Abstract Theshapeandfunctionofanaxonisdependentonitscytoskeletonincludingmicrotubulesneuro28lamentsandactinNeuro28lamentsaccumulateabnormallyinaxonsinmanyneurologicaldisordersAnearlyeventofs ID: 839524
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1 AnalysisofaStochasticModelofAxonalCytosk
AnalysisofaStochasticModelofAxonalCytoskeletonSegregationinDiseasesXigeYangDepartmentofMathematics,OhioStateUniversityAdvisor:ChuanXue Abstract Theshapeandfunctionofanaxonisdependentonitscytoskeleton,includingmicrotubules,neurolamentsandactin.Neurolamentsaccumulateabnormallyinaxonsinmanyneurologicaldisorders.Anearlyeventofsuchaccu-mulationisastrikingradialsegregationofmicrotubulesandneurolaments.Thissegregationphenomenonhasbeenobservedforover30yearsnow,buttheunderlyingmechanismisstillpoorlyunderstood.Inthisposterwepresentastochasticmodelthatexplainedthisphenomenonandpreliminaryanalysisofthemodel. BiologicalBackground Axonalcytoskeleton.Theaxonalcytoskeletonisanintracellularpolymersystemthatconsistsofthreelamentouscomponents:microtubules(MTs),neurolaments(NFs),andactin(smallerandthusneglectable).Axonaltransport.Thistransportismediatedbymolecularmotorscalledkinesinanddynein.OrganellesoftenmovemorefrequentlyandmorepersistentlythanNFsandMTs,thelattermoveshorterdistancesandspendmostoftheirtimepausing.NeurodegenerativeDiseases.Neurolamentshasbeenobservedtoaccumulateabnormallyinaxonsandcauseaxonalswellinginmanyneurodegenerativediseasesandtoxicneuropathiesincludingamyotrophiclateralsclerosis,hereditaryspasticparaplegiaetc.Astrikingradialsegregationofmicrotubulesandneurolamentsoftenproceedstheaxonalswelling.IDPNisatoxinthatcausessimilarsymptoms. Figure1:Cross-sectionaldynamicsafterIDPNinjection PriorWork Questionstobeunderstood:HowdomicrotubulesandneurolamentssegregateafterIDPNapplication?Whydoesthesegregationphenomenaoccursonatimescaleofhoursandisreversible?HowissegregationrelatedtoimpairmentofNFtransportandaxonalswelling? Figure2:MTs,NFsandOrganellesThestochasticmodel.Xueetal.[1]developeda2DstochasticmodelthatdescribesthedynamicsofMTs,NFsandorganellesinanaxonalcross-section,dxki=Fki=kdt+kdWki:(1)herexkiisthepositionofthei-thparticleofk-type,(k=1;2;3representingforMTs,NFsandorganellsrespectively)Fkiisthetotalforceactingonthatparticle Figure3:MT-MTandMT-NFbindingModelresults:segregationandremixingexplained Figure4:Numericalsimulationofcross-sectionalpolymermove-mentafterIDPNinjectionandwashout.Bluecircles:organells;Darkdots:MTs;Shallowdots:NFs. PreliminaryResults Asimpliedmodel.WeconsiderasimpliedmodelthatreplacestheindirectinteractionofMTsthroughorganellesbydirectinteractionofMTsthroughspringforces.Thisismotivatedbytheresultsofthestochasticmodel,yielding(2)below: Figure5:ModelSimplication8]TJ ; -4;.61; Td; [00;]TJ ; -4;.61; Td; [00;]TJ ; -4;.61; Td; [00;:dxi=Pj1 (R(xjxi)+sijS(xjxi))dt+dWi;sijjumpsbetween0and1withrates 01(xjxi)and 10(xjxi)independently:(2)ThedCKequation.Sincemodel(2)involvesalargenumberofequations,wewanttoderivethecorrespongdingdierantialChapman-Kolmogorov(dCK)equation.Assumestatedynamicsvarymuchfasterthanpolymerlocations,usingfollowingnotations:x:=(x1;x2;:::;xn):theconcatenatedpositionvectorwithnparticles.s:=fsij;i6=jg:thebindingstateoftheparticlesystemp(s;x;t):theprobabilityforthesystem(2)tohavepositionxandstatesattimetwithproperinitialconditions.Du:=2 2:thediusioncoecient.F(s;x):=Pi;j;i6=j1 [R(xixj)+sijS(xixj)]w(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)ASpecialCase: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 DeneprojectionoperatorP:=fs1T,sops=Pps+(IP)ps.It'seasytocheckPps=(x;t)fs,(IP)ps=Vs.ActingPon(3),collectingleadingtermsinweget:AVs=Fsfrxfsg+2 2xfsrx[FTsfs]fs@ @t=rx(FTsfs)+rx(Brx)WhereA(x);B(x)aresquarematrixeswhosedimensionsmatchthatofVs.NumericalJustication.Numericalresultssuggestthatwecanapproximate(2)withdxki=Xj1 (R(xjxi)+k1S(xjxi))dt+dWi(4)wherek1= 01=( 01+ 10).Wesimulatedthemovementof20MTsin1Dintervalof2mlong.Graphsarelistedbelow,indicatingthatmodel(3)and(4)matchquitewell. Figure6:Left:20samplepathsofMTpositions.Right:MeanandstandarddeviationofMTpositions FutureWorks ExtendabovecasetoincludebothMTsandNFsthatinteractthroughrandomMT-MTandMT-NFbinding.Extendtheanalysistoallowforrandomarrivalanddepartureofparticles.Furthernumericaljusticationofthemodelastherstbulletpoint. Reference [1]C.Xue,S.B.Brown.AStochasticMultiscaleModelThatExplainstheSegregationofAxonalMicrotubulesandNeurolamentsinNeurologicalDiseases.PLoSComputBiol,2015. Acknowledgement ThisworkisfundedbyUSNSF1312966and1553637.