Abstract Theshapeandfunctionofanaxonisdependentonitscytoskeletonincludingmicrotubulesneuro28lamentsandactinNeuro28lamentsaccumulateabnormallyinaxonsinmanyneurologicaldisordersAnearlyeventofs ID: 839524
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1 AnalysisofaStochasticModelofAxonalCytosk
AnalysisofaStochasticModelofAxonalCytoskeletonSegregationinDiseasesXigeYangDepartmentofMathematics,OhioStateUniversityAdvisor:ChuanXue Abstract Theshapeandfunctionofanaxonisdependentonitscytoskeleton,includingmicrotubules,neuro lamentsandactin.Neuro lamentsaccumulateabnormallyinaxonsinmanyneurologicaldisorders.Anearlyeventofsuchaccu-mulationisastrikingradialsegregationofmicrotubulesandneuro laments.Thissegregationphenomenonhasbeenobservedforover30yearsnow,buttheunderlyingmechanismisstillpoorlyunderstood.Inthisposterwepresentastochasticmodelthatexplainedthisphenomenonandpreliminaryanalysisofthemodel. BiologicalBackground ˆAxonalcytoskeleton.Theaxonalcytoskeletonisanintracellularpolymersystemthatconsistsofthree lamentouscomponents:microtubules(MTs),neuro laments(NFs),andactin(smallerandthusneglectable).ˆAxonaltransport.Thistransportismediatedbymolecularmotorscalledkinesinanddynein.OrganellesoftenmovemorefrequentlyandmorepersistentlythanNFsandMTs,thelattermoveshorterdistancesandspendmostoftheirtimepausing.ˆNeurodegenerativeDiseases.Neuro lamentshasbeenobservedtoaccumulateabnormallyinaxonsandcauseaxonalswellinginmanyneurodegenerativediseasesandtoxicneuropathiesincludingamyotrophiclateralsclerosis,hereditaryspasticparaplegiaetc.Astrikingradialsegregationofmicrotubulesandneuro lamentsoftenproceedstheaxonalswelling.IDPNisatoxinthatcausessimilarsymptoms. Figure1:Cross-sectionaldynamicsafterIDPNinjection PriorWork ˆQuestionstobeunderstood:ˆHowdomicrotubulesandneuro lamentssegregateafterIDPNapplication?ˆWhydoesthesegregationphenomenaoccursonatimescaleofhoursandisreversible?ˆHowissegregationrelatedtoimpairmentofNFtransportandaxonalswelling? Figure2:MTs,NFsandOrganellesˆThestochasticmodel.Xueetal.[1]developeda2DstochasticmodelthatdescribesthedynamicsofMTs,NFsandorganellesinanaxonalcross-section,dxki=Fki=kdt+kdWki:(1)herexkiisthepositionofthei-thparticleofk-type,(k=1;2;3representingforMTs,NFsandorganellsrespectively)Fkiisthetotalforceactingonthatparticle Figure3:MT-MTandMT-NFbindingˆModelresults:segregationandremixingexplained Figure4:Numericalsimulationofcross-sectionalpolymermove-mentafterIDPNinjectionandwashout.Bluecircles:organells;Darkdots:MTs;Shallowdots:NFs. PreliminaryResults ˆAsimpli edmodel.Weconsiderasimpli edmodelthatreplacestheindirectinteractionofMTsthroughorganellesbydirectinteractionofMTsthroughspringforces.Thisismotivatedbytheresultsofthestochasticmodel,yielding(2)below: Figure5:ModelSimpli cation8���&#x]TJ ;� -4;.61; Td;&#x [00;&#x]TJ ;� -4;.61; Td;&#x [00;&#x]TJ ;� -4;.61; Td;&#x [00;:dxi=Pj1 (R(xj�xi)+sijS(xj�xi))dt+dWi;sijjumpsbetween0and1withrates 01(xj�xi)and 10(xj�xi)independently:(2)ˆThedCKequation.Sincemodel(2)involvesalargenumberofequations,wewanttoderivethecorrespongdingdierantialChapman-Kolmogorov(dCK)equation.Assumestatedynamicsvarymuchfasterthanpolymerlocations,usingfollowingnotations:ˆx:=(x1;x2;:::;xn):theconcatenatedpositionvectorwithnparticles.ˆs:=fsij;i6=jg:thebindingstateoftheparticlesystemˆp(s;x;t):theprobabilityforthesystem(2)tohavepositionxandstatesattimetwithproperinitialconditions.ˆDu:=2 2:thediusioncoe cient.ˆF(s;x):=Pi;j;i6=j1 [R(xi�xj)+sijS(xi�xj)]ˆw(sjs0;x):thejumpratefromstates0tostateswithinitialpositionvectorx.ThecorrespondingdCKequationyields:@p(s;x;t) @t+rx[F(s;x)p(s;x;t)]=x[2 2p(s;x;t)]+1 Xs06=s[w(sjs0;x)p(s0;x;t)�w(s0js;x)p(s;x;t)](3)ˆASpecialCase:ModelReduction.Bymeansofquasi-steady-stateanalysis,wecanwritep(s;x;t)=(x;t)f(s;x)+V(s;x;t).Here(x;t)=Psp(s;x;t)andf(s;x)isthequasi-stationarydistributionofs.Alsodenotevstobetherowvector(v(s;x;t))Tsinwhichviseitherp;f;ForV.VisthenoisetermsatisfyingPsV(s;x;t)=0 De neprojectionoperatorP:=fs1T,sops=Pps+(I�P)ps.It'seasytocheckPps=(x;t)fs,(I�P)ps=Vs.ActingPon(3),collectingleadingtermsinweget:AVs=Fsfrxfsg+2 2xfs�rx[FTsfs]fs@ @t=�rx(FTsfs)+rx(Brx)WhereA(x);B(x)aresquarematrixeswhosedimensionsmatchthatofVs.ˆNumericalJusti cation.Numericalresultssuggestthatwecanapproximate(2)withdxki=Xj1 (R(xj�xi)+k1S(xj�xi))dt+dWi(4)wherek1= 01=( 01+ 10).Wesimulatedthemovementof20MTsin1Dintervalof2mlong.Graphsarelistedbelow,indicatingthatmodel(3)and(4)matchquitewell. Figure6:Left:20samplepathsofMTpositions.Right:MeanandstandarddeviationofMTpositions FutureWorks ˆExtendabovecasetoincludebothMTsandNFsthatinteractthroughrandomMT-MTandMT-NFbinding.ˆExtendtheanalysistoallowforrandomarrivalanddepartureofparticles.ˆFurthernumericaljusti cationofthemodelasthe rstbulletpoint. Reference [1]C.Xue,S.B.Brown.AStochasticMultiscaleModelThatExplainstheSegregationofAxonalMicrotubulesandNeuro lamentsinNeurologicalDiseases.PLoSComputBiol,2015. Acknowledgement ThisworkisfundedbyUSNSF1312966and1553637.