Paradoxical suppression of polyspecic broadly neutralizing antibodies in the presence - PDF document

Paradoxicalsuppressionofpoly-speci“cbroadlyneutralizingantibodiesinthepresenceofstrain-speci“cneutralizingantibodiesfollowingHIVinfectionStancaM.Ciupe,PatrickDeLeenheer,ThomasB.KeplerDepartmentofMathematics,UniversityofLouisianaatLafayette,433MaximDoucet,Lafayette,LA70504,UnitedStates Contentslistsavailableatjournalhomepage:www.elsevier.com/locate/yjtbiJournalofTheoreticalBiology0022-5193/$-seefrontmatter2011ElsevierLtd.Allrightsreserved. Correspondingauthor.Tel.:+13374825287.E-mailaddress:msc6503@louisiana.edu(S.M.Ciupe). JournalofTheoreticalBiology277(2011)55…66 toinducethroughvaccination.Thefailuremaybeduetohostregulatoryconstraints(Haynesetal.,2005),incorrectepitopeconformation(Mooreetal.,2006),HIVinductionofpolyclonalBcellactivationandterminaldifferentiation(Levesqueetal.,2009and/orBcellcompetition(DeemandLee,2003;Heyman,2003WhilemanydifferentBcellsclonescanrecognizeagivenHIVvirusstrain,onlythoseofhighaf“nity(strain-speci“c)respondinlargenumberstoproduceneutralizingantibodies.Foraseriesofdiscreterandominfectionsovertime(continuousimmunization),competitionamongBcellclonesmayleadtothephenomenonoforiginalantigenicsin,whereBcellsproducedinresponsetoa“rstviralinfectioncansuppressthecreationofnewimmunecellsinresponsetoasecondinfectionwitharelatedstrain(DeemandLee,2003).Forachronicinfectionwithamutatingvirus,theoriginalantigenicsinmaybelimited,sincethereisenoughtimefortheimmunesystemtocreateBcellsagainstthenewstrain.However,thereisatimedelayintheproductionofeachstrain-speci“cneutralizingantibodythatmaycausethatvirusstraintoexpandathighlevelsbeforetheantibodycancontrolit(Burtonetal.,2004;Richmanetal.,2003).Mostimportantly,thecontinuouspresenceofstrain-speci“cantibodiesmayleadtosuppressionoftheless“tpoly-speci“cBcellclonescapableofproducingbroadneutralizingantibodies.Thelimitationinnumberofbroadlyneutralizinganti-bodiesmayrepresentthegreatestweaknessoftheimmunesystem.Antibody-mediatedimmunesuppressionhasbeenobservedduringpassiveadministrationofantibodiesaswell.Inthissituation,Bcellsarepreventedfromstimulationthroughareductionofavailableantigenicdeterminants(Heyman,2003Finally,studiesofHepatitisCchronicinfectionshaveshownthatstrain-speci“cantibodiesmayinhibitthedevelopmentofpoly-speci“cantibodiesbypreventingthemfromrecognizingantigenZhangetal.,2004Toinvestigatethecompetitionamongstrain-speci“candpoly-speci“cantibodies,wedevelopedmathematicalmodelsofvirus…antibodyinteractionsduringbothimmunizationandnaturalinfectionwithHIV.Westartwiththeassumptionthattheimmunesystemproducesbothstrain-speci“candpoly-speci“c,crossreactive,neutralizingantibodies.Thestrain-speci“candpoly-speci“cneutralizingantibodiestargetvariable(uniquetoeachvariant)orconserved(sharedamongvariants)epitopes,respectively,onthevirusenvelope.ThegoverninghypothesisisthatwhileBcellsproducingboth(strain-speci“candpoly-speci“c)neutralizingantibodiesareactivatedduringtheinfec-tion,thoseproducingpoly-speci“cbroadlyneutralizingantibo-diesaremadeinef“cientandconsequentlykeptatundetectablelevels.ThisprocessismediatedbytheircompetitionwiththeBcellsthatproducemore“tstrain-speci“cantibodieswithwhichtheyshareantigenicstimulation,kineticprolongation,spaceinthelymphnodesandTcellconjugates.WeuseinformationfrompreviousmodelingstudiesofHIVviralinfection(Hoetal.,1995;NowakandMay,2000;Perelsonetal.,1996,1997),cellularimmuneresponses(Ciupeetal.,2006;Staffordetal.,2000),antibodyformation(OpreaandPerelson,1996;Tomarasetal.,2008)andcompetition(Antiaetal.,1998;Boeretal.,2001;Borghansetal.,1999;LeenheerandPilyugin,)toderiveandanalyzemodelsoftheinteractionbetweenvirusandneutralizingantibodies.Ouraimistodeterminetheparameterregimesthatleadtoantibodyfailureandviralpersis-tence,andtopredictwaystoreversethesephenomena.Thepaperisstructuredasfollows.InSection2wedevelopandanalyzethemathematicalmodeldescribingtheinteractionbetweenfamiliesofBlymphocytesproducingpoly-speci“candstrain-speci“cneutralizingantibodiesfollowingcontinuousimmunizationwithseveralHIVvariants.InSection3weexpandthemodeltoaccountfornaturalinfectionandviralevolution;theiranalysisispresentedintwoappendices.InSection4wepresentnumericalresultsofthetwomodels.Weconcludewithadiscussion.2.ModelofantibodyresponsesfollowingcontinuousimmunizationLetbevirusesofspeci“city1bestrain-speci“cneutralizingantibodiesofspeci“city1,andbethepoly-speci“cbroadlyneutralizingantibody.Virusesareintroducedintothebodyattimes,anddonotmutate.Wecoarse-grainthevirallife-cycle,aggregatingtheprocessesofinfection,integrationandhost…cellviralproductionintoasimplereplicationmodelinwhichvirusesreplicatewithdifferentviral“tnessper-capitarates.Wetreatthedynamicsofantibodyproductionsimilarly,assumingthatanti-bodyconcentrationisinquasi-equilibriumwiththeBcellpopula-tionthatproducesthem,andwithoutrepresentingthecomponentsubprocessessuchasactivation,differentiationandantibodysecre-tion.Theconcentrationofantibodyspeci“ctoviralstraindenoted,andthatofpoly-speci“cantibody.Weonlyconsiderthefractionoftheproducedantibodiesthathasneutralizingfunction.Inthepresenceofneutralizingantibodiesvirusesareremovedatratesbythestrain-speci“candpoly-speci“cneutralizingantibodies,respectively.WeassumethattheremovalratesareindependentofstrainandthatStrain-speci“cneutralizingantibodiesareelicitedatratetheviralstraintowhichtheyarespeci“c.Poly-speci“cneutraliz-ingantibodiesareelicitedatratebyallviralstrains.WedenotethedifferencesbetweenBcellsproliferationanddeathrates,effectivelytreatingtheantibodyatquasi-equilibriumwiththesecellsassurvivingatthatrate.Finally,allBcellscompetewitheachother(withinandbetweenclones)forantigen,spaceinthelymphnodes,andconjugateT-cellhelp.ThestrengthofthiscompetitionisgovernedbyparameterThedynamicsofthesystemisdescribedbythefollowing dVidtiKAiK0A0ÞVi, dAidt¼lViþAiðabATÞ, InSection2.1weinvestigatethesystemdynamicsforthecasewherestrain-speci“cBcellsareabsent.InSection2.2weexplorethedynamicswhenbothpoly-speci“candstrain-speci“canti-bodiesareproducedinresponsetoinfection.2.1.Virusdynamicsduringpoly-speci“cantibodyresponsesLetusconsiderthecasewherevirusesintroducedintothebodyattimes,independentofeachother.Theimmunesystemreactsbyproducingpoly-speci“cantibodies,,atrate,whichneutralizeallvirusstrainsatrate.Forsimplicity,weassumethatallviralstrainsareequallyadaptedtothehostandtheyreplicateatthesamerateindependentofthestrain.System(1)becomes dVdtK0A0ÞV, S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 2.1.1.Steady-stateanalysisWeinvestigatethelong-runbehaviorofsystem(2)whenallvirusesarepresentinthepopulation.Toeliminatethedisconti-nuitiesinthemodel,weconsiderthatthesystemstartsattimeleadingtoinitialconditions0andThesteady-statesofsystem(2)canbedividedintothreeclasses:1.Theno-infectionsteady-state2.Theviralclearancesteady-state3.Thechronicinfectionhyperplane 1, V2, whichexistsfor T¼Xni¼1 Vi¼ rK0l0b Letusstudytheasymptoticbehaviorofthesteady-states.TheJacobianmatrixcorrespondingtooursystemis ... V10rK0 ... ............... A0K0 ... Proposition1.Theinfectionfreesteady-stateSisalwaysTheviralclearancesteady-stateSisasymptoticallystableif andunstableiftheinequalityisreversed thechronicinfectionhyperplaneSexistsandeachofitssteady-stateshaszeroeigenvalues(a)Thecharacteristicequationforthesteady-statehaspositiveeigenvalues.Therefore,theinfectionfreesteady-stateisalwaysunstable.(b)Thecharacteristicequationforthesteady-state haseigenvalues ,theeigenvaluesarenegativeandtheviralclearancesteady-stateisasymptoticallystable.Iftheinequalityisreversed,theeigenvaluesarepositiveandtheviralclearancesteady-stateisunstable.Inotherwords,whentheantibodyresponseatsteady-statesexceedstheviralproduction,theviruswillbeclearedotherwisetheviruspersists.(c)Thecharacteristicequationforthesteady-state rK0þrb haseigenvalues a2b  rK04rb Althoughthelatterpairofeigenvaluesarenegativefortherearealwayszeroeigenvalues,andhencewecannotdecidestabilityofthechronichyperplanebasedonlinearanalysis.2.1.2.GlobalstabilityProposition2.Whenrisgloballyasymptoticallystable.Considerthefunction K0 A20l0ZA0 A0 1 A0 Notethatforpositive...ispositivesemi-de“niteandzeroonlyif.......More- A0K0 A20A0 K0 A0l0bðA0 A0Þ2¼Xni¼1VirK0 A0K0A02 A0þ A20A0 K0 A0l0bðA0 A0Þ2¼Xni¼1ViðrK0 A0ÞXiVi K0A0ðA0 A20Þ K0 A0l0bðA0 Whenissatis“edisnegativesemi-de“nite,andthelargestinvariantsetwhere0is....Since0,andisaproperfunction(i.e.foreach,thesetiscompact)wehavethatallsolutionsof(2)arebounded.Therefore,fromLasallesinvarianceprinciple,isgloballyasymptoticallystable.Proposition3.Whenrisgloballyasymptoticallysatis“essystem dTdt¼TðrK0A0Þ, Considerthefunction TZT T 1 T 1ttþZA0 A0ðK0trÞdt,ð16Þwhere A0¼r=K0and Notethatispositivesemi-de“nitefor0,andzeroat T, .Moreover, T0TþA0ðabA0ÞÞðK0A0rÞ0A0rÞðl0 TþA0ðabA0ÞÞ0A0rÞð A0ðab whereweusedthat Tþ A0ðab 0.Noticethatisaproductoftwofactors,eachofwhichhasasinglerootat .For ,thesefactorshaveoppositesigns,andthus0.Moreover,thelargestinvariantsetwhere0is T, .Since0,andisaproperfunction(foreachthesetiscompact)wehavethatallsolutionsof(15)arebounded.FromLasallesinvarianceprinciple,theisgloballyattractive.S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 2.2.Virusdynamicsinthepresenceofcompetitionbetweenstrain-speci“candpoly-speci“cantibodyresponsesLetusconsiderthegeneralcasegivenbythesystem(1)whereimmunizationwithviruses...attimes...leadstoproductionofbothpoly-speci“candstrain-speci“cneutralizingantibodies,...respectively.Asbefore,weassumethatallviralstrainsreplicateatthesamerateindependentofthestrain.Westudytheviruseslongtermbehaviorasstrain-speci“candpoly-speci“cneutralizingantibodiescompetewitheachotherforresources.2.2.1.Steady-stateanalysisAsbefore,weeliminatethediscontinuitiesinthemodel,byconsideringsystemsinitialconditionfortime,whereallvirusesarealreadypresentinthepopulation.Thesteady-statescanbedividedintothreeclasses:1.Theinfectionfreesteady-state...2.Theclearancehyperplane... A1, ... An, A0Þ,where 0for...,and 0areanyvector,respectivelynumber,suchthat .Thissetofsteady-statesistheintersectionofahyperplaneofdimension,andthenon-negativeorthant3.Let...,andassumethat1.Thentherearechronicsteady-states 1, ... Vn, A1, ... An, A0Þwithð Vi, 0forall Vj¼ 0forallandonlyif mK0l0þKlmðlþl0Þ Thenonzerocomponentsofsuchsteady-statesaregivenby A¼ Ai¼ lrl0þKl, Vi¼ rl0þKlb forall A0¼ Proposition4.Theinfectionfreesteady-stateSisalwaysTheviralclearancehyperplaneShaszeroeigenvalues.Foreverym...forwhichtherearecorrespondingsteady-statesSofformhavingexactlymvirusstrainsandcorrespondingantibodiespresentandtheyareallunstable.Moreoverthereisauniquesteady-stateofformhavingexactlymnvirusstrainsandcorrespondingantibodiesanditisasymptoticallystable.TheJacobianmatrixcorrespondingtothelinearizedsys-temis ... V10X2K ... .............................. VnK0 Vnl0Y1b ... A1b A10lb ... A2b ........................... Anb ... Anl0l00b A0b ... wherefor A0þK AiÞ,Yi¼abð ATþ AiÞ,Z¼abð ATþ (a)Thecharacteristicequationofthesteady-statehaspositiveeigenvalues.There-foretheinfectionfreesteady-stateisalwaysunstable.(b)Thecharacteristicequationofthehyperplane A0K Þg¼haseigenvalues A0K isunstablewhen AiþK foratleastone.Thestabilityofanysteady-statethatbelongstothehyperplanecannotbedeterminedfromlinearanalysis.(c)If(18)holdsfor1,thenitalsoholdsforallForeach,therewillbesteady-stateswithexactlystrainsandcorrespondingantibodiespresentthatsatisfyequalities(19).Weinvestigatethelocalstabilityofthechronicsteady-statesforwhichstrainspersistandstrainsarecleared.Thesteady-stateshavethecharacteristicequation ............... rKlmK0l0þKlmðyxÞm1fðx1ÞyÞvþmzwg,ð24Þwherex¼L2 Aþb ATaÞLþK Aðb ATaÞ,y¼b AL,z¼ AfbLþK0ðb ATaÞg,w¼ A0ðLaþb ATÞ,v¼n AðLaþb whichsimpli“esto rKlmK0l0þKlmðLþb ATaÞðL2 ATaÞLþK Aðb ATaÞÞm1ð AL2 ATaþmb Aþb A0ÞLþrðb The“rstareposi-tive.Thereforethechronicsteady-statesforwhichpersistandstrainsareclearedarealwaysunstable.Finally,wewillshowthattheuniquesteady-statewithvirusstrainsandcorrespondingantibodiespresentisasymptoti-callystable.S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 ,satisfying(19)for,hasthecharacteristicequation........................... Aþb ATaÞLþK Aðb ATaÞ,y¼b AL,z¼ AfbLþK0ðb ATaÞg,w¼ A0ðLaþb ATÞ,v¼n AðLaþb whichsimpli“esto ATaÞðL2 ATaÞLþK Aðb ATaÞÞn1ð AL2 ATaþnb Aþb A0ÞLþrðb Onecanshowthatalleigenvalueshavenegativerealpartsprovided .Therefore,ifthechronicsteady-stateexiststhenitisstable.Inotherwords,whentheviralproductionexceedsthecombinedremovalbyantibodiestheviruseswillpersist.3.ModelofantibodyresponsesfollowingnaturalinfectionModel(1)assumesthatthevirusesareintroducedintothehostatrandomtimesthrough,forexample,acontinuousimmunization.WeareinterestedinhowtheseresultschangeinanindividualchronicHIVinfection,wherethevirusmutatesovertime.Assumingthataprimaryinfectionwithleadstoproductionofstrain-speci“candpoly-speci“cneutralizingantibodies(forexample,thepoly-speci“cproductionmaybecausedbypreviousvaccination),andthatmutatesovertimeintostrains)atrates0,whichstimulatestrain-speci“cimmunecellstoproduceantibodies,model(1)becomes Vidt¼riXnj¼1mjViðþK0A0Þ, dAidt¼lViþAiðabATÞ, 1,0(0)(0)(0)0forisamutationmatrixwithnon-negativeoff-diagonalentriesandcolumnsentriesthatadduptoone.WeconsidertwosituationsdescribingHIVevolutionovertime:(A)Theinitialvirusstrainmutatestoproducevirusstrainatrate0,whichmutatestoproducevirus0andsoforth.Themutationisirreversible,andthemutationmatrixdescribingthissituationhasthe(B)Virusesmutateforwardandbackwardrandomlywiththeaddi-tionalassumptionthatthemutationmatrixirreducible.Notethathasasimpledominanteigenvalue1withacorre-spondingnon-negativeeigenvector0suchthat.Incaseisanentry-wisepositivevector(bythePerron…Frobeniustheorem)andincase(2)Asbeforeweassumethatallvirusesreplicateatthesamerate.Weperformstabilityanalysisofmodel(30)forthemutationmatrixQsatisfyingcondition(A).Whentheantibodyresponseispoly-speci“callvirusesareclearedwhenandpersistotherwise.Thisrelationisindependentonthemutationmatrixorthenumberofvirusespresent(seeAppendixA).Whentheantibodyresponsesarepoly-speci“candstrain-speci“cwestudyviruseslongtermbehaviorforthecaseandmutationmatrixsatisfyingcondition(A),toproducevirusatrateandthemutationisirreversible.System(30)becomes dV1dt¼rð1mÞV1V1ðKA1þK0A0Þ, dV2dt¼rmV1þrV2V2ðKA2þK0A0Þ, dA1dt¼lV1þA1ðabATÞ, dA2dt¼lV2þA2ðabATÞ, Weshowthatforvirusesarecleared.Conversely,whenatleastonevirusstrainpersists.Similarlyfor,bothvirusespersist,whereasshowninAppendixB.Notethatdependsontherelativeratioofthepoly-speci“candstrain-speci“cantibodyproductionandviralremovalrates.4.NumericalresultsPreviousstudies(Ciupeetal.,2006;Staffordetal.,2000)haveconsideredaninitialHIVloadof10virionsperl,correspond-ingtothepresenceofasmallnumberofvirionsintheinoculum.Weincreasethisestimateto10virionsperltoaccountfora O¼ K0Kþ ll0 K0Kþ ll0 0K1þ ll0 K0Kþ3 ll02m2 K0Kþ  K0Kþ3 ll02m2 K0Kþ ll08ð1mÞ22 K0Kþ ll0 0Kþ S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 strongercontinuousimmunization.Theviraldynamicsisnotsensitivetothisvalue,aswecanshownumerically.Whenwevarytheinitialinoculum“veordersofmagnitude,thepeakofthevirusshiftstotherightbyonlyoneday.Virusesreplicateataneffectiverateof25virionsperdayCiupeetal.,2006;Staffordetal.,2000).Inresponse,Bcellsbecomeactivatedanddifferentiateintoantibodyproducingplasmacells.Typicalantibodyaf“nitiesfortheelicitedantigenare10HollingerandLiang,2001).SinceeachHIVvirionhasmanypotentialbindingsitesandaf“nitymaturationmayoccur,weassumetheavidityofspeci“candpoly-speci“canti-bodiestobeashighas6perdayforstrain-speci“cantibodyand3perdayforthepoly-speci“cantibody.UsingtheAvogadrosnumberwecancoverttheavidityratesmeasuredininversemolarsintoremovalratesmeasuredinlpermoleculeasfollows: Litermole¼ 106mlmole¼ 106ml61023molecules¼ Usingthisconversionweobtainremovalratesoflperantibodymoleculeperdayand5lperantibodymoleculeperdaybythestrain-speci“candpoly-speci“cantibodyInitially,atthetimeofimmunization,thereareneitherstrain-speci“cnorpoly-speci“cneutralizingantibodiespresent,0moleculesperl.OneBcellsecretesbetween10and10antibodymoleculespersecond,correspondingto8and8antibodymoleculesperday(Bachmannetal.,1994AssumingthatoneBcellisactivatedbyoneviralepitope,weconsideranaverageantibodyproductionrateofmoleculesperdaypervirion(regardlessofantibodytype).Finally,weassumethatthedifferencebetweenantibody(Bcell)prolif-erationrateanddeathrateis1.4perday(Hodgkinetal.,1996;Janewayetal.,2001)andthatthesourceBcellscompetewitheachotherataratevaryingbetween2.5lperantibodymoleculeperday.Forthecontinuousimmunizationmodel(1),numericalresultsfortheinteractionbetweenfourvirusstrains,introducedinthebodyattimes10and15,inthesolepresenceofpoly-speci“cantibodyarepresentedinFig.1.Poly-speci“cneutralizingantibodyisproducedimmediatelyafterinfectionwithvirus,expandsatafastrate,andreachesitspeakatthesametimeasthevirus.Aslightdecreasetoitssteady-statevalueoccursthreedayslater.Whenandallsubsequentviralinfectionsdecayexponentially(leftpanel).Whenvirusespersistsandreachdifferentsteady-statevalues.Whileweknowthesteady-statepositionofthetotalvirusload,wecannotdeterminethepositionofindividualviralsteady-stateswhichishighlydependentoninitialconditionsandinoculationtimes.Iftwoorallvirusesareintroducedattimezeroandreplicateatthesamerate,thentheyhaveidenticaldynamics(notshown).When,however,virusesareintroducedlaterthanvirusandantibodytheyreachsteady-statesvaluesthatmaybedifferentthanthesteady…statevalueofvirusWhenbothpoly-speci“candstrain-speci“cneutralizinganti-bodiesarepresentedthenbothstrain-speci“candpoly-speci“cneutralizingantibodiesareproducedimmediatelyafterinfectionwithvirus.Theyexpandatafastrate,andreachhighsteady-statevalues.Whenandallsubsequentvirusstrainsdecayexponentially(Fig.2,leftupperpanel).Theintroductionofnewvirusstrainsleadstoproductionofcorrespondingstrain-speci“cantibodieswhichexpandtolowersteady-statevaluesduetocompetitionwithexistentanti-body.Whileweknowthetotalantibodyvalue,,wecannotpredictindividualantibodyvaluesatsteady-statewhicharehighlydependentofinitialconditionsandinoculationtimes.Ifoneorallstrain-speci“cantibodyareintroducesatthesametimeasthepoly-speci“cantibodythentheirsteady-statelevelsareidentical.Theotherstrain-speci“cantibodiesarepreventedfromexpandingduetotheircompetitionwithhighloadsatthetimeoftheirappearance(Fig.2,left,lowerpanel).,allvirusespersistandundergodampedoscillations.Thisisduetotheexistenceofcomplexeigenvalueswithnegativerealpartsin(29)(Fig.2,right,upperpanel).Strain-speci“cantibodiesreachthesamesteady-statewhichislowerthanthatofpoly-speci“cantibody(Fig.2left,lowerpanel).Thecombinedantibodyresponseisinef“cientincontrollingtheinfection.Puttingtogethertheanalyticalresultsregardingvirusdynamicsinthepresenceofpoly-speci“cneutralizingantibodiesaloneandinthepresenceofcompetingpoly-speci“candstrain-speci“cneutralizingantibodies,we“ndthatforany2for abK04r4 ab virusesareclearedinthepresenceofpoly-speci“cneutralizingantibodiesbutpersistinthepresenceofadditionalimmuneresponses,intheformofstrain-speci“cneutralizingantibodiesFig.3Forthenaturalinfectionmodel(30),weconsideraninitialviralloadof10virionsperl,whichmutatesovertimeatrate,varyingbetween1%and80%toproduceasecondvirus.Thevirusiseliminatedatrate10lperantibodymoleculeperdaybythestrain-speci“cneutralizingantibodiesand5lperantibodymoleculeperdaybythepoly-speci“cneutralizingantibodies(HollingerandLiang,2001).Therearenostrain-speci“candpoly-speci“cneutralizingantibodiesinthebodyatthetimeofinfection,i.e.A0moleculesperl.Anti-bodiesareproducedatratemoleculesperdaypervirionregardlessoftheantibodytype(Bachmannetal.,1994andcompetewitheachotheratratevaryingbetween10lperantibodymoleculeperday.Ratesareasintheimmunizationmodel.Numericalresultsfortheinteractionbetweentwovirusstrainsinthesolepresenceofpoly-speci“cneutralizingantibodiesandstepwisemutationmatrix,,arepresentedinFig.4.Infectionandmutantstrainleadstoimmediateproductionof Virus RNA per µ l (solid lines)Polyspecific antibody per µ l (dotted lines) 01020304050051015202530 1051001051010 V4V1V2V3V4V3A0V1V2A0 Fig.1.Virusdynamics(solidlines)inthepresenceofpoly-speci“cbroadlyneutralizingantibodies(dottedline):(leftpanel)fortheclearancecondition;(rightpanel)forthechroniccondition.Virusesareintroducedasindependentimmunizationsevery“vedays.Parameterusedinthesimulations,and(leftpanel);(rightpanel).S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 poly-speci“cantibodywhoclearsbothviruseswhen(leftpanel).When,onlythedominantvirusloadpersists(rightpanel).Theclearanceconditionsareindependentofthemutationrate,whichonlycontrolstherateatwhichthedominantvirusgetsestablished.Whenbothstrain-speci“candpoly-speci“cneutralizingantibo-diesarepresentandformutationmatrix,,satisfyingcondition(A),infectionwithandmutantstrainleadstoimmediateproduc-tionofbothstrain-speci“candpoly-speci“cantibodywhoclearbothviruseswhenFig.5,leftpanel).Poly-speci“candstrainspeci“cantibodiesareproducedimmediatelyafterinfectionandourmodelpredictsthetotalanti-bodyvalueneededforclearance,,butnottheindividualantibodyvaluesatsteady-state.When,andthemutationratesatis“esthenasinglechronicinfectionestablisheswherethedominantviruspersistsandthesuboptimalvirusiscleared.Thepoly-speci“canddominantvirus-speci“cantibodieshavethesamedynamicwhen,whiletheotherstrain-speci“cantibodyisdegradedandleavesthebodyFig.5,middlepanel).Whenandthemutationratesatis“esthenbothvirusstrainspersist.Thestrain-speci“cantibodyreachsteady-statelevelssmallerthanthe 5 10 15 20 25 1040 1020 100 1010 Virus RNA per µ l 0 10 20 30 40 5 1010 Days after inoculationPolyspecific antibody per µ l (A0)Strainspecific antibody per µ l (Ai) 010203040010203040 V1 V2 V4V1 V2A1 A2A2 A3 A4A0=A1A0 Fig.2.Virusdynamics(solidlines)inthepresenceofbothpoly-speci“cneutralizingantibodies(dottedline)andstrain-speci“cneutralizingantibodies(dashedlines):(leftpanel)fortheclearancecondition;(rightpanel)forthechroniccondition.Virusesareintroducedasindependentimmunizationsevery“vedays.ParametersareasinFig.1(leftpanel)and(rightpanel). 51001051010 Days after inoculation 020406005101520 10101051001051010 Virus RNA per µ l (solid lines)Strainspecific antibody per µ l (dashed lines)Nonspecific antibody per µ l (dotted lines)A0A0 Fig.3.Paradoxicalresultsshowing:(leftpanel)viralclearance(solidlines)inthesolepresenceofpoly-speci“cneutralizingantibodyresponses(dottedline),(rightpanel):viralpersistence(solidlines)inthepresenceofstronger(yetcompeting)poly-speci“c(dashedlines)andstrain-speci“cneutralizingantibodyresponses(dottedline)forparameterssatisfyingareasinFig.2 0 105 1010 Days after infectionVirus RNA per µ l (solid lines)Polyspecific antibody per µ l (dotted lines) 01234502468 2A0A0V1V1 V2 Fig.4.Virusdynamics(solidlines)followingnaturalinfectioninthepresenceofbroadlyneutralizingpoly-speci“cantibodies(dottedline)whenvirussmutationisdescribedbycondition(A):(leftpanel)fortheclearancecondition(rightpanel)forthechroniccondition.Parameterusedinthesimula-tionsare5and(leftpanel);(rightpanel).S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 poly-speci“cantibodyandtheircombinedavidityisinef“cientincontrollingthevirus(Fig.5,rightpanel).Thatparameterregion, abK04r4 abK0O,mo describestheinterestingsituationwherevirusesareclearedinthepresenceofpoly-speci“cimmuneresponsesandpersistinthepresenceofadditional(yetcompeting)immuneresponses,intheformofstrain-speci“cneutralizingantibodies(seeFig.65.DiscussionWedevelopedmathematicalmodelsofHIVviraldynamicsthataccountforthecounterintuitivehypothesisthatadditionalimmuneresponses(intheformofstrain-speci“cantibodyresponses)maybedetrimentaltothehostandleadtoviralpersistence.Weshowthatthisphenomenonmayoccurevenwhensuccessfulpreviousvaccinationleadstoproductionofpoly-speci“cbroadlyneutralizingantibodies.Forparameterregimesforwhichthepoly-speci“cbroadlyneutralizingantibodyresponse(whenoperatingalone,thatis,unaccompaniedbystrain-speci“cBcells)aresuf“cientforclearanceofthevirus,thepresenceofadditionalstrain-speci“cantibodiesleadstoviralreboundandintheendviralchronicity.Inthe“rstmodel,wedescribedhost…virusinteractionduringimmunizationwithseveralvirusstrainsandfoundthatforparameterssatisfying abK04r4 ab virusesgoextinctinthepresenceofonlypoly-speci“cbroadlyneutralizingantibodies,andpersistwhenbothstrain-speci“candpoly-speci“cneutralizingantibodiesarepresent(seeFig.3).Theresultsarebasedontheassumptionthatvirusesreplicateatthesamerate.Thisassumptionisjusti“edbyrecent“ndingsArnottetal.(2010)thathaveshownthatvirusesobtainedshortlyafterinitialinfectionfromindividualswhowerenottakingantiretroviraltherapy(ART)haveahigher“tnesslevelthanwaspreviouslybelievedandthereforejustifyingourassumptionthatcloselyinducedviralstrainscanhavethesamereplicationrates.However,sinceitisknownthatinthelong-run“tnesschangesandvirusesbecomemore(less)“tintheabsence(presence)ofARTwehaverunsimulationstoseehowthischangeaffectsourresults.Wehavefoundtheexistenceofparameterregimesforwhichfourvirusesreplicatingatdifferentratesgoextinctinthepresenceofonlypoly-speci“cbroadlyneutralizingantibodies,andpersistwhenbothstrain-speci“candpoly-spe-ci“cneutralizingantibodiesarepresent(seeFig.8).Analyticalresultsthatsustainthisobservationwillbepresentedelsewhere.Foranaturalinfectionwithavirusthatmutatesovertimeata,when abK04r4 virusesgoextinctinthepresenceofonlypoly-speci“cbroadlyneutralizingantibodies,andpersistwhenbothstrain-speci“candpoly-speci“cneutralizingantibodiesarepresent(seeFig.6).Theresultsaremaintainedwhenviruseshavedifferent“tnessrates(notshown).Theseresultsallowustoadvancetheideathatoneofthereasonsfortheabsence(orinef“ciency)ofpoly-speci“cbroadlyneutralizingantibodies,invivo,isthecompetitionbetweenplasmalymphocyesthatproducesthemwiththeplasmalym-phocytesproducingstrain-speci“cneutralizingantibodies.Parti-cularly,evenifsuchpoly-speci“cantibodiesarebeinginduced(saythroughsuccessfulvaccines),theymaybekeptatlowenoughlevelsbythemore“tstrain-speci“cantibodies.Para-doxically,thepresenceofadditional,speci“c,immuneresponseisdetrimentaltothehostandallowsfortheestablishmentofchronicinfections.Thesuppressionofpoly-speci“cimmunecellsbythemore“timmunecellshasbeendocumentedbothinHIVandHCVinfections(Heyman,2003;Zhangetal.,2004),buthasneverbeenimpliedasthereasonforviralchronicity.OurstudyimpliesthatcompetitionbetweenimmunecellsleadingtosuppressionofBcellscapableofinducingbroadlyneutralizingresponsesmaybesuf“cientforHIVtobecomechronic.The 1 2 3 105 100 105 1010 Viral RNA per µ l (solid lines)Polyspecific antibody per µ l (dotted lines)Strainspecific antibody per µ l (dashed lines) 5 10 Days after infection 5 10 A1A1A1V1V2A0=A2A0=A2A2A0V1 V2V2V1 Fig.5.Virusdynamics(solidlines)followingnaturalinfectioninthepresenceofbothpoly-speci“cneutralizingantibodies(dottedline)andstrain-speci“cneu-tralizingantibodies(dashedlines)whenvirussmutationisdescribedbycondition(A):(leftpanel)forcondition;(middlepanel)forthesingle-infectionchronicconditions;(rightpanel)forthemultiple-infectionchronicconditions.ParameterareasinFig.4,and5(leftpanel);8(middlepanel);5(rightpanel). 1 2 3 4 5 10010510 Viral RNA per µ l (solid lines)Polyspecific antibody per µ l (dotted lines)Strainspecific antibody per µ l (dashed lines) 0 5 10 15 20 V1V2V2A0V1A1A2A0 Fig.6.Paradoxicalresultsshowing:(leftpanel)viralclearance(solidlines)inthesolepresenceofbroadlyneutralizingpoly-speci“cantibodyresponses(dottedline),and(rightpanel)whenvirussmutationisdescribedbycondition(A):viralpersistence(solidlines)inthepresenceofstronger(yetcompeting)poly-speci“c(dashedlines)andstrain-speci“cneutralizingantibodyresponses(dottedline)fornaturalinfectionandparameterssatisfying.Weconsidernaturalinfectionwithvirusstrain,whichmutatesovertimeatrate.ParametersusedinsimulationsareS.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 implicationisevenmoreworrisomeifweconsiderthatthecontemporarystrain-speci“cantibodiesusuallyrecognizeandinhibitprecedingbutnotcurrentviralstrains(Burtonetal.,2004;Richmanetal.,2003;Weietal.,2003).Ourmodelassumesthatneutralizingantibodiestothecurrentdominantviralstrainareproducedimmediatelyaftervirusinfection,ormutation.Adelayintheirproductionbytwoweeksresultsinachronicinfectionwithanincreasedviralsetpointbyoneorderofmagnitude(resultsnotshown).Theanalyticalresultsforsystems(1)and(30)whenbothantibodytypesarepresentarebasedonlocalanalysis.Wehypothesizedthattheconditionsforlong-termviralclearanceandpersistenceareindependentofinitialconditions.Ourconclusionsarebolsteredbynumericalresults;futureworkisneededtoproveglobalstabilityanalytically.Inthemodelofnaturalinfectionwherethevirusmutates,theanalyticalresultsarepresentedfortheparticularcaseoftwovirus…twostrain-speci“cantibodypopulations.Numerically,wecanshowthatviralclearanceinthepresenceofbroadlyneutra-lizingpoly-speci“cantibodiesandviralpersistenceinthepre-senceofbothstrain-speci“candpoly-speci“cneutralizingantibodiesinthesameparameterregimeoccursforanynumberofnewviralstrains.Moreover,theanalyticalresultsforthemutationmatrixsatisfyingcondition(A)canbeobtainedforamutationmatrixsatisfyingcondition(B),aslongas,isirredu-cible(resultsnotshown).Theconditionofviralclearanceinthesolepresenceofthepoly-speci“cantibodyisgivenby.Since,viralchronicityinthepresenceofcompetingstrain-speci“candpoly-speci“cneutralizingantibodieshappenswhenthedoublechronicinfectionsteady-state(butnotthesingleinfectionone)exists(parametersdescribedincase3).Ifwecaninduceapoly-speci“cneutralizingantibodyofequalaviditytothatofstrain-speci“cantibody,thenforknownvirusaverageproductionandantibodylifespanwecandeterminethecorrelationbetweenthepoly-speci“cantibodylevelsandtheminimumavidityratesneededforthisconditiontofail.AsseeninFig.7,clearanceinthepresenceofpoly-speci“cbroadlyneutralizingantibodyalone(regions1and4)ismaintainedforcompetingpoly-speci“candstrain-speci“cantibodieswhenpoly-speci“cantibodylevelsaremuchhigherthanstrain-speci“cantibodylevels(region2).Whentheirlevelsaresimilarthenwecaninsureclearanceinthecompetitionmodelbyincreasingtheavidityofthepoly-speci“cbroadlyneutralizingantibody(region4).6.ConclusionsInthisstudy,weinvestigatedthehypothesisthatpoly-speci“cbroadlyneutralizingantibodiesdevelopalongsidestrain-speci“cneutralizingantibodies.Usingamathematicalmodelingapproach,wehavedeterminedtheparameterregionswherecompetitionbetweenpoly-speci“candstrain-speci“cantibodies 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 22.533.544.555.566.5x 109 /0K0 /(a) Region 2Region 3Region 4Region 1 Fig.7.Minimalvaluesofpoly-speci“cantibodyavidityrateforwhichvirusesareclearedbypoly-speci“c(solidline)andcompetingpoly-speci“candstrain-speci“cbroadlyneutralizingantibodiesformutationrates3asfunctionsof.Forhighpoly-speci“c/strain-speci“cantibodyratioclearancebypoly-speci“cantibodiesaloneissuf“cientforclearancebycompetingpoly-speci“candstrain-speci“cneutralizingantibodiesregardlessofmutationrates.Forsimilarvaluesofpoly-speci“cthanstrain-speci“cantibody,clearancebypoly-speci“cantibodyalonedoesnotinsureclearancebycompetingpoly-speci“candstrain-speci“cbroadlyneutralizingantibodiesforwhichweneedanincreaseintheavidityratesofpoly-speci“cantibodytocontroltheinfection.ParametersareasinFig.6 µ l (solid lines)Polyspecific antibody per l (dashed lines)Strainspecific antibody per µ l (dotted lines) 50 1020 1010 Days after inoculation 050100150200010203040 6 108 1010 10 20 30 40 100 105 1010 105 10 20 30 40 106 108 1010 A1 A2 A3V2V1V4V3V1 V2 V3 V4A0A0 Fig.8.Paradoxicalresultsshowing:(leftpanel)viralclearance(solidlines)inthesolepresenceofpoly-speci“cneutralizingantibodyresponses(dottedline),and(rightpanel):viralpersistence(solidlines)inthepresenceofstronger(yetcompeting)poly-speci“c(dashedlines)andstrain-speci“cneutralizingantibodyresponses(dottedline)forviruseswithdifferentreplicationrates.ParametersareasinFig.3S.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 canhelpthevirusescapeandpredictedwaysofpreventingit,providinginsightintotheultimaterolesofantibodyresponsesincontrollingHIVinfection.Wepredictthatinapreventivevaccinationdesign,onecanprevent/delayviralchronicitybyinducingpoly-speci“cantibodiesofhighavidity,increasingtheoverallpoly-speci“cantibodylevelsandbyspeedingtherateatwhichvirusesmutate.S.M.C.acknowledgessupportfromNSFGrantDMS-1022865.WethanktheanonymousreviewersfortheircommentsandAppendixAVirusevolutionduringpoly-speci“cantibodyresponse:Weassumethatprimaryinoculationleadstoinfectionbyasinglevirusstrain.Thisvirusmutatesovertimeunderpressurefrompoly-speci“cimmuneresponse,,givingrisetodistinctvirusstrainsatratesgivenbythemutationmatrix,satisfyingcondition(A)or(B).Asbefore,weconsiderthereplicationratetobeindependentofthevirus.Themodeldescribingthevirus-hostinteractionbecomes dVdtK0A0InÞV, 0and0,for0,wecande“ne),for.Thenthefollowingresultfollows.PropositionA.1.Thedynamicsofareequivalenttothedynamicsof dfdt¼rðQInÞf, dTdtK0A0ÞT, formutationmatrixQsatisfyingconditionNotethat 0and11.Weseethatthereisabijectivecorrespondencebetweensolutionsof(38)with0,andsolutionsof(39).Steady-stateanalysis:Noticethatthelasttwoequationsof(39)aredecoupledfromthe“rst.Westudythestabilityofsteady-statesofthisplanarsubsystemnext.System(38)hasthreesteady-states:1.Theinfectionfreesteady-state2.Theviralclearancesteady-state3.Thechronicsteady-state¼ðð,whichexistsforPropositionA.2.Theinfectionfreesteady-stateSisalwaysIfrthenSitisGASwithrespecttoinitialconditionsIfrthenSitisGASwithrespecttoinitialconditionsTheproofissimilartothatofPropositions2and3.TheoremA.3.IfrtheneverysolutionwithTÞÞ¼ð rK0l0b rK0a wherezisthedominanteigenvectorofthesimpleeigenvaluethematrixQTheprooffollowsfromProposition6ifweshowthat.Fromtheeigenvectorexpansionwehavethatwheretherealpartofallsisnegativeandandsince11forall,itfollowsthatCorollaryA.4.UndertheconditionsofTheoremeverysolutionÞÞ¼ rK0l0b  ThisisimmediatefromTheorem1sincePropositionA.5.IfrtheneverysolutionThisisimmediatefromProposition6since0AppendixBVirusevolutionduringbothstrain-speci“candpoly-speci“cantibodyresponses:Letusconsiderthegeneralcasegivenbysystem(30),wherevirusstrainmutatesovertimeunderthepressureofbothspeci“candpoly-speci“cimmuneresponsestoproduce1geneticallydistinctvirusstrains)andtheircorrespondingstrain-speci“cneutralizingantibodies,.Moreover,themutationmatrix,,satis“escondition(A).Similarresultsfollowforcondition(B)andwillnotbeshownhere.Steady-stateanalysis:Westudyviruseslongtermbehaviorforthecase2andmutationmatrixsatisfyingcondition(A),mutatestoproducevirusatrateandthemutationisirreversible.System(30)becomes dV1dt¼rð1mÞV1V1ðKA1þK0A0Þ, dV2dt¼rmV1þrV2V2ðKA2þK0A0Þ, dA1dt¼lV1þA1ðabATÞ, dA2dt¼lV2þA2ðabATÞ, dA0dt¼l0ðV1þV20ðabATÞ,ð46ÞwithV1(0)¼V1,0,V2(0)¼A1(0)¼A2(0)¼A0(0)¼0. Indeed,for2,wehavethat0,andthus1Thisimpliesthat0andthusthatS.M.Ciupeetal./JournalofTheoreticalBiology277(2011)55…66 System(46)hasatmostfourtypesofnon-negativesteady-1.Theinfectionfreesteady-state2.Theviralclearancehyperplane A1, A2, ,suchthat 3.Thechronicsingleinfectionsteady-state rK0l0þKlb rðlþl0ÞK0l0þKla0, rlK0l0þKl, whichexistswhen4.Thechronicco-infectionsteady-state 1, V2, A1, A2, A0Þ,wherex¼ K0Kþ ll0,y¼2 K0Kþ ll0, A0¼  A1¼ rð1mÞK K0K A0, A2¼ K0Kþ ll0 0 rð1mÞK, V1¼ 1lb1þ ll0 0a ð1mÞK K0K A0 V2¼ 1lb1þ ll0 0a 0Kþ ll0 0 existswhen 1:When2:When3:When4:Whenthenallfoursteady-statesexist.PropositionB.1.Theinfectionfreesteady-stateSisalwaysTheviralclearancehyperplaneShaszeroeigenvalues.Thechronicsingleinfectionsteady-stateSisasymptoticallystableforparameterssatisfyingconditionaandunstableforparameterssatisfyingconditionThechronicco-infectionsteady-stateSisasymptoticallystablewheneveritexistsforparameterssatisfyingconditionsTheJacobianmatrixcorrespondingtothelinearizedsys-temis V10K0 V1rmY20K V2K0 V2l0ab ATb A1b A1b A10lb A2ab ATb A2b A2l0l0b A0b A0ab ATb A00BBBB@1CCCCA,ð50ÞwithY1¼rð1mÞK0 A0K A1,Y2¼rK0 A0K (a)Thecharacteristicequationofhaspositiveeigenvalues.There-fore,theinfectionfreesteady-stateisalwaysunstable.(b)Thecharacteristicequationofhyperplane A0K A2ÞÞðLðrð1mÞK0 A0K ÞÞ¼haseigenvalues A0K A2andL5¼rð1mÞK0 A0K .Therefore,isunstablefor A0þK A2,1=ð1mÞðK0 A0þK .Sincesomeeigenvaluesarezero,thestabilitycannotbedeterminedfromlinearanalysis.Wewillshownumericallythehyperplaneisasymptoticallystableforparameterssatisfyingcondition(A).(c)Thechronicsingleinfectionsteady-statehasthechar-acteristicequation .Theeigenvalue isnegativewhenandpositiveotherwise.Therestoftheeigenvalues rðl0þlÞK0l0þKl,L4,5¼ arenegativewhen.Thishappenseverytimesteady-state(d)Let .Wewillshowthatwhenthechronicsteady-stateexists(respectively,whenispositive)itisasymptoticallystable.Thecharacteristicequationcorrespond-ingtothechronicsteady-stateis ATaÞðL4þa1l3þa2L2þa3Lþa4,ð57Þwherea1¼2oþb ATþrm V1 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