Molecular Similarity in Medicinal Chemistry Miniperspective

Transcription

1 pubs.as.org/jm Moleular Similarity in Mediinal Chemistry Miniperspetive Gerald Maggiora,*,, Martin Vogt, Dagmar Stumpfe, and Ju rgen Bajorath*, College of Pharmay and BIO5 Institute, University of Arizona, 1295 North Martin, P.O. Box , Tuson, Arizona 85721, United States Translational Genomis Researh Institute, 445 North Fifth Street, Phoenix, Arizona 85004, United States Department of Life Siene Informatis, B-IT, LIMES Program Unit Chemial Biology and Mediinal Chemistry, Rheinishe Friedrih-Wilhelms-Universitaẗ, Dahlmannstrasse 2, D Bonn, Germany ABSTRACT: Similarity is a subjetive and multifaeted onept, regardless of whether ompounds or any other objets are onsidered. Despite its intrinsially subjetive nature, attempts to quantify the similarity of ompounds have a long history in hemial informatis and drug disovery. Many omputational methods employ similarity measures to identify new ompounds for pharmaeutial researh. However, hemoinformatiians and mediinal hemists typially pereive similarity in different ways. Similarity methods and numerial readouts of similarity alulations are probably among the most misunderstood omputational approahes in mediinal hemistry. Herein, we evaluate different similarity onepts, highlight key aspets of moleular similarity analysis, and address some potential misunderstandings. In addition, a number of pratial aspets onerning similarity alulations are disussed. INTRODUCTION Moleular similarity is one of the most heavily explored and exploited onepts in hemial informatis and is also a entral theme in mediinal hemistry. 1 3 Many omputational similarity methods have been (and ontinue to be) introdued. 1,2 Why do we apparently are so muh about similarity in the moleular world? Simply put, omparing ompounds and their properties, espeially ativity, is one of the most frequent exerises in hemial and pharmaeutial researh but often for rather different reasons. In mediinal hemistry, questions are asked suh as the following: Can a similar follow-up andidate ompound be identified for a liability-assoiated lead? Is a andidate too similar to a ompetitor s ompound to establish an intelletual property position? How an we omplement our ompound olletion with different (i.e., dissimilar) ompounds? Providing answers to these and other questions requires the assessment of similarity (or dissimilarity) in one way or another. As will be disussed throughout this review, three basi omponents are required to onstrut suitable omputational measures of moleular similarity: (1) a representation whose omponents enode the moleular and/or hemial features relevant for similarity assessment, (2) a potential weighting of representation features, and (3) a similarity funtion (also alled a similarity oeffiient) that ombines the information ontained in the representations to yield an appropriate similarity. This value usually lies between 0 and 1, where 1 results from the omplete identity of the moleular representations (but not neessarily the ompounds). Representation features typially are different types of moleular desriptors. A weighting sheme will be required if ontributions of these features should be differently prioritized for similarity assessment (otherwise, if all seleted features should be equally onsidered, no weighting is required). Appliations in hemial informatis that involve systemati omparisons of ompounds and the quantifiation of their similarity provide a stimulating intelletual setting for method development. Quantitative readouts of similarity are also of pratial relevane in, for example, the identifiation of new andidate ompounds on the basis of known atives via virtual sreening, 4,5 for whih similarity searhing is one of the most popular approahes. 6,7 Why is similarity assessment a ompliated problem? Two ompounds that share a ommon substruture an be deteted unambiguously, or all ompounds sharing this substruture an be retrieved from a ompound database. However, as illustrated in Figure 1, it annot be said with ertainty if two ompounds are similar to eah other, what their degree of similarity might be and how similarity should be assessed. In this ase, the ath is that it is diffiult to rationalize relationships that are prinipally subjetive in nature. First and foremost, similarity like beauty is more or less in the eye of the beholder. The diffiulty of the problem inreases further when attempting to desribe similarity relationships in a formally onsistent manner and to quantify them with aid of omputational methods, as Reeived: September 12, 2013 Published: Otober 23, Amerian Chemial Soiety 3186

2 Figure 1. Similarity pereption and onepts. Two exemplary vasular endothelial growth fator reeptor 2 ligands are shown, and different ways to assess their similarity are illustrated. further detailed below. Although similarity is diffiult to rationalize and quantify, omputational deision support in similarity assessment is nevertheless often requested in mediinal hemistry; unfortunately, it fails more often than not. Why is this so? Herein, different similarity onepts and omputational approahes for similarity assessment are disussed. In addition, an attempt is made to rationalize why there is often a disrepany between omputational and mediinal hemial views of similarity and address some ommon misunderstandings. Finally, the use and interpretation of similarity alulations in the pratie of mediinal hemistry are disussed. DO SIMILAR STRUCTURES HAVE SIMILAR PROPERTIES? In the ontext of a seminal book publiation 8 that appeared in the early 1990s when moleular similarity analysis first beame popular, the similarity property priniple (SPP) emerged, whih stated that similar ompounds should have similar properties, the most frequently studied property being biologial ativity. Although this fundamental priniple sounds simple enough, it is very diffiult to apture methodologially. At the heart of the problem is the requirement to learly define and onsistently aount for similarity. As illustrated in Figure 2, ompounds that might not be onsidered similar often share similar ativity (horizontal ompound relationship) or other property values. In ontrast, ompounds that likely would be onsidered very similar might not do so (vertial ompound relationship), learly illustrating the limitations of the SPP. Struture ativity relationship (SAR) disontinuity, i.e., small hemial modifiations that lead to signifiant hanges in biologial ativity, represents a major limitation of the SPP. The extreme form of SAR disontinuity is provided by ativity liffs A key aspet assoiated with the SPP that strongly influenes nearly all onsiderations of similarity in hemial informatis and mediinal hemistry is that moleular similarity values are rarely of interest per se. Rather, they are used as a basis for orrelating similarity, however assessed, with ompounddependent properties suh as biologial ativity. Despite its fundamental importane, this aspet is surprisingly often not onsidered in omputational similarity analysis. 3187

3 Figure 2. Similarity versus ativity. Three vasular endothelial growth fator reeptor 2 ligands are shown that represent different (vertial vs horizontal) similarity ativity (poteny) relationships. SIMILARITY HAS MANY DIFFERENT MEANINGS It is evident that similarity is a widely used onept that is of relevane for reognizing and organizing all omponents of the physial environment as well as many other aspets of life. However, even in the more narrowly onfined moleular world, similarity may have many different meanings or interpretations depending on our individual perspetive. Hene, if the ultimate aim is to formally desribe similarity in a onsistent manner despite its intrinsi limitations, it is of ritial importane to first distinguish between different similarity riteria and onepts, as illustrated in Figure 1. Chemial or Moleular Similarity? Although the terms hemial and moleular similarity are often used synonymously, this may not be entirely aurate. Chemial similarity is based primarily on the physiohemial harateristis of ompounds (e.g., solubility, boiling point, log P, moleular weight, eletron densities, dipole moments, et.) while moleular similarity fouses primarily on the strutural features (e.g., shared substrutures, ring systems, topologies, et.) of ompounds and their representation. Physiohemial properties and strutural features are typially aounted for by different types of desriptors. Suh desriptors are generally defined as mathematial funtions or models of hemial properties or moleular struture. For hemial similarity assessment, reation information and different funtional groups an also be onsidered. In the urrent work, the fous is more on moleular than hemial similarity. 2D versus 3D Similarity. Similarity an be evaluated on the basis of 2D and 3D moleular representations. 2D similarity methods rely on information dedued from moleular graphs. Diret graph omparisons 12 and graph similarity alulations are omputationally demanding and not widely applied in moleular similarity analysis at present. By ontrast, moleular desriptors that apture graph information suh as fragment 13 or topologial atom environment fingerprints 14 are very popular. Fingerprints are generally defined as bit string 13 or feature set 14 representations of moleular struture and properties. Suh moleular representations an be effiiently ompared omputationally, thus enabling similarity alulations on a large sale. Beause ompounds are inherently threedimensional and their moleular onformations have generally higher information ontent than their orresponding moleular graphs, one might antiipate that 3D similarity, whih involves the omparison of moleular onformations and assoiated properties, 15,16 should be generally preferred to 2D similarity. However, this is not the ase for two prinipal reasons. First, hemists are trained on the basis of moleular graphs (i.e., 2D strutural representations) and in general are more omfortable with basing their onsiderations on graphs than on the 3D strutures of ompounds. Moleular graphs typially used by hemists often also ontain onformational and stereohemial information. Seond, given the unertainties assoiated with identifying biologially ative onformations in vast onformational ensembles of test ompounds, 2D approahes are typially more robust, despite their relative simpliity, and often yield superior results in SAR analysis and ativity predition. 17,18 Many urrent similarity methods preferentially utilize 2D moleular representations; most, however, do not ontain any stereohemial information, whih limits their ability to properly treat enantiomeri ompounds. Sine suh ompounds have idential atom onnetivity, their similarity values will be unity if stereoinsensitive moleular representations are used. Furthermore, as will be disussed below in detail, similarity alulations on the basis of 2D moleular representations have a number of other intrinsi limitations. In the following, we will base our disussion of similarity alulations and similarity measures on 2D approahes, in partiular, fingerprint similarity searhing, for several reasons. As pointed out above, hemists are generally more familiar with 2D than 3D representations of ompounds and onsider similarity mostly on the basis of 2D moleular graphs. Furthermore, many of the onlusions drawn from the analysis of simple similarity searhing readily apply to more omplex similarity methods. In this ontext, our preferene for 2D similarity assessment should not be interpreted as a disregard of 3D similarity onepts and methods. Given the mediinal hemistry fous of our presentation, we mostly adhere to 2D similarity onsiderations herein. Moleular versus Biologial Similarity. Another similarity onept that requires onsideration is the biologial similarity of ompounds, whih departs from the oneptual framework of the SPP. Instead, the usual strutural or physiohemial property desriptors are replaed by the 3188

4 Figure 3. Complex similarity relationships. Cylooxygenase (COX) inhibitors and their ativity profiles are ompared. HSL stands for hormonesensitive lipase. ativities of the ompounds against a panel of referene targets, generally proteins, that provide biologial signatures 19,20 analogous to the struture- or property-based representations extensively disussed herein. In this ase, the ativity profiles orresponding to the biologial signatures of the ompounds are ompared using an appropriate similarity funtion as a measure of pairwise similarity, irrespetive of the strutural features of the ompounds. Hene, in this ase, biologial similarity is assessed in target spae rather than hemial spae. For SAR analysis and mediinal hemistry programs, biologial similarity is generally more diffiult to implement than struture- or property-based representations beause speifi ativity values might not be available for ompounds of interest. In addition to their use as moleular similarity measures, biologial signatures an also provide an approximate measure of ompound promisuity. 21 For example, summing the individual values in a binary biologial signature (ative = 1 or inative = 0) yields the number of targets against whih the assoiated ompound exhibits ativity. Global versus Loal Similarity. A very important riterion for similarity analysis is distinguishing between global and loal similarity views. For example, the omparison of pharmaophore models in drug design fouses only on seleted atoms, groups, or funtionalities that are known or hypothesized to be responsible for ativity. This represents a loal view of similarity, in ontrast to the more global view typially found in hemial informatis, where ompounds are onsidered in their entirety. In the latter ase, the alulated property or strutural desriptors typially used to ompute moleular similarities are generally derived from strutural information assoiated with entire ompounds. For example, if we translate the strutural information of a ompound into a fragment fingerprint, a global moleular representation is obtained. This whole-ompound view of similarity is harateristi of the perspetive of hemoinformatiians. Mediinal Chemistry. In addition to loal and global views, however, speial attention must also be paid to a mediinal hemist s perspetive in this ontext. Consider, for example, the set of well-known ylooxygenase (COX) inhibitors ompared in Figure 3. All of these inhibitors are approved drugs exept lumiraoxib, whih lost its United States approval in If we apply a whole-ompound view, ompounds suh as the ibuprofen enantiomers, ibuprofen and paraetamol, or dilofena and lumiraoxib, appear visibly similar. From a mediinal hemistry point of view, however, this assessment may not be generally agreed upon sine small hemial differenes an lead to important hanges in speifiity profiles (e.g., dilofena vs lumiraoxib) or ompounds ontaining different funtional groups an be synthesized or derivatized in different ways (e.g., ibuprofen vs paraetamol). Hene, a mediinal hemist s view of similarity might again be more loal in nature and/or take hemial reation information diretly into aount. Moreover, these COX inhibitors are involved in highly omplex similarity ativity relationships that also annot easily be separated from a mediinal hemistry perspetive. For example, the (R)-( )-enantiomers of ibuprofen and naproxen are inative, but under physiologial onditions the (R)-( )-enantiomer of ibuprofen is onverted into the ative (S)-(+)-enantiomer by the enzyme 2- arylpropionyl-coa epimerase. Furthermore, paraetamol and lumiraoxib are seletive for COX-2, but the other inhibitors are ative against both COX-1 and COX-2, the former ativity giving rise to gastrointestinal side effets. Moreover, naproxen alone is also ative against hormone-sensitive lipase. Suh examples illustrate that onsiderations of hemial and funtional riteria might readily alter the pereption of global moleular resemblane. Clearly, suh similarity onsiderations fall into a gray zone, as they are influened by subjetive riteria as well as the experiene of the investigator, and hene, there is no generally aepted way to judge suh similarity relationships. Aordingly, relations between the ognitive and omputational aspets of moleular similarity are disussed in more detail in the following setion. 3189

5 Figure 4. Similarity assessment through pattern reognition. Exemplary omputer- and human-based pattern reognition proesses for similarity assessment are illustrated. COGNITIVE VERSUS COMPUTATIONAL ASPECTS OF MOLECULAR SIMILARITY While similarity as pereived by trained mediinal hemists is deidedly not the same as similarity obtained by omputational means, there are some aspets of the two that are omparable. For example, in both ases, some type of symboli representation is required to haraterize the strutural information of the ompounds being ompared, although in the former ase the representation is not expliitly stated. Regardless of their details, however, both types of symboli representation must make moleular information omprehensible in suh a way that strutural/feature patterns an be identified and reognized. In general terms, pattern reognition refers to the ability to detet reurrent themes, organization priniples, relationships, and rules in large data sets, 22 an essential requirement for deision making by humans as well as for omputational learning. 22,23 The identifiation of patterns within data forms a basis for lassifiation and diretly applies to our moleular world. More than anything else, the reognition of moleular patterns, based on human or omputational exploration, provides a basis for arriving at deisions as to whether two ompounds are similar to eah other or not. Sine data omplexity generally sales with the number of patterns that an be disovered, it quikly beomes impossible for humans to onsider them in a omprehensive manner. Therefore, humans intuitively, and often unonsiously, redue patterns to simpler ones that ontain the essential feature(s) of the original pattern. But unlike appliations of omputational pattern reognition, the preise nature of these key patterns in human pattern reognition is unknown. For instane, to ross a road safely, we need to reognize patterns assoiated with moving objets and/or engine noise but are not required to understand whih type of ar or motorbike is approahing. This intuitive redutionist approah to pattern reognition is learly refleted by deision-making by mediinal hemists, as further disussed below. Seleting key patterns regardless of whether they are mathematially defined or expressed in terms of vague onsious or subonsious mental onstruts is the most ruial element in any assessment of moleular similarity. The key patterns used by humans or omputers will generally vary from individual to individual or from algorithm to algorithm, a situation that most likely will yield results with varying degrees of agreement for the same set of data. This follows beause the representations used by humans and by omputers, whih most likely are signifiantly different, are ruial omponents in determining what an be understood about relationships of objets to eah other, whether they are physial objets, onepts, ideas, or ompounds. Despite the ommon searh for key patterns, the use of representations to determine similarity in mahine omputation ompared to human pereption of similarity by mediinal hemists differs signifiantly, 2 as shematially illustrated in Figure 4. In the ase of mahine omputation, algorithms have been developed for onstruting suitable representations of the strutural information in ompounds and for evaluating similarity funtions or oeffiients assoiated with these representations. 4,24,25 However, sine there is no unique or invariant way to represent moleular and hemial information, onstruting representations suitable for a given task or goal depends on what is the task or goal. As noted earlier and disussed further below, mathematial funtions that are designed to reflet the degree of moleular similarity typially yield values that lie on the unit interval [0, 1] of the real line. But as is also disussed below, the form of these funtions also influenes the similarity values beause they usually differ even when idential representations are used, although in some ases they are linearly or monotonially related. 2 Role of Chemial Intuition and Experiene. Although well-defined, omputed values may not aount for the degree of similarity in a way that is onsistent with the pereptions of mediinal hemists beause human pereption of similarity is a muh more ompliated, varied, and subtle task (vide supra). Moreover, the ognitive algorithms by whih mediinal hemists pereive similarity are largely unknown, although some reent work has begun to address this question These studies learly show that hemial intuition and 3190

6 experiene play major roles in deision making in mediinal hemistry. Surprisingly, there is typially little onsensus between experiened mediinal hemists in judging preferred ompounds and assessing favorable or unfavorable moleular features Furthermore, it has been shown that pereption of moleular strutures is strongly ontext-dependent; i.e., depending on the order in whih we view ompounds and how they are grouped, different onlusions are drawn. 27 This points to a potential advantage of omputational similarity assessment beause ompound representations or patterns are onstant and ontext-independent. It has also been shown that mediinal hemists often have diffiulties omprehending the nature and meaning of the parameters they might have onsidered and the sientifi riteria upon whih deisions on ompounds are based. 28 Mediinal hemists typially base their ompound deisions on very few patterns or parameters, fewer than they believe, 28 a fat that learly reflets the pattern-redution approah referred to above. Deision parameters generally result from feature redution and pattern redution, whih also provides a foundation of mahine learning approahes. 22,23 Computational methods suh as neural networks, 29 or support vetor mahines, 30 are essentially designed for pattern-based similarity assessment, whih requires training data the use of whih also renders these omputational modeling efforts ontext-dependent. The resulting omputational models have the often ited box blak harater, whih means that they annot be interpreted in hemial terms. In some ways, this provides an interesting analogy to mediinal hemists who do not realize upon whih parameters their ompound deisions might be based. 28 Although it may not be possible to rationalize our judgments, we are typially more ontent with our own deisions than those obtained omputationally that, in many ases, an be diffiult to interpret. Aordingly, mahine learning methods suh as deision trees 31 or emerging hemial patterns 32 are often favored in pratie beause they yield interpretable patterns, even though they may be based on rather abstrat representations of moleular and hemial information. In light of the above, it is lear that judgments of moleular similarity an be influened by a number of ognitive aspets. Lastly, with regard to the SPP, it should be re-emphasized that mere assessment of moleular similarity is generally not the ultimate goal. Rather, in many ases, it is the identifiation of similar ompounds that, based on the SPP, are presumed to have similar properties (espeially biologial ativities) to known referene or target ompounds. This adds additional layers of omplexity to our pereption of similarity and an further ompliate our judgments. Similarity Coeffiients. The question then arises as to whether it is reasonable to assume that any rationalization of similarity, or that any onsistent omputational representation and omparison of ompounds that yields a numerial readout, will inrease our own onsensus and be superior to subjetive deisions. The Tanimoto oeffiient (T) 24,33 is introdued to help answer this and related questions and to provide an illustration of how moleular similarity an be quantified. Although it may not be the best proedure, it is by far the most popular and, beause of its ease of implementation and speed, is in widespread use today in hemial informatis and omputational mediinal hemistry. As detailed in the sequel, a variety of other similarity measures, 24,25 most of whih did not originate in hemial informatis but in other sientifi fields (suh as statistis, eology, and psyhology), have also been used to ompare speifi moleular representations. The Tanimoto oeffiient is generally defined by T(A, B) = a + b (1) where a and b are the number of features present in ompounds A and B, respetively, and is the number of features shared by A and B. Hene, T quantifies the fration of features ommon to A and B to the total number of features of A or B, where the term in the denominator orrets for double ounting of the features. Another perhaps more intuitive way to interpret Tanimoto similarity is based on an alternative form of the denominator on eq 1, i.e., a + b = ( a ) + ( b ) + (2) Here the terms (a ) and (b ) are the number of features unique to A or B, respetively. Substituting eq 2 into eq 1 yields the numerially equivalent form of T, T(A, B) = ( a ) + ( b ) + (3) Dividing numerator and denominator by (a ) +(b ) gives Ra (, b, ) T(A, B) = 1 + Ra (, b, ) (4) where R( a, b, ) = ( a ) + ( b ) (5) whih an be interpreted as the ratio of the number of features shared by A and B to the number of their unique features. As A and B beome more similar, the number of shared features approahes the number of features in A and B (i.e., a,b) and the number of unique features in both ompounds approahes zero (i.e., (a ) 0 and (b ) 0) beause in the limit the number of shared features and number of features in A and B beome equal (i.e., a = b = ). Thus, their ratio goes to infinity, (i.e., R(a,b,) ), whih in the limit gives T(A,B) = 1. Conversely, as A and B beome less similar, the number of shared features approahes zero and onsequently all of the features of A and B are unique, and thus, the ratio of these features also goes to zero (i.e., 0, (a ) a, (b ) b, and R(a,b,) 0); thus, in the limit, T(A,B) = 0. In the intermediate region where the number of shared features is greater than zero but less than the lesser of the number of features in A and B (i.e., 0 < < min(a,b)) and where the number of unique features is less than the total number of possible features (i.e., (a ) +(b ) <a + b), the Tanimoto similarity will lie between the extremes of the unit interval of the real line, i.e., 0 < T(A,B) < 1. One way to think about this is to note that as the number of shared features between two ompounds inreases, their number of unique features must orrespondingly derease. Thus, there is interplay between the number of shared features and the number of unique features exemplified by their ratio R(a,b,). The alulation of Tanimoto similarity is typially based on representations alled moleular fingerprints, 4,6,7 whih an be viewed as lassial sets or binary vetors whose elements have values of 1 or 0 orresponding, respetively, to the presene or absene of speifi features (e.g., moleular 3191

7 fragments). In some ases, elements with value 1 are alled on-bits and those with value 0 are alled off-bits, hene, the desription of moleular fingerprints as bit strings or bit vetors. Note that the moleular fingerprints desribed above do not aount for multiple ourrenes of the different features, only whether they our at least one in a given ompound. However, feature ounts an be added to fingerprints by using integer values to represent features instead of a binary format. Fingerprints of different design and omplexity are available, 7 as further disussed below. For similarity searhing, fingerprints are among the original and to this date most popular desriptors. Dissimilarity an be quantified in a omplementary manner suh that small values indiate similarity and large values dissimilarity. Aordingly, a dissimilarity measure an be derived from the T by taking the appropriate omplement known as the Soergel distane (Sg), 24 i.e., Sg(A, B) = 1 T(A, B) = 1 a + b (6) that an be rewritten as ( a + b ) Sg(A, B) = 1 = a + b a + b ( a ) + ( b ) = a + b (7) As noted above, the denominators in eqs 1 and 3, a + b and (a ) +(b ) +, respetively, represent the number of features that our in either A or B, and the T an then be rationalized as the perentage of shared features, whereas the Soergel distane orresponds to the perentage of features unique to A or B given by (a ) and (b ), respetively. Another similarity measure that is growing in usage is the Tversky oeffiient (Tv), 34 whih is given by Tv αβ, (A, B) = α( a ) + β( b ) + The denominator is losely related to that given for Tanimoto similarity in eq 3 exept for the two parameters α and β that weight the number of features unique to A or B, (a ) and (b ), respetively. As defined by Tversky, 34 α and β are nonnegative. In hemial informatis and omputational mediinal hemistry appliations, these parameters are typially hosen to lie within the unit interval [0, 1] of the real line. In either ase, zero and unity bound the value of Tv. The larger α is ompared to β, the more weight is put on the unique features of referene ompound A and the less on database ompound B and vie versa. Thus, in the ase of Tv, whose values also range from 0 to 1, the similarity values hange as the two weights vary. This makes it possible to study the relative importane of ommon and unique features for ompound ranking with respet to the referene and database ompounds. As disussed further below, the weighting sheme an be applied to introdue asymmetry into similarity alulations. For the speial ase α = β = 1, where the unique features of both ompounds are weighted equally, Tv is idential to T. In the ase where α = β = 0.5, Tv is idential to the Die oeffiient (D) 24 D(A, B) = 1 (8) ( a + b) 2 (9) written here in a form that learly shows that the denominator is the arithmeti mean of the number of features in A and B. Sine 1 / 2 (a + b) (a + b), it follows that T(A,B) D(A,B), as illustrated by the distributions depited in Figure 6. Both similarity oeffiients are symmetri, sine the similarity of A with respet to B is the same as the similarity of B with respet to A. In fat, any Tv in whih α = β yields a symmetri similarity oeffiient suh that Tv α=β (A,B) = Tv α=β (B,A). Tversky similarity oeffiients with two unequal weighting fators (α β) are, on the other hand, asymmetri, their degree of asymmetry depending on the relative magnitudes of the weighting fators. Similarity oeffiients an be lassified aording to their ompound ranking harateristis. Coeffiients that always produe the same ranking of ompounds, although their absolute similarity values might differ, are said to be monotoni. For example, Tv α,β (A,B) and Tv α,β (A,B) are monotonially related if the parameters have the same ratio so that α = kα and β = kβ. These oeffiients an be onverted into eah other by the monotoni funtion Tv (A, B) = α, β k Tv αβ(a, B) k, (10) whih an be verified by elementary algebrai transformations. Thus, normalization of the parameters imposes no restrition on the ranking and, hene, the generality of Tv. In the following, the sum of the weighting parameter values is restrited to unity, i.e., α + β = 1. Replaing β in eq 8 by 1 α yields Tv α(a,b) = α( a ) + (1 α)( b ) + = αa + (1 α) b (11) Thus, Tversky similarity now only depends on the single parameter α. Note that differenes in the numerial distribution of the normalized Tv and T are to a large extent due to the fat that the T orresponds to a non-normalized Tv under the ondition α + β = 2. Furthermore, as learly shown in Figure 6, Tv (A, B) = T(A, B) Tv (A, B) α= 1, β= 1 α= 1/2, β= 1/2 = D(A, B) (12) An extreme form of Tv ours when the referene ompound A is weighted (α = 1) and the database ompound is not (β = 0), in whih ase eq 8 beomes Tv α= 1, β= 0(A, B) = a (13) In this ase, the Tversky similarity oeffiient provides a measure of how similar A is to B, whih an be interpreted as the fration of the features in the referene ompound A that are mathed by database ompound B. Interhanging the values of the weighting fators so that now α = 0 and β = 1 plaes the entire weighting on the database ompound B and gives Tv α= 0, β= 1(A, B) = b (14) whih in this ase an be interpreted as the fration of the database ompound B that is similar to the referene ompound A. These two forms of Tv represent extreme forms of Tversky similarity oeffiients. 3192

8 Inreasing moleular size or omplexity generally leads to inreasing fingerprint bit densities, whih are defined for a given ompound A as ρ (A) = FP number of on bits total number of fingerprint bits (15) Suh inreases in the bit density ρ FP (A) have a statistial tendeny to yield higher similarity values for larger ompounds, 35 a well-known ompliation in similarity searhing 7 and a ause of apparent asymmetry in distributions of similarity values. 36 Moleular omplexity effets an be balaned or eliminated in different ways, for example, by equally taking into aount bits that are set on or off in similarity alulations 37,38 or by ombining binary fingerprint representations with their omplements, i.e., adding the omplement to the original bit string, thereby produing a onstant fingerprint bit density for ompounds of any size. 39 Calulating Tanimoto, Tversky, or Die similarity has an assumed advantage that numerial values an now be used to distinguish similarity relationships in a onsistent manner. How does this numerial approah from hemial informatis relate to, and perhaps influene, the more subjetive assessment of similarity in mediinal hemistry? Are alulated similarity values suitable to replae hemial intuition and judgment? Computed versus Intuitive Similarity. There are a number of issues that arise when omparing omputed similarity values with those assigned by mediinal hemists. One issue is that the similarity sale employed by mediinal hemists is not uniform. The following argument, whih depends on the omplementary nature of similarity and dissimilarity, illustrates this point. In omputations the degree of dissimilarity is typially taken as the omplement of similarity: dissimilarity = 1 similarity Hene, the more dissimilar two ompounds are, the less similar they are to eah other and vie versa. Importantly, suh omplementary behavior between omputed similarity and dissimilarity values does not, however, apply in the ase of human pereption. For example, humans an better assess similarity the more similar ompared objets are to eah other. By ontrast, as objets beome less and less similar, a point is reahed where it is generally diffiult for humans to assess their degree of similarity or dissimilarity. Reall that in the former ase one is dealing with features that are ommon to both ompounds, whereas in the latter ase one is dealing features that are unique to eah of the ompounds. This follows from the basi psyhophysis of human pereption beause it is easier for humans to make omparative judgments of objets with ommon features than between objets whose features are unique. Sine omputed similarity values do not suffer from these problems, a divergene between human pereptions and omputed values of similarity likely arises. In most ases, this is not a problem for mediinal hemists who typially want to synthesize and test ompounds that are similar to known atives. Then, high alulated similarity values have an intuitive meaning. However, if similarity values are dereasing in size, boundaries between similarity and dissimilar beome rather diffuse and one is often unable to interpret suh values. The question of symmetry vs asymmetry of similarity, as formally disussed above, should also be onsidered from an intuitive perspetive. Tversky similarity, whih originated in psyhology (not informatis), is oneptually based on a number of asymmetri harateristis that are assoiated with human pereptions of similarity. An example given by Tversky involves a omparison of Korea and China; the similarity of Korea to China is usually onsidered to be greater than the similarity of China to Korea. This view, whih is rather general, suggests that relative size, however aounted for, has a signifiant influene on the pereived asymmetry of the similarity of entities, inluding ompounds, when ompared by humans. Moreover, this an also be interpreted in terms of eqs 13 and 14, sine the fration of Korea that is similar to China is definitely not the same as the fration of China that is similar to Korea. Often it is not onsidered that the Tversky similarity oeffiient is parametrized to aount for asymmetri aspets of similarity by apturing the asymmetri harateristis inherent in many different types of objets under omparison. To understand, in light of the above, how human pereption of the similarity of two ompounds might be asymmetri, it is neessary to distinguish the ompounds being ompared. Let us onsider an ordered pair in whih A is a referene ompound and B a database ompound. If the referene A is a small ompound and a substruture of a larger ompound, A is rather similar to B. This follows beause A is a lose math to a part of B. However, if the situation is reversed, i.e., B is now used as the referene and A is the database ompound, the similarity will be lower beause most of B differs from A. This is a moleular example of the size effet desribed above in the ase of the pereived asymmetri similarity omparisons of Korea and China. Equations 13 and 14 and the aompanying disussion fully support this analysis. In T alulations, this pereived asymmetri similarity relationship is not refleted, but Tv alulations offer this possibility as a onsequene of appropriate weighting. Importantly, pereived relative size-dependent asymmetri similarity is distint from representation-dependent moleular size or omplexity effets mentioned above, whih systematially bias similarity alulations by produing large values for larger and topologially more omplex ompounds. Human Pereption. The assessment of similarity on the basis of human pereption is onsiderably more ompliated than refleted by the examples given above beause a number of other onsious and subonsious fators also play a role. For example, a key fator in similarity assessment is the ability of humans, in general, and mediinal hemists, in partiular, to intuitively redue the omplexity of the problem at hand (vide supra). This need to redue omplexity largely depends on the fat, as pointed out by numerous psyhologists, that humans an only hold a relatively small number of things in their working memory at any point in time. 40,41 Working memory is that part of memory that atively holds multiple piees of transitory information that an be manipulated by verbal and nonverbal tasks, suh as reasoning and omprehension, and makes the results of these tasks available for further information-proessing. In the ase of mediinal hemists this means that only strutural features pereived to be most essential, or some simplified representation of them, might be retained and onsidered for similarity assessment, very onsistent with the results obtained by Kuthukian et al., 28 indiating the partly unonsious use of only one or two hemial parameters by mediinal hemist in ompound evaluation and deision making. Understanding these riteria, whih will undoubtedly differ from mediinal hemist to 3193

9 Figure 5. Frequeny of fingerprint features. The relative frequeny of ourrene of the 150 most frequent features of (a) MACCS and (b) ECFP4 is alulated for a random subset of 1 million ZINC database ompounds. mediinal hemist, is a nontrivial task. Thus, omputed similarity values and judgments by mediinal hemists are both influened by dependenies on moleular size and omplexity, but the effet is muh more pronouned and diffiult to predit in the ase of mediinal hemists assessments of similarity. The inonsisteny of humans when onfronted with omplex deision tasks 42 is well refleted by generally observed hanges in mediinal hemists judgment about the quality of the same ompounds when presented in different orders (vide supra). 27 It is evident that mediinal hemists are often left with onsious or subonsious impressions, whih they fold into their assessments of similarity in some impliit way, being intuitively aware of the omplexity of the problem at hand, whih then automatially leads to a redutionist approah in deision making. It is therefore not surprising that similarity alulations are attrative in mediinal hemistry beause they redue omplex moleular omparisons to a simple numerial readout. Then, however, the key question beomes what suh omputed values atually mean. CHARACTERISTICS OF SIMILARITY CALCULATIONS In the following setion, we highlight opportunities and limitations of similarity alulations in light of the above disussion. Thereby, we evaluate the apparent attrativeness of numerial similarity measures as a omplement, or replaement, of human pereption and study relationships between alulated and pereived similarities. Similarity Property Priniple Revisited. A ritially important aspet to realize is that most similarity methods do not expliitly take biologial ativity into aount. Thus, similarity values generally reflet the similarity of hosen moleular representations. Yet this is hardly of interest in mediinal hemistry. Instead, hemoinformatiians and mediinal hemists typially attempt to bridge between alulated similarity and biologial ativity, well in aord with the SPP disussed above. In fat, the key question asked in this ontext typially is Whih T value reliably indiates that ompound B has the same ativity as referene ompound A? In other words, How similar must A and B be to have the same ativity? This is the major attration of reduing omplex similarity relationships to simple numbers and the soure of some profound misunderstandings of similarity alulation. The 0.85 Myth. In a seminal study quantifying hemial neighborhood behavior, investigators from Tripos established, using their in-house fingerprints and sets of ative ompounds, that a T value of 0.85 refleted a high probability that two ompounds shared the same ativity. 43 For more than 15 years, this T value has propagated in the literature as a general threshold for bioativity and has been applied in many pratial appliations, although the value is not reliable when other moleular representations are used for similarity alulations. 4,7,44 Neighborhood behavior and alulated similarity values are strongly dependent on hosen moleular representations and similarity measures. 4 While this is generally wellknown, it is often underappreiated in mediinal hemistry even today. The often-observed use of putative T threshold values of biologial ativity reflets ommon misunderstandings of similarity alulations. In the following, we present and disuss exemplary similarity alulations to highlight several harateristi features. Fingerprints of Different Design. In the following, two oneptually different fingerprints are ompared that are popular in omputational mediinal hemistry. The moleular aess system (MACCS) fingerprint, 13 also termed MACCS strutural keys, is a prototypi fragment-based fingerprint that onsists of 166 strutural fragments with 1 10 non-hydrogen atoms and is one of the original and most popular similarity searh tools. 6,7 Its design is simple. Eah bit position is assigned to one partiular strutural fragment or key and its presene or absene in a ompound is deteted. By ontrast, we use the extended onnetivity fingerprint (ECFP) with bond diameter four (ECFP4) that urrently is one of the most popular fingerprints for similarity searhing. 14 ECFPs aount for the loal bond topologies, whih desribe the onnetivity of atoms in the neighborhood of eah nonhydrogen atom in a moleule. The size of the neighborhood depends on the so-alled bond diameter given by the maximum number of bonds onsidered. The ECFP design is muh more omplex than MACCS beause many different atom environment features an be generated. Different from MACCS, ECFP4 onsists of sets of ompound-speifi features whose overlap is quantified as a measure of moleular similarity. Although many different atom environments an in priniple exist, feature sets derived for individual ompounds are often relatively small (e.g., ontaining less than 100 features), depending on their topology. Similarity Value Distributions. Although the definition of T yields an interpretable value as the perentage of fingerprint features shared between two ompounds, it is very diffiult to judge whether a given T value indiates the 3194

10 Figure 6. Similarity oeffiient distributions. Distributions of similarity values resulting from 10 million omparisons of randomly hosen ZINC ompounds are reported for the Tanimoto and Die oeffiient and the (a) MACCS and (b) ECFP4 fingerprint. Figure 7. Comparison of similarity oeffiients. For two thrombin inhibitors Die, Tanimoto, and Tversky oeffiients are ompared using MACCS and ECFP4. Tversky similarity alulations were arried out using different parameter settings. presene of signifiant similarity or not. This is the ase beause the oeffiient value does not tell us anything about the speifi features under omparison. For instane, many MACCS bit positions refer to strutural features that are often found in ompounds, whereas ECFP4 systematially enodes atom environments, many of whih are infrequently found in ompound data sets. For this reason, ECFP4 T values are generally smaller than MACCS T values. This differene in feature frequenies is illustrated in Figure 5 that reports the relative frequenies of the 150 most frequently deteted MACCS and ECFP4 features in ompounds randomly seleted from the ZINC (version 12) database. 45 MACCS and ECFP4 fingerprints were alulated with the Moleular Operating Environment (MOE). 46 Overall the ZINC subset ontained different ECFP4 features, but only 632 of these features ourred in more than 1% of the ompounds. Considering the sparseness of most ECFP4 features, it is not surprising that some moleules that are struturally similar ontain a signifiant number of unique 3195 features. Importantly, the differenes in feature distribution between MACCS and ECFP4 lead to very different distributions of similarity oeffiient values. To illustrate these differenes similarity values were alulated for randomly hosen pairs of ZINC ompounds. The results are shown for MACCS and ECFP4 T and D alulations in Figure 6, where it is lear that the D distributions are shifted toward higher values and are less symmetrial than the omparable T distributions. These effets are due to the normalization (α + β = 1) of the D and an be rationalized based on the disussions assoiated with eqs 9 and 11. Similar effets are, in general, observed for Tv, yielding distributions very similar to those of the D, regardless of the value of the parameter α. The figure shows that different ombinations of fingerprints and similarity oeffiients produe different similarity value distributions, further emphasizing the ritially important point that alulated similarity has no absolute meaning.

11 Figure 8. Similarity searhing using different fingerprints and similarity oeffiients. By use of ompound A from Figure 7 as a referene, similarity values were alulated for 1 million ZINC ompounds and 25 thrombin inhibitors (inluding ompound B from Figure 7) using the Tanimoto and Tversky (α = 0.1 and α = 0.9) oeffiients and the (a) MACCS and (b) ECFP4 fingerprints. The similarity oeffiient is plotted as a funtion of the rank (reported on on a logarithmi sale). The positions of the 25 thrombin inhibitors are marked on eah urve. Although the global distribution of Tv values does not signifiantly depend on the settings of α, this parameter determines how similarity relative to a given referene moleule is pereived. If more weight is put on features (bit settings) of the referene moleule (i.e., if α > 0.5), different similarity relationships evolve. Compounds that ontain most of the referene features plus some additional ones are onsidered to be more similar to the referene moleule than ompounds that ontain fewer of the referene features but also fewer additional features, although the perentage of shared features might be the same for both moleules. How different representations and similarity oeffiients affet omputed similarity values is illustrated in Figure 7, using two exemplary thrombin inhibitors taken from the ChEMBL (version 15) 47 database. Both moleules ontain more ECFP4 features than MACCS features, but the number of shared features is lower for ECFP4, as expeted on the basis of the feature distributions disussed above. Consequently, the different oeffiients produe signifiantly lower similarity values for ECFP4. D is inreased ompared to T as shown in the disussions related to eqs 9 and 12. Beause D is idential to normalized Tv with α = 1, its value an be numerially ompared to the asymmetrial Tv values with parameters α = 0.1 and α = 0.9, respetively. It an be observed that Tv dereases for α = 0.1 and inreases for α = 0.9. In the first ase, more weight is put on the features exlusive to moleule B, and in the seond ase, less weight is put on these features. Thus, the influene of these features on the similarity value is either inreasing or dereasing ompared to D. Changing α has an effet on omputed similarity values. More importantly, however, the parameter also influenes how similarity is pereived in a searh when database ompounds are ranked in the order of dereasing similarity to a referene moleule. Here, the absolute value of similarity is not of interest, espeially if the value annot be interpreted in a meaningful way. Rather, the rank positions of ompounds with the desired properties determine the usefulness of a similarity oeffiient. Figure 8 illustrates the effet that the hoie of different similarity oeffiients has on the ranking of ompounds in a similarity searh. Moleule A in Figure 7 was taken as a referene, and ZINC ompounds together with moleule B and 24 other thrombin inhibitors were searhed and ranked using different similarity oeffiients. In Figure 8, the ranks are displayed on the x-axis from low to high ranks on a logarithmi sale. On the y-axis, the orresponding oeffiient values are reported. For eah similarity oeffiient, the position of the 25 thrombin inhibitors is marked. The graphs illustrate that ompound ranks signifiantly vary depending on the oeffiient and representation used. In this example, MACCS in ombination with Tv and α = 0.9 yields the largest number of thrombin inhibitors within the top 1000 database ompounds (orresponding to 0.1% of the sreened database). However, it is stressed that no general onlusions about the relative performane of individual oeffiients and fingerprints an be drawn from a single example given the strong ompound lass dependene of similarity alulations (vide infra). Similarity Threshold Values. Considering the global distributions of similarity values, it is of interest to derive threshold values that indiate a statistially signifiant level of similarity. Signifiane analysis of similarity values an be used, for instane, to determine if similarities between ompounds sharing a property like biologial ativity might simply our by hane or if ompound similarity is likely to be assoiated with the shared property. For this purpose, onventional p-values an be alulated. For example, a T threshold value at a signifiane level of p = 0.01 would indiate a probability of 1% that the T value alulated for two randomly hosen ompounds meets or exeeds the threshold. Threshold values an be estimated from the distribution of a large sample of similarity values obtained by randomly seleting pairs of ompounds and alulating their similarity oeffiient. The umulative distribution funtion F(t) of the values then relates a similarity value t to the ratio of similarity values less than or equal to t, and the signifiane is given by p =1 F(t). If suh threshold values are generally appliable in the ontext of similarity searhing, i.e., if a similarity value exeeding a threshold value is a rare event and thus indiates signifiant similarity, they must be largely independent of the seleted referene ompound. It is emphasized at this point that only alulated similarity values and their statistis are onsidered; aounting for ompound ativity aording to the SPP is addressed in the next subsetion. 3196