An accurate approximation for the expected site frequency spectrum in a Galton–Watson process under an infinite sites mutation model John L. Spouge.pdf

该【An accurate approximation for the expected site frequency spectrum in a Galton–Watson process under an infinite sites mutation model John L. Spouge 】是由【探春文档】上传分享，文档一共【9】页，该文档可以免费在线阅读，需要了解更多关于【An accurate approximation for the expected site frequency spectrum in a Galton–Watson process under an infinite sites mutation model John L. 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】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。TheoreticalPopulationBiology127(2019)7–15ContentslistsavailableatScienceDirectTheoreticalPopulationBiologyjournalhomepage:ate/tpburateapproximationfortheexpectedsitefrequencyspectruminaGalton–,NationalLibraryofMedicine,NationalInstitutesofHealth,Bethesda,MD,USAarticleinfoabstractArticlehistory:Ifvirusesorotherpathogensinfectasinglehost,,theexpectednumberofcellsinfectedbyaAvailableonline12March2019singleinfectedcell,:determineR0,theKingmancoalescentorcontinuous-timebirth-and-,Galton–Watsonprocessthetwomodelscanmaketheinferencefromthesitefrequencyspectrum(SFS),thecountofmutationsViralreproductionnumberthatappearinexactlyksequences(k=1,2,...,M).Inthecaseofviruses,however,ifR0islargeandaninfectedcellburstswhilepropagatingvirus,thetwomodelsaresuspect,,thisarticledevelopsanapproximationfortheSFSofadiscrete-timebranchingprocesswithsynchronousgenerations(.,aGalton–Watsonprocess).Whenevaluatedinsimulationswithanasynchronous,non-Markovianmodel(aBellman–Harrisprocess)withparametersintendedtomimictheburstingviralreproductionofHIV,theapproximationprovedsuperiortoapproximationsderivedfromtheKingmancoalescentorcontinuous-timebirth-and-,,itsuggeststheutilityofanalyzingtheSFSofsequencesderivedfrompatientandanimaltrialsofviraltherapies,becauseinsomecases,theinitialR0maybeabletoindicatesubtletherapeuticprogress,(/licenses/by//).,theinitialR0,thebasicreproductionnumberatthestartofinfection,couldinprincipleprovideameasureforThetheoryinthisarticleisstronglymotivatedbytheprac-,essoftheinvasivePresently,,ypeofviralinfectionofasinglehost,theinvasivepopulationoftendatumhasdirectpertinence,.,,urs(Peguetal.,2013;Strboetal.,evendescendsfromasinglefounder,.,about80%ofhuman2013;al.,2016).Animaltrialstestingviraltherapiesimmunodeficiencyvirus(HIV)infectionshaveasinglefounderhavethereforedevelopedingeniousdesigns(Regoesetal.,2005;(Keeleetal.,2008;Haalandetal.,2009;Loveetal.,2016).al.,2015),primarilytosqueezebinarydatafortheinitialbasicreproductionnumberR0,inviralinfectiontheex-statisticalpowerrequiredtodetectsubtlechangesininfectabil-(ietal.,2010),ess,<,ifR01,,,theinvadershaveasmallbutpositivechanceofUnfortunately,essible,survival,andifR0islarge,,becausetheinitialviralpopulationistypicallylatent,.,≈6(Staffordetal.,2000)orR0≈8E-mailaddress:******@.(Ribeiroetal.,2010),.,theseestimatespertaintoviremiahttps:///.-5809/(/licenses/by//).(2019)7–15(.,esdetectableinblood)(Fiebigetal.,2003),mutationsingp120.(Thethreesignificantfiguresrepresentanwhichstartsonaverageabout10daysafterthefoundersinfectunrealisticprecisionbutretainconsistencywiththeliterature.)(KahnandWalker,1998).icssuggestsomemethodsforestimat-bloodisabout20viruses/ml(Kosakaetal.,2017),soaviremiaingR0fromtheSFSingp120sequencedata,butthemethodsareimpliesthatthetotalbloodvolume(5L),,therefore,theviralconsiderthefollowingidealizedmodelofaninitialHIVinfection,,ifcur-(R0≈6orR0≈8)ofthebasicreproductionthecell,releasingabout10–10viralparticles(al.,2007),numberinviremiawerepertinentduringtheinitialinfection,a(DeBoeretal.,2010).Asnotedabove,=8,soatypicalviralparticlehasaminisculeNote,however,thattheinvasiverouteswiththehighestesti-,ifviralpar-matesofper-actHIVtransmissionriskare(per10,000exposures)ticlesinfectindependently,thecountofinfecteddaughtercellsbloodtransfusion(9250),mother-to-childtransmission(2260),approximatelyfollowsaPoissondistributionwhosemeanequalsandreceptiveanalintercourse(138).OtherrouteshaveanHIVtheunknowninitialR0(see,.,(Arratiaetal.,1989a,b).Thus,transmissionrisklessthan63per10,000exposures(seeTable1thevirusesfromtheinitialinfectedcellinfectZ1daughtercells,inPateletal.,2014).BecauseHIVtransmissionissouncertain,(Loveetal.,2016),2daysbeingareasonableapproximationviremiasuggest.(Markowitzetal.,2003).Wethereforemodelthetime-intervalsAlthoughdetectionthresholdspreventdirectmeasurementofbetweenthelysisofamothercellandherinfecteddaughtertheinitialR0,ellsasindependentrandomvariateswithagammadistribution,acidsofpresent-dayhumansretainfootprintsfromthepop-approximatingthemeanby2daysandthestandarddeviationbyulationhistory(.,(DurrettandLimic,2001;Durrett,2008).=2?(,viralsequencessampledduringearlyviremiamaybeHIVreplicationtime,,).ThegammadistributionapproximatingtheInparticular,-moresampledsequencescontaininformationabouttheinitialeters(n,λ)=(,)(.,itsmeannλ?1=2andvariance?≈).Thereproductivecyclebeginsanewwiththelysismotivatetheanalysis,thetheorypresentedherehasbroadappli-–Harrisprocess(,1948);callitthe‘‘Gammamodel’’,(,timerunsdownward,).andforsimplicityeachviralpopulationatthebottomhasaThetheoryinSection2exploitstheDeltamodelillustratedinsinglefounderatthetop((aGalton–Watsonbranchingprocess,indiscrete-time)tocomplications,butdonotchangetheconceptillustrated).,theZofindividuals,,sothefounderhasZ1withmeanR0,andthedeterministicreplicationtimeintheDeltadaughters,whereforillustration,weassumeZ1ismuchlargermodelis2days,(M?1)/,anytwooftheMsampledsequencesAsthemodels’namessuggest,thesynchronousgenerationsinthearelikelytodescendfromdifferentdaughters(manystandardDeltamodelthereforesubstituteaDiracdeltadistribution(.,areferencesonthe‘‘Birthdayproblem’’implicitlyprovethisstate-distributionhavingasingleatomofprobability1,adistributionment).Ifso,thesampledsequencescannotshareanymutationsastightlyconcentratedarounditsmeanaspossible),,sothefounderhasonlyZ1=2simulationsoftheGammamodel,abirth-and-’ssequencehasamutationawayconstantbirthanddeathintensitiesandtheKingmancoalescentfromthefoundersequence,,,so(ment)‘‘sitesimulationsoftheGammamodelwithparametersrelevanttoHIVfrequencyspectrum’’,.,-amutationappearsinexactlymoftheMsampledsequencesetersregimes,whichrequireseparateassessment,arebeyondthe(m=1,2,...,M).,,,blackcirclesrepresentthelysisofinfectedcells,andeachasafunctionofR0thesitefrequencyspectrum(SFS)describedancestrystartsfromasinglefounderatthetop,,wefirstestablishtheparameterrangesrelevanttoanawayfromthefounder,,thelysisofeachinfectedfrompatients,Mtypicallyrangedfrom16to30perpatientcellgivesbirthtoarandomnumberofviraldaughterswhose(Leeetal.,2009).Thegp120geneisabout2550ntlong,andprogenylysecellsatrandomtimeschosenindependentlyfroma(withcrossoversneglected)eachHIVreplicationaveragesε≈×10?5pointmutations/base/replication(ietal.,2010).,therefore,eachRNAreplicationentailsμ≈,(2019)7–,.???daughterswholysetheircellssimultaneously,insynchronousg=0,1,...),andletIg,ihaveXg,,Zg+1=?generationsattime-intervalsequaltothemeangenerationtime∑Zg?{?}Xg,,iaremutuallyindependent,identicallydis-=1tributednon-negativeintegerrandomvariates,.,(Theory)aGalton–-hesubscriptednotationR0formathematical0??gmanipulation,definer=EZ1=R0,.,EZg=>orfromcontinuous-timebirth-and-(asupercriticalGalton–Watsonprocess)isofgreatest(Methods),because(astheIntroductionindicates)??()(Results)uracyoftheLetPs=sdenotetheprobabilitygeneratingfunction(pgf)???′ofthenumberofdaughtersinG1,sor=EZ1=P(1).Thevariousapproximationswhenthesimulationsusetheparameters??relevanttoHI