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Mathematical and Statistical Methods for Identifying DNA Sequence Variants That Encode Drug Response

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MATHEMATICALANDSTATISTICALMETHODSFORIDENTIFYINGDNA SEQUENCEVARIANTSTHATENCODEDRUGRESPONSE By MINLIN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2005

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Copyright2005 by MinLin

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Idedicatethisworktomyhusband,parentsandbrother.

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ACKNOWLEDGMENTS Firstandforemost,IwanttoexpressmydeepestappreciationtoD r.Rongling Wu,whohasbeenawonderfuladvisor,colleagueandfriend.Ia mextremely gratefulforhisguidance,patience,insightandencouragem ent.Hisendlessideas andenthusiasmforresearchamazeandinspireme.Healwayshadcon dencein meandwholeheartedlysupportedmethroughoutmydoctoralstu dy.Withouthis understandingandaid,thisdissertationwouldnothavebeenpo ssible.Iwouldalso liketothankthemembersofmycommittee,Dr.HartmutDerendo rf,Dr.Ramon Littell,Dr.KennethPortierandDr.RonaldRandlesfortheg enerousgiftsoftheir timeaswellasknowledge.Theirreadingofandcommentingon mydissertation wereveryhelpful. Specialthanksgotomyfamily,whohaveunconditionallyand constantly supportedmeovertheyears.Myparents,JinhuandRuilin,andmy brotherHai havemorefaithinmyabilityandeventualsuccessesthanImyself everdid.Ireally appreciateeverythingtheyhavedoneforme.Mostimportantl y,Iwanttothank myhusband,Xiang,forhisunwaveringcondenceinmeandforno tlettingmegive up.Hislove,patience,encouragementandrespecthelpedmege tthroughallthose toughtimes. Last,Iwouldliketothankallmyfriends,insideandoutsidethew orldof statistics,fortheirhelpandconcern. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFTABLES.................................viii LISTOFFIGURES................................x ABSTRACT....................................xi CHAPTER 1INTRODUCTION..............................1 1.1BasicGenetics.............................1 1.1.1GenesandChromosomes...................1 1.1.2GenotypeandPhenotype...................2 1.1.3MolecularGeneticMarkers..................3 1.2LinkageAnalysis...........................3 1.3LinkageDisequilibriumAnalysis...................5 1.4FunctionalMapping..........................7 1.5HapMap................................8 1.6SequenceMapping:FromQTLtoQTN...............10 1.7PharmacogenomicsandDrugResponse...............12 1.8StructureandOrganization.....................13 2MODELFORQTNMAPPINGDRUGRESPONSEWITHHAPMAP.14 2.1Introduction..............................14 2.2Theory.................................15 2.2.1Notation............................15 2.2.2LikelihoodFunctions.....................18 2.2.3AnIntegrativeEMAlgorithm................21 2.2.4HypothesisTests........................22 2.2.5 R -SNPSequenceModel....................23 2.3Application..............................24 2.4Discussion...............................26 3AJOINTMODELFORSEQUENCINGDRUGEFFICACYANDTOXICITY....................................33 3.1Introduction..............................33 v

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3.2BasicPrincipleForSequenceMapping...............35 3.3AnIntegrativeSequenceandFunctionalMappingFramewor kfor DrugEcacyandToxicity.....................39 3.3.1TheMultivariateNormalDistribution............39 3.3.2TheMappingFramework...................42 3.3.3ComputationalAlgorithm...................45 3.3.4ModelforanArbitraryNumberofSNPs..........46 3.4HypothesisTests...........................47 3.5Results.................................48 3.6Discussion...............................51 4MODELFORDETECTINGSEQUENCE-SEQUENCEINTERACTIONS FORCOMPLEXDISEASES.......................55 4.1Introduction..............................55 4.2TheModel...............................56 4.2.1Notation............................56 4.2.2EpistaticEects........................58 4.2.3LikelihoodFunctions.....................60 4.2.4AnIntegrativeEMAlgorithm................67 4.3HypothesisTests...........................68 4.4Results.................................69 4.5Discussion...............................72 5MODELFORDETECTINGSEQUENCE-SEQUENCEINTERACTIONS FORDRUGRESPONSE.........................75 5.1Introduction..............................75 5.2Theory.................................76 5.2.1TheNormalMixtureModel..................76 5.2.2EpistaticEects........................77 5.2.3LikelihoodFunctions.....................79 5.2.4ModellingtheMean-covarianceStructures......... .84 5.2.5AnIntegrativeEM-simplexAlgorithm............87 5.3HypothesisTests...........................88 5.4AWorkedExample..........................90 5.5MonteCarloSimulation.......................93 5.6Discussion...............................95 6MODELLINGTHEGENETICETIOLOGYOFPHARMACOKINTICPHARMACODYNAMICLINKSWITHTHEARMAPROCESS...98 6.1Introduction..............................98 6.2HaplotypingaComplexTrait....................100 6.3HaplotypingtheIntegratedPK-PDProcess.............1 04 6.3.1TheLikelihoodFunctions...................104 6.3.2ModellingtheMeanVector..................108 vi

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6.3.3ModellingtheCovarianceMatrix...............110 6.3.4ComputationalAlgorithms..................116 6.3.5ModelforanArbitraryNumberofSNPs..........117 6.4HypothesisTests...........................117 6.5Results.................................119 6.6Discussion...............................123 7CONCLUSIONSANDPROSPECTS....................127 7.1Summary...............................127 7.2FutureDirections...........................129 7.2.1Gene-EnvironmentInteraction................129 7.2.2Case-ControlStudy......................129 7.2.3Dose-DependencyofAllometricScalingPerformance... .130 7.2.4MissingDataProblem.....................130 APPENDIX ADERIVATIONOFASYMPTOTICCOVARIANCEMATRIX......132 BDERIVATIONOFMLESUSINGEMALGORITHM...........137 REFERENCES...................................142 BIOGRAPHICALSKETCH............................150 vii

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LISTOFTABLES Table page 2{1Possiblediplotypeconfgurationsofninegenotypesattwo SNPsand theirhaplotypecompositionfrequencies................ 17 2{2Log-likelihoodratio(LR)teststatisticsofdierenthapl otypemodels andMLEsofpopulationandquantitativegeneticparameters within the f 2ARgene.............................27 2{3Testingresultsfortwodrugresponseparameters,HandEC 50 ,andtotalgenetic,additiveanddominanteectsundertheoptimal haplotypemodel................................28 2{4MLEsofSNPallelefrequencyandlinkagedisequilibriuman dparametersdescribingthethreedynamiccurvesbasedonthesigmoid al Emaxmodel................................29 3{1Possiblediplotypeconfgurationsofninegenotypesattwo SNPswhich aectdrugecacyandtoxicity....................38 3{2MLEsofpopulationgeneticparameters,thecurveparamet ersandmatrixstructuringparametersforecacyandtoxicityresponses.... ..50 4{1Possiblediplotypeconfgurationsofninegenotypesattwo SNPsand theirhaplotypecompositionfrequencies................ 61 4{2MLEsofSNPpopulationandquantitativegeneticparamete rsassociatedwithphenotypicvariationinBMI................71 5{1Possiblediplotypesandtheirfrequeneciesforeachofnin egenotypes attwoSNPswithinaQTN,haplotypecompositionfrequenciesfor eachgenotypeandgenotypicvaluevectorsofcompositegenot ypes.81 5{2Likelihoodratiosfor16possiblecombinationsofassumedre ference haplotypeswithonefromcandidategene f 1ARandthesecondfrom candidategene f 2AR..........................92 5{3MLEsofparameterswithintwoindependentcandidategen es f 1AR and f 2AR................................93 5{4MLEsoftotalgenetic,additive,dominantandinteracti oneectsundertheoptimalhaplotypemodel....................95 viii

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5{5MLEsofparametersofSNPallelefrequencyandlinkagedise quilibriumandparametersdescribingtheninedynamiccurves..... .96 6{1Possiblediplotypeconfgurationsofninegenotypesattwo SNPswhich aectpharmacokinetics(PK)andpharmacodynamics(PD)... ..103 6{2MLEsofSNPpopulationquantitativegeneticparametersf orpharmacokineticsandpharmacodynamicsresponses.............12 2 ix

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LISTOFFIGURES Figure page 1{1Twenty-threepairsofchromosomesinthehumangenome... ....2 1{2AshortstretchofDNAfromfourversionsofthesamechromosome regionindierentpeople........................9 2{1Proflesofheartrateinresponsetodierentconcentratio nsofdobutamineforthreecompositegenotypes.................28 3{1Estimatedresponsecurveseachcorrespondingtooneofthree compositegenotypesundertheheritability( H 2 )of0.1and0.4inacomparisonwiththehypothesizedcurves.................54 5{1Proflesofheartrateinresponsetodierentdosagesofdobu tamine forninecompositegenotypes.....................94 6{1Landscapesofdrugeectsvaryingasafunctionofdosagean dtime forthreehypothesizedcompositegenotypes..............12 0 6{2Estimatedresponsecurveseachcorrespondingtooneofthree compositegenotypesforPKandPDinacomparisonwiththehypothesizedcurves..............................124 x

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MATHEMATICALANDSTATISTICALMETHODSFORIDENTIFYINGDNA SEQUENCEVARIANTSTHATENCODEDRUGRESPONSE By MinLin August2005 Chair:RonglingWuMajorDepartment:Statistics Substantialvariabilityexistsamongdierentpatientsinph armacological responsetomedications.Drugresponseistypicallyacomplextra itthatiscontrolledbyanetworkofmultifariousgenesaswellasbiochem ical,developmental andenvironmentalfactors.Theidenticationofgeneticfac torsthatcontribute toamong-persondierentiationhasbeenoneofthemostimport antanddicult tasksforpharmacogeneticresearchanddrugdiscovery.Withth ereleaseofthe haplotypemap,orHapMap,constructedfortheentirehumangen omebasedon high-throughputsinglenucleotidepolymorphisms(SNPs),thed etectionofspecic DNAsequenceaectingresponsestodrugscannowbemadepossible. Inthisdissertation,Iwillproposeaseriesofstatisticalmodels andalgorithms formappingandidentifyinggeneticvariantsthatareassocia tedwiththedynamic featuresofdrugresponse.FoundedontheSNP-basedhaplotypebl ockingtheory, thesemodelsareconstructedwithinthecontextofmaximumlik elihoodand implementedwithaclosed-formsolutionfortheEMalgorithmt oestimatethe populationgeneticparametersofSNPs.Thesimplexalgorithmi susedtoestimate thecurveparametersthatdescribethepharmacodynamicand/ orpharmacokinetic xi

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changesandthecovariancematrixstructuringparameters.Th eincorporationof clinicallyimportantmathematicalfunctionsfordrugrespo nsenotonlymakesmy modelsmorepowerfulforgenedetection,butalsoallowsfora numberofhypothesis testsattheinterplaybetweengeneactions/interactionsand pharmacological actions.MonteCarlosimulationstudiesbasedonvariousschemes havebeen performedtoinvestigatedierentstatisticalaspectsofmymod els.Thedetection ofsignicantDNAsequencevariantsfordrugresponseinworkedex ampleshas validatedtheusefulnessofthemodels.Potentialapplication stopharmacogenetic researchhavebeendiscussedforeachofmymodels.Itcanbeantici patedthatmy modelswillhavemanyimplicationsforelucidatingthedeta iledgeneticarchitecture ofdrugresponseandultimatelydesigningpersonalizedmedicat ionsbasedoneach patient'sgeneticblueprint. xii

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CHAPTER1 INTRODUCTION 1.1BasicGenetics 1.1.1GenesandChromosomes Geneticsisthestudyofheredityorinheritance.Geneticshe lpstoexplainhow traitsareinheritedfromparentstotheirospring.Parents passontraitstotheir youngthroughgenetransmission.Thefundamentalphysicalandf unctionalunitof heredityisa gene ,whichwasrstrevealedbyGregorMendel'speaexperimentsa nd mathematicalmodelin1865.Genesarecomposedofdeoxyribon ucleicacid(DNA), adouble-strandhelixofnucleotides.Eachnucleotidecontai nsadeoxyribosering, aphosphategroup,andoneoffournitrogenousbases:adenine(A) ,guanine(G), cytosine(C),andthymine(T).Innature,basepairsformonlyb etweenAandT andbetweenGandCduetotheirchemicalcongurations.Itist heorderofthe basesalongDNAthatcontainsthehereditaryinformationthat willbetransmitted fromonegenerationtothenext. AsingleDNAmoleculecondensedintoacompactstructureinacell nucleus iscalled chromosome .Thechromosomesoccurinsimilar,orinhomologous,pairs, wherethenumberofpairsisconstantforeachspecies.Inhumans, thereare twenty-threepairsofchromosomes,carryingtheentiregenet iccode,inthenucleus ofeverycellinthebody.Foreachpair,onechromosomeisinhe ritedfromthe motherandtheotherfromthefather.Theentirecollectiono fthesechromosomes isreferredtoasthehuman genome .Oneofthechromosomepairsinthegenomeis thesexchromosomes(denotedby X and Y )thatdeterminegeneticsex.Theother pairsare autosomes thatguidetheexpressionofmostothertraits(Figure1-1). 1

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2 Figure1{1:Twenty-threepairsofchromosomesinthehumange nome 1.1.2GenotypeandPhenotype AgeneissimplyaspeciccodingsequenceofDNAandmayoccurinal ternativeformscalled alleles .Asinglealleleforeachgeneisinheritedfromeachparent, termedthematernalandpaternalallelerespectively.Thepa irofallelesconstructs the genotype ,whichistheactualgeneticmakeup.Ifagivenpairconsistsof similar alleles,theindividualissaidtobe homozygous forthegeneinquestion;whileifthe allelesaredissimilar,theindividualissaidtobe heterozygous .Forexample,ifwe havetwoallelesatagivengeneofanindividual,say A and a ,therearetwokinds ofhomozygotes,namely AA and aa ,andonekindofheterozygote,namely Aa Therefore,threedierentgenotypes, AA;Aa and aa ,areformedwithasinglepair ofalleles. Incomparison, phenotype representsalltheobservablecharacteristicsof anindividual,suchasphysicalappearance(eyecolor,height ,etc.)andinternal physiology(disease,drugresponse,etc.).

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3 1.1.3MolecularGeneticMarkers Moleculargenetic markers arereadilyassayedphenotypesthathaveadirect1:1correspondencewithDNAsequencevariationataspecic locationinthe genome.Inprinciple,theassayforageneticmarkerisnotaec tedbyenvironmentalfactors.GeneticmarkersareDNAsequencepolymorphismsand havemany dierenttypes.Restrictionfragmentlengthpolymorphisms(R FLPs)aretherst geneticmarkersthatwerewidelyusedforgenomicmappingand populationstudies. Thepolymerasechainreaction(PCR)providesausefulwaytoob taingeneticmarkers.Ampliedfragmentlengthpolymorphisms(AFLPs)areoneofth ePCR-based anonymousmarkers. OneofthefruitsoftheHumanGenomeProjectisthediscoveryof millions ofDNAsequencevariantsinthehumangenome.Themajorityofth esevariants aresinglenucleotidepolymorphisms(SNPs),whichcompriseappr oximately80% ofallknownpolymorphisms,andtheirdensityinthehumangenom eisestimated tobeonaverage1per1000basepairs(TheInternationalHapMap Consortium 2003).SNPs,asthenewestmarkers,havebeenthefocusofmuchatt entionin humangeneticsbecausetheyareextremelyabundantandwellsuitedforautomated large-scalegenotyping.AdensesetofSNPmarkersopensupthepo ssibilityof studyingthegeneticbasisofcomplexdiseasesbypopulationapp roaches,although SNPsarelessinformativethanothertypesofgeneticmarkersb ecauseoftheir biallelicnature.SNPsaremorefrequentandmutationallysta ble,makingthem suitableforassociationstudiestomapdisease-causingmutations, especiallyuseful inpersonalizedmedicinefortheirassociationwithdiseasesuscep tibility,drug treatmentresponseandnutritionalneeds. 1.2LinkageAnalysis SincethepublicationoftheseminalmappingpaperbyLandera ndBotstein (1989),therehasbeenalargeamountofliteratureconcerni ngthedevelopmentof

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4 statisticalmethodsformappingcomplextraits(reviewedinJ ansen2000;Hoechele 2001).Althoughtheideaofassociatingacontinuouslyvaryingp henotypewitha discretetrait(marker)datesbacktotheworkofSax(1923),i twasLanderand Botstein(1989)whorstestablishedanexplicitprincipleforl inkageanalysis.They alsoprovidedatractablestatisticalalgorithmfordissectinga quantitativetrait intotheirindividualgeneticlocuscomponents,referredto asquantitativetraitloci (QTL).TheaimofQTLmappingistoassociategeneswithquantit ativephenotypic traits.Forexample,wemightbeinterestedinQTLthataectth eresponsetoa givendrug,sowemightbelookingforregionsonachromosometh atareassociated withdrugresponse. ThesuccessofLanderandBotsteinindevelopingapowerfulmeth odfor linkageanalysisofacomplextraithasrootsintwodierentd evelopments.First, therapiddevelopmentofmoleculartechnologiesinthemidd le1980sledtothe generationofavirtuallyunlimitednumberofmarkersthatsp ecifythegenome structureandorganizationofanyorganism(Draynaetal.1984 ).Second,almost simultaneously,improvedstatisticalandcomputationaltechn iques,suchastheEM algorithm(Dempsteretal.1977),madeitpossibletotacklecom plexgeneticand genomicproblems. GeneticmappingofQTLliesintheideathatgeneticmarkersc anbeclose tothegeneofinterest.LanderandBotstein's(1989)modelfor intervalmapping ofQTLisregardedasappropriateforanideal(simplied)situ ation,inwhichthe segregationpatternsofallmarkerscanbepredictedontheba sisoftheMendelian lawsofinheritance,atraitunderstudyisstrictlycontrolle dbyoneQTLona chromosomeandtheexpectedeectofsuchhypotheticalQTLise stimatedfrom thegenotypesatmarkerlocirankingtheinterval.Thiswork wasextendedand improvedbymanyresearchers(Haleyetal.1994;JansenandStam 1994;Zeng 1994;Xu1996),withsuccessfulidenticationofso-called\outc rossing"QTLin

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5 real-lifedatasetsfrompigs(Anderssonetal.1994)andpine(Kn ottetal.1997).A generalframeworkforQTLanalysiswasrecentlyestablishedby Wuetal.(2002b) andLinetal.(2003). Anintervalmappingapproachcannotadequatelyuseinformati onfrom allpossiblemarkersonthegenome.Zeng(1993,1994)proposeda so-called compositeintervalmapping techniquetoincreasetheprecisionofQTLdetectionby controllingthechromosomalregionoutsidethemarkerinterv alunderconsideration. Thisapproach,alsodevelopedindependentlybyJasen(1993)a ndJansenand Stam(1994),hasbeenwidelyadoptedinpractice.Statistica lly,compositeinterval mappingisacombinationofintervalmappingbasedontwogive nrankingmarkers andapartialregressionanalysisonallmarkersexceptforthet woonesbracketing theQTL.However,thechoiceofsuitablemarkerlocithatservea scovariatesisstill anopenproblem. Aninterestingapproach,called multipleintervalmapping ,isproposedbyKao etal.(1999)andistheextensionofintervalmappingbyusingm ultiplemaker intervalssimultaneouslytotmultipleputativeQTL.Inthis method,theQTL locationscanbeusedtoinferthepositionsbetweenmarkersev enwithsome missinggenotypedata,andcanallowustotakebothmainandint eractioneects intoaccountinmappingthemultipleQTL. 1.3LinkageDisequilibriumAnalysis Themostimportantgoalingeneticresearchistoidentifyandc haracterizethe actualgenesthatareresponsibleforphenotypicvariation.T husfar,onlyahandful ofgenesthatdeterminevariationincommerciallyimportan ttraitshavebeen described.Thereasonfortheidenticationofrelativelyfew genescanbeattributed tolimitationsofthetechniquesusedtodetectgenes.Inthela stdecadelinkage analysis-basedmappingapproacheshavebeeninstrumentalinde tectingQTLfora widevarietyoftraitsindierentorganisms.Butlinkageanal ysistypicallydenes

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6 thelocationofaQTLtowithina20-30cMchromosomalinterval {perhaps1%of aspecies'genome.Giventhataround70,000functionalgenesa reestimatedina typicalgenome,thereareroughly700genesthatarethought toexistundereach QTL\bump"(Slateetal.2002).Thus,identifyingthegene(or genes)inruencing thetraitofinterestbasedonlinkageanalysisisamonumentalt ask. Morerecently,analternativeapproachbasedonlinkagediseq uilibrium,i.e.,the non-randomco-segregationofallelesatlinkedloci,hasbee nshowntobepowerful foraidinggenediscovery(TerwilligerandWeiss1998).Theba sicpremisebehind linkagedisequilibriummappingisthataparticularallelea tamarkerwilltendto co-segregatewithoneallelicvariantofthegeneofinterest, providedthemarker andgeneareverycloselylinked.LDmappingpotentiallyhast woadvantages overconventionallinkagemapping.Therstisthatitmaybel ogisticallyeasier. Intheory,breedingschemessuchasbackcrossesorfull-sibmatin gsmaynotbe required,makingexperimentaldesignmorestraightforwarda ndsavingconsiderable time.Thesecond,probablygreater,advantageoeredbyLDma ppingisthat QTLmaybemappedtoverysmallregionsthusaidingdiscoveryof theunderlying gene(s).InordertoperformecientLDmapping,markersmustb emappedata densitycompatiblewiththedistancesthatLDextendsinthepo pulation.Currently severalconsortiaandlaboratorieshaveundertakentodevelo pdensemapsofsingle nuclearpolymorphism(SNP)markersforawidevarietyofspecie s.However,in ordertopredicthowmanySNPswillberequiredforLDmapping, theextentof linkagedisequilibriummustrstbeestablished.LDhasbeenestim atedinhumans (Kruglyak1999)andalsoHolsteincattle(Farniretal.2000). Thedisadvantageoflinkagedisequilibriummappingisthatth eassociation betweenmarkerlociarealsoaectedbyevolutionaryforcessu chasmutation,drift, selectionandadmixture.Thisdisadvantagecanbeovercomeby amappingstrategy

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7 combininglinkageandlinkagedisequilibrium,suchasthatde velopedinWuand Zeng(2001)andWuetal.(2002a). 1.4FunctionalMapping Innature,manytraits,suchasgrowth,AIDSprogressionanddrugr esponse, aredynamicandshouldbemeasuredinalongitudinalway.Althou ghtheelucidationoftherelationshipbetweengeneticcontrolanddevelop mentforlongitudinal traitsisstatisticallyapressingchallenge,someofthekeydic ultieshavebeen overcomebyR.Wuandcolleagues(Maetal.2002;Wuetal.2002 b,2003,2004a, 2004b,2004c).Theyhaveproposedageneralstatisticalframew ork,referredto as functionalmapping ,whichmapsgenome-widespecicQTLrelatedtothe developmentalpatternofacomplextrait. Thebasicrationaleoffunctionalmappingliesintheconnect ionbetweengene actionorenvironmentaleectsanddevelopmentparametric ornonparametricmodelsofphenotype.FunctionalmappingcandetectdynamicQTL thatareresponsible forabiologicalprocessmeasuredatanitenumberoftimepoin ts.Anumber ofmathematicalmodelshavebeenestablishedtodescribethede velopmental processofabiologicalphenotype.Forexample,aseriesofgro wthequationshave beenderivedtodescribegrowthinheight,sizeorweight(vonB ertalany1957; Richards1959)thatoccurwhenevertheanabolicormetaboli crateexceedstherate ofcatabolism.Basedonfundamentalprinciplesbehindbiolog icalorbiochemical networks,Westetal.(2001)havemathematicallyprovedtheun iversalityofthese growthequations.Withmathematicalfunctionsincorporate dintotheQTLmappingframework,functionalmappingestimatesparametersth atdetermineshapes andfunctionsofaparticularbiologicalnetwork,insteadof directlyestimatingthe geneeectsatallpossibletimepoints.Becauseofsuchconnectio nsamongthese pointsthroughmathematicalfunctions,functionalmapping strikinglyreducesthe

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8 numberofparameterstobeestimatedand,hence,displaysincr easedstatistical power. Fromastatisticalperspective,functionalmappingisaproble mofjointlymodellingmean-covariancestructuresinlongitudinalstudies,a nareathathasrecently receivedconsiderableattentioninthestatisticalliteratur e(Pourahmadi1999,2000; DanielsandPourahmadi2002;PanandMackenzie2003;WuandP ourahmadi 2003).However,asopposedtogenerallongitudinalmodelling ,functionalmapping integratestheparameterestimationandtestprocesswithinab iologicallymeaningfulmixture-basedlikelihoodframework.Functionalmappin gisthusadvantageous intermsofbiologicalrelevancebecausebiologicalprincip lesareembeddedinto theestimationprocessofQTLparameters.Theresultsderivedfr omfunctional mappingwillbeclosertobiologicalrealms. 1.5HapMap SeveralrecentempiricalstudiessuggestthatSNPsarenotevenl ydistributed overthegenomeintermsoftheextentofLDandthatthestructu reof haplotype (alineararrangementofnonallelesatlinkedloci;Figure1 -2)onachromosome canbebrokenintoaseriesofdiscretehaplotypeblocks(Dalye tal.2001;Patil etal.2001;Dawsonetal.2002;Gabrieletal.2002;Phillipse tal.2002).Ineach haplotypeblock,consecutivesitesareincomplete(ornearly complete)LDwith eachotherandthereislimitedhaplotypediversityduetolim ited(coldspot)intersiterecombination.Adjacentblocksareseparatedbysitesthat showevidenceof historicalrecombination(hotspot).Ithasgenerallybeenassu medthatthepresence ofhaplotypeblocksprovidesevidenceforne-scalevariati oninrecombination rates,withblockscorrespondingtoregionsofreducedrecomb ination,separatedby recombinationhotspots.Basedonastudyofthewholechromosome2 1(Patiletal. 2001),35,989observedSNPscanbeclassiedintodierentblock swithverylow

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9 Figure1{2:AshortstretchofDNAfromfourversionsofthesamechr omosome regionindierentpeople.ShownareSNPs( a ),haplotypes( b )andtagSNPs( c ). ThreeSNPsareshownwherevariationoccurs,surroundedbymucho ftheDNA sequenceidenticalinthesechromosomes.Ahaplotypecontainsa particularcombinationofallelesatnearbySNPs.ThepositionsofthethreeSNPssh owninpanel a arehighlighted.GenotypingjustthethreetagSNPsoutofthe2 0SNPsthat extendacross6,000basesofDNAissucienttoidentifythesefour haplotypes uniquely.AdoptedfromTheInternationalHapMapConsortium(2 003). haplotypediversityand80%ofthevariationinthischromosom ecanbedescribed byonlythreeSNPsperblock. Giventheblock-likepatternofLDdistributioninthegenome ,itshouldbe moreecienttolocateallelicvariantsforacomplexhumand iseasetraitbasedon haplotypeblocksthanindividualSNPstowithinastretchofDNA thatisamenable topositionalcloningtechniques.Becauseofthereportedlowh aplotypediversity withinblocksthereisapossibilitythatveryfewhaplotype-t agSNPs(htSNPs)need beexaminedtodetectcommonvariantsinvolvedinhumandisea ses(Figure1-2 c ; WallandPritchard2003). Withthereleaseofahaplotypemapofthehumangenome,theHapM ap, whichdescribesthecommonpatternsofhumanDNAsequencevaria tionbasedon SNPs(TheInternationalHapMapConsortium2003),ithasbeenpo ssibletond

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10 specicDNAsequencesthatencodehealth,disease,andresponsesto drugsand environmentalfactors.AsshowninFigure1-2 b ,peoplewhocarryhaplotype1 maybemoresusceptibletoagivendrugthanthosewhocarryother haplotypes. ItisshownthatthefourchromosomesinFigure1-2 b areadequatelydetermined bythreetagSNPsorhtSNPs(Figure1-2 c ).Thus,associationstudiesbetween haplotypeanddrugresponsecanbeundertakenonthebasisofthe sehtSNPs becauseifaparticularchromosomehasthepatternA-T-Catthese threetagSNPs, thispatternmatchesthepatterndeterminedforhaplotype1 .Thedetectionofa muchfewernumberofhtSNPsshowedfacilitateassociationstudie sforcommon diseasesandultimatelywillenhanceourabilitytochoosetarg etsfortherapeutic intervention. 1.6SequenceMapping:FromQTLtoQTN ThebasicprincipleforQTLmappingisthecosegregationofthe allelesat aQTLwiththoseatoneorasetofknownpolymorphicmarkersgeno typedon agenome.IfaQTLiscosegregatingwithmolecularmarkers,the geneticeects ofQTLandtheirgenomicpositionscanbeestimatedfromthemar kers.This approachisrobustandpowerfulforthedetectionofmajorQTL andpresentsthe mostecientwaytoutilizemarkerinformationwhenmarkerma psaresparse. However,thisapproachislimitedintwoaspects.First,becauset hemarkersand QTLbracketedbythemarelocatedatdierentgenomicpositio ns,thesignicant linkageofaQTLdetectedwithgivenmarkerscannotprovidea nyinformation aboutthesequencestructureandorganizationofQTL.Second, theinferenceof theQTLpositionsusingthenearbymarkersissensitivetomarker informativeness, markerdensityandmappingpopulationtype.Asaresult,onlyaf ewQTLdetected frommarkershavebeensuccessfullycloned(Fraryetal.2000), despiteaconsiderablenumberofQTLreportedintheliterature.Therefore, geneticinformation providedbyQTLmappingapproachisnotpreciseenough.

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11 Amoreaccurateandusefulapproachforthecharacterization ofgenetic variantscontributingtoquantitativevariationistodire ctlyanalyzeDNAsequences associatedwithaparticulardisease.IfastringofDNAsequenceisk nownto increasediseaserisk,thisriskcanbereducedbythealterationo fthisDNA sequencestringusingaspecializeddrug.Thecontrolofthisdise asecanbemade moreecientifallpossibleDNAsequencesdeterminingitsvaria tionareidentied intheentiregenome.Anewterm, quantitativetraitnucleotides orQTN,hasbeen denedtodescribethesequencepolymorphismsthatcausephenot ypicvariationin aquantitativetrait. Therecentdevelopmentofthehumangenomeproject,withits massive amountsofDNAsequencedataavailableforthehumangenome(In ternational HapMapConsortium2003),hasprovidedfuelforidentifyingQT Nforcomplex traitssuchasdrugresponse.ThehaplotypemaporHapMapconstruc tedbysingle nucleotidepolymorphisms(SNPs),beingthemostcommontypeofv ariantin theDNAsequence,hasfacilitatedthecompleteidentication ofspecicsequence variantsresponsibleforcomplexdiseases.Alineararrangement ofalleles(i.e., nucleotides)atdierentSNPsonasinglechromosome,orpartofa chromosome, iscalleda haplotype .ThecosegregationofSNPallelesonhaplotypesleadsto non-randomassociation,i.e.,linkagedisequilibrium(LD),b etweentheseallelesin thepopulation.EmpiricalanalysesofLDforSNPshaveshowntha tnearbySNPs inthehumangenometendtodisplayhighlysignicantlevelsof LDandareoften distributedinblock-likepatterns,ratherthandisplayingra ndomorevenspaced distributionasoriginallypredicted(Patiletal.2001;Daw sonetal.2002;Gabriel etal.2002).SNPswithinhaplotypeblocksaremuchmorestrong lyassociatedwith eachotherthanthosebetweendierentblocks.Haplotypediver sitywithineach blockcanbewellexplainedbyonlyanitenumberofSNPs,calle d tagSNPs or representativeSNPs .TheexistenceofthesetagSNPsmeansthatitisnotnecessary

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12 toassociateadiseasewithallSNPsintheDNAsequenceinordertound erstand thecompletegeneticcontrolofthediseaseordrugresponse.The reiscurrentlya pressingdemandforstatisticalmodelsthatcharacterizetheha plotypestructure withinQTNforcomplexdiseasesorcomplexprocesses. 1.7PharmacogenomicsandDrugResponse Incurrentpharmacogeneticresearch,increasingly,attempt shavebeenmade toidentifycandidategenesthatinruencepharmacological responses(Johnson 2003;WattersandMcLeod2003).Theseincludegenesinvolved indrugtransport (e.g.,polymorphismsinthegeneencodingP-glycoprotein1a ndtheplasma concentrationofdigoxin),genesinvolvedindrugmetaboli sm(e.g.,polymorphisms inthegeneencodingthiopurineS-methyltransferaseandthio purinetoxicity)and genesencodingdrugtargets(e.g.,polymorphismsinthegene encodingthe 2 adrenoceptorandresponseto 2 -adrenoceptoragonists)(Johnson2003).With advancedmoleculargenotypingtechnologies,anumberofpol ymorphicsites(such assinglenucleotidepolymorphismsorSNPs)withinornearthesec andidategenes canbegenotyped.SNPs,especiallySNPsthatoccuringeneregula toryorcoding regions(cSNPs),canbeassociatedwithphenotypictraitstodet ectgeneticvariants causingpharmacologicalresponsevariability.Linkagedisequ ilibriummappingbased onthenonrandomassociationbetweendierentgenesinapopul ationhasproven tobeapowerfulmeansforhigh-resolutionmappingofgenesfo rcomplextraits (WuandCasella2005).Morerecently,aclosed-formsolutionba sedontheEM algorithmforestimatingtheallelefrequenciesoffunction algeneticvariantsand theirdisequilibriawithSNPshasbeenderived(Louetal.2003 ).Itispossiblethat thisalgorithmcanbeusedtomapQTLaectingtheextenttowhi chanindividual respondstoaparticularpharmacologicalaction. Ahostofphysiologically-basedmathematicalmodelshavebeen builttodescribethepharmacokineticandpharmacodynamicprocessesofa drug(Hochhaus

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13 andDerendorf1995).Beyondnarrativedescriptions,thesemod elshaveprovided aprecisecharacterizationofdrugeectsandtheoreticalpr edictionofdrugresponsivenessacrossabroadrangeofdoselevelsorawideperiodof times.These mathematicalmodelscanalsobeincorporatedintotheframew orkforfunctional mappingtopreciselycharacterizegeneticvariantsthatcon tributetovariationin drugresponse. 1.8StructureandOrganization Theoverallpurposeofthisdissertationistodeveloppowerful statistical modelsfordetectingDNAsequencevariantsthataectdieren taspectsofdrug response.InChapter2,ageneralframeworkwasderivedtodeci pherthegenetic machineryofpharmacodynamicprocessesofadrugattheDNAseque ncelevel. Chapter3illustratesajointstatisticalmodelofthegeneticc ontrolfordrugecacy andtoxicity.Chapter4lookstodetectepistaticcontrolove rcomplextraitsthat areexpressedassequence-sequenceinteractions.Inchapter5,se quence-sequence epistaticmodelsareextendedtostudydrugresponseasadynamic process. Chapter6describesahyperspacemodelforcharacterizingthe dierentiationinthe geneticregulationofpharmacokineticsandpharmacodynam ics.Thelastchapter summarizestheresultsandprovidesfurtherstatisticalmethod ologicalresearchin pharmacogeneticsandpharmacogenomicsofdrugresponse.

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CHAPTER2 MODELFORQTNMAPPINGDRUGRESPONSEWITHHAPMAP 2.1Introduction Althoughpharmacogeneticsorpharmacogenomics,thestudyofi nherited variationinpatients'responsestodrugs,isstillinitsinfancy .Atremendous accumulationofdataforgeneticmarkersandpharmacodynam ictestshasmade itoneofthehottestandmostpromisingareasinbiomedicalscien ce(Evansand Relling1999,2004;Roses2000;EvansandJohnson2001;Evansa ndMcLeod2003; Goldsteinetal.2003;Weinsilboum2003;FreemanandMcLeod20 04).Thecentral themeofpharmacogeneticsistoassociateinterpatientvaria bilityindrugresponse withspecicgenomicsiteswiththeaidofpowerfulstatisticalt ools.Traditional approachesforsuchassociationstudiesarebasedonthestatistica linferenceof putativegeneticlociorquantitativetraitloci(QTL)ofin teresttoaphenotype fromknownlinkageorlinkagedisequilibriummaps(Lynchand Walsh1998;Wu andCasella2005).Withthecompletionofahaplotypemap(HapM ap)constructed fromDNAsequencevariationdata(TheInternationalHapMapCon sortium2003, 2004;DeloukasandBentley2004),ithasnowbecomepossibleto characterize concretenucleotidecombinationsthatencodeacomplextra it. Liuetal.(2004)derivedanewstatisticalmethodforelucidat ingDNA sequencevariationincomplexdiseasesfromHapMap.Comparedto complex diseases,genesfordrugresponsearerelativelysimplerandgenet icallyeasierto studybecausediseaseshaveundergonealongevolutionarypressur ewhereasthe useofparticulardrugsismuchmorerecent.However,drugrespo nseisstatistically morediculttoanalyzebecauseitisadynamicprocessandshoul dbequantied 14

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15 acrossmultipledierentlevelsofdrugconcentrationordosa geandduringatime course.Statisticalmodellingofsuchlongitudinaltraitshas beenachallenging issuegiventhecomplexitiesoftheirautocorrelationstructu re.Morerecently,an innovativetheoreticalframeworkhasbeenconstructedwith inwhichclinically meaningfulpharmacodynamicmodelsareincorporatedintot hecontextofgenetic mappingfordrugresponse(Gongetal.2004).Thebasictenetoft hisframework isthemathematicalmodellingofdosage-dependentdrugeec tsthatareinruenced bygeneticdeterminantsandenvironmentalfactorsandthei rinteractions.Notonly theframeworkdoesdisplaymanystatisticaladvantages,suchasm orepowerand greaterestimationprecision,duetoareducednumberofunkno wnparameters, butalsoisbiologicallyorclinicallymorerelevantgivenit scloseassociationto fundamentalpharmacologicalprinciples. AnovelstatisticalmodelfordeterminingspecicDNAsequencest hatare associatedwiththephenotypicvariationofdrugresponseispre sented.Thismodel isderivedonthebasisofmultilocushaplotypeanalysisusinga nitenumberoftag SNPs.Aclosed-formsolutionforestimatingtheeectsofhaploty pes,haplotype frequencies,allelefrequenciesandthedegreesofLDofvari ousordersamongtag SNPsunderlyingtheresponsetodrugsisderivedandsimulationst udiesperformed totestthestatisticalbehaviorsofthishaplotype-basedsequen cemappingmodel.A workedexampleisusedtovalidatethemodel,inwhichaDNAseque ncevariantis detectedwhichsignicantlyaectstheshapeoftheheartrate curveinresponseto dierentdosagesofdobutamine. 2.2Theory 2.2.1Notation Supposethereisarandomsampledrawnfromanaturalhumanpopu lation atHardy-Weinbergequilibrium.Inthissample,anumberofSNPs aregenotyped genomewideinordertoidentifyDNAsequencesresponsibleforco mplexdiseases

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16 ordrugresponse.Recentstudieshaveshownthatthehumangenome hasa haplotypeblockstructure(Patiletal.2001;Dawsonetal.200 2;Gabrielet al.2002),suchthatitcanbedividedintodiscreteblocksofli mitedhaplotype diversity.Ineachblock,asmallfractionofSNPs,referredtoas \tagSNPs,"can beusedtodistinguishalargefractionofthehaplotypes.Conside r R ( R> 1) tagSNPsforahaplotypeblock.Eachofthese R SNPshastwoallelesdenoted by A rk r ( k r =1 ; 2; r =1 ; R ),withallelefrequenciesdenotedby p ( r ) k r forthe r thSNP.Iusesuperscriptsandsubscriptstodistinguishbetweendie rentSNPs anddierentalleleswithinSNPs,respectively.TheseSNPsform2 R possible haplotypesexpressedas A 1k 1 A 2k 2 A Rk R ,whosefrequenciesaredenotedby p k 1 k 2 k R Thehaplotypefrequenciesarecomposedofallelefrequencie sateachSNPand linkagedisequilibriaofdierentordersamongSNPs(Louetal .2003).Therandom combinationofmaternalandpaternalhaplotypesgenerates 2 R 1 (2 R +1)diplotypes expressedas[ A 1k 1 A 2k 2 A Rk R ][ A 1l 1 A 2l 2 A Rl R ](1 k 1 l 1 2 ; ; 1 k R l R 2).These R SNPsform3 R observablemultilocuszygoticgenotytpes, generallyexpressedas A 1k 1 A 1l 1 =A 2k 2 A 2l 2 = =A Rk R A Rl R .WhenatmostoneSNPis heterozygous,thediplotypeisconsistenttoitszygoticgenot ype.However,when twoormoreSNPsareheterozygous,thegenotypewillhavedier entdiplotypes and,therefore,thenumberofmultilocusgenotypeswillbef ewerthanthenumber ofdiplotypes.Forexample,genotype A 11 A 11 =A 21 A 21 hasonediplotype[ A 11 A 21 ][ A 11 A 21 ], asdoesgenotype A 11 A 11 =A 21 A 22 ,thediplotype[ A 11 A 21 ][ A 11 A 22 ].However,foradouble heterozygousgenotype A 11 A 12 =A 21 A 22 ,Ihavetwodiplotypes[ A 11 A 21 ][ A 12 A 22 ]and [ A 11 A 22 ][ A 12 A 21 ].Let P [ k 1 k 2 k R ][ l 1 l 2 l R ] and P k 1 l 1 =k 2 l 2 = =k R l R denotethediplotypeand genotypefrequencies,respectively,and n k 1 l 1 =k 2 l 2 = =k R l R denotetheobservationsof variousgenotypes. Table2-1listsallpossiblegenotypesanddiplotypesattwoSNPs genotyped fromasampleofsize n .Twohaplotypescomprisingadiplotypecomefromfour

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17Table2{1:Possiblediplotypecongurationsofninegenotype sattwoSNPsandtheirhaplotypecompositionfrequencies RelativediplotypeHaplotypecompositionGenotypic GenotypeDiplotypeDiplotypefrequencyfreq.withingenot ypes A 11 A 21 A 11 A 22 A 12 A 21 A 12 A 22 Observationmeanvector A 11 A 11 =A 21 A 21 [ A 11 A 21 ][ A 11 A 21 ] P [11][11] = p 211 11000 n 11 = 11 u 2 A 11 A 11 =A 21 A 22 [ A 11 A 21 ][ A 11 A 22 ] P [11][12] =2 p 11 p 12 1 1 2 1 2 00 n 11 = 12 u 1 A 11 A 11 =A 22 A 22 [ A 11 A 22 ][ A 11 A 22 ] P [12][12] = p 212 10100 n 11 = 22 u 0 A 11 A 12 =A 21 A 21 [ A 11 A 21 ][ A 12 A 21 ] P [11][21] =2 p 11 p 21 1 1 2 0 1 2 0 n 12 = 11 u 0 A 11 A 12 =A 21 A 22 ( [ A 11 A 21 ][ A 12 A 22 ] [ A 11 A 22 ][ A 12 A 21 ] ( P [11][22] =2 p 11 p 22 P [12][21] =2 p 12 p 21 ( $1 $ 1 2 $ 1 2 (1 $ ) 1 2 (1 $ ) 1 2 $n 12 = 12 ( u 1 u 0 A 11 A 12 =A 22 A 22 [ A 11 A 22 ][ A 12 A 22 ] P [12][22] =2 p 12 p 22 10 1 2 0 1 2 n 12 = 22 u 0 A 12 A 12 =A 21 A 21 [ A 12 A 21 ][ A 12 A 21 ] P [21][21] = p 221 10010 n 22 = 11 u 0 A 12 A 12 =A 21 A 22 [ A 12 A 21 ][ A 12 A 22 ] P [21][22] =2 p 21 p 22 100 1 2 1 2 n 22 = 12 u 0 A 12 A 12 =A 22 A 22 [ A 12 A 22 ][ A 12 A 22 ] P [22][22] = p 222 10001 n 22 = 22 u 0 $ = p 11 p 22 p 11 p 22 + p 12 p 21 where p 11 p 12 p 21 and p 22 arethehaplotypefrequenciesof A 11 A 21 A 11 A 22 A 12 A 21 ,and A 12 A 22 ,respectively.Haplotype A 11 A 21 isassumedasthereferencehaplotype.

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18 possiblehaplotypes, A 11 A 21 A 11 A 22 A 12 A 21 and A 12 A 22 ,withrespectivefrequencies.The diplotypefrequenciescanbeexpressedintermsofthehaploty pefrequencies(Table 2-1).Twodiplotypes[ A 11 A 21 ][ A 12 A 22 ]and[ A 11 A 22 ][ A 12 A 21 ]ofadoubleheterozygote A 11 A 12 =A 21 A 22 havefrequencies p 11 p 22 and p 12 p 12 ,respectively.Thus,therelative frequenciesofthesetwodiplotypesforthisdoubleheterozy goteareafunctionof haplotypefrequencies.Table2-1alsogivestherelativeexpe ctedfrequenciesof haplotypescontainedinagivengenotype.Allgenotypes,exce ptforthedouble heterozygote,containoneortwoknownhaplotypes.Forexamp le,genotype A 11 A 11 =A 21 A 21 hasonehaplotype A 11 A 21 ,whereas A 11 A 11 =A 21 A 22 hasonehalfhaplotype A 11 A 21 andonehalfhaplotype A 11 A 22 .Thedoubleheterozygotecontainsfourpossible haplotypes,withtherelativefrequencies p 11 p 22 p 11 p 22 + p 12 p 21 forhaplotypes A 11 A 21 and A 12 A 22 and p 12 p 21 p 11 p 22 + p 12 p 21 forhaplotypes A 11 A 22 and A 12 A 21 2.2.2LikelihoodFunctions Thecompletedataarediplotypecongurationsatagivenseto fSNPsforeach genotypeandpatients'drugeectsatdierentdoages,wherea stheobserveddata arethegenotypesoftheseSNPsandtheoutcomesofdrugeects.T heconnection betweenthegenotypesandthediplotypesareviewedasthemi ssingdata. Thehaplotypefrequencies,identiedin n p =( p 11 ;p 12 ;p 21 ;p 22 ),belongtothe setof populationgeneticparameters thatcanbeestimatedusingthenineobserved genotypes( G )fortwoSNPs(Table2-1).Thelog-likelihoodfunctionofunk nown haplotypefrequenciesgivenobservedgenotypescanbewritt eninmultinomialform, i.e., log L ( n p j G ) / 2 n 11 = 11 log p 11 + n 11 = 12 log(2 p 11 p 12 )+2 n 12 = 12 log p 12 + n 12 = 11 log(2 p 11 p 21 )+ n 12 = 12 log[2( p 11 p 22 + p 12 p 21 )] + n 12 = 22 log(2 p 12 p 22 )+2 n 21 = 21 log p 21 + n 21 = 22 log(2 p 21 p 22 ) +2 n 22 = 22 log p 22 (2.1)

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19 Iintendtoassociatediplotypeswithinterpatientvariation indrugresponse basedonobserveddrugresponesmeasuredatdierentdosages( y )andSNP genotypes( G ).Generallyspeaking,agiventwo-SNPgenotype, A 1k 1 A 1l 1 =A 2k 2 A 2l 2 canbepartitionedintotwopossiblediplotypes,[ A 1k 1 A 2k 2 ][ A 1l 1 A 2l 2 ]and[ A 1k 1 A 2l 2 ] [ A 1l 1 A 2k 2 ].Statistically,thisisamixturemodelproblemwithtwocom ponents(i.e., diplotyes)havingdierentproportions.Thelog-likelihood functionofobserveddata isformulatedas log L ( n p ; n q j y ; G )= n X i =1 log[ $ i f [ k 1 k 2 ][ l 1 l 2 ] ( y i )+(1 $ i ) f [ k 1 l 2 ][ l 1 k 2 ] ( y i )] ; (2.2) where n p iscontainedwithinthemixtureproportion, $ i = P [ k 1 k 2 ][ l 1 l 2 ] i P [ k 1 k 2 ][ l 1 l 2 ] i + P [ k 1 l 2 ][ l 1 k 2 ] i ; and n q isasetof quantitativegeneticparameters thatspecifythemultivariatenormaldistribution, f ,whichincludesdiplotype-specicparameters,i.e.,phenot ypic meansoftwodierentdiplotypesatdierentdosages( u [ k 1 k 2 ][ l 1 l 2 ] and u [ k 1 l 2 ][ l 1 k 2 ] ), andparameterscommontobothdiplotypes,i.e.,the(co)vari ancematrixamong dosages( ). Supposethereexistsaparticularhaplotype, A 1k 1 A 2k 2 ,labelledby A ,whichis dierentfromtheotherthreehaplotypes,collectivelylabe lledby a ,initseect ondrugresponse.Theresultantdiplotypesarethusequivalent tothree composite genotypes AA Aa and aa .Thephenotypicmeanofeachofthreecomposite genotypesthatcontainsthetwodistinctgroupsofhaplotype sisdenotedby u j for compositegenotype j ( j =2for AA ,1for Aa and0for aa ).Consideringdrug responseatdierentconcentrations,each u j canbetbyaclinicallymeaningful pharmacodynamicmodel.OnesuchmodelistheE max modelthatspeciesthe relationshipbetweendrugconcentration(C)anddrugeect( E)(Giraldo2003).

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20 Thismodelisbasedontheequation E j (C)=E 0 j + E max j C H j EC H j 50 j +C H j ; (2.3) whereE 0 j istheconstantorbaselinevalueforthedrugresponseparameter ,E max j istheasymptotic(limiting)eect,EC 50 j isthedrugconcentrationthatresultsin 50%ofthemaximaleect,andH j istheslopeparameterthatdeterminestheslope oftheconcentration-responsecurve.ThelargerH j ,thesteeperthelinearphase ofthethelog-concentration-eectcurve.Byestimatingthe securveparameters separatelyfordierentgenotypes,onecandeterminehowtheD NAsequence variantsinruencedrugresponsebasedontheshapedierencesam ongthethree curves. Asalongitudinaltrait,the(co)variancematrixofdrugrespo nsecanbe structuredbymanystatisticalmodels,suchasarst-orderautore gressive[AR(1)] model(Gongetal.2004),whichstatesthatthevariance( 2 )isconstantover dierentconcentrationsandthatthecorrelationofresponse betweendierent concentrationsdecreasesproportionally(in )withincreasedconcentrationinterval. Assumingthathaplotype A 11 A 21 isdierentfromtheotherhaplotypes(Table 2-1),thelog-likelihoodfunctioncanbeexpandedtoinclud eallpossibleSNP genotypes,nowexpressedas log L ( n p ; n q j y ; G )= n 11 = 11 X i =1 log f 2 ( y i ) + n 11 = 12 + n 12 = 11 X i =1 log f 1 ( y i ) + n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 X i =1 log f 0 ( y i ) + n 12 = 12 X i =1 log[ $f 2 ( y i )+(1 $ ) f 0 ( y i )] ; (2.4)

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21 whereunknownvector n q nowcontainsthecurveandmatrix-structuringparameters,arrayedby(E 0 j ; E max j ; EC 50 j ; H j ; 2 ; ). 2.2.3AnIntegrativeEMAlgorithm Aclosed-formsolutionforestimatingtheunknownparametersw iththeEM algorithmisderivedinwhichhaplotypefrequenciesareexp ressedasafunctionof allelicfrequenciesandLD.Foratwo-SNPhaplotype,use p k 1 k 2 = p (1)k 1 p (2)k 2 +( 1) k 1 + k 2 D; (2.5) where D isthelinkagedisequilibriumbetweenthetwoSNPs.Thus,onceha plotype frequenciesareestimated,Icanestimateallelicfrequencie sandLDbysolving equation(2.5).Theestimatesofhaplotypefrequenciesareb asedontheloglikelihoodfunctionofequation(2.1),whereastheestimate sofdiplotypecurve parametersand(co)variance-structuringparametersareba sedonthelog-likelihood functionofequation(2.4).Thesetwodierenttypesofparam eterscanbeestimated usinganintegrativeEM-simplexalgorithm. IntheEstep,theexpectedvalueof $ i forsubject i havingdoubleheterozygousgenotypecarryingdiplotype[ A 11 A 21 ][ A 12 A 22 ]iscalculatedusing $ [11][22] i = p 11 p 22 p 11 p 22 + p 12 p 21 (2.6) Notethatforalltheothergenotypes,thisprobabilitydoesno texist. IntheMstep,theprobabilitiescalculatedinthepreviousit erationareusedto estimatethehaplotypefrequenciesusing ^ p 11 = 2 n 11 = 11 + n 11 = 12 + n 11 = 22 + P n 12 = 12 i =1 $ [11][22] i 2 n ; (2.7) ^ p 12 = 2 n 11 = 22 + n 11 = 12 + n 12 = 22 + P n 12 = 12 i =1 (1 $ [11][22] i ) 2 n ; (2.8) ^ p 21 = 2 n 22 = 11 + n 12 = 11 + n 22 = 12 + P n 12 = 12 i =1 (1 $ [11][22] i ) 2 n ; (2.9) ^ p 22 = 2 n 22 = 22 + n 22 = 12 + n 22 = 11 + P n 12 = 12 i =1 $ [11][22] i 2 n ; (2.10)

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22 TheseestimatedfrequenciesareembeddedtotheMstepforestima ting n q derived fromthesimplexalgorithm(Zhaoetal.2004).Iterationsoft heEandMstepis continueduntiltheestimatesoftheparametersconvergetost ablevalues.The asymptoticvarianceoftheseparameterscanbeestimatedbycal culatingLouis' observedinformationmatrix(Louis1982)(APPENDIXA).2.2.4HypothesisTests Twomajorhypothesesaretestedinthefollowingsequence:(1)t heassociation betweentwoSNPsbytestingtheirLD,and(2)thedierenceofag ivenhaplotype fromtheotherhaplotypesinitseectondrugresponse.Thenul landalternative hypothesesontheLDbetweentwogivenSNPsare: 8><>: H 0 : D =0 H 1 : D 6 =0 (2.11) Thelog-likelihoodratioteststatisticforthesignicanceofL Discalculatedby comparingthelikelihoodvaluesunderthe H 1 (fullmodel)and H 0 (reducedmodel) hypothesesandproduces LR 1 = 2[log L ( e p (1)1 ; e p (2)1 ;D =0 ; e n q j G ) log L ( b n p ; b n q j G )](2.12) wherethetildeandhatdenotetheMLEsofunknownparameters under H 0 and H 1 TheLR 1 teststatisticisconsideredtoasymptoticallyfollowa 2 distributionwith onedegreeoffreedom.TheMLEsofallelicfrequenciesunder H 0 canbeestimated usingtheEMalgorithmdescribedabove,butwiththeconstraint p 11 p 22 = p 12 p 21 imposed. Diplotypeorhaplotypeeectsonacomplextraitcanbetested usingthenull andalternativehypothesesexpressedas 8><>: H 0 :(E 0 j ; E max j ; EC 50 j ; H j )=(E 0 ; E max ; EC 50 ; H) ; forall j =2 ; 1 ; 0 H 1 :Atleastoneequalityin H 0 doesnothold (2.13)

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23 Thelog-likelihoodratioteststatistic(LR 2 )underthesetwohypothesescanbe similarlycalculated.TheLR 2 mayasymptoticallyfollowa 2 distributionwith eightdegreesoffreedom.However,theapproximationofa 2 distributionmay beinappropriatewhensomeregularityconditionsareviolat ed.Thepermutation testapproachdoesnotrelyuponthedistributionoftheLR 2 andmaybeused todeterminethecriticalthresholdfordeterminingtheeec tofDNAsequence variationondrugresponse.2.2.5 R -SNPSequenceModel Theideaforsequencingdrugresponsebasedonatwo-SNPmodelcanb eextended toincludeanarbitrarynumberofSNPswhosesequencesareassocia tedwith thephenotypicvariation.Consider R SNPsthatform3 R observablemultilocus zygoticgenotypes,generallyexpressedas A 1k 1 A 1l 1 =A 2k 2 A 2l 2 = =A Rk R A Rl R .These genotypesarecollapsedfromatotalof2 R 1 (2 R +1)diplotypesexpressedas [ A 1k 1 A 2k 2 A Rk R ][ A 1l 1 A 2l 2 A Rl R ](1 k 1 l 1 2 ; ; 1 k R l R 2).Akey issueforthemulti-SNPsequencingmodelishowtodistinguishamon g2 r 1 dierent diplotypesforthesamegenotypeheterozygousat r loci.Therelativefrequenciesof thesediplotypescanbeexpressedintermsofhaplotypefrequen cies.Theintegrative EMalgorithmcanbeemployedtoestimatetheMLEsofhaplotype frequencies. Louetal.(2003)providedageneralformulaforexpressinghap lotypefrequenciesin termsofallelefrequenciesandlinkagedisequilibriaofdi erentorders.TheMLEsof thelattercanbeobtainedbysolvingasystemofequations. Inthemulti-SNPsequencingmodel,Ifacemanyhaplotypesandh aplotype pairs.AnAIC-basedmodelselectionstrategyhasbeenframedtodet erminethe haplotypethatismostdistinctfromtheresthaplotypesinexpl ainingquantitative variation.

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24 2.3Application Arealexampleforageneticstudyofcardiovasculardiseaseisuse dtodemonstratetheusefulnessofourmodel.Cardiovasculardisease,princ ipallyheartdisease andstroke,istheleadingkillerforbothmenandwomenamonga llracialand ethnicgroups.Dobutamineisamedicationthatisusedtotreat congestiveheart failurebyincreasingheartrateandcardiaccontractility, withactionsontheheart similartotheeectofexercise.Dobutamineisalsocommonlyuse dtoscreenfor heartdiseaseinthoseunabletoperformanexercisestresstest.Iti sthislatteruse forwhichthestudyparticipantsreceiveddobutamineinthis study.Itisasynthetic catecholaminethatprimarilystimulates -adrenergicreceptors( AR),whichplay animportantroleincardiovascularfunctionandresponsestod rugs(Johnsonand Terra2002;Ranadeetal.2002;Nabel2003). Boththe 1ARand 2ARgeneshaveseveralpolymorphismsthatarecommoninthepopulation.Twocommonpolymorphismsarelocateda tcodons49 (Ser49Gly)and389(Arg389Gly)forthe 1ARgeneandatcodons16(Arg16Gly) and27(Gln27Glu)forthe 2ARgene(Nabel2003).Thepolymorphismsineach ofthesetworeceptorgenesareinlinkagedisequilibrium,whi chsuggeststheimportanceoftakingintoaccounthaplotypes,ratherthanasing lepolymorphism, whendeningbiologicfunction.Thisstudyattemptstodetec thaplotypevariants withinthesecandidategeneswhichdeterminetheresponseofhe artratetovarying concentrationsofdobutamine. Agroupof163menandwomeninagesfrom32to86yearsoldparti cipated inthisstudy.Patientshadawiderangeoftesting(untreated) heartrate.Each ofthesesubjectswasgenotypedforSNPmarkersatcodons49and3 89within the 1ARgeneandatcodons16and27withinthe 2ARgene.Dobutamine wasinjectedintothesesubjectstoinvestigatetheirresponsein heartratetothis drug.Thesubjectsreceivedincreasingdosesofdobutamine,un tiltheyachieved

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25 targetheartrateresponseorpredeterminedmaximumdose.Thed oselevelsused were0(baseline),5,10,20,30and40mcg|min,ateachofwhich heartratewas measured.Thetimeintervalof3minutesisallowedbetweentw osuccessivedoses forsubjectstoreachapateauinresponsetothatdose.Onlythose( 107)inwhom therewereheartratedataatallthesixdoselevelswereinclud edfordataanalyses. Byassumingthatonehaplotypeisdierentfromtherestoftheha plotypes, IhopetodetectaparticularDNAsequenceassociatedwiththeresp onseofheart ratetodobutamine.Thephenotypicdatafordrugresponsewere normalized aspercentagestoremovethebaselineeect,whichisduetobe tween-subject dierencesinheartratepriortothetest.Atthe 1ARgene,Ididnotndany haplotypethatcontributedtointer-individualdierence inheartrateresponse.A signicanteectwasobservedforhaplotypeGly16(G){Glu27( G)withinthe 2AR gene(Table2-2).Thelog-likelihoodratio(LR)teststatistic sbasedonequation (2.12)forthedierenceofGGfromtheother3haplotypeswas 30.03,whichis signicantat P =0 : 021basedonthecriticalthresholddeterminedfrom1000 permutationtests.TheLRvalueswhenselectinganyhaplotyper atherthanGGas areferencegavenosignicantresults( P =0 : 16 0 : 40).Iusedasecondtesting criterionbasedontheareaundercurve(AUC)totestthehaploty peeect.This testsupportstheresultfromthersttest. Themaximumlikelihoodestimates(MLEs)ofthepopulationgen eticparameters,suchashaplotypefrequencies,allelefrequenciesand linkagedisequilibrium betweenthetwoSNPs,withinthe 2ARgenewereobtainedfromourmodel.As indicatedbytheasymptoticvarianceoftheMLEsbasedonLouis' (1982)approach, theseestimatesdisplayreasonableprecision.Theallelefreque ncieswithinthis geneareestimatedas0.62forGly16atcodon16and0.40forGlu 27atcodon27. TheMLEofthelinkagedisequilibriumbetweenthetwoSNPsis0. 1303.These

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26 suggestthatthetwoSNPsidentiedwithinthe 2ARgenedisplayaprettyhigh heterozygosityandlinkagedisequilibrium. TheMLEsofthequantitativegeneticparameterswereobtain ed,alsowithreasonableestimationprecision(Table2-2).Usingtheestimatedresp onseparameters, Idrewtheprolesofheartrateresponsetoincreasingdoselevel sofdobutaminefor threecompositegenotypescomprisingofhaplotypesGGandnon -GG(symbolized by GG)(Figure2-1).Thecompositehomozygote[GG][GG]displaye dconsistently higherheartrateacrossalldoselevels,especiallyathigherdo selevelsthanthe compositehomozygote[ GG][ GG].Butthecompositeheterozygotehadconsistently thelowestcurveatalldoselevelstested.IusedAUCtotestinwhich geneaction mode(additiveordominant)haplotypesaectdrugresponsecu rvesforhearrate. Thetestingresultssuggestthatbothadditiveanddominanteec tsareimportant indeterminingtheshapeoftheresponsecurve(Table2-3),toge theraccountingfor about14%oftheobservedvariationindrugresponse.Ididnotde tectevidencefor haplotypestohaveaneectoncurveparameters,HandEC 50 ,fortheheartrate response. Iperformedsimulationstudiestoinvestigatethestatisticalpr opertiesofour model.Thedataweresimulatedbymimickingtheexampleusedab oveinorderto determinethereliabilityofourestimatesinthisrealappli cation.Onehaplotype wasassumedtobedierentfromtheotherthree.Thedatasimulat edunder thisassumptionweresubjecttostatisticalanalyses,pretendingt hathaplotype distinctionisunknown.Asexpected,onlyunderthecorrectha plotypedistinction couldthehaplotypeeectbedetectedandtheparametersbea ccuratelyand preciselyestimated(Table2-4). 2.4Discussion Singlenucleotidepolymorphisms(SNPs)arepowerfultoolsfor studyingthe structureandorganizationofthehumangenome(Patiletal.2 001;Dawsonetal.

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27 Table2{2:Log-likelihoodratio(LR)teststatisticsofdiere nthaplotypemodels andthecorrespondingmaximumlikelihoodestimates(MLEs)ofp opulationgenetic (SNPallelefrequenciesandlinkagedisequilibria)andquant itativegeneticparameters(drugresponseand(co)variance-structuringparameters) inasampleof107 subjectswithinthe 2ARgene.TheasymptoticvarianceoftheMLEsaregivenin theparentheses. CompositeReferencehaplotype[ A 1k 1 A 2k 2 ] genotypeParameters[AC][AG][GC] [GG] LR 1 12.1419.3211.18 30.03 P value0.340.160.40 0.02 Populationgeneticparameters b p 11 0.380.380.62 0.62(0.04) b p 21 0.600.400.60 0.40(0.04) b D 0.130.050.05 0.13(0.01) Quantitativegeneticparameters [ A 1k 1 A 2k 2 ][ A 1k 1 A 2k 2 ] b E 0 0.110.020.10 0.11(0.02) b E max 0.370.420.37 0.75(0.26) c EC 50 23.7232.6021.35 42.10(15.90) b H1.932.482.34 1.73(0.29) [ A 1k 1 A 2k 2 ][ A 1k 1 A 2k 2 ] b E 0 0.100.110.10 0.10(0.01) b E max 0.370.390.36 0.39(0.06) c EC 50 25.8738.2525.24 29.27(4.90) b H1.951.692.05 2.01(0.25) [ A 1k 1 A 2k 2 ][ A 1k 1 A 2k 2 ] b E 0 0.110.110.11 0.10(0.01) b E max 0.500.440.56 0.39(0.04) c EC 50 31.0926.8735.05 23.57(2.68) b H1.992.011.79 2.04(0.20) b 0.890.880.89 0.88(0.01) b 2 0.010.010.01 0.01(7e-4) TheLR 1 testsforthesignicanceofhaplotypeeectbasedonhypothesis (14).The optimalhaplotypemodeldetectedonthebasisoftheLRtestisi ndicatedinboldface.TherearetwoallelesArg16(A)andGly16(G)atcodon16an dtwoalleles Gln27(C)andGlu27(G)atcodon27.

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28 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 90 100 Dobutamine concentration (mcg)Heart rate (%)[GG][GG] [GG][GG] [GG][GG] Figure2{1:Prolesofheartrateinresponsetodierentconce ntrationsofdobutamine(indicatedbydots)forthreecompositegenotypes(foreg round)identiedat twoSNPswithinthe 2ARgene.Theprolesof107studiedsubjectsfromwhich thethreedierentcompositegenotypesweredetectedarealso shown(background). Table2{3:Testingresultsfortwodrugresponseparameters,Hand EC 50 ,andtotalgenetic,additiveanddominanteectsbasedonAUCin107sub jectsunderthe optimalhaplotypemodel[GG] TestHEC 50 GeneticAdditiveDominant LR0.764.1817.647.2519.83P value > 0.05 > 0.05 < 0 : 001 < 0 : 01 < 0 : 01

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29Table2{4:Maximumlikelihoodestimates(MLEs)ofSNPallelefr equencyandlinkagedisequilibriumandparametersdescribingthethreedynamiccurvesbasedonthesigmoidalEmaxmodel. Thenumbersinparenthesesarethesquaredrootsofthe meansquareerrorsoftheMLEsbasedon1000simulationreplicat es. Composite Referencehaplotype[ A 1k 1 A 2k 2 ] genotypeParameters[ A 11 A 21 ][ A 11 A 22 ][ A 12 A 21 ][ A 12 A 22 ] Populationgeneticparametersp 11 =0 : 62 0.62(0.03) 0.62(0.03)0.61(0.03)0.61(0.03) p 21 =0 : 40 0.40(0.03) 0.40(0.03)0.40(0.03)0.40(0.04) D =0 : 13 0.13(0.01) 0.13(0.01)0.13(0.01)0.13(0.01) Quantitativegeneticparameters [ A 1k 1 A 2k 2 ][ A 1k 1 A 2k 2 ]E 0 =0 : 11 0.11(0.02) 0.10(0.04)0.10(0.06)0.10(0.02) E max =0 : 75 0.75(0.24) 0.42(0.37)0.39(0.43)0.40(0.36) EC 50 =42 : 10 41.97(14.24) 25.08(20.27)24.36(21.71)24.83(18.77) H=1.73 1.72(0.29) 2.29(0.85)1.94(0.98)2.12(0.57) [ A 1k 1 A 2k 2 ][ A 1k 1 A 2k 2 ]E 0 =0 : 10 0.10(0.01) 0.10(0.01)0.10(0.05)0.10(0.01) E max =0 : 39 0.40(0.07) 0.40(0.07)0.43(0.25)0.39(0.06) EC 50 =29 : 27 30.23(5.17) 27.70(5.93)30.96(18.22)27.05(5.38) H=2.01 2.02(0.27) 2.01(0.27)2.48(1.05)2.05(0.26) [ A 1k 1 A 2k 2 ][ A 1k 1 A 2k 2 ]E 0 =0 : 10 0.10(0.02) 0.10(0.01)0.10(0.01)0.10(0.01) E max =0 : 39 0.39(0.05) 0.46(0.11)0.43(0.06)0.51(0.18) EC 50 =23 : 57 24.00(3.26) 30.56(8.87)28.50(5.74)33.35(13.26) H=2.04 2.05(0.24) 1.94(0.24)1.97(0.18)1.91(0.27) =0 : 88 0.88(0.01) 0.89(0.01)0.89(0.01)0.89(0.01) 2 =7e-3 7e-3(7e-4) 8e-3(9e-4)8e-3(1e-3)8e-3(9e-4) Thecorrecthaplotypemodelandthecorrespondingestimatesa reindicatedinboldface.

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30 2002;Gabriel2002).Therecentlydevelopedhaplotypemapo rHapMap(The InternationalHapMapConsortium2003)providesaninvaluabl eresourcefor understandingthestructure,organizationandfunctionofth ehumangenome. Theunderstandingofthersttwoaspects,genomestructureandor ganization, havebeenlessproblematicinpartbecausefewerstatisticsareu sed,butthe associationbetweenspecicgenomicsitesanddiseaseriskordrugr esponseisa pressingchallengeincurrentpharmacogeneticandpharmacog enomicstudies.The modelpresentedhere,aimedatdetectingspecicDNAsequenceva riantsfordrug response,representsatimelyeorttoacceleratetheresearcha tidentifyinggenesof interest. ThepresentedmodelisfoundedonrecentlydiscoveredtagSNPsi nthe genome,andallowsforafastscanfortheassociationbetweenvar iationinDNAsequenceandtraits(Patiletal.2001;Dawsonetal.2002;Gabri el2002).Thismodel hasthreeadvantages.First,itsolidiesthegeneticbasisforq uantitativevariation bydirectlycharacterizingspecicDNAsequencespredisposedto drugresponse. Thetraditionalstatisticalmodelsforgeneticmappingattem pttopostulatethe positionofhypothesizedQTLthatarelinkedwithknownmarker sgenotypedfrom thegenome.TheQTLdetectedfromthesemodelsareregardedas \hypothesized" becauseitisnotpossibletoknowtheirDNAsequencesand,therefo re,physiological function.Asopposedtothetraditional\indirect"approach, thismodelpresents a\direct"approach.Atpresent,theutilityofthedirectapp roachislimitedto sequencingfunctionalpartsofcandidategeneswithknownbi ochemicalorphysiologicalfunction.WiththereleaseofHapMap,thismodelmakes thedirectapproach bothusefulandecientinsearchingforcausalvariantsthroug houtthewhole genome.Second,thismodelisstatisticallysimpleandcomputa tionallyfast.The mostdicultpartofthemodelestimationisconstructingdiplo typecongurations forheterozygousgenotypesattwoormoreSNPs.Theestimationo fpopulation

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31 geneticparametersisbasedonamultinomiallikelihoodfunc tionoftheobserved genotypedata,whereastheestimationofquantitativegenet icparametersbased onamixture-basedlikelihoodfunctionincludingdierentd iplotypes.Thesetwo likelihoodfunctioncanbeeasilyintegratedtoauniedestim ationframework implementedwiththeEMalgorithm. Finally,thismodelisrobustandrexible,andabletoaccommo datedierent geneticandexperimentalsettings.Resultsfromthesimulation studyindicate thattheassociationbetweenDNAsequenceandphenotypecanbewe lldetected whenthetraithasamodestheritabilitylevel(0.14)oramode stsamplesize(107) isused.Thismodelcanalsoobtainfairlypreciseestimationofp arameterswhen diplotypesdisplayoverdominanceinthesituationwithmodest heritabilityand samplesize.Thespecicutilityofthismodeltoarealexamplef romagenetic studyleadstothesuccessfuldetectionofaDNAsequence(haplotyp e)atcodons 16and27genotypedwithinthe 2ARcandidategeneforitssignicantimpacton responseinheartratetodobutamine.Thishaplotype,composed oftheGly16form ofcodon16andtheGlu27formofcodon27,tendstoincreasehea rtratewhenit iscombinedwithitselforanyotherhaplotypes,andaccountfo rabout14%ofthe totalobservedvariationindrugresponse. Althoughthesimulationandexamplewerebasedon2-SNPanalyses,t he sequencingmodelusedwasdevelopedtoallowforthedetection ofsequencevariants involvinganynumberofSNPswithinahaplotypeblock.Inaddi tiontoitsusein studyinggeneticassociationsinnaturalpopulations,theseque ncingmodelcanbe extendedtostudythegeneticfactorscontributingtovariat ionindrugresponse incontrolledcrossessuchasthebackcrossorF 2 asusedinmouse.Itcanalsobe modiedtoestimatetheeectsofsequence-sequenceinteracti onondrugresponse. Itispossiblethatahaplotypewithinacandidategeneinterac tswithhaplotypes fromothercandidategenes.ThecharacterizationofspecicD NAsequencevariants

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32 fordrugresponseshouldallowthedevelopmentofteststopredic twhichdrugsor vaccineswouldbemosteectiveinindividualswithparticul argenotypesforgenes aectingdrugmetabolism.

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CHAPTER3 AJOINTMODELFORSEQUENCINGDRUGEFFICACYANDTOXICITY 3.1Introduction Theadministrationofaspecicdrugtopatientscanproducetw odierent responses,desiredtherapeuticeects(ecacy)andadverseeec ts(toxicity). Evidenceisincreasingforobservedinruencesofgeneticdie rencesonthesetwo responses(reviewedinEvansandJohnson2001;JohnsonandEvans 2002;Evans andMcLead2003;Weinshilboum2003).However,thegeneticcon trolofboth ecacyandtoxicityistypicallycomplex,withmultiplegen esinteractingwith variousbiochemical,developmentalandenvironmentalfac torsincoordinatedways todeterminetheoverallphenotypes(Johnson2003;Wattersa ndMcLeod2003). Withtheadventofrecentgenomictechnologies,inter-indiv idualdierencesindrug responsecannowbeexplainedbyDNAsequencevariantsingenesth atencode themetabolismanddispositionofdrugsandthetargetsofdrugt herapy(suchas receptors)(EvansandRelling1999;McLeodandEvans2001).T ocomprehensively understandthegeneticbasesofecacyandtoxicityand,ultim ately,designindividualizedmedicationswithmaximumfavorableeectsandm inimumunfavorable eects,approachesmustbedevelopedinwhichnewspecicgenes foreachresponse canbeidentied. Twoapproachesthathavebeendevelopedtodetectgenesfora complextrait. Therstapproachistheindirectinferenceofcausalgeneticl ociorquantitative traitloci(QTL)basedontheirco-segregatingmarkers(Lande randBotstein1989; WuandCasella2005).TheQTLdetectedfromthisapproachishy pothetical whoseDNAstructureandorganizationisunknown.High-throughp uttechnologies 33

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34 ofsinglenucleotidepolymorphisms(SNPs)haveprovidedapower fulmethod forsequencingcandidategenesthathavebeenknowntoaectc omplexdiseases ordrugresponse.Therecentdevelopmentofthehaplotypemapo rHapMap constructedbyanonymousSNPs(TheInternationalHapMapConsor tium2003) hasmadeitpossibletogenome-widescanfortheexistenceanddist ributionof functionalSNPsbasedonassociationanalysisandnarrowdownthe genomic regionsthatharborcausalSNPs.Motivatedbythesedevelopment s,Liuetal. (2004)proposedasecondapproachthatcandirectlyassociateDNA sequence variantswiththephenotypicvariation.Thisapproachhasp owertodetecttheDNA sequencewhereindividualsdieratasingleDNAbase. Unlikeusualcomplextraits,drugresponsehassomedynamiccharac teristic inwhichindividualsrespondtovaryingdrugdosagesorconcen trations.Drug responsecanbethereforeregardedasfunction-valuedorlong itudinaltraits.The geneticarchitectureoffunction-valuedtraitscanbestudi edusingthemarkerbased functionalmapping model,developedbyR.Wuandcolleagues(Maet al.2002;Wuetal.2002b,2003,2004a,2004b).Functionalma ppingidenties dynamicQTLresponsibleforabiologicalprocessthatneedbeme asuredata nitenumberoftimepoints.Inmodellingfunctionalmapping ,fundamental principlesbehindbiologicalorbiochemicalnetworksdescr ibedbymathematical functionsareincorporatedintoaQTLmappingframework.Fu nctionalmapping estimatesparametersdeterminingshapesandfunctionsofapa rticularbiological network,ratherthandirectlyestimatesgeneeectsatallpo ssiblepointswithinthe network.Becauseoftheconnectionofthesepointsthroughmat hematicalfunctions, functionalmappingstrikinglyreducesthenumberofparamet erstobeestimated and,hence,displaysincreasedstatisticalpower. Takingadvantagesofsequence-basedassociationstudiesandfun ctionalmapping,Iattempttoproposeanewmodelthatcancharacterizeth eDNAsequence

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35 structureofdrugresponseandcomparethegeneticdierencesb etweenecacy andtoxicityatthesingleDNAbaselevel.Inthenextsections,Ir stintroduce basicprinciplesforsequenceandfunctionalmappingandthen formulateaunifying likelihoodfunctionforthedetectionofspecicDNAsequencev ariantsthatdeterminedrugecacyandtoxicity.Thesewillbefollowedbythein vestigationofthe statisticalpropertiesofthismodelthroughextensivesimulat ionstudies. 3.2BasicPrincipleForSequenceMapping TraditionalQTLmappingstudiestherelationshipbetweenput ativeQTL genotypesandphenotypes,whereassequencemappingassociates congurationsof SNPgenotypes(i.e.,diplotypes)withphenotypes.Sequencema ppingreliesupon thethecharacterizationofSNPsfromtheentirehumangenome .Recentstudies haveshownthatthehumangenomehasahaplotypeblockstructur e(Patiletal. 2001;Gabrieletal.2002),suchthatitcanbedividedintodisc reteblocksoflimited haplotypediversity.Ineachblock,asmallfractionofSNPs,ref erredtoas\tag SNPs",canbeusedtodistinguishalargefractionofthehaplotype s. Forsimplicity,IrstconsidertwoSNPswithinahaplotypeblock thatare co-segregatingwiththelinkagedisequilibriumof D inahumanpopulationat Hardy-Weinbergequilibrium.EachSNPhastwoalleles1and2wi ththerelative proportionsof p (1)1 and p (1)2 aswellas p (2)1 and p (2)2 ,respectively,wherethesuperscriptstandsfortheidenticationofSNPand p (1)1 + p (1)2 =1and p (2)1 + p (2)2 =1. ThesetwoSNPsform4possiblehaplotypes11,12,21and22whosefre quenciesare expressedas p r 1 r 2 = p (1)r 1 p (2)r 2 +( 1) r 1 + r 2 D; where r 1 ;r 2 =1 ; 2denotetheallelesofthetwoSNPs,respectively, P 2r 1 =1 P 2r 2 =1 p r 1 r 2 =1(LynchandWalsh1998).Ifthehaplotypefrequenciesarekn own,then theallelicfrequenciesandlinkagedisequilibrium,arraye dbythepopulationgenetic parametervector n p = f p (1)r 1 ;p (2)r 2 ;D g ,canbesolvedwiththeaboveequation.

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36 Therandomcombinationofmaternalandpaternalhaplotypes generates 10distinctdiplotypesexpressedas[11][11], ,[22][22]whicharesortedinto 9genotypes11/11, ,22/22(Table3-1).Thedoubleheterozygoticgenotype 12/12containstwopossiblediplotypes[11][22]and[12][21] .Iuse P [ r m 1 r m 2 ][ r p 1 r p 2 ] (= p r m 1 r m 2 p r p 1 r p 2 )and P r 1 r 0 1 =r 2 r 0 2 todenotethediplotypeandgenotypefrequencies, respectively,and n r 1 r 0 1 =r 2 r 0 2 todenotetheobservationsofvariousgenotypes( G ), where m and p describethematernalandpaternaloriginsofhaplotypesand 1 r 1 r 0 1 2,1 r 2 r 0 2 2.Thefrequenciesandobservationsofallgenotypes, exceptforgenotype12/12,areequivalenttothoseofthecorr espondingdiplotypes. Iintendtoassociatediplotypeswithinter-patientvariatio ninaquantitative traitbasedonobservedphenotypicvalues( Y )assumedtobenormallydistributed andSNPgenotypes( G )assumedtobemultinomiallydistributed.Withoutlossof generality,Iassumethathaplotype11isdierentfromtherest ofthehaplotypes, cumulativelyexpressedas 11,intriggeringaneectonthephenotype.Icallsuch adistincthaplotype11the reference haplotype.Thereferenceandnon-reference haplotypesgeneratesthreecombinationscalledthe compositegenotypes .The genotypicmeansofthecompositegenotypes, j ( j =2for[11][11],1for[11][ 11] and0for[ 11][ 11]),andcommonresidualvariancewithinthecompositegenot ype, 2 ,thatbelongtoquantitativegeneticparametersarearraye dby n q = f j ; 2 g Iconstructedtwolog-likelihoodfunctions,oneinamultinom ialformandthe otherinamixturemodelform,toestimatethepopulationandq uantitativegenetic parametersthatare,respectively,expressedas log L ( n p jG )=Constant (3.1) +2 n 11 = 11 log p 11 + n 11 = 12 log(2 p 11 p 12 )+2 n 11 = 22 log p 12 + n 12 = 11 log(2 p 11 p 21 )+[ $n 12 = 12 log(2 p 11 p 22 )+(1 $ ) n 12 = 12 log(2 p 12 p 21 )] + n 12 = 22 log(2 p 12 p 22 )+2 n 22 = 11 log p 21 + n 22 = 12 log(2 p 21 p 22 )+2 n 22 = 22 log p 22 ;

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37 and log L ( n p ; n q jY ; G )= n 11 = 11 X i =1 log f 2 ( y i ) + n 11 = 12 + n 12 = 11 X i =1 log f 1 ( y i ) + n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 X i =1 log f 0 ( y i ) + n 12 = 12 X i =1 log[ $f 1 ( y i )+(1 $ ) f 0 ( y i )] ; (3.2) $ = p 11 p 22 p 11 p 22 + p 12 p 21 ; (3.3) and f j ( y i )= 1 p 2 exp ( y i j ) 2 2 2 : Iderivedaclosed-formsolutionforestimatingtheunknownpar ameterswith theEMalgorithm.Theestimatesofhaplotypefrequenciesare basedontheloglikelihoodfunctionofequation(3.1),whereastheestimate sofcompositegenotypic meansandresidualvariancearebasedonthelog-likelihoodfu nctionofequation (3.2).Thesetwotypesofparameterscanbeestimatedusinganin tegrativeEM algorithm. IntheEstep,theexpectednumber( $ )ofdiplotype[11][22]foradouble heterozygousgenotypeisestimatedusingequation(3.3),whe reastheposterior probability( i )withwhichsubject i carryingthedoubleheterozygousgenotypeis diplotype[11][22]iscalculatedby i = $f 1 ( y i ) $f 1 ( y i )+(1 $ ) f 0 ( y i ) : (3.4)

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38Table3{1:Possiblediplotypecongurationsofninegenotype sattwoSNPswhichaectdrugecacyandtoxicity RelativediplotypeHaplotypecompositionGenotypicmeanv ector GenotypeDiplotypeDiplotypefrequencyfreq.withingenot ypes11122122ObservationEcacy x Toxicity z 11 = 11[11][11] P [11][11] = p 211 11000 n 11 = 11 m 2 x m 0 z 11 = 12[11][12] P [11][12] =2 p 11 p 12 1 1 2 1 2 00 n 11 = 12 m 1 x m 1 z 11 = 22[12][12] P [12][12] = p 212 10100 n 11 = 22 m 0 x m 2 z 12 = 11[11][21] P [11][21] =2 p 11 p 21 1 1 2 0 1 2 0 n 12 = 11 m 1 x m 0 z 12 = 12 ( [11][22][12][21] ( P [11][22] =2 p 11 p 22 P [12][21] =2 p 12 p 21 ( $1 $ 1 2 $ 1 2 (1 $ ) 1 2 (1 $ ) 1 2 $n 12 = 12 ( m 1 x m 0 x ( m 0 z m 1 z 12 = 22[12][22] P [12][22] =2 p 12 p 22 10 1 2 0 1 2 n 12 = 22 m 0 x m 1 z 22 = 11[21][21] P [21][21] = p 221 10010 n 22 = 11 m 0 x m 0 z 22 = 12[21][22] P [21][22] =2 p 21 p 22 100 1 2 1 2 n 22 = 12 m 0 x m 0 z 22 = 22[22][22] P [22][22] = p 222 10001 n 22 = 22 m 0 x m 0 z $ = p 11 p 22 p 11 p 22 + p 12 p 21 where p 11 p 12 p 21 and p 22 arethehaplotypefrequenciesof[11],[12],[21],and[22], respectively.

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39 IntheMstep,theprobabilitiescalculatedinthepreviousit erationareusedto estimatethehaplotypefrequenciesusing ^ p 11 = 2 n 11 = 11 + n 11 = 12 + n 12 = 11 + $n 12 = 12 2 n ; (3.5) ^ p 12 = 2 n 11 = 22 + n 11 = 12 + n 12 = 22 +(1 $ ) n 12 = 12 2 n ; (3.6) ^ p 21 = 2 n 22 = 11 + n 12 = 11 + n 22 = 12 +(1 $ ) n 12 = 12 2 n ; (3.7) ^ p 22 = 2 n 22 = 22 + n 22 = 12 + n 12 = 22 + $n 12 = 12 2 n ; (3.8) Thequantitativegeneticparametersareestimatedusing ^ 2 = P n 11 = 11 i =1 y i n 11 = 11 ; (3.9) ^ 1 = P ni =1 y i + P n 12 = 12 i =1 i y i n + P n 12 = 12 i =1 i ; (3.10) ^ 0 = P ni =1 y i + P n 12 = 12 i =1 (1 i ) y i n + P n 12 = 12 i =1 (1 i ) ; (3.11) ^ 2 = 1 n n n 11 = 11 X i =1 ( y i ^ 2 ) 2 + n X i =1 ( y i ^ 1 ) 2 + n X i =1 ( y i ^ 0 ) 2 + n 12 = 12 X i =1 i ( y i ^ 1 ) 2 +(1 i )( y i ^ 0 ) 2 o ; (3.12) where_ n = n 11 = 12 + n 12 = 11 and n = n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 .Iterations includingtheEandMstepsarerepeatedamongequations(3.4) {(3.12)untilthe estimatesoftheparametersconvergetostablevalues. 3.3AnIntegrativeSequenceandFunctionalMappingFramewo rkfor DrugEcacyandToxicity 3.3.1TheMultivariateNormalDistribution Ecacyandtoxicitydescribehowpatientsrespondtodierent dosesor concentrationsofdrugs.Statistically,theserepresentalong itudinalproblemwhose underlyinggeneticdeterminantscanbemappedusingthefunc tionalmapping strategy.Here,Iintegratetheideasforsequencemappingandf unctionalmapping

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40 todirectlycharacterizeDNAsequencevariantsthatarerespon sibleforecacyand toxicityprocesses. Considerthesamesampleasdescribedaboveinwhichpopulationg enetic parametersforSNPshavebeendened.Foreachpatient,druge ectsaremeasured at C hallmarkdoseorconcentrationlevels.Whileadrugisexpecte dtodisplay favorableeects,itmayalsobetoxic.Inthisstudy,sometypica lphysiological parametersrerectingbothecacyandtoxicityarelongitud inallymeasured.Let x i =[ x i (1) ; ;x i ( C )]betheecacyeectvectorand z i =[ z i (1) ; ;z i ( C )]be thetoxicityeectvectorforadrugadministratedtopatient i .Iuse y i =( x i ; z i ) todenotethejointvectorcombiningthesetwotypesofdrugre sponse.Becausethe phenotypicmeasurementsofdrugecacyandtoxicityarecont inuouslyvariable,it isreasonabletoassumethat y followsamultivariatenormaldistribution,asusedin generalquantitativegeneticstudies(LynchandWalsh1998). Tojointlymaptheecacyandtoxicity,twolog-likelihoodf unctionsdescribed inequations(3.1)and(3.2)shouldbeconstructedforobserved SNPmarkerand longitudinalphenotypicdata.Theonlydierencefromsingl etraitmappingis thatthelog-likelihoodfunction(3.2)needstoincorporat eamultivariatenormal distributionforpatient i whocarriescompositegenotype j ,expressedas f j ( y i ; m j ; )= 1 (2 ) C j j 1 = 2 exp 1 2 ( y i m j ) 1 ( y i m j ) T ; (3.13) where y i =[ x i (1) ; ;x i ( C ) ;z i (1) ; ;z i ( C )]isa2 C -dimensionalvector ofobservationsforecacyandtoxicitymeasuredat C dosagesand m j = [ jx (1) ; ; jx ( C ) ; jz (1) ; ; jz ( C )]isavectorofexpectedvaluesforcompositegenotype j atdierentdoses.Ataparticulardose c ,therelationshipbetween theobservationandexpectedmeancanbedescribedbyalinearr egressionmodel, x i ( c )= J X j =1 ij jx ( c )+ e ix ( c ) ; (3.14)

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41 z i ( c )= J X j =1 ij jz ( c )+ e iz ( c ) ; (3.15) where ij and ij aretheindicatorvariablesdenotedas1ifaparticularcomp osite genotype j isconsideredforindividual i and0otherwise, J isthenumberof compositegenotypesand e ix ( c )and e iz ( c )aretheresidualerrorsthatareiid normalwiththemeanofzeroandthevariancesof 2 x ( c )and 2 z ( c ),respectively. Theerrorsattwodierentdoses, c 1 and c 2 ,arecorrelatedwiththecovariances of x ( c 1 ;c 2 )forecacy, z ( c 1 ;c 2 )fortoxicityand xz ( c )and xz ( c 1 ;c 2 )between ecacyandtoxicity.Allthesevariancesandcovariancescomp risethestructureof the(co)variancematrix inequation(3.13),expressedas = 0B@ x xz zx z 1CA ; (3.16) where x and z arecomposedof 2 x ( c )and x ( c 1 ;c 2 ),and 2 z ( c )and z ( c 1 ;c 2 ) (1 c 1 ;c 2 C ),respectively;and xz and zx arecomposedof xz ( c )and xz ( c 1 x ;c 2 z )(1 c 1 x 6 = c 2 z C ),and zx ( c )and xz ( c 1 z ;c 2 x )(1 c 1 z 6 = c 2 x C ), respectively. Tosolvethelikelihoodfunctionsimplementedwithresponseda tameasured atmultipledoses,onecanextendthetraditionalintervalmap pingapproachto accommodatethemultivariatenatureofdose-dependenttrai ts.However,this extensionislimitedintwoaspects:(1)Individualexpectedme ansofdierent compositegenotypesatalldoseandallelementsinthematrix needtobe estimated,resultinginsubstantialcomputationaldiculties whenthevectorand matrixdimensionislarge;(2)Theresultfromthisapproachma ynotbeclinically meaningfulbecausetheunderlyingmedicalprinciplefordru gresponseisnot incorporated.Thus,someclinicallyinterestingquestionscan notbeaskedand answered.

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42 3.3.2TheMappingFramework Wuetal.(2002b,2004a,2004b)andMaetal.(2002)haveconstr uctedanovel statisticalframeworkforfunctionalmappingofQTLthataec tgrowthcurves. Here,thisframeworkisextendedtojointlymodeltwodieren tlongitudinaltraits, aimedatsimultaneouslycharacterizingthegeneticvariants responsibleforecacy andtoxicity.Twotasksaretakenwithinthisframeworkbymod ellingthemean vectorandthecovariancematrix. Modellingthemeanvector: Thedose-dependentexpectedvaluesof compositegenotype j canbemodelledfordrugecacybythesigmoidEmax model(Giraldo2003)andfordrugtoxicitybythepowerfunct ion(McClishand Roberts2003).TheE max modelpostulatesthefollowingrelationshipbetweendrug concentration(C)anddrugeect( ): jx (C)=E 0 j + E max j C H j EC H j 50 j +C H j ; forcompositegenotype j; (3.17) whereE 0 istheconstantorbaselinevalueforthedrugresponseparameter ,E max is theasymptotic(limiting)eect,EC 50 isthedrugconcentrationthatresultsin50% ofthemaximaleect,andHistheslopeparameterthatdetermi nestheslopeofthe concentration-responsecurve.Drugtoxicityisdescribedbyt hepowerfunctionof doseexpressedas jz (C)= j C j ; forcompositegenotype j; (3.18) where determinestheshapeofthedose-responserelationshipand adjuststhe dose-relatedgaintowhichthatshapeconforms. Itispossiblethatecacyandtoxicityhavedierentreferenc ehaplotypes,sotheircompositegenotypesshouldbetreateddierentl y.Asaresult, equations(3.17)and(3.18)togethercontain6curveparame tersforadened compositegenotype j 1 forecacyand j 2 fortoxicity,whicharearrayedby

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43 n m j 1 j 2 =(E 0 j 1 ; E max j 1 ; EC 50 j 1 ; H j 1 ; j 2 ; j 2 ).Ifdierentcompositegenotypes havedierentcombinationsoftheseparameters,thisimplies thatthissequence playsaroleingoverningthedierentiationofecacyandto xicity.Thus,bytesting forthedierenceof n m j 1 j 2 amongdierentgenotypes,Icandeterminewhether thereexistsaspecicsequencevariantthatconfersaneecton thesetwodrug responses. Modellingthestructureofthecovariancematrix: Manystatistical approacheshavebeenproposedtomodelthestructureofthecov ariancematrix forlongitudinaltraitsmeasuredatmultipletimepoints(Di ggleetal.2002).Here, thematrix-structuringmodelswillbederivedonthebasisofa commonlyused approach,structuredantedependence(SAD)(ZimmermanandN u~nez-Anton2001). AccordingtotheSADmodel,anobservationataparticulardosage c depends onthepreviousones,withthedegreeofdependencedecayingw ithtimelag.For drugecacyandtoxicity,thedose-dependentresidualerrors forsubject i described byEquations(3.14)and(3.15)canbeexpressed,intermsofthe rst-orderSAD (SAD(1))model,as e ix ( c )= x e ix ( c 1)+ x e iz ( c 1)+ ix ( c ) ; e iz ( c )= z e iz ( c 1)+ z e ix ( c 1)+ iz ( c ) ; (3.19) where x (or z )and x (or z )aretheunrestrictedantedependenceparameters inducedbytrait x (or z )itselfandbytheothertrait z (or x ),respectively,and xi ( c )and xi ( c )arethe\innovation"errorsforthetwotraits,respectively ,normally distributedas N (0 ; 2 x ( c ))and N (0 ; 2 z ( c ))(Nu~nez-AntonandZimmerman2000; ZimmermanandNu~nez-Anton2001).ThebivariateSAD(1)mod elforsubject i describedbyequation(3.20)canbeexpressedinmatrixnotatio nas e i = 1 i ; (3.20)

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44 where e i = f e 1 (1) ; ;e i ( C ) g T i = f i (1) ; ; i ( C ) g T ,and = 0B@ x xz zx z 1CA ; x = 0BBBBBBBBBBBBBB@ 100 0 x 10 0 0 x 1 0 00 x 0 ... ... ... . ... 000 1 1CCCCCCCCCCCCCCA ; z = 0BBBBBBBBBBBBBB@ 100 0 z 10 0 0 z 1 0 00 z 0 ... ... ... . ... 000 1 1CCCCCCCCCCCCCCA ; xz = 0BBBBBBBBBBBBBB@ 000 0 x 00 0 0 x 0 0 00 x 0 ... ... ... . ... 000 0 1CCCCCCCCCCCCCCA ; zx = 0BBBBBBBBBBBBBB@ 000 0 z 00 0 0 z 0 0 00 z 0 ... ... ... . ... 000 0 1CCCCCCCCCCCCCCA : Thecovariancematrixof e i underthebivariateSAD(1)model(21)canbe obtainedas = 1 ( 1 ) T ; (3.21) where = 0B@ x xz zx z 1CA ;

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45 x = 0BBBBBBB@ 2 x (1)00 0 0 2 x (2)0 0 ... ... ... . ... 000 2 x ( C ) 1CCCCCCCA ; z = 0BBBBBBB@ 2 z (1)00 0 0 2 z (2)0 0 ... ... ... . ... 000 2 z ( C ) 1CCCCCCCA : and xz = zx = 0BBBBBBB@ x z (1)00 0 0 x z (2)0 0 ... ... ... . ... 000 x z ( C ) 1CCCCCCCA : Inthismodelling,theinnovativevarianceisassumedtobecon stantoverdierent dosages.AsshownbyJarezicetal.(2003),theSAD(1)modelwith thisassumptioncanstillallowforboththevarianceandcorrelationtoc hangewiththedose level.Inaddition,Iassumethatthecorrelation( )ininnovativeerrorsbetweenthe twotraits x and z isstableoverdosage.Withtheseassumptions,Ineedtoestimate anarrayofparameterscontainedin n v =( 2 x ; 2 z ; x ; z ;' x ;' z ; )thatmodelthe structureofthecovariancematrixundertheSADmodel. InthecontextofthebivariateSADmodel,thecross-correlatio nfunctions canbeproventobeasymmetrical,i.e., Corr xz ( c 1 x ;c 2 z ) 6 = Corr zx ( c 1 z ;c 2 x ).This favorablefeatureofthebivariateSADmodelmakesitusefulfo runderstandingthe geneticcorrelationbetweendierenttraits.Inpractice,i nnovativevarianceand correlationcanbemodelledbyapolynomialandexponential function,respectively (Nu~nez-AntonandZimmerman2000).3.3.3ComputationalAlgorithm IimplementedtheEMalgorithm,originallyproposedbyDempst eretal. (1977),toobtainthemaximumlikelihoodestimates(MLEs)oft hreegroupsof unknownparametersintheintegratedsequenceandfunctionm appingmodel,

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46 thatis,themarkerpopulationparameters( n p ),thecurveparameters( n m j 1 j 2 ) thatmodelthemeanvector,andtheparameters( n v )thatmodelthestructureof thecovariancematrix.Theseunknownsaredenotedby n =( n p ; n m j 1 j 2 ; n v ).A detaileddescriptionoftheEMalgorithmwasgiveninWuetal. (2002b,2004b)and Maetal.(2002). Asdescribedforsingletraitmapping,IimplementtheEsteptoca lculatethe expectednumber( $ )ofdiplotype[11][22]containedinthedoubleheterozygot e 12/12andtheposteriorprobability( i )ofdoubleheterozygoticpatient i who carriesdiplotype[11][22].IntheMstep,Iusethecalculated $ and i valuesto estimatethehaplotypefrequenciesusingequations(3.5){(3 .8).Butinthisstep,I encounteraconsiderabledicultyinderivingthelog-likel ihoodequationsfor n m j 1 j 2 and n v becausetheyarecontainedincomplexnonlinearequations.Zh aoetal. (2004)implementedthesimplexmethodasadvocatedbyNeldera ndMead(1965) totheestimationprocessoffunctionalmapping,whichcanstri kinglyincrease computationaleciency.Inthischapter,thesimplexalgori thmisembeddedinthe EMalgorithmabovetoprovidesimultaneousestimationofhapl otypefrequencies andcurveparametersandmatrix-structuringparameters.3.3.4ModelforanArbitraryNumberofSNPs Theideaforsequencingdrugresponsehasbeendescribedforatwo -SNP model.Itispossiblethatthetwo-SNPmodelistoosimpletochara cterizegenetic variantsforvariationindrugresponse.Ihaveextendedthism odeltoincludean arbitrarynumberofSNPswhosesequencesareassociatedwithdrug response variation.Akeyissueforthemulti-SNPsequencingmodelishowt odistinguish among2 r 1 dierentdiplotypesforthesamegenotypeheterozygousat r loci.The relativefrequenciesofthesediplotypescanbeexpressedinte rmsofhaplotype frequencies.TheintegrativeEMalgorithmcanbeemployedto estimatetheMLEs ofhaplotypefrequencies.Bennett(1954)providedageneral formulaforexpressing

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47 haplotypefrequenciesintermsofallelefrequenciesandli nkagedisequilibriaof dierentorders.TheMLEsofthelattercanbeobtainedbysolvi ngasystemof equations. 3.4HypothesisTests Dierentfromtraditionalmappingapproaches,thefunction almappingfor function-valuedtraitsallowsfortestsofanumberofbiolog icallyorclinicallymeaningfulhypotheses.Thesehypothesesincludetestsfortheexisten ceofsignicant DNAsequencevariants,forthegeneticeectonthemaximal(asym ptotic)eect (E max ),forthedrugconcentrationthatresultsin50%ofthemaxima leect,andfor theslopesthatdeterminesthesteepnessoftheconcentrationresponsecurve. Theexistenceofspecicgeneticvariantsaectingdrugecac yandtoxicity canbetestedbyformulatingnullandalternativehypotheses, 8>><>>: H 0 : n m j 1 j 2 n m ;j 1 ;j 2 =2 ; 1 ; 0 H 1 :atleastoneoftheequalitiesabovedoesnothold ; (3.22) where H 0 correspondstothereducedmodel,inwhichthedatacanbetby a singledrugresponsecurve,and H 1 correspondstothefullmodel,inwhichthere existdierentdynamiccurvestotthedata.Theteststatisticf ortestingthe hypothesesinequation(3.23)iscalculatedasthelog-likel ihoodratio(LR)ofthe reducedtothefullmodel: LR= 2[log L ( e n j y ; G ) log L ( b n j y ; G )] ; (3.23) where e n and b n denotetheMLEsoftheunknownparametersunder H 0 and H 1 ,respectively.TheLRisasymptotically 2 -distributedwith12degreesof freedom.Anempiricalapproachfordeterminingthecritical thresholdisbasedon permutationtests,asadvocatedbyChurchillandDoerge(1994 ).Byrepeatedly shuingtherelationshipsbetweenmarkergenotypesandpheno types,aseriesofthe

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48 maximumlog-likelihoodratiosarecalculated,fromthedist ributionofwhichthe criticalthresholdisdetermined. Icanalsotestforthesignicanceofthegeneticeectondruge cacyor toxicityataparticularconcentrationlevel( c )ofinterest,expressedas 8>><>>: H 0 : j 1 x ( c )= x ( c ) ; j 2 z ( c )= z ( c ) H 1 :Atleastoneoftheeqaulitiesabovedoesnothold ; (3.24) whichisequivalenttotestingthedierenceofthefullmodel withnorestrictionand thereducedmodelwitharestrictionin H 0 .Similarrestrictionscanbetakentotest thegeneticeectonindividualcurveparameters,suchasE max ,EC 50 ,H, and Thetestsoftheseparametersareimportantforthedesignofper sonalizeddrugsto controlparticulardiseases. 3.5Results IperformMonteCarlosimulationexperimentstoexaminethest atistical propertiesofthemodelproposedforgeneticmappingofdruge cacyandtoxicity. ThesimulationwillbebasedonthebivariateSAD(1)model(equa tion3.21).For computationalsimplicity,Ionlyconsiderstheantedependen ceparametersinduced bythetraititself,i.e.,setting x = z =0.Irandomlychoose200individualsfrom ahumanpopulationatHardy-Weinbergequilibriumwithrespec ttohaplotypes. LetusconsidertwoSNPswithallelefrequenciesandlinkagedi sequilibriumgiven inTable3-1.Thediplotypesderivedfromfourhaplotypesat thesetwoSNPsaect drugecacyandtoxicity.HereIassumethatthesetwotypesofdru gresponsehave dierentreferencehaplotypes,i.e.,11forecacyand12for toxicity.Thus,each drugresponsecorrespondstoadierentsetofthreecompositegen otypes. Theecacy(E 0 ,E max ,EC 50 ,H)andtoxicitycurveparameters( and )for therespectivethreecompositegenotypes,giveninTable3-1,a redeterminedin therangesofempiricalestimatesoftheseparametersfrompha rmacologicalstudies

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49 (Sowinskietal.1995;McClishandRoberts2003).Iuseageneral clinicaldesign, withdrugeectsmeasuredat9dierentconcentrationsofdru g,0,5,10,15,20,25, 30,35and40,expressedas c =1 ; 2 ; ; 9(Figure3-1),forthissimulationstudy. Usingthegeneticvarianceduetothecompositegenotypicdier enceinthearea undercurve,Icalculatetheresidualvariancesunderdiere ntheritabilitylevels ( H 2 =0 : 1and0.4).Theseresidualvariances,plusgivenresidualcorrel ations,form astructuredresidualcovariancematrix (equation3.22).Thephenotypicvaluesof drugeectfor200randompatientsaresimulatedbythesummati onsofgenotypic valuespredictedbythecurvesandresidualerrorsfollowing multivariatenormal distributions,with MVN (0 ; ). ThepopulationgeneticparametersoftheSNPscanbeestimated withreasonablyhighprecisionusingtheclosed-formsolutionapproach( Table3-2).The estimatesoftheseparametersareonlydependentonsamplesizea ndarenotrelated tothesizeofheritability.Figure3-1illustratesdierentf ormsofthedrugecacy andtoxicitycurvesfromthethreecompositegenotypeswitha comparisonbetween thehypothesizedandestimatedcurves.Theestimatedcurvesare consistentwith thehypothesizedcurves,especiallywhentheheritabilityisl arger(0.4),suggesting thatthemodelcanprovidereasonableestimatesofdrugresponse curves.The parametersfortheecacyandtoxicitymodelsofeachcomposi tegenotypecanbe estimatedaccuratelyandprecisely(Table3-2).Asexpected,t heestimationprecision(assessedbysquarerootofMSEs)increasesremarkablywhentheh eritability increasesfrom0.1to0.4.TheestimatesoftheSADparametersth atmodelthe structureofthecovariancematrix alsodisplayreasonablyhighprecision(Table 3-2).Butitseemsthattheirestimationprecisionisindepende ntofthesizeof heritability. Ineachof100simulations,Icalculatethelog-likelihoodrat ios(LR)forthe hypothesistestofthepresenceofageneticvariantaectingbo thdrugresponses.

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50 Table3{2: MaximumlikelihoodestimatesofSNPpopulationgeneticpar ameters(allele frequenciesandlinkagedisequilibrium),thecurveparame tersandmatrix-structuringparametersforecacyandtoxicityresponses.Thenumbersinp arenthesesarethesquare rootofMSEsoftheestimates. CompositeHeritability ParametersgenotypeTruevalue0.10.4 Populationgeneticparameters p (1)1 0.600.60(0.02)0.60(0.02) p (2)1 0.700.70(0.02)0.70(0.02) D 0.080.08(0.01)0.08(0.02) Curveparameters:Ecacy E 0 [11][11]0.270.27(0.01)0.27(3e-3)[11][ 11]0.250.25(0.01)0.25(3e-3) [ 11][ 11]0.230.23(0.01)0.23(4e-3) E max [11][11]0.900.93(0.11)0.92(0.08)[11][ 11]0.850.85(0.12)0.84(0.06) [ 11][ 11]0.800.80(0.16)0.79(0.11) EC 50 [11][11]24.0025.57(6.14)24.78(3.66)[11][ 11]25.0025.46(7.94)25.17(4.98) [ 11][ 11]26.0027.53(14.67)26.26(10.71) H [11][11]1.101.10(0.09)1.10(0.07)[11][ 11]1.001.03(0.11)1.04(0.05) [ 11][ 11]0.900.94(0.12)0.93(0.08) Curveparameters:Toxocity [12][12]0.100.10(0.02)0.10(0.01)[12][ 12]0.130.13(0.01)0.13(2e-3) [ 12][ 12]0.160.16(4e-3)0.16(2e-3) [12][12]0.600.60(0.08)0.60(0.04)[12][ 12]0.500.50(0.01)0.50(6e-3) [ 12][ 12]0.400.40(8e-3)0.40(4e-3) Matrix-structuringparameters x 0.300.28(0.04)0.29(0.02) z 0.400.39(0.03)0.40(0.02) 0.700.70(0.01)0.70(0.01) 2 x 8e-38e-3(3e-4)1e-31e-3(5e-5) 2 z 3e-33e-3(1e-4)6e-46e-3(2e-5)

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51 TheLRvaluesineachsimulationunderbothheritabilityleve lsarestrikinglyhigher thanthecriticalthresholdestimatedfrom100replicatesofsi mulationsunderthe nullhypothesisthatthereisnodrugresponse-associatedgeneti cvariant.This suggeststhatthismodelhasenoughpowertodetectthegenetic variantundergiven SNPs,curveandmatrix-structuringparametersforthesimulati on.Withthesame setofsimulateddata,thepowertodetectsignicantgeneticva riantsisreducedby 10%whendrugecacyortoxicityismodelledseparately. Icarryoutanadditionalsimulationstudytoexaminethestatist icalbehavior ofthismodelwhen3dierentSNPsareused.This3-SNPmodelusest hesame parametersasassumedunderthe2-SNPmodel,exceptfortheincl usionofmore SNPs.Theresultsfromthe3-SNPmodelarebroadlyconcordantwit hthosefrom the2-SNPmodel(resultsnotgiven),althoughmorepopulation geneticparameters needtobeestimatedintheformerthanlatter. 3.6Discussion Drugresponseiscomplexinnature.First,nodrugisuniversally eective; ecacyratesfordrugtherapyofmostdiseasesrangesfrom25to8 0%(Spearet al.2001).Second,alldrugsthatproduceanecaciousrespon semayalsoproduce adverseeects(Johnson2002).Third,bothdrugecacyandtox icitythatvaryin humanpopulationsaredeterminedbymultiplegeneticanden vironmentalfactors thatinteractwithoneanotherincomplicatedways(Watters andMcleod2003).It ispossiblethatthegeneticpolymorphismsassociatedwithafavo rableresponseto therapymayormaynotbeassociatedwithtoxicityrisk(Nebert19 97;Wilsonetal. 2001). Thecomplexityofdrugresponseisenforcedbyitsdynamicchar acteristicwith highdimensionandcorrelatedstructure.Thegeneticanalysis ofdrugresponse, therefore,needsmoresophisticatedstatisticaltoolsthathav ecapacitytoextract usefulinformationfromlongitudinaldata.Inthischapter, Ihaveproposeda

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52 novelstatisticalmodelforcharacterizingthegeneticvaria ntsthatgoverndrug responseinahumanpopulation.Thismodelisconstructedwitha nitemixture modelframeworkfoundedonthetenetsofthesequence(Lineta l.2005)and functionalmapping(Maetal.2002;Wuetal.2002b,2004a,20 04b).Thismodel hasparticularpowertodiscernthediscrepancybetweenthege neticmechanisms underlyingdrugecacyandtoxicity. Comparedtotraditionalgeneticmappingapproaches,thismo delisboth geneticallyandstatisticallymerited.Itoersadirecttool todetectDNAsequences thatcodedrugresponse.ThehaplotypemaporHapMapconstructed fromsingle nucleotidepolymorphisms(SNPs)(TheInternationalHapMapCon sortium2003) makestheapplicationofthismodelpracticallypossible.Inp articular,Iadopt clinallymeaningfulmathematicalfunctions(Giraldo2003 ;McClishandRoberts 2003)whenmodellingdose-dependentdrugresponse.Thishastw oadvantages. First,itfacilitatesstatisticalanalysisandstrengthenspowe rtodetectsignicant geneticvariantsbecausefewerparametersareneededtobeest imated,asopposed totraditionalmultivariateanalysis.Second,theresultsitp roducesarecloseto biologicalrealmgiventhatthemathematicalfunctionsused arefoundedonarm understandingofpharmacology(reviewedinDerendorfandMe ibohm1999).The statisticalpowerofthismodelisfurtherincreasedbytheatte mpttostructurethe covariancematrixwithautoregressivemodels(Diggleetal.2 002). Ihaveperformeddierentsimulationstudiestoinvestigateth estatistical behaviorofthismodel.Theresultssuggestthatitcanprovidea ccurateandprecise estimatesoftheresponsecurvesforbothecacyandtoxicityev enwhenthe heritabilityismodest.Dierentdrugresponsecurves,assimula tedfordierent compositegenotypes(Figure3-1),canoerscienticguidanc efordeterminingthe optimaldosagethatbalancesfavorableecacyandunfavorab letoxicitybasedon individual'sgeneticbackground.Itshouldbenotedthatthi smodelwasderived

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53 onthebasisofasimpleclinicaldesigninwhichasinglecohortis assumed,with equallyconcentrationintervalsforeverypatient.Thismo delcanbeextendedto acase-controlstudybyallowinghaplotypefrequenciesandha plotypeeectstobe dierentbetweenthecaseandcontrolgroups.Aseriesofhypoth esistestsregarding between-groupdierencescanbeformulatedtodetectspeci chaplotypesthatare responsiblefordrugresponse.Mymodelcanalsobeextendedtocon sidervarious measurementschedules,inparticularwithunevenconcentrati onlagsvaryingfrom patienttopatient.Fortheseirregularmeasurementschedules, itisneededto formulateindividualizedlikelihoodfunctionswhentheme anvectorandcovariance matrixaremodelled(seeNu~nez-AntonandWoodworth1994). Incouplingwithcontinuingadvancesinmolecularbiologic altechnologyand theavailabilityofacompletereferencesequenceoftheenti rehumangenome(Patil etal.2001;Dawsonetal.2002;Gabrieletal.2002),thismode lwillassistinthe discoveryandcharacterizationofthegenesthatinruencedr ugresponse,andlead toabetterunderstandingofhowgenesfunctionandthesubseque ntdevelopmentof newapproachestothediagnosisandtreatmentofcommondisease sbymedications. Oncethenetworkofgenesthatgoverndrugresponsesinhumansi sdened,itwill thenbepossibletomoreaccuratelyoptimizedrugtherapybased oneachpatient's abilitytometabolize,transportandrespondtomedications.

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54 0 10 20 30 40 0 20 40 60 80 100 Drug concentration (mcg)Efficacy (%)A 0 10 20 30 40 0 20 40 60 80 100 Drug concentration (mcg)Toxicity (%)B 0 10 20 30 40 0 20 40 60 80 100 Drug concentration (mcg)Efficacy (%)C 0 10 20 30 40 0 20 40 60 80 100 Drug concentration (mcg)Toxicity (%)D Figure3{1:Estimatedresponsecurves(dash)eachcorresponding tooneofthree compositegenotypesundertheheritability( H 2 )of0.1( A and B )and0.4( C and D ),inacomparisonwiththehypothesizedcurves(solid)usedtosim ulateindividualcurvesfordrugecacy( A and C )andtoxicity( B and D ),respectively. Ninedierentconcentrations,0,5,10,15,20,25,30,35and40 ,atwhichresponses aremeasured,areused.Theconsistencybetweentheestimatedand hypothesized curvessuggeststhatthismodelcanprovidethepreciseestimati onofthegenetic controloverresponsecurvesinpatients.Thehomozygotecompo sedoftworeferencehaplotypesis[11][11]forecacyand[12][12]forto xicity(blue),theheterozygotecomposedofonereferenceandonenon-referenceha plotypeis[11][ 11] forecacyand[12][ 12]fortoxicity(red),andthehomozygotecomposedoftwo non-referencehaplotypesis[ 11][ 11]forecacyand[ 12][ 12]fortoxicity(green).

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CHAPTER4 MODELFORDETECTINGSEQUENCE-SEQUENCEINTERACTIONSFOR COMPLEXDISEASES 4.1Introduction Interactionsbetweendierentgenes,coinedthe epistasis ,havelongbeenrecognizedtoplayacentralroleinshapingthegeneticarchitectu reofaquantitative trait(Whitlocketal.1995;Wolfetal.2000).Recentgeneti cstudiesfromvast quantitiesofmoleculardatahavealsoindicatedthatepistasi sisofparamount importanceinthepathogenesisofmostcommonhumandiseases,such ascancer orcardiovasculardisease(Moore2003).Theevidenceforthisi sthenonlinear relationshipdetectedbetweengenotypeandphenotype.Thed eciphermentofinterconnectednetworksofgenesandtheirassociationswithdi seasesusceptibility hasbecomeapressingdemandforadetailedunderstandingofthe geneticbasisfor diseaseprocesses. Themostcommonandpowerfulapproachesfordetectinggenome -wide epistasisarebasedongeneticmappingthatassociatesphenotypi cvariationof atraitwithalinkagemapconstructedbypolymorphicmarkers (Landerand Botstein1989).Signicantassociationsimplytheexistenceof theunderlyingloci (calledquantitativetraitlociorQTL)thatareresponsiblef orvariationinthe trait.EpistaticQTLareconsideredtooccuriftheeectofone QTLdependson theexpressionofotherQTL.Whiletraditionalgeneticmappin gcanonlymake anindirectinferenceaboutQTLactionsandinteractions,an ewlydeveloped mappingapproachbasedonthehaplotypemap(HapMap)construct edfromsingle nucleotidepolymorphisms(SNPs)(TheInternationalHapMapCon sortium2003) candirectlycharacterizespecicDNAsequencevariantsforap henotype(Liuetal. 55

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56 2004).Beyondhypothesizedchromosomalsegmentsidentiedby QTLmapping, thisso-calledsequencemappingapproach,relieduponthethe characterizationof SNPsfromtheentirehumangenome,canprobeconcretenucleot oxicsites,i.e., quantitativetraitnucleotides(QTN),thatcontributetova riationinaquantitative trait. Inthischapter,Iextendthepreviousstatisticalmodeltodet erminethe epistasisbetweenspecicDNAsequencesresponsibleforthephenot ypicvariation ofdiseaserisk.Thisextendedmodelisderivedthroughtheanal ysisofpair-wise multilocushaplotypesconstructedbyanitenumberoftagSNP s.Ideriveda closedformsolutionforestimatingtheactionandinteraction eectsofhaplotypes, haplotypefrequencies,allelefrequenciesandthedegreeso fLDofvariousorders amongtagSNPsunderlyingthedisease.Simulationstudieswerep erformedtotest thestatisticalbehaviorofthishaplotype-basedepistaticmap pingmodel.Iuseda workedexampleforahumanobesitystudytodemonstratetheusefu lnessofthe model. 4.2TheModel 4.2.1Notation ConsideranaturalhumanpopulationatHardy-Weinbergequili briumfrom whicharandomsampleofsize n isdrawn.InordertoidentifyDNAsequences responsibleforacomplexdisease,IgenotypeanumberofSNPsgeno mewideand constructahaplotypemap.Recentmolecularsurveyssuggesttha tthehuman genomecontainsmanydiscretehaplotypeblocksthataresites ofcloselylocated SNPs(Dalyetal.2001;Patiletal.2001;Gabrieletal.2002). Eachblockmay haveafewcommonhaplotypeswhichaccountforalargepropor tionofchromosomalvariation.Betweenadjacentblockstherearelargeregi ons,called hotspots ,in whichrecombinationeventsoccurwithhighfrequencies.Sev eralalgorithmshave beendevelopedtoidentifyaminimalsubsetofSNPs,i.e.,\taggi ng"SNPs,that

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57 cancharacterizethemostcommonhaplotypes(Zhangetal.200 2b,2002c;Kimmel andShamir2005).Inthisstudy,Iassumethatthenumberandtype oftagging SNPswithineachhaplotypeblockhasbeendetermined.Therat ionalebehind theepistaticmodelbeingdevelopedisthattheeectofagive nDNAsequencein onehaplotypeblockonacomplexdiseaseismaskedorenhancedby oneormore sequencesinotherblocks. Supposethereare R and S ( R;S> 1)taggingSNPsfortwoarbitrary haplotypeblocks R and S ,respectively.TwoallelesofatagSNP r or s fromblock R or S canbedenotedby U r k r ( k r =1 ; 2; r =1 ; R )and V s l s ( l s =1 ; 2; s =1 ; S ), respectively.Iuse p Rk r and p Sl s todenoteallelefrequenciesatthecorresponding SNP.AlltheSNPswithineachofthetwoblocksform2 R or2 S possiblehaplotypes expressedas U 1 k 1 U 2 k 2 U R k R and V 1 l 1 V 2 l 2 V S l S ,respectively.Thecorresponding haplotypefrequencieswithineachblockaredenotedby p Rk 1 k 2 k R and p Sl 1 l 2 l S whicharecomposedofallelefrequenciesateachSNPandlinkag edisequilibriaof dierentordersamongSNPs(LynchandWalsh1998).Ageneralex pressionfor therelationshipsbetweenhaplotypefrequenciesandallele frequenciesandlinkage disequilibriawasgivenbyBennett(1954).Louetal.(2003)d erivedaclosed-form EMalgorithmtoestimatehaplotypefrequencies,whichcanbef urtherusedto estimateallelefrequenciesandlinkagedisequilibriabasedo ntheseestablished relationships. Therandomcombinationofmaternalandpaternalhaplotypes generates 2 R 1 (2 R +1) diplotypes expressedas[ U 1 k 1 U 2 k 2 U R k R ][ U 1 k 0 1 U 2 k 0 2 U R k 0 R ](1 k 1 k 0 1 2 ; ; 1 k R k 0 R 2)forblock R and2 S 1 (2 S +1)diplotypesexpressedas [ V 1 l 1 V 2 l 2 V S l S ][ V 1 l 0 1 V 2 l 0 2 V S l 0 S ](1 l 1 l 0 1 2 ; ; 1 l R l 0 R 2)forblock S Iusethebracketstoseparatematernal(former)andpaternalh aplotypes(latter) foragivendiplotype.UnlesstherearetwoormoreSNPsthatareh eterozygous, observable zygoticgenotypes willbethesameasdiplotypes.Thus,thenumbers

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58 ofzygoticgenotypes,3 R or3 S ,willbelessthanthenumberofdiplotypesand thedierencebetweenthesetwonumbersisstatisticallyviewe das missingdata Theobservedzygoticgenotypesareexpressedas U 1 k 1 U 1 k 0 1 =U 2 k 2 U 2 k 0 2 = =U R k R U R k 0 R or V 1 l 1 V 1 l 0 1 =V 2 l 2 A 2l 0 2 = =V S l S V S l 0 S forthetwoblocks,respectively.Let P R [ k 1 k 2 k R ][ k 0 1 k 0 2 k 0 R ] and P R k 1 k 0 1 =k 2 k 0 2 = =k R k 0 R denotethediplotypeandgenotypefrequencies,respectively ,for block R .Thecorrespondingexpressionsare P S [ l 1 l 2 l R ][ l 0 1 l 0 2 l 0 S ] and P S l 1 l 0 1 =l 2 l 0 2 = =l S l 0 S for block S Becausephenotypicvariationinacomplexdiseasecanbeexplai nedby haplotypediversity,aparticularhaplotypecanbeassumedtob edierentfrom otherhaplotypesforagivenphenotype(Bader2001).Liueta l.(2004)denedsuch adistincthaplotypeas referencehaplotype .Theyfurtherdenedthediplotypes formedbyreferenceand/ornon-referencehaplotypesas compositegenotypes Althoughitcannotbedirectlyobserved,thedierencebetwee nthereferenceand non-referencehaplotypescanbeinferredfromobservedzygo ticgenotypeswiththe EMalgorithm.4.2.2EpistaticEects Let A A and B B bethereferenceandnon-referencehaplotypesatblock R and S ,respectively.Inthegivensample,thesetwoblocksformnined ierent compositegenotypesexpressedas AABB AAB B AA B B A ABB A AB B A A B B A ABB A AB B and A A B B .Traditionalquantitativegenetictheories canbeusedtomodelthegeneticeectsofthecompositegenotyp es(Lynchand Walsh1998).Thegenotypicvalue( j R j S )ofajointcompositegenotypeatthetwo haplotypeblockscanbedecomposedintoninedierentcompon entsasfollows: j R j S = Overallmean(4.1) +( j R 1) R +( j S 1) S Additiveeects +[1 ( j R 1) 2 ] R +[1 ( j S 1) 2 ] S Dominanteects

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59 +( j R 1)( j S 1) I Additive additiveeect +( j R 1)[1 ( j S 1) 2 ] J Additive dominanteect +[1 ( j R 1) 2 ]( j S 1) K Dominant additiveeect +[1 ( j R 1) 2 ][1 ( j S 1) 2 ] L Dominant dominanteect ; where j R ;j S = 8>>>>><>>>>>: 2for AA or BB 1for A A or B B 0for A A or B B standforthecompositegenotypesatblocks R and S ,respectively, : and : arethe additiveanddominanteectsatthecorrespondingblock,resp ectively,and I J K and L aretheadditive additive,additive dominant,dominant additiveand dominant dominantepistaticeectsbetweenthetwoblocks,respectivel y. Thestatisticalmodelbeingdevelopedhereaimstodiagnoseref erencehaplotypesateachblockandfurtherprovideestimatesoftheaddit ive,dominantand epistaticeectsofanykindbetweenthetwoblocks.Foragiven referencehaplotype,IwillderivetheEMalgorithmwithinthemaximumlikel ihoodcontextto estimatethegenotypicvalueofeachcompositegenotype.Bysol vingagroupof regularequationsasshownbyequation(4.1),theoverallmea n,additive,dominant andfourkindsofepistaticeectsbetweentwoblockscanbeest imated,i.e., = 1 4 ( A A B B + AA B B + A ABB + AABB ) R = 1 4 ( AABB A A B B + AA B B A ABB ) S = 1 4 ( A ABB A A B B AA B B + AABB ) R = 1 4 (2 A A B B A A B B AA B B A ABB AABB +2 A ABB ) S = 1 4 (2 A AB B A A B B AA B B A ABB AABB +2 AAB B )

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60 I = 1 4 ( AABB AA B B A ABB + A A B B ) J = 1 4 (2 AAB B AABB 2 A AB B + A A B B AA B B + A ABB ) K = 1 4 (2 A ABB 2 A A B B + A A B B + AA B B A ABB AABB ) L = 1 4 (4 A AB B + A A B B + AA B B + A ABB + AABB 2 A A B B 2 A ABB 2 A AB B 2 AAB B )(4.2) Theseestimatesarethemaximumlikelihoodestimates(MLEs)base donthe invariantpropertyofthemaximumlikelihoodmethod.4.2.3LikelihoodFunctions Thismodelisuniqueinthattwodierentlikelihoodfunctio nscanbeintegratedtoestimatehaplotypefrequencies(populationgenet icparameters)and genotypicvaluesofcompositegenotypesandresidualvarianc e(quantitativegeneticparameters).Fortwodierenthaplotypeblocks R and S ,betweenwhichno linkagedisequilibriaexist(Dalyetal.2001;Gabrieletal.2 002),across-blockhaplotypefrequenciescanbecalculatedastheproductoftheco rrespondinghaplotype frequenciesfromadierentblock,expressedas p ( k 1 k 2 k R )( l 1 l 2 l S ) = p Rk 1 k 2 k R p Sl 1 l 2 l S ; (4.3) wheretheparenthesesareusedtoseparatetwodierentblocksf oragivenacrossblockhaplotype.Withtheseacross-blockhaplotypefrequenci es,expectedacrossblockdiplotypefrequenciesandacross-blockgenotypefrequ enciescanbecalculated, respectively,underHardy-Weinbergequilibrium. Withacross-blockdiplotypeandgenotypefrequencies,thelik elihoodfunction basedonacross-blockgenotypeobservations, n = f n ( k 1 k 0 1 =k 2 k 0 2 = =k R k 0 R )( l 1 l 0 1 =l 2 l 0 2 = =l S l 0 S ) g canbeconstructed.Tosimplifythepresentationofmymodel,Iw illrstassume twotaggingSNPsforeachblock.Table4-1listsallpossiblegeno typesanddiplotypesaswellastheirfrequenciesattwoSNPsgenotypedfromb lock R .Each

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61Table4{1:Possiblediplotypecongurationsofninegenotype sattwoSNPsandtheirhaplotypecompositionfrequencies RelativediplotypeHaplotypecomposition GenotypeDiplotypeDiplotypefrequencyfreq.withingenot ypes U 1 1 U 2 1 U 1 1 U 2 2 U 1 2 U 2 1 U 1 2 U 2 2 ObservationGenotypicmean U 1 1 U 1 1 =U 2 1 U 2 1 [ U 1 1 U 2 1 ][ U 1 1 U 2 1 ] P [11][11] = p 211 11000 n 11 = 11 2 = + a U 1 1 U 1 1 =U 2 1 U 2 2 [ U 1 1 U 2 1 ][ U 1 1 U 2 2 ] P [11][12] =2 p 11 p 12 1 1 2 1 2 00 n 11 = 12 1 = + d U 1 1 U 1 1 =U 2 2 U 2 2 [ U 1 1 U 2 2 ][ U 1 1 U 2 2 ] P [12][12] = p 212 10100 n 11 = 22 0 = a U 1 1 U 1 2 =U 2 1 U 2 1 [ U 1 1 U 2 1 ][ U 1 2 U 2 1 ] P [11][21] =2 p 11 p 21 1 1 2 0 1 2 0 n 12 = 11 1 = + d U 1 1 U 1 2 =U 2 1 U 2 2 ( [ U 1 1 U 2 1 ][ U 1 2 U 2 2 ] [ U 1 1 U 2 2 ][ U 1 2 U 2 1 ] ( P [11][22] =2 p 11 p 22 P [12][21] =2 p 12 p 21 ( 1 1 2 1 2 (1 ) 1 2 (1 ) 1 2 n 12 = 12 ( 1 = + d 0 = a U 1 1 U 1 2 =U 2 2 U 2 2 [ U 1 1 U 2 2 ][ U 1 2 U 2 2 ] P [12][22] =2 p 12 p 22 10 1 2 0 1 2 n 12 = 22 0 = a U 1 2 U 1 2 =U 2 1 U 2 1 [ U 1 2 U 2 1 ][ U 1 2 U 2 1 ] P [21][21] = p 221 10010 n 22 = 11 0 = a U 1 2 U 1 2 =U 2 1 U 2 2 [ U 1 2 U 2 1 ][ U 1 2 U 2 2 ] P [21][22] =2 p 21 p 22 100 1 2 1 2 n 22 = 12 0 = a U 1 2 U 1 2 =U 2 2 U 2 2 [ U 1 2 U 2 2 ][ U 1 2 U 2 2 ] P [22][22] = p 222 10001 n 22 = 22 0 = a = p 11 p 22 p 11 p 22 + p 12 p 21 where p 11 p 12 p 21 and p 22 arethehaplotypefrequenciesof U 1 1 U 2 1 U 1 1 U 2 2 U 1 2 U 2 1 ,and U 1 2 U 2 2 ,respectively,and j R = P j S =2 j S =0 j R j S

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62 diplotypeiscomposedoftwohaplotypes,onefromthemotheran dtheotherfrom thefather.Thediplotypefrequenciescanbeexpressedinterm softhehaplotype frequencies(Table4-1).Thesamegenotypemaycontaintwodi erentdiplotypes, dependingonitsheterozygosity.Table4-1alsogivestherela tivefrequencieswith whichagenotypecarriesaparticularhaplotype.Suchrelat ivefrequencieswill beusefulforpartitioningobservedgenotypesintounderlyin gdiplotypesinthe constructionoflikelihoodfunctionbasedonthephenotype. Consideringtwoblocks R and S atthesametime,Iwillhavethejoint haplotypefrequencies,arrayedby n p =( p Rk 1 k 2 ;p Sl 1 l 2 ).Thelog-likelihoodfunctionof theseunknownhaplotypefrequenciesgivenobservedgenotype s( n )canbewritten asamultinomialform,i.e., log L ( n p j n ) / (4.4) 2 2 X k 1 =1 2 X k 2 =1 2 X l 1 =1 2 X l 2 =1 n ( k 1 k 1 =k 2 k 2 )( l 1 l 1 =l 2 l 2 ) (log p Rk 1 k 2 +log p Sl 1 l 2 ) + 2 X k 1 =1 2 X k 2 =1 2 X l 1 =1 n ( k 1 k 1 =k 2 k 2 )( l 1 l 1 =l 2 l 0 2 ) [2log p Rk 1 k 2 +log(2 p Sl 1 l 2 p Sl 1 l 0 2 )] + 2 X k 1 =1 2 X k 2 =1 2 X l 2 =1 n ( k 1 k 1 =k 2 k 2 )( l 1 l 0 1 =l 2 l 2 ) [2log p Rk 1 k 2 +log(2 p Sl 1 l 2 p Sl 0 1 l 2 )] + 2 X k 1 =1 2 X k 2 =1 n ( k 1 k 1 =k 2 k 2 )( l 1 l 0 1 =l 2 l 0 2 ) [2log p Rk 1 k 2 +log(2 p Sl 1 l 2 p Sl 0 1 l 0 2 +2 p Sl 1 l 0 2 p Sl 1 l 0 2 )] + 2 X k 1 =1 2 X l 1 =1 2 X l 2 =1 n ( k 1 k 1 =k 2 k 0 2 )( l 1 l 1 =l 2 l 2 ) [log(2 p Rk 1 k 2 p Rk 1 k 0 2 )+2log( p Sl 1 l 2 )] + 2 X k 1 =1 2 X l 1 =1 n ( k 1 k 1 =k 2 k 0 2 )( l 1 l 1 =l 2 l 0 2 ) [log(2 p Rk 1 k 2 p Rk 1 k 0 2 )+log(2 p Sl 1 l 2 p Sl 1 l 0 2 )] + 2 X k 1 =1 2 X l 2 =1 n ( k 1 k 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 2 ) [log(2 p Rk 1 k 2 p Rk 1 k 0 2 )+log(2 p Sl 1 l 2 p Sl 0 1 l 2 )] + 2 X k 1 =1 n ( k 1 k 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 0 2 ) [log(2 p Rk 1 k 2 p Rk 1 k 0 2 )+log(2 p Sl 1 l 2 p Sl 0 1 l 0 2 +2 p Sl 1 l 0 2 p Sl 0 1 l 2 )]

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63 + 2 X k 2 =1 2 X l 1 =1 2 X l 2 =1 n ( k 1 k 0 1 =k 2 k 2 )( l 1 l 1 =l 2 l 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 2 )+2log( p Sl 1 l 2 )] + 2 X k 2 =1 2 X l 1 =1 n ( k 1 k 0 1 =k 2 k 2 )( l 1 l 1 =l 2 l 0 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 2 )+log(2 p Sl 1 l 2 p Sl 1 l 0 2 )] + 2 X k 2 =1 2 X l 2 =1 n ( k 1 k 0 1 =k 2 k 2 )( l 1 l 0 1 =l 2 l 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 2 )+log(2 p Sl 1 l 2 p Sl 0 1 l 2 )] + 2 X k 2 =1 n ( k 1 k 0 1 =k 2 k 2 )( l 1 l 0 1 =l 2 l 0 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 2 )+log(2 p Sl 1 l 2 p Sl 0 1 l 0 2 +2 p Sl 1 l 0 2 p Sl 0 1 l 2 )] + 2 X l 1 =1 2 X l 2 =1 n ( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 1 =l 2 l 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 0 2 +2 p Rk 1 k 0 2 p Rk 0 1 k 2 )+2log p Sl 1 l 2 ] + 2 X l 1 =1 n ( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 1 =l 2 l 0 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 0 2 +2 p Rk 1 k 0 2 p Rk 0 1 k 2 )+log(2 p Sl 1 l 2 p Sl 1 l 0 2 )] + 2 X l 2 =1 n ( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 0 2 +2 p Rk 1 k 0 2 p Rk 0 1 k 2 )+log(2 p Sl 1 l 2 p Sl 0 1 l 2 )] + n ( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 0 2 ) [log(2 p Rk 1 k 2 p Rk 0 1 k 0 2 +2 p Rk 1 k 0 2 p Rk 0 1 k 2 )+log(2 p Sk 1 k 2 p Sl 0 1 l 0 2 +2 p Sl 1 l 0 2 p Sl 0 1 l 2 )] where1 k 1 k 0 1 2 ; 1 k 2 k 0 2 2 ; 1 l 1 l 0 1 2 ; 1 l 2 l 0 2 2. Assumingthatdiplotypesareassociatedwithphenotypicvariati onina disease,Iformulatealikelihoodforunknownpopulation( n p )andquantitativegeneticparameters( n q )givenobservedphenotypes( y )andSNPgenotypes( n ).Generallyspeaking,agivenfour-SNPgenotypefromtwobloc ks, ( U 1 k 1 U 1 k 0 1 =U 2 k 2 U 2 k 0 2 )( V 1 l 1 V 1 l 0 1 =V 2 l 2 V 2 l 0 2 ),canbepartitionedintofourpossiblediplotypes,[( U 1 k 1 U 2 k 2 )( V 1 l 1 V 2 l 2 )][( U 1 k 0 1 U 2 k 0 2 )( V 1 l 0 1 V 2 l 0 2 )],[( U 1 k 1 U 2 k 2 )( V 1 l 1 V 2 l 0 2 )][( U 1 k 0 1 U 2 k 0 2 )( V 1 l 0 1 V 2 l 2 )], [( U 1 k 1 U 2 k 0 2 )( V 1 l 1 V 2 l 2 )][( U 1 k 0 1 U 2 k 2 )( V 1 l 0 1 V 2 l 0 2 )]and[( U 1 k 1 U 2 k 0 2 )( V 1 l 1 V 2 l 0 2 )][( U 1 k 0 1 U 2 k 2 )( V 1 l 0 1 V 2 l 2 )].The log-likelihoodfunctionof n p and n p canbeformulatedonthebasisofafourcomponentmixturemodel,i.e., log L ( n p ; n q j y;G ) = n X i =1 log[ $ [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] j i f [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] ( y i ) + $ [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] j i f [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ( y i )

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64 + $ [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] j i f [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] ( y i ) + $ [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] j i f [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] ( y i )] ; (4.5) wherethemixtureproportions, $ [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] j i (4.6) = p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; $ [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] j i (4.7) = p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; $ [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] j i (4.8) = p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; $ [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] j i (4.9) = p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; representtherelativefrequenciesofthecorrespondingacro ss-blockdiplotypesthat formthesamegenotype,and f [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] ( y i ) ;f [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ( y i ), f [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] ( y i )and f [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] ( y i )aretheprobabilitydensity functionsforsubject i whohasthecorrespondingdiplotype,withthegenotypic means [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] ; [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ; [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] ; and [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] ,respectively,andthecommonresidualvariance 2 .These meansandvariancearecontainedinvector n q .Notethatthemixtureproportions areexpressedasbeingsubject-specicbecausedierentsubjects eachwithaknown genotypemayhavedierentdiplotypecompositions. Assumethathaplotypes U 1 1 U 2 1 and V 1 1 V 2 1 arethereferencehaplotypesatblocks R and S ,respectively.Asshownabove,thisleadstoninedierentacro ss-block

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65 compositegenotypes.equation(4.1)providesthestructureof anarbitraryacrossblockcompositegenotypeformedbythereferenceandnon-ref erencehaplotypes. Thelog-likelihoodfunctiondescribedbyequation(4.5)can nowbeexpressed,in specicforms,as log L ( n p ; n q j y; n )= (4.10) n (11 = 11)(11 = 11) X i =1 log f AABB ( y i )+ n (11 = 11)( ) X i =1 log f AAB B ( y i )+ n (11 = 11)( ) X i =1 log f AA B B ( y i ) + n (11 = 11)(12 = 12) X i =1 log[ $ S f AAB B ( y i )+(1 $ S ) f AA B B ( y i )] + n ( )(11 = 11) X i =1 log f A ABB ( y i )+ n ( )( ) X i =1 log f A AB B ( y i )+ n ( )( ) X i =1 log f A A B B ( y i ) + n ( )(12 = 12) X i =1 log[ $ S f A AB B ( y i )+(1 $ S ) f A A B B ( y i )] + n ( )(11 = 11) X i =1 log f A ABB ( y i )+ n ( )( ) X i =1 log f A AB B ( y i )+ n ( )( ) X i =1 log f A A B B ( y i ) + n ( )(12 = 12) X i =1 log[ $ S f A AB B ( y i )+(1 $ S ) f A A B B ( y i )] + n (12 = 12)(11 = 11) X i =1 log[ $ R f A ABB ( y i )+(1 $ R ) f A ABB ( y i )] + n (12 = 12)( ) X i =1 log[ $ R f A AB B ( y i )+(1 $ R ) f A AB B ( y i )] + n (12 = 12)( ) X i =1 log[ $ R f A A B B ( y i )+(1 $ R ) f A A B B ( y i )] + n (12 = 12)(12 = 12) X i =1 log[ $ [(11)(11)][(22)(22)] f A AB B ( y i )+ $ [(11)(12)][(22)(21)] f A A B B ( y i ) + $ [(12)(11)][(21)(22)] f A AB B ( y i )+ $ [(12)(12)][(21)(21)] f A A B B ( y i )] ; where n (11 = 11)( ) = n (11 = 11)(11 = 12) + n (11 = 11)(12 = 11) ;

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66 n (11 = 11)( ) = n (11 = 11)(11 = 22) + n (11 = 11)(12 = 22) + n (11 = 11)(22 = 11) + n (11 = 11)(22 = 12) + n (11 = 11)(22 = 22) ; n ( )(11 = 11) = n (11 = 12)(11 = 11) + n (12 = 11)(11 = 11) ; n ( )(11 = 11) = n (11 = 22)(11 = 11) + n (12 = 22)(11 = 11) + n (22 = 11)(11 = 11) + n (22 = 12)(11 = 11) + n (22 = 22)(11 = 11) ; n ( )( ) = n (11 = 12)(11 = 12) + n (11 = 12)(12 = 11) + n (12 = 11)(11 = 12) + n (12 = 11)(12 = 11) ; n ( )( ) = n (11 = 12)(11 = 22) + n (11 = 12)(12 = 22) + n (11 = 12)(22 = 11) + n (11 = 12)(22 = 12) + n (11 = 12)(22 = 22) + n (12 = 11)(11 = 22) + n (12 = 11)(12 = 22) + n (12 = 11)(22 = 11) + n (12 = 11)(22 = 12) + n (12 = 11)(22 = 22) ; n ( )( ) = n (11 = 22)(11 = 12) + n (12 = 22)(11 = 12) + n (22 = 11)(11 = 12) + n (22 = 12)(11 = 12) + n (22 = 22)(11 = 12) + n (11 = 22)(12 = 11) + n (12 = 22)(12 = 11) + n (22 = 11)(12 = 11) + n (22 = 12)(12 = 11) + n (22 = 22)(12 = 11) ; n ( )( ) = n (11 = 22)(11 = 22) + n (11 = 22)(12 = 22) + n (11 = 22)(22 = 11) + n (11 = 22)(22 = 12) + n (11 = 22)(22 = 22) + n (12 = 22)(11 = 22) + n (12 = 22)(12 = 22) + n (12 = 22)(22 = 11) + n (12 = 22)(22 = 12) + n (12 = 22)(22 = 22) + n (22 = 11)(11 = 22) + n (22 = 11)(12 = 22) + n (22 = 11)(22 = 11) + n (22 = 11)(22 = 12) + n (22 = 11)(22 = 22) + n (22 = 12)(11 = 22) + n (22 = 12)(12 = 22) + n (22 = 12)(22 = 11) + n (22 = 12)(22 = 12) + n (22 = 12)(22 = 22) + n (22 = 22)(11 = 22) + n (22 = 22)(12 = 22) + n (22 = 22)(22 = 11) + n (22 = 22)(22 = 12) + n (22 = 22)(22 = 22) ; n ( )(12 = 12) = n (11 = 12)(12 = 12) + n (12 = 11)(12 = 12) ; n (12 = 12)( ) = n (12 = 12)(11 = 12) + n (12 = 12)(12 = 11) ; n ( )(12 = 12) = n (11 = 22)(12 = 12) + n (12 = 22)(12 = 12) + n (22 = 11)(12 = 12) + n (22 = 12)(12 = 12) + n (22 = 22)(12 = 12) ;

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67 n (12 = 12)( ) = n (12 = 12)(11 = 22) + n (12 = 12)(12 = 22) + n (12 = 12)(22 = 11) + n (12 = 12)(22 = 12) + n (12 = 12)(22 = 22) ; $ R = p R11 p R22 p R11 p R22 + p R12 p R21 ; (4.11) $ S = p S11 p S22 p S11 p S22 + p S12 p S21 ; (4.12) and $ [(11)(11)][(22)(22)] ;$ [(11)(12)][(22)(21)] ;$ [(12)(11)][(21)(22)] and $ [(12)(12)][(21)(21)] arethe mixtureproportionsforthecompleteheterozygousgenotyp eatallthefourSNPs fromtwoblocksandcanbeexpressedbyEquations(4.6){(4.9). Dierentfrom Equations(4.6){(4.9),theseexpressionsignore i becauseallsubjectswiththe completeheterozygousgenotypehavethesamediplotypecomp osition. 4.2.4AnIntegrativeEMAlgorithm Iderivedaclosed-formsolutionforestimatingtheunknownpar ameterswith theEMalgorithm.Theestimatesofhaplotypefrequenciesare basedontheloglikelihoodfunctionofequation(4.4),whereastheestimate sofdiplotypegenotypic meansandresidualvariancearebasedonthelog-likelihoodfu nctionofequation (4.5).Thesetwodierenttypesofparameterscanbeestimated usinganintegrative EMalgorithm(APPENDIXB). Haplotypefrequenciescanbeexpressedasafunctionofallelic frequenciesand LD.Foratwo-SNPhaplotypewithinblock R ,Ihave p Rk 1 k 2 = p R (1) k 1 p R (2) k 2 +( 1) k 1 + k 2 D R ; (4.13) where D R isthelinkagedisequilibriumbetweenthetwoSNPsatblock R .Thus, oncehaplotypefrequenciesareestimated,Icanestimatealle licfrequenciesandLD bysolvingequation(4.13).Similarcalculationsarealsodon eforblock S .After across-blockcompositegenotypicvaluesareestimated,Icanest imatetheadditive, dominantandepistaticeectsbetweentwoblocksusingequati on(4.2).The

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68 standarderrorsoftheMLEsofthepopulationandquantitativ egeneticparameters canbeestimatedonthebasisofLouis'(1982)observedinformati onmatrix. 4.3HypothesisTests Twomajorhypothesescanbemadeinthefollowingsequence:(1) theassociationbetweendierentSNPswithineachblockbytestingtheir linkagedisequilibrium(LD),and(2)thesignicanceofanassumedreferencehaplo typeforitseect onthediseaseoutcome.TheLDbetweentwogivenSNPswithinbloc k R canbe testedusingthefollowinghypotheses: 8><>: H 0 : D R =0 H 1 : D R 6 =0 (4.14) Thelog-likelihoodratioteststatisticforthesignicanceofL Discalculatedby comparingthelikelihoodvaluesunderthe H 1 (fullmodel)and H 0 (reducedmodel) using LR R = 2[log L ( e p R (1) k 1 ; e p R (2) k 2 ;D R =0 ; e n q j n ) log L ( b n p ; b n q j n )](4.15) wherethetildeandhatdenotetheMLEsofunknownparameters under H 0 and H 1 respectively.The LR R isconsideredtoasymptoticallyfollowa 2 distributionwith onedegreeoffreedom.TheMLEsofallelicfrequenciesunder H 0 canbeestimated usingtheEMalgorithmdescribedabove,butwiththeconstraint p R11 p R22 = p R12 p R21 .A similartestcanbemadeforblock S Diplotypeorhaplotypeeectsonacomplexdiseasecanbetested usingthe nullandalternativehypothesesexpressedas 8><>: H 0 : j R j S = H 1 :atleastoneequalityin H 0 doesnothold (4.16) Thelog-likelihoodratioteststatistic( LR )underthesetwohypothesescanbe similarlycalculated.The LR mayasymptoticallyfollowa 2 distributionwith

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69 eightdegreeoffreedom.However,theapproximationofa 2 distributionmaybe inappropriatewhensomeregularityconditions,suchasnormal anduncorrelated residuals,areviolated.Thepermutationtestapproachpropose dbyChurchilland Doerge(1994),whichdoesnotrelyuponthedistributionofth e LR ,maybeusedto determinethecriticalthresholdfordeterminingtheexisten ceofaQTL. Dierentgeneticeects,suchastheadditive( R and S ),dominant( R and S )andadditive additive( I ),additive dominant( J ),dominant additive ( K )anddominant dominanteects( L )betweenblocks R and S canalsobe testedindividually.Thecriticalthresholdsfortheseindivi dualeectscanbe determinedonthebasisofsimulationstudies. 4.4Results Iusearealexamplefromanobesitystudytodemonstratethepower and usefulnessofthismodel.Numerousgeneshavebeeninvestigated aspotential obesity-susceptibilitygenes(Masonetal.1999;Chagnonetal. 2003).The 1AR and 2ARgenesaretwosuchexamples(Greenetal.1995;Largeetal.1 997)in eachofwhichthereareseveralpolymorphismscommoninthepop ulation.Two commonpolymorphismsareidentiedatcodons49and389forth e 1ARgene onchromosome10andatcodons16and27forthe 2ARgeneonchromosome5, respectively.Thepolymorphismsineachofthesetworeceptorg enesareinlinkage disequilibrium,whichsuggeststheimportanceoftakingintoa ccounthaplotypes, ratherthanasinglepolymorphism,whendeningbiologicfunc tion.Thisstudy attemptstodetecthaplotypevariantswithinthesecandidat egeneswhichdetermine humanobesitytraits. Todeterminewhethersequencevariantsatthetwopolymorphi smsfromone geneinteractwiththosefromtheothergenetoaectobesityph enotypes,agroup of163menandwomenwereinvestigatedwithagesfrom32to86ye arsoldwitha largevariationinbodyfatmass.Eachofthesepatientswasdete rminedfortheir

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70 genotypesatcodon49withtwoalleles,Ser49(A)andGly49(G), andcodon389 withtwoalleles,Arg389(C)andGly389(G),withinthe 1ARgene,aswellas atcodon16withtwoalleles,Arg16(A)andGly16(G),andcodon27 withtwo alleles,Gln27(C)andGlu27(G),withinthe 2ARgene,andmeasuredforbody massindex(BMI).TwoSNPsfromeachgenetheoreticallyform81 across-gene genotypes,but,becausethesetwogenesareindependent,thefr equenciesofthese genotypescanbeexpressedastheproductofthegenotypefrequ enciesfromeach gene.TheintegrativeEMalgorithmbasedonthelikelihoodfu nction(4.4)allows fortheestimatesoffourhaplotypefrequenciesandtheresult ingallelefrequencies andlinkagedisequilibriumateachgene(Table4-2).Highlysig nicantLDwas detectedbetweentwoSNPsforeachgene( P< 0 : 001).Smallsamplingerrorsfor theestimatesofeachpopulationgeneticparameterindicate dthattheestimatesare highlyprecise. Byassumingthatonehaplotypeisdierentfromtherestofhaplo typesat eachgene,thismodelcandetectthereferencehaplotypesth atdisplaysignicant mainandinteractioneectsontheBMItrait.Usinghaplotypes AC,AG,GC andGGasareferencehaplotypeatthe 1ARgene,respectively,inconjunction withareferencehaplotypeselectedfromAC,AG,GCorGGatthe 2ARgene, Icalculatedthecorrespondinglog-likelihood-ratio(LR)t eststatistics(0.36{ 17.90)usingequation(4.10)(Table4-2).Basedonthecritica lthresholdvalueof 15.05atthe5%signicanceleveldeterminedfrom1000permut ationtests,thetwo maximalLRvalues,15.1and17.9foracross-genereferencehapl otypes(GG)(GC) and(GC)(GC),respectively,arethoughttotriggersignican thaplotypeeectson BMI.However,becausethesetworeferencehaplotypesformdie rentnumbersof compositegenotypes,withallninefor(GG)(GC)andeightfor( GC)(GC)(Table 4-2),anoptimalreference-haplotypecombinationshouldbe selectedonthebasis

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71Table4{2:MaximumlikelihoodestimatesofSNPpopulationgen eticparameters(allelefrequenciesandlinkagedisequilibrium)andquantitativegeneticparametersassociatedwit hphenotypicvariationinBMIfor145patientswhendierent across-genereferencehaplotypesareassumed Across-blockreferencehaplotypecombinations Parameters(AC)(AC)(AC)(AG)(AC)(GC)(AC)(GG)(AG)(AC)( AG)(AG)(AG)(GC)(AG)(GG)(GC)(AC)(GC)(AG)(GC)(GC)(GC) (GG)(GG)(AC)(GG)(AG)(GG)(GC)(GG)(GG) LR 1 5.004.236.641.841.991.362.290.369.243.4417.902.2810 .7710.36 15.06 8.46 P value 0.05 Populationgeneticparametersp 2 (1) 1 0.390.390.390.390.390.390.390.390.610.610.610.610.6 10.61 0.61 0.61 p 2 (2) 1 0.630.630.630.630.370.370.370.370.630.630.630.630.3 70.37 0.37 0.37 D 2 0.130.130.130.13-0.13-0.13-0.13-0.13-0.13-0.13-0.13 -0.130.130.13 0.13 0.13 p 1 (1) 1 0.850.850.150.150.850.850.150.150.850.850.150.150.8 50.85 0.15 0.15 p 1 (2) 1 0.740.260.740.260.740.260.740.260.740.260.740.260.7 40.26 0.74 0.26 D 1 -0.040.040.04-0.04-0.040.040.04-0.04-0.040.040.04-0 .04-0.040.04 0.04 -0.04 Quantitativegeneticparameters AABB 32.7235.9721.68-----32.16---29.7324.41 24.18 AAB B 29.0729.7429.7933.21----34.6529.5827.7122.6629.4629 .58 29.28 AA B B 31.7130.7831.9730.96----23.1132.9532.8131.4224.3329 .02 28.94 28.79 A ABB 29.3631.08--28.5526.09--28.6231.3136.35-28.4826.69 36.35 A AB B 32.4129.5534.0324.0620.7527.4830.6932.7633.1929.664 0.1245.7227.5327.11 27.43 24.77 A A B B 30.7031.6229.5330.8231.2227.9624.9127.5733.4731.932 8.9931.3027.6028.66 27.84 27.93 A ABB 29.2325.8430.27-29.9030.2127.40-30.4228.8522.93-31. 18 33.80 21.68A AB B 29.2928.1228.7423.9030.6729.0731.5224.1628.4928.592 8.2327.4133.8731.03 36.19 29.34 A A B B 26.2629.6128.7228.8129.2630.6929.6930.1128.0029.412 9.5129.0532.3032.88 31.40 32.44 8.758.788.708.858.848.868.838.898.628.808.348.838.5 78.59 8.43 8.64 Note:\-"denotesthemissingofcompositegenotypesunderth ecorrespondingacross-genereferencehaplotypes.

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72 oftheAICcriterion.Ifoundthatacross-genereferencehaplot ype(GC)(GC)isthe bestforexplainingtheBMIdatainthisexample. Themissingofonecompositegenotypegeneratedbyacross-genere ference haplotype(GC)(GC)preventstheestimationofalltheadditi ve,dominantand epistaticeectsusingequation(4.2)becauseofinadequatede greesoffreedom.As anexample,Iusedacross-genereferencehaplotype(GG)(GC)to demonstratehow eachofthesegeneticeectsistested.Itisfoundthattheaddi tiveanddominant eectsexertedby(GG)(GC)arenotsignicant(datanotshown) .Ofthefour kindsofepistasisbetweenthetwogenes,onlydominant dominantgeneticeect issignicantatthe5%signicancelevel.Thistypeofepistasis reduces,byabout 20%,theBMIofthepatientwhocarriesdiplotypes[GG][ GG]atthe 2ARgene and[GC][ GC]atthe 1ARgene,comparedtotheotheracross-genediplotypes. IestimatethestandarderrorsoftheMLEsofthepopulationand quantitative geneticparametersbasedonLouis'(1982)observedinformatio nmatrix,suggesting thatallMLEshavereasonableestimationprecisionalthoughth eestimatesof quantitativegeneticparametersarenotaspreciseasthoseof populationgenetic parametersduetoasmallsamplesizeused. 4.5Discussion Foranytwounrelatedpeople,about99.9%oftheirDNAsequence sare detectedtobethesame.Itistheremaining0.1%thatcontains thegeneticvariants thatinruencehowpeopledierintheirriskofdiseaseortheirr esponsetodrugs. DiscoveringconcreteDNAsequencevariantsthatcontributeto commondiseaserisk oersoneofthebestopportunitiesforunderstandingthecomp lexcausesofdisease inhumans.Therecentdevelopmentofahaplotypemapofthehum angenome,the HapMap,bytheInternationalHapMapConsortium(2003)provide sakeyresource todescribethecommonpatternsofhumanDNAsequencevariation andndgenes thataecthealth,disease,andresponsestodrugsandenvironme ntalfactors.

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73 Ihaverecentlydevelopedaseriesofstatisticalmodelsthatca ndetectspecic DNAsequencevariantsforcomplexdiseases(Liuetal.2004)ordr ugresponse(Lin etal.2005)withtheaidsoftheinformationprovidedbytheHa pMap.Dierent fromquantitativetraitloci(QTL)denedasputativechrom osomalsegments,DNA sequencevariantsdetectedfrommymodelsareshownattheindi vidualnucleotide level,whicharethereforecalled quantitativetraitnucleotides (QTN).Althoughthe modelsforQTNmappinghavebeenstudiedbytheoreticalsimula tionsandvalidatedwithreal-worlddata,theirapplicationsmaystillbel imitedbecauseoftheir underlyingassumptionthatgenesoperateindividually.Form ostcommondiseases, suchasdiabetes,cancer,obesity,stroke,heartdisease,depressio n,andasthma, however,itislikelythatasuiteofgenesandenvironmentalf actorsareinvolvedto formacomplicatedwebofinteractions(Segreetal.2005).I nteractionsbetween dierentgenes(i.e.,epistasis)orbetweengenesandenvironm entalfactorsare thoughttobeevolutionaryforcestomaintaingeneticvaria tionandbueragainst environmentalordevelopmentalperturbations(Moore2005 ).Inthischapter,I extendmyearlierstatisticalmodelstoperformagenome-wide scanforsequencesequenceinteractionsasafundamentalcomponentofgenetic networkfordisease susceptibility.Beyondtraditionalepitstaticmodelsbasedon locuscosegregations (LynchandWalsh1998),thismodelallowsforthedirectionch aracterizationofspecicDNAsequenceinteractionsontheSNP-constructedHapMap.Asa nexample, Ihaveusedthedatafromanobesityresearchprojecttovalidate theimplications ofmyinteractivemodels.SpecicDNAsequencesfromtwodiere ntcandidate genes, 1ARand 2AR,havebeenidentiedtoaecthumanobesitytraitsinan interactivemanner. Forclarication,mypresentedtheideaforsequencingacompl extraitwith aninteractionmodelbasedontwo-DNAsequencesfromeachhaplo typeblock. Itislikelythatthetwo-SNPmodelistoosimpletocharacteriz egeneticvariants

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74 forquantitativevariation.Withthefoundationforthetwo -SNPsequencing model,Icanreadilyextendthismodeltoincludeanarbitrar ynumberofSNPs whosesequencesareassociatedwiththephenotypicvariation.I nthemulti-SNP sequencingmodel,Ifacemanyhaplotypesandhaplotypepairs. AnAIC-orBICbasedmodelselectionstrategy(BurnhamandAndersson1998)hasbe enframed todeterminethehaplotypethatismostdistinctfromtherestof haplotypesin explainingquantitativevariation.However,inpractice,a simultaneouslyanalysis oftoomanySNPswillencounterconsiderablecomputationally loadand,also,may notbenecessaryfortheexplanationofdiseasevariation.Thede terminationofa maximalnumberofSNPsforsequencingmappingofQTNshouldbein tegrated withcomputationalalgorithmsforhaplotypeblockmodelli ng(KimmelandShamir 2005). OneofthemostimportantstatisticalissuesforQTNmappingisthe derivation ofaneectiveapproachtohandlemissingcompositegenotypes.F orsomereference haplotypes,oneormorecompositegenotypesaremissingduetoal owfrequencyof theiroccurrencesalthoughtheireectsmayexist.Themissing ofthesecomposite genotypespreventsthefullestimatesofthegeneticeects.G iventhatthese compositegenotypesarenotmissingatrandom,patternmixture modelsdeveloped fortreatingnon-randomlymissingdata(Little1993,1994)ca nbeusedtoestimate thegenotypicvaluesofmissingcompositegenotypes.Thus,after theideabehind thepatternmixturemodelsisincorporatedintothesequence mappingofQTN,I areclosertoprovidebetterestimatesofsequenceactionandin teractioneectson complexdiseases.

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CHAPTER5 MODELFORDETECTINGSEQUENCE-SEQUENCEINTERACTIONSFOR DRUGRESPONSE 5.1Introduction Theincreasingnumberofgeneticstudiesforcomplextraitsan dbiological processesinhumansrequiresmoreadvancedtechniquesofstati sticalanalysis (LynchandWalsh1998).Thisisduetotworeasons.First,genesin teractina complexnetworktodetermineanalphenotype.Itisveryoft enthatacomplex traitischaracterizedbyanumberofnon-Mendelian,enviro nmentallysensitive genes,ofwhichsomeactadditively,whereasmanyothersareop erationalina multiplicativeorcompensatoryway(FrankelandSchork1996 ;Moore2003).Many currentstatisticaltechniquesusedingeneticresearchassumeth eadditivecontrol ofgenes,aimedtofacilitatedataanalysisandmodelling,whi chcertainlyprovide misleadingresultswhengeneticinteractionsorepistasisactu allyoccur. Second,almosteveryphenotypictraitcanbepartitionedint oitsmultiple continuousdevelopmentalcomponentsonatimescale.Tobett erstudythegenetic architectureofthesetraits,Ineedmeasurethetraitsatamult itudeofdiscretetime points.Adierentbutstatisticallysimilarexampleoftheseso-c alledtime-series traitsisdrugresponse,aeldthatgainedmuchattentionduet othepossible clinicalapplications,rangingfromindividualizedtherap ytonewdrugdevelopment (Arranzetal.2002).Recentstudieshavesuggestedthatavariab lenumberof polymorphismsinvariousgenesaresupposedlyinvolvedinmodu latingtheresponse and/orsideeectstodrugs(SerrettiandArtioli2003). Themotivationofthischapteristodevelopastatisticalmode lfordetecting epistaticinteractionsthatcontroldrugresponse.Thisnewmo delisconstructed 75

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76 throughcombiningtworecentdevelopmentsingeneticmappi ng.Atraditional mappingstrategyisbasedonthecosegregationofgenotypedmar kersandquantitativetraitloci(QTL)thatarebracketedbythemarkers( LanderandBotstein 1989).QTLforacomplextraitidentiedfromthisstrategypr esentsahypothesized chromosomalsegmentwhoseDNAsequenceisunknown.Thesecondstrat egy, recentlydevelopedbyLiuetal.(2004)andLinetal.(2005), canidentifyspecic DNAsequencevariantsthataectacomplextrait.Thisstrategy ,thatreliesonthe recentadventofhigh-throughoutsinglenucleotidepolymor phism(SNP)technologies,allowsforthegenomewidescanofcausalDNAsequences,calle d quantitative traitnucleotides (QTN).Linetal.(2005)havedevelopedaconceptualframewor k fordetectinginteractioneectsbetweendierentQTNfora complextrait. Consideringthedynamicfeatureofmanycomplextraits,aserie sofstatistical models,called functionalmapping ,havebeenrecentlydevelopedtocharacterize QTLthatcontributetogeneticvariationforlongitudinalt raits(Maetal.2002;Wu etal.2004a,2004b,2004c).Functionalmappingcapitalize sonthemathematical functionsthatdescribebiologicalprocessesandembedsthemw ithinthecontext ofgeneticmappingtheory.Byestimatingthemathematicalpa rametersthat determinethepatternsoflongitudinaltrajectories,funct ionalmappinghasproven ecientandeectiveforunveilingthegeneticarchitectur eofcomplextraits.Inthis chapter,IincorporatetheideaofQTNmappingwithfunction almappingtodetect epistaticinteractionsbetweendierentDNAsequencevariant sthatencodedrug response.Ialsoperformsimulationstudiestoexaminethestatisti calpropertiesof thismodel.Arealexamplewasusedtovalidatetheusefulnessof thismodel. 5.2Theory 5.2.1TheNormalMixtureModel ApproachesforQTLandQTNmappingarestatisticallysimilarint heconceptualformationofamixturemodel.ForQTLmapping,eacho bservationmust

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77 arisefromoneofmultipleQTLgenotypes,althoughtheQTLgeno typeforthis individualisunknown(LanderandBotstein1989).Thenormal mixturemodelis constructedtocontainthepossibleimpactoftheseQTLgenotype seachofwhich isassumedtofollowanormaldistribution.AsshowninLiuetal.(2 004),QTN mappingisalsobasedonanormalmixturemodelbutinwhichthep henotypic valueofanindividualthatisheterozygousfortwoormoreSNP sisthoughttoarise oneofmultiplediplotypesconstructedbysetofSNPs. Dierencesamongdierentdiplotypescanbeassumedtoresultf romthe compositionofdierenthaplotypes.Liuetal.(2004)denedt hehaplotype thatisdierentfromtherestofhaplotypesas referencehaplotype .Thoserestof haplotypesarethusdenedas non-referencehaplotype .ConsideraQTN, R ,that iscomposedofasetofSNPs.Let A and A bethereferenceandnon-reference haplotypesforthisQTN,respectively,whichthusformthreed ierent composite genotypes AA A A and A A .Inthemixturemodel,longitudinalobservationsfor eachcompositegenotype, j ,arecharacterizedbyadierentmultivariatenormal distributionwithmeanvector u j andcovariancematrix ForQTNmapping,onlydiplotypesthatareheterozygousfort woormore SNPscontaindierentcompositegenotypesbecausethesediplot ypesarenot consistentwiththeirphenotypicallyobservablegenotypes.Th erefore,themixture modelisformulatedonlyforthosediplotypes.5.2.2EpistaticEects Idevelopedaninteractivemodelaimedtodetectsequence-seq uenceepistasis. Let B and B bethereferenceandnon-referencehaplotypesatasecondQTN, B respectively.ThetwoQTN, A and B ,generateninedierentcompositegenotypes expressedas AABB AAB B AA B B A ABB A AB B A A B B A ABB A AB B and A A B B .Traditionalquantitativegenetictheoriescanbeusedtomo delthe geneticeectsofthecompositegenotypes(LynchandWalsh199 8).Thegenotypic

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78 vector( u j R j S )ofajointcompositegenotypeatthetwohaplotypeblockscan be decomposedintoninedierentcomponentsasfollows: u j R j S = u Overallmean(5.1) +( j R 1) a R +( j S 1) a S Additiveeects +[1 ( j R 1) 2 ] b R +[1 ( j S 1) 2 ] b S Dominanteects +( j R 1)( j S 1) i Additive additiveeect +( j R 1)[1 ( j S 1) 2 ] j Additive dominanteect +[1 ( j R 1) 2 ]( j S 1) k Dominant additiveeect +[1 ( j R 1) 2 ][1 ( j S 1) 2 ] l Dominant dominanteect ; where j R ;j S = 8>>>>><>>>>>: 2for AA or BB 1for A A or B B 0for A A or B B standforthecompositegenotypesatblocks R and S ,respectively, a : and b : are theadditiveanddominanteectvectorsatthecorresponding block,respectively, and i j k and l aretheadditive additive,additive dominant,dominant additiveanddominant dominantepistaticeectvectorsbetweenthetwoQTN, respectively. Ifthemaximumlikelihoodestimates(MLEs)ofthegenotypicva luevectorsat theleftsideofequation(5.1)canbeobserved,Icansolvetheve ctorsfortheoverall mean,additive,dominantandfourkindsofepistaticeectsb etweentwoQTNby u = 1 4 ( u A A B B + u AA B B + u A ABB + u AABB ) a R = 1 4 ( u AABB u A A B B + u AA B B u A ABB ) a S = 1 4 ( u A ABB u A A B B u AA B B + u AABB )

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79 b R = 1 4 (2 u A A B B u A A B B u AA B B u A ABB u AABB +2 u A ABB ) b S = 1 4 (2 u A AB B u A A B B u AA B B u A ABB u AABB +2 u AAB B ) i = 1 4 ( u AABB u AA B B u A ABB + u A A B B ) j = 1 4 (2 u AAB B u AABB 2 u A AB B + u A A B B u AA B B + u A ABB ) k = 1 4 (2 u A ABB 2 u A A B B + u A A B B + u AA B B u A ABB u AABB ) l = 1 4 (4 u A AB B + u A A B B + u AA B B + u A ABB + u AABB 2 u A A B B 2 u A ABB 2 u A AB B 2 u AAB B )(5.2) 5.2.3LikelihoodFunctions ThemixturemodelusedtomapQTNincludestheproportionsofe achmixture componentandtheprobabilitydistributionfunctionsofphe notypicobservations giventhatcomponent.Themixtureproportionsaretherelat ivefrequenciesof thosediplotypesthatarephenotypicallythesameandtheycan beexpressedas afunctionofhaploidfrequencies.Inthischapter,twoQTNar eassumedtobe independent,whichmeansthattheirjointhaplotypefreque nciesaretheproductsof twohaplotypefrequencieseachfromaQTN. Supposethereare R and S SNPsforQTN R and S ,respectively.Thetwo alleles,1and2,ateachoftheseSNPsaresymbolizedby k 1 ;:::;k R and l 1 ;:::;l R respectively.Ahaplotypefrequencyisdenotedby p Rk 1 k 2 k R forQTN R and p Sl 1 l 2 l S forQTN S .Asstatedabove,across-QTNhaplotypefrequenciescanbecalcu lated astheproductofthecorrespondinghaplotypefrequenciesfr omadierentQTN, expressedas p ( k 1 k 2 k R )( l 1 l 2 l S ) = p Rk 1 k 2 k R p Sl 1 l 2 l S ; (5.3) wheretheparenthesesareusedtoseparatetwodierentQTNfora givenacrossQTNhaplotype.Withtheseacross-QTNhaplotypefrequencies,ex pectedacross-

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80 QTNdiplotypefrequenciesandacross-QTNgenotypefrequenci escanbecalculated, respectively,underHardy-Weinbergequilibrium(LynchandW alsh1998). Withacross-QTNdiplotypeandgenotypefrequencies,thelikel ihoodfunction basedonacross-QTNgenotypeobservations, n = f n ( k 1 k 0 1 =k 2 k 0 2 = =k R k 0 R )( l 1 l 0 1 =l 2 l 0 2 = =l S l 0 S ) g canbeconstructed.Isimplifythepresentationofthemodelbya ssumingtwoSNPs foreachQTN.InTable5-1,allpossiblegenotypesanddiplotype saregivenas wellastheirfrequenciesattwoSNPsgenotypedfromQTN R .Eachdiplotype iscomposedoftwohaplotypes,onefromthemotherandtheother fromthe father.Thediplotypefrequenciescanbeexpressedintermsof thehaplotype frequencies.Thesamegenotypemaycontaintwodierentdiplo types,depending onitsheterozygosity.Table5-1alsoprovidestherelativefr equencieswithwhicha genotypecarriesaparticularhaplotype.Suchrelativefre quencieswillbeusefulfor partitioningobservedgenotypesintounderlyingdiplotype sintheconstructionof likelihoodfunctionbasedonthephneotype. Let n p =( p Rk 1 k 2 ;p Sl 1 l 2 )bethepopulationgeneticparameterstobeestimated forQTN R and S .Linetal.(2005)formulatedalog-likelihoodfunctionoft hese unknownhaplotypefrequenciesgivenobservedgenotypes( n )inamultinomial form.Theyalsoprovidedaseriesofclosed-formsolutionforthe EMalgorithmto estimatethesehaplotypefrequencies.Tosavespace,adetailedp rocedureforthis estimationprocessisnotgiven. Whiletraditionalmodelsassumetheassociationbetweenthegen otypeand drugresponse(Gongetal.2004),thismodelcanestimatetheee ctsofdierent diplotypesonthepharmacodynamicresponseofdrugs.Aparticu larfour-SNP genotypefromtwoQTN,expressedas( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 0 2 ),canbepartitioned intofourpossiblediplotypes,[( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )],[( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )], [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )]and[( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )].Let n q bequantitative geneticparametersthatspecifythemeanvectorsofdierent diplotypesandthe

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81Table5{1:Possiblediplotypesandtheirfrequeneciesforeac hofninegenotypesattwoSNPswithinaQTN,haplotypecompositionfrequenciesforeachgenotypeandgenotypicvaluev ectorsofcompositegenotypes RelativediplotypeHaplotypecompositionCompositeGenot ypic GenotypeDiplotypeDiplotypefrequencyfreq.withingenot ypes[11][12][21][22]genotypemeanvector 11 = 11[11][11] P [11][11] = p 211 11000 AA u 2 11 = 12[11][12] P [11][12] =2 p 11 p 12 1 1 2 1 2 00 A A u 1 11 = 22[12][12] P [12][12] = p 212 10100 A A u 0 12 = 11[11][21] P [11][21] =2 p 11 p 21 1 1 2 0 1 2 0 A A u 1 12 = 12 ( [11][22][12][21] ( P [11][22] =2 p 11 p 22 P [12][21] =2 p 12 p 21 ( 1 1 2 1 2 (1 ) 1 2 (1 ) 1 2 ( A A A A ( u 1 u 0 12 = 22[12][22] P [12][22] =2 p 12 p 22 10 1 2 0 1 2 A A u 0 22 = 11[21][21] P [21][21] = p 221 10010 A A u 0 22 = 12[21][22] P [21][22] =2 p 21 p 22 100 1 2 1 2 A A u 0 22 = 22[22][22] P [22][22] = p 222 10001 A A u 0 TwoallelesforeachofthetwoSNPsaredenotedas1and2,resp ectively.GenotypesatdierentSNPsareseparatedbyaslas h. Diplotypesarethecombinationoftwobracketedmaternally andpaternallyderivedhaplotypes.Byassumingthathaplot ype[11]isthe referencehaplotype,compositegenotypesareaccordingly denedandtheirgenotypicmeanvectorsaregiven. = p 11 p 22 p 11 p 22 + p 12 p 21 where p 11 p 12 p 21 and p 22 arethehaplotypefrequenciesof[11],[12],[21],and[22], respectively. u j R = P j S =2 j S =0 u j R j S

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82 residualcovariancematrix.Thelikelihoodofunknownpopul ation( n p )and quantitativegeneticparameters( n q )givenadrugresponsemeasuredat C dosage levels, y = f y (1) ;:::;y ( C ) g ,andSNPgenotypeobservations, n ,isconstructedas log L ( n p ; n q j y ; n )= n X i =1 log[ $ [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] j i f [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] ( y i ) + $ [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] j i f [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ( y i ) + $ [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] j i f [( k 1 k 2 )( l 0 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ( y i ) + $ [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 1 l 0 2 )] j i f [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] ( y i )] ; (5.4) wherethemixtureproportions, $ [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] j i (5.5) = p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; $ [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] j i (5.6) = p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; $ [( k 1 k 0 2 )( l 1 l 2 )][( k 0 1 k 2 )( l 0 1 l 0 2 )] j i (5.7) = p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; $ [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] j i (5.8) = p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 p Rk 1 k 2 p Sl 1 l 2 p Rk 0 1 k 0 2 p Sl 0 1 l 0 2 + p Rk 1 k 2 p Sl 1 l 0 2 p Rk 0 1 k 0 2 p Sl 0 1 l 2 + p Rk 1 k 0 2 p Sl 1 l 2 p Rk 0 1 k 2 p Sl 0 1 l 0 2 + p Rk 1 k 0 2 p Sl 1 l 0 2 p Rk 0 1 k 2 p Sl 0 1 l 2 ; representtherelativefrequenciesofthecorrespondingacro ss-blockdiplotypesthat formthesamegenotype,and f [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] ( y i ) ;f [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ( y i ), f [( k 1 k 2 )( l 0 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ( y i )and f [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] ( y i )aretheprobabilitydensity functionsforsubject i whohasthecorrespondingdiplotype,with C -dimensional

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83 genotypicvaluevectors u [( k 1 k 2 )( l 1 l 2 )][( k 0 1 k 0 2 )( l 0 1 l 0 2 )] ; u [( k 1 k 2 )( l 1 l 0 2 )][( k 0 1 k 0 2 )( l 0 1 l 2 )] ; u [( k 1 k 2 )( l 0 1 l 2 )][( k 0 1 k 0 2 ) ( l 0 1 l 2 )] ; and u [( k 1 k 0 2 )( l 1 l 0 2 )][( k 0 1 k 2 )( l 0 1 l 2 )] ,respectively,and C C commonresidualcovariance matrix .Notethatthemixtureproportionsareexpressedasbeingsubjec t-specic becausedierentsubjectseachwithaknowngenotypemayhaved ierentdiplotype compositions. Assumethathaplotypes11(denotedby A )and12(denoted B )arethe referencehaplotypesatQTN R and S ,respectively.Asshownabove,thisleads toninedierentacross-QTNcompositegenotypes.equation(5.1 )providesthe structureofanarbitraryacross-QTNcompositegenotypeformed bythereference andnon-referencehaplotypes.Table5-1tabulatesthegenot ypicvaluevectors fordierentcompositegenotypes.Foragivencompositegenoty pe j R j S ,Ihavea multivariatenormaldistributionexpressedas f j R j S ( y i ; n q )= 1 (2 ) C= 2 j j 1 = 2 exp 1 2 ( y i u j R j S ) 1 ( y i u j R j S ) 0 ; (5.9) with j R ;j S =2 ; 1 ; 0.Ataparticularconcentration c ,therelationshipbetweenthe observationandexpectedmeancanbedescribedbyaregressionmo del(Zhaoetal. 2005), y i ( c )= 2 X j R =0 2 X j S =0 ij R j S u j R j S ( c )+ e i ( c ) ; (5.10) = 2 X j R =0 2 X j S =0 ij R j S u j R j S ( c ) + r X c 0 =1 2 X j R =0 2 X j S =0 ( c;c c 0 )[ y i ( c c 0 ) ij R j S j R j S ( c c 0 )]+ i ( c )(5.11) where ij R j S istheindicatorvariabledenotedas1ifacompositegenotype j R j S isconsideredforindividual i and0otherwise; e i ( c )istheresidualerror(i.e., theaccumulativeeectofpolygenesanderrors)thatcontain stwocomponents, r th-orderantedependentvariance(Gabriel1962;Nu~nez-An tonandZimmerman

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84 2000), P r t 0 =1 P 2j R =1 P 2j S =1 ( c;c c 0 )[ y i ( c c 0 ) ij R j S j R j S ( c c 0 )],where r = min( r;c 1)and ( c;c c 0 )'sareunrestrictedantedependenceparameters,and independentnormalrandomvariable, i ( c ),withmeanzeroandtime-dependent variance, 2 ( c ),termed innovationvariance 5.2.4ModellingtheMean-covarianceStructures Traditionalgeneticmappingformultipletraitsattemptst oestimateeach elementinthemeanvectorandthecovariancematrix.Thisma ynotbeecient andeectivefortworeasons.First,whendosagelevel C ishigh,anexponentially increasingnumberofparametersneedtobeestimated.Second, thisapproachdoes notconsiderbiologicalprinciplesunderlyingpharmacodyn amicresponses.The mainstayofmodellingdrugresponseistheHill,orsigmoidEmax,e quation,which postulatesthefollowingrelationshipbetweendrugconcentr ation(C)anddrugeect (E)(Giraldo2003) E=E 0 + E max C H EC H50 +C H ; (5.12) whereE 0 istheconstantorbaselinevalueforthedrugresponseparameter ,E max is theasymptotic(limiting)eect,EC 50 isthedrugconcentrationthatresultsin50% ofthemaximaleect,andHistheslopeparameterthatdetermi nestheslopeofthe concentration-responsecurve.ThelargerH,thesteeperthelin earphaseofthethe log-concentration-eectcurve.Whentheeectisacontinu ousvariable,estimates ofE max ,EC 50 andHareusuallyobtainedbyextendedleastsquaresoriterativ ely reweightedleastsquareswhenthereissucientdataforanalysi sofindividual subjects.Whensparsedataarepooledfrommultiplepatients,the napopulation analysisisabetterapproach.Dierentfromsuchatraditiona ltreatment,Iwill estimatethesecurveparametersseparatelyfordierentcompo sitegenotypes. Thecompositegenotype-specicmeanvectorsinequation(5.9 )willbe modelledbytheE max model,expressedas

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85 u j R j S = f u j R j S (1) ;:::;u j R j S ( C ) g = (" E 0 j R j S + E max j R j S C H j R j S 1 EC H j R j S 50 j R j S +C H j R j S 1 # ;:::; E 0 j R j S + E max j R j S C H j R j S C EC H j R j S 50 j R j S +C H j R j S C #) ; (5.13) where j R j S =(E 0 j R j S ; E max j R j S ; EC 50 j R j S ; H j R j S )isthemathematicalparameters thatdescribethedrugresponseproleforcompositegenotype j R j S .Thus,based onequation(5.13),theestimateswillbeconcentratedon j R j S ratherthanon u j R j S .Thismodellingofthemeanvectorshastwoadvantages:(1)cl inically meaningfulcurvesareusedingeneticmappingsothattheresult swillbecloserto biologicalrealm,and(2)thenumberofparameterstobeestim atedisreduced,thus increasingthepowerofthemodeltodetectsignicantQTNandt heirinteractions. Itisnotparsimonioustoestimatealltheelementsinthewithi n-subject covariancematrixamongdierentconcentrationlevelsbec ausesomestructure existsfortime-dependentvariancesandcorrelations.Thestr uctureoftheresidual covariancematrixin(5.10)canbemodelledbytherst-order autoregressive [AR(1)]model(Diggleetal.,2002),expressedas 2 (1)= = 2 ( C )= 2 forthevariance,and ( c 1 ;c 2 )= 2 j c 2 c 1 j forthecovariancebetweenanytwoconcentrationlevels c 1 and c 2 ,where0 << 1 istheproportionparameterwithwhichthecorrelationdeca yswithtimelag.The parametersthatmodelthestructureofthe(co)variancematr ixarearrayedin n v Toremovetheheteroscedasticproblemoftheresidualvariance ,whichviolates abasicassumptionofthesimpleAR(1)model,twoapproachescanbe used.The

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86 rstapproachistomodeltheresidualvariancebyaparametric functionoftime, asoriginallyproposedbyPletcherandGeyer(1999).Butthis approachneedsto implementadditionalparametersforcharacterizingtheag e-dependentchangeofthe variance.ThesecondapproachistoembedCarrollandRupert' s(1984)transformboth-sides(TBS)modelintothegrowth-incorporatednitem ixturemodel(Wu etal.2004b),whichdoesnotneedanymoreparameters.Bothem piricalanalyses withrealexamplesandcomputersimulationssuggestthattheTB S-basedmodel canincreasetheprecisionofparameterestimationandcomputa tionaleciency. Furthermore,theTBSmodelpreservesoriginalbiologicalme ansofthecurve parametersalthoughstatisticalanalysesarebasedontransform eddata. TheTBS-basedmodeldisplaysthepotentialtorelaxtheassumpti onofvariancestationarity,butthecovariancestationarityissueremai nsunsolved.Zhaoet al.(2005)usedZimmermanandNu~nez-Anton's(1997)structu redantedependence (SAD)modeltoapproachtheage-specicchangeofcorrelation intheanalysisof longitudinaltraitsbasedon(5.5).Usingmatrixnotation,the errortermin(5.4) canbeexpressedas e = A (5.14) where e =[ e (1) ; ;e ( C )] 0 =[ (1) ; ; ( C )] 0 andfortheSAD(1)model A = 0BBBBBBBBBB@ 1000 1 100 ... . C 1 1 C 2 1 1 1 1CCCCCCCCCCA (5.15) Thevariance-covariancematrixofthelongitudinaltraiti sthenexpressedas = A A 0 ; (5.16)

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87 where istheinnovationvariance-covariancematrixandisexpresse das: = 0BBBBBBB@ 2 (1)00 0 0 2 (2)0 0 ... ... ... . ... 000 2 ( C ) 1CCCCCCCA : Theclosedformsfortheinverseanddeterminantofmatrix helptoestimatethe parameters, n v =( 1 ; ; r ; 2 ),thatmodelthematrix.Thevector j R j S and n v formthequantitativegeneticparameters n q AsasimpliedexamplewiththeSAD(1)model,undertheassumption that innovationvarianceisconstantacrossdierenttimepoints,J arezicetal.(2003) derivedtheanalyticalformsforvarianceandcovariancefu nctionsamongtimedependentmeasurements,expressed,respectively,as 2 ( c )= 1 2 c 1 2 2 ; (5.17) ( c 1 ;c 2 )= c 2 c 1 1 2 c 1 1 2 2 ;c 2 c 1 ; (5.18) forequallyspacedrepeatedmeasurements.Itcanbeseenthatalt houghconstant innovationvariancesareassumed,theresidualvariancecanch angewithdosage level(Jarezicetal.2003).Also,forthesimplestSADmodel,th ecorrelation functionisnon-stationarybecausethecorrelationdoesnotd ependonlyonthedose interval c 2 c 1 butalsodependsonthestartandendpointsoftheinterval, c 1 and c 2 5.2.5AnIntegrativeEM-simplexAlgorithm Iderivedaclosed-formsolutionforestimatingthepopulation geneticparameters n p withtheEMalgorithm(Linetal.2005).TheNelder-Meadsimple x algorithm,originallyproposedbyNelderandMead(1965),can beusedtoestimate thequantitativegeneticparameters n q =( j R j S ; n v ).Itisadirectsearchmethod

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88 fornonlinearunconstrainedoptimization.Itattemptstomi nimizeascalar-valued nonlinearfunctionusingonlyfunctionvalues,withoutanyde rivativeinformation (explicitorimplicit).Thealgorithmuseslinearadjustment oftheparametersuntil someconvergencecriterionismet.Simulationstudieshavepr oventhatsimplex algorithmconvergestothesamesolutionmorerapidlythanthe EMalgorithmin theclassicalfunctionalmapping. Haplotypefrequenciescanbeexpressedasafunctionofallelic frequenciesand linkagedisequilibria(LD).Foratwo-SNPhaplotypeforQTN R ,Ihave p Rk 1 k 2 = p R (1) k 1 p R (2) k 2 +( 1) k 1 + k 2 D R ; (5.19) where D R isthelinkagedisequilibriumbetweenthetwoSNPsatQTN R .Thus, oncehaplotypefrequenciesareestimated,Icanestimatealle licfrequenciesandLD bysolvingequation(5.19).Similarcalculationsarealsodon eforblock S .After across-QTNcompositegenotypicvaluesareestimated,Icanestim atetheadditive, dominantandepistaticeectsbetweentwoQTNusingequation( 5.2). 5.3HypothesisTests Withmyepistaticmodel,Icanmakeanumberofhypothesistestsr egarding thegeneticcontrolofoveralldrugresponsetoaspectrumofdosa gesandother clinicallyimportantevents.Twomajorhypothesescanbemade inthefollowing sequence:(1)theassociationbetweendierentSNPswithineach QTNbytesting theirlinkagedisequilibria(LD),and(2)thesignicanceofa nassumedacross-QTN referencehaplotypeforitseectondrugresponse.TheLDbetw eentwogiven SNPswithinQTN R canbetestedusingthefollowinghypotheses: 8><>: H 0 : D R =0 H 1 : D R 6 =0 (5.20) Thelog-likelihoodratioteststatisticforthesignicanceofL Discalculatedby

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89 comparingthelikelihoodvaluesunderthe H 1 (fullmodel)and H 0 (reducedmodel) using LR R = 2[log L ( e p Rk 1 ; e p Rk 2 ;D R =0 ; e n q j n ) log L ( b n p ; b n q j n )](5.21) wherethetildeandhatdenotetheMLEsofunknownparameters under H 0 and H 1 respectively.The LR R isconsideredtoasymptoticallyfollowa 2 distributionwith onedegreeoffreedom.TheMLEsofallelicfrequenciesunder H 0 canbeestimated usingtheEMalgorithmdescribedabove,butwiththeconstraint p R11 p R22 = p R12 p R21 .A similartestcanbemadeforblock S Diplotypeorhaplotypeeectsonacomplexdiseasecanbetested usingthe nullandalternativehypothesesexpressedas 8><>: H 0 : j R j S = ( j R ;j S =2 ; 1 ; 0) H 1 :atleastoneequalityin H 0 doesnothold (5.22) Thelog-likelihoodratioteststatistic( LR )underthesetwohypothesescanbe similarlycalculated.The LR mayasymptoticallyfollowa 2 distributionwith 36degreesoffreedom.However,theapproximationofa 2 distributionmaybe inappropriatewhensomeregularityconditions,suchasnormal anduncorrelated residuals,areviolated.Thepermutationtestapproachpropose dbyChurchilland Doerge(1994),whichdoesnotrelyuponthedistributionofth e LR ,maybeusedto determinethecriticalthresholdfordeterminingtheexisten ceofaQTL. Dierentgeneticeects,suchastheadditive( a R and a S ),dominant( b R and b S )andadditive additive( i ),additive dominant( j ),dominant additive( k ) anddominant dominanteects( l )betweenblocks R and S canalsobetested individually.Thecriticalthresholdsfortheseindividuale ectscanbedetermined onthebasisofsimulationstudies. Thismodelallowsforthetestsofdierentoutcomesfordrugr esponse.For overalldrugresponsecurves,hypothesistestsbasedonmathemati calparameters

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90 canbeinformativeforthecharacterizationofgeneticcont rol.Onealternativetest canbeperformedontheareaundercurve(AUC).TheAUCcanbecal culatedby takingintegralforeachcompositegenotype,expressedas AUC j R j S = Z C C C 1 E 0 j R j S + E max j R j S c H j R j S EC H j R j S 50 j R j S + c H j R j S # dc: ThenullhypothesisbasedontheAUCcanbeformulatedasAUC j R j S =AUC. Thepermutationtestscanbeusedtodeterminethecriticalval ueforAUC-related hypothesistests. 5.4AWorkedExample Toshowhowthemodelworksinpractice,Iuseittoanalyzeareal example fromapharmacogeneticalstudyfordrugresponse.Numerousgene shavebeen investigatedaspotentialobesity-susceptibilitygenes(Mason etal.1999;Chagnon etal.2003).The 1ARand 2ARgenesaretwosuchexamples(Greenetal.1995; Largeetal.1997)ineachofwhichthereareseveralpolymorph ismscommoninthe population.Twocommonpolymorphismsareidentiedatcodon s49(Ser49Gly) and389(Arg389Gly)forthe 1ARgeneandatcodons16(Arg16Gly)and27 (Gln27Glu)forthe 2ARgene,respectively.Thepolymorphismsineachofthese tworeceptorgenesareinlinkagedisequilibrium,whichsugge ststheimportanceof takingintoaccounthaplotypes,ratherthanasinglepolymorp hism,whendening biologicalfunction.Thisstudyattemptstodetecthaplotyp evariantswithinthese candidategeneswhichdeterminedrugresponse. Todeterminewhethersequencevariantsatthetwopolymorphi smsfromone geneinteractwiththosefromtheothergenetoaectdrugrespo nse,agroupof 163menandwomenwereinvestigatedwithagesfrom32to86year soldwitha largevariationinheartratesinresponsetodierentdosagele velsofdobutamine whichisamedicationdesignedtopreventcongestiveheartfai lure.Dobutamine wasinjectedintothesesubjectstoinvestigatetheirresponsein heartratetothis

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91 drug.Thesubjectsreceivedincreasingdosesofdobutamine,un tiltheyachieved targetheartrateresponseorpredeterminedmaximumdose.Thed oselevelsused were0(baseline),5,10,20,30and40mcg|min,ateachofwhich heartratewas measured.Thetimeintervalof3minutesisallowedbetweentw osuccessivedoses forsubjectstoreachapateauinresponsetothatdose.Onlythose( 98)inwhom therewereheartratedataatallthesixconcentrationlevels wereincludedfordata analyses. Eachofthesepatientswasdeterminedfortheirgenotypesatc odon49with twoalleleSer49(A)andGly49(G)andcodon389withtwoallele sArg389(C) andGly389(G)withinthe 1ARgeneonchromosome10,aswellasatcodon16 withtwoallelesArg16(A)andGly16(G)andcodon27withtwoall elesGln27 (C)andGlu27(G),withinthe 2ARgeneonchromosome5,andmeasuredfor theresponseinheartratetodobutamine.TwoSNPsfromeachgene theoretically form81across-genegenotypes,but,becausethesetwogenesarein dependent,the frequenciesofthesegenotypescanbeexpressedastheproducto fthegenotype frequenciesfromeachgene. Byassumingthatonehaplotypeisdierentfromtherestofhaplo typesat eachgene,thismodelcandetectthereferencehaplotypesth atdisplaysignicant mainandinteractioneectsondrugresponse.UsinghaplotypesA C,AG,GC andGGasareferencehaplotypeatthe 1ARgene,respectively,inconjunction withareferencehaplotypeselectedfromAC,AG,GCandGGatth e 2ARgene, Icalculatethecorrespondinglog-likelihood-ratio(LR)te ststatistics(Table5-2) basedonthehypothesestestformulatedbyequation(5.22).Bec auseofunobserved genotypesatacandidategene,somecompositegenotypesdonot existforseveral reference-haplotypecombinations.ThemaximalLRvalue(62 .09)withallthe 9compositegenotypeswasobtainedwhenGCasthereferenceha plotypeofthe 1ARgeneiscombinedwithGGasthereferencehaplotypeofthe 2ARgene.

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92 Table5{2:Likelihoodratiosfor16possiblecombinationsofa ssumedreferencehaplotypeswithonefromcandidategene 1ARandthesecondfromcandidategene 2AR 1AR 2ARACAGGCGG AC37.1335.9232.9116.23 AG24.7317.5021.279.08GC18.2516.3422.214.32 GG42.7140.08 62.09 ( P =0.04)29.01 Themaximumlikelihoodratiovalueisdetectedwhen[GG]at 2ARand[GC]at 1ARareusedasthereferencehaplotypes. TheLRvaluesfortheothercombinationsrangefrom4.32to42 .71.Theoptimal reference-haplotypecombinationfordrugresponseisstatisti callytestedwith1000 permutationtestswhichobtainedthecriticalthresholdvalu eof60.51atthe5% signicancelevel.Thistestsuggeststhatthereexistsignicant haplotypeeectsat thesetwocandidategenesonheartratecurves. TheintegrativeEM-simplexalgorithmbasedonthelikelihood functionallows fortheestimatesoffourhaplotypefrequencieswithineachg eneandtheresulting allelefrequenciesandlinkagedisequilibriumateachgene. Table5-3showsthe maximumlikelihoodestimatesofallelefrequenciesandlink agedisequilibriafor SNPswithin 1ARand 2AR,aswellasdrugresponsecurveparametersand (co)variance-structuringparametersfornineacross-geneco mpositegenotypesunder theoptimalreference-haplotypecombinationmodel.Highly signicantLDwas detectedbetweentwoSNPsforeachgene( P< 0 : 001). StatisticaltestsforindividualgeneticeectsforDNAsequenc evariants basedonequation(5.2)suggestthattheadditiveanddominante ectsateach ofthecandidategenesarehighlysignicant.Fourkindsofep istasisbetweenthe twogenesareallsignicantatthe5%signicancelevel(Table 5-4).Figure5-1 displaystheproleofheartratetoincreasingconcentration levelsofdobutamine

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93 Table5{3:Maximumlikelihoodestimatesofpopulationgenet icparameters(allele frequenciesandlinkagedisequilibria)forSNPswithintwoin dependentcandidate genes 1ARand 2ARaswellasquantitativegeneticparameters(drugresponse and(co)variance-structuringparameters)fornineacross-gen ecompositegenotypes inasampleof107subjects Populationgeneticparameters p 2 (1) 1 p 2 (2) 1 D 2 p 1 (1) 1 p 1 (2) 1 D 1 0.620.390.130.150.730.04 Across-geneQuantitativegeneticparameters compositegenotypeE 0 E max EC 50 H 2 [GG][GG] = [GC][GC]0.120.3013.682.85 [GG][GG] = [GC][ GC]0.161.0042.693.44 [GG][GG] = [ GC][ GC]0.110.5330.701.89 [GG][ GG] = [GC][GC]0.080.4123.363.43 [GG][ GG] = [GC][ GC]0.100.4136.501.85 [GG][ GG] = [ GC][ GC]0.110.3425.142.13 [ GG][ GG] = [GC][GC]0.080.2719.262.63 [ GG][ GG] = [GC][ GC]0.100.4225.451.94 [ GG][ GG] = [ GC][ GC]0.100.3823.262.15 7e-30.88 Note:Thereferencehaplotypesatthe 2ARand 1ARgenesare[GG]and[GC], respectively.fornineacross-genecompositegenotypesonthebasisoftheestim atedresponse parametersinTable5-3.Considerableover-crossingamongdi erentcurvessuggests sequence-sequenceinteractionswithin 1ARand 2AR. 5.5MonteCarloSimulation Thestatisticalpropertiesofthemodelareexaminedthroughsi mulation studies.Considerarandomsamplecomposedof200individualsfro mahuman populationatHardy-Weinbergequilibriumwithrespecttohap lotypes.Two independentQTN,eachwithtwobiallelicSNPs,areassumedtobeseg regatingin thepopulation.Inordertoinvestigatethereliabilityofth eestimatesinthereal application,allelefrequenciesandlinkagedisequilibria betweentheSNPs,drug responsecurveparameters,matrix-structuringstructureparam etersaregivenin

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94 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 90 100 [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] [GG][GG]/[GC][GC] Dobutamine concentration (mcg)Heart rate (%) Figure5{1:Prolesofheartrateinresponsetodierentdosage sofdobutaminefor ninecompositegenotypes(foreground)identiedforSNPswit hintwogenes.The prolesof98studiedsubjectsfromwhichtheninedierenttwo -genecomposite genotypesweredetectedarealsoshown(background).Table5-5bymimickingtheexampleusedabove.Thediplotypes derivedfrom across-QTNhaplotypesaectdrugresponsemeasuredforeachsubje ctat6dierent concentrationlevels.Byonehaplotypeasthereferencehapl otypeforeachQTN,I willhaveatotalofninecompositegenotypes. ThepopulationgeneticparametersoftheSNPscanbeestimated withreasonablyhighprecisionusingtheclosed-formsolutionapproach( Table5-5).The parametersforthepharmacodynamicmodelsofeachcomposite genotypecanbe estimatedaccuratelyandprecisely(Table5-5).Asexpected,t hecurveparameters werepreciselyestimated(assessedbyMSE),suggestingthatthemode lcanprovide

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95 Table5{4:Maximumlikelihoodestimates(MLEs)oftotalgenet ic,additive,dominantandinteractioneectsforAUCandtheirsignicancetests undertheoptimal haplotypemodel[(GG)(GC)] GeneticeectsMLELR( P value) a 2 1.0718.83( < 0.05) b 2 -0.1325.50( < 0.05) a 1 -1.0127.92( < 0.05) b 1 0.5530.05( < 0.05) i 0.7318.81( < 0.05) j -0.4228.24( < 0.05) k 0.2324.34( < 0.05) l -1.7636.59( < 0.05) reasonableestimatesofdrugresponsecurves.Theestimatesofthe AR(1)parametersthatmodelthestructureofthe(co)variancematrix alsodisplayreasonably highprecision. Ineachof100simulations,Icalculatedthelog-likelihoodra tiosforthe hypothesistestofthepresenceofageneticvariantaectingbo thdrugresponses. Thissuggeststhatthemodelhasenoughpowertodetectthegene ticvariantunder givenSNPs,curveandmatrix-structuringparametersforthisp articularsimulation. 5.6Discussion Giventherecentupsurgeininterestinpharmacogeneticsandp harmacogenomics,thereisapressingneedforthedevelopmentofstatistica lmodelsfor unravelingthegeneticetiologyofpharmacologicalvariat ion.QTLmapping,by superimposingrealbiologicalphenotypesongenomesequence, structuralpolymorphisms,andgeneexpressiondata,canprovideanunbiasedviewoft henetworkof geneactionsandinteractionsthatbuildacomplexphenotyp elikedrugresponse. Throughanintegratedapproach,studiescanmovetocharacte rizethebestdrug, thebestdosageofthedrugandthebestinjectiontimeforindivi dualpatientsbased ontheirgeneticstructure.Byincorporatinggenetictestsin totheprescription process,physiciansmightimproveoutcomesforpatients.

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96 Table5{5:Maximumlikelihoodestimates(MLEs)ofSNPallelefr equencyandlinkagedisequilibriumandparametersdescribingtheninedynami ccurvesbasedonthe sigmoidalEmaxmodel.TheMSEsarebasedon100simulationrepli cates. Populationgeneticparameters p R (1) 1 p R (2) 1 D R p S (1) 1 p S (2) 1 D S Given0.620.390.130.150.730.04MLE0.620.390.130.160.720.04p MSE0.020.020.010.020.020.01 Across-QTNcompositeQuantitativegeneticparameters genotypesE 0 E max EC 50 H 2 CurveparametersAABB Given0.120.313.682.85 MLE0.110.3013.902.84p MSE0.040.142.341.12 AAB B Given0.161.0042.693.44 MLE0.160.9941.993.42p MSE0.031.519.211.23 AA B B Given0.110.5330.71.89 MLE0.110.5229.051.92p MSE0.020.085.270.37 A ABB Given0.080.4123.363.43 MLE0.090.4224.273.52p MSE0.040.134.031.30 A AB B Given0.10.4136.51.85 MLE0.100.4136.451.85p MSE0.010.228.800.62 A A B B Given0.10.4136.51.85 MLE0.100.4136.451.85p MSE0.010.228.800.62 A ABB Given0.080.2719.262.63 MLE0.080.2718.962.62p MSE0.040.133.670.45 A AB B Given0.10.4225.38451.94 MLE0.100.4124.951.97p MSE0.020.073.700.38 A A B B Given0.10.3823.262.15 MLE0.100.3823.462.17p MSE0.010.043.130.25 (Co)variancematrixstructureparameters Given 7e-30.88 MLE 7e-30.88 p MSE 5e-40.02

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97 Genetically,drugresponseisacomplextrait,involvingmult iplebiochemical pathwayseachcontrolledbydierentinteractinggenes(Ma rchinietal.2005). TraditionalQTLmappingcanprobetheimportantchromosomal segmentsfordrug response,butisdiculttoidentifythecausalDNAsequencesofth esesegments. Thecompletionofthehumangenomeprojectmakesitpossibleto genome-wide associateDNAsequencevariantswithcomplexphenotypes.Themod elproposed inthischapterhascapacitytocharacterizetheeectsofge neticfactorsandtheir epistaticeectsondrugresponseattheDNAsequencelevel. Thismodelhasbeenexaminedthroughsimulationstudies.Itcan detect signicantgeneticvariantsthatcontroldrugresponseandpro videreasonably preciseestimatesofthegeneticparameters.Itsapplicationt oarealexample indicatesthatthismodelwillbepracticallyuseful. Thereareseveralwaysinwhichthismodelmaybeextended.For simplicity, mypresentationisbasedontheinteractionbetweentwoSNPseque ncesfromeach QTN.Itislikelythatthetwo-SNPmodelistoosimpletocharacte rizegenetic variantsforquantitativevariation.Withthefoundationf orthetwo-SNPsequencing model,thismodelcanbeextendedtoincludeanarbitrarynum berofSNPswhose sequencesareassociatedwiththephenotypicvariation.Asshown inLinetal. (2005a),themulti-SNPsequencingmodelwillencounterthepr oblemofmany haplotypesandhaplotypepairs.AnAIC-orBIC-basedmodelselect ionstrategy (BurnhamandAndersson1998)hasbeenframedtodeterminetheha plotypethatis mostdistinctfromtherestofhaplotypesinexplainingquantit ativevariation.

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CHAPTER6 MODELLINGTHEGENETICETIOLOGYOFPHARMACOKINTICPHARMACODYNAMICLINKSWITHTHEARMAPROCESS 6.1Introduction Thegeneticstudyofpharmacokinetics(PK)andpharmacodyna mics(PD) hasreceivedaconsiderableinterestbecauseofitsfundamenta limportancein designingpersonalizedmedicationsbasedonpatients'genetic architecture(Evans andMcLeod2003;Weinshilboum2003).However,eachofthesetwo processesis complexintermsofthenetworkoftheunderlyinggenesandth eirinteractions withvariousenvironmentalanddevelopmentalfactors.Fort hisreason,statistical modellingoftheobserveddataforthesetwoprocesseshasbeenth oughttobe crucialforthecharacterizationofdetailedgeneticinfor mationaboutcomplextraits (LynchandWalsh1998). Unlikegeneralcomplextraits,PKandPDareofdynamicnature( Hochhaus andDerendorf1995)whichfurthercomplicatestheiranalysi s.Todetectthegenetic factorsforPKandPD,moreadvancedstatisticalmodelsarenee ded.Although theelucidationoftherelationshipbetweengeneticcontrol anddrugresponseis statisticallyapressingchallenge,someofkeydicultieshaveb eenovercomebyR. Wuandcolleagues(Maetal.2002;Wuetal.2002b,2003,2004a ,2004b).They haveproposedageneralstatisticalframework,i.e., functionalmapping ,togenomewidemapspecicQTLthatdeterminethedevelopmentalpatter nofacomplex trait. Thebasicrationaleoffunctionalmappingliesintheconnect ionbetweengene actionorenvironmentaleectsanddrugresponsebyparametri cornonparametricmodels.FunctionalmappingmapsdynamicQTLthatarerespo nsiblefora 98

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99 biologicalorclinicalprocessthatismeasuredatanitenumb eroftimepoints orconcentrations.Anumberofmathematicalmodelshavebeen establishedto describethePKandPDprocessofdrugresponse. Theintensityofabiologicalresponseproducedbyadrugisrela tedtothe concentrationofthedrugatthesiteofactionwhichisinturn aectedbyavariety offactorsincludinggenesandenvironments.Therearefourp harmacokineticphases whichallaectdrugconcentrationatthesiteofaction.These phasesare:(1) absorption{theabilityofadrugtoenterthebloodstream,(2) distribution,which isaectedbythephysiochemicalpropertiesofthedrug,card iacoutputandblood row,thebloodbrainbarrieranddrugreservoirs,(3)biotransf ormationreactions thatalterthechemicalstructureofadrug,and(4)excretion istheremovalof drugsandbiotransformationproductsfromthebody.Anumber ofmathematical functionshavebeenproposedtodescribethePKprocessbyasetof parameters thatarerelatedtophsyiological,clinical,physicalorchem icalpropertiesofdrug (HochhausandDerendorf1995). Tomorepreciselydeterminedrugresponse,PKisintegratedwit hPDto quantifytherelationshipbetweenpharmacodynamicactions andthedoselevelof drugaswellasthetimeafterthedrugisadministrated(Hochha usandDerendorf 1995).Theintegrationofthesetwoprocesseswillbelikelytog iverisetothe therapeuticallyrelevantresultsrelatedtodrugresponse.In thischapter,Iproposed astatisticalmodelfordetectingandidentifyingthegenetic variantsthatcontrol theintegrativePKandPDprocesses.Thismodelcapitalizesont heinformationof DNAsequencegeneratedbysinglenucleotidepolymorphisms(SNPs) andembeds mathematicalfunctionsofPKandPDprocessesintoamixturemo delframework. Thismodeldisplaysuniqueadvantagesintwoareas.First,them odelcandetect specicDNAsequencevariantsratherthantraditionallydene dquantitative traitloci(QTL).Theutilityofthismodelcanbemoreeecti veinthediscovery

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100 ofdrugs.Second,thismodelisstatisticallyvigorousinthati toersecient estimationofparametersthatmodelthemeanvectorsandcova riancematrix. Inthischapter,Idevelopacovariance-structuringmodelba sedonautoregressivemovingaverage(ARMA)modelsandmakeuseoftheclosed-fo rmofthe covarianceoftheARMAerrorsanditsinverseanddeterminant( vanderLeeuw 1994;Haddad2004),aimedtoincreasetheeciencyofthemodel .Inparticular, theARMA-basedmappingmodelprovidesageneralplatformforvi gorousmodellingofthecovariancematrix.Iperformsimulationstudies toinvestigatethe statisticalbehaviorofthemodel. 6.2HaplotypingaComplexTrait InLinetal.(2005),anewhaplotype-basedmappingapproachh asbeen derivedtodetectDNAsequencevariantsforabiologicalproce ss,suchasdrug response.Thisnewapproachreliesuponthethecharacterizat ionofSNPsfromthe entirehumangenome.Accordingtoresultsfromrecentstudies,t hehumangenome canbepartitionedintodiscretehaplotypeblocks(Patileta l.2001;Dawsonetal. 2002;Gabrieletal.2002),ineachofwhichalimitednumbero fSNPs,referredto as\tagSNPs",canexplainalargefractionofthehaplotypediv ersity. Toclearlydescribethismodel,letusrstconsidertwoSNPswith inahaplotypeblockthatareassociatedwiththelinkagedisequilibri umof D inahuman populationatHardy-Weinbergequilibrium.Therearetwoall eles1and2withthe relativeproportionsof p (1)1 and p (1)2 fortherstSNPaswellas p (2)1 and p (2)2 forthe secondSNP,with p (1)1 + p (1)2 =1and p (2)1 + p (2)2 =1.ThesetwoSNPsform4possible haplotypes11,12,21and22whosefrequenciesareexpressedas p r 1 r 2 = p (1)r 1 p (2)r 2 +( 1) r 1 + r 2 D; where r 1 ;r 2 =1 ; 2denotetheallelesofthetwoSNPs,respectively, P 2r 1 =1 P 2r 2 =1 p r 1 r 2 =1(LynchandWalsh1998).Ifthehaplotypefrequenciesarekn own,thenthe

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101 allelicfrequenciesandlinkagedisequilibrium,arrayedby thepopulationgenetic parametervector n p = f p (1)r 1 ;p (2)r 2 ;D g ,canbesolvedwiththeaboveequation. Therandomcombinationofmaternalandpaternalhaplotypes generates 10distinctdiplotypesexpressedas[11][11], ,[22][22]whicharesortedinto 9genotypes11/11, ,22/22(Table6-1).Thedoubleheterozygoticgenotype 12/12containstwopossiblediplotypes[11][22]and[12][21] .Iuse P [ r m 1 r m 2 ][ r p 1 r p 2 ] (= p r m 1 r m 2 p r p 1 r p 2 )and P r 1 r 0 1 =r 2 r 0 2 todenotethediplotypeandgenotypefrequencies, respectively,and n r 1 r 0 1 =r 2 r 0 2 todenotetheobservationsofvariousgenotypes( G ), where m and p describethematernalandpaternaloriginsofhaplotypesand 1 r 1 r 0 1 2,1 r 2 r 0 2 2.Thefrequenciesandobservationsofallgenotypes, exceptforgenotype12/12,areequivalenttothoseofthecorr espondingdiplotypes. Iintendtoassociatediplotypeswithinter-patientvariatio ninaquantitative traitbasedonobservedphenotypicvalues( Y )assumedtobenormallydistributed. Withoutlossofgenerality,Iassumethathaplotype11isdiere ntfromtheother haplotypes,cumulativelyexpressedas 11,intriggeringaneectonthephenotype. Suchadistincthaplotype11iscalledthe referencehaplotype .Thereference andnon-referencehaplotypesgeneratethreecombinations calledthe composite genotypes .Thegenotypicmeansofthecompositegenotypes, j ( j =2for[11][11], 1for[11][ 11]and0for[ 11][ 11]),andcommonresidualvariancewithinthecomposite genotypes, 2 ,belongtoquantitativegeneticparametersandareidenti edin n q = f j ; 2 g Hence,twolog-likelihoodfunctionscanbeconstructed,onei namultinomialformandanotherinamixturemodelform,toestimatethep opulationand quantitativegeneticparametersthatare,respectively,ex pressedas log L ( n p jG )=Constant(6.1) +2 n 11 = 11 log p 11 + n 11 = 12 log(2 p 11 p 12 )+2 n 11 = 22 log p 12

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102 + n 12 = 11 log(2 p 11 p 21 )+[ $n 12 = 12 log(2 p 11 p 22 )+(1 $ ) n 12 = 12 log(2 p 12 p 21 )] + n 12 = 22 log(2 p 12 p 22 )+2 n 22 = 11 log p 21 + n 22 = 12 log(2 p 21 p 22 )+2 n 22 = 22 log p 22 ; and log L ( n p ; n q jY ; G )= n 11 = 11 X i =1 log f 2 ( y i ) + n 11 = 12 + n 12 = 11 X i =1 log f 1 ( y i ) + n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 X i =1 log f 0 ( y i ) + n 12 = 12 X i =1 log[ $f 1 ( y i )+(1 $ ) f 0 ( y i )] ; (6.2) $ = p 11 p 22 p 11 p 22 + p 12 p 21 ; (6.3) and f j ( y i )= 1 p 2 exp ( y i j ) 2 2 2 : Iderivedaclosed-formsolutionforestimatingtheunknownpar ameterswith theEMalgorithm(Linetal.2005).Theestimatesofhaplotype frequenciesare basedonthelog-likelihoodfunctionofequation(6.1),wher eastheestimatesof compositegenotypicmeansandresidualvariancearebasedonth elog-likelihood functionofequation(6.2).Thesetwotypesofparameterscan beestimatedusing anintegrativeEMalgorithm. IntheEstep,theexpectednumber( $ )ofdiplotype[11][22]foradouble heterozygousgenotypeisestimatedusingequation(6.3),whe reastheposterior probability( i )withwhichsubject i carryingthedoubleheterozygousgenotypeis diplotype[11][22]iscalculatedby i = $f 1 ( y i ) $f 1 ( y i )+(1 $ ) f 0 ( y i ) : (6.4)

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103Table6{1:Possiblediplotypecongurationsofninegenotype sattwoSNPswhichaectpharmacokinetics(PK)andpharmacodynamics(PD) Parametersfor RelativediplotypeHaplotypecompositiongenotypicmeanv ector GenotypeDiplotypeDiplotypefrequencyfreq.withingenot ypes11122122ObservationPKPD 11 = 11[11][11] P [11][11] = p 211 11000 n 11 = 11 k e 0 (E max 2 ; EC 50 2 ) 11 = 12[11][12] P [11][12] =2 p 11 p 12 1 1 2 1 2 00 n 11 = 12 k e 1 (E max 1 ; EC 50 1 ) 11 = 22[12][12] P [12][12] = p 212 10100 n 11 = 22 k e 2 (E max 0 ; EC 50 0 ) 12 = 11[11][21] P [11][21] =2 p 11 p 21 1 1 2 0 1 2 0 n 12 = 11 k e 0 (E max 1 ; EC 50 1 ) 12 = 12 ( [11][22][12][21] ( P [11][22] =2 p 11 p 22 P [12][21] =2 p 12 p 21 ( $1 $ 1 2 $ 1 2 (1 $ ) 1 2 (1 $ ) 1 2 $n 12 = 12 ( k e 0 k e 1 ( (E max 1 ; EC 50 1 ) (E max 0 ; EC 50 0 ) 12 = 22[12][22] P [12][22] =2 p 12 p 22 10 1 2 0 1 2 n 12 = 22 k e 1 (E max 0 ; EC 50 0 ) 22 = 11[21][21] P [21][21] = p 221 10010 n 22 = 11 k e 0 (E max 0 ; EC 50 0 ) 22 = 12[21][22] P [21][22] =2 p 21 p 22 100 1 2 1 2 n 22 = 12 k e 0 (E max 0 ; EC 50 0 ) 22 = 22[22][22] P [22][22] = p 222 10001 n 22 = 22 k e 0 (E max 0 ; EC 50 0 ) $ = p 11 p 22 p 11 p 22 + p 12 p 21 where p 11 p 12 p 21 and p 22 arethehaplotypefrequenciesof[11],[12],[21],and[22], respectively.ThePKandPDare assumedtobeaectedbydierentreferencehaplotypes,[12 ]forthePKand[11]forthePD.

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104 IntheMstep,theprobabilitiescalculatedinthepreviousit erationareusedto estimatethehaplotypefrequenciesusing ^ p 11 = 2 n 11 = 11 + n 11 = 12 + n 12 = 11 + $n 12 = 12 2 n ; (6.5) ^ p 12 = 2 n 11 = 22 + n 11 = 12 + n 12 = 22 +(1 $ ) n 12 = 12 2 n ; (6.6) ^ p 21 = 2 n 22 = 11 + n 12 = 11 + n 22 = 12 +(1 $ ) n 12 = 12 2 n ; (6.7) ^ p 22 = 2 n 22 = 22 + n 22 = 12 + n 12 = 22 + $n 12 = 12 2 n ; (6.8) Thequantitativegeneticparametersareestimatedusing ^ 2 = P n 11 = 11 i =1 y i n 11 = 11 ; (6.9) ^ 1 = P ni =1 y i + P n 12 = 12 i =1 i y i n + P n 12 = 12 i =1 i ; (6.10) ^ 0 = P ni =1 y i + P n 12 = 12 i =1 (1 i ) y i n + P n 12 = 12 i =1 (1 i ) ; (6.11) ^ 2 = 1 n n n 11 = 11 X i =1 ( y i ^ 2 ) 2 + n X i =1 ( y i ^ 1 ) 2 + n X i =1 ( y i ^ 0 ) 2 + n 12 = 12 X i =1 i ( y i ^ 1 ) 2 +(1 i )( y i ^ 0 ) 2 o ; (6.12) where_ n = n 11 = 12 + n 12 = 11 and n = n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 .Iterations includingtheEandMstepsarerepeatedamongequations(6.2) {(6.12)untilthe estimatesoftheparametersconvergetostablevalues. 6.3HaplotypingtheIntegratedPK-PDProcess 6.3.1TheLikelihoodFunctions Considerthesamesampledpopulationofsize n asdescribedabovefromwhich SNPshavebeengenotyped.Usingthispopulation,Iwilltestthea ssociationbetweenSNPdiplotypesanddrugresponse.Becausetheeectofadru gvarieswith itsdosagethroughapharmacodynamicprocess,drugeectsshoul dbemeasured atamultitudeofdiscretedoselevels.Experiencingfourpharm acokineticphases,

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105 absorption,distribution,biotransformationreactionsande xcretion,drugconcentrationinthebodywilldecaywithtime.Thus,thedegreeofdr ugconcentration todecaywithtimewill,intheultimate,aectdrugeect.In theotherword,drug eectisdeterminedsimultaneouslybytheoriginaldosageandt hetimeafterthe drugwasadministrated.Let y i ( d;t )betheobserveddrugeectforpatient i ata particulardosage d ( d =1 ;:::; D)andtime t ( t =1 ;:::; T). Todetecthaplotypeeectsondrugresponsetodierentdosaged uringatime course,Ineedtojointlyformulatemultinomial-andGaussianm ixturemodel-based likelihoodsbasedontheSNPandlongitudinaldata.Whilethem ultinomial-based likelihoodhasthesameformasequation(6.1),themixture-b asedlikelihoodneeds toincorporatelongitudinalinformation,whichisexpressed ,foratwo-SNPmodel, as log L ( n p ; n q jY ; G )= n 11 = 11 X i =1 log f 2 ( y i ) + n 11 = 12 + n 12 = 11 X i =1 log f 1 ( y i ) + n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 X i =1 log f 0 ( y i ) + n 12 = 12 X i =1 log[ $f 1 ( y i )+(1 $ ) f 0 ( y i )] ; (6.13) where y i = h y i (1 ; 1) ;:::;y i (1 ; T) | {z } dose1 ;:::;y i (D ; 1) ;:::;y i (D ; T) | {z } doseD i isthevectorforlongitudinaldrugeectsmeasuredunderall dosagesandat alltimesforpatient i .Becausethephenotypicmeasurementsofdrugeectare continuouslyvariable,itisreasonabletoassumethat y i followsamultivariate normaldistribution,asusedingeneralquantitativegenetic studies(Lynchand Walsh1998).Themultivariatenormaldistributionforpatien t i whocarries

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106 compositegenotype j isexpressedas f j ( y i ; u j ; )= 1 (2 ) DT j j 1 = 2 exp 1 2 ( y i u j ) 1 ( y i u j ) 0 ; (6.14) with u j = h u j (1 ; 1) ;:::;u j (1 ; T) | {z } dose1 ;:::;u j (D ; 1) ;:::;u j (D ; T) | {z } doseD i (6.15) beingavectorofexpectedvaluesforcompositegenotype j atdierentdosesand times. Ataparticulardose d andtime t ,therelationshipbetweentheobservationand expectedmeancanbedescribedbyalinearregressionmodel, y i ( d;t )= 2 X j =0 ij u j ( d;t )+ e i ( d;t ) ; (6.16) where ij aretheindicatorvariablesdenotedas1ifaparticularcomp ositegenotype j isconsideredforindividual i and0otherwiseand e i ( d;t )istheresidualerrors thatareiidnormalwithmeanzeroandvariance 2 ( d;t ).Thecovariancematrix inequation(6.14)isfactorizedintodierentblocks,expre ssedas = 0BBBB@ 1 ::: 1D ... . ... D1 ::: D 1CCCCA ; (6.17) where 1 ;:::; D arethe(T T)covariancematricesamongdierenttime pointsunderDdierentdrugdoselevels,respectively,and d 1 d 2 istheacross-dose covariancematrixamongdierenttimepointsbetweendosage d 1 and d 2 Forthelikelihoodfunction(6.13), n p istheunknownvectorforpopulationgeneticparametersincludingSNPhaplotypefrequenciesand n q containsquantitative geneticparametersincludingthemeanvectorsofdrugeect sfordierentcomposite

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107 genotypesatdierentdosagesandtimepointsandthecovaria ncematrixamong dierentdosagesandtimepoints. Althoughitisnotperfect,theassumptionofindependenceamon gDdierent dosagescanfacilitatethemodellingandanalysis.Withthisassu mption,the likelihoodofequation(6.13)canbere-writtenas log L ( n p ; n q jY ; G ) = n 11 = 11 X i =1 log f 2 [ y i (1)]+ ::: +log f 2 [ y i (D)] + n 11 = 12 + n 12 = 11 X i =1 log f 1 [ y i (1)]+ ::: +log f 1 [ y i (D)] + n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 X i =1 log f 0 [ y i (1)]+ ::: +log f 0 [ y i (D)] + n 12 = 12 X i =1 log f $f 1 [ y i (1)]+(1 $ ) f 0 [ y i (1)] g + ::: +log f $f 1 [ y i (D)] +(1 $ ) f 0 [ y i (D)] g ; (6.18) where y i (1)=[ y i (1 ; 1) ;:::;y i (1 ; T)] ; ... y i (D)=[ y i (D ; 1) ;:::;y i (D ; T)] ; withdosage-specicmeanvectorsforcompositegenotype j speciedby 8>>>>>>>>>><>>>>>>>>>>: u j (1)=[ u j (1 ; 1) ;:::;u j (1 ; T)] ; ... u j ( d )=[ u j ( d; 1) ;:::;u j ( d; T)] ... u j (D)=[ u j (D ; 1) ;:::;u j (D ; T)] ; (6.19)

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108 anddosage-speciccovariancematricesdenotedby 1 ;:::; d ;:::; D ; (6.20) respectively. Iwillnotdirectlyestimatetheelementsinthemeanvectoran dcovariance matrix,ratherthanestimatetheparametersthatmodeltheme an-covariancestructures.Thiswilldependontwofactors.First,theparametersth atmodeldynamic drugeectsshouldbebiologicallymeaningful.Second,stati sticalmodellingofthe mean-covariancestructureswillbeinformativeandcanbeea silyimplementedin computingprograms.6.3.2ModellingtheMeanVector Thedose-andtime-dependentexpectedvaluesofdrugeectfo rcomposite genotype j canbetusingajointmodelofPKandPD(HochhausandDerendorf 1995).ThePDmodelconcernstherelationshipbetweendrugco ncentration(C) anddrugeect(E),whichcanbemathematicallyexpressedbyth eE max model E(C)= E max C EC 50 +C ; (6.21) whereE max istheasymptotic(limiting)eectandEC 50 isthedrugconcentration thatresultsin50%ofthemaximaleect. ThePKmodeldealswiththeconcentration-timeproles,whic hcanbe describedinseveraldierentways.ThePKandPDprocessescanbel inked directlyorindirectly.Thedirectlinkappliestoasituatio ninwhichthemeasured concentration(ortheplasmaconcentration c p )isproportionaltodrugconcentration attheeectsite(biophase).Theplasmaconcentrationforasing ledoseinthe directlinkiscalculatedfordierentcases(HochhausandDer endorf1995),which

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109 include c p = 8>>>>><>>>>>: d V d e k e t forintravenous(IV)bolus dk a V d ( k a k e ) ( e k e t e k e t )forrst orderabsorption k o k e V d ( e k e 1) e k e t forzero orderabsorption (6.22) where d isthebioavailabledose, V d isthevolumeofdistribution(whichisa constantforaknowndrug), k e istheeliminationrateconstant, k a isarst-order absorptionrateconstant, k o isazero-orderabsorptionrateconstant, isthe durationofzero-orderabsorption( = t duringabsorptionandconstantinthe postabsorptionphase)and t isthetimeafterthelastdosewasadministrated. TheindirectapproachforlinkingPKandPDassumesthattherei sahypotheticaleectcompartmentanditisbasedontheconcentra tion( c e )inthe compartmenthypothesizedtocausethedrugeect.Therelatio nshipbetween c e andtimecanbedescribedby c e = 8>>>>>>>>>>>>><>>>>>>>>>>>>>: dk co V d ( k co k e ) ( e k c t e k co t )forintravenous(IV)bolus dk a k co V d e k c t ( k a k e )( k co k e ) + e k a t ( k e k a )( k co k a ) + e k co t ( k e k co )( k a k co ) forrst orderabsorption k o k e V d ( k co k e ) [ k co ( e k e 1) e k e t k c ( e k co 1) e k co t ] forzero orderabsorption (6.23) where k co istherateconstantfordistributingdrugfromthecompartmen tinwhich drugconcentrationcausetheeect. ThePKandPDprocessescanbejointlymodelledbysubstitutingph armacokineticmodel(6.22)or(6.23)topharmacodynamicmodel( 6.21).Withsucha substitution,theeect-time-dosagerelationshipcanbederiv ed.Formodel(6.22), themeasuredconcentrationisused,whereasformodel(6.23), theconcentration hypothesizedtocausethedrugeectisused.Forexample,anint egrativePK-PD

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110 modelinthecaseofanIVbolusinjectionwithoutaneectcomp artmentcanbe expressedas E( d;t )= E max de k e t EC 50 V d + de k e t : (6.24) Thisexpressionwillbeintegratedtothemeanvectorofdruge ectsatdierent timepointsfordierentdosagesinequation(6.21).Foragiv encompositegenotype j ( j =0 ; 1 ; 2),Iwillhavetheunknownvectorforthreeparameters n m j = ( E max j ;EC 50 j ;k ej )todescribetime-dependentdrugeectsforaparticulardosa ge d expressedas u j ( d )= E max j de k ej t 1 EC 50 j V d + de k ej t 1 ;:::; E max j de k ej t T EC 50 j V d + de k ej t T : Ifdierentcompositegenotypeshavedierentcombinations oftheseparameters,thisimpliesthatthissequenceplaysaroleingoverning thedierentiationof thePK-PDlink.Thus,bytestingforthedierenceof n mj amongdierentcompositegenotypes,Icandeterminewhetherthereexistsaspecicseq uencevariantthat confersaneectonthePK-PDlink.6.3.3ModellingtheCovarianceMatrix TheARandstructuredantedependence(SAD)modelshavebeenpro posed tomodelthestructureofthecovariancematrixforlongitudi naltraitsmeasured atmultipletimepoints(ZimmermanandNu~nez-Anton2001; Diggleetal.2002). Inthischapter,Iwillincorporatetheautoregressivemoving average(ARMA) modeltoconstructthecovariancestructure.Therearethreem ajoradvantagesfor ARMAmodeltoapproximatethecovariancefunction.First,the closed-formof theautocovariancematrixfortheARMAprocesshasbeenderive d(Haddad2004), whichallowsfortheexpressionofautocovariancefunctionas afunctionoftheAR ( 1 ;:::; r )andMAcoecients( 1 ;:::; s ).Second,thederivationoftheinverse anddeterminantofthecovariancematrixfortheARMAprocessb yHaddad(2004)

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111 haslargelyfacilitatedthecomputationofthelikelihoodf unction(6.18).Third,as shownabove,theARMAmodelisageneralformofthecommonlyused ARmodel, whichmakestheARMA-basedstructuringapproachmoreusefulinpr actice. Givenadosage d ,theresidualerrorattime t inequation(6.16)dependsonits previouserrors(deterministic)andonarandomdisturbance( opportunistic).Forall thefollowingnotationwithdosage,suchas e i ( d;t ),Iwillomitthesymbol d unless itisspecied.Ifthedependenceof e i ( t )ontheprevious r errorsisfurtherassumed tobelinear,Icanwrite e i ( t )= 1 e i ( t 1)+ 2 e i ( t 2)+ ::: + r e i ( t r )+ e i ( t ) ; (6.25) whereconstants( 1 ;:::; r )arecalled autoregressive (AR) coecients ,and e i ( t ) isthedisturbanceattime t andisusuallymodelledasalinearcombinationof zero-mean,uncorrelatedrandomvariablesorazero-meanwh itenoiseprocess, i ( t ), i.e., e i ( t )= i ( t )+ 1 i ( t 1)+ 2 i ( t 2)+ ::: + s i ( t s ) ; (6.26) inwhich ( t )isawhitenoiseprocesswithmean0andvariance 2 ifandonlyif E [ ( t )]=0, E [ 2 ( t )]=0forall t ,and E [ ( t 1 ) ; ( t 2 )]=0if t 1 6 = t 2 ,where E denotes theexpectation.Theconstants( 1 ;:::; s )arecalledthe movingaverage (MA) coecients .Combiningequations(6.25)and(6.26)Ihave e i ( t ) 1 e i ( t 1) ::: r e i ( t r )= i ( t )+ 1 i ( t 1)+ ::: + s i ( t s ) ; (6.27) whichisdenedasazero-meanARMAprocessoforder r and s ,orARMA( r;s ). ForanonzerostationaryARMAprocess,twosequencesatdierentt imepoints,say t and k ,canbeconnectedthroughthebackwardshiftoperator B ,i.e., B k e i ( t )= e i ( t k ) :

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112 Denetheautoregressivepolynomial ( x )andmovingaveragepolynomial ( x )as ( x )=1 1 x 2 x 2 ::: r x r ; (6.28) ( x )=1+ 1 x + 2 x 2 + ::: + s x s ; (6.29) withtheassumptionthat ( x )and ( x )havenocommonfactors.equation(6.27) canbewritteninform ( B ) e ( t )= ( B ) ( t ) : (6.30) Twospecialcases: When s =0onlytheARpartremainandequation(6.27) reducestoapureautoregressiveprocessoforder r denotedbyAR( r ).Similarly, if r =0,Iobtainapuremovingaverageprocessoforder s ,MA( s ).Forthesetwo cases,Ihave e i ( t ) 1 e i ( t 1) ::: r e i ( t r )= ( B ) e ( t )= ( t )forAR( r )(6.31) e i ( t )= i ( t )+ 1 i ( t 1)+ ::: + s i ( t s )= ( B ) ( t )forMA( s ) : (6.32) Whenneither r nor s iszero,anARMA( r;s )modelissometimesreferredtoasa \mixedmodel". SeveralecientmethodscanbeusedtocomputetheexactARMAco variance, d ,foraparticulardosage d .Here,IusevanderLeeuw's(1994)generalmatrix representationinclosed-formtodescribetheARMAcovariance. Followingvander Leeuw(1994),IdenetwospecialtypesofToeplitzmatrices.T herstoneisa square(T T)ToeplitzmatrixwhichisgivenfortheARandMAparametersi s, respectively,by R = 264 R 1 O R 2 R 3 375 ; S = 264 S 1 O S 2 S 3 375 (6.33)

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113 where R 1 = 266666666664 100 ::: 0 1 10 ::: 0 2 1 1 ::: 0 ... ... ... . ... r 1 r 2 r 3 ::: 1 377777777775 r r S 1 = 266666666664 100 ::: 0 1 10 ::: 0 2 1 1 ::: 0 ... ... ... . ... s 1 s 2 s 3 ::: 1 377777777775 s s ; R 2 = 2666666666666666666664 r r 1 r 2 ::: 1 0 r r 1 ::: 2 00 r ::: 3 ... ... ... . ... 000 ::: r 000 ::: 0 ... ... ... . ... 000 ::: 0 3777777777777777777775 (T r ) r S 2 = 2666666666666666666664 r r 1 r 2 ::: 1 0 r r 1 ::: 2 00 r ::: 3 ... ... ... . ... 000 ::: r 000 ::: 0 ... ... ... . ... 000 ::: 0 3777777777777777777775 (T s ) s R 3 = 2666666666666666666664 100 ::: 00 ::: 0 1 10 ::: 00 ::: 0 2 1 1 ::: 00 ::: 0 ... ... ... . ... ... . ... r r 1 r 2 ::: 10 ::: 0 r r 1 r 2 ::: 1 1 ::: 0 ... ... ... . ... ... . ... 000 r r 0 r 0 1 ::: 1 3777777777777777777775 (T r ) (T r )

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114 S 3 = 2666666666666666666664 100 ::: 00 ::: 0 1 10 ::: 00 ::: 0 2 1 1 ::: 00 ::: 0 ... ... ... . ... ... . ... s s 1 s 2 ::: 10 ::: 0 s s 1 s 2 ::: 1 1 ::: 0 ... ... ... . ... ... . ... 000 s s 0 s 0 1 ::: 1 3777777777777777777775 (T s ) (T s ) : TheseconddenedmatrixhasdimensionT r fortheARparametersand T s fortheMAparameters,whoseupperpartsareasquare( r r )or( s s ) Toeplitzmatrix.Forthesetwotypesofparameters,Idene,re spectively, U = 264 U 1 O 375 ; V = 264 V 1 O 375 ; (6.34) where U 1 = 266666664 r r 1 ::: 1 0 r ::: 2 ... ... . ... 00 ::: r 377777775 r r ; V 1 = 266666664 s s 1 ::: 1 0 s ::: 2 ... ... . ... 00 ::: s 377777775 s s Giventheerrorvectors, e =[ e ( t 1) ;e ( t 2) ;:::;e ( t r +1) ;e ( t r )] T ; =[ ( s 1) ; ( s 2) ;:::; ( t s +1) ; ( t s )] T ; Idenethecorrespondingauxiliaryvectorsby e =[ e ( r +1) ;e ( r +2) ;:::;e ( 1) ;e (0)] T ; =[ ( s +1) ; ( s +2) ;:::; ( 1) ; (0)] T :

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115 Ithenwriteequation(6.27)inmatrixform [ UR ] 264 ee 375 =[ VS ] 264 375 or Re = V S U e ; fromwhichaseriesofmatrixoperationscanbeperformedtopr ovethatthe covariancematrix d correspondingtotheARMA( r;s )errorspecicationis solutiontotheequation(vanderLeeuw1994), R d R T = SS T + VV T +[ UO ] d [ UO ] T [ VO ] S T T d [ UO ] T [ UO ] R 1 S [ VO ] T : (6.35) AsshowninvanderLeeuw(1994),iftheinvertibilityconditio nfortheARpart holds,thecovarianceequation(6.35)hasanuniquesolution, whichis d = R 1 [ SS T +( RV SU )( P T1 P 1 U T1 U 1 ) 1 ( RV SU ) T ] R T : (6.36) Bysubstituting R = I and Q = O inequation(6.35),thecovariancematrixforthe MA( s )modelisobtainedas d = SS T + VV T : (6.37) Bysubstituting S = O and V = O ,nextpremultiplyingbothsidesofequation (6.36)andpostmultiplyingbyitstranspose,thecovariancema trixfortheAR( r ) modelisderivedas d =( R T R UU T ) 1 : (6.38) Basedontheexpressionfortheinverseofthesumoftwomatrices(R ao1973),I obtaintheinverseof d as 1 d = R T S T [ I T M ( M T M + R T1 R 1 U 1 U T1 ) 1 M T ] S T R ; (6.39)

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116 with M = S 1 RV U : Giventhatthevalueofthedeterminantof S 1 R isequalto one,Ihave j d j = I T S 1 264 R 1 V 1 S 1 U 1 O 375 ( R T1 R 1 U 1 U T1 ) 1 264 R 1 V 1 S 1 U 1 O 375 T S 1 (6.40) = I r +( R T1 R 1 U 1 U T1 ) 1 ( R 1 V 1 S 1 U 1 ) T e S T1 e S 1 ( R 1 V 1 S 1 U 1 ) ; (6.41) where e S 1 isthe(T r )matrix,consistingoftherst r columnsof S 1 .Forthe AR( r )model,thedeterminantisreducedto ( R T1 R 1 U 1 U T1 ) 1 ; whichisindependentofT. Withtheabovederivations,itcanbeseenthattheparameterst hatmodelthe covariancematrixarearrayedby n v = f 1 ( d ) ;:::; r ( d ) ; 1 ( d ) ;:::; s ( d ) ; 2 g d D d = d 1 forDdoselevelswhentheARMA( r;s )modelisimplemented.Theunknownvector canreducetotheAR( r )orMA( s )model,dependingonpracticalproblems. 6.3.4ComputationalAlgorithms IimplementedtheEMalgorithm,originallyproposedbyDempst eretal. (1977),toobtainthemaximumlikelihoodestimates(MLEs)oft hreegroupsof unknownparametersintheintegratedsequenceandfunctionm appingmodel, thatis,themarkerpopulationparameters( n p ),thecurveparameters( n m j )that modelthemeanvector,andtheparameters( n v )thatmodelthestructureofthe covariancematrix.Theseunknownsaredenotedby n =( n p ; n m j ; n v ).Adetailed descriptionoftheEMalgorithmwasgiveninWuetal.(2002b,2 004b)andMaet al.(2002). Asdescribedforsingletraitmapping,IimplementtheEsteptoca lculatethe expectednumber( $ )ofdiplotype[11][22]containedinthedoubleheterozygot e

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117 12/12andtheposteriorprobability( i )ofdoubleheterozygoticpatient i who carriesdiplotype[11][22].IntheMstep,Iusethecalculated $ and i valuesto estimatethehaplotypefrequenciesusingequations(6.5){(6 .8).Butinthisstep,I encounteraconsiderabledicultyinderivingthelog-likel ihoodequationsfor n m j and n v becausetheyarecontainedincomplexnonlinearequations.Zh aoetal. (2004)implementedthesimplexmethodasadvocatedbyNeldera ndMead(1965) totheestimationprocessoffunctionalmapping,whichcanstri kinglyincrease computationaleciency.Inthischapter,thesimplexalgori thmisembeddedinthe EMalgorithmabovetoprovidesimultaneousestimationofhapl otypefrequencies andcurveparametersandmatrix-structuringparameters.6.3.5ModelforanArbitraryNumberofSNPs Theideaforsequencingdrugresponsehasbeendescribedforatwo -SNP model.Itispossiblethatthetwo-SNPmodelistoosimpletochara cterizegenetic variantsforvariationindrugresponse.Ihaveextendedthism odeltoincludean arbitrarynumberofSNPswhosesequencesareassociatedwithdrug response variation.Akeyissueforthemulti-SNPsequencingmodelishowt odistinguish among2 w 1 dierentdiplotypesforthesamegenotypeheterozygousat w loci. Therelativefrequenciesofthesediplotypescanbeexpressedi ntermsofhaplotype frequencies.TheintegrativeEMalgorithmcanbeemployedto estimatetheMLEs ofhaplotypefrequencies.Bennett(1954)providedageneral formulaforexpressing haplotypefrequenciesintermsofallelefrequenciesandli nkagedisequilibriaof dierentorders.TheMLEsofthelattercanbeobtainedbysolvi ngasystemof equations. 6.4HypothesisTests Thismodelallowsforthetestsofanumberofbiologicallyorc linicallymeaningfulhypothesesattheinterfacebetweengeneactionsandd rugresponse.The existenceofspecicgeneticvariantsaectinganintegrated PK-PDprocesscanbe

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118 testedbyformulatingthefollowinghypotheses, 8>><>>: H 0 : n m j n m ;j =2 ; 1 ; 0 H 1 :atleastoneoftheequalitiesabovedoesnothold ; (6.42) where H 0 correspondstothereducedmodel,inwhichthedatacanbetby a singledrugresponsecurve,and H 1 correspondstothefullmodel,inwhichthere existdierentdynamiccurvestotthedata.Theteststatisticf ortestingthe hypothesesinequation(6.42)iscalculatedasthelog-likel ihoodratio(LR)ofthe reducedtothefullmodel: LR= 2[log L ( e n j y ; G ) log L ( b n j y ; G )] ; (6.43) where e n and b n denotetheMLEsoftheunknownparametersunder H 0 and H 1 respectively.TheLRisasymptotically 2 -distributedwith8degreesoffreedom.An empiricalapproachfordeterminingthecriticalthresholdi sbasedonpermutation tests,asadvocatedbyChurchillandDoerge(1994).Byrepeate dlyshuingthe relationshipsbetweenmarkergenotypesandphenotypes,aseri esofthemaximum log-likelihoodratiosarecalculated,fromthedistributio nofwhichthecritical thresholdisdetermined. AnalternativeapproachfortestingtheexistenceofDNAsequence variants thatareresponsiblefortheintegratedPK-PDprocesscanbebase donthevolume underthelandscape( VUL ).The VUL foralandscapeforagivencomposite genotype j canbecalcualtedby VUL j = Z d D d 1 Z t T t 1 E max j de k ej t EC 50 j V d + de k e jt dt dd = Z d D d 1 E max j k ej ln 1+ d EC 50 j V d dd as t 1 0and t T !1 (6.44) ThenullhypothesisfortheexistenceofasignicantDNAsequence variantbased

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119 onthe VUL isexpressedas VUL j VUL .AsimilarLRteststatisticcanbe calculatedusingequation(6.43)forstatisticaltesting. AlthoughIhereassumethesamePK( k e )andPDparameters(E max andEC 50 ) fordierentdoselevels,thismodelcanbegeneralizedtoallo wtheseparametersto varyoverdosage.Itispossiblethatdierentreferencehaplot ypesareoperational forpharmacodynamiceectsunderdierentdosage,whichist hecauseofhaplotype dosageinteractioneects.Foranypairofdosages, d 1 and d 2 ,thishaplotype dosageinteractioncanbetestedbyformulatingthehypothesis testsofdrugeects: 8>><>>: H 0 :E 0 ( d 1 ;t ) E 0 ( d 2 ;t )=E 1 ( d 1 ;t ) E 1 ( d 2 ;t )=E 2 ( d 1 ;t ) E 2 ( d 2 ;t ) H 1 :atleastoneoftheequalitiesabovedoesnothold ; (6.45) whereE j ( d 1 ;t )orE j ( d 2 ;t )istheeect-timeproleforcompositegenotype j under dosage d 1 or d 2 ,respectively,whichisdescribedbyequation(6.24).Thepar ameters underthe H 0 of(6.44)canbeestimatedbyposingappropriateconstraintsas shown inLinetal.(2005).Ifthenullhypothesisisrejected,thism eansthatthesame haplotypemaytriggerdierentimpactsondrugeectunderd ierentdosages. Similarly,Icanmakeahypothesistestabouthaplotype timeinteractions. Thenullhypothesisforthistestisbasedontheeect-dosagepro lewiththe relationshiplikeE 0 ( d;t 1 ) E 0 ( d;t 2 )=E 1 ( d;t 1 ) E 1 ( d;t 2 )=E 2 ( d;t 1 ) E 2 ( d;t 2 )for twogiventimepoints t 1 and t 2 .Inaddition,individualcurveparameters,suchas E max ,EC 50 and k e ,canbetested.Thetestsoftheseparametersareimportantfor thedesignofpersonalizeddrugstocontrolparticulardiseases. 6.5Results Toexaminethestatisticalpropertiesofthegeneticmodelfor theintegrated PK-PDprocess,Iperformasimulationstudyinwhichalandscapeof drugeect asafunctionofdoselevelandtimeissimulated.Figure6-1isa typicalexample generatedbyequation(6.24)inwhichthreelandscapeseachp resentingadierent

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120 0 10 20 30 40 50 10 0 10 1 10 2 10 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (h) Dose (mg)Effect (%) Figure6{1:Landscapesofdrugeectsvaryingasafunctionof dosageandtimefor threehypothesizedcompositegenotypes.compositegenotypeover-crosses,suggestingthatthereareinter actionsbetween haplotypesanddosagesandtimes. ThesimulationwillbebasedontheARMA( r;s )model(6.30).Forcomputationalsimplicity,IonlyconsidertheAR(1)parameterssotha tonly and 2 areusedtomodelthestructureoftheresidualcovariancematri x.Atotalof200 unrelatedindividualsareassumedtorandomlysamplefromahum anpopulation atHardy-Weinbergequilibriumwithrespecttohaplotypes.Tab le6-1tabulates thedistributionofthefrequenciesofhaplotypesconstructe dbytwoSNPsinterms ofallelefrequenciesandlinkagedisequilibrium.Thediplo typesderivedfromfour haplotypesatthesetwoSNPsaecttheintegratedPKandPDproc ess.Astwo

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121 dierentprocesses,itispossiblethatPKandPDarecontrolledby dierentDNA sequencevariants.Byassumingarbitrarilydierentreference haplotypesforthe PK(say12)andPD(say11),Icanspecifythemeanvectorsforeach composite genotype(Table6-1). LongitudinalPKandPDdataaresimulatedat6timepointsfor4 doselevels underthemultivariatenormaldistributionusingthePK( k e )andPDcurveparameters(E max andEC 50 )fortherespectivethreecompositegenotypes,asgiven inTable6-2.Thesecurveparametersaredeterminedintheran gesofempirical estimatesoftheseparametersfrompharmacologicalstudies(Ho chhausandDerendorf1995).Figure6-1illustratesthedierencesintheeec t-concentration-time landscapeamongthreecompositegenotypes.Usingthegeneticvar ianceduetothe compositegenotypicdierenceinthevolumeunderlandscape, Icalculatetheresidualvariancesunderdierentheritabilitylevels( H 2 =0 : 1and0.4).Theseresidual variances,plusgivenresidualcorrelations,formastructured residualcovariance matrix ThepopulationgeneticparametersoftheSNPscanbeestimated withreasonablyhighprecisionusingtheclosed-formsolutionapproach( Table6-2).The estimatesoftheseparametersareonlydependentonsamplesizea ndarenotrelatedtothesizeofheritability.Thismodelallowsfortheid enticationofcorrect referencehaplotypesforthePKandPDprocesses.Byconsidering allpossible combinationofreferencehaplotypesforthesetwoprocesses,Ic alculatetheLR teststatisticsusingequation(6.43)foreachcombination.The maximumLRvalue correspondstothecaseinwhichthereferencehaplotypesfort hesetwoprocesses areconsistentwiththegivenones(seeTable6-2). Figure6-2describesdierentshapesofthePKandPDfromtheth reecompositegenotypeswithacomparisonbetweenthegivenandestima tedcurvesunder aselecteddoselevel.Theestimatedcurvesareconsistentwithth egivencurves,

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122 Table6{2:MaximumlikelihoodestimatesofSNPpopulationgen eticparameters (allelefrequenciesandlinkagedisequilibrium),thecurve parametersandmatrixstructuringparametersforpharmacokineticsandpharmacod ynamicsresponses.The numbersinparenthesesarethesquarerootofMSEsfortheestima tes. CompositeHeritability ParametersgenotypeTruevalue0.10.4 Populationgeneticparameters p (1)1 0.600.60(0.02)0.60(0.02) p (2)1 0.600.60(0.03)0.60(0.02) D 0.080.08(0.01)0.08(0.01) Curveparameters:Pharmacokinetics k e [12][12]0.200.20(0.02)0.21(0.02)[12][ 12]0.300.30(2e-3)0.30(9e-4) [ 12][ 12]0.400.40(3e-3)0.40(1e-3) Curveparameters:Pharmacodynamics E max [11][11]0.600.60(6e-3)0.60(3e-3)[11][ 11]0.800.80(5e-3)0.80(2e-3) [ 11][ 11]1.001.00(7e-3)0.10(3e-3) EC 50 [11][11]0.010.01(2e-4)0.01(1e-4)[11][ 11]0.020.02(2e-4)0.02(1e-4) [ 11][ 11]0.030.03(3e-4)0.03(1e-4) Matrix-structuringparameters Dose d =1 0.700.70(0.02)0.69(0.10) 2 9e-59e-5(1e-5)1e-51e-5(4e-6) Dose d =10 0.700.70(0.02)0.71(0.09) 2 2e-32e-3(2e-3)3e-44e-4(3e-4) Dose d =100 0.700.70(0.02)0.72(0.06) 2 0.020.02(0.01)3e-33e-3(4e-3) Dose d =1000 0.700.70(0.02)0.72(0.07) 2 0.050.05(0.04)0.010.01(0.01)

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123 suggestingthatthismodelcanprovidereasonableestimatesofP KandPDcurves. TheparametersforthePKandPDmodelsofeachcompositegenot ypecanbe estimatedaccuratelyandprecisely(Table6-2).Asexpected,t heestimationprecision(assessedbyMSEs)increasesremarkablywhentheheritability increasesfrom 0.1to0.4.TheestimatesoftheAR(1)parametersthatmodelthe structureof thecovariancematrix alsodisplayreasonablyhighprecision(Table6-2).Butit seemsthattheirestimationprecisionisindependentofthesize ofheritability. Ineachof100simulations,Icalculatethelog-likelihoodrat ios(LR)for thehypothesistestofthepresenceofageneticvariantaectin gbothPKand PDprocesses.TheLRvaluesineachsimulationunderbothheritab ilitylevels arestrikinglyhigherthanthecriticalthresholdestimatedfr om100replicates ofsimulationsunderthenullhypothesisthatthereisnoPK/PD -associated geneticvariant.Thissuggeststhatthismodelhasenoughpowe rtodetectthe geneticvariantundergivenSNPs,curveandmatrix-structurin gparametersforthe simulation. 6.6Discussion Thegeneticcontrolofpharmacokinetics(PK)andpharmacod ynamics(PD)is oneofthemostimportantaspectsincurrentpharmacogenomico rpharmacogenetic research(WattersandMcleod2003).Historically,thiskindof studyhasbeen complicatedbytwofactors.First,therewasnoadequateinfor mationavailable abouttheDNAstructureandorganizationofthehumangenome.T herecent releaseofthehaplotypemap(HapMap)constructedbySNPs(TheIn ternational HapMapConsortium2003)hasmadeitpossibletoidentifypolymor phicsitesatthe DNAsequencelevelandfurtherassociatetheDNAsequencevariants withcomplex diseasesordrugresponse.Second,drugresponseisadynamicproce ss,aectedby timeanddrugconcentration.Itisinsucienttostudythegene ticbasisfordrug

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124 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 [12][12] [12][12] [12][12] Time (h)Concentration (mg/l) 0 20 40 60 80 100 0 20 40 60 80 100 [11][11] [11][11] [11][11] Concentration (ng/ml)Effect (%) 0 10 20 30 40 50 0 20 40 60 80 100 Diplotype [11][12] Time (h)Effect (%) Figure6{2:Estimatedresponsecurves(dash)eachcorresponding tooneofthree compositegenotypesforPK( A ),PD( B )andintegratedPK-PD( C )underdose 100mg,inacomparisonwiththehypothesizedcurves(solid)used tosimulateindividualcurves.Theconsistencybetweentheestimatedandhypo thesizedcurves suggeststhatthismodelcanprovidethepreciseestimationofth egeneticcontrol overresponsecurvesinpatients.eectsatasingletimepointanddosagebecausethisislikelyto providebiased resultsaboutthegeneticregulationofpharmacodynamicact ionsofadrug. Inthischapter,Ihaveproposedanovelstatisticalmodelforch aracterizing geneticcontrolmechanismsthatunderliePKandPDprocesses.By considering theinruencesofbothtimeanddosage,Ihavepushedacurve-base dlongitudinal problemtoitslandscapeextension(seeFigure6-1).Thismodel isconstructedwith anitemixturemodelframeworkfoundedonthetenetsofthese quence(Linet al.2005)andfunctionalmapping(Maetal.2002;Wuetal.200 2b,2004a,2004b).

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125 Thismodelhasparticularpowertodiscernthediscrepancybet weenthegenetic mechanismsforPKandPD.WiththeaidofHapMap,thismodelallo wsforthe genome-widescanofgeneticvariantsthatresponsibleforthese twodierentbut physiologicallyrelatedprocesses. Unliketraditionalstatisticalmodelsinsimilarareas,mymodel candirectly detectDNAsequencesthatcodePKandPDprocesses.Thus,geneticin formation extractedfromobserveddatabythismodelismoreinformativ eandprecisethan thatfromtraditionalapproaches.Inparticular,clinallym eaningfulmathematical functions(HochhausandDerendorf1995;Giraldo2003;McCli shandRoberts2003) havebeenintegratedtoastatisticalframework,thusdisplaysi gnicantadvantages. First,itfacilitatesstatisticalanalysisandstrengthenspowe rtodetectsignicant geneticvariantsbecausefewerparametersareneededtobeest imated,asopposed totraditionalmultivariateanalysis.Second,theresultsitp roducesarecloseto biologicalrealmgiventhatthemathematicalfunctionsused arefoundedonarm understandingofpharmacology(reviewedinDerendorfandMe ibohm1999).Based ondierencesinthePK-PDlandscape,asshowninFigure6-1,Ic antestforthe geneticcontrolregardinghowpharmacokineticactionscha ngeovertime(thetime surface)andhowpharmacodynamiceectsvaryoverdosage(the dosagesurface)by estimatingtheslopesofeachsurface.Thestatisticalpowerofth ismodelisfurther increasedbytheattempttostructurethecovariancematrixwi thautoregressive models(Diggleetal.2002). Thismodelhasbeenrelieduponthemodellingandanalysisoft heresidual covariancematrix.Inthischapter,Ihaveforthersttimein corporatedatime seriesmodel{autoregressivemovingaverage(ARMA)model{intot hemixture modelcontext.TheARMAmodelhasbeencommonlyusedineconome tricresearch aimedtoapproachthetime-dependentinruenceoferrors(va nderLeeuw1994). Thismodelisveryrexibleinthatanyorderofapolynomialth atmodeldynamic

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126 errorscanbedetermined.Furthermore,thereisasurprisingl ysimpleformfor itsdescriptionofthecovariancematrix.Theclosedformsoft heinverseand determinantofthematrixcangreatlyenhancecomputationa leciency(vander Leeuw1994;Haddad2004).WiththeARMAmodel,theAR(1)modelth athas beenusedtomodelthecovariancestructure(Maetal.2002;Wue tal.2004a, 2004b;Linetal.2005)canbeviewedasaspecialcase. Asusual,simulationstudieshavebeenemployedtoinvestigateth estatistical behaviorofthismodel.Theresultssuggestthatitcanprovidea ccurateand preciseestimatesoftheresponsecurvesforbothPKandPDevenwh enthe heritabilityismodest.Dierentdrugresponsecurves,assimula tedfordierent compositegenotypes(Figure6-1),canoerscienticguidanc efordeterminingthe optimaldoseandoptimaldosageformthatdisplayfavorabledru geectsbasedon individual'sgeneticbackground.Itshouldbenotedthatthi smodelwasderived onthebasisofasimpleclinicaldesigninwhichasinglecohortis assumed,with equallyconcentrationintervalsforeverypatient.Thismo delcanbeextendedto acase-controlstudybyallowinghaplotypefrequenciesandha plotypeeectstobe dierentbetweenthecaseandcontrolgroups.Aseriesofhypoth esistestsregarding between-groupdierencescanbeformulatedtodetectspeci chaplotypesthat areresponsiblefordrugresponse.Thismodelcanalsobeextended toconsider variousmeasurementschedules,inparticularwithunevenconc entrationlags varyingfrompatienttopatient.Fortheseirregularmeasurem entschedules,itis neededtoformulateindividualizedlikelihoodfunctionsw henthemeanvectorand covariancematrixaremodelled(seeNu~nez-AntonandWoodw orth1994).With theseextensions,mymodelwillassistinthediscoveryandcharacte rizationofa networkofgenesthatinruencedrugresponseintermsofpharma cokineticsand pharmacodynamics.

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CHAPTER7 CONCLUSIONSANDPROSPECTS 7.1Summary TheinformationabouthowDNAsequencevariesacrossthehumang enome iscrucialforunravellingthegeneticbasisofdrugresponse.T heoverallobjective ofthisdissertationistodevelopstatisticalmodelsfordirect lydetectingand characterizingspecicDNAsequencevariantsthatareresponsib lefordrugresponse undervariousproblemsettings.Thebasictenetsofthesemodels aretointegrate mathematicalaspectsofdrugresponseandSNP-basedhaplotypebl ockingtheory intoastatisticalmixturemodel-basedmappingframework. InChapter2,aconceptualmodelwasderivedasthegeneralfr ameworkfor myseriesofstudies.Thismodelwasemployedtoapharmacogenet icstudyof cardiovasculardisease,leadingtothedetectionofaparticul arhaplotypethat exhibitsadierenteectonheartratecurvecomparedtothe restofhaplotypes. Thissimplemodelwasthenextendedtoincludetwodierentbu trelatedprocesses, drugecacyanddrugtoxicityinChapter3.Simulationstudie ssuggestthatthe extendedmodelhaspowertotestwhetherthereexistdierentD NAsequence variantsthatgovernthesetwodrugresponsesandhowdierentd ierentvariants aectecacyandtoxicityinacoherentway. Becausegeneticinteractionsarethoughttobeaubiquitousb iologicalphenomenonandplayapivotalruleincontrollingmanyimportan thumandiseasesas wellasdrugresponse.InChapter4and5,twomodelswerederive dforthecharacterizationofdierentDNAsequencesthatinteractinacoor dinatedmannerto regulatethecomplexdiseaseandthedynamicprocessofdrugee cts,respectively. 127

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128 Themodelsprovidequantitativeframeworksfortestingthea dditive additive, additive dominant,dominant additiveanddominant dominantinteraction eectsbetweendierentQTNsequencesfromhaplotypeblocks. Thesignicant sequence-sequenceinteractionswereeectivelydetectedin aworkedexample. Chapter6providesahigh-dimensionalstatisticalmodelforun veilingthe geneticsecretsfordrugresponsebyintegratingpharmacokine tics(PK)and pharmacodynamics(PD)intoageneticmappingframework.Si mulationstudies wereperformedtotesthowthespecichaplotypesaectdiere ntlytheeectof bodyonthedrug(PK)andtheeectofdrugonthebody(PD)simul taneously. Perhaps,themostsignicantstrengthofmydissertationisinitsn ature ofcuttingedgeresearchtowardahistoricallycomplicatedbi omedicalproblem. Thetheoreticalmodelsproposedinmydissertationwerederive donthebasis ofbiomedicalprinciplesandhavepotentiallymuchbroader applications.The statisticalcontributionsofmydissertationincludetheelega ntderivationfora closedformsolutionforestimatingallelicfrequenciesandli nkagedisequilibria forsequencevariantsthatcontroldrugresponsewithinthecon textoftheEM algorithmaswellastheintegrationoftheEM-simplexalgori thmforestimatingthe geneticactionandinteractionsofthesevariants. Inmydissertation,Ialsoattempttomakeacontributiontophar macogeneticandpharmacogenomicresearch.Dierentfromcandidat egeneapproaches, mymodelscanbedirectlyemployedtoidentifyDNAsequenceswi thoutprior knowledgeaboutpharmacodynamicmechanismsofdrugeects.I nparticular,my modelswillallowforthegenome-widesearchforgeneticvari antsthatcontroldrug response.Also,becausethederivationsofmymodelsarebasedonth ebiochemical principlesbehinddrugresponse,theresultsfromthemshouldbe biologicallymore relevantandclinicallymoreeective.

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129 Byintegratingmathematical,statisticalandcomputerlangu agesintothe geneticcontrolofpharmacodynamicandpharmacokineticpr ocesses,mymodels provideaquantitativeandtestableframeworkforaskinganda ddressingbiological problemsattheinterplaybetweengeneactionanddrugrespon se. 7.2FutureDirections Althoughthisdissertationhasforthersttimedocumentedbasic modelsfor thegeneticstudyofdrugresponse,thereisstillmuchroomforfu rtherderivations andmodelling.Below,Ipinpointseveralareasinwhichfurth ereortcanmakea visibleprogress.7.2.1Gene-EnvironmentInteraction Awealthofevidencehassuggestedthattheinteractionsbetwe engenesand variousenvironmentalordevelopmentalsignalscanexertpr onouncedeectson drugresponse.Afurtherinteractionmodel,incorporatedwit henvironmental impactsondierentiationindrugresponse,canbederivedtop rovideatestable frameworkforunravellingthemechanismsofhowsequencevari antsinteractwith environmenttomediatedrugresponse. Ahigh-dimensionalgene-environmentinteractionmodelcan bederivedto characterizedierentiatedgeneticmechanismsfortherespo nsivenessofthesame drugtodierentenvironmentalfactors.Theresultsfromsuchi nteractionmodels canhelptodesignanoptimaldrugandoptimaldosefortreating adiseasebased oncoordinatedinteractionsbetweenapatient'sgeneticar chitectureandthe environmentofthepatient.7.2.2Case-ControlStudy Genesfordrugresponsemaybesegregatingdierentlybetweenn ormal (control)andaected(case)populations.Thispopulation-d ependentsegregation patternsofDNAsequencevariantscanbeincorporatedintoaser iesofprocedures

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130 totesthowDNAvariantsoperatedierentlyincaseorcontrolpo pulationsto governdrugresponse. Thecase-controlmodelcandetectthedierencesofthefrequ enciesandeects ofcausalhaplotypesforaQTNbetweendierentpopulations.T hisinformation helpstopreciselycomparethepopulation-dependentdiere ncesinDNAstructure andorganizationofQTNthoughttoregulatedrugresponse.7.2.3Dose-DependencyofAllometricScalingPerformance Theuseofadrugwouldgiverisetoalternationsinapatient'sp hysiological metabolismwhichhasbeenfoundtouniversallyscaleasamultip leofquarters ofbodymass(Westatal.1997,1999).Theperformanceofallome tricscaling ofdoseasapowerofbodyweightremainsuntestedinmanycases.How ever,its integrationwithageneticmappingstudylikethiswouldprod uceexcitingresultsin pharmacogeneticresearch. Themotivationofsizetobeincorporatedintoananalysisofdr ugeect resultsfromknownbiologicalprinciples.Agreatmanyphysiol ogical,structural andtimerelatedvariablesscalepredictablywithinandbetw eenspecieswith weightexponentsof0.75,1and0.25,respectively.Westatal. (1997,1999)have usedfractionalgeometrytomathematicallyexplainthisphe nomenon.The3/4 powerlawformetabolicrateswasderivedfromageneralmode lthatdescribes howessentialmaterialsaretransportedthroughspace-lledfr actionalnetworksof branchingtubes.Thesedesignprinciplesareindependentofde taileddynamicsand explicitmodelsandshouldapplytovirtuallyallorganisms.7.2.4MissingDataProblem Inpharmacogeneticresearch,missingdataarealmostalwaysapr oblem.Data canbemissingcompletelyatrandom(MCAR),missingatrandom(MAR) ,or missingnotatrandom(MNAR).Statisticalmodelshavebeenavaila bleforhandling missingdatawithMCAR,buthavenotbeensucientlydevelopedfo rtheother

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131 twomissingmechanisms.Itappearsthatpattern-mixturemodels canbeusedto solvesomeproblemsrelatedtoMARandMNAR.Theimplementationof patternmixturemodelsinthegeneticmappingwithDNAsequencevarian tswouldbe powerfultocharacterizethegeneticregulationofdrugresp onsewithnon-ignorable data.

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APPENDIXA DERIVATIONOFASYMPTOTICCOVARIANCEMATRIX AsseeninLouis(1982),theobservedinformation( I obs )canbeobtainedby subtractingthemissinginformation( I mis )fromthecompleteinformation( I com ). UsingLouis'notation,wedenotethecompletedataby x =( x 1 ;x 2 ; ;x n ) T theobserveddataby y =( y 1 ;y 2 ; ;y n ) T andtheparametervectorby .The likelihoodoftheincompletedataisexpressedas f Y ( y j )= Z R f X ( x j ) d ( x ) ; (A1) where R = f x : y ( x )= y g and ( x )isadominatingmeasure.Forequation(A1), wehavethescore, @ log f Y ( y j ) @ = Z R f X ( x j ) R R f X ( x j ) d ( x ) @ log f X ( x j ) @ d ( x ) ; (A2) andtheHessianmatrix, @ 2 log f Y ( y j ) @@ T = Z R f X ( x j ) R R f X ( x j ) d ( x ) @ 2 log f X ( x j ) @@ T d ( x ) + Z R f X ( x j ) R R f X ( x j ) d ( x ) @ log f X ( x j ) @ d ( x ) @ log f X ( x j ) @ T d ( x ) Z R f X ( x j ) R R f X ( x j ) d ( x ) @ log f X ( x j ) @ d ( x ) Z R f X ( x j ) R R f X ( x j ) d ( x ) @ log f X ( x j ) @ T d ( x ) : (A3) When x 1 ;x 2 ; ;x n areindependentbutnotnecessarilyidenticallydistributed and y i ( x )= y i ( x i ),wehave R = R 1 R 2 R n .Tomakethescore(A2) andtheHessianmatrix(A3)moretractable,weneedtouseFubini's theorem. ByincorporatingFubini'stheorem,thescoreandtheHessianmat rixcannowbe 132

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133 expressedas @ log f Y ( y j ) @ = n X i =1 Z R i f X ( x i j ) R R i f X ( x i j ) d ( x i ) @ log f X ( x i j ) @ d ( x i )(A4) and @ 2 log f Y ( y j ) @@ T = n X i =1 Z R i f X ( x i j ) R R i f X ( x i j ) d ( x i ) @ 2 log f X ( x i j ) @@ T d ( x i ) + n X i =1 Z R i f X ( x i j ) R R i f X ( x i j ) d ( x i ) @ log f X ( x i j ) @ @ log f X ( x i j ) @ T d ( x i ) n X i =1 Z R i f X ( x i j ) R R i f X ( x i j ) d ( x i ) @ log f X ( x i j ) @ d ( x i ) Z R i f X ( x i j ) R R i f X ( x i j ) d ( x i ) @ log f X ( x i j ) @ T d ( x i ) # : (A5) Basedon(A4)and(A5),thecomplete I com andmissinginformation I mis can becalculatedusingLouis'formulae,fromwhichtheobservedin formation I obs is estimated,i.e. I obs = I com I mis : Inatwo-SNPcase,thecomplete-datalog-likelihoodfunction ofpopulation geneticparameters, n p =( p (1)1 ;p (2)1 ;D ) T ,isgivenas log L ( n p j G )=Constant +2 n 11 = 11 log p 11 + n 11 = 12 log(2 p 11 p 12 )+2 n 11 = 22 log p 12 + n 12 = 11 log(2 p 11 p 21 )+ Z log(2 $p 11 p 22 )+( n 12 = 12 Z )log(2(1 $ ) p 12 p 21 ) + n 12 = 22 log(2 p 12 p 22 )+2 n 22 = 11 log p 21 + n 22 = 12 log(2 p 21 p 22 ) +2 n 22 = 22 log p 22 ; (A6) where Z denotestheobservationofdiplotype[ A 11 A 21 ][ A 12 A 22 ]withindoubleheterozygousgenotype A 11 A 12 =A 21 A 22 and Z hasbinormialdistributionwithsuccessive probability $ = p 11 p 22 p 11 p 22 + p 12 p 21 and Z isunobserved.Accordingtoequation(2.5),

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134 haplotypefrequenciesare p 11 = p (1)1 p (2)1 + D; p 12 = p (1)1 (1 p (2)1 ) D; p 21 =(1 p (1)1 ) p (2)1 D; p 22 =(1 p (1)1 )(1 p (2)1 )+ D: ontheotherhand,thepopulationgeneticparameterscanbeso lvedonceweknow theestimatesofhaplotypefrequencies,thatare b p (1) 1 = b p 11 + b p 12 (A7) b p (2) 1 = b p 11 + b p 21 (A8) b D = b p 11 b p 2 11 b p 11 b p 12 b p 11 b p 21 b p 12 b p 21 (A9) Let M 1 (3 1) = @ log L ( n p j G ) @ n p = @ log L ( n p j G ) @p (1)1 ; @ log L ( n p j G ) @p (2)1 ; @ log L ( n p j G ) @D T ,then M 1 11 = 2 n 11 = 11 p (2)1 p 11 + n 11 = 12 [ p (2)1 p 12 + p 11 (1 p (2)1 )] p 11 p 12 + 2 n 11 = 22 (1 p (2)1 ) p 12 + n 12 = 11 ( p (2)1 p 21 p 11 p (2)1 ) p 11 p 21 + Z [2 p (2)1 p 22 2(1 p (2)1 ) p 11 $ S 1 ] p 11 p 22 ( n 12 = 12 Z )( p (2)1 p 11 p 21 $ S 1 ) p 12 p 21 + n 12 = 22 (1 p (2)1 )( p 22 p 12 ) p 12 p 22 2 n 22 = 11 p (2)1 p 21 n 22 = 12 [ p (2)1 p 22 + p 21 (1 p (2)1 )] p 21 p 22 2 n 22 = 22 (1 p (2)1 ) p 22 ; M 1 21 = 2 n 11 = 11 p (1)1 p 11 + n 11 = 12 [ p (1)1 p 12 p 11 (1 p (1)1 )] p 11 p 12 2 n 11 = 22 p (1)1 p 12 + n 12 = 11 [ p (1)1 p 21 + p 11 (1 p (1)1 )] p 11 p 21 + Z [2 p (1)1 p 22 2(1 p (1)1 ) p 11 $ S 2 ] p 11 p 22 ( n 12 = 12 Z )( p (1)1 p 11 p 12 $ S 2 ) p 12 p 21 n 12 = 22 [ p (1)1 ( p 22 p 12 )+ p 12 ] p 12 p 22 + 2 n 22 = 11 (1 p (1)1 ) p 21 + n 22 = 12 (1 p (1)1 )( p 22 p 21 ) p 21 p 22 2 n 22 = 22 (1 p (1)1 ) p 22 ;

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135 M 1 31 = 2 n 11 = 11 p 11 + n 11 = 12 ( p 12 p 11 ) p 11 p 12 2 n 11 = 22 p 12 + n 12 = 11 ( p 21 p 11 ) p 11 p 21 + Z [2( p 22 + p 11 ) $ S 3 )] p 11 p 22 ( n 12 = 12 Z )[1+ $ S 3 ] p 12 p 21 n 12 = 22 ( p 22 p 11 ) p 12 p 22 2 n 22 = 11 p 21 n 22 = 12 ( p 22 p 21 ) p 21 p 22 + 2 n 22 = 22 p 22 ; whereS 1 = p (2)1 ( p 22 p 12 )+(1 p (2)1 )( p 21 p 11 ),S 2 = p (1)1 ( p 22 p 21 )+(1 p (1)1 )( p 12 p 11 ) andS 3 = p 11 + p 22 p 12 p 21 Let M 2 (3 3) = @ 2 log L ( n p j G ) @ n p @ n Tp ,therefore I com p =E f M 2 g n p = ^ n p and I mis p =E n ( M 1 )( M 1 ) T o n p = ^ n p n E( M 1 ) n p = ^ n p E( M 1 ) Tn p = ^ n p o where G =( n 11 = 11 ;n 11 = 12 ; ;n 22 = 22 )isthevectorofobservationsand b n p arethe MLEsofpopulationparameterscalculatedusingequations(A7 ){(A9). Similarly,wealsocanestimatetheasymptoticvariance-covar iancematrix forquantitativegeneticparameters( n q ).Intwo-SNPcase,thecomplete-data log-likielihoodfunctionof n q isexpressedas log L ( n q j Y com ; G ; n p )= n 11 = 11 X i =1 log f 2 ( y i ) + n 11 = 12 + n 12 = 11 X i =1 log f 1 ( y i ) + n 11 = 22 + n 12 = 22 + n 22 = 11 + n 22 = 12 + n 22 = 22 X i =1 log f 0 ( y i ) + n 12 = 12 X i =1 Z i log[ $f 1 ( y i )] + n 12 = 12 X i =1 (1 Z i )log[(1 $ ) f 0 ( y i )] ;

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136 where Z i istheindicatorvariabledenotedas1ifindividual i hasdiplotype [ A 11 A 21 ][ A 12 A 22 ]withingenotype A 11 A 12 =A 21 A 22 and0otherwise. Z i hasBernoulli distributionwithsuccessiveprobability $ and Z i isunobserved.Hence, I com q =E @ 2 log L ( n q j Y com ; G ; n p ) @ n q @ n Tq j Y obs n p = ^ n p ; n q = ^ n q I mis q =E @ log L ( n q j Y com ; G ; n p ) @ n q @ log L ( n q j Y com ; G ; n p ) @ n q T j Y obs ) n p = ^ n p ; n q = ^ n q ( E @ log L ( n q j Y com ; G ; n p ) @ n q j Y obs n p = ^ n p ; n q = ^ n q E @ log L ( n q j Y com ; G ; n p ) @ n q j Y obs Tn p = ^ n p ; n q = ^ n q ) where b n q aretheMLEsofquantitativeparametersderivedfromsimplex algorithm.

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APPENDIXB DERIVATIONOFMLESUSINGEMALGORITHM Inwhatfollows,wederiveacomputationalprocedureforimpl ementingtheEM algorithmtoestimatehaplotypefrequencies,compositegenot ypicmeansandthe residualvariance.TheMLEsofallelefrequencies,linkagedi sequilibriaandsequence actionandinteractioneectscanbeobtainedthroughsolvin gthecorresponding equations. IntheEstep,calculatetheexpectednumbers, $ R and $ S ,ofaparticular diplotypewithinthegenotypethatisheterozygousforboth SNPsatblocks R and S ,respectively,usingequations(4.11){(4.12).Meanwhile,c alculatetheposterior probabilitiesofaheterozygoussubject i thatcarriesaparticularacross-block diplotype(andthereforecompositegenotype)byusing RA ABB j i = p R11 p R22 f A ABB ( y i ) p R11 p R22 f A ABB ( y i )+ p R12 p R21 f A ABB ( y i ) ; (B1) RA AB B j i = p R11 p R22 f A AB B ( y i ) p R11 p R22 f A AB B ( y i )+ p R12 p R21 f A AB B ( y i ) ; (B2) RA A B B j i = p R11 p R22 f A A B B ( y i ) p R11 p R22 f A A B B ( y i )+ p R12 p R21 f A A B B ( y i ) ; (B3) SAAB B j i = p S11 p S22 f AAB B ( y i ) p S11 p S22 f AAB B ( y i )+ p S12 p S21 f AA B B ( y i ) ; (B4) SA AB B j i = p S11 p S22 f A AB B ( y i ) p S11 p S22 f A AB B ( y i )+ p S12 p S21 f A A B B ( y i ) ; (B5) S A AB B j i = p S11 p S22 f A AB B ( y i ) p S11 p S22 f A AB B ( y i )+ p S12 p S21 f A A B B ( y i ) ; (B6) [(11)(11)][(22)(22)] j i = p R11 p R22 p S11 p S22 f [(11)(11)][(22)(22)] ( y i ) i ; (B7) [(11)(12)][(22)(21)] j i = p R11 p R22 p S12 p S21 f [(11)(12)][(22)(21)] ( y i ) i ; (B8) 137

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138 [(12)(11)][(21)(22)] j i = p R12 p R21 p S11 p S22 f [(12)(11)][(21)(22)] ( y i ) i ; (B9) [(12)(12)][(21)(21)] j i = p R12 p R21 p S12 p S21 f [(12)(12)][(21)(21)] ( y i ) i : (B10) where i = p R11 p R22 p S11 p S22 f [(11)(11)][(22)(22)] ( y i )+ p R11 p R22 p S12 p S21 f [(11)(12)][(22)(21)] ( y i )+ p R12 p R21 p S11 p S22 f [(12)(11)][(21)(22)] ( y i )+ p R12 p R21 p S12 p S21 f [(12)(12)][(21)(21)] ( y i ).IntheMstep,the expecteddiplotypenumberscalculatedwithequations(4.1 1)and(4.12)areusedto estimatethehaplotypefrequenciesusing ^ p R11 = 1 2 n [2 n (11 = 11)( = ) + n (11 = 12)( = ) + n (12 = 11)( = ) + n (12 = 12)( = ) $ R ; (B11) ^ p R12 = 1 2 n [2 n (11 = 22)( = ) + n (11 = 12)( = ) + n (12 = 22)( = ) + n (12 = 12)( = ) $ R ; (B12) ^ p R21 = 1 2 n [2 n (22 = 11)( = ) + n (22 = 12)( = ) + n (12 = 11)( = ) + n (12 = 12)( = ) $ R ; (B13) ^ p R22 = 1 2 n [2 n (22 = 22)( = ) + n (22 = 12)( = ) + n (12 = 22)( = ) + n (12 = 12)( = ) $ R ; (B14) ^ p S11 = 1 2 n [2 n ( = )(11 = 11) + n ( = )(11 = 12) + n ( = )(12 = 11) + n ( = )(12 = 12) $ S ; (B15) ^ p S12 = 1 2 n [2 n ( = )(11 = 22) + n ( = )(11 = 12) + n ( = )(12 = 22) + n ( = )(12 = 12) $ S ; (B16) ^ p S21 = 1 2 n [2 n ( = )(22 = 11) + n ( = )(22 = 12) + n ( = )(12 = 11) + n ( = )(12 = 12) $ S ; (B17) ^ p S22 = 1 2 n [2 n ( = )(22 = 22) + n ( = )(22 = 12) + n ( = )(12 = 22) + n ( = )(12 = 12) $ S ; (B18) thatarederivedfromthelikelihoodfunction(4.4).Also,the posteriorprobabilities calculatedwithequations(4.13){(4.22)areusedtoestimate theacross-block compositegenotypicvalueswithequationsderivedfromthel ikelihoodfunction,i.e., ^ AABB = n (11 = 11)(11 = 11) X i =1 y i n (11 = 11)(11 = 11) ; (B19) ^ AAB B = n (11 = 11)( ) + n (11 = 11)(11 = 12) X i =1 y i + n (11 = 11)(12 = 12) X i =1 RAAB B j i y i n (11 = 11)( ) + n (11 = 11)(11 = 12) + n (11 = 11)(12 = 12) X i =1 RAAB B j i ; (B20)

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139 ^ AA B B = n (11 = 11)( ) X i =1 y i + n (11 = 11)(12 = 12) X i =1 (1 RAAB B j i ) y i n (11 = 11)( ) + n (11 = 11)(12 = 12) X i =1 (1 RAAB B j i ) ; (B21) ^ A ABB = n ( )(11 = 11) + n (11 = 12)(11 = 11) X i =1 y i + n (12 = 12)(11 = 11) X i =1 SAAB B j i y i n ( )(11 = 11) + n (11 = 12)(11 = 11) + n (12 = 12)(11 = 11) X i =1 SAAB B j i ; (B22) ^ A AB B = N A AB B D A AB B ; where (B23) N A AB B = n ( )(12 = 12) + n ( )(12 = 12) X i =1 RA AB B j i y i + n (12 = 12)( ) + n (12 = 12)( ) X i =1 SA AB B j i y i + n (12 = 12)(12 = 12) X i =1 [(11)(11)][(22)(22)] j i y i ; D A AB B = n ( )(12 = 12) + n ( )(12 = 12) X i =1 RA AB B j i + n (12 = 12)( ) + n (12 = 12)( ) X i =1 SA AB B j i + n (12 = 12)(12 = 12) X i =1 [(11)(11)][(22)(22)] j i ; ^ A A B B = N A A B B D A A B B ; where (B24) N A A B B = n ( )( ) X i =1 y i + n (12 = 12)( ) X i =1 RA A B B j i y i + n (12 = 12)(12 = 12) X i =1 ( [(11)(12)][(22)(21)] j i + [(12)(11)][(21)(22)] j i ) y i ; D A A B B = n ( )( ) + n (12 = 12)( ) X i =1 RA A B B j i + n (12 = 12)(12 = 12) X i =1 ( [(11)(12)][(22)(21)] j i + [(12)(11)][(21)(22)] j i ) ; ^ A ABB = n ( )(11 = 11) X i =1 y i + n (12 = 12(11 = 11)) X i =1 (1 SAAB B j i ) y i n ( )(11 = 11) + n (11 = 12)(11 = 11) + P n (12 = 12)(11 = 11) i =1 SAAB B j i ; (B25)

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140 ^ A AB B = N A AB B D A AB B ; where (B26) N A AB B = n ( )( ) X i =1 y i + n ( )(12 = 12) X i =1 S A AB B j i y i + n (12 = 12)(12 = 12) X i =1 ( [(22)(21)][(11)(12)] j i + [(21)(22)][(12)(11)] j i ) y i ; D A AB B = n ( )( ) + n ( )(12 = 12) X i =1 R A AB B j i + n (12 = 12)(12 = 12) X i =1 ( [(22)(21)][(11)(12)] j i + [(21)(22)][(12)(11)] j i ) ; ^ A A B B = n ( )( ) X i =1 y i + n (12 = 12)(12 = 12) X i =1 [(12)(12)][(21)(21)] j i y i n ( )( ) X i =1 + n (12 = 12)(12 = 12) X i =1 [(12)(12)][(21)(21)] j i ; (B27) ^ 2 = 1 n n n (11 = 11)(11 = 11) X i =1 ( y i ^ AABB ) 2 + n (11 = 11)( ) + n (11 = 11)(11 = 12) X i =1 ( y i ^ AAB B ) 2 + n (11 = 11)(12 = 12) X i =1 RAAB B j i ( y i ^ AAB B ) 2 + n (11 = 11)( ) X i =1 ( y i ^ AA B B ) 2 + n (11 = 11)(12 = 12) X i =1 (1 RAAB B j i )( y i ^ AA B B ) 2 + n ( )(11 = 11) + n (11 = 12)(11 = 11) X i =1 ( y i ^ A ABB ) 2 + n (12 = 12)(11 = 11) X i =1 SAAB B j i ( y i ^ A ABB ) 2 + n ( )(12 = 12) + n ( )(12 = 12) X i =1 RA AB B j i ( y i ^ A AB B ) 2 + n (12 = 12)( ) n (12 = 12)( ) X i =1 SA AB B j i ( y i ^ A AB B ) 2 + n (12 = 12)(12 = 12) X i =1 [(11)(11)][(22)(22)] j i ( y i ^ A AB B ) 2 + n ( )( ) X i =1 ( y i ^ A A B B ) 2 + n (12 = 12)( ) X i =1 RA A B B j i ( y i ^ A A B B ) 2

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141 + n (12 = 12)(12 = 12) X i =1 ( [(11)(12)][(22)(21)] j i + [(12)(11)][(21)(22)] j i )( y i ^ A A B B ) 2 + n ( )(11 = 11) + n (11 = 12)(11 = 11) X i =1 ( y i ^ A ABB ) 2 + n (12 = 12)(11 = 11) X i =1 SAAB B j i ( y i ^ A ABB ) 2 + n ( )( ) X i =1 ( y i ^ A AB B ) 2 + n ( )(12 = 12) X i =1 R A AB B j i ( y i ^ A AB B ) 2 + n (12 = 12)(12 = 12) X i =1 ( [(22)(21)][(11)(12)] j i + [(21)(22)][(12)(11)] j i )( y i ^ A AB B ) 2 + n ( )( ) X i =1 ( y i ^ A A B B ) 2 + n (12 = 12)(12 = 12) X i =1 [(12)(12)][(21)(21)] j i ( y i ^ A A B B ) 2 o ; (B28) where n ( k 1 k 0 1 =k 2 k 0 2 )( = ) = 2 X l 1 =1 2 X l 2 =1 2 X l 0 1 =1 2 X l 0 = 1 n ( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 0 2 ) and n ( = )( l 1 l 0 1 =l 2 l 0 2 ) = 2 X k 1 =1 2 X k 2 =1 2 X k 0 1 =1 2 X k 0 = 1 n ( k 1 k 0 1 =k 2 k 0 2 )( l 1 l 0 1 =l 2 l 0 2 ) ; with1 k 1
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BIOGRAPHICALSKETCH MinLinwastherstborntoJinhuLinandRuilinLiuinBeijing, China.She hasoneyoungerbrother,Hai.MinwenttoCapitalUniversityofM edicalSciences inBeijing,whereshereceivedherbachelor'sdegreeinclini calmedicineinJulyof 1996.WiththehonorforenteringtheGraduateProgramwithe xamexemption, MinstartedhergraduatestudyatPekingUnionMedicalCollegei nSeptember of1996andobtainedaMasterofSciencedegreeinanticancerp harmacology& microbialpharmacythreeyearslater. In1999,MinmovedtoGainesville,Florida,topursueherdocto raldegreein molecularbiologyattheUniversityofFlorida.Byfollowingh erlong-timeinterest intheeldofstatistics,Mindecidedtoreorienthercareer.Sh ewasacceptedas agraduatestudentintheDepartmentofStatisticsattheUniver sityofFloridain Mayof2000.AfterearningherMasterofStatisticsdegreeinAugu stof2002,Min enteredPh.D.program,workingasagraduateassistantunderth esupervisionof Dr.RonglingWu.SheisexpectedtoreceiveherPh.D.degreei nAugustof2005. Aftergraduating,MinwillstarthernewpositionasanAssistantPro fessorwiththe DepartmentofBiostatisticsandBioinformaticsatDukeUniver sity,inDurham,NC. 150


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MATHEMATICAL AND STATISTICAL METHODS FOR IDENTIFYING DNA
SEQUENCE VARIANTS THAT ENCODE DRUG RESPONSE














By
MIN LIN














A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by
Min Lin
















I dedicate this work to my husband, parents and brother.
















ACKNOWLEDGMENTS

First and foremost, I want to express my deepest appreciation to Dr. Rongling

Wu, who has been a wonderful advisor, colleague and friend. I am extremely

grateful for his guidance, patience, insight and encouragement. His endless ideas

and enthusiasm for research aniaze and inspire me. He ahr-l- 0 had confidence in

me and wholeheartedly supported me throughout my doctoral study. Without his

understanding and aid, this dissertation would not have been possible. I would also

like to thank the nienters of my coninittee, Dr. Hartnmut Derendorf, Dr. Ramon

Littell, Dr. K~enneth Portier and Dr. Ronald Randles for the generous gifts of their

time as well as knowledge. Their reading of and coninenting on my dissertation

were very helpful.

Special thanks go to my family, who have unconditionally and constantly

supported me over the years. My parents, Jinhu and Ruilin, and my brother Hai

have more faith in my ability and eventual successes than I myself ever did. I really

appreciate everything they have done for me. Most importantly, I want to thank

my husband, Xiang, for his unwavering confidence in me and for not letting me give

up. His love, patience, encouragement and respect helped me get through all those

tough times.

Last, I would like to thank all my friends, inside and outside the world of

statistics, for their help and concern.


















TABLE OF CONTENTS
page

ACK(NOWLEDGMENTS .......... .. iv

LIST OF TABLES ......... .. .. viii

LIST OF FIGURES ......... .. .. x

ABSTRACT ......... .... .. xi

CHAPTER

1 INTRODUCTION . ...... ..... 1

1.1 Basic Genetics ......... .. 1
1.1.1 Genes and ChInin....in.!! -; . ... ... .. 1
1.1.2 Genotype and Phenotype .... ... .. 2
1.1.3 Molecular Genetic Markers .... .. :3
1.2 Linkage Analysis ......... .. :3
1.3 Linkage Disequilibriunt Analysis .... ... .. 5
1.41 Functionlal M'apping ........ .. 7
1.5 Haphiap ............ .... ... .... 8
1.6 Sequence Mapping: Front QTL to QTN ... ... .. 10
1.7 Pharniacogenontics and Drug Response ... ... .. 12
1.8 Structure and Organization ...... .. 1:3

2 MODEL FOR QTN MAPPING DRITG RESPONSE WITH HAPMAP 14

2.1 Introduction ......... ... 14
2.2 Theory ......... ... .. 15
2.2.1 Notation ......... ... 15
2.2.2 Likelihood Functions ...... .. 18
2.2.3 An Integrative EM Algorithm ... ... .. 21
2.2.4 Hypothesis Tests . .... .. 22
2.2.5 R-SNP Sequence Model .... .... 2:3
2.3 Application ........ . 24
2.4 Discussion ........ . .. 26

:3 A JOINT MODEL FOR SEQUENCING DRITG EFFICACY AND TOX-
ICITY. ............ .......... 3:3

:3.1 Introduction ......... ... 3:3











:3.2 Basic Principle For Sequence Mapping ... .. . .. :35
:3.3 An Integrative Sequence and Functional Mapping Framework for
Drug Efficacy and Toxicity .... .. :39
:3.3.1 The Multivariate Normal Distribution .. .. .. .. :39
:3.3.2 The Mapping Framework .... ... .. 42
:3.3.3 Computational Algorithm .... ... .. .. 45
:3.3.4 Model for an Arbitrary Number of SNPs .. .. .. 46
:3.4 Hypothesis Tests ......... ... 47
:3.5 Results ........... ......... 48
:3.6 Discussion ........ . .. 51

4 MODEL FOR DETECTING SEQITENCE-SEQITENCE INTER ACTIONS
FOR COMPLEX DISEASES . ...... .. 55

4.1 Introduction ......... .. .. 55
4.2 The Model ......... . .. 56
4.2.1 Notation ......... ... 56
4.2.2 Epistatic Effects . ..... .. 58
4.2.3 Likelihood Functions ...... .. 60
4.2.4 An Integrative EM Algorithm .... .. .. 67
4.3 Hypothesis Tests ......... ... 68
4.4 Results ......... . .. .. 69
4.5 Discussion ......... . .. 72

5 MODEL FOR DETECTING SEQITENCE-SEQITENCE INTER ACTIONS
FOR DRITG RESPONSE ........ .. .. 75

5.1 Introduction ......... .. .. 75
5.2 Theory ............ .. ... ...... 76
5.2.1 The Nornial Mixture Model .... .. .. .. 76
5.2.2 Epistatic Effects . ..... .. 77
5.2.3 Likelihood Functions ..... .... 79
5.2.4 Modelling the Mean-covariance Structures .. .. .. .. 84
5.2.5 An Integrative EM-sintplex Algorithm .. .. .. 87
5.3 Hypothesis Tests ......... ... 88
5.4 A Worked Example ........ .. 90
5.5 Monte Carlo Simulation . ...... .. 9:3
5.6 Discussion ......... .. .. 95

6 MODELLING THE GENETIC ETIOLOGY OF PHARMACOK(INTIC-
PHARMACODYNAMIC LINKS WITH THE ARMA PROCESS .. 98

6.1 Introduction ........ .. .. .. .. 98
6.2 Haplotyping a Complex Trait .. .. .. 100
6.3 Haplotyping the Integrated PK(-PD Process .. .. .. .. 104
6.3.1 The Likelihood Functions ... .. ... .. 104
6.3.2 Modelling the Mean Vector .... .... .. 108











6.3.3 Modelling the Covariance Matrix .. .. . .. 110
6.3.4 Computational Algorithms .. .... .. 116
6.3.5 Model for an Arbitrary Number of SNPs .. .. .. .. 117
6.4 Hypothesis Tests ......... ... .. 117
6.5 Results ......... . .. 119
6.6 Discussion ......... . .. 123

7 CONCLUSIONS AND PROSPECTS .... ... .. 127

7.1 Summary ......... . .. 127
7.2 Future Directions ....... .. .. 129
7.2.1 Gene-Environment Interaction ... .. .. 129
7.2.2 Case-Control Study .... ... .. .. 129
7.2.3 Dose-Dependency of Allometric Scaling Performance .. 130
7.2.4 Missing Data Problem ..... .. . 130

APPENDIX

A DERIVATION OF ASYMPTOTIC COVARIANCE MATRIX .. .. 132

B DERIVATION OF MLES USING EM ALGORITHM .. .. . 137

REFERENCES ......... . .. .. 142

BIOGRAPHICAL SK(ETCH .....__. ... .. 150
















LIST OF TABLES
Table page

2-1 Possible dipll..ri pe configurations of nine ._ 11..i pes at two SNPs and
their haplotype composition frequencies ... .. .. .. 17

2-2 Log-likelihood ratio (LR) test statistics of different haplotype models
and MLEs of population and quantitative genetic parameters within
the /32AR gene .. ... ... 27

2-3 Testing results for two drug response parameters, H and EC5o, and to-
tal genetic, additive and dominant effects under the optimal haplo-
type model ......... . 28

2-4 MLEs of SNP allele frequency and linkage disequilibrium and para-
meters describing the three dynamic curves based on the sigmoidal
Emax model. ......... .. 29

:31 Possible dipll..ri pe configurations of nine ._ 11..i pes at two SNPs which
affect drug efficacy and toxicity ..... .. :38

:32 MLEs of population genetic parameters, the curve parameters and matrix-
structuring parameters for efficacy and toxicity responses .. .. 50

4-1 Possible diplotype configurations of nine ._ 11..i pes at two SNPs and
their haplotype composition frequencies ... .. .. .. 61

4-2 MLEs of SNP population and quantitative genetic parameters associ-
ated with phenotypic variation in BMI ... .. .. 71

5-1 Possible diplotypes and their frequenecies for each of nine genotypes
at two SNPs within a QTN, haplotype composition frequencies for
each genotype and genotypic value vectors of composite ,_ 11..(i pes. 81

5-2 Likelihood ratios for 16 possible combinations of assumed reference
haplotypes with one from candidate gene /31AR and the second from
candidate gene /32AR ......... .. 92

5-3 MLEs of parameters within two independent candidate genes /31AR
and 32AR ...... ...... ......... 9:3

5-4 MLEs of total genetic, additive, dominant and interaction effects un-
der the optimal haplotype model ..... .... 95










5-5 MLEs of parameters of SNP allele frequency and linkage disequilib-
rium and parameters describing the nine dynamic curves .. .. 96

6-1 Possible diplotype configurations of nine 7.~ nd v(pes at two SNPs which
affect pharmacokinetics (PK() and pharmacodynamics (PD) .. .. 103

6-2 MLEs of SNP population quantitative genetic parameters for pharma-
cokinetics and pharmacodynamics responses .. .. .. 122

















LIST OF FIGURES
Figure page

1-1 Twenty-three pairs of chromosomes in the human genome .. .. .. 2

1-2 A short stretch of DNA front four versions of the same chromosome
region in different people . ..... .. 9

2-1 Profiles of heart rate in response to different concentrations of dobu-
tantine for three composite genotypes ... ... .. 28

3-1 Estimated response curves each corresponding to one of three com-
posite genotypes under the heritability (H2) of 0.1 and 0.4 in a com-
parison with the hypothesized curves ... ... .. 54

5-1 Profiles of heart rate in response to different dosages of dobutantine
for nine composite genotypes ...... .. 94

6-1 Landscapes of drug effects varying as a function of dosage and time
for three hypothesized composite genotypes. .. .. .. 120

6-2 Estimated response curves each corresponding to one of three com-
posite genotypes for PK( and PD in a comparison with the hypoth-
esized curves ........ .. .. 124















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Phike..phlli-

MATHEMATICAL AND STATISTICAL METHODS FOR IDENTIFYING DNA
SEQUENCE VARIANTS THAT ENCODE DRUG RESPONSE

By
Min Lin

August 2005

C'I I!r: Rongling Wu
Major Department: Statistics

Substantial variability exists among different patients in pharmacological

response to medications. Drug response is typically a complex trait that is con-

trolled by a network of multifarious genes as well as biochemical, developmental

and environmental factors. The identification of genetic factors that contribute

to among-person differentiation has been one of the most important and difficult

tasks for pharmacogenetic research and drug discovery. With the release of the

haplotype map, or HapMap, constructed for the entire human genome based on

high-throughput single nucleotide polymorphisms (SNPs), the detection of specific

DNA sequence affecting responses to drugs can now be made possible.

In this dissertation, I will propose a series of statistical models and algorithms

for mapping and identifying genetic variants that are associated with the dynamic

features of drug response. Founded on the SNP-based haplotype blocking theory,

these models are constructed within the context of maximum likelihood and

implemented with a closed-form solution for the EM algorithm to estimate the

population genetic parameters of SNPs. The simplex algorithm is used to estimate

the curve parameters that describe the pharmacodynamic and/or pharmacokinetic










changes and the covariance matrix structuring parameters. The incorporation of

clinically important niathentatical functions for drug response not only makes my

models more powerful for gene detection, but also allows for a number of hypothesis

tests at the interplay between gene actions/interactions and pharmacological

actions. 1\onte Carlo simulation studies based on various schemes have been

performed to investigate different statistical aspects of my models. The detection

of significant DNA sequence variants for drug response in worked examples has

validated the usefulness of the models. Potential applications to pharniacogenetic

research have been discussed for each of my models. It can he anticipated that my

models will have many implications for elucidating the detailed genetic architecture

of drug response and ultimately designing personalized medications based on each

patient's genetic blueprint.
















CHAPTER 1
INTRODUCTION

1.1 Basic Genetics

1.1.1 Genes and Chromosomes

Genetics is the study of heredity or inheritance. Genetics helps to explain how

traits are inherited from parents to their offspring. Parents pass on traits to their

young through gene transmission. The fundamental physical and functional unit of

heredity is a gene, which was first revealed by Gregor Mendel's pea experiments and

mathematical model in 1865. Genes are composed of deoxyribonucleic acid (DNA),

a double-strand helix of nucleotides. Each nucleotide contains a deoxyribose ring,

a phosphate group, and one of four nitrogenous bases: adenine (A), guanine (G),

cytosine (C), and thymine (T). In nature, base pairs form only between A and T

and between G and C due to their chemical configurations. It is the order of the

bases along DNA that contains the hereditary information that will be transmitted

from one generation to the next.

A single DNA molecule condensed into a compact structure in a cell nucleus

is called chromosome. The chromosomes occur in similar, or in homologous, pairs,

where the number of pairs is constant for each species. In humans, there are

twenty-three pairs of chromosomes, carrying the entire genetic code, in the nucleus

of every cell in the body. For each pair, one chromosome is inherited from the

mother and the other from the father. The entire collection of these chromosomes

is referred to as the human genome. One of the chromosome pairs in the genome is

the sex chromosomes (denoted by X and Y) that determine genetic sex. The other

pairs are autosomes that guide the expression of most other traits (Figure 1-1).




















a 7 8 9 10 11 12






10 20 21 22

autosomes siex chromosomes


Figure 1-1: Twenty-three pairs of chromosomes in the human genome


1.1.2 Genotype and Phenotype

A gene is simply a specific coding sequence of DNA and may occur in alterna-

tive forms called alleles. A single allele for each gene is inherited from each parent,

termed the maternal and paternal allele respectively. The pair of alleles constructs

the.i i.* l..which is the actual genetic makeup. If a given pair consists of similar

alleles, the individual is said to be it.I,~~ .;,;;. .;, i. for the gene in question; while if the

alleles are dissimilar, the individual is said to be hete,. ..;,I int For example, if we

have two alleles at a given gene of an individual,;i wA and a, there are two kinds

of homozygotes, namely AA and aa, and one kind of heterozygote, namely Aa.

Therefore, three different genotypes, AA, Aa and aa, are formed with a single pair

of alleles.

In comparison, Iph. ,..-liti'.: represents all the observable characteristics of

an individual, such as physical appearance (eye color, height, etc.) and internal

physiology (disease, drug response, etc.).









1.1.3 Molecular Genetic Markers

Molecular genetic markers are readily .I-- li-. Il phenotypes that have a di-

rect 1:1 correspondence with DNA sequence variation at a specific location in the

genome. In principle, the .I-- li- for a genetic marker is not affected by environmen-

tal factors. Genetic markers are DNA sequence polymorphisms and have many

different types. Restriction fragment length polymorphisms (RFLPs) are the first

genetic markers that were widely used for genomic mapping and population studies.

The polymerase chain reaction (PCR) provides a useful way to obtain genetic mark-

ers. Amplified fragment length polymorphisms (AFLPs) are one of the PCR-based

anonymous markers.

One of the fruits of the Human Genome Project is the discovery of millions

of DNA sequence variants in the human genome. The us! in ~r~y of these variants

are single nucleotide polymorphisms (SNPs), which comprise approximately H I' .

of all known polymorphisms, and their density in the human genome is estimated

to be on average 1 per 1000 base pairs (The International HapMap Consortium

2003). SNPs, as the newest markers, have been the focus of much attention in

human genetics because they are extremely abundant and well-suited for automated

large-scale ... n..~i ping. A dense set of SNP markers opens up the possibility of

studying the genetic basis of complex diseases by population approaches, although

SNPs are less informative than other types of genetic markers because of their

biallelic nature. SNPs are more frequent and mutationally stable, making them

suitable for association studies to map disease-causing mutations, especially useful

in personalized medicine for their association with disease susceptibility, drug

treatment response and nutritional needs.

1.2 Linkage Analysis

Since the publication of the seminal mapping paper by Lander and Botstein

(1989), there has been a large amount of literature concerning the development of









statistical methods for mapping complex traits (reviewed in Jansen 2000; Hoechele

2001). Although the idea of associating a continuously varying phenotype with a

discrete trait (marker) dates back to the work of Sax (1923), it was Lander and

Botstein (1989) who first established an explicit principle for linkage analysis. They

also provided a tractable statistical algorithm for dissecting a quantitative trait

into their individual genetic locus components, referred to as quantitative trait loci

(QTL). The aim of QTL mapping is to associate genes with quantitative phenotypic

traits. For example, we might be interested in QTL that affect the response to a

given drug, so we might be looking for regions on a chromosome that are associated

with drug response.

The success of Lander and Botstein in developing a powerful method for

linkage analysis of a complex trait has roots in two different developments. First,

the rapid development of molecular technologies in the middle 1980s led to the

generation of a virtually unlimited number of markers that specify the genome

structure and organization of any organism (Drayna et al. 1984). Second, almost

simultaneously, improved statistical and computational techniques, such as the EM

algorithm (Dempster et al. 1977), made it possible to tackle complex genetic and

genomic problems.

Genetic mapping of QTL lies in the idea that genetic markers can be close

to the gene of interest. Lander and Botstein's (1989) model for interval I'arrtl~l., l

of QTL is regarded as appropriate for an ideal (simplified) situation, in which the

segregation patterns of all markers can be predicted on the basis of the Mendelian

laws of inheritance, a trait under study is strictly controlled by one QTL on a

chromosome and the expected effect of such hypothetical QTL is estimated from

the 7.~ nd v(pes at marker loci flankingf the interval. This work was extended and

improved by ]rn lw: researchers (Haley et al. 1994; Jansen and Stam 1994; Zeng

1994; Xu 1996), with successful identification of so-called "outcrossingt QTL in









real-life data sets from pigs (Andersson et al. 1994) and pine (K~nott et al. 1997). A

general framework for QTL analysis was recently established by Wu et al. (2002b)

and Lin et al. (200:3).

An interval mapping approach can not adequately use information from

all possible markers on the genome. Zeng (1993, 1994) proposed a so-called

composite interval I't.;; J : .9l technique to increase the precision of QTL detection by

controlling the chromosomal region outside the marker interval under consideration.

This approach, also developed independently by Jasen (199:3) and Jansen and

Stam (1994), has been widely adopted in practice. Statistically, composite interval

mapping is a combination of interval mapping hased on two given flankingf markers

and a partial regression analysis on all markers except for the two ones bracketing

the QTL. However, the choice of suitable marker loci that serve as covariates is still

an open problem.

An interesting approach, called multiple interval I'arly J:,:ll is proposed by K~ao

et al. (1999) and is the extension of interval mapping by using multiple maker

intervals simultaneously to fit multiple putative QTL. In this method, the QTL

locations can he used to infer the positions between markers even with some

missing ... n..~i epe data, and can allow us to take both main and interaction effects

into account in mapping the multiple QTL.

1.3 Linkage Disequilibrium Analysis

The most important goal in genetic research is to identify and characterize the

actual genes that are responsible for p in e! ripic variation. Thus far, only a handful

of genes that determine variation in commercially important traits have been

described. The reason for the identification of relatively few genes can he attributed

to limitations of the techniques used to detect genes. In the last decade linkage

as & l-h;;-ased mapping approaches have been instrumental in detecting QTL for a

wide variety of traits in different organisms. But linkage analysis typically defines









the location of a QTL to within a 20-30cM chromosomal interval -perhaps 1 of

a species' genome. Given that around 70,000 functional genes are estimated in a

typical genome, there are roughly 700 genes that are thought to exist under each

QTL "bump" (Slate et al. 2002). Thus, identifying the gene (or genes) influencing

the trait of interest based on linkage analysis is a monumental task.

More recently, an alternative approach based on linkage disequilibrium, i.e., the

non-random co-segregation of alleles at linked loci, has been shown to be powerful

for aiding gene discovery (Terwilliger and Weiss 1998). The basic premise behind

linkage disequilibrium mapping is that a particular allele at a marker will tend to

co-segregate with one allelic variant of the gene of interest, provided the marker

and gene are very closely linked. LD mapping potentially has two advantages

over conventional linkage mapping. The first is that it may be logistically easier.

In theory, breeding schemes such as backcrosses or full-sib matings may not be

required, making experimental design more straightforward and saving considerable

time. The second, probably greater, advantage offered by LD mapping is that

QTL may be mapped to very small regions thus aiding discovery of the underlying

gene(s). In order to perform efficient LD rs Ilpplilr_ markers must be mapped at a

density compatible with the distances that LD extends in the population. Currently

several consortia and laboratories have undertaken to develop dense maps of single

nuclear polymorphism (SNP) markers for a wide variety of species. However, in

order to predict how many SNPs will be required for LD nar lpplilr_ the extent of

linkage disequilibrium must first be established. LD has been estimated in humans

(K~ruglyak 1999) and also Holstein cattle (Farnir et al. 2000).

The disadvantage of linkage disequilibrium mapping is that the association

between marker loci are also affected by evolutionary forces such as mutation, drift,

selection and admixture. This disadvantage can be overcome by a mapping strategy









combining linkage and linkage disequilibrium, such as that developed in Wu and

Zeng (2001) and Wu et al. (2002a).

1.4 Functional Mapping

In nature, many traits, such as growth, AIDS progression and drug response,

are dynamic and should be measured in a longitudinal way. Although the elucida-

tion of the relationship between genetic control and development for longitudinal

traits is statistically a pressing challenge, some of the key difficulties have been

overcome hv R. Wu and colleagues (:\! et al. 2002; Wu et al. 2002b, 2003, 2004a,

2004h, 2004c). They have proposed a general statistical framework, referred to

as functional I'nll.:,: llr which maps genome-wide specific QTL related to the

developmental pattern of a complex trait.

The basic rationale of functional mapping lies in the connection between gene

action or environmental effects and development parametric or nonparametric mod-

els of phenotype. Functional mapping can detect dynamic QTL that are responsible

for a biological process measured at a finite number of time points. A number

of mathematical models have been established to describe the developmental

process of a biological phenotype. For example, a series of growth equations have

been derived to describe growth in height, size or weight (von Bertalanffy 1957;

Richards 1959) that occur whenever the anabolic or metabolic rate exceeds the rate

of catabolism. Based on fundamental principles behind biological or biochemical

networks, West et al. (2001) have mathematically proved the universality of these

growth equations. With mathematical functions incorporated into the QTL map-

ping framework, functional mapping estimates parameters that determine shapes

and functions of a particular biological network, instead of directly estimating the

gene effects at all possible time points. Because of such connections among these

points through mathematical functions, functional mapping strikingly reduces the









number of parameters to be estimated and, hence, d~-1i ph increased statistical

power.

From a statistical perspective, functional mapping is a problem of jointly mod-

elling mean-covariance structures in longitudinal studies, an area that has recently

received considerable attention in the statistical literature (Pourahmadi 1999, 2000;

Daniels and Pourahmadi 2002; Pan and Mackenzie 2003; Wu and Pourahmadi

2003). However, as opposed to general longitudinal modelling, functional mapping

integrates the parameter estimation and test process within a biologically meaning-

ful mixture-based likelihood framework. Functional mapping is thus advantageous

in terms of biological relevance because biological principles are embedded into

the estimation process of QTL parameters. The results derived from functional

mapping will be closer to biological realms.

1.5 HapMap

Several recent empirical studies -II__- -r that SNPs are not evenly distributed

over the genome in terms of the extent of LD and that the structure of ii l'l''i,''

(a linear arrangement of nonalleles at linked loci; Figure 1-2) on a chromosome

can be broken into a series of discrete haplotype blocks (Daly et al. 2001; Patil

et al. 2001; Dawson et al. 2002; Gabriel et al. 2002; Phillips et al. 2002). In each

haplotype block, consecutive sites are in complete (or nearly complete) LD with

each other and there is limited haplotype diversity due to limited (coldspot) inter-

site recombination. Adli ... n i blocks are separated by sites that show evidence of

historical recombination (hotspot). It has generally been assumed that the presence

of haplotype blocks provides evidence for fine-scale variation in recombination

rates, with blocks corresponding to regions of reduced recombination, separated by

recombination hotspots. Based on a study of the whole chromosome 21 (Patil et al.

2001), 35,989 observed SNPs can be classified into different blocks with very low










a SNPs

Chromosomel
Chromosome 2
ChromnosomeS
Chromosome4


b Haplotypes






c Tag SNPs


SAACA~ QC CCA.... T T C o OOC3 ... AGcT C8roACCO....
AA~ICA~LC OeC A.... TTrC oAGoTrC .... ASTC GA A eC B....
AALCATOCCA.... TTCGGGOTC.... ASTCA ACCO....
AACAICOCCA.... fTYORGOQTC.... AOTCGACCO....




Haplotype~ ~ ~ EbE 1 TAATC TA A- Sensitive to drag
Haplatype2 TTGiATTGCGCCAACAGTAWAT
Haplotype sce G CATC dT G6t GATACiT G GT G; ~Insenitie todr

AT
oc a


Figure 1-2: A short stretch of DNA from four versions of the same chromosome
region in different people. Shown are SNPs (a), haplotypes (b) and tag SNPs (c).
Three SNPs are shown where variation occurs, surrounded by much of the DNA
sequence identical in these chromosomes. A haplotype contains a particular combi-
nation of alleles at nearby SNPs. The positions of the three SNPs shown in panel
a are highlighted. G. noJ viping just the three tag SNPs out of the 20 SNPs that
extend across 6,000 bases of DNA is sufficient to identify these four haplotypes
uniquely. Adopted from The International HapMap Consortium (2003).


haplotype diversity and MI' of the variation in this chromosome can be described

by only three SNPs per block.

Given the block-like pattern of LD distribution in the genome, it should be

more efficient to locate allelic variants for a complex human disease trait based on

haplotype blocks than individual SNPs to within a stretch of DNA that is amenable

to positional cloning techniques. Because of the reported low haplotype diversity

within blocks there is a possibility that very few haplotype-tag SNPs (htSNPs) need

be examined to detect common variants involved in human diseases (Figure 1-2c,

Wall and Pritchard 2003).

With the release of a haplotype map of the human genome, the HapMap,

which describes the common patterns of human DNA sequence variation based on

SNPs (The International HapMap Consortium 2003), it has been possible to find










specific DNA sequences that encode health, disease, and responses to drugs and

environmental factors. As shown in Figure 1-2b, people who carry haplotype 1

may be more susceptible to a given drug than those who carry other haplotypes.

It is shown that the four chromosomes in Figure 1-2b are adequately determined

by three tag SNPs or htSNPs (Figure 1-2c). Thus, association studies between

haplotype and drug response can he undertaken on the basis of these htSNPs

because if a particular chromosome has the pattern A-T-C at these three tag SNPs,

this pattern matches the pattern determined for haplotype 1. The detection of a

much fewer number of htSNPs showed facilitate association studies for coninon

diseases and ultimately will enhance our ability to choose targets for therapeutic

intervention.

1.6 Sequence Mapping: From QTL to QTN

The basic principle for QTL mapping is the cosegregation of the alleles at

a QTL with those at one or a set of known polymorphic markers genotyped on

a genome. If a QTL is cosegfregfatingf with molecular markers, the genetic effects

of QTL and their genontic positions can he estimated front the markers. This

approach is robust and powerful for the detection of 1! in r~ QTL and presents the

most efficient way to utilize marker information when marker maps are sparse.

However, this approach is limited in two aspects. First, because the markers and

QTL bracketed hv them are located at different genontic positions, the significant

linkage of a QTL detected with given markers cannot provide any information

about the sequence structure and organization of QTL. Second, the inference of

the QTL positions using the nearby markers is sensitive to marker inforniativeness,

marker density and mapping population type. As a result, only a few QTL detected

front markers have been successfully cloned (Frary et al. 2000), despite a consid-

erable number of QTL reported in the literature. Therefore, genetic information

provided by QTL mapping approach is not precise enough.









A more accurate and useful approach for the characterization of genetic

variants contributing to quantitative variation is to directly analyze DNA sequences

associated with a particular disease. If a string of DNA sequence is known to

increase disease risk, this risk can be reduced by the alteration of this DNA

sequence string using a specialized drug. The control of this disease can be made

more efficient if all possible DNA sequences determining its variation are identified

in the entire genome. A new term, quantitative trait nucleotides or QTN, has been

defined to describe the sequence polymorphisms that cause phenotypic variation in

a quantitative trait.

The recent development of the human genome project, with its massive

amounts of DNA sequence data available for the human genome (International

HapMap Consortium 2003), has provided fuel for identifying QTN for complex

traits such as drug response. The haplotype map or HapMap constructed by single

nucleotide polymorphisms (SNPs), being the most common type of variant in

the DNA sequence, has facilitated the complete identification of specific sequence

variants responsible for complex diseases. A linear arrangement of alleles (i.e.,

nucleotides) at different SNPs on a single chromosome, or part of a chromosome,

is alld aimi.-linThe cosegregation of SNP alleles on haplotypes leads to

non-random association, i.e., linkage disequilibrium (LD), between these alleles in

the population. Empirical analyses of LD for SNPs have shown that nearby SNPs

in the human genome tend to display highly significant levels of LD and are often

distributed in block-like patterns, rather than displaying random or even spaced

distribution as originally predicted (Patil et al. 2001; Dawson et al. 2002; Gabriel

et al. 2002). SNPs within haplotype blocks are much more strongly associated with

each other than those between different blocks. Haplotype diversity within each

block can be well explained by only a finite number of SNPs, called tag SNPs or

representative SNPs. The existence of these tag SNPs means that it is not necessary









to associate a disease with all SNPs in the DNA sequence in order to understand

the complete genetic control of the disease or drug response. There is currently a

pressing demand for statistical models that characterize the haplotype structure

within QTN for complex diseases or complex processes.

1.7 Pharmacogenomics and Drug Response

In current pharmacogenetic research, increasingly, attempts have been made

to identify candidate genes that influence pharmacological responses (Johnson

2003; Watters and McLeod 2003). These include genes involved in drug transport

(e.g., polymorphisms in the gene encoding P-glycoprotein 1 and the plasma

concentration of digoxin), genes involved in drug metabolism (e.g., polymorphisms

in the gene encoding thiopurine S-methyltransferase and thiopurine toxicity) and

genes encoding drug targets (e.g., polymorphisms in the gene encoding the P2-

adrenoceptor and response to P2-adrenoceptor agonists) (Johnson 2003). With

advanced molecular n.0 vr~iping technologies, a number of polymorphic sites (such

as single nucleotide polymorphisms or SNPs) within or near these candidate genes

can be genotyped. SNPs, especially SNPs that occur in gene regulatory or coding

regions (cSNPs), can be associated with phenotypic traits to detect genetic variants

causing pharmacological response variability. Linkage disequilibrium mapping based

on the nonrandom association between different genes in a population has proven

to be a powerful means for high-resolution mapping of genes for complex traits

(Wu and Casella 2005). More recently, a closed-form solution based on the EM

algorithm for estimating the allele frequencies of functional genetic variants and

their disequilibria with SNPs has been derived (Lou et al. 2003). It is possible that

this algorithm can be used to map QTL affecting the extent to which an individual

responds to a particular pharmacological action.

A host of physiologically-based mathematical models have been built to de-

scribe the pharmacokinetic and pharmacodynamic processes of a drug (Hochhaus










and Derendorf 1995). Beyond narrative descriptions, these models have provided

a precise characterization of drug effects and theoretical prediction of drug re-

sponsiveness across a broad range of dose levels or a wide period of times. These

mathematical models can also be incorporated into the framework for functional

mapping to precisely characterize genetic variants that contribute to variation in

drug response.

1.8 Structure and Organization

The overall purpose of this dissertation is to develop powerful statistical

models for detecting DNA sequence variants that affect different aspects of drug

response. In ('!s Ilter 2, a general framework was derived to decipher the genetic

machinery of pharmacodynamic processes of a drug at the DNA sequence level.

C'!s Ilter 3 illustrates a joint statistical model of the genetic control for drug efficacy

and toxicity. ('! .pter 4 looks to detect epistatic control over complex traits that

are expressed as sequence-sequence interactions. In chapter 5, sequence-sequence

epistatic models are extended to study drug response as a dynamic process.

('!, Ilter 6 describes a hyperspace model for characterizing the differentiation in the

genetic regulation of pharmacokinetics and pharmacodynamics. The last chapter

summarizes the results and provides further statistical methodological research in

pharmacogenetics and pharmacogenomics of drug response.















CHAPTER 2
MODEL FOR QTN MAPPING DRUG RESPONSE WITH HAPMAP

2.1 Introduction

Although pharmacogenetics or pharmacogenomics, the study of inherited

variation in patients' responses to drugs, is still in its infancy. A tremendous

accumulation of data for genetic markers and pharmacodynamic tests has made

it one of the hottest and most promising areas in biomedical science (Evans and

Rellingf 1999, 2004; Roses 2000; Evans and Johnson 2001; Evans and McLeod 200:3;

Goldstein et al. 200:3; Weinsilboum 200:3; Freeman and McLeod 2004). The central

theme of pharmacogenetics is to associate interpatient variability in drug response

with specific genomic sites with the aid of powerful statistical tools. Traditional

approaches for such association studies are based on the statistical inference of

putative genetic loci or quantitative trait loci (QTL) of interest to a ph.! n,~ J1 ipe

from known linkage or linkage disequilibrium maps (Lynch and Walsh 1998; Wu

and Casella 2005). With the completion of a haplotype map (Haphiap) constructed

from DNA sequence variation data (The International Haphiap Consortium 200:3,

2004; Deloukas and Be al. v- 2004), it has now become possible to characterize

concrete nucleotide combinations that encode a complex trait.

Liu et al. (2004) derived a new statistical method for elucidating DNA

sequence variation in complex diseases from Haphiap. Compared to complex

diseases, genes for drug response are relatively simpler and genetically easier to

study because diseases have undergone a long evolutionary pressure whereas the

use of particular drugs is much more recent. However, drug response is statistically

more difficult to analyze because it is a dynamic process and should be quantified










across multiple different levels of drug concentration or dosage and during a time

course. Statistical modelling of such longitudinal traits has been a challenging

issue given the complexities of their autocorrelation structure. 1\ore recently, an

innovative theoretical framework has been constructed within which clinically

meaningful pharmacodynamic models are incorporated into the context of genetic

mapping for drug response (Gong et al. 2004). The basic tenet of this framework

is the mathematical modelling of dosage-dependent drug effects that are influenced

by genetic determinants and environmental factors and their interactions. Not only

the framework does display m? Ilw: statistical advantages, such as more power and

greater estimation precision, due to a reduced number of unknown parameters,

but also is biologically or clinically more relevant given its close association to

fundamental pharmacological principles.

A novel statistical model for determining specific DNA sequences that are

associated with the phenotypic variation of drug response is presented. This model

is derived on the basis of multilocus haplotype analysis using a finite number of tag

SNPs. A closed-form solution for estimating the effects of haplotypes, haplotype

frequencies, allele frequencies and the degrees of LD of various orders among tag

SNPs underlying the response to drugs is derived and simulation studies performed

to test the statistical behaviors of this haplotype-based sequence mapping model. A

worked example is used to validate the model, in which a DNA sequence variant is

detected which significantly affects the shape of the heart rate curve in response to

different dosages of dobutamine.

2.2 Theory

2.2.1 Notation

Suppose there is a random sample drawn from a natural human population

at Hardy-Weinherg equilibrium. In this sample, a number of SNPs are genotyped

genome wide in order to identify DNA sequences responsible for complex diseases









or drug response. Recent studies have shown that the human genome has a

haplotype block structure (Patil et al. 2001; Dawson et al. 2002; Gabriel et

al. 2002), such that it can be divided into discrete blocks of limited haplotype

diversity. In each block, a small fraction of SNPs, referred to as I Ig SNPs," can

be used to distinguish a large fraction of the haplotypes. Consider R (R > 1)

tag SNPs for a haplotype block. Each of these R SNPs has two alleles denoted
by'4 A7L (k,~ = 1,2; r = 1, -R), wLithllee1.1 frequenc111;1ies denotedi by11. pl for_ the

rth SNP. I use superscripts and subscripts to distinguish between different SNPs

and different alleles within SNPs, respectively. These SNPs form 2R possible

hapltyps epresedas A A -- A, whose frequencies are denoted by pkkg---ky.

The haplotype frequencies are composed of allele frequencies at each SNP and

linkage disequilibria of different orders among SNPs (Lou et al. 2003). The random

combination of maternal and paternal haplotypes generates 2 -1(2R + 1) diplotypes

expressed as [AA A -A ][Ai, A2 R \ ]( < kI < 11 < 2,.. -, < kgR <

In < 2). These R SNPs form 3R observable multilocus zygotic 11..i vtpes,

generally expressed as A] All /A Al2 / -- / At When at most one SNP is

heterozygous, the diplotype is consistent to its zygotic genotype. However, when

two or more SNPs are heterozygous, the genotype will have different diplotypes

and, therefore, the number of multilocus genotypes will be fewer than the number

of diplotypes. For example, .11 .*1v ipe A ]Al /AIA I has one diplotype [A] A2 ][A] Af]
as does 11.(ir pe A]A1/"AIA the dfillleri pe [A]Af] [A]A2 ]. However, for a double

heterozygous genotype A1A /"AA I have two diplotypes [A1A ]n [AjA ] an

[A:A ][AjA ]. Let P,,,~ ,, I,~.....,] and Pkl~l/k212/---/knly denote the diplotype and

:- nod vipe frequencies, respectively, and nkil~l/kgl.lg/--kl, denote the observations of

various ,_ 11. *1 vpes.

Table 2-1 lists all possible ._ 11.11vpes and diplotypes at two SNPs ._ 11.11vped

from a sample of size n. Two haplotypes comprising a dipled vlipe come from four

















Table 2-1: Possible diplotype configurations of nine 7.~ nd v(pes at two SNPs and their haplotype composition frequencies


Relative diplotype


Haplotype composition


Genotypic


A]A A]A Observation mean vector


Genotype Diplotype Diplotype frequency freq. within genotypes A:A~ A:A


A:A:/A A
A]A1/"A A
A(A(/A A
A1Aj/A A
A(Aj/A A

A:Aj/A A
A]A1/"A A
Aj j/A A
A]A1/"A A


[A:AI ][A:A
[A]A ][AA ] '


[A:A ] [A:A ]

[A: A ] [A~A ]
[A 1A ]I[AAli ]

[A]A ][AA ] '

[A1A ][AA ] '

[A]A ][AA ] '


= p ,
2piipl2
= p2
2piip21
= 2piip22
2pl2921
2pl2922
= P1
2p21922
= 2 a


P[11][12] =
P1[12][1]


P[1~1[21] =


P[12][22] =
P~1[21][2]
P[21][22] =
P[22] [22]


n11/12
811/22
n I 711


ul2/12


n12/22


U1



U1

(.0
HO
HO
HO
HO


1 1/ 1\
2Zw 2(1 w) 1 \- w)I w


s0 j- pp p where pnl, pl2, 921 and p22 are the haplotype frequencies of A1A A1A A1A and A1A respectively. Haplotyvpe
A:A~ is assumed as the reference haplotype.


r;i7
1--Zi7


1







18

possible haplotypes, A1A A1A A1A and A1A2 ,with respective frequencies. The

diplotype frequencies can be expressed in terms of the haplotype frequencies (Table
2-1). Two diplotypes [A1A2] [AjA ] and [A(A ] [A Af] ofC a doublle heterozygote

A:Aj/A A~ have frequencies p,,p22 and pl2912, TOSpectively. Thus, the relative

frequencies of these two diplotypes for this double heterozygote are a function of

haplotype frequencies. Table 2-1 also gives the relative expected frequencies of

haplotypes contained in a given genotype. All genotypes, except for the double

heterozygote, contain one or two known haplotypes. For example, genotype
A]A1/ "A2 has one haplotype A]A whereas A]A1/ "A2 has one half haplotype

A]A2 and one half haplotype A] A The double heterozygote contains four possible

haplotypes, with the relative frequencies ;;; 2~i~~ for haplotypes A1A2 a~nd. A1A

and -'"" for haplotypes A1A2 and A1A
2.2.2 Likelihood Functions

The complete data are diplotype configurations at a given set of SNPs for each

_~Ir fu(i pe and patients' drug effects at different doages, whereas the observed data

are the genotypes of these SNPs and the outcomes of drug effects. The connection

between the 7.~ nc.1vipes and the diplotypes are viewed as the missing data.

The haplotype frequencies, identified in 02, = (pl, pl2,p21,p22), belong to the

set of population genetic parameters that can be estimated using the nine observed

_~Ir fu(i pes (G) for two SNPs (Table 2-1). The log-likelihood function of unknown

haplotype frequencies given observed 7.~ nc.1vipes can be written in multinomial form,

i.e.,


log L(O2,|G) oc 2nlllll log pll + ull/12 log(2pllpl2) + 2n12/12 log pl2

+n12/11 log(2pllp21) + 12/12 log [2(pllp22 + 12921~

+n12/22 log(2pl2922) + 2n21/21 log p21 + 21/22 log(2p21922)

+2n22/22 log p2 (2.1)









I intend to associate diplotypes with interpatient variation in drug response

based on observed drug responses measured at different dosages (y) and SNP

..r,, //1\ vpes (G). Generally speakin, a,, given tw-SP entype, AA Al /nA Aa

can be partitioned into two possible diplotypes, [A1A A ][AII AF ] and [AA Af

[A,1 A ]. Statistically, this is a mixture model problem with two components (i.e.,

diplotyes) having different proportions. The log-likelihood function of observed data

is formulated as



i= 1

where 02, is contained within the mixture proportion,





and 02, is a set of quantitative genetic parameters that specify the multivariate nor-

mal distribution, f, which includes diplotype-specific parameters, i.e., phenotypic

means of two different diplotypes at different dosages (u,,, l,-,] and u,,. l, ),)

and parameters common to both diplotypes, i.e., the (co)variance matrix among

dosages (E).
Suppose, there exists, apaticular, haplotype, AA A labelled by A, which is

different from the other three haplotypes, collectively labelled by a, in its effect

on drug response. The resultant diplotypes are thus equivalent to three composite

i. t.'lar 41,~ AA, Aa and aa. The phenotypic mean of each of three composite

_t Irs..i pes that contains the two distinct groups of haplotypes is denoted by uj for

composite genotype j (j = 2 for AA, 1 for Aa and 0 for aa). Considering drug

response at different concentrations, each uj can be fit by a clinically meaningful

pharmacodynamic model. One such model is the Emax model that specifies the

relationship between drug concentration (C) and drug effect (E) (Giraldo 2003).










This model is based on the equation

E,, CHj
E,(C) = Eoy + a ,(2.3)
EC.,( + CHj

where Eoj is the constant or baseline value for the drug response parameter, Emaxy

is the .I-i-inphs'tic (limiting) effect, EC-,,. is the drug concentration that results in

501' of the maximal effect, and Hj is the slope parameter that determines the slope

of the concentration-response curve. The larger Hj the steeper the linear phase

of the the log-concentration-effect curve. By estimating these curve parameters

separately for different :-- nod (irpes, one can determine how the DNA sequence

variants influence drug response based on the shape differences among the three

curves.

As a longitudinal trait, the (co)variance matrix of drug response can be

structured by many statistical models, such as a first-order autoregressive [AR(1)]

model (Gong et al. 2004), which states that the variance (0.2) 1S COnStant OVer

different concentrations and that the correlation of response between different

concentrations decreases proportionally (in p) with increased concentration interval.

Assuming that haplotype A:A~ is different from the other haplotypes (Table

2-1), the log-likelihood function can be expanded to include all possible SNP

:- n..i irpes, now expressed as


log L;(O, n,|y, G) = Clog f2 fi
i= 1

Slog f (y,)
i= 1

C log fo(yi)
i= 1
12/12,
Slog [w f2 yfi 01 7) fiY)], (2.4)
i= 1










where unknown vector 02, now contains the curve and matrix-structuring parame-

ters, arrated I by (Eo,,,EmaxyEC -,,. Hj a, 0.2

2.2.3 An Integrative EM Algorithm

A closed-form solution for estimating the unknown parameters with the EM

algorithm is derived in which haplotype frequencies are expressed as a function of

allelic frequencies and LD. For a two-SNP haplotype, use


(k1k (2 + 9 ki+k'D, (2.5)


where D is the linkage disequilibrium between the two SNPs. Thus, once haplotype

frequencies are estimated, I can estimate allelic frequencies and LD by solving

equation (2.5). The estimates of haplotype frequencies are based on the log-

likelihood function of equation (2.1), whereas the estimates of dipkJ v ipe curve

parameters and (co)variance-structuring parameters are based on the log-likelihood

function of equation (2.4). These two different types of parameters can be estimated

using an integrative EM-simplex algorithm.

In the E step, the expected value of we~ for subject i having double heterozy-
gous genotype carrying diplotype [A)A ][AjA"'; ] is calculate using
1111L'211 11' L922~~ UII

w,,p__ =(2.6)


Note that for all the other genotypes, this probability does not exist.

In the M step, the probabilities calculated in the previous iteration are used to

estimate the haplotype frequencies using

2nlllll 81/1 n11/22 I nllU. 01" 11
911 = (2.7)
2n
2811/22 (1/1 12220 I .
pl =(28
2n
2n22/11 pn12/11 822/1 Z 1'1- ).
p21 =l~lnr~a t= (2.9)
2n
2n22/22 + 822 ~~l +C2/12 822/1 I U.0-
922 (2.10)
2n









These estimated frequencies are embedded to the M step for estimating C2, derived

from the simplex algorithm (Zhao et al. 2004). Iterations of the E and M step is

continued until the estimates of the parameters converge to stable values. The

.I-i-ingdol'tic variance of these parameters can be estimated by calculating Louis'

observed information matrix (Louis 1982) (APPENDIX A).

2.2.4 Hypothesis Tests

Two 1!! .1.i- hypotheses are tested in the following sequence: (1) the association

between two SNPs by testing their LD, and (2) the difference of a given haplotype

from the other haplotypes in its effect on drug response. The null and alternative

hypotheses on the LD between two given SNPs are:


(2.11)
Hi :H D OO

The log-likelihood ratio test statistic for the significance of LD is calculated by

comparing the likelihood values under the H1 (full model) and Ho (reduced model)

hypotheses and produces


LR1 = -2[log L~py pi D = 0, O2,|G) log L (O;, 1 ,|G)] (2.12)

where the tilde and hat denote the MLEs of unknown parameters under Ho and H1.

The LR1 test statistic is considered to .I-i-inid1'' ulcally follow a X2 distribution with

one degree of freedom. The MLEs of allelic frequencies under Ho can be estimated

using the EM algorithm described above, but with the constraint pllp22 = 12921

imposed.

Diplotype or haplotype effects on a complex trait can be tested using the null

and alternative hypotheses expressed as

Ho : (Eoy, Emaxy, EC -,,., Hj) = (Eo, Emax, ECso, H), for all j = 2, 1, O
(2.13)
H1 At least one equality in Ho does not hold









The log-likelihood ratio test statistic (LR2) under these two hypotheses can he

similarly calculated. The LR2 may .l-vi-!llla tically foll0W a X2 distribution with

eight degrees of freedom. However, the approximation of a X2 distribution may

be inappropriate when some regularity conditions are violated. The permutation

test approach does not rely upon the distribution of the LR2 and may be used

to determine the critical threshold for determining the effect of DNA sequence

variation on drug response.

2.2.5 R-SNP Sequence Model

The idea for sequencing drug response based on a two-SNP model can he extended

to include an arbitrary number of SNPs whose sequences are associated with

the phenotypic variation. Consider R SNPs that form 3R ObSerVable multilocus

zygo;,,,tic ,, genotypes generally ~ exrese as 4 1 4/" 4 /~ -n / 4R. These

_. Ir..i ipes are collapsed from a total of 2R-1(2R + 1) diplotypes expressed as

4n1 42..~1" '12 -4 < kl < lI < 2, i 1 < kR < R < 2). AZ key

issue for the multi-SNP sequencing model is how to distinguish among 2r-1 different

diplotypes for the same genotype heterozygous at r loci. The relative frequencies of

these diplotypes can he expressed in terms of haplotype frequencies. The integrative

EM algorithm can he emploi- .1 to estimate the MLEs of haplotype frequencies.

Lou et al. (2003) provided a general formula for expressing haplotype frequencies in

terms of allele frequencies and linkage disequilibria of different orders. The MLEs of

the latter can he obtained by solving a system of equations.

In the multi-SNP sequencing model, I face many haplotypes and haplotype

pairs. An AIC-based model selection strategy has been framed to determine the

haplotype that is most distinct from the rest haplotypes in explaining quantitative

variation.









2.3 Application

A real example for a genetic study of cardiovascular disease is used to demon-

strate the usefulness of our model. Cardiovascular disease, principally heart disease

and stroke, is the leading killer for both men and women among all racial and

ethnic groups. Dobutamine is a medication that is used to treat congestive heart

failure by increasing heart rate and cardiac contractility, with actions on the heart

similar to the effect of exercise. Dobutamine is also commonly used to screen for

heart disease in those unable to perform an exercise stress test. It is this latter use

for which the study participants received dobutamine in this study. It is a synthetic

catecholamine that primarily stimulates /3-adrenergic receptors (73AR), which ph i

an important role in cardiovascular function and responses to drugs (Johnson and

Terra 2002; Ranade et al. 2002; Nahel 200:3).

Both the /31AR and /32AR genes have several polymorphisms that are com-

mon in the population. Two common polymorphisms are located at codons 49

(Ser49Gly) and :389 (Arg:389Gly) for the /31AR gene and at codons 16 (Argl6Gly)

and 27 (Gln27Glu) for the /32AR gene (Nahel 200:3). The polymorphisms in each

of these two receptor genes are in linkage disequilibrium, which so__~--- -is the im-

portance of taking into account haplotypes, rather than a single polymorphism,

when defining biologic function. This study attempts to detect haplotype variants

within these candidate genes which determine the response of heart rate to varying

concentrations of dobutamine.

A group of 16:3 men and women in ages from :32 to 86 years old participated

in this study. Patients had a wide range of testing (untreated) heart rate. Each

of these subjects was .-- n..~i iped for SNP markers at codons 49 and :389 within

the /31AR gene and at codons 16 and 27 within the /32AR gene. Dobutamine

was injected into these subjects to investigate their response in heart rate to this

drug. The subjects received increasing doses of dobutamine, until they achieved









target heart rate response or predetermined maximum dose. The dose levels used

were 0 (baseline), 5, 10, 20, 30 and 40 mcg-min, at each of which heart rate was

measured. The time interval of 3 minutes is allowed between two successive doses

for subjects to reach a pateau in response to that dose. Only those (107) in whom

there were heart rate data at all the six dose levels were included for data analyses.

By assuming that one haplotype is different from the rest of the haplotypes,

I hope to detect a particular DNA sequence associated with the response of heart

rate to dobutamine. The p in v! ripic data for drug response were normalized

as percentages to remove the baseline effect, which is due to between-subject

differences in heart rate prior to the test. At the P1AR gene, I did not find any

haplotype that contributed to inter-individual difference in heart rate response. A

significant effect was observed for haplotype Glyl6(G)-Glu27(G) within the P2AR

gene (Table 2-2). The log-likelihood ratio (LR) test statistics based on equation

(2.12) for the difference of GG from the other 3 haplotypes was 30.03, which is

significant at P = 0.021 based on the critical threshold determined from 1000

permutation tests. The LR values when selecting any haplotype rather than GG as

a reference gave no significant results (P = 0.16 0.40). I used a second testing

criterion based on the area under curve (AUC) to test the haplotype effect. This

test supports the result from the first test.

The maximum likelihood estimates (MLEs) of the population genetic para-

meters, such as haplotype frequencies, allele frequencies and linkage disequilibrium

between the two SNPs, within the P2AR gene were obtained from our model. As

indicated by the .. vinpull'tic variance of the MLEs based on Louis' (1982) approach,

these estimates di pl ovi reasonable precision. The allele frequencies within this

gene are estimated as 0.62 for Glyl6 at codon 16 and 0.40 for Glu27 at codon 27.

The MLE of the linkage disequilibrium between the two SNPs is 0.1303. These










-II_ -- -r that the two SNPs identified within the /32AR gene di-p~l~w a pretty high

heterozygosity and linkage disequilibrium.

The 1\LEs of the quantitative genetic parameters were obtained, also with rea-

sonable estimation precision (Table 2-2). Using the estimated response parameters,

I drew the profiles of heart rate response to increasing dose levels of dobutamine for

three composite genotypes comprising of haplotypes GG and non-GG (symbolized

by GG) (Figure 2-1). The composite 1......... i--zate [GG][GG] di;11lai- II consistently

higher heart rate across all dose levels, especially at higher dose levels than the

composite homozygote [GG][GG]. But the composite heterozygote had consistently

the lowest curve at all dose levels tested. I used AUC to test in which gene action

mode (additive or dominant) haplotypes affect drug response curves for hear rate.

The testing results sell__ -1 that both additive and dominant effects are important

in determining the shape of the response curve (Table 2-3), together accounting for

about 1 1' of the observed variation in drug response. I did not detect evidence for

haplotypes to have an effect on curve parameters, H and EC5o, for the heart rate

response .

I performed simulation studies to investigate the statistical properties of our

model. The data were simulated hv mimicking the example used above in order to

determine the reliability of our estimates in this real application. One haplotype

was assumed to be different from the other three. The data simulated under

this assumption were subject to statistical analyses, pretending that haplotype

distinction is unknown. As expected, only under the correct haplotype distinction

could the haplotype effect he detected and the parameters he accurately and

precisely estimated (Table 2-4).

2.4 Discussion

Single nucleotide polymorphisms (SNPs) are powerful tools for studying the

structure and organization of the human genome (Patil et al. 2001; Dawson et al.












Table 2-2: Log-likelihood ratio (LR) test statistics of different haplotype models
and the corresponding maximum likelihood estimates (MLEs) of population genetic
(SNP allele frequencies and linkage disequilibria) and quantitative genetic parame-
ters (drug response and (co)variance-structuring parameters) in a sample of 107
subjects within the P2AR gene. The .-i-mptotic variance of the MLEs are given in
the parentheses.


[AA, A ,]
[GG]
30.03
0.02


0.62(0.04)
0.40(0.04)
0.13(0.01)


Composite
genotype


Reference haplotype


[AC]
12.14
0.34


[AG]
19.32
0.16


[GC]
11.18
0.40


0.6;2
0.60
0.05


0.10
0.37
21.35
2.34
0.10
0.36
25.24
2.05
0.11
0.56
35.05
1.79

0.89
0.01


Parameters
LRI
P value


Population genetic parameters
p; 0.38 0.38
0.60 0.40
D 0.13 0.05

Quantitative genetic parameters
Eo 0.11 0.02
Emax 0.37 0.42
ECso 23.72 32.60
H 1.93 2.48
Eo 0.10 0.11
Emax 0.37 0.39
E so 25.87 38.25
H 1.95 1.69
Eo 0.11 0.11
Emax 0.50 0.44
E so 31.09 26.87
H 1.99 2.01


[Al A2 ,][A1A2 A]




[Al A2 ,][A1A2 A]




[A] A2 1][A1A2 A]


0.11(0.02)
0.75(0.26)
42.10(15.90)
1.73(0.29)
0.10(0.01)
0.39(0.06)
29.27(4.90)
2.01(0.25)
0.10(0.01)
0.39(0.04)
23.57(2.68)
2.04(0.20)

0.88(0.01)
0.01(7e-4)


0.89
0.01


0.88
0.01


The LR1 tests for the significance of haplotype effect based on hypothesis (14). The
optimal haplotype model detected on the basis of the LR test is indicated in bold-
face. There are two alleles Argl6 (A) and Glyl6 (G) at codon 16 and two alleles
Gln27 (C) and Glu27 (G) at codon 27.












100

90 [GG][GG]
[GG][GG]
80 [GG][GG]

70-

60) -

j'50 -

b0 40-

30 -

20-

10i


0 5 10 15 20 25 30 35 40

Dobutamine concentration (mcg)


Figure 2-1: Profiles of heart rate in response to different concentrations of dobuta-
mine (indicated by dots) for three composite genotypes (foreground) identified at
two SNPs within the P2AR gene. The profiles of 107 studied subjects from which
the three different composite ad~ vr~ipes were detected are also shown (background).






Table 2-3: Testing results for two drug response parameters, H and ECso, and to-
tal genetic, additive and dominant effects based on AUC in 107 subjects under the
optimal haplotype model [GG]


Test H ECso Genetic Additive Dominant
LR 0.76 4.18 17.64 7. 25 19.83
P value >0.05 >0.05 < 0.001 < 0.01 < 0.01


















Parameters [A)Af]
Population genetic parameters
p1 = 0.62 0.62(0.03)
p = 0.40 0.40(0.03)
D = 0.13 0.13(0.01)


Refeencehaplotype [AA A ,]


Table 2-4: Maximum likelihood estimates (MLEs) of SNP allele frequency and linkage disequilibrium and parameters describ-
ing the three dynamic curves based on the sigmoidal Emax model. The numbers in parentheses are the squared roots of the
mean square errors of the MLEs based on 1000 simulation replicates.


Composite
genotype


LA[ A ]

0.62(0.03)
0.40(0.03)
0.13(0.01)


0.10(0.04)
0.42(0.37)
25.08(20.27)
2.29(0.85)
0.10(0.01)
0.40(0.07)
27.70(5.93)
2.01(0.27)
0.10(0.01)
0.46(0.11)
30.56(8.87)
1.94(0.24)

0.89(0.01)
8e-3(9e-4)


LA[ Af]

0.61(0.03)
0.40(0.03)
0.13(0.01)


0.10(0.06)
0.39(0.43)
24.36(21.71)
1.94(0.98)
0.10(0.05)
0.43(0.25)
30.96(18.22)
2.48(1.05)
0.10(0.01)
0.43(0.06)
28.50(5.74)
1.97(0.18)

0.89(0.01)
8e-3(le-3)


LA[ A ]

0.61(0.03)
0.40(0.04)
0.13(0.01)


0.10(0.02)
0.40(0.36)
24.83(18.77)
2.12(0.57)
0.10(0.01)
0.39(0.06)
27.05(5.38)
2.05(0.26)
0.10(0.01)
0.51(0.18)
33.35(13.26)
1.91(0.27)

0.89(0.01)
8e-3(9e-4)


Quantitative
Eo 0.11
Emax 0.75
ECso = 42.10
H=-1.73
Eo 0.10
Emax 0.39
ECso = 29.27
H=-2.01
Eo 0.10
Emax 0.39
ECso 23.57
H1-2.04


genetic parameters
0.11(0.02)
0.75(0.24)
41.97(14.24)
1.72(0.29)
0.10(0.01)
0.40(0.07)
30.23(5.17)
2.02(0.27)
0.10(0.02)
0.39(0.05)
24.00(3.26)
2.05(0.24)


[Al A ,][A1A2 A ]



[Al A ,][A1A2 A ]



[Al A ,][A1A2 A ]


p = 0.88
0.2 =7e-3


0.88(0.01)
Te-3(7e-4)


The correct haplotype model and the corresponding estimates are indicated in boldface.










2002; Gabriel 2002). The recently developed haplotype map or HapMap (The

International HapMap Consortium 2003) provides an invaluable resource for

understanding the structure, organization and function of the human genome.

The understanding of the first two aspects, genome structure and organization,

have been less problematic in part because fewer statistics are used, but the

association between specific genomic sites and disease risk or drug response is a

pressing challenge in current pharmacogenetic and pharmacogenomic studies. The

model presented here, aimed at detecting specific DNA sequence variants for drug

response, represents a timely effort to accelerate the research at identifying genes of

interest.

The presented model is founded on recently discovered tag SNPs in the

genome, and allows for a fast scan for the association between variation in DNA se-

quence and traits (Patil et al. 2001; Dawson et al. 2002; Gabriel 2002). This model

has three advantages. First, it solidifies the genetic basis for quantitative variation

by directly characterizing specific DNA sequences predisposed to drug response.

The traditional statistical models for genetic mapping attempt to postulate the

position of hypothesized QTL that are linked with known markers genotyped from

the genome. The QTL detected from these models are regarded as "hypoth.~ -i.. 4

because it is not possible to know their DNA sequences and, therefore, physiological

function. As opposed to the traditional "indirect" approach, this model presents

a "direct" approach. At present, the utility of the direct approach is limited to

sequencing functional parts of candidate genes with known biochemical or physio-

logical function. With the release of HapMap, this model makes the direct approach

both useful and efficient in searching for causal variants throughout the whole

genome. Second, this model is statistically simple and computationally fast. The

most difficult part of the model estimation is constructing diphi v1 ipe configurations

for heterozygous 7.~ nc.1vipes at two or more SNPs. The estimation of population










genetic parameters is based on a multinomial likelihood function of the observed

:- Ir..i vpe data, whereas the estimation of quantitative genetic parameters based

on a mixture-based likelihood function including different diplotypes. These two

likelihood function can he easily integrated to a unified estimation framework

implemented with the E1\ algorithm.

Finally, this model is robust and flexible, and able to accommodate different

genetic and experimental settings. Results from the simulation study indicate

that the association between DNA sequence and phenotype can he well detected

when the trait has a modest heritability level (0.14) or a modest sample size (107)

is used. This model can also obtain fairly precise estimation of parameters when

diplotypes display overdominance in the situation with modest heritability and

sample size. The specific utility of this model to a real example from a genetic

study leads to the successful detection of a DNA sequence (haplotype) at codons

16 and 27 ... n..~i eped within the /32AR candidate gene for its significant impact on

response in heart rate to dobutamine. This haplotype, composed of the Gly16 form

of codon 16 and the Glu27 form of codon 27, tends to increase heart rate when it

is combined with itself or any other haplotypes, and account for about 1 1' of the

total observed variation in drug response.

Although the simulation and example were based on 2-SNP an~ lli-- the

sequencing model used was developed to allow for the detection of sequence variants

involving any number of SNPs within a haplotype block. In addition to its use in

studying genetic associations in natural populations, the sequencing model can he

extended to study the genetic factors contributing to variation in drug response

in controlled crosses such as the backcross or F2 aS used in mouse. It can also be

modified to estimate the effects of sequence-sequence interaction on drug response.

It is possible that a haplotype within a candidate gene interacts with haplotypes

from other candidate genes. The characterization of specific DNA sequence variants









for drug response should allow the development of tests to predict which drugs or

vaccines would be most effective in individuals with particular genotypes for genes

affecting drug metabolism.















CHAPTER :3
A .JOINT MODEL FOR SEQUENCING DRITG EFFICACY AND TOXICITY

3.1 Introduction

The administration of a specific drug to patients can produce two different

responses, desired therapeutic effects (efficacy) and adverse effects (toxicity).

Evidence is increasing for observed influences of genetic differences on these two

responses (reviewed in Evans and .Johnson 2001; .Johnson and Evans 2002; Evans

and McLead 200:3; Weinshilbount 200:3). However, the genetic control of both

efficacy and toxicity is typically complex, with multiple genes interacting with

various biochemical, developmental and environmental factors in coordinated r- 0-~

to determine the overall phenotypes (.Johnson 200:3; Watters and McLeod 200:3).

With the advent of recent genomic technologies, inter-individual differences in drug

response can now he explained by DNA sequence variants in genes that encode

the nietabolism and disposition of drugs and the targets of drug therapy (such as

receptors) (Evans and Relling 1999; McLeod and Evans 2001). To comprehensively

understand the genetic hases of efficacy and toxicity and, ultimately, design indi-

vidualized medications with nmaxiniun favorable effects and nxininiun unfavorable

effects, approaches must he developed in which new specific genes for each response

can he identified.

Two approaches that have been developed to detect genes for a complex trait.

The first approach is the indirect inference of causal genetic loci or quantitative

trait loci (QTL) hased on their co-segregating markers (Lander and Botstein 1989;

Wu and Casella 2005). The QTL detected front this approach is hypothetical

whose DNA structure and organization is unknown. High-throughput technologies










of single nucleotide polymorphisms (SNPs) have provided a powerful method

for sequencing candidate genes that have been known to affect complex diseases

or drug response. The recent development of the haplotype map or Haphiap

constructed by .Ilr1..vsymous SNPs (The International Haphiap Consortium 2003)

has made it possible to genome-wide scan for the existence and distribution of

functional SNPs based on association analysis and narrow down the genomic

regions that harbor causal SNPs. Motivated by these developments, Liu et al.

(2004) proposed a second approach that can directly associate DNA sequence

variants with the phenotypic variation. This approach has power to detect the DNA

sequence where individuals differ at a single DNA hase.

Unlike usual complex traits, drug response has some dynamic characteristic

in which individuals respond to varying drug dosages or concentrations. Drug

response can he therefore regarded as function-valued or longitudinal traits. The

genetic architecture of function-valued traits can he studied using the marker-

hased functional I'nll.:,:lln model, developed by R. Wu and colleagues (il .. et

al. 2002; Wu et al. 2002b, 2003, 2004a, 2004h). Functional mapping identifies

dynamic QTL responsible for a biological process that need be measured at a

finite number of time points. In modelling functional mapping, fundamental

principles behind biological or biochemical networks described by mathematical

functions are incorporated into a QTL mapping framework. Functional mapping

estimates parameters determining shapes and functions of a particular biological

network, rather than directly estimates gene effects at all possible points within the

network. Because of the connection of these points through mathematical functions,

functional mapping strikingly reduces the number of parameters to be estimated

and, hence, di pl .ni~ increased statistical power.

Taking advantages of sequence-based association studies and functional map-

ping, I attempt to propose a new model that can characterize the DNA sequence









structure of drug response and compare the genetic differences between efficacy

and toxicity at the single DNA base level. In the next sections, I first introduce

basic principles for sequence and functional mapping and then formulate a unifying

likelihood function for the detection of specific DNA sequence variants that deter-

mine drug efficacy and toxicity. These will be followed by the investigation of the

statistical properties of this model through extensive simulation studies.

3.2 Basic Principle For Sequence Mapping

Traditional QTL mapping studies the relationship between putative QTL

:- Irled vpes and h in II! v! rpes, whereas sequence mapping associates configurations of

SNP ;_ n..~ivpes (i.e., diple-lvpes)s with phenotypes. Sequence mapping relies upon

the the characterization of SNPs from the entire human genome. Recent studies

have shown that the human genome has a haplotype block structure (Patil et al.

2001; Gabriel et al. 2002), such that it can be divided into discrete blocks of limited

haplotype diversity. In each block, a small fraction of SNPs, referred to as 1.

SNPs", can be used to distinguish a large fraction of the haplotypes.

For simplicity, I first consider two SNPs within a haplotype block that are

co-segregating with the linkage disequilibrium of D in a human population at

Hardy-Weinberg equilibrium. Each SNP has two alleles 1 and 2 with the relative

proportions of pil and p l) as well as pt2) and p 2); TOSpectively, where the super-

script stands for thle identificationl of SNP and p~l () = 1 anld p(2) 2") =

These two SNPs form 4 possible haplotypes 11, 12, 21 and 22 whose frequencies are

expressed as

plrZr = pr p +(1rzrD

where rl, T2 1, 2 denote the alleles of the twvo SN~rs, respectively C1, CE=

perr, = 1 (Lynch and Walsh 1998). If the haplotype frequencies are known, then

the allelic frequencies and linkage disequilibrium, arrmy. Al by the population genetic

parameter vector 62, = {pj@:, p ,D}, can be solved with the above equation.









The random combination of maternal and paternal haplotypes generates

10 distinct diplotypes expressed as [11] [11], [22] [22] which are sorted into

9 genotypes 11/11, 22/22 (Table 3-1). The double heterozygotic genotype

12/12 contains two possible diplotypes [11] [22] and [12] [21]. I use P i r]

(= prlnrpPrgry) and Przrt/rrg to denote the diplotype and 7.~ nc.1vipe frequencies,

respectively, and neri/r~lr'2 to denote the observations of various genotypes (0),
where m and p describe the maternal and paternal origins of haplotypes and

1 < rl < r', < 2, 1 < r2 < r' < 2. The frequencies and observations of all genotypes,

except for genotype 12/12, are equivalent to those of the corresponding diplotypes.

I intend to associate diplotypes with inter-patient variation in a quantitative

trait based on observed phenotypic values (Y) assumed to be normally distributed

and SNP genotypes (0) assumed to be multinomially distributed. Without loss of

generality, I assume that haplotype 11 is different from the rest of the haplotypes,

cumulatively expressed as 11, in tri r-~-;ingl~ an effect on the phenotype. I call such

a distinct haplotype 11 the reference haplotype. The reference and non-reference

haplotypes generates three combinations called the composite .i. ,:olor.~ The

:- Ir..i vpic means of the composite genotypes, py (j = 2 for [11] [11], 1 for [11] [1]

and 0 for [11][11]), and common residual variance within the composite ._ nc.1vipe,

O2, that belong to quantitative genetic parameters are arrmy. A by n2, = {pyl, O2.2

I constructed two log-likelihood functions, one in a multinomial form and the

other in a mixture model form, to estimate the population and quantitative genetic

parameters that are, respectively, expressed as


log L(O2,|Q) = Constant (3.1)

+2nlllll log pll + ull/12 log(2iirpl2) + 2n11/22 log pl2

+812/1 log(2pllpp21 + [ 12/12 log(2pllp22) (1 12/12nll l log(2pl2921~

+n12/22 log(2pl2922) + 2n22/11 log p21 + 22/12 log(2p21922) + 2n22/22 log p22,









and

log L(O,, ,|}, 0) = logf2,
i= 1

+ log ft(ys)j
i= 1


C log fo(vs)
i= 1

+ log[w~fr(ys) + (1 i) fo(yi)], (3.2)
i= 1



p11922
w = (3.3)
p11922 + 12921

and

fy (yi) = exp-.
o-2;( 2o-2

I derived a closed-form solution for estimating the unknown parameters with

the EM algorithm. The estimates of haplotype frequencies are based on the log-

likelihood function of equation (3.1), whereas the estimates of composite ._ I n..i vpic

means and residual variance are based on the log-likelihood function of equation

(3.2). These two types of parameters can be estimated using an integrative EM

algorithm.

In the E step, the expected number (wi) of dfilh~i vpe [11] [22] for a double

heterozygous genotype is estimated using equation (3.3), whereas the posterior

probability (IL) with which subject i carrying the double heterozygous ._ I nc.1vpe is

diplotype [11] [22] is calculated by



w fl (yi)
nI = .(3.4)
w~fl(yi) + (1 wi) fo(Yi)'




















Table 3-1: Possible diplotype configurations of nine genotypes at two SNPs which affect drug efficacy and toxicity


Relative diplotype


Haplotype composition


Genotypic mean vector


Genotype Diplotype

11/11 [11] [11]

11/12 [11][12]

11/22 [12 [12]

12/11 [11 [21]

12/12 [11~a]i] [22]
ia/[12 [21~a]

12/22 [12] [22]

22/11 [21] [21]

22/12 [21 [22]


Diplotype frequency freq. within genotypes 11


12 21 22 Observation Efficacy x Toxicity z


= p ,
2piipl2


2piip21

= 2pir p22
= 2pl2921

2pl2922

= P1
2p21922

= 2 a


P[11][12] =

P[12] [12]



P[11] [21]

P[11] [22]
P[12] [21]


1




r;i7
1--Zi7

1




1


n11/12

811/22
n I 711


m22

mit

mot

mlz


1 1/ \
2Zw (\1 -w) (\1 -w) w i


"22/11


mot

mot

mot

mot


=pllp ps where pn,, pl2, p21 and p22 are the haplotype frequencies of [11], [12], [21], and [22], respectively.


nl~/ lx mi02o
al2/12i








39

In the M step, the probabilities calculated in the previous iteration are used to

estimate the haplotype frequencies using

2nlllll + 11/12 + 12/11 Z712/12
ply (3.5)
2n
2n11/22 81/1 n12/22 + 812/12 -~i)la l
912 (3.6)
2n
2n22/11 n12/11 822/1 Z7112/12
2n
2n22/22 82/1 n12/22 12/12a+ i~ia
2n

The quantitative genetic parameters are estimated using

i=1 Yi
1-2 = ~,(3.9)
n11/11

C~1i=


o = (3.11)



i= 1 i= 1 i= 1


+ E [n(y il)"(- th)2ui 0] ), (3.12)
i= 1

where n = n11/12 + 12/11 and n = n11/22 81/2 22/11 n~ll+ ~~ 82/1 n22/22. IteratiOUS

including the E and M steps are repeated among equations (3.4)-(3.12) until the

estimates of the parameters converge to stable values.

3.3 An Integrative Sequence and Functional Mapping Framework for
Drug Efficacy and Toxicity

3.3.1 The Multivariate Normal Distribution

Efficacy and toxicity describe how patients respond to different doses or

concentrations of drugs. Statistically, these represent a longitudinal problem whose

underlying genetic determinants can be mapped using the functional mapping

strategy. Here, I integrate the ideas for sequence mapping and functional mapping









to directly characterize DNA sequence variants that are responsible for efficacy and

toxicity processes.

Consider the same sample as described above in which population genetic

parameters for SNPs have been defined. For each patient, drug effects are measured

at C hallmark dose or concentration levels. While a drug is expected to display

favorable effects, it may also be toxic. In this study, some typical physiological

parameters reflecting both efficacy and toxicity are longitudinally measured. Let

xi = [xi(1), xi(C)] be the efficacy effect vector and zi = [ze(1), ze(C)] be

the toxicity effect vector for a drug administrated to patient i. I use yi = (xi, zi)

to denote the joint vector combining these two types of drug response. Because the

phenotypic measurements of drug efficacy and toxicity are continuously variable, it

is reasonable to assume that y follows a multivariate normal distribution, as used in

general quantitative genetic studies (Lynch and Walsh 1998).

To jointly map the efficacy and toxicity, two log-likelihood functions described

in equations (3.1) and (3.2) should be constructed for observed SNP marker and

longitudinal phenotypic data. The only difference from single trait mapping is

that the log-likelihood function (3.2) needs to incorporate a multivariate normal

distribution for patient i who carries composite genotype j, expressed as


fj (yi; mj, C) = ()C 12exp 2 (yi mj )E- (yi m3) (.3


where y, = [xi(1), xi(C), z (1), zi(C)] is a 2C-dimensional vector

of observations for efficacy and toxicity measured at C dosages and mj

[pyz(1),, -,pz(C), Iyz(1),., -,pz(C)] is a vector of expected values for com-

posite genotype j at different doses. At a particular dose c, the relationship between

the observation and expected mean can be described by a linear regression model,


xs(c) (ispy~c) + ei,(c), (3.14)
j=1










zi(c) i zc i>) (3.15)
j= 1

where (sy and (sy are the indicator variables denoted as 1 if a particular composite

:- n..~i vpe j is considered for individual i and 0 otherwise, J is the number of

composite genotypes and ei,(c) and eiz(c) are the residual errors that are iid
normal with the mean of zero and the variances of a2/c) and ej2c), repec~ftively.

The errors at two different doses, cl and c2, arT COrrelated with the covariances

of Uz(c1, C2) foT efflCaCy, Uz C1, C2) foT tOXicity and az(c) and az(cl, c2) between

efficacy and toxicity. All these variances and covariances comprise the structure of

the (co)variance matrix E in equation (3.13), expressed as


E, = z (3.16)



where Ez and Cz are composed of aj(c) and a,(cl, c2), and a 2(c) and az(cl, c2)

(1 < cl, c2 I ), TOSpectively; and Ezz and Ezz are composed of az(c) and

Uzz(clz, c2z) (1 I Clz C2z < C), and azz(c) and az(ciz, ca,) (1 I Cl,~z C )m ,

respectively.

To solve the likelihood functions implemented with response data measured

at multiple doses, one can extend the traditional interval mapping approach to

accommodate the multivariate nature of dose-dependent traits. However, this

extension is limited in two aspects: (1) Individual expected means of different

composite :-- nod vipes at all dose and all elements in the matrix E need to be

estimated, resulting in substantial computational difficulties when the vector and

matrix dimension is large; (2) The result from this approach may not be clinically

meaningful because the underlying medical principle for drug response is not

incorporated. Thus, some clinically interesting questions cannot be asked and

answered.









3.3.2 The Mapping Framework

Wu et al. (2002b, 2004a, 2004b) and Ma et al. (2002) have constructed a novel

statistical framework for functional mapping of QTL that affect growth curves.

Here, this framework is extended to jointly model two different longitudinal traits,

aimed at simultaneously characterizing the genetic variants responsible for efficacy

and toxicity. Two tasks are taken within this framework by modelling the mean

vector and the covariance matrix.

Modelling the mean vector: The dose-dependent expected values of

composite genotype j can be modelled for drug efficacy by the sigmoid Emax

model (Giraldo 2003) and for drug toxicity by the power function (jlcClish and

Roberts 2003). The Emax model postulates the following relationship between drug

concentration (C) and drug effect (p):

E, CHj
I-lj(C) = Eoj + a"X" for composite genotype j, (3.17)
EC., "+ CHj

where Eo is the constant or baseline value for the drug response parameter, Emax is

the .I-i-inpul'tic (limiting) effect, ECso is the drug concentration that results in 50'.~

of the maximal effect, and H is the slope parameter that determines the slope of the

concentration-response curve. Drug toxicity is described by the power function of

dose expressed as


IPjz(C) = CqC 9, for composite genotype j, (3.18)


where p determines the shape of the dose-response relationship and a~ adjusts the

dose-related gain to which that shape conforms.

It is possible that efficacy and toxicity have different reference haplo-

types, so their composite -c1 1.11irpes should be treated differently. As a result,

equations (3.17) and (3.18) together contain 6 curve parameters for a defined

composite -c11 Ir(irpe jl for efficacy and j2 foT tOXiCity, Which are an1 li-. I by









Om,~j = (Eoyz, Emaxyz, EC-,,.,, Hj,, ass,, pjz). If different composite genotypes

have different combinations of these parameters, this implies that this sequence

ptI i-s a role in governing the differentiation of efficacy and toxicity. Thus, by testing

for the difference of Omlj among different 7.~ nc.1vipes, I can determine whether

there exists a specific sequence variant that confers an effect on these two drug

responses.

Modelling the structure of the covariance matrix: ?1 dIv: statistical

approaches have been proposed to model the structure of the covariance matrix

for longitudinal traits measured at multiple time points (Diggle et al. 2002). Here,

the matrix-structuring models will be derived on the basis of a commonly used

approach, structured antedependence (SAD) (Zimmerman and N~inez-Ant6n 2001).

According to the SAD model, an observation at a particular dosage c depends

on the previous ones, with the degree of dependence decaying with time lag. For

drug efficacy and toxicity, the dose-dependent residual errors for subject i described

by Equations (3.14) and (3.15) can be expressed, in terms of the first-order SAD

(SAD(1)) model, as


eiz (c) = #zei,(c 1) + cpeiz(c 1) + eiz(c),

eiz (c) = #zeiz(c 1) + (Peiz(c 1) + eiz(c), (3.19)


where 4, (or #z) and cp (or zp) are the unrestricted antedependence parameters

induced by trait x (or z) itself and by the other trait z (or x), respectively, and

exi(c) and exi(c) are the !~~~i.- II !. su" errors for the two traits, respectively, normally

distributed as NV(0, v,2(c)) and NV(0, v,2(c)) (N~inez-Ant6n and Zimmerman 2000,

Zimmerman and N~inez-Ant6n 2001). The bivariate SAD(1) model for subject i

described by equation (3.20) can be expressed in matrix notation as


ei = #- Eg, (3.20)










where ei = {e (1), e (C)}


-- 0

-- 0

- 0

-- 0



--- 1


1 0

1

0 o

0 0



0 0


0

0 -

0

0



0

under the bivariate


0

-- 0

-- 0

-- 0



-- 0


(21) can be


00

0


\u u u --

The covariance matrix of es

obtained as


SAD(1) model


E = Ee1C( -) ,


(3.21)


where


, s= e( 1),- -, () } n








45

v 2(1) 0 0 --- 0v 2(1) 0 0 0





0 0 0 -~ c i C 0 0 0 v(C)

and

vauzp(1) 0 0 0





0 0 0 vyz~p(C)

In this modelling, the innovative variance is assumed to be constant over different

dosages. As shown by Jaffriizic et al. (2003), the SAD(1) model with this assump-

tion can still allow for both the variance and correlation to change with the dose

level. In addition, I assume that the correlation (p) in innovative errors between the

two traits x and z is stable over dosage. With these assumptions, I need to estimate

an array of parameters contained in 02, = (v,2, v,2, 4,, #z, cp, zp, p) that model the

structure of the covariance matrix under the SAD model.

In the context of the bivariate SAD model, the cross-correlation functions

can be proven to be .-i-mmetrical, i.e., Corrz(ciz, Caz) COTr,,zz, Ca,C2). This

favorable feature of the bivariate SAD model makes it useful for understanding the

genetic correlation between different traits. In practice, innovative variance and

correlation can be modelled by a polynomial and exponential function, respectively

(N~inez-Ant6n and Zimmerman 2000).

3.3.3 Computational Algorithm

I implemented the EM algorithm, originally proposed by Dempster et al.

(1977), to obtain the maximum likelihood estimates (MLEs) of three groups of

unknown parameters in the integrated sequence and function mapping model,









that is, the marker population parameters (02,), the curve parameters (Omyjlzy)

that model the mean vector, and the parameters (02,) that model the structure of

the covariance matrix. These unknowns are denoted by 02 = (02,, Omjlz 0,). A

detailed description of the EM algorithm was given in Wu et al. (2002b, 2004b) and

Ma et al. (2002).

As described for single trait mapping, I implement the E step to calculate the

expected number (wi) of diplotype [11] [22] contained in the double heterozygote

12/12 and the posterior probability (nI) of double heterozygotic patient i who

carries dipk J vpep [11] [22]. In the M step, I use the calculated wi and nIs values to

estimate the haplotype frequencies using equations (3.5)-(3.8). But in this step, I

encounter a considerable difficulty in deriving the log-likelihood equations for Omy,jlj

and 02, because they are contained in complex nonlinear equations. Zhao et al.

(2004) implemented the simplex method as advocated by Nelder and Mead (1965)

to the estimation process of functional mapping, which can strikingly increase

computational efficiency. In this chapter, the simplex algorithm is embedded in the

EM algorithm above to provide simultaneous estimation of haplotype frequencies

and curve parameters and matrix-structuring parameters.

3.3.4 Model for an Arbitrary Number of SNPs

The idea for sequencing drug response has been described for a two-SNP

model. It is possible that the two-SNP model is too simple to characterize genetic

variants for variation in drug response. I have extended this model to include an

arbitrary number of SNPs whose sequences are associated with drug response

variation. A key issue for the multi-SNP sequencing model is how to distinguish

among 2'-1 different diplotypes for the same genotype heterozygous at r loci. The

relative frequencies of these diplotypes can be expressed in terms of haplotype

frequencies. The integrative EM algorithm can be emploi-v I to estimate the MLEs

of haplotype frequencies. Bennett (1954) provided a general formula for expressing









haplotype frequencies in terms of allele frequencies and linkage disequilibria of

different orders. The MLEs of the latter can be obtained by solving a system of

equations .

3.4 Hypothesis Tests

Different from traditional mapping approaches, the functional mapping for

function-valued traits allows for tests of a number of biologically or clinically mean-

ingful hypotheses. These hypotheses include tests for the existence of significant

DNA sequence variants, for the genetic effect on the maximal (.l-i-mptotic) effect

(Emax), for the drug concentration that results in 50'; of the maximal effect, and for

the slopes that determines the steepness of the concentration-response curve.

The existence of specific genetic variants affecting drug efficacy and toxicity

can be tested by formulating null and alternative hypotheses,


Ho : Omy,jl Om,, 31, 32 = 2, 1, O
(3.22)
H1 : at least one of the qualities above does not hold,

where Ho corresponds to the reduced model, in which the data can be fit by a

single drug response curve, and H1 corresponds to the full model, in which there

exist different dynamic curves to fit the data. The test statistic for testing the

hypotheses in equation (3.23) is calculated as the log-likelihood ratio (LR) of the
reduced to the full model:


LR = -2 [log L(O2 y, 0) log L(O2 y, 0)], (3.23)


where 02 and 02 denote the MLEs of the unknown parameters under Ho and

H1, respectively. The LR is .I-i-n!!111'i ;cally X2-distributed with 12 degrees of

freedom. An empirical approach for determining the critical threshold is based on

permutation tests, as advocated by Chat, 1..! I and Doerge (1994). By repeatedly

shuffling the relationships between marker genotypes and phenotypes, a series of the









maximum log-likelihood ratios are calculated, from the distribution of which the

critical threshold is determined.

I can also test for the significance of the genetic effect on drug efficacy or

toxicity at a particular concentration level (c*) of interest, expressed as


Ho : 1-j1z(c*) = -l(c*), Iyjz (c*) = pz(c*)
(3.24)
H1 : At least one of the eqaulities above does not hold,

which is equivalent to testing the difference of the full model with no restriction and

the reduced model with a restriction in Ho. Similar restrictions can be taken to test

the genetic effect on individual curve parameters, such as Emax, ECso, H, a~ and P.

The tests of these parameters are important for the design of personalized drugs to

control particular diseases.

3.5 Results

I perform Monte Carlo simulation experiments to examine the statistical

properties of the model proposed for genetic mapping of drug efficacy and toxicity.

The simulation will be based on the bivariate SAD(1) model (equation 3.21). For

computational simplicity, I only considers the antedependence parameters induced

by the trait itself, i.e., setting cp, = cpz = 0. I randomly choose 200 individuals from

a human population at Hardy-Weinberg equilibrium with respect to haplotypes.

Let us consider two SNPs with allele frequencies and linkage disequilibrium given

in Table 3-1. The diplotypes derived from four haplotypes at these two SNPs affect

drug efficacy and toxicity. Here I assume that these two types of drug response have

different reference haplotypes, i.e., 11 for efficacy and 12 for toxicity. Thus, each

drug response corresponds to a different set of three composite genotypes.

The efficacy (Eo, Emax, ECso, H) and toxicity curve parameters (a~ and P) for

the respective three composite genotypes, given in Table 3-1, are determined in

the ranges of empirical estimates of these parameters from pharmacological studies










(Sowinski et al. 1995; McClish and Roberts 200:3). I use a general clinical design,

with drug effects measured at 9 different concentrations of drug, 0, 5, 10, 15, 20, 25,

:30, :35 and 40, expressed as c = 1, 2, 9 (Figure :3-1), for this simulation study.

Using the genetic variance due to the composite genotypic difference in the area

under curve, I calculate the residual variances under different heritability levels

(H2 = 0.1 and 0.4). These residual variances, plus given residual correlations, form

a structured residual covariance matrix E (equation :3.22). The phenotypic values of

drug effect for 200 random patients are simulated by the suninations of genotypic

values predicted by the curves and residual errors following multivariate normal

distributions, with MVVN(0, E).

The population genetic parameters of the SNPs can he estimated with rea-

sonably high precision using the closed-fornt solution approach (Table :3-2). The

estimates of these parameters are only dependent on sample size and are not related

to the size of heritability. Figure :3-1 illustrates different forms of the drug efficacy

and toxicity curves front the three composite genotypes with a comparison between

the hypothesized and estimated curves. The estimated curves are consistent with

the hypothesized curves, especially when the heritability is larger (0.4), -11__- -r;a:

that the model can provide reasonable estimates of drug response curves. The

parameters for the efficacy and toxicity models of each composite genotype can he

estimated accurately and precisely (Table :3-2). As expected, the estimation preci-

sion (assessed by square root of AISEs) increases remarkably when the heritability

increases front 0.1 to 0.4. The estimates of the SAD parameters that model the

structure of the covariance matrix E also display reasonably high precision (Table

:3-2). But it seems that their estimation precision is independent of the size of

heritability.

In each of 100 simulations, I calculate the log-likelihood ratios (LR) for the

hypothesis test of the presence of a genetic variant affecting both drug responses.










Table 3-2: Maximum likelihood estimates of SNP population genetic parameters (allele
frequencies and linkage disequilibrium), the curve parameters and matrix-structuring pa-
rameters for efficacy and toxicity responses. The numbers in parentheses are the square
root of MSEs of the estimates.


Composite
genotype


Heritability


True value


Parameters


0.4

0.60(0.02)
0.70(0.02)
0.08(0.02)


0.27(3e-3)
0.25(3e-3)
0.23(4e-3)

0.92(0.08)
0.84(0.06)
0.79(0.11)

24.78(3.66)
25.17(4.98)
26.26(10.71)

1.10(0.07)
1.04(0.05)
0.93(0.08)


Population genetic parameters
0.60 0.60(0.02)
0.70 0.70(0.02)
0.08 0.08(0.01)


Curve parameters:
[11] [11] 0.27
[11] [H] 0.25
[11] [11] 0.23


Efficacy
0.27(0.01)
0.25(0.01)
0.23(0.01)

0.93(0.11)
0.85(0.12)
0.80(0.16)

25.57(6.14)
25.46(7.94)
27.53(14.67)

1.10(0.09)
1.03(0.11)
0.94(0.12)


Eo



Emax


[11] [11]
[11] [11]
[H] [1]

[11] [11]
[11] [11]
[H] [1]

[11] [11]
[11] [11]
[11] [11]


0.90
0.85
0.80


E1so


24.00
25.00
26.00

1.10
1.00
0.90


Curve parameters:
[12] [12] 0.10
[12] [12] 0.13
[1] [12] 0.16


SToxocity
0.10(0.02)
0.13(0.01)
0.16(4e-3)

0.60(0.08)
0.50(0.01)
0.40(8e-3)

parameters
0.28(0.04)
0.39(0.03)
0.70(0.01)
8e-3(3e-4)

3e-3(le-4)


0.10(0.01)
0.13(2e-3)
0.16(2e-3)

0.60(0.04)
0.50(6e-3)
0.40(4e-3)


0.29(0.02)
0.40(0.02)
0.70(0.01)

le-3(5e-5)

6e-3(2e-5)


[12 [12]
[12 [12]
[12 [12]


0.60
0.50
0.40


Matrix-structuring
0.30
0.40
0.70
8el-3
lel-3
3e-3
6e-4










The LR values in each simulation under both heritability levels are strikingly higher

than the critical threshold estimated from 100 replicates of simulations under the

null hypothesis that there is no drug response-associated genetic variant. This

-II- -_ -r- that this model has enough power to detect the genetic variant under given

SNPs, curve and niatrix-structuring parameters for the simulation. With the same

set of simulated data, the power to detect significant genetic variants is reduced by

~1'.when drug efficacy or toxicity is modelled separately.

I carry out an additional simulation study to examine the statistical behavior

of this model when :3 different SNPs are used. This :3-SNP model uses the same

parameters as assumed under the 2-SNP model, except for the inclusion of more

SNPs. The results front the :3-SNP model are broadly concordant with those front

the 2-SNP model (results not given), although more population genetic parameters

need to be estimated in the former than latter.

3.6 Discussion

Drug response is complex in nature. First, no drug is universally effective;

efficacy rates for drug therapy of most diseases ranges front 25 to M'I (Spear et

al. 2001). Second, all drugs that produce an efficacious response may also produce

adverse effects (Johnson 2002). Third, both drug efficacy and toxicity that vary in

human populations are determined by multiple genetic and environmental factors

that interact with one another in complicated v- 0-<~ (Watters and 1\kleod 200:3). It

is possible that the genetic polymorphisms associated with a favorable response to

therapy may or may not he associated with toxicity risk (Nehert 1997; Wilson et al.

2001).

The complexity of drug response is enforced by its dynamic characteristic with

high dimension and correlated structure. The genetic analysis of drug response,

therefore, needs more sophisticated statistical tools that have capacity to extract

useful information front longitudinal data. In this chapter, I have proposed a









novel statistical model for characterizing the genetic variants that govern drug

response in a human population. This model is constructed with a finite mixture

model framework founded on the tenets of the sequence (Lin et al. 2005) and

functional mapping (11 I et al. 2002; Wu et al. 2002b, 2004a, 2004h). This model

has particular power to discern the discrepancy between the genetic mechanisms

underlying drug efficacy and toxicity.

Compared to traditional genetic mapping approaches, this model is both

genetically and statistically merited. It offers a direct tool to detect DNA sequences

that code drug response. The haplotype map or Haphiap constructed from single

nucleotide polymorphisms (SNPs) (The International Haphiap Consortium 200:3)

makes the application of this model practically possible. In particular, I adopt

clinally meaningful mathematical functions (Giraldo 200:3; McC'!-l- and Roberts

200:3) when modelling dose-dependent drug response. This has two advantages.

First, it facilitates statistical analysis and strengthens power to detect significant

genetic variants because fewer parameters are needed to be estimated, as opposed

to traditional multivariate analysis. Second, the results it produces are close to

biological realm given that the mathematical functions used are founded on a firm

understanding of pharmacology (reviewed in Derendorf and Meibohm 1999). The

statistical power of this model is further increased hv the attempt to structure the

covariance matrix with autoregressive models (Diggle et al. 2002).

I have performed different simulation studies to investigate the statistical

behavior of this model. The results -II__- -r that it can provide accurate and precise

estimates of the response curves for both efficacy and toxicity even when the

heritability is modest. Different drug response curves, as simulated for different

composite genotypes (Figure :3-1), can offer scientific guidance for determining the

optimal dosage that balances favorable efficacy and unfavorable toxicity based on

individual's genetic background. It should be noted that this model was derived










on the basis of a simple clinical design in which a single cohort is assumed, with

equally concentration intervals for every patient. This model can he extended to

a case-control study by allowing haplotype frequencies and haplotype effects to be

different between the case and control groups. A series of hypothesis tests regarding

between-group differences can he formulated to detect specific haplotypes that are

responsible for drug response. 1\y model can also be extended to consider various

measurement schedules, in particular with uneven concentration lags varying from

patient to patient. For these irregular measurement schedules, it is needed to

formulate individualized likelihood functions when the mean vector and covariance

matrix are modelled (see N~inez-Anton and Woodworth 1994).

In coupling with continuing advances in molecular biological technology and

the availability of a complete reference sequence of the entire human genome (Patil

et al. 2001; Dawson et al. 2002; Gabriel et al. 2002), this model will assist in the

discovery and characterization of the genes that influence drug response, and lead

to a better understanding of how genes function and the subsequent development of

new approaches to the diagnosis and treatment of common diseases by medications.

Once the network of genes that govern drug responses in humans is defined, it will

then he possible to more accurately optimize drug therapy based on each patient's

ability to metabolize, transport and respond to medications.








54






1001 1 100
A B
80~ 80

60 60

40 / 40
W 1-
20 20

0 0
0 10 20 30 40 0 10 20 30 40
Drug concentration (mcg) Drug concentration (mcg)

100, 1 100

80 ~ c- 80

60 60

40 40
W 1-
20 20

0 0
0 10 20 30 40 0 10 20 30 40
Drug concentration (mcg) Drug concentration (mcg)



Figure :31: Estimated response curves (dash) each corresponding to one of three
composite genotypes under the heritability (H2) of 0.1 (A and B) and 0.4 (C and
D), in a comparison with the hypothesized curves (solid) used to simulate indi-
vidual curves for drug efficacy (A and C) and toxicity (B and D), respectively.
Nine different concentrations, 0, 5, 10, 15, 20, 25, :30, :35 and 40, at which responses
are measured, are used. The consistency between the estimated and hypothesized
curves el__o- -;- that this model can provide the precise estimation of the genetic
control over response curves in patients. The honiozygote composed of two ref-
erence haplotypes is [11] [11] for efficacy and [12] [12] for toxicity (blue), the het-
erozygote composed of one reference and one non-reference haplotype is [11] [11]
for efficacy and [12] [12] for toxicity (red), and the honiozygote composed of two
non-reference haplotypes is [11] [T11 for efficacy and [12] [12] for toxicity (green).















CHAPTER 4
MODEL FOR DETECTING SEQUENCE-SEQUENCE INTER ACTIONS FOR
COMPLEX DISEASES

4.1 Introduction

Interactions between different genes, coined the epi~stasi~s, have long been recogf-

nized to pIlai- a central role in shaping the genetic architecture of a quantitative

trait (Whitlock et al. 1995; Wolf et al. 2000). Recent genetic studies from vast

quantities of molecular data have also indicated that epistasis is of paramount

importance in the pathogfenesis of most common human diseases, such as cancer

or cardiovascular disease (Moore 2003). The evidence for this is the nonlinear

relationship detected between .-- n..~i ipe and ph.! Mul ~ irpe. The decipherment of in-

terconnected networks of genes and their associations with disease susceptibility

has become a pressing demand for a detailed understanding of the genetic hasis for

disease processes.

The most common and powerful approaches for detecting genome-wide

epistasis are based on genetic mapping that associates phenotypic variation of

a trait with a linkage map constructed by polymorphic markers (Lander and

Botstein 1989). Significant associations imply the existence of the underlying loci

(called quantitative trait loci or QTL) that are responsible for variation in the

trait. Epistatic QTL are considered to occur if the effect of one QTL depends on

the expression of other QTL. While traditional genetic mapping can only make

an indirect inference about QTL actions and interactions, a newly developed

mapping approach hased on the haplotype map (Haphiap) constructed from single

nucleotide polymorphisms (SNPs) (The International Haphiap Consortium 2003)

can directly characterize specific DNA sequence variants for a phenotype (Liu et al.










2004). Beyond hypothesized chromosomal segments identified by QTL mapping,

this so-called sequence mapping approach, relied upon the the characterization of

SNPs from the entire human genome, can probe concrete nucleotoxic sites, i.e.,

quantitative trait nucleotides (QTN), that contribute to variation in a quantitative

trait.

In this chapter, I extend the previous statistical model to determine the

epistasis between specific DNA sequences responsible for the phenotypic variation

of disease risk. This extended model is derived through the as~ lli--; of pair-wise

multilocus haplotypes constructed by a finite number of tag SNPs. I derived a

closed form solution for estimating the action and interaction effects of haplotypes,

haplotype frequencies, allele frequencies and the degrees of LD of various orders

among tag SNPs underlying the disease. Simulation studies were performed to test

the statistical behavior of this haplotype-based epistatic mapping model. I used a

worked example for a human obesity study to demonstrate the usefulness of the

model.

4.2 The Model

4.2.1 Notation

Consider a natural human population at Hardy-Weinherg equilibrium from

which a random sample of size n is drawn. In order to identify DNA sequences

responsible for a complex disease, I .-- n..~i vpe a number of SNPs genomewide and

construct a haplotype map. Recent molecular surveys -II__- -1 that the human

genome contains many discrete haplotype blocks that are sites of closely located

SNPs (Daly et al. 2001; Patil et al. 2001; Gabriel et al. 2002). Each block may

have a few common haplotypes which account for a large proportion of chromoso-

mal variation. Between .ll11 Il-ent blocks there are large regions, called hot~spots, in

which recombination events occur with high frequencies. Several algorithms have

been developed to identify a minimal subset of SNPs, i.e., I I__!size, SNPs, that







57

can characterize the most common haplotypes (Zhang et al. 2002b, 2002c; K~immel

and Shamir 2005). In this study, I assume that the number and type of' I_---ill

SNPs within each haplotype block has been determined. The rationale behind

the epistatic model being developed is that the effect of a given DNA sequence in

one haplotype block on a complex disease is masked or enhanced by one or more

sequences in other blocks.

Suppose there are R and S (R, S > 1) I_;~! SNPs for two arbitrary

haplotype blocks R and S, respectively. Two alleles of a tag SNP r or a from block

R or S can be denoted by UFr (kr = 1, 2; r = 1, R) and 1M (ls = 1, 2; s = 1, S),

respectively. I use pY, andII pl to denote allele frequencies at the corresponding

SNP. All the SNPs within each of the two blocks form 2R or 2s possible haplotypes
express, ed asl Uj2 Uf UR and -:1 -~ s,~~ respectively. Th~e corresponding



which are composed of allele frequencies at each SNP and linkage disequilibria of

different orders among SNPs (Lynch and Walsh 1998). A general expression for

the relationships between haplotype frequencies and allele frequencies and linkage

disequilibria was given by Bennett (1954). Lou et al. (2003) derived a closed-form

EM algorithm to estimate haplotype frequencies, which can be further used to

estimate allele frequencies and linkage disequilibria based on these established

relationships.

The random combination of maternal and paternal haplotypes generates

2 -1(2R + 1) diplotypes expressed as [riUa Uf -rr -i Uk R ] (1 < kl < k', <

2, 1 < kR < klR < 2) for block R and 2s-1(2s + 1) diplotypes expressed as

[1g172 S 1V -~S -rl/ r ] 1 < 1< < 2, 1 1r / 2 L 7 1
I use the brackets to separate maternal (former) and paternal haplotypes (latter)

for a given diplotype. Unless there are two or more SNPs that are heterozygous,

observable .;,I i1.-Ho il. ,:--1ti.. 4 will be the same as diplotypes. Thus, the numbers









of zygotic ._ I n..i vpes, 3R or 3s, will be less than the number of diplotypes and

the difference between these two numbers is statistically viewed as missing data.

The~ observed z~ygotic genot~ypes are expressed asD UJ~ UlI/U2 /2/ R ~k Or

Hig, /gAz%,/'/MsT/s for the two blocks, respectively. Let P Ikg---?i ||l !I .--kgl]an

Pfi~k',/kgk /---/kakic, denote the diplotype and nc.1vpe frequlencies, respectively, for
block 71. The corresponding expressions are PfiS.. I, .. and PS8, ./. for

block S.

Because phenotypic variation in a complex disease can be explained by

haplotype diversity, a particular haplotype can be assumed to be different from

other haplotypes for a given phenotype (Bader 2001). Liu et al. (2004) defined such

a distinct haplotype as reference ir'l''l;,'1./'. They further defined the diplotypes

formed by reference and/or non-reference haplotypes as composite .i. ,:'altil''

Although it cannot be directly observed, the difference between the reference and

non-reference haplotypes can be inferred from observed zygotic genotypes with the

EM algorithm.

4.2.2 Epistatic Effects

Let A, A and B, B be the reference and non-reference haplotypes at block

R and S, respectively. In the given sample, these two blocks form nine different

composite genotypes expressed as AABB, AABB, AABB, AABB, AABB,

AABB, AABB, AABB and AABB. Traditional quantitative genetic theories

can be used to model the genetic effects of the composite genotypes (Lynch and

Walsh 1998). The genotypic value (pays,) of a joint composite ._ I nc.1vpe at the two

haplotype blocks can be decomposed into nine different components as follows:


pay, = Overall mean (4. 1)

+ (jn 1)ax + (jS 1)aS Additive effects

+ [1 (jn 1)2]P _1 ( 2]PS Dominant effects










+ (j 1(je 1)IAdditive x additive effect

+ (jR 1) [1 (js 1)2]J Additive x dominant effect

+ [1- (a )2]ja -1)K Dominant x additive effect

+[1 (jR 1)21- _j 12]L Dominant x dominant effect,

where

2 for AA or BB

jR, ]s = 1 for AA or BB

0 for AA or BB

stand for the composite genotypes at blocks R and S, respectively, a.~ and P. are the

additive and dominant effects at the corresponding block, respectively, and I, J, K

and L are the additive x additive, additive x dominant, dominant x additive and

dominant x dominant epistatic effects between the two blocks, respectively.

The statistical model being developed here aims to diagnose reference haplo-

types at each block and further provide estimates of the additive, dominant and

epistatic effects of any kind between the two blocks. For a given reference haplo-

type, I will derive the EM algorithm within the maximum likelihood context to

estimate the genotypic value of each composite genotype. By solving a group of

regular equations as shown by equation (4.1), the overall mean, additive, dominant

and four kinds of epistatic effects between two blocks can be estimated, i.e.,










Ps = (2p-AABB A -AABB AABB AABB I-AABB + 2p-AABB)










I = -(p~AABB A-AABB AABB + -AABB)

J = -(2p1AABB I-AABB 2p-AABB p g-AAB p-AABB + -AABB)

K = -(2p~AABB 2p-AABB + AABB A -AABB AABB I-AABB)

L = -(4p-AABB + -AABB + AABB + -AABB + -AABB 2p-AABB

-2p~AABB 2p g ges 2p-AABB) (4.2)


These estimates are the maximum likelihood estimates (MLEs) based on the

invariant property of the maximum likelihood method.

4.2.3 Likelihood Functions

This model is unique in that two different likelihood functions can be inte-

grated to estimate haplotype frequencies (population genetic parameters) and

:- Ir..i vpic values of composite genotypes and residual variance (quantitative ge-

netic parameters). For two different haplotype blocks R and S, between which no

linkage disequilibria exist (Daly et al. 2001; Gabriel et al. 2002), across-block hap-

lotype frequencies can be calculated as the product of the corresponding haplotype

frequencies from a different block, expressed as


p~k1g---g)(lly--ls)2 .kg--kg (12--1 '

where the parentheses are used to separate two different blocks for a given across-

block haplotype. With these across-block haplotype frequencies, expected across-

block diplotype frequencies and across-block genotype frequencies can be calculated,

respectively, under Hardy-Weinberg equilibrium.

With across-block diplotype and genotype frequencies, the likelihood function

based on across-block genotype observations, n = {R(k,~k(/kgk /---/kak'p)(111 (7._/---/ -...))

can be constructed. To simplify the presentation of my model, I will first assume

two' I__;! _; SNPs for each block. Table 4-1 lists all possible ._ I nc.1vpes and diplo-

types as well as their frequencies at two SNPs ._ I n..i vped from block R. Each



















Table 4-1: Possible diplotype configurations of nine 7.~ nd v(pes at two SNPs and their haplotype composition frequencies


Relative diplotype


Haplotype composition


Genotype



Uz U /rU22 2
Uf U1 12

u~u~lu2
UfU21 2 ~








U21 2/122

U21 2122 22


Diplotype Diplotype frequency freq. within genotypes U~ ~ U U U22 ~ ~21 2~1 22 Observation Genotypic mean


1r] 2 1r] 2
1] 1 22



[Ul2 [U22 U21


2V1 12 21 12



2V1 22 21 2L 2


P[11][11] = 1 l

P[11][12] = 2pllpl2

P[12][12] = 2 ,

P[11][21] = 2pllp21

P 1 1[1 1] = 2lpl~1[22 ] = 2p llp 22

P[12][21] = 2pl2921

Pvp[1] 2] = 2pl222

P _, p = 2 p 2 9 2

P [ 2 2 ] [ 2 2] 2


1




~r


~-a



=~+d
=~-a

~-a

~-a

~-a

~-a


n11/11 ~2






812/12
nl~/l~~10-


1jT 21 / 2 L1 Z


n12/22


= p where pl pl2~ P21 and p22 are the haplotype frequencies of U~li U U U2 21Uf,2 and U2]1 22 respectively, and
ij~=Cs=2










diplotype is composed of two haplotypes, one from the mother and the other from

the father. The diplotype frequencies can be expressed in terms of the haplotype

frequencies (Table 4-1). The same genotype may contain two different diplotypes,

depending on its heterozygosity. Table 4-1 also gives the relative frequencies with

which a .-- no~~(vpe carries a particular haplotype. Such relative frequencies will

be useful for partitioning observed genotypes into underlying diplotypes in the

construction of likelihood function based on the phenotype.

Considering two blocks R and S at the same time, I will have the joint

haplotype frequencies, aswea by. VY =Y (p1~ Ik) plS~). The log-likelihood function of

these unknown haplotype frequencies given observed genotypes (n) can be written

as a multinomial form, i.e.,


log L(O2,|n) oc (4.4)
2 222 I,,


k1=1 kg1=1 1 2=1 1=1
222


k1= 1 g= 1 11=1
222
IC C (k~iki/kgk))(l;i/21,) [2 log, p Ik o (pi, 12
k1=1 kg=112=1


R ikiki~/kgkg)(lil',/ ) [210g p Ik~lg log (2p II,#1', 2 9 1
k1= 1 kg =
222

I C C C n(k Ic I ki/kgk)(111/1212) [log(21,p~ kg? Ik( + 210g (p)1
k1=111=112=1
2 2

R~kiki/kgk()(1111 _.)[o (p kI( log (2p it,#1zln)
ki=1li=1
2 2

R~kiki/kgk')(lil /122) [log(2p IkI()+lg(2 I 2)1
k1=112=1


R~kiki/kgk()(lil'zy _.)[o (p Ik k)+lg 2 i e2pp2 1 12
k1= 1









222

kg=111=11 =1
2 2

kg=111=1
2 2

Tj~ji l(lk;k/kgk2)(111',/1212) [Log(2p ~Ikg?~ kg)~ + log (2Zp It,#
kg=112=1


(kik /kg )(ll',/ _. ) [lg(2pIkgID:b kg lg(2p if 7~1' 2p rbiqh )


(kik /kgk, )(11/112 [o(p, kg? + 2pk Ik i kg) 2; log p ,l3



11= 1


12=1

ll(l I;kik /k;)(lil 7. _.~ ) [Log(2p~Jl IkgB sk 2p ik clk zk? ) +lg2~g 1 pi


where 1 < kl < k' < 2, 1 < k2 < k' < 2, 1 <11 < l' < 2, 1 < 12 < l' < 2.

Assuming that diplotypes are associated with phenotypic variation in a

disease, I formulate a likelihood for unknown population (02,) and quantita-

tive genetic parameters (09,) given observed phenotypes (y) and SNP geno-

types (n). Generally -II p. & ya given four-SNP genotype from two blocks,

(ri /~i U ) ( / 2I:), can be partitioned into four possible diplo-
tyes [( U) ]( )\M] (k




log-likelihood function of 02 and 02, can be formulated on the basis of a four-

component mixture model, i.e.,


log L(O2,, O2,|y, G)


i= 1















where the mixture proportions,


"to (112) 1 ] i(4.6)

2 2 kg 11



R 1. I. 1(1 1 i)II ~ )( 1 ~II (4.7)


pp Ikg? p1 k kP111 kp~ 12 k~~r/lz 11 kggl Ik 11',P zkg 12


5 1. (1 2)II ) 1(4.8)
R R R S R SR SR R,
pp Ikg? p12 k V I29 1 k 2l~pt Ik nri1p~2p kg l't YIk 11',s~lp z 12


5 1. (1 ')II ) 1(4.9)


R Ikg? 112 ksI l' +p~Ik2F11 12 Ik2 IRPslR kgpt Il Ik 11',P~ ,k 1

represent the relative frequencies of the corresponding across-block dipk.1v ipes that



f." 7 .7)(lIl: .||1 pk)(lil )] Wi) and f, r ii~l )( ,; |1t k)(lils)] Wi) are the probability density

functions for subject i who has the corresponding diplotype, with the genotypic



1#[(kik )(111/2 1', |azi_)(l'zl2)], TOSpectively, and the common residual variance O.2. These

means and variance are contained in vector fly. Note that the mixture proportions

are expressed as being subject-specific because different subjects each with a known

:- Ir..i vpe may have different diplotype compositions.

Assume that haplotypes U, U,2 and VllV,2 are the reference haplotypes at blocks

71 and S, respectively. As shown above, this leads to nine different across-block










composite genotypes. equation (4.1) provides the structure of an arbitrary across-

block composite ._ no~~(vpe formed by the reference and non-reference haplotypes.

The log-likelihood function described by equation (4.5) can now be expressed, in

specific forms, as


log L(O2,, O2,|y, n) =

lo ]OfAABB WiJ) IC0g SAABD Wi) 10g ]OfAADD
i= 1 i= 1 i= 1

I C log~w z~fAABD2, Wi (I~ )AA Wi~
i= 1

lo1g f4AABB) Ci 10g JlAABD iJ)+ l0g JAADD Wi)i
i= 1 i= 1 i= 1


i= 1
(**)(1/11 (**(*) (**)(**)
loIg fA~AB(ys)+ lo1g f B (y )) + log fAABB(y )
i= 1 i= 1 i= 1



Slog [w f44,4(y ) (1 w ) f pso(yi)]
i= 1
n(12/12)(111)
Slog[~f w"o, fAABB mRi fAABB Wi
i= 1
n(12/12)(*)
I~ logpw fAABD(2 ~ (Ii BA Wi)( 2
i= 1

Ilo [w fAAD Wi D Wi)(
i= 1


(4.10)


(yi>


)


t i[(12)(11 ||. I, s., ...|AAB(i Zi O[(12)(1..||. ~I ., I ,| flAADD i~l


where


(11/11)(*) R(11/11)(11/12) (11/11)(12/11),












+ (11/11)(11/22) ,





t (11/12)(11/11) ,


n 1/1). .) ~


11(11/11)(12/22) 71(11/11)(22/11) 71(11/11)(22/12)


li(12/11)(11/

11(12/22)(11/




11(11/12)(12/


11),

11)


71(22/11)(11/11) 71(22/12)(11/11)


11)t 7(12/


71(12,


11)(11,


121)12/121)


12)(22/22)

11)(22/12)

22)(11/12)

22)(11/12)

12)(12/11)

22)(11/22)

22)(22/22)

22)(22/12)

11)(22/11)

12)(12/22)

22)(11/22) +

22)(22/22),

12)(12/12) +



12)(11/12) +



22)(12/12),


22)

22),

12)

11)

11)

22)

22)

22)

12)

11)

22) +




12)

11)

12) +


S(11,

1(12,

S(11,




1(22,

1(22,

7 (11,




7 (11,

71(12,

S(22,

1(22,

7 (22,


71(12/11)(12/22) 7 1(12/11)(22/11) -


~11)(11/

11i)(22;

22a)(11;

22a)(12;

22a)(12;

22a)(12;

22a)(11;

22a)(22;

11i)(22;

12a)(22;

22a)(12;


71(22,

71(22,




71(11,

71(12,

71(22,

71(22,

71(22,

71(22,


12)

11)


11)(11,

22)(12,




22)(22,

22)(12,

11)(11,

11)(22,

12)(22,

22)(22,


12)(11/

11)(12/


22)(22/12)

22)(22/11)

11)(12/22)

12)(11/22)

12)(22/22)

22)(22/12)


71(*)(12/12)

71(12/12)(*)

7 (**)(12/12)


11i)(12/

12a)(12/

22a)(12;


71(22/11)(12/12) 7 1(22/12)(12/12)










R(2/2)** n (12/12)(11/22) + (12/12)(12/22) + (12/12)(22/11) + (12/12)(22/12)








ws p 1 2 (4.12)


and Zi[(11)(1 1 ,~~~~~, ,__,__, war)(1 ,__,,_,, ,wi;[ _,1, ,_,,,__,~~~~~~ and wi[(12)(: ~I.|.In.l~I .| are the

mixture proportions for the complete heterozygous ._ nc.1vipe at all the four SNPs

from two blocks and can be expressed by Equations (4.6) -(4.9). Different from

Equations (4.6) -(4.9), these expressions ignore i because all subjects with the

complete heterozygous genotype have the same diplotype composition.

4.2.4 An Integrative EM Algorithm

I derived a closed-form solution for estimating the unknown parameters with

the EM algorithm. The estimates of haplotype frequencies are based on the log-

likelihood function of equation (4.4), whereas the estimates of dilll"'ivpe ._ I nc.1vpic

means and residual variance are based on the log-likelihood function of equation

(4.5). These two different types of parameters can be estimated using an integrative

EM algorithm (APPENDIX B).

Haplotype frequencies can be expressed as a function of allelic frequencies and

LD. For a two-SNP haplotype within block R, I have


p Ikg(1) R(2) + kl+k'DR, (4.13)


where DR is the linkage disequilibrium between the two SNPs at block R. Thus,

once haplotype frequencies are estimated, I can estimate allelic frequencies and LD

by solving equation (4.13). Similar calculations are also done for block S. After

across-block composite ._ nc.vi~Pc values are estimated, I can estimate the additive,

dominant and epistatic effects between two blocks using equation (4.2). The







6;8

standard errors of the MLEs of the population and quantitative genetic parameters

can be estimated on the basis of Louis' (1982) observed information matrix.

4.3 Hypothesis Tests

Two 1!! li1 .r hypotheses can be made in the following sequence: (1) the associ-

ation between different SNPs within each block by testing their linkage disequilib-

rium (LD), and (2) the significance of an assumed reference haplotype for its effect

on the disease outcome. The LD between two given SNPs within block R can be

tested using the following hypotheses:

Ho : DR = 0
(4.14)
H1 DR / 0

The log-likelihood ratio test statistic for the significance of LD is calculated by

comparing the likelihood values under the H1 (full model) and Ho (reduced model)

usmng


LRR = -2[log L~ij$ ), pR(2), D" = 0, 62,|n) log L(ip, 6j,|n)] (4.15)


where the tilde and hat denote the MLEs of unknown parameters under Ho and H1,

respectively. The LRR is considered to .I-i int111nd cally follow a X2 distribution with

one degree of freedom. The MLEs of allelic frequencies under Ho can be estimated
using the EM algorithm described above, but with- the- constraint p'' R 29 1. A

similar test can be made for block S.

Diplotype or haplotype effects on a complex disease can be tested using the

null and alternative hypotheses expressed as

Ho : I-ljns = I-
(4.16)
H1 at least one equality in Ho does not hold

The log-likelihood ratio test statistic (LR) under these two hypotheses can be

similarly calculated. The LR may .I- i- illIsi 1 ;cally follow a X2 distribution with










eight degree of freedom. However, the approximation of a X2 distribution may be

inappropriate when some regularity conditions, such as normal and uncorrelated

residuals, are violated. The permutation test approach proposed by C'lInI 1..!I and

Doerge (1994), which does not rely upon the distribution of the LR, may be used to

determine the critical threshold for determining the existence of a QTL.

Different genetic effects, such as the additive (con and a~s), dominant (fPR and

/3s) and additive x additive (I), additive x dominant (J), dominant x additive

(K) and dominant x dominant effects (L) between blocks R and S can also be

tested individually. The critical thresholds for these individual effects can he

determined on the basis of simulation studies.

4.4 Results

I use a real example from an obesity study to demonstrate the power and

usefulness of this model. Numerous genes have been investigated as potential

obv-~i-;its-~ n .tibility genes (11 I-ton et al. 1999; C'1!s Is.. is et al. 200:3). The /31AR

and /32AR genes are two such examples (Green et al. 1995; Large et al. 1997) in

each of which there are several polymorphisms common in the population. Two

common polymorphisms are identified at codons 49 and :389 for the /31AR gene

on chromosome 10 and at codons 16 and 27 for the /32AR gene on chromosome 5,

respectively. The polymorphisms in each of these two receptor genes are in linkage

disequilibrium, which -II_--- -is the importance of taking into account haplotypes,

rather than a single polymorphism, when defining biologic function. This study

attempts to detect haplotype variants within these candidate genes which determine

human obesity traits.

To determine whether sequence variants at the two polymorphisms from one

gene interact with those from the other gene to affect obesity phenotypes, a group

of 16:3 men and women were investigated with ages from :32 to 86 years old with a

large variation in body fat mass. Each of these patients was determined for their









_. Ired irpes at codon 49 with two alleles, Ser49 (A) and Gly49 (G), and codon 389

with two alleles, Arg389 (C) and Gly389 (G), within the P1AR gene, as well as

at codon 16 with two alleles, Argl6 (A) and Glyl6 (G), and codon 27 with two

alleles, Gln27 (C) and Glu27 (G), within the P2AR gene, and measured for body

mass index (BMI). Two SNPs from each gene theoretically form 81 across-gene

_t vIredirpes, but, because these two genes are independent, the frequencies of these

_t vIredirpes can be expressed as the product of the genotype frequencies from each

gene. The integrative EM algorithm based on the likelihood function (4.4) allows

for the estimates of four haplotype frequencies and the resulting allele frequencies

and linkage disequilibrium at each gene (Table 4-2). Highly significant LD was

detected between two SNPs for each gene (P < 0.001). Small sampling errors for

the estimates of each population genetic parameter indicated that the estimates are

highly precise.

By assuming that one haplotype is different from the rest of haplotypes at

each gene, this model can detect the reference haplotypes that display significant

main and interaction effects on the BMI trait. Using haplotypes AC, AG, GC

and GG as a reference haplotype at the P1AR gene, respectively, in conjunction

with a reference haplotype selected from AC, AG, GC or GG at the P2AR gene,

I calculated the corresponding log-likelihood-ratio (LR) test statistics (0.36

17.90) using equation (4.10) (Table 4-2). Based on the critical threshold value of

15.05 at the 5' significance level determined from 1000 permutation tests, the two

maximal LR values, 15.1 and 17.9 for across-gene reference haplotypes (GG)(GC)

and (GC)(GC), respectively, are thought to trigger significant haplotype effects on

BMI. However, because these two reference haplotypes form different numbers of

composite genotypes, with all nine for (G G) (GC) and eight for (G C) (GC) (Table

4-2), an optimal reference-haplotype combination should be selected on the basis





























Table 4-2: Maximum likelihood estimates of SNP population genetic parameters (allele frequencies and linkage disequi-

librium) and quantitative genetic parameters associated with phenotypic variation in BMI for 145 patients when different

across-gene reference haplotypes are assumed





Across-block reference haplotype combinations

Parameters (AC)(AC) (AC)(AG) (AC)(GC) (AC)(GG) (AG)(AC) (AG)(AG) (AG)(GC) (AG)(GG) (GC)(AC) (GC)(AG) (GC)(GC) (GC)(GG) (GG)(AC) (GG)(AG) (GG)(GC) (GG)(GG)

LR1 5.00 4.23 6.64 1.84 1.99 1.36 2.29 0.36 9.24 3.44 17.90 2.28 10.77 10.36 15.06 8.46
P value 0.05

Population genetic parameters
p 2()0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61
2(2)0.63 0.63 0.63 0.63 0.37 0.37 0.37 0.37 0.63 0.63 0.63 0.63 0.37 0.37 0.37 0.37
g42 0.13 0.13 0.13 0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 0.13 0.13 0.13 0.13
p 11 0.85 0.85 0.15 0.15 0.85 0.85 0.15 0.15 0.85 0.85 0.15 0.15 0.85 0.85 0.15 0.15
p 1(2) 0.74 0.26 0.74 0.26 0.74 0.26 0.74 0.26 0.74 0.26 0.74 0.26 0.74 0.26 0.74 0.26
D 1 -0.04 0.04 0.04 -0.04 -0.04 0.04 0.04 -0.04 -0.04 0.04 0.04 -0.04 -0.04 0.04 0.04 -0.04

Quantitative genetic parameters
PLAABB 32.72 35.97 21.68- ---- 32.16--- 29.73 24.41 24.18
pLAABB 29.07 29.74 29.79 33.21 ---- 34.65 29.58 27.71 22.66 29.46 29.58 29.28
pLAABB 31.71 30.78 31.97 30.96 ---- 23.11 32.95 32.81 31.42 24.33 29.02 28.94 28.79
pLAABB 29.36 31.08- 28.55 26.09-- 28.62 31.31 36.35- 28.48 26.69 36.35
pLAABB 32.41 29.55 34.03 24.06 20.75 27.48 30.69 32.76 33.19 29.66 40.12 45.72 27.53 27.11 27.43 24.77
pLAABB 30.70 31.62 29.53 30.82 31.22 27.96 24.91 27.57 33.47 31.93 28.99 31.30 27.60 28.66 27.84 27.93
PLAABB 29.23 25.84 30.27- 29.90 30.21 27.40- 30.42 28.85 22.93- 31.18 33.80 21.68
PLAABB 29.29 28.12 28.74 23.90 30.67 29.07 31.52 24.16 28.49 28.59 28.23 27.41 33.87 31.03 36.19 29.34
PLAABB 26.26 29.61 28.72 28.81 29.26 30.69 29.69 30.11 28.00 29.41 29.51 29.05 32.30 32.88 31.40 32.44
8.75 8.78 8.70 8.85 8.84 8.86 8.83 8.89 8.62 8.80 8.34 8.83 8.57 8.59 8.43 8.64

Note: "-" denotes the missing of composite genotypes under the corresponding across-gene reference haplotypes.









of the AIC criterion. I found that across-gene reference haplotype (GC)(GC) is the

best for explaining the BMI data in this example.

The missing of one composite .-- n..~i vpe generated by across-gene reference

haplotype (GC)(GC) prevents the estimation of all the additive, dominant and

epistatic effects using equation (4.2) because of inadequate degrees of freedom. As

an example, I used across-gene reference haplotype (GG)(GC) to demonstrate how

each of these genetic effects is tested. It is found that the additive and dominant

effects exerted by (GG)(GC) are not significant (data not shown). Of the four

kinds of epistasis between the two genes, only dominant x dominant genetic effect

is significant at the 5' significance level. This type of epistasis reduces, by about

211'.~ the BMI of the patient who carries diplotypes [GG][GG] at the P2AR gene

and [GC][GC] at the P1AR gene, compared to the other across-gene diplotypes.

I estimate the standard errors of the MLEs of the population and quantitative

genetic parameters based on Louis' (1982) observed information matrix, -II---- -r;11.-

that all MLEs have reasonable estimation precision although the estimates of

quantitative genetic parameters are not as precise as those of population genetic

parameters due to a small sample size used.

4.5 Discussion

For any two unrelated people, about 99.91' of their DNA sequences are

detected to be the same. It is the remaining 0.1 that contains the genetic variants

that influence how people differ in their risk of disease or their response to drugs.

Discovering concrete DNA sequence variants that contribute to common disease risk

offers one of the best opportunities for understanding the complex causes of disease

in humans. The recent development of a haplotype map of the human genome, the

HapMap, by the International HapMap Consortium (2003) provides a key resource

to describe the common patterns of human DNA sequence variation and find genes

that affect health, disease, and responses to drugs and environmental factors.










I have recently developed a series of statistical models that can detect specific

DNA sequence variants for complex diseases (Liu et al. 2004) or drug response (Lin

et al. 2005) with the aids of the information provided by the Haphiap. Different

from quantitative trait loci (QTL) defined as putative chromosomal segments, DNA

sequence variants detected from my models are shown at the individual nucleotide

level, which are therefore called quantitative imait nucleotides (QTN). Although the

models for QTN mapping have been studied by theoretical simulations and vali-

dated with real-world data, their applications may still be limited because of their

underlying assumption that genes operate individually. For most common diseases,

such as diabetes, cancer, obesity, stroke, heart disease, depression, and asthma,

however, it is likely that a suite of genes and environmental factors are involved to

form a complicated web of interactions (Segre et al. 2005). Interactions between

different genes (i.e., epistasis) or between genes and environmental factors are

thought to be evolutionary forces to maintain genetic variation and buffer against

environmental or developmental perturbations (Moore 2005). In this chapter, I

extend my earlier statistical models to perform a genome-wide scan for sequence-

sequence interactions as a fundamental component of genetic network for disease

susceptibility. Beyond traditional epitstatic models based on locus cosegregations

(Lynch and Walsh 1998), this model allows for the direction characterization of spe-

cific DNA sequence interactions on the SNP-constructed Haphiap. As an example,

I have used the data from an obesity research project to validate the implications

of my interactive models. Specific DNA sequences from two different candidate

genes, /31AR and /32AR, have been identified to affect human obesity traits in an

interactive manner.

For clarification, my presented the idea for sequencing a complex trait with

an interaction model based on two-DNA sequences from each haplotype block.

It is likely that the two-SNP model is too simple to characterize genetic variants










for quantitative variation. With the foundation for the two-SNP sequencing

model, I can readily extend this model to include an arbitrary number of SNPs

whose sequences are associated with the phenotypic variation. In the multi-SNP

sequencing model, I face many haplotypes and haplotype pairs. An AIC- or BIC-

hased model selection strategy (Burnham and Andersson 1998) has been framed

to determine the haplotype that is most distinct from the rest of haplotypes in

explaining quantitative variation. However, in practice, a simultaneously analysis

of too many SNPs will encounter considerable computationally load and, also, may

not he necessary for the explanation of disease variation. The determination of a

maximal number of SNPs for sequencing mapping of QTN should be integrated

with computational algorithms for haplotype block modelling (K~immel and Shamir

2005).

One of the most important statistical issues for QTN mapping is the derivation

of an effective approach to handle missing composite genotypes. For some reference

haplotypes, one or more composite genotypes are missing due to a low frequency of

their occurrences although their effects may exist. The missing of these composite

:- n..~i epes prevents the full estimates of the genetic effects. Given that these

composite genotypes are not missing at random, pattern mixture models developed

for treating non-randomly missing data (Little 1993, 1994) can he used to estimate

the .-- n..~i vpic values of missing composite genotypes. Thus, after the idea behind

the pattern mixture models is incorporated into the sequence mapping of QTN, I

are closer to provide better estimates of sequence action and interaction effects on

complex diseases.















CHAPTER 5
MODEL FOR DETECTING SEQITENCE-SEQITENCE INTERACTIONS FOR
DRITG RESPONSE

5.1 Introduction

The increasing number of genetic studies for complex traits and biological

processes in humans requires more advanced techniques of statistical analysis

(Lynch and Walsh 1998). This is due to two reasons. First, genes interact in a

complex network to determine a final phenotype. It is very often that a complex

trait is characterized by a number of non-Mendelian, environmentally sensitive

genes, of which some act additively, whereas many others are operational in a

niultiplicative or compensatory way (Frankel and Schork 1996; Moore 2003). Alany

current statistical techniques used in genetic research assume the additive control

of genes, aimed to facilitate data analysis and modelling, which certainly provide

misleading results when genetic interactions or epistasis actually occur.

Second, almost every phenotypic trait can he partitioned into its multiple

continuous developmental components on a time scale. To better study the genetic

architecture of these traits, I need measure the traits at a multitude of discrete time

points. A different but statistically similar example of these so-called tinte-series

traits is drug response, a field that gained much attention due to the possible

clinical applications, ranging from individualized therapy to new drug development

(Arranz et al. 2002). Recent studies have el__o-r-- I that a variable number of

polymorphisms in various genes are supposedly involved in modulating the response

and/or side effects to drugs (Serretti and Artioli 2003).

The motivation of this chapter is to develop a statistical model for detecting

epistatic interactions that control drug response. This new model is constructed









through combining two recent developments in genetic mapping. A traditional

mapping strategy is based on the cosegregation of genotyped markers and quan-

titative trait loci (QTL) that are bracketed by the markers (Lander and Botstein

1989). QTL for a complex trait identified from this strategy presents a hypothesized

chromosomal segment whose DNA sequence is unknown. The second strategy,

recently developed by Liu et al. (2004) and Lin et al. (2005), can identify specific

DNA sequence variants that affect a complex trait. This strategy, that relies on the

recent advent of high-throughout single nucleotide polymorphism (SNP) technolo-

gies, allows for the genomewide scan of causal DNA sequences, called quantitative

trait nucleotides (QTN). Lin et al. (2005) have developed a conceptual framework

for detecting interaction effects between different QTN for a complex trait.

Considering the dynamic feature of many complex traits, a series of statistical

models, called functional notrlyl..t..y have been recently developed to characterize

QTL that contribute to genetic variation for longitudinal traits (11 .. et al. 2002; Wu

et al. 2004a, 2004b, 2004c). Functional mapping capitalizes on the mathematical

functions that describe biological processes and embeds them within the context

of genetic mapping theory. By estimating the mathematical parameters that

determine the patterns of longitudinal trajectories, functional mapping has proven

efficient and effective for unveiling the genetic architecture of complex traits. In this

chapter, I incorporate the idea of QTN mapping with functional mapping to detect

epistatic interactions between different DNA sequence variants that encode drug

response. I also perform simulation studies to examine the statistical properties of

this model. A real example was used to validate the usefulness of this model.

5.2 Theory

5.2.1 The Normal IVixture Model

Approaches for QTL and QTN mapping are statistically similar in the con-

ceptual formation of a mixture model. For QTL mapping, each observation must










arise from one of multiple QTL ,_ 11..(i pes, although the QTL genotype for this

individual is unknown (Lander and Botstein 1989). The normal mixture model is

constructed to contain the possible impact of these QTL, .11. Irirpes each of which

is assumed to follow a normal distribution. As shown in Liu et al. (2004), QTN

mapping is also hased on a normal mixture model but in which the phenotypic

value of an individual that is heterozygous for two or more SNPs is thought to arise

one of multiple diplotypes constructed by set of SNPs.

Differences among different diplotypes can he assumed to result from the

composition of different haplotypes. Liu et al. (2004) defined the haplotype

that is different from the rest of haplotypes as reference ir'l''l;,'1./'. Those rest of

haplotypes are thus defined as non-reference hir'7-''/ ti. Consider a QTN, 71, that

is composed of a set of SNPs. Let A and ,4 he the reference and non-reference

haplotypes for this QTN, respectively, which thus form three different composite

i. ,:t.*/1~ li. A 4, 4 4 and A4,. In the mixture model, longitudinal observations for

each composite 11. .~~i pe, j, are characterized by a different multivariate normal

distribution with mean vector uj and covariance matrix E.

For QTN ]rn ppllfir only diplotypes that are heterozygous for two or more

SNPs contain different composite ,_ 11..(i pes because these diplotypes are not

consistent with their ph II! l v! ripically observable genotypes. Therefore, the mixture

model is formulated only for those diplotypes.

5.2.2 Epistatic Effects

I developed an interactive model aimed to detect sequence-sequence epistasis.

Let B and B be the reference and non-reference haplotypes at a second QTN, B,

respectively. The two QTN, A and B, generate nine different composite ,_ 11..(i ipes

expressed as A4,BB, A4,BB, 4 4BB, A4,BB, A4,BB, A4,BB, A4,BB, 4 4BB

and A4,BB. Traditional quantitative genetic theories can he used to model the

genetic effects of the composite ,_ 11..(i pes (Lynch and Walsh 1998). The .-- 11.(i vpiC










vector (uj, ys) of a joint composite -,~ 11..(i pe at the two haplotype blocks can he

decomposed into nine different components as follows:


Overall mean (5.1)

Additive effects

Dominant effects

Additive x additive effect

Additive x dominant effect

Dominant x additive effect

Dominant x dominant effect,


+ (jn 1)aR + (js 1)as

+ [1 (jR 1)2]bR + [1 -- (.is 1)2]bs



+ (jR 1) [1 (.is 1)2]j

+ [1 (jR 1)2](.s -- 1)k

+ [1 (jR 1)2][ _j 12]l


where


for A4, or BB

for A4, or BB

for A4, or BB


stand for the composite genotypes at blocks R. and S, respectively, a. and b~ are

the additive and dominant effect vectors at the corresponding block, respectively,

and i, j, k and I are the additive x additive, additive x dominant, dominant x

additive and dominant x dominant epistatic effect vectors between the two QTN,

respectively.

If the maximum likelihood estimates (1\LEs) of the .-- 11.(i vpiC value vectors at

the left side of equation (5.1) can he observed, I can solve the vectors for the overall

mean, additive, dominant and four kinds of epistatic effects between two QTN by


u = -(uAABB + u44pp+ ugAAB+ u44s)

aR = -(U44BB U44 + 44B- UAABB)

as = -(UAABB UA49- u449+ u4AAB)


2

.7w ~S. = 1

0










bR = -(2uAABB uAABB HAABB HAABB HAABB + 2uAABB)

bs = -(2uaABB u~AAB uAABB HAABB HAABB + 2uAABB)

i = -(uAABB uAABB HAABB U uAABB)

j = -(2uAABB HAABB 2u~AAB + uaAAs uAABB H AABB)

k = -(2uAABB 2uAABB H+ y HAB AABB HAABB HAABB)

l = -(4uAABB Ugy HAApp H AABB H AABB -2 uAABB-2AAB

-2uAABB 2uAABB 2uAABB) (5.2)


5.2.3 Likelihood Functions

The mixture model used to map QTN includes the proportions of each mixture

component and the probability distribution functions of phenotypic observations

given that component. The mixture proportions are the relative frequencies of

those diplotypes that are phenotypically the same and they can be expressed as

a function of haploid frequencies. In this chapter, two QTN are assumed to be

independent, which means that their joint haplotype frequencies are the products of

two haplotype frequencies each from a QTN.

Suppose there are R and S SNPs for QTN R and S, respectively. The two

alleles, 1 and 2, at each of these SNPs are symbolized by kl,..., kR and li,...,1R,

respectively. A haplotype frequency is denoted by pjka---kK for Q~TN R andc pfits...i

for QTN S. As stated above, across-QTN haplotype frequencies can be calculated

as the product of the corresponding haplotype frequencies from a different QTN,

expressed as


p(kl kg---kR)( lily---Is) = Ikg---kn? 1 /2 ---15 (5.3)


where the parentheses are used to separate two different QTN for a given across-

QTN haplotype. With these across-QTN haplotype frequencies, expected across-








80

QTN diplotype frequencies and across-QTN :*1 .11vipe frequencies can be calculated,

respectively, under Hardy-Weinberg equilibrium (Lynch and Walsh 1998).

With across-QTN diplotype and ._ 11.11vpe frequencies, the likelihood function

based on across-QTN genotype observations, n = {7(k(~k'/kgk(/---/kak'p)(111' ,!/ _. /--.))

can be constructed. I simplify the presentation of the model by assuming two SNPs

for each QTN. In Table 5-1, all possible 11. 11v(ipes and diplotypes are given as

well as their frequencies at two SNPs ._ 11.11vped from QTN 71. Each diplotype

is composed of two haplotypes, one from the mother and the other from the

father. The diplotype frequencies can be expressed in terms of the haplotype

frequencies. The same ,_ Ir 1 vpe may contain two different diplotypes, depending

on its heterozygosity. Table 5-1 also provides the relative frequencies with which a

:- Irn..i pe carries a particular haplotype. Such relative frequencies will be useful for

partitioning observed ._ 11.11vpes into underlying diplotypes in the construction of

likelihood function based on the phneotype.

Let Y0, = (]~lp Ig2112) be the population genetic parameters to be estimated

for QTN R and S. Lin et al.(2005) formulated a log-likelihood function of these

unknown haplotype frequencies given observed genotypes (n) in a multinomial

form. They also provided a series of closed-form solution for the EM algorithm to

estimate these haplotype frequencies. To save space, a detailed procedure for this

estimation process is not given.

While traditional models assume the association between the genotype and

drug response (Gong et al. 2004), this model can estimate the effects of different

diplotypes on the pharmacodynamic response of drugs. A particular four-SNP

:- Ir1..ivpe from two QTN, expressed as (klk:/kak )(1 11/1~2~1), can1 be partitioned

into four possible diplotypes, [(klk2 1 2l)][(k'k)ll )] ( 21(][k!i('l

[(kiki)>(lil2)][(kik2 11) and- [(i i)(i) [(kik 2a)].lal Let 02, be quantitative

genetic parameters that specify the mean vectors of different diplotypes and the

















Table 5-1: Possible diplotypes and their frequenecies for each of nine genotypes at two SNPs within a QTN, haplotype com-
position frequencies for each genotype and genotypic value vectors of composite genotypes


Relative diplotype
freq. within genotypes


Haplotype composition
[11] [12] [21] [22]


Composite
genotype


Genotypic
mean vector


Genotype Diplotype Diplotype frequency


P11[ 1 = p ,

P[11][12] = 2pllpl2
P1[12] [12 9 2
P[11][21] = 2pllp21

P[111~] 2lpaI[22] = 2plap22

P[12] [21] = 2pl2921
P[12] [22] = 2pl222

P 4 = 2p21922
P,,[22] [22 9 2


11/11
11/12
11/22
12/11

12/12

12/22
22/11
22/12
22/22


[11] [11]




[11][21]





[12] [22]
[21 [21]
[21 [22]


1 0


0 1


0 0 AA
0 0 AA
0 0 AA


u2-
ul.
uo.
ul.

ul.
uo.

uo.
uo.
uo.
uo.


0 AA


1AA
2 AA
AA

AA


1




1


1 1/ 1
-xT -(1 -L x) -1 x)


0 O
0 0
0 0


0
1

0


Two alleles for each of the two SNPs are denoted as 1 and 2, respectively. Genotypes at different SNPs are separated by a slash.
Diplotypes are the combination of two bracketed maternally and paternally derived haplotypes. By assuming that haplotype [11] is the
reference haplotype, composite genotypes are accordingly defined and their genotypic mean vectors are given. xr = 2 22~ where
p11, p12, p21 and p22 are the haplotype frequencies of [11], [12], [21], and [22], respectively. uj,. uCjis









residual covariance matrix. The likelihood of unknown population (02,) and

quantitative genetic parameters (09,) given a drug response measured at C dosage

levels, y = {y(1),..., y(C)}, and SNP genotype observations, n, is constructed as


log L(62,, O, |y n) = lo w,)ibI., )lz,]i[kk)11)| ('l,]f
i= 1
"b 1. '-( 1a1 )II. ~ )(1;12) )]|if[(kikg)(lil || I ~,, I)(lzl2)] fi)





(5.4)


where the mixture proportions,


)(1 2)II )lil']|i(5.5)





1. !. ( 1 li)II. )( 1 ~II (5.6)

2 2 k2 11k 1



I )1 1)II' ( 1(5.7)


kgl z p129 k V kp 111 kp 12 k ~~1122 kg lrZ' Ik p 11',lp kp gly


Ziir )(1 li)II. 2) 1 II/.~ 1 (5.8)

S k 112 k 1


represent the relative frequencies of the corresponding across-block dipk vr ipes that

form the same genotype, and f[(k, kg)( lil. ,II, pk )( ll~) )] i), [(klkg)(l lll' )|| i' ,)(lil2)] fi >

f[(k, kg)( l'zl_. II''Ik )( 192)]1 ( i) and f,,i ~ )(be,;||. kg', )(lils) ] fi) are the probability density

functions for subject i who has the corresponding diplotype, with C-dimensional







83

:- nod vipic value vectors u[(kkg)(lil_ I. ?l, i, I)ll),U(~ g(i' | ('l),U(~ g('l_.I

(1;12)], and ul,i,i )(111igil ,' p)(1;12)], respectively, and C x C common residual covariance
matrix E. Note that the mixture proportions are expressed as being subject-specific

because different subjects each with a known genotype may have different diplotype

compositions .

Assume that haplotypes 11 (denoted by A) and 12 (denoted B) are the

reference haplotypes at QTN 71 and S, respectively. As shown above, this leads

to nine different across-QTN composite ,11.*1v(ipes. equation (5.1) provides the

structure of an arbitrary across-QTN composite ,_ 11.*1vrpe formed by the reference

and non-reference haplotypes. Table 5-1 tabulates the genotypic value vectors

for different composite ,_ 11.*1vrpes. For a given composite genotype jnjs, I have a

multivariate normal distribution expressed as


(5.9)


with jn, js = 2, 1, 0. At a particular concentration c, the relationship between the

observation and expected mean can be described by a regression model (Zhao et al.

2005),


yi(c) s suzc ei(c), (F

jR=0 jS =0

r* 2 2
I C(nc, c -c') [s(c c) (nrsiL/ (c c')] + es(c) (I
c'=1 jn=0 js =0

where (44,j, is the indicator variable denoted as 1 if a composite genotype jnjs

is considered for individual i and 0 otherwise; ei(c) is the residual error (i.e.,

the accumulative effect of polygenes and errors) that contains two components,

rth-order antedependent variance (Gabriel 1962; Ntifiez-Ant6n and Zimmerman


5.10)


5.11)


Sia,, (Yi: O,) = (2x)C/ 1/2I exp(Yi -uris) C 'Yi -uxis) ,










i2000), CI:= (c, c~= -(l c)y (c -c' %3-S- (q,,pgS(c c) Iwhere r*=

min(r, c 1) and ~(c, c c')'s are unrestricted antedependence parameters, and

independent normal random variable, es(c), with mean zero and time-dependent

variance, O.2(c), termed innovation variance.

5.2.4 Modelling the Mean-covariance Structures

Traditional genetic mapping for multiple traits attempts to estimate each

element in the mean vector and the covariance matrix. This may not be efficient

and effective for two reasons. First, when dosage level C is high, an exponentially

increasing number of parameters need to be estimated. Second, this approach does

not consider biological principles underlying pharmacodynamic responses. The

In .;1!-r li- of modelling drug response is the Hill, or sigfmoid Emax, equation, which

postulates the following relationship between drug concentration (C) and drug effect

(E) (Giraldo 2003)

E aCH
E = Eo + mx(.2
EC~ HO H ) '.

where Eo is the constant or baseline value for the drug response parameter, Emax is

the .I-i-mpll')icti (limiting) effect, ECso is the drug concentration that results in 50'.~

of the maximal effect, and H is the slope parameter that determines the slope of the

concentration-response curve. The larger H, the steeper the linear phase of the the

logf-concentration-effect curve. When the effect is a continuous variable, estimates

of Emax, ECso and H are usually obtained by extended least squares or iteratively

reweighted least squares when there is sufficient data for analysis of individual

subjects. When sparse data are pooled from multiple patients, then a population

as~ l1i--;-; is a better approach. Different from such a traditional treatment, I will

estimate these curve parameters separately for different composite genotypes.

The composite nal vr~ipe-specific mean vectors in equation (5.9) will be

modelled by the Emax model, expressed as













uss= { q,j, (1), uj,j, (C) }
maJJ 1 mayg
(I +EC,'"'" + C,* EC,,R3 +C
(5.13)


where 8 4,j = (F,-,. EmaxyxisEC ,,. .,H,j, Hy) is the mathematical parameters

that describe the drug response profile for composite ._ I noJ vpe jnjs. Thus, based

on equation (5.13), the estimates will be concentrated on 84,j, rather than on

ujsj,. This modelling of the mean vectors has two advantages: (1) clinically

meaningful curves are used in genetic mapping so that the results will be closer to

biological realm, and (2) the number of parameters to be estimated is reduced, thus

increasing the power of the model to detect significant QTN and their interactions.

It is not parsimonious to estimate all the elements in the within-subject

covariance matrix among different concentration levels because some structure

exists for time-dependent variances and correlations. The structure of the residual

covariance matrix in (5.10) can be modelled by the first-order autoregressive

[AR(1)] model (Diggle et al., 2002), expressed as


0.2(1 a2( a2


for the variance, and

O C1, C2) = 0.2 |c -ci|I

for the covariance between any two concentration levels cl and c2, Where 0 < p < 1

is the proportion parameter with which the correlation decays with time lag. The

parameters that model the structure of the (co)variance matrix are an1 li-. I in O2,.

To remove the heteroscedastic problem of the residual variance, which violates

a basic assumption of the simple AR(1) model, two approaches can be used. The









first approach is to model the residual variance by a parametric function of time,

as originally proposed by Pletcher and G. v. r (1999). But this approach needs to

implement additional parameters for characterizing the age-dependent change of the

variance. The second approach is to embed Carroll and Rupert's (1984) transform-

both-sides (TBS) model into the growth-incorporated finite mixture model (Wu

et al. 2004b), which does not need any more parameters. Both empirical analyses

with real examples and computer simulations -II__- -1 that the TBS-based model

can increase the precision of parameter estimation and computational efficiency.

Furthermore, the TBS model preserves original biological means of the curve

parameters although statistical analyses are based on transformed data.

The TBS-based model di physi~ the potential to relax the assumption of vari-

ance stationarity, but the covariance stationarity issue remains unsolved. Zhao et

al. (2005) used Zimmerman and N~inez-Ant6n's (1997) structured antedependence

(SAD) model to approach the age-specific change of correlation in the analysis of

longitudinal traits based on (5.5). Using matrix notation, the error term in (5.4)

can be expressed as

e = Ae (5.14)

where e = [e(1), ,e(C)]', e = [e(1), e(C)]' and for the SAD(1) model

1 0 0 0

~1 1 0 0

A =l (5.15)



\ C-2 1

The variance-covariance matrix of the longitudinal trait is then expressed as


E = AE,A/,


(5.16)









where E, is the innovation variance-covariance matrix and is expressed as:


e2(1) 0 0 0

0 a2(2) 0 0



0 0 0 2(C

The closed forms for the inverse and determinant of matrix E help to estimate the

parameters, 02, = ( 1, r, 62), that model the matrix. The vector Ojay, and 02,

form the quantitative genetic parameters 09,.

As a simplified example with the SAD(1) model, under the assumption that

innovation variance is constant across different time points, Jaffriizic et al. (2003)

derived the analytical forms for variance and covariance functions among time-

dependent measurements, expressed, respectively, as


2 (C) = 2 (5.17)

1 ~2cl
a(c1, c2) = c-c1y 2, C2 C1, (5.18)


for equally spaced repeated measurements. It can be seen that although constant

innovation variances are assumed, the residual variance can change with dosage

level (Jaffriizic et al. 2003). Also, for the simplest SAD model, the correlation

function is non-stationary because the correlation does not depend only on the dose

interval C2 C1 but also depends on the start and end points of the interval, cl and



5.2.5 An Integrative EM-simplex Algorithm

I derived a closed-form solution for estimating the population genetic para-

meters 02, with the EM algorithm (Lin et al. 2005). The Nelder-Mead simplex

algorithm, originally proposed by Nelder and Mead (1965), can be used to estimate

the quantitative genetic parameters 09, = (83xi, Ov~). It is a direct search method









for nonlinear unconstrained optimization. It attempts to minimize a scalar-valued

nonlinear function using only function values, without any derivative information

(explicit or implicit). The algorithm uses linear adjustment of the parameters until

some convergence criterion is met. Simulation studies have proven that simplex

algorithm converges to the same solution more rapidly than the EM algorithm in

the classical functional mapping.

Haplotype frequencies can be expressed as a function of allelic frequencies and

linkage disequilibria (LD). For a two-SNP haplotype for QTN R, I have


pjkg = (1) R~(2) k)i'+k'D'U, (5.19)

where DR is the linkage disequilibrium between the two SNPs at QTN R. Thus,

once haplotype frequencies are estimated, I can estimate allelic frequencies and LD

by solving equation (5.19). Similar calculations are also done for block S. After

across-QTN composite genotypic values are estimated, I can estimate the additive,

dominant and epistatic effects between two QTN using equation (5.2).

5.3 Hypothesis Tests

With my epistatic model, I can make a number of hypothesis tests regarding

the genetic control of overall drug response to a spectrum of dosages and other

clinically important events. Two 1! in r~ hypotheses can be made in the following

sequence: (1) the association between different SNPs within each QTN by testing

their linkage disequilibria (LD), and (2) the significance of an assumed across-QTN

reference haplotype for its effect on drug response. The LD between two given

SNPs within QTN R can be tested using the following hypotheses:

Ho : DR = 0
(5.20)
H1 DR / 0

The log-likelihood ratio test statistic for the significance of LD is calculated by