To be published in Biophysical Journal Single-Part

TobepublishedinBiophysicalJournalSingle-ParticleTrackingBrownianDynamicsofViscoelasticMaterialsHongQianDepartmentsofAppliedMathematicsandBioengineeringUniversityofWashington,Seattle,WA98195September20,1999ABSTRACTAunifyingtheoreticalframeworkforanalyzingstochasticdatafromsingle-particletracking(SPT)inviscoelasticmaterialsispresented.Ageneralizationofthebead-springmodelforlinearpolymersisdevelopedfromamolecularpointofviewandfromthestandpointofphenomenologicallinearviscoelasticity.Thehydrodynamicinterac-tionintheformerisidentiedasthedashpotsinthelatter.Inelementaryterms,theintimatecorrespondencebetweentime-correlationoftheuctuationmeasurementsandtransientrelaxationkineticsafterperturbationisdiscussed,andthecentralroleofuctuation-dissipationrelationisemphasized.Theworkpresentedhereprovidesabridgebetweenthemicroscopicandthemacroscopicviewsoflinearviscoelasticbio-logicalmaterials,andisapplicabletomembraneproteindiusion,linearDNAchaindynamics,aswellasmechanicsofintracellularcytoskeletalnetworks.Runningtitle:SPTStudyofViscoelasticityKeywords:Bead-and-spring/creepfunction/uctuation-dissipation/macromolec-ularmechanics/nano-biochemistry/video-enhancedopticalmicroscopy1INTRODUCTIONRecentlydevelopedmicro-rheologybasedontrackingthemovementsofindividualBrownianparticlesprovidesanovelapproachforstudyingviscoelasticmaterialsandbiologicaltissuesatasubcellularandmolecularlevel(Gittesetal.,1997,Masonetal.,1997).Themethodologybelongstoaclassofopticaltechniqueswhichquantitativelyfollowthemovementofsmallnoninvasiveopticalmarkerswithhighspatialresolution(nm),knownassingle-particletracking(SPT)amongmanyothernames(Geertsetal.,1987,Gellesetal.,1988,GrossandWebb,1988,Qianetal.,1991,Amblardetal.,1996,QianandElson,1999,seealsoareviewbySaxtonandJacobson,1997,onitswideapplicationstocellmembranedynamics).Interpretingthequantitativeyetstochasticdatademandsatheoreticalframework.Twoparallelapproaches,amolecularandaphenomenological,arepossible.Inthispaper,wedevelopontheonehandamolecularapproachbasedonanaturalextensionofthesimplediscretepoly-mertheory,whichtreatsapolymergelasbeadsconnectedbyspringsinanaqueoussolutionwithhydrodynamicinteractions(DoiandEdwards,1986).Thephenomeno-logicaltreatmentoflinearviscoelasticityinmechanics,ontheotherhand,isbasedontheconceptofmemoryfunctionsknownascreepandrelaxation(Fung,1965,Ferry,1980).Combiningthesetwoapproaches,weshowthatthecreepfunctioninthephe-nomenologicaltheorycanbederivedintermsofthemolecularmodel.Consequently,thestochasticmovementofaparticleinaviscoelasticmaterialcanbeanalyzedeitherasaBrownianmotionofoneparticleinanN-particlesystemwithspatiallycorrelatedwhitenoise(N-dimensionalLangevinequation),orasingleparticlewithanon-whitenoise(generalizedLangevinequation)wherethenoisesrepresenttherandomcolli-sionsbetweenthebeadsandthesolventmolecules(Brownianforces).Basedonthisapproach,thepresentworkprovidesaunifyingtheoryforSPT,demonstratesacor-respondencebetweenthetime-correlationofthenoveluctuationmeasurementsandthetraditionalrelaxationkineticsafterperturbation(theOnsager’shypothesisoflin-earirreversibility),establishesanequivalencebetweentheOnsager’shypothesisandtheuctuation-dissipationrelation,anddevelopsabridgebetweenthemicroscopicandthemacroscopicviewsoflinearviscoelasticity.TheapproachinthisworkisapplicabletoSPTinviscousuidswithandwithoutdrift(Qianetal.,1991),SPTofsinglepolymerchains(QianandElson,1999),aswellasSPTinpolymergelnetworks.Thepaperisorganizedasfollows.Inthenexttwosectionswedevelopourmainanalysesbasedonthemolecularapproachandthephenomenologicalapproach,re-spectively.Itwillbeshownthatuctuation-dissipationrelationleadstotheOnsager’shypothesis,andviceversa.ThefourthsectiondealswithSPTmeasurementsinvis-2coelasticliquids(complexuids)inwhichtheBrownianmotionisnonstationary.Hencemean-squaredisplacementratherthanthetimecorrelationfunctionhastobeintroduced.ThelastsectionprovidesacomprehensivediscussiononSPT,itstheories,applications,relationswithotherwork,anditsmathematicalfoundation.MOLECULARVISCOELASTICITY:SPRINGSANDDASHPOTSThesimplestmoleculartheoryofviscoelasticityassumesthatthemicroscopicdy-namicsofapolymernetworkisdescribedbyasystemofsphericalBrownianparticlesinanaqueoussolution,connectedbyspringsandinteractingwitheachotherviahy-drodynamicinteractions(DoiandEdwards,1986).LettheelementkijofamatrixKbethestinessconstantofthespringconnectingbeadsiandj,andhijofamatrixHbethehydrodynamicinteractionbetweenbeadsiandj.Hisknownasthemobilitymatrix.TheNewton’sequationforoverdampedmotionofasystemofparticlesinviscoussolution,therefore,isdXdt=H(KX+F)(1)whereX=(x1;x2;:::;xN)Tarethecoordinatesofbeads1,2,...,N.F=(f1;f2;:::;fN)Tarewhitenoiselikerandomforcessatisfyinghfi(t)fj(t0)i=ij(t0t).Tisacovari-ancematrix,tobediscussedlater,characterizingthespatialcorrelationbetweentherandomforces.Therandomforcesrepresenttheincessantrandomcollisionsbetweenthebeadsandthesolventmolecules.Aself-containedbriefdiscussionofrandomforcinginBrowniandynamicscanbefoundinKlapperandQian(1998).FormoregeneraldiscussionsseeF