A network inference method for large-scale unsupervised identification of novel drug-drug interactions
AA Network Inference Method for Large-ScaleUnsupervised Identification of Novel Drug-DrugInteractions
Roger Guimera` * , Marta Sales-Pardo Institucio´ Catalana de Recerca i Estudis Avanc¸ats (ICREA), Barcelona, Catalonia, Spain, Departament d’Enginyeria Quı´mica, Universitat Rovira i Virgili, Tarragona,Catalonia, Spain
Abstract
Characterizing interactions between drugs is important to avoid potentially harmful combinations, to reduce off-targeteffects of treatments and to fight antibiotic resistant pathogens, among others. Here we present a network inferencealgorithm to predict uncharacterized drug-drug interactions. Our algorithm takes, as its only input, sets of previouslyreported interactions, and does not require any pharmacological or biochemical information about the drugs, their targetsor their mechanisms of action. Because the models we use are abstract, our approach can deal with adverse interactions,synergistic/antagonistic/suppressing interactions, or any other type of drug interaction. We show that our method is able toaccurately predict interactions, both in exhaustive pairwise interaction data between small sets of drugs, and in large-scaledatabases. We also demonstrate that our algorithm can be used efficiently to discover interactions of new drugs as part ofthe drug discovery process.
Citation:
Guimera` R, Sales-Pardo M (2013) A Network Inference Method for Large-Scale Unsupervised Identification of Novel Drug-Drug Interactions. PLoSComput Biol 9(12): e1003374. doi:10.1371/journal.pcbi.1003374
Editor:
Nathan D. Price, Institute for Systems Biology, United States of America
Received
April 8, 2013;
Accepted
October 17, 2013;
Published
December 5, 2013
Copyright: (cid:2)
Funding:
This work was supported by a James S. McDonnell Foundation Research Award, Spanish Ministerio de Economı´a y Comptetitividad (MINECO) GrantFIS2010-18639, European Union Grant PIRG-GA-2010-277166 (to RG), European Union Grant PIRG-GA-2010-268342 (to MSP), and European Union FET Grant317532 (MULTIPLEX). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests:
The authors have declared that no competing interests exist.* E-mail: [email protected]
Introduction
Understanding interactions between drugs is becoming increas-ingly important. A recent large-scale study of older adults (ages57–85) in the U.S. found that 29% of them use five or moreprescription medications concurrently, and that as many as 4%may be at risk of having a major adverse drug-drug interaction [1].For this reason, the evaluation of drug interactions is ‘‘an integralpart of drug development and regulatory review prior to its marketapproval’’ [2], and institutions like the FDA put much effort indeveloping guidelines for in vitro and in vivo studies, as well as fordeveloping in silico models and methods.Potentially beneficial effects of drug interactions, on the otherhand, are equally important. Indeed, some drugs show synergisticeffects against their targets, which not only increases the efficacy oftreatments but may also improve the selectivity and reduce off-target effects [3]. Antagonistic interactions can be used to study themechanisms of action of drugs [4], and even suppressinginteractions between drugs, in which one drug inhibits the actionof the other, have been found to be potentially very relevant in thefight against antibiotic-resistant pathogens [5].More broadly, it is becoming increasingly clear that druginteractions leading to network effects at a systems level are the normin pharmacology, rather than the exception [6–11]. According tosome, these network effects may even be at the root of the dismal resultsof attempts to develop single-target drugs, and of the simultaneousdecline of drug development productivity [7]. Therefore, networkpharmacology is emerging as a new paradigm in drug discovery. However, despite the conceptual appeal of abstract networkapproaches to drug development, one may argue that thecontributions of network analysis have so far been relativelymodest. Indeed, most of these contributions have been related topointing out network properties that make certain proteins morelikely to be good targets [8], for example connector versus non-connector enzymes [12,13], or central versus peripheral proteins[11]. These contributions notwithstanding, there is little in theform of actual, concrete, examples where network analysis hasresulted in a clear application to the discovery of new drugs or tothe study of the effects of existing drugs.Here we present one such application. In particular, we use theinformation that is encoded in networks of reported druginteractions to predict uncharacterized interactions. Because themodels we use are abstract, our approach can deal with adverseinteractions as well as synergistic/antagonistic/suppressing inter-actions or any other type of drug interaction. We show that ourmethod is able to accurately predict drug interactions, and that itcan be used efficiently to discover interactions of new drugs as partof the drug discovery process.
Results
A network approach for the inference of unknown druginteractions
For specific drug pairs, interactions can be predicted in silico from mechanistic or flux balance models of the pathways and rocesses in which their targets are involved [6,14]. However, thisapproach is difficult to generalize and is, therefore, inappropriatefor large-scale identification of interactions and for the identifica-tion of interactions between drugs whose mechanisms are not fullyunderstood. Another approach is to use statistical models based onmolecular and pharmacological data [15] but, again, such data isnot always available. Finally, there are mechanism-independentmethods to predict multidrug interactions based on maximumentropy approaches, but these require knowledge of pairinteractions [16], which is what we aim to uncover here.As in other biological problems, network theory [11,17,18]provides a useful, although abstract, alternative to mechanistic andmolecular modeling. In a network representation of druginteractions, each node represents a drug and each link representsan interaction between the corresponding pair of drugs. Interac-tions of different types (for example, synergistic versus antagonistic)are represented by links of different types (Fig. 1A).Drug interaction networks contain explicit information aboutthe interactions that are known, but also about implicitinformation about interactions that have never been tested; thequestion we are concerned with is how to extract this informationfrom the network. Here, we present a network-based approach topredict an interaction r ij between drugs i and j from a network N O of known drug interactions (which includes i and j but no explicitinformation about their interaction r ij ). Our approach dealsrigorously with the information contained in the network bymeans of Bayesian model averaging [19] (Methods). The approachis completely unsupervised and parameter-free.Within our Bayesian model averaging approach, the onlyrelevant modeling question is what family of models can accuratelydescribe the network of drug interactions. In this regard, it is wellestablished that pairwise drug interactions are largely determinedby the cellular functions targeted by the drugs [20–22]. In networkterms, this means that the interaction r ij is determined by thecellular functions s i and s j of i and j , respectively; in other words,nodes can be partitioned into groups (by cellular function) suchthat the interaction between any pair of nodes depends only on thegroups to which they belong (Fig. 1B–C). Stochastic block modelsare a family of network models that mathematically formalize the idea of group-dependent interactions [23–25]. Although originallyproposed in the context of social interactions, stochastic blockmodels are increasingly used to describe the structure of complexnetworks in general [19,26] and for network inference [19](Methods). Again, after this choice of plausible models theresulting algorithm is completely unsupervised and parameter-free (Methods).To benchmark the performance of our algorithm, we considertwo alternative heuristic approaches. The first benchmark is basedon the idea that similar drugs have similar interactions. In thisspirit, we set r ij ~ r i ’ j ’ , where i ’ (respectively, j ’ ) is a drug whoseknown interactions are as similar as possible to those of i ( j ), and r i ’ j ’ is a known interaction (Methods). Second, we consider anapproach based on the Prism algorithm, which was developed toidentify groups of drugs (or genes) with similar interactions to otherdrugs [20,27]. Instead of averaging over all possible partitions ofdrugs into groups as done in our Bayesian model averagingapproach, we take the partition proposed by Prism and use thatpartition to make the prediction (Methods).Additionally, we consider as a baseline the simplest possiblealgorithm for predicting r ij , which is to use the overall rate of eachinteraction type in the network. For example, if 60% of knowninteractions in a network are synergistic ( S ) and 40% areantagonistic ( A ), then we set r ij ~ S with 60% probability and r ij ~ A with 40% probability. This baseline captures the fact that itis harder to make a prediction when the ratio of S = A interactionsis 60/40 than when the ratio is, for example, 95/5. Figure 1. Stochastic block models for the prediction ofunknown drug interactions. ( A ) Consider a hypothetical situationin which all of the interactions between drugs A { H are known withthe exception of the interaction between B and G , which is, in reality,antagonistic. There are many partitions of the drugs into groups. Thepartition in ( B ) has high explanatory power (low value of H ( P ) in Eqs.(5) and (6)), since most drug interactions between a pair of groups areof the same type. Therefore, the predictions of this partition have alarge contribution to the estimation of the probability of theunknown interaction. Conversely, the partition depicted in ( C ) haslittle explanatory power (high value of H ( P ) ) and has a smallcontribution to the estimation of the probability of the unknowninteraction.doi:10.1371/journal.pcbi.1003374.g001 Author Summary
Over one in four adults older than 57 in the US take five ormore prescriptions at the same time; as many as 4% are atrisk of a major adverse drug-drug interaction. Potentiallybeneficial effects of drug combinations, on the other hand,are also important. For example, combinations of drugswith synergistic effects increase the efficacy of treatmentsand reduce side effects; and suppressing interactionsbetween drugs, in which one drug inhibits the action ofthe other, have been found to be effective in the fightagainst antibiotic-resistant pathogens. With thousands ofdrugs in the market, and hundreds or thousands beingtested and developed, it is clear that we cannot rely onlyon experimental assays, or even mechanistic pharmaco-logical models, to uncover new interactions. Here wepresent an algorithm that is able to predict suchinteractions. Our algorithm is parameter-free, unsuper-vised, and takes, as its only input, sets of previouslyreported interactions. We show that our method is able toaccurately predict interactions, even in large-scale data-bases containing thousands of drugs, and that it can beused efficiently to discover interactions of new drugs aspart of the drug discovery process. alidation on exhaustive pairwise interaction data
We start by testing the algorithms described above against twoexperiments in which all pairwise interactions between a small setof drugs were exhaustively tested [20,28]. In the first experiment,Yeh and coworkers tested the effect of all pairwise combinations of21 antibiotics on
E. coli ’s growth [20]. They classified eachinteraction as synergistic, additive, antagonistic or suppressing. Inthe second experiment, Cokol and coworkers studied the effect ofall pairwise combinations of 13 anti-fungal drugs on the growth of
S. cerevisiae [28]. They classified interactions as synergistic, additiveor antagonistic (except for some interactions that were unresolved).To study the performance of the algorithms, we simulatesituations in which not all pairwise interactions are known. Inparticular, we simulate a situation in which only a fraction of allinteractions are observed, and then try to predict the unobservedinteractions (repeated random sub-sampling validation). In eachcase, we measure the fraction of predictions that are exactlycorrect (exact classification), as well as the fraction of predictionsthat deviate from the experimental observation by at most one level ( + classification). For example, miss-predicting a synergisticinteraction as additive is considered correct by the + classifica-tion metric, but miss-predicting a synergistic as antagonistic orsuppressing (or vice versa), or an additive as suppressing (or viceversa) is considered incorrect.In Fig. 2 we show the results of the validation. As expected, thestochastic block model, the neighbor-based and the Prism-baedpredictions have accuracies well above the baseline, even when asmany as 80% of the interactions are unobserved. In the majority ofcases, the stochastic block model is significantly and consistentlymore accurate than the neighbor-based and the Prism-basedpredictions with one exception: when the fraction of observedinteractions is high ( § %) in the Cokol dataset, in which theneighbor-based prediction is best. Note that as soon as the numberof interaction types grows (from 3 in Cokol to 4 in Yeh) or thefraction of observed interactions decreases, the stochastic blockmodel becomes more accurate. Moreover, even when theneighbor-based exact predictions are more accurate, Validation on evolving databases of drug interactions
Next, we test our algorithm against the existence of adversedrug interactions in two drug interaction databases: the databaseavailable through the web site Drugs.com and the DrugBankdatabase [29,30]. For the Drugs.com database, we restrict ouranalysis to major adverse interactions between generic drugs; forthe DrugBank, we consider all interactions.We consider two snapshots of each of the databases. For theDrugs.com database, we collected the first snapshot in May 10,2010, and the second one in February 22, 2012. A total of 1,518drugs are listed in both snapshots. There are 32,074 druginteractions present in both instances of the network; n N ~ novel interactions present in the 2012 dataset but not in the 2010dataset, and n S ~ spurious interactions present in the 2010dataset but not present in the 2012 dataset. For the DrugBankdataset, the first snapshot corresponds to January 2009, and thesecond to April 2012. A total of 1,012 drugs are listed in bothsnapshots; there are 9,113 drug interactions present in bothinstances of the network, with n N ~ and n S ~ .We evaluate to what extent could our network algorithms havepredicted which interactions needed to be added to each of thefirst snapshots (that is, to what extent can the algorithms uncovernovel interactions), and which ones needed to be removed (that is,to what extent can they detect spurious interactions). As we showin Fig. 3, the algorithm based on stochastic block models is able toaccurately uncover spurious and, especially, novel interactions. Incontrast, neighbor-based and Prism-based predictions performonly marginally better than the baseline.First, we measure the area under the receiver operatingcharacteristic (AUROC) curve (Fig. 3A–B) [31]. In the case ofuncovering novel interactions, the AUROC gives the probabilitythat an interaction randomly chosen from those that were addedto the first snapshot has a higher score than one randomly chosenfrom the set of interactions that were never added to the network.For the Drugs.com database, we find this probability to be 0.87 forthe stochastic block model, 0.53 for neighbor-based predictions,and 0.52 for Prism-based predictions. For the DrugBank dataset,these probabilities are 0.71, 0.52 and 0.53, respectively.Similarly, when dealing with spurious interactions, the AUROCgives the probability that an interaction randomly chosen fromthose that were removed from the 2010 snapshot has a lower scorethan one randomly chosen from the set of interactions that werenot removed from the network. For the Drugs.com database, wefind this probability to be 0.73 for the stochastic block model, 0.51for neighbor-based predictions, and 0.45 for Prism-based predic-tions. For the DrugBank dataset, these probabilities are 0.61, 0.50and 0.50, respectively.It is also interesting to analyze the sensitivity-specificity curves(Fig. 3C–F). Consider first the results for the Drugs.com database.For the most pressing case of uncovering previously unreportedmajor drug interactions (Fig. 3C), we find that at 95% sensitivity,the stochastic block model has a specificity of 62%, that is, that wecould have built, in 2010, a list of potential interactions containing95% of the interactions that were actually added to the database,and excluding 62% of those that were never added. Conversely, at95% specificity we obtain a sensitivity of 45%, that is, a listcontaining only 5% of the interactions that were never added tothe network would have included close to half of all theinteractions that were actually added to the database. Whileresults for spurious interactions and for the DrugBank dataset aremore modest, our method, unlike the neighbor-based or thePrism-based algorithms, has significant predictive power in all thecases we study. Application to drug discovery
Finally, we demonstrate that our algorithm can be used todiscover interactions of novel drugs as part of the drug discoveryprocess. In particular, consider a lab that has developed a newdrug D which is known to have a harmful interaction with anotherdrug H . Ideally, the lab wants to identify all other drugs f H , H , . . . H m g that also have harmful interactions with D . Sincein principle there are as many potential interactions as drugs in themarket (more than 1,000, according to the Drugbank andDrugs.com databases), it would be extremely costly to test allpossible interactions experimentally. Considering that the typicaldrug interacts with approximately 20–40 other drugs (in rugBank and Drugs.com, respectively), random testing forinteractions would require 35–55 experiments to uncover a singleharmful interaction.Lacking any knowledge about D (other than its interaction with H ), our algorithm can guide experiments by identifying thosedrugs that are most likely to interact with D . In particular, wecould use the stochastic block model inference approach to predictthe most likely interaction, test it in the lab, and iterate the process adding, at each iteration, whatever interaction information the labassay gave.To test whether such an approach would work in practice, wehave simulated the discovery of two drugs whose interactions arein fact known and reported in the 2012 snapshot of DrugBank—acetophenazine and cinacalcet (these drugs were selectedrandomly among those with 10 to 20 interactions). For each ofthese drugs, we proceed exactly as if no data were available in the Figure 2. Performance of drug interaction inference methods on exhaustive pair interaction data.
We test the algorithms against resultsof two experiments in which all pairwise interactions between a small set of drugs were tested: [28] ( A , C and E ; interactions are synergistic, additiveor antagonistic) and [20] ( B , D and F ; interactions are synergistic, additive, antagonistic or suppressing). We simulate situations in which only afraction f of all interactions are observed, and then try to predict the unobserved interactions (repeated random sub-sampling validation). In eachcase, we measure the fraction of predictions that are exactly correct ( A and B ), as well as the fraction of predictions that deviate from theexperimental observation by at most one level ( C and D ). For example, miss-predicting a synergistic interaction as additive is considered correct bythe + classification metric, but miss-predicting a synergistic interactions as antagonistic or suppressing (or vice versa), or an additive one assuppressing (or vice versa) is considered incorrect. Error bars indicate the standard error of the mean and are usually smaller than the symbols. ( E and F ) Relative improvement of the stochastic block model predictions over the neighbor-based predictions. If c is the frequency of correct classification,we define the relative improvement as ( c SBM { c X ) = ( c X { c B ) atabase except for one seed interaction H , which we alsochoose at random. From the seed interaction and interaction datafor all drugs other than D , we use the stochastic block modelapproach to infer the next most likely interaction of D , check ifthe interaction truly exist, add this information to the network,and iterate.As we show in Fig. 4, the results are very promising. Foracetophenazine, the 16 iterations we carry out are enough todiscover 11 of the 15 interactions that are reported in DrugBank.For cinacalcet, we are able to uncover 8 of the 12 reportedinteractions. As mentioned above, these numbers need to be compared with the approximately 55 experiments that would benecessary to uncover a single interaction without any guidance. Discussion
There is a pressing need to elucidate and understandinteractions between drugs. With thousands of drugs in themarket, and hundreds or thousands being tested and developed, itis clear that we cannot rely only on experimental assays to uncoverinteractions. Therefore, we need to develop computational data-mining methods to guide experimental analysis.There are many possible approaches to predict drug interac-tions computationally. One is to mine patient data that arecollected as part of post-marketing surveillance. However, this
Figure 3. Performance of drug interaction inference methodson an evolving database of major adverse drug interactions.
Left: Drugs.com database; right: DrugBank dataset. (
A–B ) Area underthe receiver operating characteristic (AUROC) curve. For novelinteractions the AUROC gives the probability that an interactionrandomly chosen from those that were added to the first snapshot hasa higher score than one randomly chosen from the set of interactionsthat were never added to the network. Similarly, for spuriousinteractions the AUROC gives the probability that an interactionrandomly chosen from those that were removed from the first snapshothas a lower score than one randomly chosen from the set ofinteractions that were not removed from the network. (
C–F )Sensitivity-specificity curves for novel (
C–D ) and spurious interactions(
E–F ). Sensitivity is defined as the ratio of true positives to all realpositives (true positives plus false negatives). Specificity is defined asthe ratio of true negatives to all real negatives (true negatives plus falsepositives).doi:10.1371/journal.pcbi.1003374.g003
Figure 4. Inference of drug interactions as part of the process ofdrug discovery and development.
For each of the two drugs (( A )acetophenazine and ( B pproach is problematic because of confounding factors that maynot be properly accounted for in existing reporting systems [32].Another approach is to use models based on molecular andpharmacological data [15].Our approach is complementary to these efforts, and exploitsthe information that is encoded in the network of known druginteractions—since known interactions are the result of certain(known or unknown) ‘‘pharmacological rules’’, we can infer‘‘rules’’ from known interactions and then use the inferred ‘‘rules’’to, in turn, predict unreported interactions (as we show in theSupporting Text S1 and Fig. S2, the inferred ‘‘rules’’ correlatestrongly with drug structure, category and target). Although thenetwork approach has been frequently invoked as a new paradigmin pharmacology [7,8] and there are large-scale databases thatcompile and report drug interactions [10,30], this is, to the best ofour knowledge, the first attempt to use network inference topredict drug interactions.The network inference algorithm we have presented is veryabstract and does not take into consideration any informationother than reported interactions. It may be necessary in the futureto complement the method with chemical, biological and/orpharmacological information. However, one advantage of ourabstract approach is that, precisely because it is abstract, it can beeasily extended to other kinds of pharmacological interaction datathat can be represented as networks. For example, it isstraightforward to extend our approach to predict associationsbetween drugs and adverse side effects from pharmacosafety networks [33], protein- and target-drug interactions [34,35], or associationsbetween drugs and therapies [15] and drugs and diseases [36],which may help to guide drug repositioning. Our approach caneven be used to predict gene-disease associations [37] and,therefore, to uncover novel targets.Another interesting extension of our approach is to predictmultidrug interactions (that is, interactions between groups ofthree or more drugs), which are relevant to cancer treatmentamong others. Although it seems that knowledge of pairinteractions may be enough to describe higher-order interactions[16], within our framework tertiary interactions could also bemodeled using three-dimensional stochastic block models in whichthe probability Q ( s A , s B , s C ) that three drugs A , B and C interactdepends only on the groups s A , s B and s C to which they belong.The generalization to interactions between any number of drugs isstraightforward. All in all, we think that our approach opens thedoor to new ways of looking at and making predictions frompharmacological networks. Methods
Dataset collection
For the Yeh et al. dataset, we collected the data on pairwisecombinations of 21 antibiotics from Figs. 3 and 4a of [20]. For theCokol et al.
Estimation of link type probability using stochastic blockmodels
The fundamental assumption of our approach is that thestructure of the drug interaction network can be satisfactorilyaccounted for by a model M , which is unknown but belongs to afamily M of models, that is, a group of models that can beparametrized in some consistent way. Then, the probability that r ij ~ R given the observed network N O is [19] p ( r ij ~ R D N O ) ~ ð M dM p ( r ij ~ R D M ) p ( M D N O ), ð Þ To estimate this integral we rewrite it, using Bayes theorem, as[19,38] p ( r ij ~ R D N O ) ~ Ð M dM p ( r ij ~ R D M ) p ( N O D M ) p ( M ) Ð M dM p ( N O D M ) p ( M ) : ð Þ Here, p ( N O D M ) is the probability of the observed interactionsgiven a model and p ( M ) is the a priori probability of a model,which we assume to be model-independent p ( M ) ~ const .For the family of stochastic block models, each model M ~ ( P , Q , . . . , Q K ) is completely determined by a partition P of drugs into groups and the group-to-group interaction proba-bility matrices Q R . Here, K is the total number of interaction types(for example, if interactions can be synergistic, additive orantagonistic, then K ~ ) and, for a given partition P , the matrixelement Q R ( a , b ) is the probability that a drug in group a and adrug in group b interact with each other (these matrices verify that P r Q r ( a , b ) ~ for all pairs of groups ( a , b ) ). Thus, if i belongs togroup s i and j to group s j we have that [38] p ( r ij ~ R D M ) ~ Q R ( s i , s j ) ; ð Þ and p ( N O D M ) ~ P a ƒ b P r Q r ( a , b ) nr ( a , b ) , ð Þ where n r ( a , b ) is the number of interactions of type r between druggroups a and b .The integral over all models in M can be separated into a sumover all possible partitions of the drugs into groups, and an integralover all possible values of each Q r ( a , b ) . Using this together withEqs. (2), (3) and (4), and under the assumption of no priorknowledge about the models ( p ( M ) ~ const : ), we have p ( r ij ~ R j R O ) ~ Z X P ð S d Q . . . ð S d Q K Q R ( s i , s j ) P a ƒ b P r Q r ( a , b ) nr ( a , b ) ; ð Þ where the integral is over all Q r ( a , b ) within the subspace S thatsatisfies the normalization constraints P r Q r ( a , b ) ~ , and Z is thenormalizing constant (or partition function). These integralsfactorize into terms corresponding to all pairs ( a , b ) [38], eachwith the general form dQ ( Q ) n ð { Q dQ ( Q ) n (cid:2) (cid:2) (cid:2) ð { Q { ... { QK { dQ K { ( Q K { ) nK { (1 { Q { . . . { Q K { ) nK ~ n ! n ! . . . n K ! ( n z n z . . . z n K z K { ! : Using these expressions in Eq. (5), one obtains p ( r ij ~ R D N O ) ~ Z X P n R ( s i , s j ) z n ( s i , s j ) z K (cid:2) (cid:3) exp( { H ( P )), ð Þ where the sum is over all partitions of the drugs, n ( s i , s j ) ~ P r n r ( s i , s j ) is the total number of known interactionsbetween groups s i and s j , and H ( P ) is a function that depends onthe partition only H ( P ) ~ X a ƒ b ln( n ( a , b ) z K { ! { X Kr ~ ln n r ( a , b ) ð Þ ! " : ð Þ This sum can be estimated using the Metropolis algorithm [19,39]as detailed next.
Implementation details
The sum in Eq. (6) cannot be computed exactly because thenumber of possible partitions is combinatorially large, but can beestimated using the Metropolis algorithm [19,39]. This amounts togenerating a sequence of partitions in the following way. From thecurrent partition P , select a random drug and move it to arandom new group giving a new partition P . If H ( P ) v H ( P ) ,always accept the move; otherwise, accept the move only withprobability e H ( P ) { H ( P ) .By doing this, one gets a sequence of partitions f P i g such that[39] p ( r ij ~ R D N O ) & N X P [ f Pi g n R ( s i , s j ) z n ( s i , s j ) z K , ð Þ where N is the number of samples in f P i g .In practice, it is useful to ‘‘thin’’ the sample f P i g , that is, toconsider only a small fraction of evenly spaced partitions so as toavoid the computational cost of sampling very similar partitionswhich provide very little additional information. Moreover, oneneeds to make sure that sampling starts only when the sampler is‘‘thermalized’’, that is, when sampled partitions are drawn fromthe desired probability distribution (which in our case is given by e { H ( P ) = Z ). Our implementation automatically determines areasonable thinning of the sample, and only starts sampling whencertain thermalization conditions are met. Therefore, the wholeprocess is completely unsupervised. The source code of our imple-mentation of the algorithm is publicly available from http://seeslab.info/downloads/drugraph/ and http://github.com/seeslab/drugraph.As often happens in Metropolis sampling, in general it is betterto run many short independent sampling processes that a singlevery long sampler. Results reported here are obtained using 50independent sampling processes of 200 (conveniently thinned)partitions each. These sampling processes can be run in parallel, taking on the order of 1–2 days to complete on high-end CPUs forthe largest network considered here (with over 1,500 drugs).Sampling an equivalent 10,000 partitions with a single run cantake 2–3 weeks. Prism-based prediction of interactions
The Prism algorithm [27] was originally developed to identifygroups of genes that interact monochromatically, that is, that havethe same type of interactions with genes in other groups. Yeh andcoworkers then introduced Prism II [20] to identify groups ofdrugs relaxing the requirement for perfect monochromaticity.Our implementation of Prism II is as follows. Each drug isinitially placed in a group by itself. Then, groups are sequentiallymerged until all drugs belong to a single group. At each step, wemerge the two groups with the smallest ‘‘distance’’ to each other.The distance F ab between groups a and b is F ab ~ min i [ a , j [ b f d ij g { T D S ab : ð Þ Here, the normalized drug-drug distance d ij between drugs i and j is d ij ~ K { ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP k r ik { r jk (cid:5) (cid:6) N ij s , ð Þ with N ij the number of interactions reported for both i and j . Thechange of monochromaticity entropy D S ab is D S ab ~ S ( m ab z m aa z m bb ) { S ( m ab ) { S ( m aa ) { S ( m bb ) z X s = ab S ( m as z m bs ) { S ( m as ) { S ( m bs ) (cid:7) (cid:8) ð Þ where m ab ~ ( m { ab , m z ab ) is a vector with the number of synergistic( ) and antagonistic ( + ) interactions between groups a and b , and S ( m ) ~ m z z m { M z z M { ( p z log p z z p { log p { ) ð Þ with p z = { ~ m z = { = ( m z z m { ) and M z = { ~ P ab m z = { ab : By itself, the Prism II algorithm returns a tree of nested druggroupings. To make interaction predictions, we need to: (i) set thefree parameter T ; (ii) cut the tree at a certain level to get a singlepartition of the drugs into groups (a process that needs to beunsupervised); and (iii) given those groups, determine theprobability of each type of interaction. To cut the tree, we choosethe partition with the smallest number of groups among those withtotal monochromaticity entropy S ~ P a , b S ( m ab ) that satisfies S v : S m ax , where S m ax is the partition that corresponds toputting all drugs in a single group. Additionally, we set T ~ toget results consistent with those reported in Ref. [20] (we alsochecked that these parameters yield good results for the Cokoldataset, and that the results do not improve using other values of T ; see Supporting Text S1 and Fig. S1).Finally, once the groups are defined, we estimate the probability p Prism ( r ij ~ R D N O ) as Prism ( r ij ~ R D N O ) ~ n R ( s i , s j ) n ( s i , s j ) , ð Þ where n R ( s i , s j ) and n ( s i , s j ) are defined as above.With our implementation, the Prism-based algorithm takes 1–2days on high-end CPUs to generate interaction predictions for thelarge networks considered here (with over 1,000 drugs). Neighbor-based prediction of interactions
Given a network of drug interactions, we define the interactionsimilarity s ik between drugs i and k as the fraction of interactionswith other drugs that are equal for i and k , over the total numberof interactions that are reported for both drugs. In particular s ii ~ , and s ik ~ if two drugs do not have any equal interactionwith others.To predict the interaction r ij between drugs i and j , we order allpossible drug pairs ( k i , k j ) by decreasing value of the product ofsimilarities to the query drugs s ik i | s jk j . We then select the pair ( k i ~ i ’ , k j ~ j ’ ) with the highest product for which the interaction r i ’ j ’ is known, and use that value as our prediction of r ij . Note thatwe may have i ’ ~ i , that is, we may use the known interactionbetween i and a drug j ’ that is very similar to j to predict r ij . Supporting Information
Figure S1
The accuracy of the Prism-based method, asmeasured by the AUROC, does not improve consistently andsignificantly by choosing values of T other than T ~ , as used inthe main text.(TIFF) Figure S2
Drug groups and drug mechanisms of action fromstochastic block models. For each drug pair in the Cokol et al. dataset (
A–B ), the Yeh et al. dataset (
C–D ) and the DrugBank 2012snapshot ( E ), we calculate the probability that any two drugsbelong to the same drug group (see Section 2). We call thisprobability the co-classification probability. ( A ) and ( C ) The matrix of co-classification probabilities for the Cokol et al. dataset ( A ), andYeh et al. dataset ( C ), ordered so that large co-classificationprobabilities appear close to the diagonal [40]. Dashed lines are aguide to the eye. The mechanism of action of each drug isindicated by color bars on top of drug abbreviations (( A ) Cyan:ergosterol metabolism; dark red: acting on serine/threonine; otherdrugs were intentionally selected with different targets andmechanisms of action. ( C ) Dark red: protein synthesis, 30S; cyan:protein synthesis, 50S; red: folic acid biosynthesis; pink: DNAgyrase; dark blue: cell wall; yellow: aminoglycoside, proteinsynthesis, 30S). Co-classification boxes correspond, to a largeextent, to mechanisms of action. ( B ) and ( D ) The reported druginteractions show clear patterns once they are ordered accordingto the co-classification probability. For example in the Yeh et al. dataset, most interactions between the group f AMK,STR,TOB g and the group f TMP,SLF,NIT,CPR,LOM g are synergistic. ( E )We use information in DrugBank to analyze the overlap (orfunctional similarity) in substructure, category and target betweenpairs of drugs (see Section 3). We plot these quantities as a functionof the co-classification probability of the corresponding drug pairs(we average over drug pairs with similar co-classificationprobability; error bars represent the standard error of the meanand are generally smaller than the symbols). Drugs with higher co-classification probability are significantly more likely to sharesubstructures, categories and targets.(TIFF) Text S1
Sensitivity analysis for the Prism-based algorithm.Discussion on drug groups and drug mechanisms. Discussion onthe estimation of the co-classification probability using stochasticblock models. Analysis of drug similarities for drugs in theDrugBank.(PDF)
Author Contributions
Conceived and designed the experiments: RG MSP. Performed theexperiments: RG. Analyzed the data: RG MSP. Contributed reagents/materials/analysis tools: RG MSP. Wrote the paper: RG MSP.
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