0, 0, 0) represent the feature that corre-sponds to aanch. Let gk(θ , γ ,φ,ψ) represent the xyzposition of the atom that corresponds to the kth fea-ture. This position is expressed only in terms of thevariables in the kinematic formulation (4).

Given θ , γ , φ, and ψ , the total amount of errorbetween the prescribed feature positions and the ac-tual feature positions can be measured as

d(θ , γ ,φ,ψ) =N−1∑i= 1

∥∥Gi − gi(θ , γ ,φ,ψ)∥∥, (5)

in which ‖·‖ denotes the Euclidean norm.

THE ENERGY FUNCTION

In a sense, the energy function measures the like-lihood that the molecule will achieve a conforma-tion in nature (lower energy states are more likelyto occur). It is common in molecular modeling33 to

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CONFORMATIONAL SEARCH AND DATABASE SCREENING

use an empirical energy function; we use the SYBYLsystem (Tripos Inc.30) energy function:

e(θ) =∑

bonds

12 Kb(R− R′)2

+∑ang

12 Ka(α − α′)2

+∑

torsions

Kd[1+ cos(nθ − θ ′)]

+∑

i,j

{4εij

[(σij

rij

)12

−(σij

rij

)6]+ qiqj

εrij

}. (6)

In the above, the first sum is taken over all bonds,the second over all bond angles, the third over allrotatable bonds, and the last sum of is taken over allpairs of atoms. Kb, Ka, and Kd are force constants,ε is the dielectric constant, and n is a periodicityconstant. R, α, and θ are the measured values ofthe bond lengths, bond angles, and torsional anglesin conformation θ , while R′, α′, and θ ′ are equilib-rium (or preferred) values for these bond lengths,bond angles, and torsional angles. rij measures thedistance of atom centers in θ . The parameters σij, εij,and qi are the Lennard–Jones radii, well depth, andpartial charge for each atom in the system.

The expression for the energy of a molecule mayappear quite complicated, especially due to inter-actions between each pair of atoms; however, theenergy depends only on the conformation, θ , be-cause the anchor orientation parameters are onlyused for pharmacophore purposes. Notice also thatthe first and the second term of the energy functionare constant with our assumptions.

Our algorithms require no assumptions on theparticular form of the energy function. Thus, a va-riety of other energy models could be successfullysubstituted.

THE GENERAL TASK

The general task is to find conformations thatsatisfy both pharmacophore and energy considera-tions. For a given molecule in the database and agiven pharmacophore, two important questions areasked:

Question 1: Can this molecule achieve a low-energy conformation that satisfies the givenpharmacophore?

Question 2: What are the distinct low-energyconformations that satisfy the pharma-cophore?

Question 1 decides whether the molecule isworth considering as a candidate in the drug de-sign process. Using the concepts defined in thissection, this question can be formulated as deter-mining whether there exist values for θ , γ , φ, and ψ ,such that d(θ , γ ,φ,ψ) and e(θ ) are below some fixedthresholds.

The answer to Question 2 provides additionalinformation for pharmaceutical chemists. This isgenerally referred to as the conformational coverageproblem. Techniques for analyzing the amount ofcoverage obtained from a small set of representativeconformations appear in refs. 23 – 25. There mightbe several low-energy conformations that satisfy thepharmacophore, but each could place the other non-feature atoms in very different locations. In general,having alternative low-energy conformations is use-ful because in many cases it is not the lowest-energyconformation that results in docking, but anotherlow-energy conformation.

These distinct conformations may also be usedto refine a pharmacophore model. A new, hypoth-esized feature might only be satisfied in some ofthese conformations. By looking at multiple confor-mations for several molecules, it might be possibleto select a single conformation from each moleculethat also satisfies the new feature. One difficulty inproviding a set of distinct conformations is that thenotion of “distinct” is not well defined. The differ-ence between two conformations can be comparedby defining a metric, such as the RMS distance ofthe atomic displacements between the two confor-mations. In a later section, a clustering technique isdescribed that ensures that the reported clusters suf-ficiently differ according to this metric.

Randomized Conformational Searchwith Constraints

THE APPROACH

In this section we focus our attention on Ques-tion 1 above. In the next section we show how tointegrate the approach with a database screeningsystem that can answer both Questions 1 and 2.Figure 4 indicates the three main steps involved inthe computation. The first step generates a randomconformation, θ , and anchor orientation, given byγ , φ, and ψ . The second step attempts to reducethe kinematic error (5) by performing a randomizedgradient descent. If the second step is successful,the third step is reached; otherwise, the first stepis repeated. The third step attempts to reduce the

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FIGURE 4. An overview of constrained conformationalsearch.

energy while keeping the kinematic error within anacceptable range. If the energy falls below a pre-scribed threshold, then the method reports success.Otherwise, the first step is repeated. In practice,a limit is set of the maximum number of allow-able failures before the algorithm terminates. It isimportant to note that if the algorithm terminatesafter any number of failures, it cannot be concludedwith certainty that the molecule does not admit alow-energy conformation that satisfies a given phar-macophore. This is the tradeoff that is typicallymade for an efficient, randomized algorithm, as op-posed to a costly, systematic approach that carefullyconsiders all possibilities. Randomized techniquesare popular for conformational search problems thatdo not involve pharmacophore constraints (see, e.g.,refs. 21, 34 – 37).

An alternative way to approach the constrainedsearch problem would be to obtain an explicit char-acterization of the set of conformations that satisfythe pharmacophore. This generally involves char-acterizing the solutions to the inverse kinematicsproblem. The equations of the form (4) that expressthe pharmacophore can be converted to a poly-nomial system. Each trigonometric function canbe replaced by a ratio of polynomials by usingstereographic projection, and the problem confor-mations that satisfy the equations lie in an alge-braic variety.38 Efficient elimination techniques fromcomputational algebraic geometry have been devel-

oped for numerating the solution set for problemin which there are a finite number of solutions.31, 39

For the problem discussed in this article, the systemof equations is generally underconstrained, whichleads to a complicated, multidimensional solutionset. Although it is straightforward to obtain a para-metric representation of some algebraic varieties(such as a sphere or a torus), it not generally pos-sible to employ elimination techniques to find aparameterization of any algebraic variety.38 Thisconsideration, and the need for efficiency, led to thechoice of a numerical, randomized technique, as op-posed to performing symbolic computations withpolynomial systems.

DISTANCE MINIMIZATION

The algorithms for reducing the kinematic er-ror and for decreasing the energy are shown inFigures 5 and 6, respectively. The first algorithm,DECREASE_KINEMATIC_ERROR, resembles theminimization technique used in the UNITY-3Dpackage (which is included in SYBYL, from TriposInc.30) The algorithm accepts an initial conformationand anchor orientation parameters, and iterativelyadjusts these until the total kinematic error falls be-low some acceptable limit, dtol. Our discussion in thelast section shows how to compute the kinematicerror d. In each iteration, the RANDOM_NEIGHfunction slightly perturbs each of the conforma-tion and orientation parameters. The size of thisneighborhood was determined experimentally inadvance. There are two ways in which this algo-rithm can fail: (1) the counter k reaches its maximumvalue, kmax, which means that too many iterationshave been executed, or (2) the counter f reaches itsmaximum value fmax, which means that too manyfailures to reduce the error have occurred. This sec-ond failure essentially detects that the minimizationis trapped in a local minimum. Rather than try toescape this minimum, a new conformation is sam-pled.

ENERGY MINIMIZATION

If the kinematic error is successfully reduced,the resulting conformation and base orientationare passed to DECREASE_ENERGY. This algorithmproceeds in the same manner as the previous one,except that two different criteria must be monitored.It attempts to reduce the energy, while ensuringthat the kinematic error distance, d, does not in-crease beyond dtol. For this algorithm there are threecounters, each of which can halt the algorithm if

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FIGURE 5. This algorithm iteratively attempts to reduce the kinematic error.

its limit is reached. The counter k records the to-tal number of iterations, f records the number ofconsecutive failures to maintain the pharmacophorewithout reducing energy, and g records the numberof consecutive failures to reduce the energy. ethr isthe desired energy threshold.

This algorithm exploits that fact that some toler-ance is allowed for matching the pharmacophore.Figure 7a illustrates that a perturbation of the con-formation and anchor orientation parameters onlyslightly moves the feature atoms. If each of thefeature atoms remains within acceptable tolerance,

FIGURE 6. This algorithm iteratively attempts to reduce the energy while keeping the kinematic error withintolerance limits.

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FIGURE 7. (a) After perturbing the molecule, it must be determined whether the feature atoms remain withinacceptable tolerance from their prescribed positions; (b) the search technique can be considered as a randomizedtraversal inside of a thick surface.

then the new parameters are evaluated based onenergy. The speed of the algorithm is directly re-lated to the feature tolerance. If the tolerance islarger, then larger variations can be considered inRANDOM_NEIGH. This causes the molecule tomove more quickly toward an energy minimum. Asshown in Figure 7b, the “thickness” of the set ofacceptable parameters greatly facilitates a random-ized traversal. In the current implementation, theneighborhood size in RANDOM_NEIGH is an em-pirically chosen constant.

Note that the choice of anchor atom prohibitsthe specification of a tolerance for the feature at theorigin. This can be overcome by defining three posi-tional offset variables for the anchor (in addition tothe rotation parameters); however, this would causesome loss of computational performance.

An Integrated DatabaseScreening System

A prototype database screening system has beenimplemented and tested on a small molecular data-base. The principle module in our system is theconstrained conformational search technique de-scribed in the last section. We have not performedextensive experimentation and evaluation on largedatabases because we are not evaluating pharma-cophore models and representations, as is usuallydone in this area. Instead, our experiments aredesigned to illustrate the capabilities of our com-putational techniques on pharmacophore matchingproblems that are sufficiently complicated and di-verse.

We assume that our techniques will be appliedto a small set of molecules that were initially fil-tered from a database using 2D information. Thiscould lead to small set of several hundred or fewermolecules that contains both successful candidates

and many false positives. Standard indexing tech-niques are used to isolate these candidates fromthe full database. Our system is designed to fur-ther reduce the set of candidates by performingkinematics-based conformational search.

ITERATIVE SWEEPING ACROSS THE SETOF CANDIDATES

Suppose that one would like to use the ap-proach described in Figure 4 to search a databasefor molecules that have a low-energy conformationthat satisfies a given pharmacophore. Initially, 2Dconstraints can be used to quickly discard any mole-cules that do not contain all of the feature atoms.Our work pertains to the remaining set of mole-cules. The primary question to answer is how manyiterations of our approach should be applied to eachmolecule? For a single molecule, each time a newsample conformation and anchor orientation is cho-sen and fails to lead to success, the likelihood thatthe molecule will ever succeed is decreased. How-ever, after any number of iterations, it is impossibleto conclude that the molecule will never succeed(assuming that other constraints have already fil-tered out molecules that do not satisfy nongeomet-ric constraints).

One sensible way to use this method is to sweepacross the database, using only one iteration of theapproach in Figure 4 for each molecule. Once thelast molecule has been processed, a second sweepcan be made across the database. The sweeps acrossthe database can be repeated until some predeter-mined criteria are met (e.g., a certain number ofsuccesses have been found or the maximum num-ber of sweeps has been reached). Each iteration ofthis method is completely independent; therefore,there is no difficulty in switching frequently be-tween molecules. Furthermore, the whole processcan be easily parallelized. Because the likelihoodthat a molecule will succeed decreases with each it-

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CONFORMATIONAL SEARCH AND DATABASE SCREENING

eration, it makes sense to avoid focusing multipleiterations on a single molecule before continuing.

The result is an “any-time” algorithm in the sensethat the solutions gradually improve over time, andthere is no natural termination point. Typically, thefirst sweep might turn up a small number of suc-cessful molecules. These results could be analyzedwhile the program continues to identify other suc-cessful molecules over time. If the goal is onlyto identify successful molecules, then a successfulmolecule is removed from the search set before thenext sweep. The computation will gradually focusin this case on the more difficult molecules. Onetradeoff is that saving search information after eachiteration could yield costly storage requirements, ifa very large set of candidates is considered.

CONFORMATION CLUSTERING

The goal might alternatively be to characterizethe set of possible solutions for each of the suc-cessful molecules. (This is Question 2 from before.)In this case, clustering can be performed incremen-tally for each solution that is generated.40 A metricm(θ1, θ2) can be defined that quantifies the differ-ence between two conformations. In the databasescreening system, m(θ1, θ2) is defined as the RMSof the displacements of the atoms between the twoconformations. A threshold, mthr, is set as the maxi-mum distance allow for two different conformationsto be considered as part of the same cluster. Analternative metric, which measures distance in theconformation space (by comparing torsion angles),also yields reasonable performance.

The incremental clustering approach proceedsare follows. The “Report Success” step in Figure 4is replaced by an operation that updates the clusterrecord using the new conformation (the anchor ro-tation is also retained). Each cluster is representedby a single conformation that has the lowest en-ergy compared with any other known conformationwithin distance mmax. If the cluster record is empty,then the first update generates a single cluster forthe given conformation. Suppose the cluster recordcontains several conformations when an update isrequested for a new conformation. If there are anyother conformations that are within a distance mmax

that have lower energy, then the new conformationis discarded (a better, similar conformation alreadyexists). Otherwise, the new conformation is addedto the cluster record. Any existing clusters that arewithin a distance mmax and have higher energy aredeleted. This has the effect of making the “represen-

tative” of each cluster the conformation that has theleast energy.

EXPERIMENTS

The database screening system was implementedin Gnu C on an SGI Indy and on a Pentium Pro200Mhz PC running Linux. A database of moleculeswas provided from Pfizer Limited. This databaseincludes six different inhibitors of thermolysin41, 42

and six different inhibitors of ACE.43, 44 Table I re-ports the number of atoms and the number ofrotatable bonds for each of these molecules. Allmolecules are very flexible, as they contain 3 to13 torsional degrees of freedom. Some of the in-hibitors of thermolysin are shown in Figures 8 and 9,and some of ACE are shown in Figures 10 and 11.Each molecule appears in a random conformationin our database. For both sets of molecules, thedocked conformations are known by previous phar-macophore identification studies. We use subsets ofthe true pharmacophore as queries in our databasefor testing purposes.

For the thermolysin inhibitors, we tried a querywith four features of the known pharmacophore,and for the ACE inhibitors we ran a query withthree features of the pharmacophore. In both caseswe let our program complete 20 iterations and setthe cluster distance mmax to 1.5 Å. The maximumfeature tolerance was set to 0.5 Å. It was gener-ally found that a single iteration of the methodin Figure 4 could be performed in about 5–10 s.A sufficient clustering record for a single moleculerequired about 5–20 min. Our results are shown inTable II for the thermolysin inhibitors and in Ta-ble III for the ACE inhibitors. Column 1 of thesetables shows our molecules. We report in columns2 and 3 the number of clusters found and the min-imum energy conformation in all of these clustersas a proof of the fact that the conformations that

TABLE I.The Thermolysin and ACE Inhibitors Used in OurExperiments.

Thermolysin Inhibitors Ace Inhibitors

Molecule Atoms Rot. Bonds Atoms Rot. Bonds

mol1 69 10 48 8mol2 66 11 50 8mol3 22 3 31 7mol4 42 8 47 8mol5 64 13 45 6mol6 63 12 30 3

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FIGURE 8. The inhibitors of theormolysin used in our experiments.

we find are indeed of low energy. In general, theenergies of the conformations found are within 2–7 kcal/mol of the energy of the known dockedconformation.

Our clustering scheme ensures that we recordconformations that are significantly different fromeach other. In Figure 12 we show the representa-tives of the clusters that are generated for mol5 of

FIGURE 9. The 3D models of the inhibitors of theormolysin. Each molecule is at a random conformation.

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FIGURE 10. The inhibitors of ACE used in our experiments.

the thermolysin inhibitors. The pharmacophore fea-tures are circled in white. Notice how well these arematched in contrast with the rest of the atoms in the

molecule. Clusters for mol5 of the ACE inhibitorsare shown in Figure 13. A similar observation holdshere.

FIGURE 11. The 3D models of the inhibitors of ACE. Each molecule is at a random conformation.

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TABLE II.The Results of a Four-Feature Thermolysin Search.

Thermolysin Number of Minimum Energy FoundInhibitors Clusters (kcal/mol)

mol1 17 13.44mol2 14 20.30mol3 3 20.94mol4 7 25.44mol5 12 27.72mol6 14 26.84

Our implementation allows us to answer the twoquestions posed earlier. Question 1 is answeredby our randomized constrained minimization tech-nique. Question 2 is answered by iterating our con-strained minimization and by clustering the results.

TABLE III.The Results of a Three-Feature ACE Search.

Ace Number of Minimum Energy FoundInhibitors Clusters (kcal/mol)

mol1 13 42.42mol2 11 43.12mol3 15 37.88mol4 13 42.25mol5 5 39.40mol6 2 36.38

In each iteration of the algorithm a new randomconformation of the molecule is generated and sub-sequently minimized. As discussed at the end ofthe second section, the position of nonfeature atomsmay help the pharmaceutical chemist select among

FIGURE 12. Twelve clusters were obtained for mol5 of the thermolysin inhibitors.

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FIGURE 13. Five clusters were obtained for mol5 of the ACE inhibitors.

possible ligands or modify existing ligands in searchfor a better drug.

Our previous work with randomized techniqueshas shown4, 8 that if we continue iterating our al-gorithm we increase our chances of covering theconformational space of the molecule, and hence,our chances of providing exhaustive informationabout the constrained conformations of the mole-cule. As a proof of concept, in the examples usedin this section we obtained conformations that arefairly close to the known docked conformations. Forthe five thermolysin inhibitors, we obtained confor-mations whose RMS distances from the correspond-ing docked conformation were 0.50, 2.96, 0.59, 0.81,2.40, and 2.56 Å correspondingly. In the ACE case,we obtained conformations whose RMS distancesfrom the corresponding docked conformation were1.26, 1.79, 0.94, 2.03, 1.87, and 1.98 Å. Notice thatthe more features we use, the closer we can get tothe docked conformation (four features for the ther-molysin inhibitors vs. three features for the ACEinhibitors). The ACE inhibitors have less flexibility,which helps to reduce the fitted RMS distance. Incertain cases, we get very close to the docked con-formation (certain thermolysin inhibitors), which isan indication of good conformational space cover-age. Without doubt, more extensive experiments areneeded to fully evaluate the system, but our prelim-inary results are encouraging.

Discussion

A database screening system has been presentedthat can help expedite the drug design process. Thesystem identifies molecules that are able to satisfya given pharmacophore, and are, therefore, reason-able candidates for further investigation. The keyto this screening system is a randomized confor-mational search approach that considers both thekinematic error imposed by the pharmacophoreconstraints and the energy. The simplicity of the ap-proach should enable straightforward extensions toother classes of molecules, such as those with flexi-ble rings.

We point out, however, that a variety of can-didate evaluation models are used by other re-searchers. Our approach focuses on one particularmodel (a rigid pharmacophore), and evaluates thecorrectness of our computational approach. Moreextensive experimentation would be needed to eval-uate our approach in comparison to other eval-uation models proposed in pharmaceutical drugdesign and computational biology. One possibleconstraint that could be added to our models is ex-clusion spheres. These would serve as additionalkinematic constraints that could be incorporated di-rectly into the search.

A difficulty with the current approach is theselection of the step size for the random neighbor-

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hoods in the kinematic error descent and the energydescent. Although the same value was used overa wide variety of molecules, it seems that perfor-mance can be greatly improved by giving more care-ful attention to this selection. For example, largerstep sizes might be appropriate if the conformationis near a kinematic singularity. This could compen-sate for the fact that large displacements near asingularity lead to small displacements of the fea-ture atom. A quaternion parameterization of theanchor orientation, instead of a Euler angle para-meterization, might also improve performance forsimilar reasons. Performance improvement mightalso be obtained by constraining random neighbor-hood samples to lie in the tangent space to theconstraints.

Another limitation of our approach is the localminimum problem, which could cause false neg-atives. By generating multiple conformations foreach molecule, and by appropiately focusing thedatabase screening in an “any time” manner, wehave avoided focusing matching efforts on a single,local optimization. This avoids many local mini-mum problems. Over time, we expect the search togradually find a global minimum and sufficientlycover the conformation space of each molecule, butit is difficult to guarantee that this will be done effi-ciently.

Acknowledgments

This work started in the context of a grant withPfizer Limited and Stanford University. At that time,Paul Finn was with Pfizer Limited and Steve LaVallewas at Stanford University. This work was contin-ued and completed under different funding. The au-thors are grateful to David Hsu, Thomas Lengauer,Rajeev Motwani, and Suresh Venkatasubramanianfor their helpful comments. The authors also thankJaney Nicholas and Miguel Teodoro for their help.The authors greatly appreciate the careful readingand numerous helpful suggestions provided by theanonymous reviewers of this article.

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