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Functional Mapping of Dynamic Systems

Permanent Link: http://ufdc.ufl.edu/UFE0041189/00001

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Title: Functional Mapping of Dynamic Systems
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Luo, Jiangtao
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dynamic, functional, qtl
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The dynamic pattern of viral load in a patient s body critically depends on the host s genes. For this reason, the identification of those genes responsible for virus dynamics,although difficult, is of fundamental importance to design an optimal drug therapy based on patients genetic makeup. Here, we present a differential equation (DE)model for characterizing specific genes or quantitative trait loci (QTLs) that affect viral load trajectories within the framework of a dynamic system. The model is formulated with the principle of functional mapping, originally derived to map dynamic QTLs,and implemented with a Markov chain process. The DE-integrated model enhances the mathematical robustness of functional mapping, its quantitative prediction about the temporal pattern of genetic expression, and therefore its practical utilization and effectiveness for gene discovery in clinical settings. The model was used to analyze simulated data for viral dynamics, aimed to investigate its statistical properties and validate its usefulness. With an increasing availability of genetic polymorphic data, the model will have great implications for probing the molecular genetic mechanism of virus dynamics and disease progression. This thesis consists of five chapters. In chapter one we briefly summarize the importance of study the dynamic system from the genetic viewpoint. Chapter two proposes the general framework for virus dynamic models. We focus on drug resistance with parameters having Bayesian structure in chapter two.In chapter three, we discuss the EM algorithm of mixture models used in the previous chapters. Some useful results have been given with strict math proof, which guarantees the correctness of the algorithm. Final chapter, chapter five, we talk about the ongoing research and future work.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jiangtao Luo.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hager, William W.
Local: Co-adviser: Wu, Rongling.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041189:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041189/00001

Material Information

Title: Functional Mapping of Dynamic Systems
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Luo, Jiangtao
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dynamic, functional, qtl
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The dynamic pattern of viral load in a patient s body critically depends on the host s genes. For this reason, the identification of those genes responsible for virus dynamics,although difficult, is of fundamental importance to design an optimal drug therapy based on patients genetic makeup. Here, we present a differential equation (DE)model for characterizing specific genes or quantitative trait loci (QTLs) that affect viral load trajectories within the framework of a dynamic system. The model is formulated with the principle of functional mapping, originally derived to map dynamic QTLs,and implemented with a Markov chain process. The DE-integrated model enhances the mathematical robustness of functional mapping, its quantitative prediction about the temporal pattern of genetic expression, and therefore its practical utilization and effectiveness for gene discovery in clinical settings. The model was used to analyze simulated data for viral dynamics, aimed to investigate its statistical properties and validate its usefulness. With an increasing availability of genetic polymorphic data, the model will have great implications for probing the molecular genetic mechanism of virus dynamics and disease progression. This thesis consists of five chapters. In chapter one we briefly summarize the importance of study the dynamic system from the genetic viewpoint. Chapter two proposes the general framework for virus dynamic models. We focus on drug resistance with parameters having Bayesian structure in chapter two.In chapter three, we discuss the EM algorithm of mixture models used in the previous chapters. Some useful results have been given with strict math proof, which guarantees the correctness of the algorithm. Final chapter, chapter five, we talk about the ongoing research and future work.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jiangtao Luo.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hager, William W.
Local: Co-adviser: Wu, Rongling.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041189:00001


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IamgratefultomysupervisorycommitteemembersDr.HartmutDerendorf,Dr.JayadeepGopalakrishnan,andDr.RonaldRandlesfortheirvaluablecritiques,suggestions,andremarks.SpecialthanksaregiventomysupervisorsDr.WilliamHagerandDr.RonglingWufortheirguidanceandgreathelp.Inthepastyearstheyhavedistilledtheirresearchexperiencetomeineverypossibleway.Everystepofmyprogresshasslatedtheirhardwork.IamveryindebtedtoDungPhanandDr.XiqiangZhengfortheirunselshhelp.IamdeeplyobligatedtoProfessorDavidWilsonforhishelpandencouragement.ThanksaregiventomywifeYanpinandmysonBinjie.Iowemuchtotheirefforts,uponwhichIrely.OntheheavenIwanttothankmygrandmother,mother,andbrothersincetheysacricedeverythingtheyhadformyeducation.Finally,Ithankmyfatherforsupportingmyeducationsinmyearlylifeandprovidingmestubbornnesstonishmygraduateschool. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 9 LISTOFSYMBOLS .................................... 10 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Problems .................................... 13 1.2GeneticControlofPhenotypes ........................ 16 1.3DissertationGoals ............................... 18 2ADIFFERENTIALEQUATIONMODELFORFUNCTIONALMAPPINGOFAVIRUS-CELLDYNAMICSYSTEMS ........................ 19 2.1Introduction ................................... 19 2.2DynamicModelsofVirusLoad ........................ 20 2.2.1DifferentialEquations .......................... 20 2.2.2MarkovProperties ........................... 21 2.3FunctionalMapping .............................. 23 2.3.1GeneticDesign ............................. 23 2.3.2Likelihood ................................ 24 2.3.3EstimationandAlgorithm ....................... 26 2.4HypothesisTesting ............................... 27 2.4.1TheSignicanceofQTL ........................ 27 2.4.2GeneticMechanisms .......................... 28 2.4.3PhysiologicalControlofQTL ..................... 28 2.5ApplicationtoSimulatedData ......................... 29 2.6Discussion ................................... 31 3BAYESIANINFERENCEFORGENETICMAPPINGSOFDRUGRESISTANCE 37 3.1Introduction ................................... 37 3.2DynamicModelsofDrugResistance ..................... 38 3.2.1DifferentialEquations .......................... 38 3.2.2BayesianMarkovModelforDrugResistance ............ 39 3.3GeneticMappingforDrugResistance .................... 42 3.3.1LikelihoodFunction ........................... 42 5

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....................... 44 3.3.3TestingtheSignicanceofQTLandDrugResistance ........ 45 3.4ComputerSimulationandDiscussion ..................... 45 4EMALGORITHMFORSOLVINGMIXTUREMODELSINCOMPLEXGENETICTRAITS ........................................ 50 4.1Introduction ................................... 50 4.2Algorithm .................................... 50 4.2.1GeneticDesign ............................. 50 4.2.2Likelihood ................................ 51 4.2.3Algorithm ................................ 53 4.2.4MainResults .............................. 61 4.3DSEDataAnalysis ............................... 71 4.4NumericalExperiment ............................. 72 4.5ConclusionandDiscussion .......................... 74 5ONGOINGRESEARCHANDFUTUREWORK .................. 75 REFERENCES ....................................... 77 BIOGRAPHICALSKETCH ................................ 87 6

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Table page 2-1JointgenotypefrequenciesatthemarkerandQTLintermsofgametichaplotypefrequencies,fromwhichtheconditionalprobabilitiesofQTLgenotypesgivenmarkergenotypescanbecalculatedaccordingtoBayes'theorem. ....... 33 2-2TheMLEsofparametersthatdenevirus-hostdynamicsforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.1.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. .................................. 34 2-3TheMLEsofparametersthatdenevirus-hostdynamicsforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.4.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. .................................. 35 3-1TheMLEsofparametersthatdenethedynamicsofviraldrugresistanceforthreedifferentQTLgenotypesandtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.05.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. ........................ 48 3-2TheMLEsofparametersthatdenethedynamicsofviraldrugresistanceforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.1.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. ........................ 49 4-1EstimatesoftheParametersforvecodonsinDSEdata ............ 72 4-2TheMLEsofparametersforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulationassumingthattheheritabilityoftheassumedQTLisH2=0.1.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. ............... 73 4-3TheMLEsofparametersforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulationassumingthattheheritabilityoftheassumedQTLisH2=0.4.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. ............... 73 7

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Figure page 1-1Leaf,stem,androotpartsofaplant.M:biomass,L:length,D:cross-sectionalarea,:porosityofrootsandstem.Adaptedfrom[ 132 ]byZensandWebb(2002). ........................................ 15 2-1Estimatedandtruecurvesforasystemofviralinfectionincludinguninfectedcells .......................................... 36 9

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9 41 43 49 67 78 97 125 129 131 139 ]).Anewchallengethatnowfacesmethodologicaldevelopmentishowtodissectaphenotypictraitintoitsbiologicalcomponentsandthenreorganizethesecomponentsintoanewphenotypebenecialtohumans.(a)Fromstatictodynamicmapping:Oneofthemostinterestingtopicsingeneticstudiesistouseanddevelopdynamicmodelstocomparethedifferencesofgeneticcontrolatdifferentstagesofcomplextraits(see[ 42 45 46 64 71 72 82 126 ]).Unlikethetraditionalstaticmodelsthatanalyzephenotypictraitsatindividualtimepoints,thecentralmotivationofdynamicmodelsliesinthestudyofthetemporalpatternofgeneticvariationforaquantitativetraitinatimecourse[ 1 ]andtheidenticationofspecicgenes(i.e.,quantitativetraitlociorQTLs)thatdeterminesuchatime-dependentchangeofthetrait[ 113 ].ThesemodelshavebeeninstrumentalfordetectingandmappingdynamicQTLsforindividualstraits,suchasstemgrowthandrootgrowthinforesttrees[ 120 133 ],plantheightinrice[ 137 ],tillernumberincreaseinrice[ 20 ],biomassgrowthinsoybeans[ 54 ],bodymassgrowthinmice[ 117 136 ],bodyheightgrowthinhumans[ 52 ],anddrugresponse[ 57 ]. 13

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1-1 );moreimportantly,itentailsthecoordinationofthesepartsthroughnaturallaws.Theselawsincludemaximizingleafsurfaceareaforphotosynthesisandminimizingthetransportdistanceforwater,nutrients,andcarbon.Thecoordinationofleaf,stem,androotbiomassforaplantcanbedescribedbyasystemofordinarydifferentialequations(ODEs): whereML,MS,andMRarethebiomassesoftheleaves(L),thestems(S),andtheroots(R),respectively,withwhole-plantbiomassMT=ML+MS+MR,andaretheconstantandexponentpowerofanorganbiomassscalingaswhole-plantbiomass,andistherateofeliminatingageingleavesandroots[ 12 ].TheinteractionsbetweendifferentpartsofaplantcanbemodeledandstudiedbyestimatingandtestingtheODEparameters(L,L,L,S,S,R,R,R). 14

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Figure1-1. Leaf,stem,androotpartsofaplant.M:biomass,L:length,D:cross-sectionalarea,:porosityofrootsandstem.Adaptedfrom[ 132 ]byZensandWebb(2002). 35 ]byHoetal.1995;[ 80 81 ]byPerelsonetal.1997,1996;[ 96 ]bySedaghatetal.2008).Abasicmodelfordescribingshort-termvirusdynamics(see[ 5 ]byBonhoefferetal.1997)isexpressedas dt=dxxvdy dt=xvaydv dt=kyuv, whereuninfectedcellsareyieldedataconstantrate,,anddieattheratedx;freevirusesinfectuninfectedcellstoyieldinfectedcellswithratexv;infectedcellsdiewith 15

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1 )and( 1 ),willbeabletomapQTLsfortrajectoriesofindividualtraitsaswellasinteractionsamongdifferentcomponentsofthesystem.BelowisadescriptionofhowQTLmappingwithinadynamicalsystemcanbeusedtoaddressfundamentalbiologicalquestions:(a)Size-shaperelationship:Sizedoesmatter,butshapemaymatterevenmoreinnature.Shapeisoneofthemostconspicuousaspectsofanorganism0sphenotypeandprovidesanintricatelinkbetweenbiologicalstructureandfunctioninchangingenvironments.Giventheparameters(L,L,L,S,S,R,R,R)forsystem( 1 ),onecanseehowmuchbiomasshasbeenallocatedtotheleaves,stem,androots.Itispossiblethatsomeplantshaveadominantmainstem,withlessleaves,whilesomeplantsallocatemorecarbontotheroots(belowground)thantheleavesandstem(above-ground).Thus,byintegratingtheODE( 1 )intoaQTLmappingframework,speciceffectsofaQTLonaplant0ssizeandformorshapecanbeestimated.Furthermore,howtheQTLgovernsthedynamicrelationshipbetweensizeandshapecanbequantied. 16

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1 )isimplementedwithanadditivetnessvariable,thiswillconstituteadynamicalsystemforstructural-functionalrelationships.GeneticmappingofQTLsforsuchrelationshipswillshedlightonthegeneticmechanismsinvolvedinbalancingvegetativeandreproductivegrowth.(c)Cause-effectrelationship:Awebofdirectedeventsformsacomplexcause-effectrelationship.Theuseofanantiviraldrugcanincreasetheamountofuninfectedcellsbyreducingtheloadoffreevirusparticlesinapatient,whichreducesthelikelihoodofthepatienttoprogressintoAIDS.Suchcause-effectrelationshipsbetweendifferenttypesofcellscanbequantiedbydifferentialequations( 1 ).IntegratedwithQTLmappingmodels,onecandeterminehowspecicQTLscontrolthedynamicchangesofdifferenttypesofcellsinthecourseoftime.(e)Sink-sourcerelationship:Inplants,thefunctionofcarbohydratesourcetosinkrelationshipsdeterminestheirproductivity.Carbohydratesaretransportedfromsupplyareas(sources)toareasofgrowthorstorage(sinks).Carbohydratesareproducedthroughphotosynthesisintheleavesandchanneledthroughthephloemtotheroots,whichactasthemaincarbohydratesinksduringgrowth.Therateofcarbohydratetransportisprimarilyruledbythesinkstrengthofplantorgans.Adynamicsystemofsink-sourcesrelationshipsiscomposedofpotentialgrowthrate,carbonlossesthroughgrowthandmaintenancerespirationprocesses,andcarbondemandrelatedtoactivereservestorage.TheidenticationofspecicQTLsthataffectthesecomponentsandthereforesink-sourcesrelationshipscanbemadepossiblebyconstructingasystemofODEsandintegratingitwiththeprincipleofQTLmapping. 17

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1 ].Amajorchallengethatfacesdrugdevelopmentanddeliveryforcontrollingviraldiseasesistodevelopaquantitativemodelforanalyzingandpredictingthedynamicsofdeclineinvirusloadduringdrugtherapyandfurtherprovidingestimatesoftherateofemergenceofresistantvirus.Thedevelopmentofsuchamodelcannowbemadepossiblewithrecentadvancesintwoseeminglyunrelatedareas.First,thecombinationbetweennovelinstrumentsandanincreasingunderstandingofmoleculargeneticshasledtothebirthofhigh-throughputgenotypingassaysforsinglenucleotidepolymorphisms(SNPs).Throughtheconstructionofahaplotypemap(HapMap)withSNPdata[ 6 ],weareabletocharacterizeconcretenucleotidesortheircombinationsthatencodeacomplexphenotype,andultimatelydocument,mapandunderstandthestructureandpatternsofthehumangenomelinkedtodrugresponse.Second,thepasttwodecadeshavewitnessedatremendousgrowthofinterestinderivingsophisticatedmathematicalmodelsforcharacterizingvirusdynamicsfrommolecularandcellularmechanismsofinteractionsbetweenvirusanddrug[ 1,7 ].Thesemodelsmostlybuiltwithdifferentialequationshavebeeninstrumentalforstudyingthefunctionofvirusandtheoriginsandpropertiesofvirusdynamics. 19

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10 ].ThebasicideaoffunctionalmappingistomapdynamicQTLforthepatternofdevelopmentalchangesintimecourse.Thepurposeofthisarticleistoproposeastatisticalstrategyforimplementingasystemofdifferentialequationsintothefunctionalmappingframework,ultimatelytomapQTLsfromthehostgenomethatdeterminethedynamicpatternofvirusloadinpatients'bodies.ThenewstrategyisfoundedonasetofrandomsamplesdrawnfromanaturalpopulationatHardy-Weinbergequilibrium.WeintegratetheMarkovchainpropertiesofdynamicdataintothemodeltofacilitatetheestimationofparametersthatdenevirusdynamics.Simulationstudieswereperformedtoinvestigatestatisticalpropertiesofthemodelandvalidateitsusefulnessandutilization. 2.2.1DifferentialEquationsAbasicmodelfordescribingshort-termvirusdynamicswasprovidedbymanyresearchers[ 7 ].Thismodelincludesthreevariables:uninfectedcells,x,infectedcells,y,andfreevirusparticles,v.Thesethreetypesofcellsinteractwitheachothertodeterminethedynamicchangesofvirusinahost'sbody,whichcanbedescribedbyasystemofordinarydifferentialequations(ODE): dt=dxxvdy dt=xvaydv dt=kyuv, whereuninfectedcellsareyieldedataconstantrate,,anddieattheratedx;freevirusinfectsuninfectedcellstoyieldinfectedcellsatratexv;infectedcellsdieatrateay;andnewvirusisyieldedfrominfectedcellsatratekyanddiesatrateuv[ 8 ].Thesystem( 2 )isdenedbysixparametersf,d,,a,k,ugandtheinitialconditionsfor 20

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dt,dy dt,anddv dtterms.Second,anybiologicaldevelopmentisrelatedtogenes,butthemodeldoesnotinvolveanygeneticcomponents.Third,thedynamicchangeofthevirusisaccompaniedbynoisewhichcannotbeneglectedinthedynamicmodeling.Fourth,thisrandomnoiseordevelopmentnoisewillbecarriedfromonestagetothenext.Itshouldbenotedthatthemodel( 2 )usedtoexplainourideainthisarticleisabasicsculptureofrealvirusinfectionasitignoresthedynamicsofimmuneresponsesandvirusmutations.Let0=t0
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wherexi(tk)N(0,2x),yi(tk)N(0,2y),andvi(tk)N(0,2v)aretheinnovationerrorsforthreevariables,x,y,andv,respectively,eachofwhichisassumedtobeiidandtime-independent.Tosimplifyourlineofanalysis,weassumethatthesethreevariablesareindependentofeachother,althoughthisassumptioncanberelaxed.Forsimplicity,weusexik,yik,andviktostandforxi(tk),yi(tk),andvi(tk),respectively.Foraconditionaldensityfunction,f(.j.),wederivetheMarkovpropertiesofthedynamicsystem( 2 )asfollows:Theorem1.1:Allthefuturevaluesofuninfectedcells,infectedcells,andfreevirusparticlesdependstatisticallyonlyontheirpresentvalues.Thatis,f(xik+1,yik+1,vik+1j(xi1,yi1,vi1),...,(xik,yik,vik))=f(xik+1,yik+1,vik+1j(xik,yik,vik)),f(xik+1j(xi1,yi1,vi1),...,(xik,yik,vik))=f(xik+1j(xik,yik,vik)),f(yik+1j(xi1,yi1,vi1),...,(xik,yik,vik))=f(yik+1j(xik,yik,vik)),f(vik+1j(xi1,yi1,vi1),...,(xik,yik,vik))=f(vik+1j(xik,yik,vik)).Theprooffollowsdirectlyfrom( 2 )andthedenitionsofxi(tk),yi(tk),andzi(tk).Fromthistheorem,wehavethefollowingresults.Corollary1.2.1:Conditionalon(xik,yik,vik),(xik1,yik1,vik1)and(xik+1,yik+1,vik+1)arestatisticallyindependent.

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16 ].Hence,Corollary1.2.1holds.TheproofsofCorollaries1.2.2,1.2.3,and1.2.4canbemadeinasimilarway.Now,wegetthefollowingtheorems:Corollary1.2.5:Conditionalon(xik,yik,vik),(xij,yij,vij)forj=0,1,...,k1and(xik+1,yik+1,vik+1)arestatisticallyindependent.Corollary1.2.6:Conditionalon(xik,yik,vik),fxi1,...,xik1g,andxik+1arestatisticallyindependent.Corollary1.2.7:Conditionalon(xik,yik,vik),fyi1,...,yik1g,andyik+1arestatisticallyindependent.Corollary1.2.8:Conditionalon(xik,yik,vik),fvi1,...,vik1g,andvik+1arestatisticallyindependent.Allthesecorollarieswillbeusedtoderivecomputingalgorithmsforsolvingasystemofdifferentialequations( 2 )embeddedinfunctionalmapping. 2.3.1GeneticDesignGeneticmappingofQTLscanbebasedonlinkageanalysisforapedigree[ 17 ]orlinkagedisequilibriumanalysisforanaturalpopulation[ 11 ].Inthisarticle,we 23

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10,15 ].BasedontheCorollariesgivenabove,themultivariatedistributioncanbespeciedbythefollowingtransitionmodel 22x(xik+1gj(xik+1))2,fj(yik+1jxik,yik,vik;j,2y)=1 22y(yik+1hj(yik+1))2,fj(vik+1jxik,yik,vik;j,2v)=1 22v(vik+1lj(vik+1))2,

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18,19 ]isimplementedtogetthemaximumlikelihoodestimates(MLE)ofallunknownparameters.Thegradientofthelog-likelihoodfunction 26

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WangandWu[ 11 ]proposedaclosedalgorithmicformtoobtaintheMLEsofhaplotypefrequenciesp11,p10,p01andp00and,therefore,allelefrequenciesofthemarker(p)andQTL(q)andtheirlinkagedisequilibrium(D)withoutproof.ThesewillbeprovedinChapter4.Genotype-specicmathematicalparametersforviraldynamicsandvariancesforthethreetypesofvirusesarecalculatedbyimplementingtheNewtonalgorithmwiththeArmijosearch[ 3 ]. 2.4.1TheSignicanceofQTLWhetherthereisaspecicQTLresponsibleforviraldynamicsdescribedbyasystemofdifferentialequations( 2 )canbetestedbyusingthefollowinghypotheses: Thelikelihoodsunderthenull(L0)andalternativehypotheses(L1)arecalculated,fromwhichalog-likelihoodratioteststatisticiscomputedbyLR=2[(logL0(~,~jx,y,z)logL1(^,^j,^jx,y,z,M)],wherethetildesandhatspresentthemaximumlikelihoodestimatesunderthenullandalternativehypotheses,respectively.Becauseofviolationoftheregularityassumption,theLRmaynotasymptoticallyfollowa2-distributionwiththedegreesoffreedomequaltothedifferenceofparameternumbersbetweenthetwohypotheses( 2 ).Forthis 27

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21 ]becausethisapproachdoesnotrelyonthedistributionofLRvalues.AfterasignicantQTLisclaimed,itssignicantassociationwiththemarkerconsideredcanbetestedbythefollowinghypotheses: forj=2,1,0.Ifallthenullhypothesesarerejected,thenthismeansthattheQTLpleiotropicallyaffectthesethreedifferentaspectsofviraldynamics.ThepleiotropiceffectoftheQTLonanypairofthreetypesofcellscanalsobetestedaccordingly.Anempiricalapproachfordeterminingthecriticalthresholdisbasedonsimulationstudies. 2 ),including (1) Theaveragelife-times,1=d,1=a,and1=u,ofuninfectedcells,infectedcells,andfreevirus,respectively, 28

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Theaveragenumberofvirusparticlesortheburstsize,k=a,yieldedoverthelifetimeofasingleinfectedcell, (3) Basicreproductiveratio,R0=k=(adu),i.e.,theaveragenumberofnewlyinfectedcellsthatarisefromanyoneinfectedcellwhenalmostallcellsareuninfected.HowaQTLaffectsthesephysiologicalaspectsofviraldynamicsseparatelyorjointlycanbetested. 8 ].Thephenotypicvaluesofthesethreevariablesareexpressedasthesumofthegenotype-specicmeansandinnovationerrorsassumedtofollowamultivariatenormaldistribution.Thephenotypicdataweresimulatedforapracticallyreasonablenumberofequallyspacedtimepoints(say22)undertwodifferentlevelsofheritability,low(0.1)andhigh(0.4).ThegeneticvarianceduetotheQTLforvirusresponseatamiddlemeasurementpointwasusedtodenetheheritability.Theresidualvariances 29

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12 ].Thedifferentialequation-incorporatedfunctionalmappingmodelwasusedtoanalyzethesimulateddata,withtheresultssuggestingthattheQTLresponsibleforthedynamicsystemofviralinfectioncanbedetectedusingamolecularmarkerinassociationwiththeQTL.Asexpected,populationgeneticparametersaboutQTLsegregationinapopulationcanwellbeestimatedwithaclosedformoftheEMalgorithmderivedin[ 11 ].ThecurveparametersforvirusresponsesofeachQTLgenotypecanbeestimatedaccuratelyandpreciselywithamodestsamplesize(100)evenforalowheritabilityofviralloads(Tables1.2and1.3).Theprecisionofallparameterscanincreasewithincreasingheritabilitylevel.Bydrawingthecurvesofviraltrajectorieswithsixparameters,thedynamicbehaviorofthesystemcanbevisualized.Figure1illustratesQTLgenotype-speciccurvesofuninfectedcells,infectedcells,andfreevirusparticlesinadynamicsystemfromarandomrunofsimulation.Itisfoundthattheshapesoftheestimatedcurvesarebroadlyconsistentwiththethoseofthetruecurves,suggestingthatthesystemcanbereasonablyestimatedwiththenewmodel.Simulationstudiesshowedthatthenewmodeldisplaysreasonablyhighpower,0.75foramodestheritability(0.1)and0.99forahighheritability(0.4),todetectasignicantQTLresponsibleforadynamicsystemofviralinfection.HypothesistestsdescribedinSections1.4.2and1.4.3provideageneralplatformforaddressingthegeneticcontrolmachineryofviraldynamics.Foragivensetofsimulationdata,itappearsthatthesetestscanbereasonablymade.Forexample,thepowerfordetectingapleiotropicQTLforthreetypesofviralcelldynamicsisadequatelyhigh(0.7)foramodestsamplesize 30

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10 ]andmathematicalmodels[ 7-9 ]providesnewinsightsintothegeneticcontrolofviruspopulationdynamics.Inthisarticle,wehaveproposedastatisticalmodelformappingquantitativetraitloci(QTLs)thataffectthedynamicpatternofviralinfection.Oneofthemeritoriousadvantagesofthenewmodel,ascomparedtoexistingfunctionalmappingmodels,liesintheorganizationofmultiplecorrelatedaspectsofviralinfectionintoadynamicsystemthroughagroupofordinarydifferentialequationsandtheimplementationofsuchaviraldynamicsystemintotheframeworkoffunctionalmapping.Toourbestknowledge,theworkpresentedhereisarstmodelofgeneticmappingwhichtreatsmultiplecomplextraitsasacomplexsystem.Thecurrentmodelisnotasimpleextensionoffunctionalmappingformultipletraits[ 22 ].Thepreviousmulti-traitmodelsdonottakeintoaccounttherelationshipsofgenotypicvaluesofdifferenttraits,althoughtheymodelacross-traitcorrelationsinresidualerrors.Thenewmodelviewsmultipletraitsasawholeinwhichdifferenttraitscoordinateeachothertodeterminethedynamicbehaviorofthesystem.Thus,byalteringonevariableortrait,othervariableswillchange,leadingtothechangeoftheentiresystem.Thegeneticmappingofgenesforadynamicsystemwillprovideapowerfulmeansforunderstandingthegeneticarchitectureofabiologicalprocess.Themathematicalstrengthofthenewmodelisthedeploymentofasystemofdifferentialequationsinageneticmappingcontext.Thesolutionofmultipledifferentialequations,especiallyhigh-dimensionalones,iscomputationallychallenging.Inthisarticle,weapplyaNewtonalgorithmwiththeEMsettingtoprovidenumericalestimates 31

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2-2 and 2-3 ).Asademonstrationofthenewmodel,weassumethatadynamicsystemiscontrolledbyasingleQTL,althoughthisassumptionistoosimpleinrealworld.Thegenome-widemodelingofmultipleQTLsthroughoutthegenomecanbeincorporatedintothecurrentmodelsetting,allowingthecharacterizationofepistaticinteractionsamongdifferentQTLs[ 23,24 ].Amulti-locuslinkagedisequilibriummodelhasbeenavailabletospecifyhigh-ordernon-randomassociationsamongmultiplelociinanaturalpopulation[ 25 ].Althoughmoreparametersareinvolvedinamulti-locusmodel,theclosedformsderivedfortheEMalgorithm[ 25 ]facilitatestheestimationofmanyparametersatthesametime.Also,amulti-locusmodelallowsthetestoftheroleofgeneticinterferenceinrecombinationeventsbetweenadjacentintervals.Althoughlinkagedisequilibriummappinghasproventobepowerfulforthehigh-resolutionofQTLs,itoftengivesspuriousresultsduetopopulationstructureandotherevolutionaryforces.Anewgeneticdesignthatsamplesasetofrandomfamilies,eachcomposedofparentsandtheiroffspring,canovercomethislimitationoflinkagedisequilibriummapping[ 25,26 ].Thisdesignallowsthesimultaneousestimationofthelinkageandlinkagedisequilibriumbetweendifferentgenes,thusmakingitpossibletoconstructagenome-widelinkagedisequilibriummapforgenediscovery.Ourmodelfocusesontheidenticationofgenesforadynamicsystemofviralchangesinahost'sbodybeforetheadministrationofananti-viraldrug.Whenthepatientsaretreatedwithadrug,theequilibriumstateofthesystemwillbeviolated,fromwhichanewequilibriumwillbegenerated.Bonhoefferetal.[ 8 ]describedaseriesofdifferentialequationsthatspecifythedynamicchangeofthesystemafterdrugtreatment.Thecurrentmodelcanbereadilyextendedtomodelthegeneticcontrolof 32

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JointgenotypefrequenciesatthemarkerandQTLintermsofgametichaplotypefrequencies,fromwhichtheconditionalprobabilitiesofQTLgenotypesgivenmarkergenotypescanbecalculatedaccordingtoBayes'theorem. 27,28 ].WiththeideapresentedinthisChapter,theycanbereadilyincorporatedintothefunctionalmappingmodel,inahopetoachievethemaximumpreventionofvirusresistancetodrugsbydetermininganoptimaladministrationdoseandtimeforindividualpatientsbasedontheirgeneticmakeups. 33

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TheMLEsofparametersthatdenevirus-hostdynamicsforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.1.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs.

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TheMLEsofparametersthatdenevirus-hostdynamicsforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.4.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs.

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Estimatedandtruecurvesforasystemofviralinfectionincludinguninfectedcells,x,infectedcells,y,andfreevirusparticles,vforthreegenotypesatasimulatedQTL,AA,Aa,andaa,underdifferentheritabilitylevels,0.1(rightpanel)and0.4(leftpanel).Thebroadconsistencybetweentheestimatedandtruecurvessuggeststhatthemodelcanprovideareasonablygoodestimateofthedynamicsystem. 36

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4 6 88 ].Reverse-transcriptaseinhibitorspreventthereversetranscriptionofviralgenomicRNAintoproviralDNAandtherebypreventtheinfectionofnewcells.Proteaseinhibitorsinuencethecleavageofviralpolyproteins,resultingintheproductionofnoninfectiousvirusparticles.Bothtypesofdrugsareeffectiveinreducingtheviralloadofinfectedindividuals.Beforetreatmentisinitiated,thevirusloadintheindividual'sbodyisinaquasisteadystate,i.e.,thevirusloadisconstantoverashortperiodoftime[ 4 ].Whenadrugisadministered,thevirusloaddeclinesdramaticallyoverseveralordersofmagnitudeafterexperiencingatransientshoulderphase.However,ifmonotherapyisused,resistantviruswillreboundrapidly,insomecaseswithinonlyafewweeksafterthestartoftherapy.Whilemathematicalmodelshavebeenwidelyusedtostudythedeclineoffreevirusintreatedpatients[ 35 77 80 81 ],agrowingbodyofinteresthasemergedinmodelingthedynamicsofviraldrugresistanceusingasystemofdifferentialequations[ 4 7 88 90 91 ].Mathematicalmodelsmaybeinstrumentalinsheddingsomelightonthepredictionoftheemergenceofdrug-resistantvirusandultimatelythedesignoflong-termtherapy.Givenagreatdealofvariationintherateandpatternofreboundofresistantvirusamongdifferenthosts[ 98 ],thereisapressingdemandontheintegrationofgeneticinformationintomathematicalmodelsfortheprecisepredictionofthedynamicsofdeclineinvirusloadduringdrugtherapyandtherateofemergenceofresistantvirus.Theincreasingavailabilityofsinglenucleotidepolymorphism(SNP)datahasmadeitpossibletocharacterizeconcretenucleotidesortheircombinationsthatencodeacomplexphenotypeand,ultimately,document,mapandunderstand 37

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34 ]).ThepurposeofthischapteristouseBayesiantoderiveastatisticalmodelforidentifyingQTLsresponsibleforviraldrugresistance.ThenewmodelisfoundedonasetofrandomsamplesdrawnfromanaturalpopulationatHardy-Weinbergequilibrium.TheBayesianMarkovchainpropertiesofdynamicdatawillbeincorporatedintothemodeltofacilitatetheestimationofparametersthatdenevirusdynamics.Weperformsimulationstudiestoinvestigatestatisticalpropertiesofthemodelandvalidateitsusefulnessandutilization.ForstandardBayesianstatisticstheory,pleasesee[ 89 ].Fordataanalysisandmodeling,ourreferenceis[ 30 ]. 3.2.1DifferentialEquationsThebasicmodelfordescribingvirusdynamicsis 2 .Theemergenceofdrugresistantvirusinthetherapycanbedescribedbyincorporatingthedifferenceofwild-typeandmutantvirusesintotheequations,whichisexpressedas dt=dx1xv12xv2dy1 wheretherearevevariables:uninfectedcells,x,cellsinfectedbywild-typevirus,y1,cellsinfectedbymutantvirus,y2,freewild-typevirus,v1,andfreemutantvirus,v2.Thesevetypesofcellsinteractwitheachothertodeterminethedynamicchangesofdrugresistantvirusinahost'sbody. 38

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3 )isdenedbynineparametersf,d,1,2,,a,k1,k2,ugandtheinitialconditionsforx,y1,y2,v1,andv2.Thedynamicpatternofthissystemcanbedeterminedandpredictedbythechangeoftheseparametersandtheinitialconditionsofx,y1,y2,v1,andv2.AgaintheapplicationforceustointroducetheMarkovmodelbyusingEulerscheme tk=dx(tk)1x(tk)v1(tk)2x(tk)v2(tk)y1(tk+1)y1(tk) tk=1(1)x(tk)v1(tk)+2x(tk)v2(tk)ay1(tk)y2(tk+1)y2(tk) tk=1x(tk)v1(tk)+2(1)x(tk)v2(tk)ay2(tk)v1(tk+1)v1(tk) tk=k1y1(tk)uv1(tk)v2(tk+1)v2(tk) tk=k2y2(tk)uv2(tk). Forlongtermtreatmentnotonlythethedrugresistanceappearsbutalsotheparameterschangedrastically.ThelongitudinalobservationsdonotfollowthemodelintroducedinChapter1.Abettermodelistoassumethattheparametersthemselvesarerandomobservationsfollowingcertaindistributions. 39

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wherexi(tk)N(0,2x),y1i(tk)N(0,2y1),y2i(tk)N(0,2y2),v1i(tk)N(0,2v1),andv2i(tk)N(0,2v2)aretheerrorsforvevariables,x,y1,y2,v1,andv2,respectively,eachofwhichisassumedtobeiidandtime-independent.Againweassumethattheseerrortermsofthevevariablesareindependentofeachother,althoughthisassumptioncanberelaxed.Thepriorsarealsoassumedtobeindependent,whichisbiologicallymeaningful.Forsimplicity,weusexik,y1ik,y2ik,v1ik,andv2iktostandforxi(tk),y1i(tk),y2i(tk),v1i(tk),andv2i(tk),respectively.JustasinChapter1,wehave:Theorem2.2.1:Allthefuturevaluesofuninfectedcells,cellsinfectedbywild-typevirus,cellsinfectedbymutantvirus,freewild-typevirus,andfreemutantvirusdepend

PAGE 42

3.3.1LikelihoodFunctionWeusethesamegeneticdesignasChapter1.Forapatientthedynamicmodelisdescribedbytherandomvectori=fi,di,1i,2i,i,ai,k1i,k2i,uigofpriordistribution.TheiforthepatientswiththesameQTLgenotypejhavethesamedistributionwithmeanfj,jd,j1,j2,j,ja,jk1,jk2,jug(j=2forAA,1forAa,or0foraa).Thedrugresistanceisdeterminedbythedistributionandstructureoftheserandomvectors.Thelikelihoodoflongitudinalviraldata(xi,y1i,y2i,v1i,v2i)=fxi(tk),y1i(tk),y2i(tk),v1i(tk),v2i(tk)gNk=0andmarkerinformationMiforpatientiisformulatedbythemixtureofBayesiantransitionalMarkovmodel,expressedas 42

PAGE 43

wherefj(xik+1,y1ik+1,y2ik+1,v1ik+1,v2ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,)=fj(xik+1jxik,y1ik,y2ik,v1ik,v2ik;j,)fj(y1ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,)fj(y2ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,)fj(v1ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,)fj(v2ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,),fj(xik+1jxik,y1ik,y2ik,v1ik,v2ik;j,2x)=1 22x(xik+1gj(xik))2,fj(y1ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,2y1)=1 22y1(y1ik+1h1j(y1ik))2,fj(y2ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,2y2)=1 22y2(y2ik+1h2j(y2ik))2,fj(v1ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,2v1)=1 22v1(v1ik+1l1j(v1ik))2,fj(v2ik+1jxik,y1ik,y2ik,v1ik,v2ik;j,2v2)=1 22v2(v2ik+1l2j(v2ik))2,=(2x,2y1,2y2,2v1,2v2), 43

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22(jj)21 22d(djjd)21 221(1jj1)2#1 222(2jj2)2#1 22(jj)21 22a(ajja)21 22k1(k1jjk1)21 22k2(k2jk2)21 22u(ujju)2, 44

PAGE 45

3 )canbetestedbyusingthefollowinghypotheses: Thelikelihoodsundereachhypothesisarecalculatedfromwhichalog-likelihoodratioteststatisticiscalculated.AfterasignicantQTLisclaimed,itssignicantassociationwiththemarkerconsideredcanbetestedbythefollowinghypotheses: forj=2,1,0.Ifallthenullhypothesesarerejected,thenthismeansthattheQTLpleiotropicallyaffectthesevedifferentaspectsofviraldynamics.ThepleiotropiceffectoftheQTLonanypairofvetypesofcellscanalsobetestedaccordingly.Anempiricalapproachfordeterminingthecriticalthresholdisbasedonsimulationstudies. 45

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21 ].Theidenticationofgenesthatcontrolthedynamicpatternofviraldrugresistancewillprovideusefulinformationtounderstandtheemergenceofdrugresistanceandbetterpredicttreatmentoutcomes.Inthischapter,wehavedevelopedastatisticalmodelformappingquantitativetraitloci(QTLs)thataffecttherateandpatternofreboundofresistantvirusafterdrugtherapy.ThismodelintegratesasystemofdifferentialequationsforthedynamicchangeofviraldrugresistanceintofunctionalmappingdevelopedbyWuandgroup[ 64 113 ],fromwhichanumberofhypothesesabouttheinterplaybetweengeneticactionsandviraldynamicscanbeformulatedandaddressed.Thecurrentmodelisnotasimpleextensionoffunctionalmappingformultipledynamictraits[ 135 ].Thepreviousmulti-traitmodelsdonottakeintoaccounttherelationshipsofgenotypicvaluesofdifferenttraits,althoughacross-traitcorrelationsduetoresidualerrorsareconsidered.Thenewmodelviewsmultipletraitsasawholeinwhichdifferenttraitscoordinateeachothertodeterminethedynamicbehaviorofthesystem.Thus,byalteringonevariableortrait,othervariableswillchange,leadingtothechangeoftheentiresystem.Thegeneticmappingofgenesforadynamicsystemwillprovideapowerfulmeansforunderstandingthegeneticarchitectureofabiologicalprocess.Themathematicalstrengthofthenewmodelisthedeploymentofasystemofdifferentialequationsinageneticmappingcontext.Thesolutionofmultipledifferentialequations,especiallyhigh-dimensionalones,iscomputationallychallenging.Inthisarticle,weapplyaNewtonalgorithmwithintheEMsettingtoprovidenumericalestimatesoftheparametersthatdenethedynamicsystem.Withthetheoremsderivedfromseveralassumptionsofindependence,thealgorithmisshownfromsimulation 46

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3-1 and 3-2 ).Asademonstrationofthenewmodel,weassumethatadynamicsystemiscontrolledbyasingleQTL,althoughthisassumptionistoosimpleinrealworld.Thegenome-widemodelingofmultipleQTLsthroughoutthegenomecanbeincorporatedintothecurrentmodelsetting,allowingthecharacterizationofepistaticinteractionsamongdifferentQTLs.Amulti-locuslinkagedisequilibriummodelhasbeenavailabletospecifyhigh-ordernon-randomassociationsamongmultiplelociinanaturalpopulation[ 114 ].Althoughlinkagedisequilibriummappinghasproventobepowerfulforthehigh-resolutionofQTLs,itoftengivesspuriousresultsduetopopulationstructureandotherevolutionaryforces.Anewgeneticdesignthatsamplesasetofrandomfamilies,eachcomposedofparentsandtheiroffspring,canovercomethislimitationoflinkagedisequilibriummapping[ 55 ].Thisdesignallowsthesimultaneousestimationofthelinkageandlinkagedisequilibriumbetweendifferentgenes,thusmakingitpossibletoconstructagenome-widelinkagedisequilibriummapforgenediscovery.Theemergenceofdrug-resistantvirusmaybeduetothepreexistenceofdrugresistantstrainsbeforetheinitiationoftherapyorthegenerationofresistantvirusduringthecourseoftreatment.Itisimportanttoidentifywhichprocessismorelikelytobetrue,drugresistantviruspreexistsbeforetheonsetoftherapyortheyareproducedbyresidualvirusreplicationduringthecourseofantiviraltreatment,becauseeachprocessrequiresdifferentdrugregimenstomaximizetheclinicalbenets[ 7 ].[ 90 91 ]developedamathematicalmodeltoinvestigateanalyticallythemechanismsunderlyingtheemergenceofdrug-resistantvariantsduringantiviraltreatment.Byincorporatingthismathematicalmodelintoourfunctionalmappingframework,itispossibletotestwhetherthereisaspecicQTLthatdetermineseachofthesetwoprocessesandhowtheycanbepredictedwithgeneticinformationoftheQTLdetected. 47

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TheMLEsofparametersthatdenethedynamicsofviraldrugresistanceforthreedifferentQTLgenotypesandtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.05.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs.

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TheMLEsofparametersthatdenethedynamicsofviraldrugresistanceforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulation,assumingthattheheritabilityofthesimulatedQTLisH2=0.1.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs.

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2 ),whicharewidelyusedingenetics.Wearenotgoingtoreviewtheliteraturesaboutthemixturemodelssincethishasbeendonebymanyauthors(see[ 24 65 66 87 ]andtheirreferences).First,wearegoingtogiveabriefreviewofgenetictheoryandEMalgorithm.ThenwediscussthetheoreticalpropertiesoftheEMalgorithmformixturemodel( 2 ).Wealsotagroupofrealgeneticdataandusecomputertodosimulationstudy. 4.2.1GeneticDesignHerewebrieyreviewtheconceptofgeneticdesignusedinChapter1.ThenaturalhumanpopulationfromwhichnindividualsaresampledisassumedtobeatHardy-Weinbergequilibrium(HWE).ApanelofSNPmarkersaregenotypedforallsubjects,aimedattheidenticationofQTLsaffectingaspecialgrowthfactor.SupposethereisafunctionalQTLofallelesAandaforthegrowth.Letqand1qdenotetheallelefrequenciesofAanda.TheQTLformsthreepossiblegenotypes,AA(symbolizedby2),Aa(symbolizedby1),andaa(symbolizedby0).WeassumethatthisQTLisassociatedwithaSNPmarkerofallelesM(inafrequencyofp)andm(inafrequencyof1p).ThedetectionofsignicantlinkagedisequilibriumbetweenthemarkerandQTLimpliesthattheQTLmaybelinkedwithand,therefore,canbegeneticallymanipulatedbythemarker.ThefourhaplotypesforthemarkerandQTLareMA,Ma,mA,andma,withrespectivefrequenciesexpressedasp11=pq+D,p10=p(1q)D,p01=(1p)qD, 50

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106 ],andfj(Yijj,j)isusuallyamultivariatenormaldistributionwithQTLgenotype-specicmeanvectorbeingfunctionofj,andcovariancematrixspeciedbyj. 51

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106 ],P(M2jp)=p2,P(M1jp)=2p(1p),P(M0jp)=(1p)2;and!2j2=(pq+D)2 2p(1p),!0j1=2(p(1q)D)((1p)(1q)+D) 2p(1p),!1j1=2(pq+D)((1p)(1q)+D)+2(p(1q)D)((1p)qD) 2p(1p);!2j0=((1p)qD)2 (1p)2.Pluggingtheaboveformulaein( 4 )andcancelingallcommonfactors,weknowthatourlikelihoodfunctionconsistsoftheproductofthefollowingthreeparts with1=2(pq+D)((1p)qD),2=(2(pq+D)((1p)(1q)+D)+2(p(1q)D)((1p)qD)),3=2(p(1q)D)((1p)(1q)+D).

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Thereforeourlikelihoodfunction( 4 )canberewrittenas 4 )withrespecttop,q,D,,andundertheconditionthatfjhasmultiplenormaldistributionorMarkovmodelconstructionwithnormaldistribution. 53

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4 )weneedtondthecriticalpointsthatsatisfy Accordingtolargesampletheorywehavethefollowingexistingtheorem 4 )existsandispositivedeniteatthetrueparametervalue0,thenthereexistsa>0suchthatforsufcientlylargen,intheneighborhoodfjjj0jj
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22 ]tosolvethemaximum-likelihoodproblemswithmissingdata.TheframeworkofusingEMalgorithmforsolvingmaximum-likelihoodestimatesofnitemixturedistributionscanbefoundin[ 24 66 87 ]andtheirreferences.HerewebrieyreviewtheEMalgorithmforourconvenience.Forthedetailpleasesee[ 22 87 ],anditsextensions[ 65 ].SupposeourobserveddataareY,whichareincomplete.AndthecompletedataareX=(Y,Z),ofwhichZcannotbeobserved.Moreover,theMLEofL(jX)ismucheasiertosolvethanthatofL(jY).IfweletK(XjY,)denotetheconditionaldistributionofXgiven(Y,Z),and thentheEMalgorithmconsistsofthefollowingtwosteps: 1. E-step.ComputeQ(jt), 2. M-step.ComputeargmaxQ(jt).Dempster,Laird,andRubin(see[ 22 ])haveprovedthatQ(jt)isincreasing,andH(jt)isdecreasing.Wu(see[ 111 ])hasprovedtheconvergence.EMalgorithmframeworkhasbeenusedtosolvetheMLEofthemixturemodelswithdistributiondensityfunction bymanyauthors(see[ 24 65 66 87 ]andtheirreferences).IntherestofthispaperwearegoingtoshowthattheEMalgorithmcanbeusedtosolveourproblem( 4 ),namely( 4 ),andwealsojustifyourchoiceofpriors. 55

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ThenZikjfork,j=0,1,2,andYareourcompletedata.Thecorrespondinglikelihoodfunctionis whichgivesustheloglikelihoodfunction NowfollowingtheEMalgorithmwehave andthenextstepistomaximize( 4 )withrespecttoourparameters.TodothiswerstndtheE(Zikj)byusingthefollowingtheorem. 4 ) 56

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Therefore, Ontheotherhand,ifwedonotincludemissingvalueZikjforYiourlikelihoodfunctionis ( 4 )and( 4 )arethetwosidesofthesamecoin.FollowingthedenitionofE-stepweget( 4 ).Therefore,intheEsteptheposteriorprobabilitywithwhichanindividualihasaspecicQTLgenotypejbasedonthemarkerinformationMkandphenotypicdataiscalculatedby 2n"m1Xi=1(22ji+1ji)+m2Xi=1(2ji+1ji)#(4) 2n"m1Xi=1(1ji+20ji)+m2Xi=10ji+(1)1ji#(4) 2n"m3Xi=1(22ji+1ji)+m2Xi=12ji+(1)1ji#(4) 57

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2n"m3Xi=1(1ji+20ji)+m2Xi=10ji+1ji#(4)with 4 )canbewrittenasQ(j(t))=2Xk=0mkXi=12Xj=0E(Zikj)logfj(Yijj,j)+m1Xi=1E(Zi22)log(p211)+E(Zi21)[log(2p11p10)]+E(Zi20)log(p210)+m2Xi=1fE(Zi12)[log(2p11p01)]+E(Zi10)[log(2p10p00)]g+m2Xi=1fE(Zi11)[log(2p11p00+2p10p01)]g+m3Xi=1E(Zi02)log(p201)+E(Zi01)[log(2p01p00)]+E(Zi00)log(p200)Undertheconstraincondition 58

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4 )withrespectp11,p10,p01,andp00,weget, 59

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ByKTTcondition,wehave and whichcompletetheproofof( 4 ),( 4 ),( 4 ),( 4 ),and( 4 )since=n.Theproofof( 4 )followsfromtheinvarianttheoremofmaximumlikelihood.NoteTheformulaswereobtainedbyWangandWu[ 106 ].Theproofhasneverbeengivenasweknow.AlsointheMstep,theotherparametersareestimatedbysolvingthefollowinglog-likelihoodequations: 60

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whereij=(j,j).Evenfornormaldistributionsitisoftenimpossibleforustosolve( 4 ),and( 4 )explicitly.Thereforewehavetoturntoiterativemethod.Hereweintroducethefollowingiterative: wheret,andtaredeterminedbytheArmijolinesearch[ 3 ].Ofcourse,themethodusedin( 4 )and( 4 )isjustconditionalNewtonAlgorithm.Awaytosavecomputationistouseonlythediagonalelements.Letussummarizeouralgorithmasfollows: 1. Usept,qt,Dt,(t),and(t)tocomputept+1,qt+1,andDt+1through( 4 4 ), 2. Usept+1,qt+1,Dt+1,(t),and(t)tocomputet+1,and(t+1)through( 4 ),and( 4 ),Wecontinuetheabovetwostepsuntilitconverges.InsteadofusingconditionalHessianmatricesaswedoin( 4 ),and( 4 )wecanhybridWangandWuformulaswithNewtonAlgorithm,anduse 61

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4 4 )areusedtocomputep,q,andD,thenthesamplingerrorsbyusing( 4 4 )tosolvethecriticalpointsof( 4 )aresmallerthanthatofNewtonmethod( 4 )ifbothofthemareconvergentProofLetusdenebinarylatentvariablesZijasP(Zij=1)=jji=$jjifj(Yijj,j) 4 )asarandomprocessY.Then whereZdenotesthecollectionofallZijfori=1,...,n,andj=0,1,2.TheHessianof( 4 4 )isconditionalonZ.ThereforeitsvarianceisVar(E(YjZ))butthatof( 4 )isVar(E(YjZ))+E(Var(YjZ)).Then,weprovetheestimatorsofp,q,andDareconsistent.WeseeitisatradeoffbetweenNewtonalgorithmandEM.EMisslowerthanNewtonbutitserrorissmaller.Second,weprovetheestimatorsofp,q,andDareconsistent. 4 ),then^p!p,^q!q,and^D!Dasn!1.Also,^pisequaltotheMLEofmultinomialdistributionn! 4 ),( 4 ),( 4 ),and( 4 ),then^p11!pq+D,^p10!p(1q)D,^p01!(1p)qD,and^p00!(1p)(1q)+D,asn!1.ProofAccordingtothedenitionof2ji,weknow fork=0,1,2.Wealsoknow 62

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4 ),and( 4 ),wehave^p=^p11+^p10=1 2n"m1Xi=1(22ji+1ji)+m2Xi=1(2ji+1ji)#+1 2n"m1Xi=1(1ji+20ji)+m2Xi=10ji+(1)1ji#=1 2n"2m1Xi=1(2ji+1ji+0ji)+m2Xi=1(2ji+1ji+0ji)#=1 2n[2m1+m2]!p2+p(1p)(asn!1)=pwhichcompletestheproofof^p!p,andalsoshowsthat^p=2m1+m2 ThentheloglikelihoodfunctionbecomeslogL(,,p,q,DjY;A)=n1log(q2)+n1Xi=1logf2(Yij2)+n2log(2q(1q))+n2Xi=1logf1(Yij1)+n3log((1q)2)+n3Xi=1logf0(Yij0)

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4.2 4 ),( 4 ),and( 4 )wehave^q=p11+p01=1 2n"m1Xi=1(22ji+1ji)+m2Xi=1(2ji+1ji)#+1 2n"m3Xi=1(22ji+1ji)+m2Xi=12ji+(1)1ji#=1 2n"2m1Xi=12ji+m2Xi=12ji+m3Xi=12ji!#+1 2n"m1Xi=11ji+m2Xi=11ji+m3Xi=11ji#,whichcanbeusedtoestimate 2n(2n1)+1 2n(n2)!Pq2+q(1q)=q(asn!1).(4)Toprove^D!PD(asn!1)weassumethatthegenotypenumbersofAA,Aa,andaaarem11,m12,andm13,respectively,amongthemarkertypeMM,then 64

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whichmeans 65

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2p(1p)f2(Yij2,2)P(M1)m22Yi=12(pq+D)((1p)(1q)+D) 2p(1p)f1(Yij1,1)P(M1)m23Yi=12(p(1q)D)((1p)qD) 2p(1p)f1(Yij1,1)P(M1)m24Yi=1(p(1q)D)((1p)(1q)+D) 2p(1p)f0(Yi)P(M1)=m21Yi=1[2(pq+D)((1p)qD)f2(Yij2,2)]m22Yi=1[2(pq+D)((1p)(1q)+D)f1(Yij1,1)]m23Yi=1[2(p(1q)D)((1p)qD)f1(Yij1,1)]m24Yi=1[(p(1q)D)((1p)(1q)+D)f0(Yij0,0)]withloglikelihoodfunctionm21log[2(pq+D)((1p)qD)]+m21Xi=1logf2(Yij2,2)+m22log[2(pq+D)((1p)qD)]+m22Xi=1logf1(Yij1,1)+m23log[2(p(1q)D)((1p)qD)]+m23Xi=1f1(Yij1,1)+m24log[2(p(1q)D)((1p)(1q)+D)]+m24Xi=1f0(Yij0,0)=m2Xi=1[2(pq+D)((1p)qD)f2(Yij2,2)+(2(pq+D)((1p)(1q)+D)+2(p(1q)D)((1p)qD))f1(Yij1,1)+2(p(1q)D)((1p)(1q)+D)f0(Yij0,0)].

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and Similarly,ifweassumethatm31,m32,andm33representthegenotypenumbersofAA,Aa,andaaamongthemarkermm,wehave FormultinomialdistributionMN(n;1,...,10)with 67

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andobservation(m11,m12,m13,m21,m22,m23,m24,m31,m32,m33,m41,m42,m43),byinvarianttheoremtheMLEforpq+Dis since(pq+D)2+(pq+D)(p(1q)D)+(pq+D)((1p)qD)+(pq+D)((1p)(1q)+D)=pq+D.Similarly,wehave dueto(pq+D)(p(1q)D)+(p(1q)D)2+(p(1q)D)((1p)qD)+(p(1q)D)((1p)(1q)+D)=p(1q)D,and 68

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because(pq+D)((1p)(1q)+D)+(p(1q)D)((1p)(1q)+D)+((1p)qD)((1p)(1q)+D)+((1p)(1q)+D)2=(1p)(1q)+D.Therefore Hence Buttheproblemiswecannotestimatem22,andm23directlysincewecanonlyestimatetheirsum.Therefore,intheiterativealgorithm( 4 )isdened,whichinvolvesthe 69

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4 )tosolvefor.Sinceinunivariatecasewecaneasilygettheunbiasedestimatorsofvarianceparameters.

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69 70 ]. 22exp(yij)2 ThereforeintheM-stepofEMalgorithmwehave 4 ) 4 )and( 4 )wehave and 22+(yij)2 4 ). 71

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EstimatesoftheParametersforvecodonsinDSEdata ParametersCodon16Codon27Codon49Codon398Codon492 72

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TheMLEsofparametersforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulationassumingthattheheritabilityoftheassumedQTLisH2=0.1.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. Table4-3. TheMLEsofparametersforthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulationassumingthattheheritabilityoftheassumedQTLisH2=0.4.ThenumbersintheparenthesesarethesquarerootsofthemeansquareerrorsoftheMLEs. 73

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ComparisontheMLEsofNewtonmethodandEMalgorithmforEstimatingtheparametersofthreedifferentQTLgenotypes,andtheassociationbetweenthemarkerandQTLinanaturalpopulationassumingthattheheritabilityoftheassumedQTLisH2=0.1.Thenumberintheparenthesesarethecomputationerrors. ParametersTrueValueNewtonEM 87 ].ThenwestrictlyfollowEMalgorithmandprovidetheE-stepinoursecondtheorem.InTheorem4.3wegivethesolutionforgeneticparametersp,q,andD.InthepastyearspeoplealwayssaythatEMalgorithmisslow.Hereweshowitwillgivesussmallercomputationerrors(Theorem4.4).Theorem4.5orconsistenttheoremshowthattheaccuracyofp,q,andDaredrivenbysamplesizeandtheaccuracyofotherparameters.Moreresearchresultsandcomputationstipswillbegiveninthefollowupresearch. 74

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1. FamilyBasedLinkageDisequilibrium:Wearegoingtopresentastatisticalalgorithmforconstructingajointlinkage-linkagedisequilibriummapbysimultaneouslyestimatingtherecombinationfractionsandlinkagedisequilibriausingmultilocusmarkerdatainanaturalhumanpopulation.Thedataareasetofrandomunrelatedfamilies,eachincludingafather,amotherandavaryingnumberofoffspring,sampledfromapopulationatHardy-Weinbergequilibrium.Thestrategyistoprovideanalgorithmandstudyitstheoreticalandpracticalproperties. 2. SequencingComplexDiseases:WearegoingtostudyspecicsequencevariantsthatareresponsiblefordiseaseriskbasedonthehaplotypestructureprovidedbyHapMap.Asanexamplewearegoingtomodelthedatafromahumanobesitystudywith155patients. 3. SoybeandataApplicationofourmethodology:AsanapplicationofChapter1wearegoingtostudySoybeanDataandmodelitsbiologicaldevelopment. (a) ThedatawerecollectedbyagroupofChinesescientists.Samplesizeis184.Thedatacontainthefollowinginformation: i. 25linkagesegments, ii. 498markers(genes)foreachsample, iii. Distancesbetweenmarkers. 75

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Soybeanfollowsthefollowinggrowthdynamicmodel: whereML,MP,MS,andarethebiomassofleaves,petioles,andstems,respectively;andMTisthetotalbiomass.AlltheMswerecollectedatdifferenttimepoints.Ourresearchgoalforthisprojectistouseourmethodandintervalfunctionalmappingandndthegenesaffectingthesoybeangrowthdynamicmodel( 5 ). 4. DecaydynamicsofHIV-1:WeplanttouseuseMarkovchainmodeltocharacterizespecicgenesorquantitativetraitloci(QTLs)thataffectvirallifetrajectorieswithintheframeworkofadynamicsystem.Thepurposeistofocusondifferentgenesfordifferentstage.Inthefuturewearegoingtostudy: 1. Bayesianmethod:UseBayesianapproachtostudylinkageandputprioronourp,andq. 2. RNArelatedproblem. 3. Protein. 76

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JiangtaoLuowasborninafarmfamilywithaverylonghistoryinChonqing,China.Luosstresstheconfussianvalueofstudyrstandworkingisessential.Afteryearsofcompetitivestudy,Jiangtaoearnedhismaster'sdegreeatInnerMongoliaUniversityinmathematics.HisrstjobwasasalectureratChongqingTeachersCollege,nowChongqingNormalUniversity.ThenhepursuedfurtherstudyatPekingUniversityandlatertaughtatGuangzhouNormalUniversity,nownamedGuangzhouUniversity.SinceJiangtaohasbeenworkingonhisCo-Ph.D.inMathematicsandStatisticsbuttheactualworkisconcurrentPh.D.degrees.Hehasvarietyofresearchinterests.Rightnowhismainfocusisstatisticalgenetics. 87