Prediction of drug cocktail effects when the number of measurements is limited

16 Pages·2017·4.04 MB·English

METHODSANDRESOURCES
Prediction of drug cocktail effects when the
number of measurements is limited
AnatZimmer,AvichaiTendler,ItayKatzir,AviMayo,UriAlon*
DepartmentofMolecularCellBiology,WeizmannInstituteofScience,Rehovot,Israel
*[email protected]
Abstract
a1111111111
a1111111111
a1111111111 Cocktailsofdrugscanbemoreeffectivethansingledrugs,becausetheycanpotentially
a1111111111 workatlowerdosesandavoidresistance.However,itisimpossibletotestalldrugcocktails
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drawnfromalargesetofdrugsbecauseofthehugenumberofcombinations.Toovercome
thiscombinatorialexplosionproblem,onecansamplearelativelysmallnumberofcombina-
tionsanduseamodeltopredicttherest.Recently,ZimmerandKatziretal.presenteda
modelthataccuratelypredictedtheeffectsofcocktailsatalldosesbasedonmeasuring
OPENACCESS pairsofdrugs.Thismodelrequiresmeasuringeachpairatseveraldifferentdosesanduses
Citation:ZimmerA,TendlerA,KatzirI,MayoA, interpolationtoreduceexperimentalnoise.However,often,itisnotpossibletomeasure
AlonU(2017)Predictionofdrugcocktaileffects eachpairatmultipledoses(forexample,inscarcepatient-derivedtumormaterialorinlarge
whenthenumberofmeasurementsislimited.
screens).Here,weaskwhethermeasurementsatonlyasingledosecanalsopredicthigh-
PLoSBiol15(10):e2002518.https://doi.org/
orderdrugcocktails.Toaddressthis,wepresentafullyfactorialexperimentaldatasetonall
10.1371/journal.pbio.2002518
drugcocktailsbuiltof6chemotherapydrugson2cancercelllines.Wedevelopaformula
AcademicEditor:SuiHuang,InstituteforSystems
thatusesonlypairmeasurementsatasingledosetopredictmuchofthevariationupto6-
Biology,UnitedStatesofAmerica
drugcocktailsinthepresentdata,outperformingcommonlyusedBlissindependenceand
Received:March8,2017
regressionapproaches.Thismodel,calledthepairsmodel,isanextensionoftheBlissinde-
Accepted:October10,2017
pendencemodeltopairs:ForMdrugs,itequalstheproductofallpaireffectstothepower1/
Published:October26,2017 (M−1).Thepairsmodelalsoshowsgoodagreementwithpreviouslypublisheddataonanti-
Copyright:©2017Zimmeretal.Thisisanopen biotictripletsandquadruplets.Thepresentmodelcanonlypredictcombinationsatthe
accessarticledistributedunderthetermsofthe samedosesinwhichthepairsweremeasuredandisnotabletopredicteffectsatother
CreativeCommonsAttributionLicense,which
doses.Thisstudyindicatesthatpair-basedapproachesmightbeabletousefullypredictand
permitsunrestricteduse,distribution,and
prioritizehigh-ordercombinations,eveninlargescreensorwhenmaterialfortestingis
reproductioninanymedium,providedtheoriginal
authorandsourcearecredited. limited.
DataAvailabilityStatement:Allrelevantdataare
withinthepaperanditsSupportingInformation
files.
Funding:EuropeanResearchCouncilExecutive Authorsummary
Agencyhttps://ec.europa.eu(grantnumber
Drugcocktailsareapromisingstrategyfordiseasessuchascancerandinfections,because
693436)receivedbyUA.Thefunderhadnorolein
studydesign,datacollectionandanalysis,decision cocktailscanbemoreeffectivethanindividualdrugsandcanovercomeproblemsofdrug
topublish,orpreparationofthemanuscript. resistance.However,findingthebestcocktailcomprisingagivensetofdrugsischalleng-
Minervafoundationhttp://www.minerva.mpg.de/ ingbecausethenumberofexperimentsneededishugeandgrowsexponentiallywiththe
weizmann/(grantnumber7125960101)received
numberofdrugs.Thisproblemisexacerbatedwhentheexperimentsareexpensiveand
byUA.Thefunderhadnoroleinstudydesign,data
thematerialfortestingisrare.Here,wepresentawaytoaddressthischallengeusinga
collectionandanalysis,decisiontopublish,or
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 1/16
Predictingdrugcocktailswithlimitedmeasurements
preparationofthemanuscript.Abisch-Frenkel
ProfessorialChairreceivedbyUA.Thefunderhad mathematicalformula,calledthepairsmodel,thatrequiresrelativelyfewexperimental
noroleinstudydesign,datacollectionand testsinordertoestimatetheeffectsofcocktailsandtopredictwhichcocktailismosteffec-
analysis,decisiontopublish,orpreparationofthe
tive.Theformuladoeswellonexperimentaldatageneratedinthisstudyusingcombina-
manuscript.IsraelSciencefoundationwww.ISF.
tionsofbetween3and6anticancerdrugs,aswellasonexistingdatathatuse
org.il(grantnumber1349/15)receivedbyUA.The
combinationsofantibiotics.
funderhadnoroleinstudydesign,datacollection
andanalysis,decisiontopublish,orpreparationof
themanuscript.
Competinginterests:Theauthorshavedeclared
Introduction
thatnocompetinginterestsexist.
Abbreviations:CbPt,Carboplatin;CisPt,Cisplatin; Cancerandantibiotictreatmentsfacetheproblemofdrugresistance.Cancerdrugsalsoface
CPT,Camptothecin;Etopo,Etoposide;LD20,lethal problemsofefficacyatthelowdosesneededtotoleratesideeffects[1–5].Onestrategytoover-
dose20%;NCZ,Nocodazole. comethesechallengesiscocktailsofseveraldrugs[5–9].Cocktailsovercamethechallengesof
resistanceandefficacyindiseasessuchasHIV[10–12]andareusedindiversemedicalcon-
texts.Effectivecocktailscan,inprinciple,bedesignedforeachindividualpatient[13].
Currently,thereareapproximately1,000availablecompoundstotreatcancer[14,15].Test-
ingallcombinationsatalldosesisimpossible,becausethenumberofexperimentsgrowsexpo-
nentiallywiththenumberofdrugsanddoses.Hence,veryeffectivecocktailsmaybehiddenin
thisvastspaceofpossiblecombinations[1,6,16,17],asrecentlydemonstratedbyHornetal.in
anextensivestudyofcolorectalcancer[5].
Currentapproachestoovercomethecombinatorialexplosionproblemofdrugcombina-
tionsusemathematicalmodelstopredicttheeffectsofcombinationsbasedonasmallnumber
ofmeasurements[18–20].ThecommonlyusedBlissmodelisareasonablefirstapproximation
butdoesnotincludesynergyandantagonismeffects.Machinelearningapproacheshavebeen
proposed,butthelargedatasetsneededtotrainthesemodelsarestillnotwidelyavailable[21–
26].TheIsserlismodelproposedbyWoodetal.showedexcellentresultsonantibioticcombi-
nationsof3and4drugs[27–29].Additionalstudiesconsideredtheinteractionsofmultiple
drugsusingfirst-andsecond-ordertermsfordrugeffects[30–38].
Recently,ZimmerandKatziretal.suggestedamodelthataccuratelypredictedthemulti-
doseresponseoftripletsandquadrupletsofantibioticsandcancerdrugs[29].Thismodel
requiresmeasurementsofeachpairofdrugsatafewdosecombinations.Itusesthismultidose
informationtofitasmoothdrugresponsecurveforalldoses,inwhicheachdrugchangesthe
effectivedoseoftheotherdrugs.Thismodelgreatlyreducesthenumberofexperiments
neededinordertoscanthespaceofdrugcocktailsatmultipledoses.
Dataatmultipledosesrequiredforthemultidosemodelof[29]isnotalwaysavailable.This
factismostpressingwhenthemeasurementsaremadeonrarematerialswhoselimitedquan-
tityallowsonlyafewtests,suchaspatient-derivedmaterialinindividualizedmedicineapplica-
tions[39–42]orinexpensivehigh-throughputscreens[43–45].Thereisthereforeneedfora
modelthatusesmeasurementsatasingledoseinordertopredicttheeffectsofhigh-order
combinations(suchamodelisexpectedtoworkonlyatthemeasureddosesandnoother
doses).Inparticular,previousmodelsweretestedonlyuptotripletsandquadrupletsofdrugs.
Experimentsonhigher-ordercombinationsarehenceofinterest[5].
Toaddressthis,wetestedallcombinationsof6chemotherapydrugsatasingledoseon2
celllines(afullyfactorialdesign).Wefindthatsynergyandantagonismarecell-linedependent
andareusuallyconsistentwiththesynergy/antagonismofthepairsthatmakeupeachcombi-
nation.Wedevelopedasimplemodelforcocktailsthatisinsensitivetoexperimentalnoise
andthatusesonlymeasurementsondrugpairsatasingledose.Inadditiontothe6-drugcom-
binations,wefurthertestedthemodelonpreviousfully-factorialdatasetsof3chemotherapy
drugsat8doses[29]andof3or4antibiotics[28],totaling1,392additionaltripletsand248
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 2/16
Predictingdrugcocktailswithlimitedmeasurements
quadruplets.Thepairsmodelpredictswelltheeffectofthesecocktails,withthelimitationthat
itcanonlypredicteffectsatthesamedoseatwhichthepairsweremeasured.
Results
Fullyfactorialexperimentson6drugsand2celllines
Weselected6cytotoxicdrugsfromseveralclinicallyemployeddrugfamilieswithdifferent
mechanismsofaction(Table1).Wechosedrugsthatareusedincombinationinsomeclinical
settings,suchasanalkylatingagentwithamicrotubulepoison.Foreachdrug,wemeasuredcell
survivalafter48husingtheneutralredassay.Asecondsurvivalassay,MTT,gaveverysimilar
results(S1Fig).Eachmeasurementwasrepeatedonatleast6biologicalrepeats,intriplicate.
Werepeatedthisfor2humancancercelllinesfromdifferentcancersandgeneticback-
grounds,H1299andHeLa.Thenon-small-celllungcarcinomalineH1299wasderivedfrom
metastaticlymphnodefromamalepatientandcharacterizedbypartialhomozygousdeletion
inthep53gene,resultinginlackofexpressionofthep53protein[46].HeLaisacervicalcancer
cellline.HeLacellsexpressp53atlowlevelsandwerederivedfromaprimarytumorfroma
femalepatient[47].
Wefirstmeasuredsingle-drugdoseresponsecurves,whichwerewelldescribedbyHill
functionswithmoderatecooperativity(Table1).Foreachdrug,wechoseadosethatshows
about80%survival,lethaldose20%(LD20;wechoseadoseofhighsurvivalinordertohave
sensitivemeasurementswhencombining6ofthedrugs).Wethenmeasuredall63combina-
tionsofthesixdrugs:6singles,15pairs,20triplets,15quadruplets,6quintuplets,andthesex-
tuplet.InthecaseofHeLacells,17ofthecombinationshadlargevariationsbetweenrepeats
andwerenotincluded.TheresultsforallmeasureddrugcombinationsareshowninTable2
(andS1Table),aswellasinteractionsandtheirconfidenceintervalsbasedonbootstrapping
thebiologicalrepeats.
Somecombinationsshowedsynergisminbothcelllines,andothers
werecell-linespecific
Wemeasuredtheinteractionsbetweenthedrugsbycomparingtheeffectsofthecombination
toBlissindependence.Theeffectofdrugi,g,isthefractionalreductionincellsurvival.In
i
othercontexts,g isthefractionalreductionincellgrowthrate[28,29,48]orothermeasuresof
i
drugimpact.Blissindependenceisamodelinwhichthecombinationeffectistheproductof
theeffectsofeachindividualdrug
gBliss ¼g g ...g ð1Þ
1::k 1 2 k
WequantifiedtheinteractionbythelogarithmicdeviationfromtheBlissmodel,
I=log(1+g −gBliss )[49](forotherapproachestocalculatesynergy,see[5]).Negative
1..k 1..k
numbersindicatesynergy,wheredrugskillmorecellsthanBlissindependence,andpositive
numbersindicateantagonism.Wecomputed95%confidenceintervalsforeachinteraction
termbybootstrappingoverthebiologicalrepeatsofeachmeasurement.Weconsiderinter-
actiontermsasnon-zerowhenI=0isnotincludedintheir95%confidenceinterval.
Wefindthat30%ofthecombinationsaresynergisticforHeLacellsand44%forH1299(S2
Fig).ThemostsynergisticcombinationforH1299wasCisplatin,Carboplatin,andNocodazole
(CisPt+CbPt+NCZ),withI=−0.36.ForHeLa,themostsynergisticcombinationwasCamp-
tothecin,Carboplatin,andEtoposide(CPT+CbPt+Etopo)withI=−0.14.
Wecomparedthesynergy/antagonismofeachcombinationbetweenthe2celllines.Fig1
(andS1Table)showseachcombinationplottedbyitsinteractioninHeLaversusH1299.
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 3/16
Predictingdrugcocktailswithlimitedmeasurements
Table1. Cytotoxicdrugsandtheirmechanismsofaction,LD20concentrations,andHillcoefficients. LD20concentrationsarethedrugconcentra-
tionsinμMthatkill20%ofthecells,andnistheHillcoefficientofthedoseresponsecurves.
H1299 HeLa
Drugname Abbreviation Target Targetanddamagemode LD20 n LD20 n
[uM] [uM]
Camptothecin CPT Topoisomerase TopoisomeraseIpoison,thatbindsandstabilizesthetopoisomerase 0.1 0.5 0.15 0.3
I I-DNAcomplex,causingDNAdamageandcelldeath.
Cisplatinum CisPt DNA CrosslinksDNAinseveraldifferentways,makingitimpossiblefor 13.8 2.2 7 2.1
rapidlydividingcellstoduplicatetheirDNAformitosisandleadingto
DNAdamageandcelldeath.
Nocodazole NCZ Tubulin Anantimitoticdrugthatbindstubulinathighaffinityandinhibits 50 0.6 0.01 0.95
microtubuleassemblythuspromotingmicrotubuledepolymerization.
Etoposide Etopo Topoisomerase BindstoandinhibitstopoisomeraseII,resultingintheaccumulation 10 0.85 6 0.5
II ofsingle-ordouble-strandDNAbreaks.Thisresultsintheinhibition
ofDNAreplicationandtranscriptionandapoptoticcelldeath.
Carboplatin CbPT DNA Thoughttoactsimilarlytocisplatinumthoughwithreducedside 200 2.2 165 1.6
effects.
Proteasome MG132 Proteasome Proteasomeinhibitor. 0.15 1.5 0.38 1.2
inhibitor
Abbreviations:CbPt,Carboplatin;CisPt,Cisplatin;CPT,Camptothecin;Etopo,Etoposide;LD20,lethaldose20%;NCZ,Nocodazole
https://doi.org/10.1371/journal.pbio.2002518.t001
Overall,thereisonlyamoderatecorrespondencebetweentheinteractionsinthe2celllines
(R=0.16).Thehigherthecombinationorder,thesmallerthecorrelationbetweenthetwocell
lines:Two-druginteractions(circles)tendtobemoresimilarbetweenthe2celllines(R=0.5)
thantriplets(triangles,R=0.1)andquadruplets(squares,R=0.08).Thisindicatesthatcell-
line-specificpredictionsareneeded,especiallyforthehigh-ordercocktailsinthissample,
althoughcareisneededwiththisinterpretationduetosmallnumbereffectsandnoise.
Thesynergy/antagonismsignofacombinationinacelllineisoften
consistentwiththatofitsconstituentpairs
Toaddressthepossibilityofusingpairstomodelthecocktails,weaskedwhetherthesynergy/
antagonismsignofpairsinacombinationisinformativewithregardstotheoverallsynergy/
antagonismsignofthecocktailinagivencellline.Forthispurpose,wecomparedtheinterac-
tion(I)valuesofeachcombinationofdrugstotheinteractionofitsconstituentpairs.The
resultsfortripletsareshowninFig2(andS1Table).
Anindicationofhigh-orderinteractionsiswhenthesignsofthepairinteractionsdonot
correspondtotheinteractionsignofthecombination.Suchhigh-orderinteractionswould
makeitdifficulttousepairstomodelhigher-ordercombinations,especiallyiftheinteraction
signofallpairsisoppositetothatofthecombination.WefindthatforHeLacellsallofthe
cocktails(of3–6drugs)thatareantagonistichadatleast1antagonisticpair(16/16),andallthe
synergisticcocktailshadatleast1synergisticpair(10/10).ForH1299,alloftheantagonistic
cocktailshadatleast1antagonisticpair(7/7),andallbut3ofthesynergisticcombinationshad
atleast1synergisticpair(19/22).The3exceptionsarealltriplets,underlinedinFig2,andmay
becandidatesforexploringthird-orderinteractions.Takingalldatatogether,thesynergy/
antagonismsignofthecombinationsisdifferentfromallofthepairsonlyrarely(6%overall)
inthissample,suggestingthatdrugpairsmaycarryusefulinformationaboutthecocktail
effects[5,28,29].
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 4/16
Predictingdrugcocktailswithlimitedmeasurements
Table2. Cellsurvivalandinteractionsofalldrugcombinationsinthepresentstudy. Viabilitywasmeasuredusingtheneutralreadassay.5%CIand
95%CImark5%and95%confidenceintervalsfrombootstrapping.InteractionisI=log(1+viability−Bliss)whereBlissistheproductofthesingle-drugviabil-
ityvalues.AntagonismandsynergyaremarkedinblueandpinkanddenotecaseswhereI>0orI<0togetherwiththedemandthatI=0isoutsidethe95%
confidenceinterval.Cocktailsinboldaresynergistic/antagonisticforbothcelllines.
H1299 HeLa
Drugcocktail viability ste Interaction 5%CI 95%CI viability ste Interaction 5%CI 95%CI
Singles CPT 0.77 0.01 0.73 0.02
CisPt 0.82 0.01 0.65 0.03
CbPt 0.77 0.01 0.27 0.04
NCZ 0.73 0.02 0.83 0.02
MG132 0.68 0.02 0.70 0.02
Etopo 0.72 0.01 0.71 0.02
Pairs CPT+CisPt 0.67 0.02 0.04 0.01 0.08 0.62 0.02 0.13 0.10 0.16
CPT+CbPt 0.58 0.02 −0.01 −0.05 0.02 0.09 0.01 −0.14 −0.19 −0.09
CPT+NCZ 0.66 0.02 0.07 0.04 0.09 0.66 0.03 0.10 0.07 0.13
CPT+MG132 0.66 0.01 0.12 0.09 0.15 0.76 0.02 0.21 0.17 0.24
CPT+Etopo 0.51 0.01 −0.05 −0.08 −0.02 0.60 0.02 0.07 0.04 0.10
CisPt+CbPt 0.73 0.02 0.08 0.05 0.11 0.11 0.02 −0.07 −0.09 −0.05
CisPt+NCZ 0.72 0.02 0.10 0.07 0.12 0.87 0.04 0.24 0.19 0.29
CisPt+MG132 0.74 0.02 0.15 0.12 0.18 0.80 0.03 0.28 0.24 0.32
CisPt+Etopo 0.68 0.02 0.08 0.06 0.11 0.63 0.01 0.15 0.11 0.18
CbPt+NCZ 0.41 0.02 −0.17 −0.23 −0.11 - - - - -
CbPt+MG132 0.58 0.02 0.06 0.03 0.09 0.25 0.03 0.05 0.03 0.07
CbPt+Etopo 0.64 0.02 0.08 0.06 0.11 0.17 0.01 −0.02 −0.06 0.01
NCZ+MG132 0.47 0.02 −0.04 −0.08 −0.01 0.78 0.02 0.15 0.11 0.19
NCZ+Etopo 0.78 0.02 0.23 0.20 0.25 0.77 0.03 0.18 0.13 0.22
MG132+Etopo 0.72 0.02 0.21 0.18 0.24 0.83 0.02 0.28 0.23 0.32
Triplets CPT+CisPt+CbPt 0.37 0.04 −0.07 −0.13 −0.01 0.08 0.00 −0.07 −0.10 −0.04
CPT+CisPt+NCZ 0.31 0.03 −0.16 -0.26 −0.07 0.60 0.02 0.19 0.13 0.23
CPT+CisPt+MG132 0.37 0.01 −0.04 −0.10 0.02 0.55 0.01 0.18 0.14 0.22
CPT+CisPt+Etopo 0.40 0.04 −0.03 −0.12 0.07 0.41 0.01 0.06 0.03 0.09
CPT+CbPt+NCZ 0.53 0.01 0.05 −0.01 0.12 - - - - -
CPT+CbPt+MG132 0.31 0.02 −0.06 −0.13 0.00 0.08 0.04 −0.07 −0.09 −0.05
CPT+CbPt+Etopo 0.32 0.03 −0.08 −0.20 0.04 0.02 0.00 −0.14 −0.19 −0.10
CPT+NCZ+MG132 0.28 0.05 −0.11 −0.18 −0.05 0.63 0.02 0.19 0.17 0.22
CPT+NCZ+Etopo 0.34 0.00 −0.08 −0.14 −0.02 0.54 0.00 0.14 0.10 0.19
CPT+MG132+Etopo 0.58 0.03 0.14 0.05 0.25 0.64 0.01 0.22 0.16 0.27
CisPt+CbPt+NCZ 0.18 0.01 −0.35 −0.45 −0.26 - - - - -
CisPt+CbPt+MG132 0.55 0.01 0.08 −0.02 0.17 0.10 0.00 −0.04 −0.06 −0.01
CisPt+CbPt+Etopo 0.52 0.02 0.02 −0.03 0.07 0.07 0.01 −0.07 −0.09 −0.04
CisPt+NCZ+MG132 0.22 0.01 −0.25 −0.42 −0.10 0.74 0.02 0.26 0.23 0.30
CisPt+NCZ+Etopo 0.68 0.01 0.20 0.15 0.25 0.64 0.01 0.20 0.17 0.24
CisPt+MG132+Etopo 0.72 0.01 0.26 0.21 0.31 0.61 0.01 0.24 0.21 0.27
CbPt+NCZ+MG132 0.15 0.01 −0.30 −0.51 −0.14 - - - - -
CbPt+NCZ+Etopo 0.29 0.01 −0.05 −0.12 0.02 - - - - -
CbPt+MG132+Etopo 0.50 0.01 0.17 0.12 0.23 0.19 0.00 0.05 0.02 0.07
NCZ+MG132+Etopo 0.30 0.01 0.02 −0.01 0.05 0.71 0.02 0.25 0.20 0.30
(Continued)
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 5/16
Predictingdrugcocktailswithlimitedmeasurements
Table2. (Continued)
H1299 HeLa
Drugcocktail viability ste Interaction 5%CI 95%CI viability ste Interaction 5%CI 95%CI
Quadruplets CPT+CisPt+CbPt+NCZ 0.14 0.00 −0.15 −0.21 −0.10 - - - - -
CPT+CisPt+CbPt+MG132 0.19 0.01 −0.07 v0.14 0.00 0.03 0.00 −0.09 −0.13 −0.05
CPT+CisPt+CbPt+Etopo 0.10 0.02 −0.22 −0.28 −0.15 0.02 0.00 −0.09 −0.11 −0.06
CPT+CisPt+NCZ+MG132 0.17 0.01 −0.06 −0.11 −0.01 0.48 0.02 0.17 0.14 0.20
CPT+CisPt+NCZ+Etopo 0.15 0.01 −0.13 −0.17 −0.09 0.41 0.01 0.12 0.09 0.16
CPT+CisPt+MG132+Etopo 0.16 0.01 −0.09 −0.15 −0.02 - - - - -
CPT+CbPt+NCZ+MG132 0.16 0.00 −0.14 −0.23 −0.07 - - - - -
CPT+CbPt+NCZ+Etopo 0.14 0.01 −0.22 −0.28 −0.15 - - - - -
CPT+CbPt+MG132+Etopo 0.25 0.01 −0.03 −0.09 0.03 0.03 0.00 −0.06 −0.09 −0.03
CPT+NCZ+MG132+Etopo 0.21 0.01 −0.08 −0.12 −0.04 0.77 0.02 0.40 0.37 0.42
CisPt+CbPt+NCZ+MG132 0.10 0.00 −0.24 −0.32 −0.16 - - - - -
CisPt+CbPt+NCZ+Etopo 0.21 0.01 −0.10 −0.17 −0.03 - - - - -
CisPt+CbPt+MG132+Etopo 0.55 0.01 0.22 0.19 0.26 0.04 0.01 −0.06 −0.09 −0.04
CisPt+NCZ+MG132+Etopo 0.37 0.01 0.08 0.03 0.13 0.80 0.01 0.40 0.37 0.43
CbPt+NCZ+MG132+Etopo 0.11 0.01 −0.20 −0.25 −0.14 - - - - -
Quintuplets CPT+CisPt+CbPt+NCZ+MG132 0.22 0.04 −0.19 −0.27 −0.11 - - - - -
CPT+CisPt+CbPt+NCZ+Etopo 0.27 0.02 0.00 −0.06 0.05 - - - - -
CPT+CisPt+CbPt+MG132+Etopo 0.43 0.01 0.06 0.00 0.13 0.00 0.00 −0.06 −0.08 −0.04
CPT+CisPt+NCZ+MG132+Etopo 0.26 0.01 0.01 −0.06 0.08 0.58 0.01 0.32 0.30 0.34
CPT+CbPt+NCZ+MG132+Etopo 0.17 0.03 −0.15 −0.25 −0.06 - - - - -
CisPt+CbPt+NCZ+MG132+Etopo 0.20 0.01 −0.09 −0.14 −0.03 - - - - -
Sextuplet CPT+CisPt+CbPt+NCZ+MG132+Etopo 0.29 0.03 0.04 −0.01 0.09 - - - - -
Abbreviations:CbPt,Carboplatin;CisPt,Cisplatin;CPT,Camptothecin;Etopo,Etoposide;LD20,NCZ,Nocodazole
https://doi.org/10.1371/journal.pbio.2002518.t002
Wescannedafamilyofmathematicalmodelstopredicthigh-order
cocktailsbasedonpairs
WenextaskedtowhatextentdataondrugpairscanimproveontheBlissformulato
predicttheeffectofcocktails.Thisrequiresmathematicalformulatoproperlyutilizepairdata.
Thereareseveralexistingapproachesforsuchpair-basedpredictions.Awidelyusedmachine
learningapproachisbasedonlogarithmicregression[21–26].Fortriplets,regressionyields
Y Y
gRegression =g g g /g g g ,andforquadrupletsgRegression ¼ g = g2[29].
123 12 23 13 1 2 3 1234 ij i
AsecondapproachemploystheIsserlisformulaintroducedbyWoodetal.basedonmaxi-
mumentropyconsiderations[28].Fortriplets,theIsserlisformulaisgIsserlis =g g +
123 1 23
g g +g g −2g g g ,andforquadruplets,gIsserlis =g g +g g +g g −2g g g g .
2 13 3 12 1 2 3 1234 12 34 13 24 14 23 1 2 3 4
TheIsserlisformulashowedexcellentagreementwithdataonantibiotictripletsandquadru-
plets[28].TheregressionandIsserlisformulashavethedesirablemathematicalpropertythat
ifallpairsareBliss(thatis,ifforallpairsg =g g),thenthepredictedcocktailisalsoBliss
ij i j
g =g g g g .WecallthispropertyBlissconservation.
123..k 1 2 3... k
Wesoughttotestadditionalmodelsthatusesingledrugsanddrugpairsasinputs,mea-
suredatasingledose.WethereforesoughtafamilyofmodelsthatgeneralizetheBlissand
regressionformulas.Wechoosethefamilyoflog-linearcombinationsofsingleandpairmea-
surements,whichsmoothlyinterpolatesbetweenBlissandregression.Fortriplets,wetested
g =(g g g )α(g g g )β,whereαandβareparameters.TopreservetheBlissconservation
123 12 23 13 1 2 3
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Predictingdrugcocktailswithlimitedmeasurements
Fig1.Acomparisonofdruginteractionsbetweenthe2celllines.Circlesstandforcocktailsofpairsofdrugs,trianglesfortriplets,
squaresforquadruplets,andstarsforcocktailsof5drugs.Theerrorbarsare95%confidenceintervalsofthemeasurements.Thenumberin
eachshapeidentifiesthecocktailaccordingtothelistontheright.CbPt,Carboplatin;CisPt,Cisplatin;CPT,Camptothecin;Etopo,
Etoposide;NCZ,Nocodazole.
https://doi.org/10.1371/journal.pbio.2002518.g001
Q Q
propertyrequiresβ=1−2α.GeneralizingtoMdrugs,wehaveg ¼ð g Það gÞbwith
1...M ij i
β=1−(M−1)α.TheBlissformulaisobtainedwhenα=0andtheregressionformulawhen
α=1.
Wescannedthevalueoftheparameterα,settingβ=1−(M−1)αtopreserveBlissconser-
vation,andcomparedtheR2valuesforthefullyfactorialdatasetsdescribedabove.Wefind
thatasimpleformulashowsnearlymaximalR2valuesandoutperformsBliss,Isserlis,and
regression(seenextsection)(Fig3,S1Data).Thisformulaincludesonlypairdata(β=0,α=
1/(M−1)),andwehencenameitthepairsmodel:
Y
gpairs ¼ð g Þ1=ðM(cid:0) 1Þ ð2Þ
1...M ij
Thus,fortriplets,thepairsmodelisthesquarerootoftheproductofthe3paireffects.One
featureofthepairsmodelisthatitislesssensitivetoexperimentalnoisethanmostother
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 7/16
Predictingdrugcocktailswithlimitedmeasurements
Fig2.Thesynergy/antagonismsignoftripletsismostlyconsistentwiththatoftheirconstituentpairs.ShownarealltripletsofHeLa
(upperpanel)andH1299(lowerpanel).Drugnamesareattheverticesofthetriangles.Thenumberinsidethetriangleisthetripletinteraction(I =
123
log(1+g −g g g )),andedgesaremarkedbythepairinteractions.Colorindicatestheinteractionsign:redisantagonistic,blueissynergistic,and
123 1 2 3
whiteisadditive(thatis,I=0isincludedinthe95%confidenceinterval).Thetrianglesareorderedfromthemostantagonistictriplettothemost
synergisticone.TheunderlinedH1299tripletistheonlytripletwherethesignofI isnotconsistentwiththesignofthe3pairinteractions,apossible
123
PLOSBiology|https://doi.org/10.1371/journal.pbio.2002518 October26,2017 8/16
Predictingdrugcocktailswithlimitedmeasurements
indicationofthird-orderinteraction.TwoothersynergisticH1299tripletsaremarginallyinconsistentinthesensethattheycontain2antagonisticpairs
and1pairthatisjudgedadditiveaccordingtothepresentexperimentalvariation.I,interaction.CbPt,Carboplatin;CisPt,Cisplatin;CPT,Camptothecin;
Etopo,Etoposide;NCZ,Nocodazole.
https://doi.org/10.1371/journal.pbio.2002518.g002
modelsinthisclass,becauseitusesonlydataforpairs;othermodelsusebothpairandsingle
drugdata,increasingthenumberofvariablesandhencethesensitivitytonoise.Assuming
independentmultiplicativeexperimentalnoiseforeachmeasurementwithstandarddeviation
pffiffiffiffiffi
σ,theBlissformulahastotalexperimentalnoiseof Ms,theregressionformulahaslarger
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
noiseofs MðM(cid:0) 1ÞþMðM(cid:0) 2Þ2,andthepairsformulahasnoiseofonlys M=2ðM(cid:0) 1Þ.
2 pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi
Fortriplets(M=3),forexample,thesenoisetermsare 3¼1:7, 6¼2:5,and 3=4¼0:9
timesσforBliss,regression,andpairs,respectively.Thepairsmodelisexpectedtobemost
usefulwhendataisnoisy.
ThepairsformulaimprovesontheBlissapproximationforhigh-order
cocktaileffects
WeplottheagreementofdifferentformulastothepresentdatasetinFig4andS1Data.The
Isserlisandregressionformulasarefarfromthedata,asevidencedbytheirnegativeR2values
Fig3.Todefinedifferentmodels,wescannedthelog-linearmodelfamilyoverαwhilesettingβ=1−(M−1)αandcalculatedthe
R2values.ThemaximalR2occursclosetoα=1/(M−1)andβ=0,whichisthedefinitionofthepairsmodel.Theleftpanelisacomparison
betweenthepairsmodel,theBlissmodel,andthemodeldefinedbytheαwiththehighestR2.TheleftpanelistheR2valuesofα=0:0.05:1.
TherowsareforM=3,4,and5,respectively.
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Predictingdrugcocktailswithlimitedmeasurements
Fig4.ThepairsmodelimprovesontheBlissapproximationandothermodelsforcancerdrugcocktails.Theplotscomparethe
experimentaldataandthepredictionoftheindicatedmodels.
https://doi.org/10.1371/journal.pbio.2002518.g004
R2=−0.79andR2=−12.1,respectively(negativeR2indicateslackofaccuracyforthedata
P P
meansinceR2 ¼1(cid:0) ðy (cid:0) fÞ2= ðy (cid:0) y(cid:22)Þ2fordatay andpredictionf).Blissindepen-
i i i i i i i i
dencehasR2=0.29.ThepairsmodelimprovesthistoR2=0.54.ThisvalueofR2isreasonable
giventheexperimentalnoiseinthesemeasurements,whichaddsupwhenconsideringcombi-
nationsof4to6drugs.Also,thehigherthenumberofdrugs,themorethepotentialforhigh-
ordereffectsabovepairs,whichcannotbecapturedbythepresentmodel.
Wetestedthepairsmodelalsoonpreviouslypublisheddatafor1,360antibiotictripletsand
quadrupletsbyWoodetal.[28]Thepairsmodelshowsgoodfittothisdata(R2=0.78),com-
parabletotheIsserlisformula(R2=0.88).Inthisdataset,errorsmaybesmallerthanthepres-
entdataset(S3andS5Figs).
Thisantibioticdataareextensiveenoughtoaskhowwellthemodelscanrankthecombina-
tionsintermsofefficacy.Efficacyrankingisofinterestifoneneedstoprioritizepotential
cocktailsbasedonmeasuringthepairs.Wefindthatthepairsmodelshows85%accuracyin
identifyingthetop10%mosteffectivetriplets(thatis,tripletswithlowestbacterialgrowth
rate),comparedto75%accuracyintheIsserlismodel,22%forBliss,and10%forrandom(S6
Fig).Theregressionmodelshowsworseaccuracythanrandom.
Thepairsmodelalsodescribesthemultidosecancerdrug-tripletdatasetbyZimmeretal.
[29],withR2=0.7comparedtoR2=−0.29,−2.03,and0.58forBliss,regression,andIsserlis
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