THEMATHEMATICALMODELSOFNUTRITIONALPLASTICITYANDTHEBIFURCATIONINANONLOCALDIFFUSIONEQUATIONByYuLiangADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofAppliedMathematics-DoctorofPhilosophy2016ABSTRACTTHEMATHEMATICALMODELSOFNUTRITIONALPLASTICITYANDTHEBIFURCATIONINANONLOCALDIFFUSIONEQUATIONByYuLiangThethesisconsistsoftwoparts.Inthepart,IinvestigatethedevelopmentalmechanismsthatregulatethenutritionalplasticityoforgansizesinDrosophilamelanogaster,thefruit.HereIfocusontheinsulin-likesignallingpathway(IIS)throughwhichthedevelopmentalnutritionissignalledtogrowingorgans.Twomathematicalmodels,anODEmodelandaPDEmodel,areestablishedbasedontheIISpathway.IntheODEmodel,thegeneexpressionofeachcomponentinIISpathwayisconsideredasmodelvariables.ByanalyzingthesteadystatesoftheODEmodelunderdierentparametersettings,thehypothesisthatthedierenceofthenutritionalplasticityamongallorgansofDrosophilamelanogasterisduetothevariationofthetotalgeneexpressionsofcomponentsinIISpathwayisvFurthermore,theforkheadtranscriptionfactorFOXO,anegativegrowthregulatorthatisactivatedwhennutritionandinsulinsignalingarelow,isakeyfactortomaintainorgdierencesinnutritional-plasticityandinsulin-sensitivity.InthePDEmodel,Iincludethecellularstructureandtransportationwithinthecell.Thetransportationofproteinsbetweenthenucleusandthecellmembraneismodelledwithanadvection-diusionpro-cess.InsimulationsofthePDEssystem,thehypothesisthattheconcentrationofFOXOdecreasesastheconcentrationofinsulinincreaseisvInthesecondpartofthethesis,IstudythebifurcationpropertiesofthenonlocalChafee-Infanteproblem:Lu+(uu3)=0:Here,insteadoftheLaplacian,LuisanintegralbyLu=Zˇ03J(yx)(u(y)u(x))dy;whereJ(x)isacontinuous,non-negative,radiallysymmetrickernelwithJ(0)>0.Itisshownthatasthescalingparameterapproacheszero,theequationhaspitchforkbifurcationsattheeigenvaluesofLandtheseeigenvaluesareclosetothoseofcJ,withcJconstant.Aconcreteexampleisconsidered,andthebifurcationresultisdemonstratedbysolvingtheequationwithNewton'sMethod.ACKNOWLEDGMENTSIwouldliketoexpressthedeepestappreciationtomyadvisor,Dr.PeterBates,forhisguid-ance,assistance,encouragement,andheartysupportinallthephasesofmydoctoralprogramatMichiganStateUniversity.Thisdissertationwouldnothavebeenpossiblewithouthisguidanceandpersistenthelp.Iwouldliketothankmycommitteemembers,Dr.KeithPromislow,Dr.MoxunTang,Dr.ChichiaChiufortakingtimetoserveonmydissertationcommitteeandfortheirusefulcommentsandsuggestions.IamthankfultoDr.AlexzenderShingletonforvaluablediscussions,duringwhichtheypro-videdmewithinspiringsuggestionsaboutmyresearch.IamgratefultoJiLi,JiayinJin,YinCao,QiliangWuandmanyothersfortheirfriendshipatMichiganStateUniversity.Atlast,butbynomeanstheleast,myheart-feltappreciationsgotomydearestfamiliesfortheircontinuousandtime-invariantsupport.Withouttheirunderstanding,thisworkcannotbedone.ivTABLEOFCONTENTSLISTOFFIGURES......................................viChapter1Introductionofthemathematicalmodelsofnutritionalplasticity....1Chapter2Backgrounds.................................42.1Sizeandshapeininsects...............................42.2Theinsulinsignalingpathway............................8Chapter3AODEmodel.................................113.1Preliminary......................................113.2TheOrdinaryDierentialEquationsmodeloftheIISpathway...........133.2.1Insulinreceptorsubsystem..........................133.2.2Chico/PI3Ksubsystem............................153.2.3Lipidssubsystem...............................163.2.4Aktsubsystem................................173.2.5dFOXOsubsystem..............................183.3Steadystateofthesystem...............................203.4Thenutritionalplasticity...............................243.5Theresults.......................................253.6Modelcoecients...................................26Chapter4APDEmodel.................................384.1Preliminary......................................384.2ThePartialDierentialEquationsmodeloftheIISpathway.............404.2.1Insulinreceptorssubsystem..........................404.2.2Chico-PI3Kcomplexsubsystem.......................454.2.3Lipidssubsystem...............................474.2.4Aktsubsystem................................484.2.5FOXOsubsystem...............................494.2.6PTPases....................................514.3Theresults.......................................514.4Modelcoecients...................................54Chapter5ThebifurcationsofanonlocalChafee-Infanteproblem..........585.1Introduction......................................585.2Thelocalbifurcation.................................595.3Aconcreteexample..................................65BIBLIOGRAPHY.......................................73vLISTOFFIGURESFigure2.1:Therelationshipbetweenreactionnormsandallometries.............6Figure2.2:ThelifecycleofDrosophilamelanogaster....................7Figure2.3:ThenutritionalregulationofbodyandorgansizeinDrosophilamelanogaster.8Figure2.4:TheIISandTOR-signalingpathwayinDrosophilamelanogaster........9Figure3.1:ThestructureofIISpathway............................30Figure3.2:TheconcentrationofactivateddFOXOv.s.theconcentrationofinsulin.....31Figure3.3:Thenumberofcellsofwingv.s.theconcentrationofinsulin...........31Figure3.4:Thenutritionalplasticityv.s.thegeneexpressionofInr.............32Figure3.5:Thenutritionalplasticityv.s.thegeneexpressionofChico............33Figure3.6:Thenutritionalplasticityv.s.thegeneexpressionofPI3K............34Figure3.7:Thenutritionalplasticityv.s.thegeneexpressionofAkt.............35Figure3.8:Thenutritionalplasticityv.s.thegeneexpressionofdFOXO...........36Figure3.9:Thereisanon-linearrelationshipbetweendFOXOexpressionandnutritionalplasticity......................................36Figure3.10:Thenutritionalplasticityv.sthegeneexpressionofInranddFOXO.......37Figure3.11:Thenutritionalplasticityv.sthegeneexpressionofAktanddFOXO......37Figure4.1:Thestructureofthecell..............................44Figure4.2:NuclearFOXOvs.Insulin............................53Figure4.3:TheSensitivityofActivatedFOXO.......................54Figure5.1:ThekernelfunctionJ(x)..............................65Figure5.2:solutionsofthenonlocalequation........................66viFigure5.3:Comparisonofthesolutionsofthenonlocalequationwith=2andtheChafee-Infanteequation..............................68Figure5.4:Comparisonofthesolutionsofthenonlocalequationwith=3andtheChafee-Infanteequation..............................69Figure5.5:Comparisonofthesolutionsofthenonlocalequationwith=4andtheChafee-Infanteequation..............................69Figure5.6:Thebifurcationofthenonlocalequation..................70Figure5.7:Thesecondbifurcationofthenonlocalequation.................70Figure5.8:Thethirdbifurcationofthenonlocalequation..................71Figure5.9:Thesolutionsofthenonlocalequationwithvarying..............71Figure5.10:Thesolutionofthenonlocalequationthebifurcationbranchwhen=1000072viiChapter1IntroductionofthemathematicalmodelsofnutritionalplasticityOneofthequalitiesoflifeisthatorganismsareabletorespondtoenvironmentalstimuli,eitherdevelopmentally,physiologicallyorbehaviorally.Thisprocess,calledphenotypicplasticity,underliessuchdiversephenomenaastheeectofdevelopmentalnutritiononadultbodysizeinanimals,themelanizationofhumanskininresponsetoUVirradiation,andescaperesponsetoperceivedpredatorsinanimalprey.Animportantaspectofphenotypicplasticityisthatthedegreeoftheresponseisappropriatetothedegreeofthestimulus.Apoormatchbetweenstimulusandresponsecanhavesevereimplicationsfororganismalformandfunction.Forexample,inhumansanaphylaxisresultsfromaninappropriatelyseverereactiontoallergens.Conversely,type2diabetesisaresultofareducedresponsetocirculatingsugarsinthebloodstream.Workoverthelast50yearshasrevealedmyriadsignalingpathwaysthattransduceenvironmentalinformationtodevelopmental,physiologicalorneurologicalprocesses.Nevertheless,whilstwehaveagoodunderstandingofthecomponentsofthesepathway,thefactorsthatregulatewhetherasignalingpathwayorattenuatesvariationinanenvironmentalsignalarelesswellelucidated.Moregenerally,wehaveapoorunderstandingofhoworganismsandthecellswithinthemareabletomanipulatesignaltransductionpathwaystoregulatethedegreeofresponsetoenvironmentalchange.1Duetotheinnatecomplexitiesofsignalingpathways,afruitfulapproachtounderstandinghowtheyamplifyorattenuateenvironmentalsignalsistomodeltheminsilico.Thesemodelscanbeusedtoexplorehowchangesinpathwaystructureandfunctionaectshowthepathwaytrans-ducesenvironmentalinformation.Theresultinghypothesescanthenbetestedinvivo,inmodelorganismswheregeneticengineeringallowsprecisemanipulationofthesignalingpathway.Theinsulin/IGF-signaling(IIS)pathwayisanidealsystemwithwhichtoexplorethefunctionofsig-nalingpathwaysthatunderliephenotypicplasticity(herereferredtoasplasticitypathways).IndevelopinganimalstheIISpathwayregulatesgrowthanddevelopmentinresponsetonutrition.InadultanimalstheIISpathwayregulates,amongotherthings,theuptakeofsugarsfromthebloodstream(invertebrates)orhemolymph(inarthropods).Inmammals,thesemetabolicandmitogenicprocessesaremediatedbytheinsulinreceptorandtheinsulin-likegrowthfactor(IGF)receptors,respectively.Incontrast,arthropodssuchasDrosophilahaveasingleinsulinreceptor.Neverthe-less,allinsulin/IGFreceptorsfeedintothesamepathway,whichisextremelyconservedamongallanimalsandhasbeenwellelucidated.,insulin-likepeptides(ILPs)arereleasedintothebloodstreamorhemolymphinresponsetodevelopmentalnutrition.Thesepeptidesthenbindtotheinsulinreceptoronthecellmembraneandinitiateasignal-transductioncascadethatultimatelyregulatestheexpressionofgenesthat,whentranslatedintoproteins,regulatecellgrowth,prolif-erationandmetabolism.Indevelopinganimals,areductioninnutritionreducessignalingthroughthepathway,whichinturninitiatesthetranscriptionofgrowthinhibitors.Theresultisareductionincellgrowthandproliferationleadingtoadecreaseinorganandbodysize.Importantly,notallorgansshowthesamegrowthresponsetochangesindevelopmentalnutritionandIIS.Inthefruit,Drosophilamelanogaster,areductionindevelopmentalnutritionhasmoreofaneectonwingsizethanongenitalsize,andthisisaconsequenceofgenitalgrowthbeinglesssensitivetochangesinIIS.Similarly,inmammalsthedevelopingbrainisrelativelyinsensitivetochangesin2nutrition,aphenomenoncalledheadsparing.Suchorgdierencesinnutritional-andinsulin-sensitivityisfundamentaltoensurethatbodyproportioniscorrectacrossarangeofadultsizes.WorkonDrosophilahasrevealedthatthereducedinsulin-sensitivityofthegenitaliaisacon-sequenceofchangesintheexpressionofkeygenesintheIISpathway,theforkheadtranscriptionfactorFOXO,anegativegrowthregulator.WhenIISishigh,FOXOisphosphorylatedbytheIISpathway.ThisdisruptsDNAbindingandcausesFOXOtotranslocatetothecytoplas-m.AdeclineinIISleadstode-phosphorylationofFOXO,whichaccumulatesinthenucleusandinitiatesthetranscriptionofgrowthinhibitors,aswellastheinsulinreceptoritself.ThegenitaliaofDrosophilaareabletolimittheirsizeresponsetochangesinnutritionandIISbyexpressingonlylowlevelsofFOXO.Consequently,adeclineindevelopmentalnutritiondoesnotresultintheactivationofgrowthinhibitorsinthegenitalia,allowingthemtomaintaingrowthevenasthegrowthofotherorgansandofthebodyasawholeisslowed.Thesestudiesdemonstratethatchangesintheexpressionoractivityofgeneswithinplasticitypathwayscanaectwhethertheyamplifyorattenuatevariationinanenvironmentalsignal.WhatisunclearisthegeneralityofthisDochangesinexpression/activityofothercomponentsinthepathwayhavethesameeect?Howdofeedbackloopswithintheplasticitypathwayaf-fecthowittransducestheenvironmentalsignal?Canwemakebroadgeneralizationsastohowchangesinaplasticitypathwaysstructureandfunctionchangessensitivityofphenotypetoenvi-ronmentalsignals?Inthisdissertation,weestablishthemathematicalmodelsoftheinsulin-likesignalingpathway,whichserveasthefundamentaltoolsforexploringthemechanismsregulatingthenutritionalplasticityoforgansinDrosophilamelanogaster.3Chapter2Backgrounds2.1SizeandshapeininsectsIntuitively,individualswithlargerbodiestendtohavelargerconstituentparts.Forinstant,largerhumanswillhavelongerarmsandlegs,biggerliversandlargerhearts.Biggerfruittendtohavelargerwingsandlegs.Thisscalingrelationshipbetweenthesizesofindividualtraitsandthesizeofthewholebodyiscalledallometry([13]).Allometriesaretraditionallymodelledusingtheallometricequationy=axb;wherexandyaretwogiventraits.ALog-transformationofthisequationproducesthelinearequationlog(y)=log(a)+blog(x),i.e.,alinearrelationbetweenthelogoftraitsizexandthelogoftraitsizey.Furthermore,theslopeofthelinearrelationisrepresentedbytherealnumberb.Thus,weareabletoclassifytheallometriesaccordingtothevalueofb.Wesaythattwotraitsscaleisometricallywhenb=1.ThisisthecaseofpalpsizeagainstbodysizeinD.melanogaster(see2.1).Traityscalehypometricallytotraitxwhenb<1,whichisthecaseofgenitalsizeagainstbodysizeinDrosophilamelanogasterin2.1.Traityscalehypermetricallytotraitxwhenb>1.Allometrynotonlydescribesthescalingrelationshipbetweentraitsandbody,butalsothescal-ingrelationshipamongtraitsthemselves.Generally,itisbytheproportionalchangeinthedimensionsofonetraitrelativetoanothertraitortooverallbodysize.More,therearethreetypesofallometries:ontogenetic,evolotionary,andstaticallometries.Ontegeneticallome-triesdescribethegrowthtrajectorieswithinasingleindividual.Theycharacterizethegrowthof4anorganrelativetothegrowthofanotherorganorgrowthofthebodyinasingleindividual.Evo-lutionaryallometriesdescribetherelativesizeofdierentorgansamongindividualsatthesamedevelopmentalstageacrossspecies.Staticallometriesaresimilartoevolutionaryallometriesbuttheydescribetherelativesizeofdierentorganswithinspecies.Whilethevariationofgrowthinevolutionaryallometriesiscausedbytheevolvingofdierentspeciesintheevolutionaryprocess,thevariationofgrowthinstaticallometriesmayduetothegeneticdierencebetweenindividuals,thedierenceintheenvironmentinwhichtheydevelopedorduetotheinteractionbetweenthetwo.Hence,themechanismsbehindthestaticallometriesaccordingtogeneticandenvironmentalfactorsareverydierent.Furthermore,boththegeneticandenvironmentalfactorscouldgeneratevariousallometricrelationshipsduetothedierentsourcesofthosegeneticandenvironmentalfactors.Forinstance,temperature,nutrition,sunlight,etc,aredierentenvironmentalfactorsthatcouldcontributetothestaticallometries.Similarly,theremaybedierentsourcesofgeneticvariation,suchthatallelicvariationatonelocusmayproduceadierentallometriesthanvaria-tionatanother.Thissuggeststhatresearchersshouldinvestigatedierentbiologymechanismsinregardstoanyfactorofthestaticallometries.Inthisdissertation,wemainlyfocusoninvestigatingthemechanismsrelatedtotheenvironmentalstaticallometries,especiallyforstat-icallometriesduetothenutritionalvariation.WeeliminatethegeneticfactorsbyreproducingDrosophilamelanogaster,fruit,withthesamegenotypeinthelabandstudyanyenvironmentalfactorsbycontrollingtheenvironmentwherethefruitareraised.Environmentalfactorsaectthestaticallometriesbecausetheyregulatetherateanddurationofcellgrowthanddivision.Thedevelopmentalresponsetotheenvironmentiscalledphenotypicplasticity.Theresponseoftraitsizeagainstthevalueofaparticularenvironmentalvariableforasinglegenotypeisthereactionnorm.Itdescribesthepatternofphenotypicvariationproducedbyasinglegenotyperearedunderarangeofenvironmentalconditions.Forinstance,malnutrition5Figure2.1:Therelationshipbetweenreactionnormsandallometries.duringdevelopmentreducesadultsizeinDrosophilamelanogaster.Fig2.1aandbshowthereactionnormsforwingareaandthoraxlengthofDrosophilamelanogasterasafunctionoflarvalnutrition.Combining2.1aandb,wegetthenutritionalstaticallometriesbetweenwingareaandthoraxlength.([13])Hence,theenvironmentalstaticallometriesoftwotraitsaredirectlyrelatedtothereactionnormsofthetwotraitswithrespecttoanenvironmentalvariable.Asoneofthetypicalholometabolousinsects,Drosophilamelanogasterbeginlifeasworm-likelarvae,moltingthroughthreelarvalinstarsbeforeundergoingcompletemetamorphosisaspupaandeventuallyclosingintotheiradultform,shownin2.2.Adulthaveastiexoskeletonandsotheycannotcontinuetogrow.Hence,adultbodysizeisentirelyregulatedbygrowthduringthepremetamorphiclarvalstages.Further,theadultorgansofDrosophilamelanogasterarenotvisibleuntilaftermetamorphosis.Infact,theygrowasimaginaldiscswithinthedevelopinglarvae,eachdisccorrespondingtoanadultstructure.Duringmetamorphosistheirimaginaldiscsdierentiateandevaginatetoformtheadultorgans.Thus,theadultorgansizeisalsodeterminedbygrowthoftheimaginaldiscsduringthelarvalstages.Thosephysiologicalprocessescontrollingthemeta-morphosisofDrosophilamelanogasterareregulatedbyseveralhormones.Atsomepointinthelarvalinstar,attainmentofaparticularbodysizeisassociatedwithareductioninthelevelsofcirculatingjuvenilehormones.Thissizeiscalledthecriticalsize.Oncecriticalsizeisattained,6Figure2.2:ThelifecycleofDrosophilamelanogaster.alarvairreversiblyinitiatesthehormonalcascadethatendsinmetamorphosis,andsothereisadelaybetweentheattainmentofcriticalsizeandtheterminationofbodygrowth.Thisperiodisreferredtoastheterminalgrowthperiod(TGP).Thus,wehavethefollowingformulasaboutthebodyororgansizeofDrosophilamelanogaster:F=CS+Rt;whereFisthebodyororgansize,CSiscriticalsize,RistherateofgrowthduringTGPand7tisthedurationofTGP.Figure2.3:ThenutritionalregulationofbodyandorgansizeinDrosophilamelanogaster.2.2TheinsulinsignalingpathwayThebodyandorgansizesinallanimalsvarywhenthedevelopmentalnutritionchanges.InDrosophilamelanogasterparticularly,thegrowthresponsetonutritionismediatedthroughseveralinter-connectedhormonalsystems.OneofthemisthroughthereleaseofdILPsinthebrain.([12])Theinsulin-likesignalingpathway(IIS)systemiscomprisedofthreepathways2.4):theIISpathway;theTargetofRamapmycin(TOR)signalingpathway,andtheAMP-dependentkinase8(AMPK)pathway.Thesepathwaysareextremelyconservedamongallanimals,andareessnetial-lyidenticalinDrosophilamelanogasterandvertebrates.However,inDrosophilamelanogasterthemetabolicandmitogenicrolesofIIS,whichinvertebratesareseparatedintoinsulin-likeandgrowth-factorsignalingrespectively,arecombinedintoasinglepathwaywithasingleinsulin-receptor(InR).AreductionindILPproductioncausesareductioninbodyandorgansize,whileincreasingthecencentrationofdILPscasusesanincreaseinbodyandorgansize.Figure2.4:TheIISandTOR-signalingpathwayinDrosophilamelanogaster.ThefreelydiusingdILPsmoleculescirculateintheextracellularofDrosophilamelanogaster,9wheretheybindtoinsulinreceptors(InR)locatedonthecellularmembraneofdividingcells.Bind-ingofdILPtoInrresultsinreceptorautophosphorylation.Thereceptormaytheneitherbindasec-onddILP,whichdoesnotaectitsphoshphorylationstate,ormaydissociatefromthedILP,whichcausesdephosphorylation.ThedInrmoleculemayalsobedephosphorylatedbyproteintyrosinephosphotases(PTP).Membrane-boundphosphorylatedreceptorscanbereversiblyinter-nalizedthroughendocytosis,wheredephosphorylationbyPTPreleasesthemintotheintracelluarpooloffreeInR.Thesefreereceptorscanthenbereversiblyreintegratedintothecellmembrane,wheretheybecomeavailabletobindingwithdILPs.InadditiontothisrecyclingofInRsfromthecellmembrane,unphosphorylatedreceptorsalsoentertheintracellularpoolthroughsynthesisandleaveitthroughdegradation.ThephosphorylatedInRrecruitsinsulinreceptorsubstrate(Chico)tothemembraneandphosphorylatesit,whereuponitformsacomplexwithPi3K.TheresultingIRS-PI3Kcomplexisanactiveproteinkinase.ThephosphorylatedIRS-PI4Kcomplexphosphrylatesphosphatidylinostiol4,5-bisphosphate,PI(4,5)P2toPI(3,4,5)P3,atthecellmembrane.Addition-ally,PI(3,4)P2isconvertedtoPI(3,4,5)P3independentlyofIRS-PI3K.TheAktmoleculebindstoPI(3,4,5)P3atthecellularmembraneandactivatestheserine/threoninekinaseAkt.ActivatedAktdetachsfromthecellmembraneandtranslocatestothenucleus(J.CellSci.114,2903-2910),whereitphosphorylatestheforkheadtranscriptionfactorFOXO.TranscriptionaltargetsofFOXOincludenegativegrowthregulators,forexample4E-BP,aswellastheinsulinreceptorInr.Phos-phorylationbyAktcauseFOXOtotranslocateoutofthenucleusandhenceloseitstranscriptionalactivity.10Chapter3AODEmodel3.1PreliminaryTogetamathematicalmodeloftheIISpathwayincludingdFOXO,wedevelopanODEsystembasedonSedaghatetla'smodel([11]).WetakeSedaghatetla'smodelwithvariablesfromx2tox17,thenincludenewvariablesfortheactivatedanddeactivateddFOXO.Forbetterexploringourquestionsandmodellingtherealityofthepathway,wemodifySedaghatetla'smodelfromseveralaspects.First,weadddegradationsforeachofthecomplexinthepathwayandincludethebasaltranscriptionstotheunphosphorylatedstatesofinsulinreceptor,Chico,PI3K,AktanddFOXO.Second,wemodelthelipids,Aktofthepathwaybyconcentrationinsteadofpercentage.Third,amathematicalmodelofdFOXOsubsystemisdevelopedbythemechanismoftheinteractionofdFOXOandAkt.ApositivefeedbackfromactivateddFOXOtoinsulinreceptorsisincludedinthemodel.Fourth,therelationbetweenactivityofPTPasesandactivatedAktismodeledbyasmoothlyexponentialfunctioninsteadofalinearfunction.TheillustrationofthenewmodelabouttheIISpathwayisshownin3.1.Herewelistthevariablesrepresentingthemoleculesconcentrationsinthesignallingpathway:ILP:I,concentrationofinsulinInR:R1(t),concentrationofunboundunphosphorylatedcell-surfacereceptors,11R2(t),concentrationofonce-boundunphosphorylatedcell-surfacereceptors,R3(t),concentrationofphosphorylatedtwice-boundcell-surfacereceptors,R4(t),concentrationofphosphorylatedonce-boundcell-surfacereceptors,R5(t),concentrationofunboundunphosphorylatedintracellularreceptors,R6(t),concentrationofphosphorylatedtwice-boundintracellularreceptors,R7(t),concentrationofphosphorylatedonce-boundintracellularreceptors.Chico:C1(t),concentrationofunphosphorylatedChico,C2(t),concentrationofphosphorylatedChico,PI3K:3(t),concentrationofdeactivatedPI3K,(t),concentrationofphosphorylatedChico-PI3Kcomplex.Lipids:P3(t)betheconcentrationofPI(3;4;5)P3,P4(t)betheconcentrationofPI(3;4)P2,P5(t)betheconcentrationofPI(4;5)P2.Akt:A(t),concentrationofdeactivatedAkt,Ap(t),concentrationofactivatedAkt.dFOXO:F(t),concentrationofactivatedFOXO,f(t),concentrationofdeactivatedFOXO.PTP:P(t):AprefactorrepresentingtherelativeactivityofPTPasesinthecell.123.2TheOrdinaryDierentialEquationsmodeloftheIISpath-way3.2.1InsulinreceptorsubsystemInthedrosophilainsulinreceptorssubsystem,eachfreedrosophilainsulinreceptorcouldbepotentiallyboundwithtwoinsulin.Unboundandboundinsulinreceptorcyclebetweencellmem-braneandcytoplasma.BoundinsulinreceptorsonthecellmembranewouldphophorylateChicoandinitiatesasignaltransductioncascade.ThesynthesisofR1:Freeinsulinreceptors(R1)onthemembranebindtoinsulin(I)andbe-comeonce-boundunphosphorylatedsurfacereceptors(R2)attheratek1.Thatreactionisreversiblewithratek1.Phosphorylatedonce-boundsurfacereceptors(R4)aredephosphorylatedbyPTPas-es,releasetheirinsulinandbecomeunboundunphosphorylatedsurfacereceptors(R1)withratek3P.Atthesametime,freesurfacereceptors(R1)passthroughthecellmembranetobecomeintracellularreceptors(R5)withratek4andtheintracellularreceptorsattachtothecellmembrane,becomingsurfacereceptorswithratek4.Finally,acertainfraction(d)ofreceptorsdegradeandarelost.Therefore,thesynthesisrateoffreereceptoronthemembrane,R1,isexpressedbyR1=k1IR1+k1R2+k3PR4+k4R5k4R1dR1:(3.2.1)ThesynthesisofR2:InadditiontotheexchangeswithR1,describedabove,theonce-boundunphosphorylatedsurfacereceptors(R2)degradeatthesameratedandarephosphorylatedtobecomephosphorylatedonce-boundsurfacereceptors(R4)attheratek3.Therefore,thesynthesis13rateofR2isR2=k1IR1k1R2k3R2dR2:(3.2.2)ThesynthesisofR3:Phosphorylatedonce-boundsurfacereceptors(R4)bindtoinsulin(I)andbecomephosphorylatedtwice-boundsurfacereceptors(R3)withratek2.Thisreactionisreversiblewithratek2.Atthesametime,phosphorylatedtwice-boundsurfacereceptors(R3)passthroughthecellmembranewithratek40tobecomephosphorylatedtwice-boundintracellularreceptors(R6).Thisprocessisreversiblewithratek40.Therefore,thesynthesisrateofR3isR3=k2IR4k2R3+k40R6k40R3dR3:(3.2.3)ThesynthesisofR4:InadditiontotheexchangeswithR1;R2;andR3,describedabove,phos-phorylatedonce-boundsurfacereceptors(R4)passthoughthemembranewithratek40,becomingphosphorylatedonce-boundintracellularreceptors(R7).Thatprocessisreversiblewithratek40.Therefore,thesynthesisrateofR4isR4=k2IR4k3PR4+k3R2+k2R3+k40R7k40R4dR4:(3.2.4)ThesynthesisrateofR5:Thesourcetermsarethebasaltranscriptiontounboundunphosphory-latedintracelluarreceptors(R5),b6andthepositivefeedbackfromtheactivateddFOXO,l=fdF1+F,whichismodeledbytheMichaelis-Mentenequation(isaconstantrepresentingtheanityofactivateddFOXOboundwiththeDNAstrand.Thestrongertheanityis,thelargerthevalueofwillbe).Thephosphorylatedtwice-boundintracellularreceptors(R6)andphosphorylatedonce-boundintracellularreceptors(R7)aredephosphorylatedandbecomeunboundunphosphorylated14intracelluarreceptors(R5).Thus,theequationforthesynthesisrateofR5isR5=b5+ldR5+k5P(R6+R7)+k4R1k4R5(3.2.5)ThesynthesisrateofR6andR7:Throughtheinteractionswithfreeinsulinreceptorsincyto-plasma(R5)andboundinsulinreceptorsinthecellmembrane,wehaveR6=k40R3k40R6k5PR6dR6(3.2.6)R7=k40R6k40R7k5PR7dR7(3.2.7)3.2.2Chico/PI3KsubsystemIntheinsulin-likesignalingpathwayofDrosophilamelanogaster,phosphorylatedboundin-sulinreceptorscouldphosphorylatechico,henceformingachico-PI3Kcomplexinthecell.ThesynthesisrateofC1:Atthecellmembranethephosphorylatedsurfacereceptors(R3andR4)phosphorylateChico,accordingtoamass-actionlawwithratek7.Also,phosphorylatedChico(C2)isdephosphorylatedbyPTPases,accordingtoamass-actionreactionwithratek7.Mean-while,unphosphorylatedChicoistranslatedfromRNAwiththeratedenotedbybcanddegradeswiththeratedenotedbydc.Therefore,thesynthesisrateofunphosphorylatedChicoisC1=bcdcC1+k7PC2k7C1(R3+R4)(3.2.8)ThesynthesisrateofC2:BesidestheinterationswithunphosphorylatedChico,thephosphory-latedChico(C2)bindwithdeactivatedPI3K(3)formingChico-PI3Kcomplex()withtherate15ofk8.Therefore,theequationtodescribethesynthesisrateofphosphorylatedChico(C2)isC2=k7C1(R3+R4)k7PC2+k8k83C2dcC2:(3.2.9)Thesynthesisrateof3:UnphosphorylatedPI3K(3)istranslatedformRNAwiththeratedenotedbybp.ThedegradationrateofunphosphorylatedPI3K(3)isdenotedbydp.Thus,theequationtodescribethesynthesisrateofunphosphorylatedPI3K(3)is3=bpdp3+k8k83C2(3.2.10)Thesynthesisrateof:Asmentionedabove,throughamass-actionreactiontheproductionrateofthephosphorylatedChico-PI3Kcomplexisdenotedbyk8andthedissociationrateisde-notedbyk8.WiththedegradationrateofphosphorylatedPI3K-Chicocomplextobedpc,thesynthesisrateofphosphorylatedChico-PI3Kcomplex()is=k8C23k8dpc:(3.2.11)3.2.3LipidssubsystemAdjacenttothecellmembrane,thephosphorylatedChico-PI3Kcomplex()convertsthesubstratephosphatidylinositol4,5-bisphosphate(PI(4;5)P2)tothesubstrateproductphosphatidylinositol3,4,5-trisphosphate(PI(3;4;5)P3).Furthermore,thereisspontaneousphosphorylationanddephosphorylationcaus-ingtransitionsbetweenthesetwostatesandbetweenPI(3;4;5)P3)andanother,PI(3;4)P2.SomeofthesearecatalyzedbyPTENandSHIP,whoseconcentrationswetaketobeconstantandareimplicitlyincludedintherateconstantsshownbelow.WeassumethatthetotalamountofPIPis16conserved.LetLP3+P4+P5bethetotalamountofPIP,theequationstodescribethesynthesisratesofP3,P4andP5areP3=k9pP5+k9bP5+k10P4k9P3k10P3;(3.2.12)P4=k10P3k10P4;(3.2.13)P5=k9P3(k9p+k9b)P5:(3.2.14)3.2.4AktsubsystemThesynthesisrateofAandAp:ThelipidPI(3;4;5)P3(P3)phosphorylatesdeactivatedAktatarateproportionaltotheconcentrationsofthelipidandAwiththerateconstantdenotedbyk11:ActivatedAktisdephosphorylatedspontaneouslyandbecomesdeactivatedAktwiththeratek11.Atthesametime,deactivatedandactivatedAkt(AandAp)degradeattherateofdA.ThedeactivatedAktistranslatedfromRNAwiththerateofbA.Hence,theequationstodescribethesynthesisrateofAandApareA=bAdAA+k11Apk11P3A(3.2.15)Ap=k11P3Ak11ApdAAp(3.2.16)173.2.5dFOXOsubsystemInthedFOXOsubsystem,dierentfromprevioussubsystem,activateddFOXO(F)isun-phosphorylateddFOXOanddeactivateddFOXO(f)isphosphorylateddFOXO.TheactivatedAktphosphorylatestheactivateddFOXO,turningitfromactivatedstatetodeactivatedstate.Ontheotherhand,thedeactivateddFOXOsimultaneouslyunphophorylatesitselfbacktoactivatedstate.TomodeltheprocessofAktphosphorylatingdFOXO,wenoticethatwhenactivatedAktin-teractswithactivateddFOXO,asmallamountoftemporarycomplex[AF]formsquickly.Then[AF]degradestofreeactivatedAktandphosphorylateddFOXOf.UsingtheMichaelis-Mentenformalism,assumingquasisteadystateforthisfastreaction,andignoringhigherordertermsofsmallquantities,wetheproductionofphosphorylateddFOXOfisproportionaltotheamountof[AF]:k12ApF(+1)F+Ap;whereistheratiooftherateatwhichthe[AF]formstotherateatwhichitdissociates.WiththedegradationrateofFandfbeingdfandthebasaltranscriptionalratetoFbeingbF,wehaveF=k12ApF(+1)F+Ap+k12f+bFdfF(3.2.17)f=k12ApF(+1)F+Apk12fdff(3.2.18)18ThereactionprocessthatdFOXOisphosphorylatedbyAktismodelledasaMichaelisMentenkinetics.,letEbethefreeenzyme,Sbethesubstrate,[ES]betheenzyme-substratecomplex,andEtbethetotalenzyme(freeenzymeplusenzyme-substratecomplex).Onehavethefollowingequation:[ES]=konESE+[ES]koff[ES]:Atequilibriumpoint,[ES]=0.Therefore,konES=koff[ES]E+koff[ES]2;E=koff[ES]2konSkoff[ES]:Let=konkoff:SincetheEtisaconstant,Et=E+[ES]givesus:[ES]=EtSS+Et:Hence,theproductionrateofS0is:qEtSS+Et:TheactivityofPTPasesismodelledbyanexponentialfunctionofactivatedAkt(Ap)([17]).P=exp(kAp)(3.2.19)193.3SteadystateofthesystemThemodelconsistsofaneighteenordinarydierentialequationswiththeconcentrationoftheinsulinasthemodelinput.ToanalysetherelationsofthecomponentsintheIISpathwaywiththemodelinput,theconcentrationoftheinsulin,IturntolookatthesteadystateoftheODEsystem.Thus,Ineedtosolvetheeighteen-equationalgebraicsystem.Thestrategytosolvethealgebraicsystemisbyconstructinganiterationmappingwhichconvergestothesolutionofthealgebraicsystem.,Iseparatethealgebraicsystemintovesubsystemsashowthemodelisestablishedintheabovesection.InthedFOXOsubsystem,IsolvetheactivateddFOXOasafunctionoftheactivatedAkt.F=(k12+k12+dF)Ap(bF+k12bFdF)(+1)2(dF+k12)(+1)+r((k12+k12+dF)Ap(bF+k12bFdF)(+1))2+4(dF+k12)(+1))((bF+k12bFdF)Ap))2(dF+k12)(+1)(3.3.1)andP=exp(0:003Ap):IntheAktsubsystem,IsolvetheactivatedAktasafunctionofthePI(3;4;5)P3.Ap=aAk11P3k11P3+k11+dA:20Inthelipidsubsystem,IsolvethePI(3;4;5)P3asafunctionofthephosphorylatedChico-Pi3kcomplex.P3=11+29=31+k9k9pk9b:IntheChicosubsystem,IsolvethephosphorylatedChico-Pi3kcomplexasafunctionoftheinsulin-boundedreceptorsonthecellmembrane.=BpB24AC2AwhereA=k8(dc+k7(R3+R4))k7P+dc+k7(R3+R4);B=k8(dc+k7(R3+R4))(ap+acbcdc+k7(R3+R4)k7P+dc+k7(R3+R4)dk8;C=k8(dc+k7(R3+R4))(ap+acbcdc+k7(R3+R4)k7P+dc+k7(R3+R4):Fortheinsulinreceptorsubsystem,Isolvetheinsulin-boundedreceptorsonthecellmembraneasafunctionoftheactivateddFOXO.ThenIcomposethosefunctionstogethertogetanitera-tionoftheactivateddFOXO.Finally,Ikeepiteratingthecompositionfunctionuntiltheiterationconverges.Inthesteadystate,thetotalamountofinsulinreceptorisequalto:R1+R2+R3+R4+R5+R6+R7=b5+ld:AssumingthedegradationrateofChico,phosphorylatedChicoandChico-PI3Kcomplexarethesame,i.e.dc=dp=dpc,onecanderivethatthetotalamountofChicoisequalto:21C1+C2+=bcdcand+=bpdp:Similarly,intheAktsubsystem,onecanderivethatthetotalamountofAktisequalto:A+Ap=bada;andinthedFOXOsubsystem,thetotalamountofdFOXOisequalto:F+f=bFdf:Ithata5=b5d;feedback=fbd;ac=bcdc;ap=bpdp;aA=badaandaF=bFdf:22Thosevariablesserveastheparametersregulatingthetotalamountofinsulinreceptors,Chico,PI3k,AktanddFOXO.InDrosophilamelanogaster,theconcentrationsofthoseproteinsaredif-ferentinvariousorgans.Thus,byintroducingthoseparameters,Icouldinvestigatetheallometriesofvariousorgansbymanipulatingthevaluesofthoseparameters.Foranyedsetofparametersa5,ac,ap,aA,aF,feedbackandmodelinput,theconcentrationofinsulinI,Icomputethecon-centrationofactivateddFOXO,F.Hence,varyingtheconcentrationofinsulinI,IgetacurveoftheconcentrationofactivateddFOXOF.Forinstance,leta5=1pM,ac=1pM,ap=0:1pM,aA=5pM,aF=5pM,feedback=0:5pM,Ihasthecurvein3.2.ThedFOXOisanegativegrowthfactor.Inordertoestablishtherelationbetweentheconcen-trationoftheinsulinandtheorgansizeofIusedempiricaldatatoestimatetherelationshipbetweendFOXOactivityandorgansize,andhencethenutritionalplasticityoforgansize.WeplottherelationshipbetweenwingSizeandactivedFOXO.ThesizeofthewingdecreasesastheconcentrationofactivedFOXOincreases.Moreover,thereisalowerlimitforthewingsize.Thesizeofthewingisneversmallerthan8105nomatterwhattheconcentrationofactivedFOXOandenvironmentalconditionsare.Thus,thenaturalwaytotherelationshipbetweenthewingsizeandtheconcentrationofactiveFOXOistousetherationalfunction.Organsize=aF+b+c;whereFrepresentstheconcentrationofactivateddFOXO,fromtheempiricaldataofwingcellnumberofIobservefromtheempiricaldatathatthelowerlimitforthewingsize,c,is8105.ThenweusetheLeastSquaremethodtotheparameteraandb.SincetherelationshipofWingsizeandactivedFOXOdiersastheenvironmentalconditionsvary,thecoecientaandbalsovary.Fromourmethods,parameterarangesfrom2:5105to3:5105andbrangesfrom230:4to0:6.Tosimplifythecomputationandanalysis,Ipickatobe3105andbtobe0:5.Hence,thesameparametersasa5=1pM,ac=1pM,ap=0:1pM,aA=5pM,aF=5pM,feedback=0:5pM,therelationbetweenthenumberofcellsofwingandtheconcentrationofinsulinisshownin3.3.3.4ThenutritionalplasticityInDrosophilamelanogaster,theresponseofthebodyandorgansizetochangesindevelop-mentalnutritioniscalledthenutritionalplasticity.MalnutritionduringdevelopmentreducesadultbodysizeofDrosophilamelanogaster.However,notalltheorgansrespondtothemalnutritiontothesameextent.Someorgans,forexample,themalegenitalia,areremarkablyresistanttochangesindevelopmentalnutrition.Likethemammalianbrain,theyaremoreorlessthesamesizeinlargeandsmallindividuals.OnehypothesisisthatthenutritionalplasticityisregulatedthroughtheIISpathway.TheamountsofgeneexpressionsofcomponentsintheIISpathwayarethefactoraect-ingthenutritionalplasticityinDrosophila.Hence,Iestablishamathematicalmodeltorepresentthenutritionalplasticity.Thenutritionalplasticityinthepaperistobethedierenceofthelogarithmicorgansizeattwoinsulinconcentrationlevels.,letOrganSize=S(I)bethefunctionoftheorgansizewithrespecttotheconcentrationofinsulin.IPlasticity=log(S(I+))logS((I))(3.4.1)whereI+andIaretwodierentlevelsofinsulinconcentration.Withoutfurtherremark,ItakeI=20pMand=1000pMinthisdissertation.243.5TheresultsIn3.4,thenutritionalplasticityisregulatedbythetotalgeneexpressionofinsulinre-ceptors.FixingthegeneexpressionofChico,PI3K,AktanddFOXOtobe1pM,0:1pM,5pMand5pMrespectively,thenutritionalplasticityincreasesasthegeneexpressionoftotalinsulinreceptorsincrease.Buttherateofincrease(thetangentlineofthecurve)decreases.In3.5,thenutritionalplasticityisregulatedbythetotalgeneexpressionofChico.FixingthegeneexpressionofInr,PI3K,AktanddFOXOtobe5pM,0:1pM,5pMand5pMrespectively,thenutritionalplasticityincreasesthendecreasesasthegeneexpressionoftotalinsulinreceptorsincrease.In3.6,thenutritionalplasticityisregulatedbythetotalgeneexpressionofPI3K.FixingthegeneexpressionofInr,Chico,AktanddFOXOtobe5pM,1pM,5pMand5pMrespectively,thenutritionalplasticityincreasesthendecreasesasthegeneexpressionoftotalinsulinreceptorsincrease.In3.7,thenutritionalplasticityisregulatedbythetotalgeneexpressionofAkt.FixingthegeneexpressionofInr,Chico,PI3KanddFOXOtobe5pM,1pM,0:1pMand5pMrespec-tively,thenutritionalplasticityincreasesthendecreasesasthegeneexpressionoftotalinsulinreceptorsincrease.In3.8,thenutritionalplasticityisregulatedbythetotalgeneexpressionofdFOXO.FixingthegeneexpressionofInr,Chico,PI3KandAkttobe5pM,1pM,0:1pMand5pMre-spectively,thenutritionalplasticityincreasesthendecreasesasthegeneexpressionoftotalinsulinreceptorsincrease.Thereisanon-linearrelationshipbetweenFOXOexpressionandnutritionalplasticity.Thisgraphiscoincidentwiththerealexperimentresults(See3.9).([15])Fromthesimulationofthe3.8,thenutritionalplasticityhasanonlinearrelationship25withthetotalgeneexpressionofdFOXO.ForthoseorganswitheitherveryhighorverylowtotalgeneexpressionofdFOXO,theyaremoreorlessinsensitivetothechangeofnutrition.ThehighnutritionalplasticityisachievedwhentotalgeneexpressionofdFOXOisatmediumlevel.Thisdoesnotchangewhenthetotalgeneexpressionofinsulinreceptosrincrease.However,maximumnutritionalplasticityincreasesasthetotalgeneexpressionofinsulinreceptorsincrease.Inaddition,thevalueoftotalgeneexpressionofdFOXOatwhichthemaximumisachievedincreasesaswell.Seethe3.10forathreedimensionalgraphofthenutritionalplasicityversusthetotalgeneexpressionofinsulinreceptorsandthetotalgeneexpressionofdFOXOwhenthegeneexpressionofChico,PI3KandAkttobe1pM,0:1pMand5pMrespectively.Inthethreedimensional3.11ofthenutritionalplasticityversusthegeneexpressionofAktanddFOXO,theridgeisinthedirectionofAkt.ThatisduetothebellcurveofthenutritionalplasticityversusthedFOXO.3.6ModelcoecientsHereliststhosemodelcoecientsthataretakenfrom[11]:k1=6105pM1min1;k1=0:2min1;k2=k1min1;k2=20min1;k3=2500min1;k3=0:2min1;k4=0:0003min1;k4=0:003min1;26k40=2:1103min1;k40=2:1104min1;k6=0:461min1;k7=4:638min1;k7=1:396min1;k8=0:707pM1min1;k8=10min1;k9=42:148min1;k9b=0:131min1;k9p=1:390min1;k10=2:961min1;k10=2:77min1;k11=2:484min1;k11=6:932min1.Thenewcoecientsoftheordinarydierentialequationsare:k=0:03;k12=30pM1min1;k12=1min1;=1;=2;d5=0:01min1;dc=0:01min1;dp=0:01min1;dpc=0:01min1;27dA=0:01min1;df=0:01min1;b5:0:110pMmin1;bc:0:110pMmin1;bp:0:110pMmin1;ba:0:110pMmin1;bF:0:110pMmin1.Theillustrationoftheabbreviationsinthemodelare:ILP:Insulin(protein)isahormonecentraltoregulatingcarbohydrateandfatmetabolisminthebody.Itisthemodelinputwhichbindstoinrandinitiatesasignaltransductioncascadeinvolvingthephosphorylationofmultipleintermediateproteins.Inr:Itisaproteinkinaseonthecellmembranewhichcanbindwithinsulin.Wheninsulinbindstotheinsulinreceptoronthecellsurface,thereceptorchangesshapesothatthekinaseregionsinsidethecellbecomeactivated.Theactivatedinsulinreceptorthenactivatesanumberofdierenttargetswithinthecell.Chico/IRS:Insulinreceptorsubstrateisaproteincontainingaphosphotyrosinebinding-domain(PTB-domain).TheinsulinreceptorcontainsaNPXpYdomain.ThePTB-domainbindswiththeNPXpYdomain.Thus,IRSbindswithinsulinreceptor.PI3K:Phosphatidylinositol3-kinasesorPI3-kinasesareafamilyofenzymesinvolvedincellularfunc-tionssuchascellgrowth,proliferation,dierentiation,motility,survivalandintracellulartrack-28ing.TheyinteractwithIRS(insulinreceptorsubstrate)throughaseriesofphosphorylationevents.PI(3,4)P2,PI(4,5)P2,PI(3,4,5)P3:Variousphosphoinositollipidsonthecellmembrane.Akt:Akt,alsoknownasProteinKinaseB(PKB),isaserine/threonineproteinkinase.AktpossessesaproteindomainknownasaPHdomain(PleckstrinHomologydomain).ThisdomainbindstoPIP3onthecellmembrane.OnceAktiscorrectlypositionedatthemembraneviabindingofPIP3,itcanthenbephosphorylatedandbecomeactivated.FOXO:FOXOisatranscriptionfactorwhichcannegativelyregulatethebodygrowth.FOXOcanbephos-phorylatedbyactivatedAktatthreeconservedresidues.PhosphorylatedFOXOexportsfromthenucleustothecytoplasm,therebyinhibitingFOXO-dependenttranscription.PTP:PTPasesareaclassofenzymesthatcanregulatetyrosinekinaseactivitybyremovingaphosphate.29Figure3.1:ThestructureofIISpathway.30Figure3.2:TheconcentrationofactivateddFOXOv.s.theconcentrationofinsulin.Figure3.3:Thenumberofcellsofwingv.s.theconcentrationofinsulin.31Figure3.4:Thenutritionalplasticityv.s.thegeneexpressionofInr.32Figure3.5:Thenutritionalplasticityv.s.thegeneexpressionofChico.33Figure3.6:Thenutritionalplasticityv.s.thegeneexpressionofPI3K.34Figure3.7:Thenutritionalplasticityv.s.thegeneexpressionofAkt.35Figure3.8:Thenutritionalplasticityv.s.thegeneexpressionofdFOXO.Figure3.9:Thereisanon-linearrelationshipbetweendFOXOexpressionandnutritionalplasticity.36Figure3.10:Thenutritionalplasticityv.sthegeneexpressionofInranddFOXO.Figure3.11:Thenutritionalplasticityv.sthegeneexpressionofAktanddFOXO.37Chapter4APDEmodel4.1PreliminaryTheinsulinandinsulin-likesignaling(IIS)pathwaypropagatesasignalfromreceptorsinthecellmembranetothenucleusvianumerousmolecules.Someofthesemoleculesresideonorcontiguoustothecellmembrane,someresideinorcontiguoustothenucleusandsomeareinthecytosol,beingtransportedbetweenthesetworegionsinasomewhatstochasticway.IncludedinthispathwayistheforkheadtranscriptionfactorFOXO,whichpromotestheexpressionofnegativegrowthregulators([12]).FOXOisnegativelyregulatedbytheinsulinsignalingpathway,anditisthereductionininsulinsignalingandtheresultingactivationofFOXOthatis,inpart,responsibleforinhibitingorgangrowthinconditionsofreducednutrition([4]).Theresultisthatorgangrowthisnutritionallyplastic.However,dierentorgansshowdierentlevelsofnutritionalplasticity,essentialtoensuringthatcertainkeyorgans,forexamplethemammalianbrain,arelargelysparedtheeectsofmalnutrition.Recentresearchhassuggestedthatthesedierencesinnutritionalplasticityaremediatedbydierencesinthestructureoftheinsulin-signalingpathwayindierentorgans.However,howchangesinthestructureoftheinsulinsignalingpathwayaectshowthepathwayregulatesgrowthwithrespecttonutritionisunclear.Hereweusemathematicalmodelingtohelpsolvethisproblem.IntheODEmodel,thevariablesoftheequationsarethecirculatingconcentrationofeachcom-ponentsofIISpathway.Wedon'tmodelthemovementofthemolecularofproteinscomponents38ofIIS.However,althoughmanyoftheproteincomponentsoftheinsulinsignalingpathwayareattachedto,orassociatedwith,thecellmembranetheyareallatsomepointtransportedthroughthecytoplasm.Allproteinsaresynthesizedattheendoplasmicreticulum(ER),whichiscontigu-ouswiththenuclearmembrane.Theproteinsarethenpackagedintovesicles,bytheERandthenbytheGolgiapparatus,beforethesevesicle,andtheproteinswithinthem,aretransportedtotheirdestination.Vesicletransportisthroughtheactionofmolecularmotors,e.g.kinesin,whichattachtothevesicleandwalkitalongthemicrotubulesthatformthecytoskeletonofthecell.Becausethetransportationprocesshasastochasticaspecttoit,duetothedistributionofmotorsandmicro-tubules(orotherscaolding)andtheprocessivityofthemotors,thisismodeledasanenhancedordirectionaldiusiveprocess.Forsimplicityweuseasphericallysymmetriccellwiththenuclearshellhavingradiusr1andthecellmembranehavingradiusr2(r1<r2).Amongotherthings,thisdiusivetransportbuildsadelayintothedynamicsdescribedin[11],whichmaybeimportantinregulation.Wealsoincludedegradationandbasaltranscriptionofmoleculesintothemodel.Asmentionedabove,thegrowthoforgansisnegativlyregulatedinpartbythetranscriptionfactorFOXO,theactivestateofwhichresidesinthenucleus.Aproteinkinase,Akt,downstreamfromtheinsulinreceptors,initsactive(phosphorylated)state,deactivatesFOXObyphosphorylatingit.PhosphorylationofFOXObyAktbothinhibitsitsactivityasatranscriptionfactorandcausesittobetransportedoutofthenucleus.TheinactiveFOXOinthecytoplasmcanbereactivatedandtargetedfornuclearlocalizationthroughphosphorylationormonoubiquitinationbyproteinsinothersignalingpathways.Importantly,activatedFOXOpromotesthetranscriptionoftheinsuinreceptor,creatinganegativefeedbackloopbetweenthetopandthebottomofthepathway([7]).Thus,oursystembecomesmorecomplexbyaddingbothmolecularspecieswithtranscriptionalfeedback,andspatialtransportofsomemolecules.Partofourpurposeforthisistodeterminethemechanismwherebyorgansizeplasticityisregulated.Webelievethatthisregulation39ofplasticityisachievedthroughtheinsulin-signalingpathwayingeneralandFOXO.Ourmodelexploresthathypothesis.4.2ThePartialDierentialEquationsmodeloftheIISpath-wayWedividethesystemintoseveralcoupledsubsystems,startingwiththe4.2.1InsulinreceptorssubsystemLetIdenotetheconcentrationofinsulin,whichisaconstantinthispaper(wetaketwovaluesofIwhencomputingsensitivity).Aninsulinreceptorinthecellmembranemaybindoneortwoinsulinmoleculesandwhenbound,autophosphorylationoccursandsoinitiatesasignaltransduc-tioncascade.Receptorsmayalsoresidetemporarilyinthecytosol,wheretheyaretransportedtothecellmembrane.Thuswehavethestatevariables:Receptors:R1(t),concentrationofunboundunphosphorylatedcell-surfacereceptors,R2(t),concentrationofonce-boundunphosphorylatedcell-surfacereceptors,R3(t),concentrationofphosphorylatedtwice-boundcell-surfacereceptors,R4(t),concentrationofphosphorylatedonce-boundcell-surfacereceptors,R5(r;t),concentrationofunboundunphosphorylatedintracellularreceptors,R6(r;t),concentrationofphosphorylatedtwice-boundintracellularreceptors,R7(r;t),concentrationofphosphorylatedonce-boundintracellularreceptors.40PTP:P(r;t):AprefactorrepresentingtherelativeactivityofPTPases(theclassofenzymesthatregulatetyrosinekinaseactivitybyremovingaphosphate)inthecell.ThisfactordependsuponthelevelofactivatedAkt,whichvariesaccordingtoapartialdierentialequationcoupledtotherestofthesystem.FOXO:F(r;t),concentrationofactivatedFOXO,f(r;t),concentrationofdeactivatedFOXO.ThesynthesisofR1:Freeinsulinreceptors(R1)onthemembranebindtoinsulin(I)andbecomeonce-boundunphosphorylatedsurfacereceptors(R2)attheratek1.Thatreactionisre-versiblewithratek1.Phosphorylatedonce-boundsurfacereceptors(R4)aredephosphorylatedbyPTPases,releasetheirinsulinandbecomeunboundunphosphorylatedsurfacereceptors(R1)withratek3P.Atthesametime,freesurfacereceptors(R1)passthroughthecellmembranetobecomeintracellularreceptors(R5)withratek4andtheintracellularreceptorsattachtothecellmembrane,becomingsurfacereceptorswithratek4([8],[9]).Finally,acertainfraction(d)ofreceptorsdegradesandislost.Therefore,thesynthesisrateoffreereceptoronthemembrane,R1,isexpressedbyR1=k1IR1+k1R2+k3PR4+k4R5(r2;t)k4R1dR1:(4.2.1)ThesynthesisofR2:InadditiontotheexchangeswithR1,describedabove,theonce-boundunphosphorylatedsurfacereceptors(R2)degradeatthesameratedandarephosphorylatedtobecomephosphorylatedonce-boundsurfacereceptors(R4)attheratek3.Therefore,thesynthesis41rateofR2isR2=k1IR1k1R2k3R2dR2:(4.2.2)ThesynthesisofR3:Phosphorylatedonce-boundsurfacereceptors(R4)bindtoinsulin(I)andbecomephosphorylatedtwice-boundsurfacereceptors(R3)withratek2.Thisreactionisreversiblewithratek2.Atthesametime,phosphorylatedtwice-boundsurfacereceptors(R3)passthroughthecellmembranewithratek40tobecomephosphorylatedtwice-boundintracellularreceptors(R6).Thisprocessisreversiblewithratek40.Therefore,thesynthesisrateofR3isR3=k2IR4k2R3+k40R6(r2;t)k40R3dR3:(4.2.3)ThesynthesisofR4:InadditiontotheexchangeswithR1;R2;andR3,describedabove,phos-phorylatedonce-boundsurfacereceptors(R4)passthoughthemembranewithratek40,becomingphosphorylatedonce-boundintracellularreceptors(R7).Thatprocessisreversiblewithratek40.Therefore,thesynthesisrateofR4isR4=k2IR4k3PR4+k3R2+k2R3+k40R7(r2;t)k40R4dR4:(4.2.4)ThesynthesisofR5:ThegeneresponsibleforcreatinginsulinreceptorsistranscribedinthenucleusandtheresultingRNApassesthroughporesinthenuclearshellenteringthecytoplasm.Onceinthecytoplasm,withthehelpofaribosome,translationstartstotheprotein(receptor)andiscompletedintheER(endoplasmicreticulum),whichiscontiguoustothenucleus.Thereceptoristhenpackagedintovesicles(thevesiclemembraneholdsthereceptorwhichisatransmembraneprotein)andistakentotheGolgiapparatus,andthenthecellularsurfacemembrane.Allviatransportvesiclesthataremovedbyoneofthemotorproteinsalongmicrotubules.Here,we42simplifythisprocessbypostulatingtheproductionoffreeinsulinreceptors(R5)onthenuclearsurface(r=r1)dueinparttotheactivatedtranscriptionfactorFOXOwhichenhancesbasaltranscription.Theresultingreceptorsareactivelytransportedtotheoutercellmembrance,butinastochasticway.Thus,R5hasaspatio-temporaldistributionthatisgovernedbyanequationwithadvectionanddiusionterms,aswellasreactionterms,exchangingbetweenotherstates,andadegradationterm.Theadvection-diusionoperatorweemployhastheformLuDur(xjxju);givingradially-directedtransporttowardsthecellmembrane.Intheequationsbelow,weexpressthisoperatorinradialcoordinatessinceweassumesphericalsymmetry.BoundaryconditionsrepresentaFOXO-dependentsourceatr1andasinkatr2,asreceptorsleavethecytoplasmtobecomeembeddedinthecellmembrane.ThewayinwhichactivatedFOXO(F)operatesincreatingfreereceptorsatthenucleusismodeledusingtheMichaelis-Mentenrelation,givingaterml=F1+F,whereisaconstantrepresentingtheanityofFOXObindingtoDNAandisaratefactor.Wemayalsoconsideramass-actionreactionrate,whichgivesqualitativelysimilarresults.Theexchangesbetweenotherstatesinthecytoplasmincludeonlyintracellularreceptorswhicharephosphorylatedandhaveoneortwoinsulinmoleculesboundbecomeunphosphorylatedataratek6P,releasingtheirinsulin,therebycontributingtoR5.Therefore,thedistributionofR5isdescribedby@R5@t=Dr2@@r(r2@R5@r)r2@@r(r2R5)dR5+k5P(R6+R7);(4.2.5)43Figure4.1:Thestructureofthecell.withtheboundaryconditionsobtainedthroughacalculation,D@R5@r(r1;t)R5(r1;t)=lb5;D@R5@r(r2;t)R5(r2;t)=k4R1k4R5;t>0:ThesynthesisofR6andR7:Asdescribedabove,thetwice-bound(R6)andonce-bound(R7)in-tracellularreceptorsmaypassthroughthemembranetobecomesurfacereceptors,andvice-versa.Alsothesereceptorsbecomeunphosphorylatedataratek6,releasingtheirinsulin,andcontribut-ingtoR5.Whileinthecytosol,weassumethatthesereceptorsareactivelytransportedtowardstheplasmamembraneinthesamewayasthefreereceptos,thatis,accordingtoanadvectiveanddiusiveprocess.Again,theirdegradationrateisgivenbyd.Therefore,thesynthesisratesofR6andR7are@R6@t=Dr2@@r(r2@R6@r)r2@@r(r2R6)dR6k5PR6;(4.2.6)44withboundaryconditionsD@R6@r(r1;t)R6(r1;t)=0;D@R6@r(r2;t)R6(r2;t)=k40R3k40R6;t>0;and@R7@t=Dr2@@r(r2@R7@r)r2@@r(r2R7)dR7k5PR7;(4.2.7)withboundaryconditionsD@R7@r(r1;t)R7(r1;t)=0;D@R7@r(r2;t)R7(r2;t)=k40R4k40R7;t>0:4.2.2Chico-PI3KcomplexsubsystemChicoisaninsulinreceptorsubstrate,whichactsasascaoldbringingtogetherothermoleculesresponsibleforthesignal.PI3Ksareafamilyofrelatedintracellularsignaltransducerenzymescapableofphosphorylatingthe3positionofalipidwheninacomplexwithChico.Phosphorylatedinsulin-boundsurfacereceptorsphosphorylateChico,leadingtotheChico-PI3Kcomplexinthecell,aproductupstreamoftheactivationofAktandthedeactivationofFOXO.Inthissubsystem,thestatevariablesareChico:C1(r;t),concentrationofunphosphorylatedChico,C2(t),concentrationofphosphorylatedChico,andPI3K:3(r;t),concentrationofdeactivatedPI3K,(t),concentrationofphosphorylatedChico-PI3Kcomplex.ThesynthesisofC1:Aswithfreereceptors,thegeneforChicoistranscribedinthenucleusandtheRNAistranslatedtounphosphorylatedChicoatalocationcontiguoustothenuclearmembrane45fromwhereitisactivelytransportedtothecellmembrane.Atthecellmembranethephosphory-latedsurfacereceptors(R3andR4)phosphorylateChicoaccordingtoamass-actionlawwithratek7.Also,phosphorylatedChico(C2)isdephosphorylatedbyPTPasesaccordingtoamass-actionreactionwithratek7.ThebasaltranscriptionrateofunphosphorylatedChicoisdenotedbybcanditdegradesataratedenotedbydc.Therefore,thesynthesisrateofunphosphorylatedChicois@C1@t=Dr2@@r(r2@C1@r)r2@@r(r2C1)dcC1;(4.2.8)withboundaryconditionsD@C1@r(r1;t)C1(r1;t)=bc;D@C1@r(r2;t)C1(r2;t)=k7PC2k7C1(R3+R4):ThesynthesisofC2:PhosphorylationofChicobysurfacereceptors(R3andR4)isdescribedabove,asisitsdephosphorylationbyPTPases.PhosphorylatedChico(C2)bindswithdeactivatedPI3K(3)accordingtomass-actionkineticsformingtheChico-PI3Kcomplex()ataratedenotedbyk8.ThedissociationofthephosphorylatedChico-PI3Kcomplexintoitstwocomponentstakesplaceataratedenotedbyk8.PhosphorylatedChicodegradesataratedenotedbydc.Therefore,thesynthesisrateofphosphorylatedChicoisC2=k7C1(r2;t)(R3+R4)+k8k7PC2k83(r2;t)C2dcC2:(4.2.9)Thesynthesisof3:ThisunphosphorylatedPI3kinaseistranslatedadjacenttothenucleusfromwhereitistransportedtothecellmembrane,aswithotherproteinsdescribedabove.ThebasaltranscriptionrateofunphosphorylatedPI3Kisdenotedbybpandthedegradationrateis46denotedbydp.AsmentionedabovethedissociationrateofthephosphorylatedIRS-PI3Kcomplexisdenotedbyk8.Therefore,thesynthesisrateofunphosphorylatedPI3Kis@3@t=Dr2@@r(r2@3@r)r2@@r(r23)dp3;(4.2.10)withboundaryconditionsD@3@r(r1;t)3(r1;t)=bp;D@3@r(r2;t)3(r2;t)=k83(r2;t)C2+k8:Thesynthesisof:Asmentionedabove,throughamass-actionreactiontheproductionrateofthephosphorylatedChico-PI3Kcomplexisdenotedbyk8andthedissociationrateisdenotedbyk8.WeusedpctodenotethedegradationrateofphosphorylatedPI3K-Chicocomplex.Therefore,thesynthesisrateofphosphorylatedChico-PI3Kcomplex()is=k8C23(r2;t)k8dpc:(4.2.11)4.2.3LipidssubsystemAdjacenttothecellmembrane,thephosphorylatedChico-PI3Kcomplex()convertsthesubstratephosphatidylinositol4,5-bisphosphate(PI(4;5)P2)tothesubstrateproductphosphatidylinositol3,4,5-trisphosphate(PI(3;4;5)P3).Furthermore,thereisspontaneousphosphorylationanddephosphorylationgivingtransitionsbetweentheset-wostatesandbetweenPI(3;4;5)P3)andanother,PI(3;4)P2.SomeofthesearecatalyzedbyPTENandSHIP,whoseconcentrationswetaketobeconstantandareimplicitlyincludedintherateconstantsshownbelow.WeassumethatthetotalamountofPIPisconserved.47LetP3(t)betheconcentrationofPI(3;4;5)P3,P4(t)betheconcentrationofPI(3;4)P2,P5(t)betheconcentrationofPI(4;5)P2.NoticethatconservationgivesLP3+P4+P5.TheequationstodescribethesynthesisratesofP3,P4andP5areP3=k9pP5+k9bP5+k10P4k9P3k10P3;(4.2.12)P4=k10P3k10P4;(4.2.13)P5=k9P3(k9p+k9b)P5:(4.2.14)4.2.4AktsubsystemAktisalsoknownasProteinKinaseB(PKB),andasthisnamesuggestsitisa(serine/threonine)proteinkinase,thatis,itactsasacatalystforproteininteractions.Itisproducedinthevicinityofthenucleus,initsinactiveorunphosphorylatedstatefromwhereitisactivelytransportedtothecellmembranewhereitbecomesphosphorylatedbythelipidPI(3;4;5)P3.ThestatevariablesforAktaredenotedby:A(r;t),concentrationofdeactivatedAkt,Ap(r;t),concentrationofactivatedAkt.ThesynthesisofAandAp:ThebasaltranscriptiontounphosphorylatedAkt(A)isdenotedbybAanditsdegradationrate(decayconstant)isdenotedbydA.WeassumethatthedegradationofactivatedAkt(Ap)occursatthesamerate.ThelipidPI(3;4;5)P3(P3)phosphorylatesinactiveAkt48atarateproportionaltotheconcentrationsofthislipidandofAwiththerateconstantdenotedbyk11:ActivatedAktisdephosphorylatedspontaneouslyandbecomesdeactivatedAktwiththeratek11.Also,activatedAktistransportedfromthecellmembranetothenucleus,whereitinteractswithactivatedFOXO,deactivatingitthroughphosphorylation([16]).Hence,theequationstodescribethesynthesisratesofAandApare@A@t=Dr2@@r(r2@A@r)r2@@r(r2A)dAA+k11Ap;(4.2.15)withboundaryconditionsD@A@r(r1;t)A(r1;t)=bA;D@A@r(r2;t)A(r2;t)=k11A(r2;t)P3;and@Ap@t=Dr2@@r(r2@Ap@r)+r2@@r(r2Ap)dAApk11Ap;(4.2.16)withboundaryconditionsD@Ap@r(r1;t)+Ap(r1;t)=0;D@Ap@r(r2;t)+Ap(r2;t)=k11A(r2;t)P3:4.2.5FOXOsubsystemAsindicatedabove,FOXOisatranscriptionfactor,codingforinsulinreceptorsamongotherproteins.ItsactivestateisunphosphorylatedbutactivatedAktphosphorylatesFOXO,makingitinactive([5]).Initsinactivestate,FOXOleavesthenucleusand,whileinthecytoplasm,spon-taneouslybecomesunphosphorylated,andistransportedbacktothenucleus([14]).WeassumethatbothstatesofFOXOdegradeinthecytoplasmatacommonratedf(see[6])andthatactive49FOXOhasabasaltranscriptionrateofbFinthenucleus.ThestatevariableforFOXOaredenotedby:F(r;t),concentrationofactivatedFOXO,f(r;t),concentrationofdeactivatedFOXO.TomodeltheprocessofAktphosphorylatingFOXOatthenuclearmembrane,wenoticethatwhenactivatedAktinteractswithactivatedFOXO,asmallamountoftemporarycomplex[AF]formsquickly.Then[AF]degradestofreeactivatedAktandphosphorylatedFOXOf.UsingtheMichaelis-Mentenformalism,assumingquasisteadystateforthisfastreaction,andignoringhigherordertermsofsmallquantities,wetheproductionofphosphorylatedFOXOfisproportionaltotheamountof[AF]:k12ApF(+1)F+Ap;whereistheratiooftherateatwhichthe[AF]formstotherateatwhichitdissociates.WethushavethesynthesisratesofFandf:@F@t=Dr2@@r(r2@F@r)+r2@@r(r2F)+k12fdfF;(4.2.17)withboundaryconditionD@F@r(r1;t)+F(r1;t)=k12ApF(r1;t)(+1)F(r1;t)+Ap(r1;t)bF;D@F@r(r2;t)+F(r2;t)=0;50and@f@t=Dr2@@r(r2@f@r)r2@@r(r2f)k12fdff;(4.2.18)withboundaryconditionD@f@r(r1;t)f(r1;t)=k12ApF(r1;t)(+1)F(r1;t)+Ap(r1;t);D@f@r(r2;t)f(r2;t)=0:4.2.6PTPasesInthemodelin[11],theactivityofPTPases,P,isdescribedasapiecewiselinearfunctionofthepercentageofactivatedAkt(theratioofactivatedAktovertotalAkt)insuchawaythatPdegeneratestozerowhenthepercentageofactivatedAktexceeds36:4%,inaccordancewithexperimentaldata.TogetsmoothnessofP(r;t),weuseanexponentialfunctioninsteadtomodeltheactivityofPTPases:P(r;t)=ekAp(r;t);(4.2.19)wherethecoecientkischosenbytothepiecewiselinearfunctionabove.4.3TheresultsItispossibletoshowthatthelargesystemofreaction-diusionequationscoupledtoODE'sthroughboundaryvalueshasauniquesolution,existingforalltime,atleastfornonnegativeinitialdata.However,thepointofinteresthereisthequalitativebehaviorofsolutions,andinparticular,whetherornotthesystemreproducesexperimentaldata.Wewillalsobeinterestedintheevolution51ofthepathwayandtowhatextentitisrobustandoptimizedinsomesense.Thesewillbetopicsoffurtherstudy.Theresultswereportherearetwofold.Fistofall,theinsulinsignallingleadstoareductioninactivatedFOXO.Theinsulin(ormoregenerally,insulin-like)signallingpathwayregulatesthegrowthofthecellthroughthenegativegrowthregulator,FOXO.,developmentalnutritionleadstothereleaseofinsulin-likepeptidesinthebloodstream.Wheninsulinishigh,FOXOisphosphorylateddownstreamalongtheIISpathway.ThisdisruptsDNAbindingandcausesFOXOtotranslocatetothecytoplasm.AdeclineininsulinleadstoanaccumulationofactiveFOXOinthenucleus,increasingthetranscrip-tionofgrowthinhibitors.Also,thetranscriptionofinsulinreceptorsincreases,whichstrengthenstheinsulinsignalandthusmoderatestheincreaseofgrowthinhibitors.Inthe2,wevarytheconcentrationofmodelinputŒinsulinandlookattheconcentrationofactivatedFOXOinthenucle-usat10minutes(F(r1;10)).TheconcentrationofactivatedFOXOdecreasesastheconcentrationofinsulinincreases.ThisagreeswithourunderstandingoftheIISpathway.Secondly,thesensitivityofactivatedFOXOisregulatedbytheexpressionoftotalFOXO.Foranimals,alltheorgansofanindividualsharethesamestructureofIISpathway.However,notallorgansshowthesamegrowthresponsetochangesindevelopmentalnutrition.Forinstance,inthefruit,Drosophilamelanogaster,areductionindevelopmentalnutritionhasmoreofaneectonwingsizethanongenitalsize,andthisisaconsequenceofgenitalgrowthbeinglesssensitivetochangesinIIS.Similarly,inmammalsthedevelopingbrainisrelativelyinsensitivetochangesinnutrition,aphenomenoncalledheadsparing.Suchorgdierencesinnutritional-andinsulin-sensitivityisfundamentaltoensurethatbodyproportioniscorrectacrossarangeofadultsizes.WorkonDrosophilamelanogasterhasrevealedthatthereducedinsulin-sensitivityofthegen-italiaisaconsequenceofchangesintheexpressionofkeygenesintheIISpathway,52Figure4.2:NuclearFOXOvs.InsulintheforkheadtranscriptionfactorFOXO([15]).Inordertoverifythehypothesiswithourmodel,wetheSensitivityofActivatedFOXOasthedierencesoftheconcentrationofactivatedFOXOinthenucleus(F(r1;10))attwoinsulinlevels:Sensitivity=F(r1;10;I1)F(r1;10;I2)wherethetwoinsulinlevelsareI1=1picomolandI2=105picomol.ThenbythedegradationrateofactivatedFOXOandmanipulatingthebasaltranscriptionrate,wechangetheexpressionofFOXO.Consequently,thesensitivityofactivatedFOXOisafunctionofthebasaltranscriptionrate.InFigure3,thesimulationshowsthatthesensitivityofactivatedFOXOincreasesasthebasaltranscriptionrateincreasesfrom0to10picomolar/min.ThusthesensitivityofactivatedFOXOtothesignalfromtheIISpathwayismanipulatedbytheexpressionofFOXOitself,whichvourhypothesis.53Figure4.3:TheSensitivityofActivatedFOXO4.4ModelcoecientsTheprefactorkintheexponentoftheequationofPTPwastakentobe0.03basedontheobservationthatactivatedAktinhibitstheactionofPTP1Bwitha25%decreaseaftermaximalinsulinstimulation.TheratioofactivatedAkttodeactivatedAktis1:10afterthemaximalinsulinsimulation([11])andthetotalsteadystateamountofAktistakentobe100pM,andsofromtheequationofPTPkshouldbe0:11log43.Theradiusofcell,r2,waschosentobe6mbasedonexperimentaldatathatgivesthecross-sectionalareaofDrosophilawingcellsrangingfrom87.59m2to279.83m2([10]).Assumingtheradiusofacellnucleusishalfthatofthecell,whichiscommon,wetookr1tobe3m.Weperformedsimulationswithothervaluesofr1andr2givingverysimilarresults.In([1]),transportbymolecularmotorsisgivenasbeingaround800nmsec1whichisabout50mmin1.Thus,wetooktobe50.WetookDtobe25,equivalenttoassumingthat5%ofthemoleculesaretransportedbydiusion.Therateoffeedback,,fromactivated54FOXOtoinsulinreceptorsisunknownandwetookittobeunity.Otherunknownparameters,takentobeunityforlackofexperimataldata,include,theanitycoecientoftheactivatedFOXObindingwiththeDNA,,theanitycoecientforactivatedAktbindingwithactivatedFOXO,k12,theotherparameterintheMichaelis-MentenreactiondeactivatingFOXO,andk12therateatwhichdeactivatedFOXOisdephosphorylatedinthecytoplasm,thusreturningtoitsactivestate.Thedegradationcoecientsareassumedtobe0:1min1andthebasaltranscripitonconstantsrangingfrom0to10pMmin1werebasedontheinitialconditionsofthemolecularconcentrationsintheoriginalODEmodel([11]).Herearethelistsofthevariableinthepartialdierentialequations:R1(t):theconcentrationofunboundunphosphorylatedcell-surfacereceptors;R2(t):theconcentrationofonce-boundunphosphorylatedcell-surfacereceptors;R3(t):theconcentrationofphosphorylatedtwice-boundcell-surfacereceptors;R4(t):theconcentrationofphosphorylatedonce-boundcell-surfacereceptors;R5(r;t):theconcentrationofunboundunphosphorylatedintracellularreceptors;R6(r;t):theconcentrationofphosphorylatedtwice-boundintracellularreceptors;R7(r;t):theconcentrationofphosphorylatedonce-boundintracellularreceptors;P(r;t):aprefactorrepresentingtherelativeactivityofPTPases;C1(r;t):theconcentrationofunphosphorylatedChico;C2(t):theconcentrationofphosphorylatedChico;3(r;t):theconcentrationofdeactivatedPI3K;(t):theconcentrationofphosphorylatedChico-PI3Kcomplex;P3(t):theconcentrationofPI(3;4;5)P3;P4(t):theconcentrationofPI(3;4)P2;P5(t):theconcentrationofPI(4;5)P2;55A(r;t):theconcentrationofdeactivatedAkt;Ap(r;t):theconcentrationofactivatedAkt;F(r;t):theconcentrationofactivatedFOXO;f(r;t):theconcentrationofdeactivatedFOXO.Thecoecientsthataretakenfrom[11]are(pM=picomolarandm=micrometer):k1=6105pM1min1;k1=0:2min1;k2=k1min1;k2=20min1;k3=2500min1;k3=0:2min1;k4=0:0003min1;k4=0:003min1;k40=2:1103min1;k40=2:1104min1;k6=0:461min1;k7=4:638min1;k7=1:396min1;k8=0:707pM1min1;k8=10min1;k9=42:148min1;k9b=0:131min1;k9p=1:390min1;k10=2:961min1;56k10=2:77min1;k11=2:484min1;k11=6:932min1.Thenewcoecientsoftheordinarydierentialequationsare:r1=3m;r2=6m;D=25m2min1;=50mmin1;k=0:03;k12=30pM1min1;k12=1min1;=1;=2;d5=0:1min1;dc=0:1min1;dp=0:1min1;dpc=0:1min1;dA=0:1min1;df=0:1min1;b5=1pMmin1;bc=1pMmin1;bp=1pMmin1;ba=1pMmin1;bF=1pMmin1.57Chapter5ThebifurcationsofanonlocalChafee-Infanteproblem5.1IntroductionInthesecondpartofthethesis,Iconsiderthenonlocaldiusionequation:Lu+(uu3)=0:whereLuisanintegralasLu=Zˇ03J(yx)(u(y)u(x))dy:BatesandZhaostudythespectraofthisoperator.Itisshownthatasthescalingparametertendstozero,thespectrumofthenonlocaloperatorconvergetothespectrumoftheLaplaceoperatorwithNeumannboundarycondition([26]).Hence,whenissmall,onemaycomparetheabovenonlocaldiusionequationwiththesteadystatesoftheChafee-Infanteequation:8>>>>>>><>>>>>>>:uxx+(uu3)=0;in0<x<ˇ;u0(0)=0;u0(ˇ)=0:58ItisknownthattheChafee-Infanteequationhasglobalbifurcations.ThetrivialsolutionofconstantzerobifurcateswhenequaltothespectrumoftheLaplaceoperatorwithNeumannboundarycondition([30]).ThenaturalquestioniswhetherthenonlocaloperatorhasthesimilarbifurcationpropertiesastheChafee-Infanteproblem.Inchapterveofthethesis,itisshownthatasthescalingparameterapproacheszero,thenonlocaldiusionequationhasthelocalpitchforkbifurcationatthespectrumofthenonlocaloperatorL.Inthelaterpartofthischapter,aconcreteexampleisdiscussed.Foraconcretekernel,IsolvethenonlocaldiusionequationwiththeNewton'sMethodandthesecondandthirdbifurcationdiagramsareplotted.Fortheconcreteexample,thebifurcationsofthenonlocaldiusionequationfailtobeglobal.Toshowthat,thecasewithbeinglargeisconsider.Itisshownthatastheparameterislarge,thesolutiononthebifurcationbranchhasajumpdiscontinuity.5.2ThelocalbifurcationLetbeanopensetwithsmoothboundaryonRn.anonlocaloperatorLontheHilbertspaceL2()asfollows:(L!)(x)=Z2J(xy)[!(y)!(x)]dy:where>0,J(xy)=nJ(xy)andJ()2Cc(Rn)withsuppJˆBR,BR=fx2Rn:jxRg;J0;J(z)=J(jzj):ItiseasytoverifythatLisaboundedlinearoperatoronL2().Inthispaper,wearelookingforthebifurcationpointsofthefollowingequation:Lu+(uu3)=0;in(5.2.1)59where2Ristheparameter.Thisequationcanalsobeformulatedtoafunctionalequation:G(u)=0(5.2.2)whereG:RX7!X,X=L2()isaBanachspace.NoticethatGu(0)=L+and(0)isthetrivialsolutionofthenonlocaldiusionequation.Inspiredbytheimplicitfunctiontheorem,weinvestigatethespectrumoftheoperatorL,i.e.whereGu(0)isnotinvertible.Infact,atwhereGu(0)hasdimensionalkernel,Gu(0)isaFredholmoperator.Lemma5.2.1.Assume2˙(L)anddim(N(Gu(0)))<1,thenGu(0)isaFredholmoperatorwithindexzero.Proof.OnecanverifythatGu(0)=L+isaself-adjointoperatoronL2().Infact,((L+)u;v)=Z(L+)uvdx=ZLuvdx+Zuvdx=ZZ2J(xy)(u(y)u(x))v(x)dydx+Zuvdx=ZZ2J(xy)u(y)v(x)dydxZZ2J(xy)u(x)v(x)dydx+Zuvdx=ZZ2J(xy)u(x)v(y)dydxZZ2J(xy)u(x)v(x)dydx+Zuvdx=ZZ2J(xy)(v(y)v(x))u(x)dydx+Zuvdx=((L+)v;u):(5.2.3)60Thus,Gu(0)isaself-adjointboundedlinearoperatoronHilbertspaceL2(),N(Gu(0))=N(Gu(0))=(Range(Gu(0)))?.Hence,codim(Range(Gu(0)))=dim(N(Gu(0))).Inaddition,Range(Gu(0))isclosed.Indeed,Gu(0)hasadimensionalkernel.SoN(Gu(0))isclosedandL2()=ker(Gu(0))isaBanachspace.themapS:L2()=ker(Gu(0))M(Range(Gu(0)))?!L2()tobe:S(x;c)=T(x)+c.Then,Sisaboundedlinearisomorphism.Hencebyopenmappingtheorem,itisatopologicalisomorphism.Thus,Range(Gu(0))˙L2()=ker(Gu(0))Lf0gandRange(Gu(0))isclosed.Therefore,Gu(0)isaFredholmoperatorwithindexzero.Let4N:D(4N)ˆL2()7!L2()betheNeumannrealizationoftheLapacianinbyD(4N)=fu2H2():@u=0;on@gwhere4Nu=4u=Pni=1@2u@x2i,u2D(4N)and=(1;:::;n)isoutwardnormalunitvectortotheboundary@.Thenwerewritethefollowingequation:8>>>>>>><>>>>>>>:cJ4u+u=0;in@u=0;on@61cJ=12RRnJ(z)jzj2dz:inanabstractway:F(u)=0(5.2.4)whereF:RD(4N)7!L2()isamapping.By([26]),onanyboundedcloseset,thespectrumoftheoperatorGu(0)convergestothespectrumofF(u)asissucientlysmall.Hence,weanalysethebifurcationpropertiesofequationGaroundanybifurcationpoints(0)oftheequationF.Andwehavethefollowingtheorem:Theorem5.2.2.Supposethat2˙(cJ4N)isasimpleeigenvalueofcJ4N.AndletB()=f2C:jgwith>0sosmallthatB\˙(cJ4N)=g.Then(a),thereexists>0sothatwhenissucientlysmallB\˙(L)=0gand0isasimpleeigenvalueofL.(b),ifweassumeN(Gu(0))=spanf!0gandZbeanycomplementofN(Gu(0))inL2(),thenthesolutionsetofG(u)=0near(0;0)consistspreciselyofthecurvesu=0andf((s);u(s)):s2I=(a;a)g,where:I7!RisaC2functionandz:I7!ZisaC1functionsuchthatu(s)=s!0+sz(s),(0)=0,z(0)=0and0(0)=00(0)=0.Proof.Accordingtothetheorem2.1of[26],wehavepart(a)that0isasimpleeigenvalueofL.ByLemma5.2.1,Gu(0)isaFredholmoperatorwithindex0,i.edim(N(Gu(0)))=codim(R(Gu(0)))=1.Gu(0)isanisomorphismfromZtoR(Gu(0)).WeapplytheLyapunov-Schmidtprocess,denotingQastheprojectionfromL2()intoR(Gu(0)).ThenG(u)=0isequivalenttoQG(u)=0and(IQ)G(u)=0.that:f(t;g)=QG(t!0+g)=062wheret2Randg2Z.Calculationshowsthatfg(0;0)=QGu(0),whichisanisomorphismfromZtoR(Gu(0)).Hence,bytheimplicitfunctiontheorem,for(t)near(0;0),thereexistsg=g(t)2C2suchthatf(t;g))=0.Sincethecodim(R(Gu(0)))=1,thereexistl2(L2())suchthatR(Gu(0))=fv2L2():<l;v>=0g.Thus,u=t!0+g(t)isthesolutionofQGu(0)=0ifandonlyif(IQ)G(t!+g(t))=0,i.e.,thescalarequation<l;G(t!+g(t))>=0.Noticethatf(t;g(t))=QG(t!0+g(t))=0istrueforall(t)near(0;0).Dierentiatingfandevaluatingat(0;0),weobtain0=5f=(Q(G+Gu[g]);QGu[!0+gt])Since(0)arethetrivialsolutionsofthenonlocaldiusionequation,G(0;0)=0andGu(0)isanisomorphismfromZtoR(Gu(0)),wecanconcludethatgt(0;0)=g(0;0)=0.h(t)=<l;G(t!+g(t))>,wecanapplyTheorem2.1of[32]toh.5h(0;0)=(h;ht)=(<l;G+Gu[g]>;<l;Gu[!0+gt]>)=(0;0):FortheHessianmatirx,wehaveHess(h)=0BBBBBBBBBBBB@hhththtt1CCCCCCCCCCCCA63whereht=ht=<l;Gu[!0+gt]+Guu[!0+gt;g+Gu[gt]]>=<l;Gu[!0]>;h=<l;G+2Gu[g]+Guu[g;g]+Gu[g]>=<l;0>=0;htt=<l;Guu[!0+gt;!0+gt]+Gu[gtt]>=<l;Guu[!0;!0]>:Insummary,Hess(h)=0BBBBBBBBBBBB@0<l;Gu[!0]><l;Gu[!0]><l;Guu[!0;!0]>1CCCCCCCCCCCCA:Forthedeterminant,wehavedetHess(h)=<l;Gu[!0]>2<0.Bytheorem2.1of[31],theso-lutionsetofG(u)=0near(0;0)isapairofintersectioncurves,(i(s);ui(s))=(i(s);ti(s)!0+g(i(s);ui(s))),withi=1;2.wherevi=(0i(0);t0i(0))arethesolutionof2<l;Gu[!0]>2v1v2+<l;Guu[!0;!0]>v22=0:Thesolution(v1;v2)=(1;0)correspondtothelineoftrivialsolutionsandthesolution(v1;v2)=(0;1)givesthe0(0)=0.645.3AconcreteexampleWeconsideraconcreteexample.Letthedimensionnequalto1andthekernelJ(x)asfollows:J(x)=8>>>>>>>>>>>>><>>>>>>>>>>>>>:0;x<1cj1cos(ˇx2);1x10;x>1withcj=12R11cos(ˇx2)x2dx:J(x)isapositiveevenfunction.Itisacosinefunctionwhenxiswithintheinterval(1;1)and0otherwise.Figure5.1:ThekernelfunctionJ(x).65Hence,wehavethefollowingonedimensionalnonlocalintegralequation:G(u)=Lu+(uu3)=2Zˇ0J(yx)(u(y)u(x))dy+(u(x)u(x)3)=0(5.3.1)Bythemathematicalanalysisoftheabovesection,whenthescalingparameterissmall,G(u)hasthepitchforkbifurcationat=n2withnbeingthepositiveinteger.WeapplytheNewton'smethodtothesolutionoftheequationonitsbifurcationbranch.,wetaketheinitialguesstobethefunctionu0=tanh(q2(xˇ2))andapplytheNewton'siteration:un+1=unGu(un)1G(un)wherefungistheconvergentsequencegeneratedbytheNewton'sMethod.Withtobe0:02andtobe1,2,3and4respectively,wegetthesimulations5.1).Figure5.2:solutionsofthenonlocalequationInthesimulationsof5.2,thebluecurvestandsforthesolutionswhen=1,whichis66atrivialsolution0.Theredcurvestandsforthesolutionwhenis2.Thegreencurvestandsforthesolutionwhenis3;theyellowcurvestandsforthesolutionwhenis4.Asincreases,thesolutionu(x)bifurcatesfromthetrivialsolution0andkukL1(0;ˇ)increasesfrom0to1.WeknownthatthespectraofsuchnonlocaloperatorsconvergetothespectrumofaLaplaceoperatorwiththeNeumannboundaryconditionasthescalingparametertendstozero.Thus,wewanttoknowifthesolutionsofthenonlocaldiusionequationis,tosomeextent,closetotheChafee-Infanteproblem.Hence,wesolvetheonedimensionalChafee-InfanteproblemwiththeboundaryconditionbelowbytheNewton'sMethodaswellandcomparethemwiththenonlocaldiusionequation.8>>>>>>><>>>>>>>:uxx+(uu3)=0;in0<x<ˇ;u0(0)=0;u0(ˇ)=0:In5.3,5.4,and5.5,theredcurvesstandfortheintegralequationsandthebluecurvesstandfortheChafee-Infanteequation.Thosesimulationsshowthatwhenthescalingparameterofthenonlocaldiuisonequationis0:02,forvariousvalues,thesolutionsofthenonlocaldiusionequationareveryclosetothesolutionsoftheChafee-Infanteequation.Furthermore,wewouldliketoseethebifurcationdiagramsofthenonlocaldiusionequation.ItiswellknownthatthesolutionsoftheChafee-Infanteequationbifurcateatn2.Inspiredby5.2,wetheinitialvalueoftheNewton'smethodandanothertwobifurcationpointsofthenonlocaldiusionequation.The5.6,5.7,and5.8showthatthenonlocaldiusionequationhasthepitch-forkbifurcationatpointsaround1,4,and9,whichverifythehypothesisthatwhenisapproachingzero,thebifurcationpointsofthenonlocaldiusionequationconvergestothebifurcationpoints67Figure5.3:Comparisonofthesolutionsofthenonlocalequationwith=2andtheChafee-Infanteequation.oftheChafee-Infanteequation.In5.9,thethreedimensionalgraphshowshowthesolutionsofthenonlocaldiusionequationonthebifurcationbranchvaryastheparameterincreasefrom0to20.Whenislessthanapointcloseto1,solutionsofthenonlocaldiusionequationareconstant.Whenisgreaterthanthebifurcationpoint,therearenon-constantincreasingoddsolutions.TheL1normofthosesolutionsincreaseasincreases.Figure5.10showstheextremecasethatwhenis10000.Whenis10000,thesimulationshowsthatanon-trivialsolutionofthenonlocaldiuisonequationexistsanditisadiscontinuousfunctionwithajumpdiscontinuityatx=0.68Figure5.4:Comparisonofthesolutionsofthenonlocalequationwith=3andtheChafee-Infanteequation.Figure5.5:Comparisonofthesolutionsofthenonlocalequationwith=4andtheChafee-Infanteequation.69Figure5.6:ThebifurcationofthenonlocalequationFigure5.7:Thesecondbifurcationofthenonlocalequation70Figure5.8:ThethirdbifurcationofthenonlocalequationFigure5.9:Thesolutionsofthenonlocalequationwithvarying71Figure5.10:Thesolutionofthenonlocalequationthebifurcationbranchwhen=1000072BIBLIOGRAPHY73BIBLIOGRAPHY[1]EricksonRP,JiaZ,GrossSP,ClareCY.Howmolecularmotorsarearrangedonacargoisimportantforvesiculartransport,PLoSComputBiol.,7,1Œ22.[2]JiaZ,KarpeevD,AransonIS,BatesPW.Simulationstudiesofself-organizationofmicro-tubulesandmolecularmotors,PhysicalReviewE.,77(5)(2008),051905.[3]BatesPW,JiaZH.Neck-linkertensionandthelocomotionofkinesinalongmicrotubules,Can.Appl.Math.Q.,(3)(2010),229Œ252.[4]EssaghirA,DifN,MarbehantCY,CoerPJ,DemoulinJB.ThetranscriptionofFOXOgenesisstimulatedbyFOXO3andrepressedbygrowthfactors,J.Biol.Chem.,284(2009),10334Œ10342.[5]KopsGJ,deRuiterND,DeVries-SmitsAM,PowellDR,BosJL,BoudewijnMT.DirectcontroloftheForkheadtranscriptionfactorAFXbyproteinkinaseB,Nature,398(1999),630Œ634.[6]MatsuzakiH,DaitokuH,HattaM,TanakaK,FukamizuA.Insulin-inducedphosphorylationofFKHR(Foxo1)targetstoproteasomaldegradation,Proc.Natl.Acad.Sci.USA,100(2003),11285Œ11290.[7]PuigO,TjianR.TranscriptionalfeedbackcontrolofinsulinreceptorbydFOXO/FOXO1,Genes&Dev.,19(2005),2435Œ2446.[8]QuonMJ,LA.Amathematicalmodelandcomputersimulationstudyofinsulinreceptorregulation,J.Theor.Biol.,150(1991),59Œ72.[9]QuonMJ,LA.Amathematicalmodelandcomputersimulationstudyofinsulin-sensitiveglucosetransporterregulation,J.Theor.Biol.,150(1991),93Œ107.[10]ResinoJ,Garca-BellidoA.Drosophilageneticvariantsthatchangecellsizeandrateofproliferationaectcellcommunicationandhencepatterning,MechanismsofDevelopment,121(2004),351Œ364.[11]SedaghatAR,ShermanA,QuonMJ.Amathematicalmodelofmetabolicinsulinsignalingpathways,Am.J.Physiol.Endocrinol.Metab.,283(2002),84Œ101.74[12]ShingletonAW.TheregulationoforgansizeinDrosophilamelanogaster:physiology,plas-ticity,patterningandphysicalforce,Organogenesis,6(2)(2010),76Œ87.[13]ShingletonAW,FrankinoWA,FlattT,NijhoutHF,EmlenD.Sizeandshape:thedevelop-mentalregulationofstaticallometryininsects.BioEssays,textbf29(6)(2007),536Œ548.[14]SmithGR,ShanleyDP.ModellingtheresponseofFOXOtranscriptionfactortomultiplepost-translationalmadebyageing-relatedsignallingpathways,PLoSONE,5(2010),1Œ18.[15]TangHY,Smith-CaldasMS,DriscollMV,SalhadarS,ShingletonAW.FOXOregulatesorphenotypicplasticityindrosophila,PLoSGenet.,7(2011),1Œ12.[16]VanDerHeideLP,HoekmanMF,SmidtMP.TheinsandoutsofFoxOshuttling:MechanismofFoxOtranslocationandtranscriptionalregulation,Biochem.J.,380(2004),297Œ309.[17]BatesPW,LiangY,ShingletonAW.Growthregulationandtheinsulinsignalingpathway,NHM,8(1)(2013),65Œ78.[18]NeufeldTP.Shrinkagecontrol:regulationofinsulin-mediatedgrowthbyFOXOtranscriptionfactors.JBiol.,2(2003),18Œ54.[19]JungerMA,RintelenF,StockerH,WassermanJD,VeghM,etal..TheDrosophilaForkheadtranscriptionfactorFOXOmediatesthereductionincellnumberassociatedwithreducedinsulinsignaling.JBiol.,2(2003),20Œ47.[20]KramerJM,DavidgeJT,LockyerJM,StaveleyBE.ExpressionofDrosophilaFOXOregu-latesgrowthandcanphenocopystarvation.BMCDevBiol.,3(2003),5Œ28.[21]PuigO,MarrMT,RuhfML,TjianR.ControlofcellnumberbyDrosophilaFOXO:down-streamandfeedbackregulationoftheinsulinreceptorpathway.GenesDev.,17(2003),2006Œ2020.[22]VanDerHeideLP,HoekmanMF,SmidtMP.TheinsandoutsofFoxOshuttling:mechanismsofFoxOtranslocationandtranscriptionalregulation.BiochemJ.,380(2004),297Œ309.[23]FuW,MaQ,ChenL,LiP,ZhangM,etal..MDM2actsdownstreamofp53asanE3ligasetopromoteFOXOubiquitinationanddegradation.JBiolChem.,284(2009),13987Œ14000.[24]KublaouiB,LeeJ,PilchPF.Dynamicsofsignalingduringinsulin-stimulatedendocytosisofitsreceptorinadipocytes.JBiolChem.,270(2009),59Œ65.75[25]MadoDH,MartensenTM,LaneMD.Insulinandinsulin-likegrowthfactor1stimulatethephosphorylationontyrosineofa160kDacytosolicproteinin3T3-L1adipocytes.BiochemJ.,252(1988),7Œ15.[26]BatesPW,ZhaoG.SpectralConvergenceandTurningPatternsforNonlocalDiusionSystems.[27]BatesPW,FifePC,RenX,WangX.Travelingwavesinaconvolutionmodelforphasetransitions.ArchiveforRationalMechanicsandAnalysis,138(2)(1997),105Œ136.[28]BatesPW,ChenX,ChmajAJ.Travelingwavesofbistabledynamicsonalattice.SIAMJournalonMathematicalAnalysis,35(2)(2003),520Œ546.[29]BatesPW,ChenX,ChmajAJ.HeteroclinicsolutionsofavanderWaalsmodelwithnonlocalinteractions.CalculusofVariationsandPartialDierentialEquations,24(3)(2005),261Œ281.[30]ChafeeN,InfanteEF.Abifurcationproblemforanonlinearpartialdierentialequationofparabolictype.ApplicableAnalysis,4(1)(1974),17Œ37.[31]ShiJ.Persistenceandbifurcationofdegeneratesolutions.JournalofFunctionalAnalysis,169(2)(1999),494Œ531.[32]LiuP,ShiJ,WangY.Imperfecttranscriticalandpitchforkbifurcations.JournalofFunctionalAnalysis,251(2)(2007),573Œ600.[33]CrandallMG,RabinowitzPH.BifurcationfromSimpleEigenvalues.JournalofFunctionalAnalysis,8(2)(1971),321Œ340.[34]Rabinow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