Traction - PDF document

TractionÞelds,moments,andstrainenergythatcellsexertontheirsurroundingsJAMESP.BUTLER,IVAMARIJATOLICBENFABRY,ANDJEFFREYJ.FREDBERGHarvardSchoolofPublicHealth,Boston,Massachusetts02115;RugjerBosÿkovic«Institute,10001Zagreb,CroatiaReceived15June2001;acceptedinÞnalform24October2001Butler,JamesP.,IvaMarijaTolic´-Nørrelykke,BenFabry,andJeffreyJ.Fredberg. CELLSEXERTTRACTIONSontheirsurroundingsinthecourseofavarietyofcellfunctionsincludingcontrac-tion,spreading,crawling,andinvasion.Thesefunc-tionsareassociatedwithcomplexmechanicalinterac- Addressforreprintrequestsandothercorrespondence:J.P.But-ler,PhysiologyProgram,HarvardSchoolofPublicHealth,665Hun- Thecostsofpublicationofthisarticleweredefrayedinpartbythepaymentofpagecharges.ThearticlemustthereforebeherebymarkedÔÔÕÕinaccordancewith18U.S.C.Section1734solelytoindicatethisfact.AmJPhysiolCellPhysiol282:C595ÐC605,2002.FirstpublishedOctober31,2001;10.1152/ajpcell.00270.2001.0363-6143/02$5.00Copyright2002theAmericanPhysiologicalSociety TheBoussinesqSolutionThetractioneldisdenedasthestress,i.e.,localforceperunitarea,imposedonthegelsurfacebyanadherentcell.Thetractioneld,inturn,determinesthedisplacementeldofthegelsurface.Ifthegelcanbeapproximatedasasemi-innitesolid(seeLimitsof),thedisplacementscanbecomputedfromthedistributionofsurfacetractionsasfollows.First,ndsthedisplacementeld,orGreensfunction,associatedwithapointtractiononthesurface.Thisisaclassicproblem,thesolutiontowhichwasfoundbyBoussinesq(seeRef.4).Second,integrationofthisfunctionoverthegiventractioneldthenyieldsthecorrespondingdisplacementeld;thisistheso-calledforwardproblem.Theproblemweaddressinthisarti-cleistheinverseproblem,namely,inferringthetrac-eldfrommeasureddisplacements.TheFTTCmethodisbasedonFourieranalysisandarisesfromtheobservationthatthedisplacementatanypointonthesurfaceduetoapointtractionsourceatanotherpointis(apartfromthedirectionofthedisplacementandthedirectionofthetraction)afunc-tiononlyofthedifference.Wedenotethedisplacementvectorat)andthetractionvector).WedenotetheGreensfunction,orkernel,mappingtractiontodisplacementbythetensor).Thedisplacementsarethengivenbythe.Inthisrepresentation,)and)are2-vectorswithelementslabeledignoredisplacementsinthedirection,andthetrac-tioninthedirection,ornormalstress,istakentobe0),thekernelisa22matrix,andintegrationover.Themajordifcultyintheinver-sionofthisequationisthatisnotdiagonalin(ifitwere,thenthesolutionwouldonlyinvolveinvertinga22matrix).Thefactthatisnotdiagonalinrealspace(i.e.,tractionsatonepointarecoupledtodisplacementsatdifferentpoints)istheoriginofwhyitsinversioninrealspacenecessarilyrequirestheconstructionandinversionofverylargematrices(asintheDWapproach).InFourierspace,thesedifcultiesdonotarise.TheBoussinesqsolutionisdiagonalinFourierspace.ThekeytotheFTTCmethodistheexploitationoftheFaltungorconvolutiontheorem,whichstatesthattheFouriertransformofaconvolutionisthesimpleprod-uctoftheFouriertransformsofthefunctionscon-volved.Theforwardproblemthenbecomes),wherethetildeoverbardenotesthe(twodimensional)FouriertransformwithwavevectorInthisformitisclearthatisdiagonalinthatthereisnocouplingbetweendifferentwavevectors.Ofremainsanondiagonal22matrixinsofarastractionsinthedirectionseparatelyinducedisplacementsinboththedirections,butthispresentsnodifcultybecauseremainsstrictlydiag-onalinspace.Itfollowsthatistrivialtocomputeisknown.ThesolutiontotheinverseproblemisthengivenbywhereFTdenotesthe(twodimensional)inverseFouriertransform.ExplicitevaluationofthekernelinFourierspace.ImplementationofEq.1requiresanexplicitformula),theFouriertransformoftheBoussinesqsolu-),where.Theforwardkernel,writteninmatrixform,forapointsourceattheoriginisgivenbyLandauandLifshitz(4),whichwhenreducedtodisplacementswithzeronormaltractionisgivenby ,inwhichisPoissonsratioandisYoungsmodulus.Thecomponentsofthismatrixaredenoted,wheretheindicesrunthroughx,y);thisnotationwillbeusedconsistentlywithallmatricesandvectors.Thusthetwo-dimensionalFouriertransformsof,andarerequired.Thesecanbederivedinseveraldifferentways.Oneespeciallyclearmethodcanbesketchedasfollows;itreliesonageneralizationoftheproblemtothreedimen-sionsandasubsequentreductiontotwodimensions.WebeginwiththesingularsolutiontoLaplacesequationinthreedimensions,),whereistheDiracdeltafunction.Wedenotethethree-dimensionalFouriertransformbyFT,whichwhenappliedtotheLaplaceequationyieldsFT),whereweseparatedoutthewavevectorfromthewavevectorsinthex,yplane(here,andinwhatisthenonnegativesquarerootof).Nownotethatthetwo-dimensionalFouriertransformisthesingleinversethree-dimensionaltransformin,eval-uatedat0,i.e.ThedesiredanswerreducestoanelementaryintegralTheotherthreetransformscanbeobtainedbysimplemanipulations.Considertheexpressionexpression(/x)r1].Thiscanbeevaluatedintwodistinctways.First,directlydifferentiatingwithrespecttoSecond,integratingbypartswithrespecttoFOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 whereweusethetransformofobtainedabove.Performingtheindicatedderivativewithrespecttoandsettingthetwoindependentevaluationsequal,weobtainFT.Bysymmetry,thetransformofcanbewrittendownbyinspection,andthetransformofcanbeobtainedbyevaluatingevaluating(/x)r1]inthesametwodistinctwaysasabove.Insummary,thedesiredtransformedmatrixisgivenby Contractionmoments.TheFourierapproachalsogivesrobustmeasuresofcertainlowordermomentsofthetractions.Thezerothordermomentofthetractionsisgivenbyandisequaltothenetforceappliedbythecelltothesubstrate.Forisolatedadherentcells,thisisknownaprioritobezero.However,registrationshiftsfromoneimagetoanotherofagivenpair(saybeforeandaftercontractileactivation)willinduceaspurioustractioneldcorrespondingtoanon-zeronetforce.ThisartifactcanbetriviallyaccountedforinFourierspacesimplybysetting0,whichguaranteesnonetforcebythecellonthesubstrate.rstordermomentsareassociatedwithcontrac-tion/dilationtractions(radiallyorientedtractions)andtractionscorrespondingtotorques(circumferentiallyorientedtractions).Thesecorrespondtothefourcom-binationsofthetractionsintheweightedbytheircoordinates.Forexample,apositivetractioninthedirection()atalocationwithapositivecoordinate(oranegativeatsome0)willcontributetoacounterclockwiserotationaltorquethroughatermproportionalto.Asavisualaid,thefourtermsareshownschematicallyinthefollowingdiagrammaticrepresentation Whensymmetrized(sincethenettorqueconferredbythecellonthesubstratemustbezero)andintegratedoverthesurface,thisisthecontraction/dilationandshearmomentmatrix.Writtenincomponentnota-tion,thismatrixisexplicitlygivenbyWeapproximatethederivativesbydiscretedifferencesspace.Because0byconstruction(toelim-inateregistrationartifact),thisexpressionthenin-volvesonlytheFouriertransformsofthetractionsatthelowestnon-zerowavenumbers,.ExEq.4reducesto Theinterpretationoftheelementsofthemomentma-,isasfollows.Thetotalcontributionofthecelltocontractingthesubstrateinthedirectionsisgivenby,respectively.M)isthecontributionofthecelltodeformationofthesubstratearisingfromvariationsintractionswithandvari-ationsintractionswith.Thereareadditionalaniso-tropiccontributionsarisingfromunequalvariationsoftractionswithandfromtractionswith,i.e.,when.Asimplewaytocharacterizethisistoapplyarotationoperatorsuchthatdiagonal,i.e.,0.Thisformputsallthetractionsofthecellintotheirprincipalaxes.Theori-entationoftheprincipaltractionscanbeobtainedfromcoordinateaxesoftheoriginalimagesandtheangleofrotationof.Thisorientationofthecellisthenclearlyindependentofthecoordinatesystem,andmaybeanimportantmeasureofdirectionalityincellmotilityassays,includingchemotaxis.Inthiscontext,theratiooftheprincipaltractionsisadirectmeasureofthetractionpolarityofthecell.Thenetmomenttendingtodilateorcontractthesubstrateisgivenbythetraceofthemomentmatrix;wethusdenethenetcontractilemomentofthecell(Hereeithercanbeusedbecausethetraceisinvariantundercoordinaterotations.)Thenetcontrac-tilemomentisacoordinateinvariantscalarmeasureofthecellscontractileStrainenergy.Thetotalenergytransferredfromthecelltotheelasticdistortionofthesubstrateisgivenandisanothermeasureofcontractilestrength.TheuseofFourieranalysismayinvolvesomeartifactualbe-haviorattheeldboundaries,whichisparticularlypertinentincomputingthestrainenergy.Wethereforeevaluatethisintegraloverthestrictinterioroftheeld,i.e.,withouttheboundarypoints.Becauseofthis,thevalueobtainedisdifferentfromtheevaluationofEq.7inFourierspacewithParsevalstheorem.ThesourceofthisdiscrepancyliesintheuseofFourieranalysisoveranitedomain,whereinperiodicbound-aryconditionsareimposed.Ingeneral,thedisplace-mentsattheedgesoftheeldofviewwillnotapprox-imatecontinuousperiodicfunctions,andthereforeFOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 therewillbeartifactualtractionspresentalongtheboundingnodesofthelatticegrid.Theseinturnwillcontributetotheestimatedstrainenergyiftheyareincludedintheintegralabove.SuchartifactsalsoarepresentintheFourierdomain,althoughtheyareingeneralspreadoverallFouriercomponents.Itisthere-foresimplertoavoidthisproblembydirectintegrationofthestrainenergydensityoverthestrictinteriorofthedomain.Limitsofapplicability.NotethatboththeFTTCmethodpresentedhereaswellastheDWmethodandthatofBalabanetal.(1)explicitlyapproximatetheelasticgelasasemi-innitemedium.Ithasbeenstatedthatthisisvalidifthedisplacementsaresmallcomparedwiththegelthickness(3),butthisisnotcorrect(WilsonTA,personalcommunication).Infact,theratioofdisplacementstogelthicknessbeingsmallisanecessary(butnotsufcient)conditionfortheapplicabilityoflinearelasticitytheory.Bycontrast,theuseofasemi-inniteelasticcontinuumtoapproximatenitethicknessgelisvalidtotheextentthatthelateraldimensionsofthecellandthelateraldistancesoverwhichdisplacementsaremeasuredarebothsmallcomparedwiththegelthickness.Asimpleexamplewillillustratethis.Considerthecaseofuniformtrac-overacircularregionofradius,ontopofanincompressibleslabofthickness,shearmodulusandwithaxedbottom.If,thenthegeliseffectivelyinnitelythick,theBoussinesqsolutionap-plies,andthedisplacementofthediskisapproxi-.Bycontrast,if,thenthemediumisapproximatelyinsimpleshear,andthedisplacementofthediskisapproximately.Thisimpliesthat,inthelattercase,thedisplacementswillbelowerthanwouldbeobservedinthesemi-innitemedium,whichinturnleadstoanunderestimateofthetractions.Theimplementationofallmethodsofcomputingtractionsfrombeaddisplacementsnaturallydividesintotwoparts.Therstistheestimationofthedis-elditselfonsomeappropriatelatticeormesh,andthesecondisthecomputationofthetractioneldforthatgivendisplacementeld.Thissectiondescribesthesetwoprocesses.EstimatingtheDisplacementFieldFromtheImagesToestimatethedisplacementeldofthesubstrate,wecompareddigitalimagesofthesameregionofthegel,takenatdifferenttimes.Imagesshowedcentmicrobeads(0.2m)embeddedinthegel,beforethecellswereplated.Inourexperiments,therewereusually1,0002,000beadsinanimage.Imageswereofthesize1,0241,280pixels.Eachbeadimageoccu-piedanareaof46pixelsindiameter,andneighboringbeadswereabout530pixelsapart.Theprocessingoftheseimagesbeganwiththecor-rectionofthepairofimagesforrelativetranslationalshifts.Hereweusedthecorrelationtheorem,whichsaysthattheFouriertransformofacorrelationoftwofunctionsisaproductoftheFouriertransformofonefunctionandthecomplexconjugateoftheFouriertransformoftheotherfunction.Weformedthenormal-izedtwo-dimensionalcross-correlationfunctionbe-tweenthetwoimages(normalizedbythesquarerootoftheproductofthemaximalvaluesoftheautocorrela-tionfunctionsofthese2images).Weidentiedthecoordinatesofthepeakofthecorrelationfunctionandtranslatedoneoftheimageswithrespecttotheotherbythatuniformdisplacement.Forthecalculationsofthecorrelationfunctions,weutilizedthetwo-dimen-sionalfastFouriertransform(FFT)algorithminMAT-LAB.Eventhoughtheimageswererepresentedbyrelativelylarge(1,0241,280)matrices,useofthecorrelationtheoremandFFTalgorithmmadetheac-tualcomputationsreasonablyfast.Havingthecorrectedimages,wedividedthemintoanumberofsmallwindowareas.Fortheseimages,wechoseawindowsizeoftypically6464pixels.Thewindowareasoverlapped;thedistancebetweenthecentersofsuccessivewindowswaschosentobe16pixels.Thedisplacementofeachwindowareawasthencalculatedbycorrelatingawindowinoneimagewiththewindowatthesamecoordinatesintheotherimage,inthewaydescribedaboveforthecorrelationbetweenthewholeimages.Thecoordinatesofthepeakofthecross-correlationfunctionbetweentwowindowswereassignedtothecenterofthewindowasthewindowdisplacementvector.Repeatingthesameprocedureoverallwindowsyieldedtheuniformdiscretizeddis-eldbetweentwoentireimages.(Thefactthatthistechniqueyieldsalatticewithuniformspac-ingmeansthatsimpleFFTalgorithmscanbeusedinthesubsequentanalysis.)Thewindowsizeof6464pixelswaschosentoguaranteethatatleastoneuorescentmarkerwaslocatedwithinthewindow,regardlessofthewindowposition.Toevaluatethepotentialsmoothingeffectofwidowsizeontherecoveredtractioneld,wealsoexperimentedwithsmallerwindowsdownto16pixels.Thisrequiredustomanuallyeliminatewin-dowsthatdidnotincludeuorescentmarkersandtosubstitutethemissingdisplacementswiththoseob-tainedfromrunsthatemployedlargerwindows.Inthecellthatwechoseasanexamplehere(seeFigs.3smallerwindowsizesdidnotappreciablyaltertherecoveredtractionImagesofdifferentgelsdifferedinbeadnumberanddistribution.Forimageswitharelativelylownumberofbeadsorlessuniformbeaddistribution,therewerewindowareasforwhichthecomputedcross-correlationwasbelowathresholdthatwesetat0.95.Forthedisplacementofsuchwindowareas,weusedvaluesobtainedbyttingtherestofthedisplacementeldbyathird-orderpolynomialandcalculatingthevaluesofthepolynomialinthepointsofthelatticeforwhichthevaluewasmissing.OurmethodofobtainingthedisplacementeldisdifferentfromtheDWapproach,whichreliesonthemeasurementofthecoordinatesofbeadsineachofthetwoimages.WeencounteredtwodifcultieswithFOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 thatmethod.First,thereistheissueofdeterminingbeadidentity;thedisplacementrequirestheinitialandnalpositionsofthesamebead.Thisisnotaproblemifbeaddensityislow,butinattemptingtoachievehigherresolutionwithhigherbeaddensities,somebeadidentitiescanbecomeambiguous;errorsherecanleadtospuriousdisplacementsand,hence,artifactualtractions.Second,theremaybeareasonsomeimageswherebeadsaresparseorevenabsent.Inthiscase,noestimatesarepossiblesavebyinterpolationfromneighboringregions.Ourmethodofmaximizingthecross-correlationofsmallwindowsbetweenthetwoimagesislesssensitivetotheseproblems.Intheplace,iftherearebeadswithunambiguousidentity,theircontrastdominatesthecross-correlationfunctionandtheestimateddisplacementsaresimilartothosecomputedbydirectpositionmeasurements.However,ambiguitiesinanyonebeadidentitycontributelesstothedisplacementeldtotheextentthatotherbeadsarepresentinthesamewindow.Thisisimportantinareaswherethebeaddensityishighandwheretheremaybeclustersofbeadswithambiguousidentities.Areaswithnobeadsalsocanshowreasonablecorrela-tionbetweenthetwoimages,dependingonthedis-placementsofotherfeaturesthatstillcarrysufcontrasttobemeasured.Suchfeaturesmayincludeheterogeneitiesinthegelandembeddedbeadsthatareoutoffocus.Insummary,theadvantagesofthecross-correlationapproacharethatitpermitssemiauto-matedestimatesofthedisplacementeldandisinsen-sitivetoambiguitiesregardingbeadidentibetweenimages.ComputingtheTractionFieldFromtheDisplacementFieldWehaveimplementedthesolutionforcomputingcelltractionsintwodistinctways.Therstmethod,un-constrainedFTTC,usesalldisplacementdatafromanimagepairobtainedasdescribedinEstimatingtheeldfromtheimages,doesnotuseanyconstraintsontherecoveredtractions,andisadirectapplicationofthemethodsdescribedin.Thesecondmethod,constrainedFTTC,isthesolutiontothemixedboundaryvalueproblem,whichignorestheeldoutsidetheboundaryofthecellandconstrainsthetractionsoutsidethecellboundarytobezero.Itisimportanttonotethatthismethodrequiresadditionalinformationbeyondthedisplacementnamely,anindependentestimateofthelocationofthecellboundary,drawnbyhand.Asdescribedbelow,thereareadvantagesanddisadvantagestobothmeth-ods;theyshouldbeviewedascomplementaryap-UnconstrainedFTTC.Hereweusethedirectsolu-tiongivenbyEqs.1.Thespecicprocedureisas)CalculatetheFouriertransformofthedis-eld(andsettheFouriercomponent0tozerotoeliminatetranslationartifact).)Multiplythetransformeddisplacementsbymapthetransformeddisplacementstotransformedtrac-)TaketheinverseFouriertransformoftheresulttoobtainthetractions.ConstrainedFTTC.Thisisamixedboundaryvalueproblem,withthedisplacementsunderthecellbeinged(bymeasurement)andwiththetractionsout-sidethecellboundaryspecied(byassumption)tobezero.TheFourierapproachabovealsocanbeusediterativelytosolvethisproblem.Itconsistsofthefollowingprocedures.)CalculatethetractioneldinthewaydescribedinUnconstrainedFTTC)Deneanewtractioneldbysettingthetractionsoutsideofthecellboundarytozero.)Calculatethedisplacementeldinducedbythistractioneld.ThisisdonebyusingtheFourierapproachinaforwarddirection:calculatetheFTofthetractioneld,multiplybytoobtainthetransformeddisplacements;theinverseFTisthenewdisplacement)Deneaneweldbyreplacingthedisplacementsofthecalculateddisplacementeldwithinthecellbound-arybytheexperimentallyobserveddisplacements.steps1untilconvergenceisreachedatsomeleveloftolerance.Thereareavarietyofcriteriathatcanbeused.Inourcasewechosetoterminatetheiterativeprocedurewhenthevariationinthemaxi-mummagnitudeofthetractionswithinthecellwaslessthan1partin10onsucceedingsteps.NotethatinbothconstrainedandunconstrainedFTTC,thereisanambiguityintheoff-diagonalele-mentsofattheNyquistfrequency,becausepositiveandnegativeNyquistfrequencycomponentsareindistinguishablebut.Thisproblemisavoidedbysettingtheoff-diagonalelementsoftobezerowheneitherisaNyquistfrequency.EXPERIMENTALMETHODSAtechniqueforpreparationofpolyacrylamidegelsheets(3)wasmodiedandusedtomakeexiblegeldisks.Amixtureofacrylamide(2%),bis-acrylamide(0.25%),anduorescentlatexbeads(diameter0.21:125dilutionbyvolume)wasaddedtoactivatedglasscoverslips.Thedropletofthesolutionwascoveredbyasmallcircularcoverslip.Afterpolymerization(45min),thecircularcoverslipwasremoved.TypeIcollagenwasattachedtothesurfaceofthegel.Geldisksweretypically50mthickandhadadiameterof12mm.Theelasticmodulus(Youngsmodulus)ofthegelwasdeterminedtobe1,200Pa;Poissonsratiowastakentobe0.48.HASMcellswereculturedinplasticdishesandse-rumdeprivedfor2daysbeforetheexperiments.Cellspassage3wereplatedonthegeldisksinaserum-freemediumandallowedtospreadandstabi-lizefor6h.Cellswerethenstimulatedwithhistamine(0.01mM)for5min.Photomicrographsweretakenofthecellsbothwithphase-contrastopticstovisualizethecellsandwith470-nmultravioletilluminationtoexcitethebeads,whichuoresceat515nm.Toassessthedistributionofbeads(andotherfeatureswithcon-trast)intheunstressedgel,thecellsweredetachedfromthesubstratewithtrypsin(2%);thisthereforeFOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 leavestheexiblegelwithnosurfacetractions.[Notethattheimageofthetraction-freegel(i.e.,)istakenaftertheimagesoftheposttreatmentdistributionofbeads.]Figure1showsaphase-contrastimageofarepre-sentativeHASMcell,culturedontheexiblepolyacryl-amidegelcoveredwithcollagentypeI,preparedasdescribedinEXPERIMENTALMETHODS,afterhistamineFigure2showsauorescenceimageofthesameofviewasinFig.1;the0.2-mbeadsembeddedinthegelareeasilyvisualized.Superposedonthisimageistheoutlineofthecellboundary,drawnbyhandfromFig.1.(ThisoutlineisdrawnsomewhatlargerthantheappearanceofthecellinFig.1.Thisistoensuretheinclusionofpotentialstressbearinginteractionsbe-tweenthecellandthesubstratethatmaynotbevisibleandtoavoidinteractionsbetweenthecellboundaryandthediscretizedlatticeonwhichthesolutionisned.See.)Thecorrespondingimageofthebeadsaftercelldetachmentwithtrypsin(pretreatmentcondition)looksvirtuallyindistinguish-ablefromthispicturebecausetheactualbeaddisplace-mentsareverysmall.Figure3showsthedisplacementeldcomputedfromthetwouorescentimagesofthebeadspre-andposttreatment,asdescribedinComputingthedisplace-eldfromtheimages.ThearrowsinFig.3showtherelativemagnitudeanddirectionofthedisplace-eldofthegelundertheadherentsmoothmusclecell.Figure3alsoiscolorcodedbytheabsolutemag-nitudeofthedisplacements.Figure4showsthetractioneldascomputedbyunconstrainedFTTC,thedirectcomputationoftrac-tionsfromtheFourierdecompositionofthedisplace-ments.Alsoshownistheboundaryofcell,althoughitisimportanttonotethatthisinformationwasnotused Fig.1.Aphase-contrastimageofahumanairwaysmoothmusclecell,culturedontheexiblepolyacrylamidegelcoveredwithcollagentypeI,5minaftertreatmentwith0.01mMhistamine.Bar,20 Fig.2.FluorescenceimageofthesameeldofviewasinFig.1,takenimmediatelyafterthelightmicroscopyimageinFig.1.Thembeadsembeddedinthegelareeasilyvisualized.Superposedonthisimageistheoutlineofthecell,drawnbyhandfromFig.1.Bar,20 Fig.3.Thedisplacementeldcomputedfromthe2imagesofthebeadspre-andposttreatment.Arrowsshowtherela-tivemagnitudeanddirectionofthedisplacementeldofthegelundertheadherentsmoothmusclecell.Colorsshowtheabsolutemagnitudeofthedisplacementsinm(seecolorbar).Calculationswereperformedonalatticewithspacingof2.67m(16pixels);thecolormapisshowntothisresolution.Forvisualclarityinthisillustrationandinthoseremaining,thedensityofarrowshasbeenthinnedtoaspacingofFOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 incomputingthetractions.ThearrowsinFig.4showtherelativemagnitudeanddirectionofthetractions,andthecolorsshowtheabsolutemagnitudeofthetractionvectors.Figure5showsthetractioneldcalculatedbycon-strainedFTTC,iteratingtheFourierapproachuntilconvergencewasreached.Notethatthetractionsarezerooutsidethecellboundarybyconstruction.AsinFig.4,thearrowsshowtherelativemagnitudeanddirection,andthecolorsshowtheabsolutemagnitudeofthetractionvectors.NoticerstthatthemapsinFigs.4and5aresimilarbutthatthereisatractionconcentrationneartheboundaryofthecellcomputedbyconstrainedFTTC(Fig.5).Thisresultsfromtherequirement,byconstruction,thatalltractionsexte-riortothecellmustbezero.Table1displaysthemomentmatricesthenetcontractilemoment,theorientationoftheprincipaltractions,andthestrainenergyexertedbythecellascomputedbyunconstrainedFTTC.NotethatthemomentsaregiveninunitsofpicoNewtonmeters(pNm)andenergyisgiveninunitsofpicoJoules(pJ).Weusetheseunitstodistinguishclearlybetweenmo-ments(forcesmultipliedbydistancesfromtheorigin)andenergy(forcesmultipliedbydisplacements),de-spitethefactthatpNmandpJareformallyequivalent.Table2displaysthesamequantitiesascomputedbyconstrainedFTTC,withthecellboundarydrawnbyhandasshowninFig.2.Allnon-zerotractionsareconstrainedtoliewithinthisboundary.Notethattherearetwomethodsforobtainingthemomentmatrices,givenbyEqs.4;theformerisobtainedbyintegrationoftherecoveredtractionsoverthe(real)space,whereasthelatterisobtainedbytheformalequivalentofthevectorderivativeofthetractionsinFourierspaceevaluatedattheoriginofspace.Thesetwoexpressionsinpracticecanbequitedifferent.Inprinciple,giventhatthemomentmatrixisnedbyanintegrationoverrealspace,itmightbethoughtthatusingEq.4directlywouldbepreferable, Fig.5.ThetractioneldcomputedfromthedisplacementeldinFig.3withtheuseofconstrainedFTTC,whichsolvesthemixedboundaryvalueproblemofprescribeddisplacementswithinthecellboundaryandzerotractionsexteriortothecell.ThisisaccomplishediterativelybyusingthesameFouriermethoduntilconvergenceisobtained.Arrowsshowtherelativemagnitudeanddirectionofthetractions.ColorsshowthemagnitudeofthetractionvectorsinPa(seecolorbar). Fig.4.ThetractioneldcomputedfromthedisplacementeldinFig.3withtheuseofunconstrainedFouriertransformtractioncytometry(FTTC),i.e.,thedirectcomputationoftractionsfromtheFourierdecompositionofthedisplacements.Alsoshownistheboundaryofthecell,althoughitisimportanttonotethatthisinformationwasnotusedincomputingthetractions.Arrowsshowtherelativemagnitudeanddirectionofthetractions.ColorsshowthemagnitudeofthetractionvectorsinPa(seecolorbar).Table1.MomentmatricescomputedbyunconstrainedFTTC2.830.923.150 Netcontractilemoment3.3pNmOrientationofprincipaltractions19Totalstrainenergy0.21pJMomentmatricescomputedbyunconstrainedFouriertransformtractioncytometry(FTTC)fromthetractioneldshowninFig.4.Noassumptionsaremaderegardingtheshapeofthecelloritsboundary.TheorientationoftheprincipaltractionsinthisandTables2and3isrelativetothe-axis.UnitsarepNm(pico-Newtonmeters)andpJ(picoJoules).Table2.MomentmatricescomputedbyconstrainedFTTC1.980.482.160 Netcontractilemoment2.9pNmOrientationofprincipaltractions21Totalstrainenergy0.12pJMomentmatricescomputedbyconstrainedFTTCfromthetractioneldshowninFig.5.Notethatinthismethod,allnon-zerotractionsareforcedtoliewithinthecellboundaryasdrawnbyhand.FOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 buttheexistenceofboundarytractionsassociatedwiththeFourierdecompositionintherstplaceimpliesapotentiallysubstantialerrorfromtheeldboundary,especiallyinunconstrainedFTTC.Bycontrast,theuseoftherstnon-zeroFouriercoefcient(Eq.5)asameasurementofthemomentmatrixhastheadvantageofmultiplyingtheentireeldbythelowestfrequencysinecomponent,thusexactlycancelingthemajordi-poleartifactarisingfromtheboundaryoftheeldofview.Thisisthemethodweuse.Asanaidtointerpretingtherelationshipbetweenthetractionmapandmoments,weshowinFig.6,,thedisplacementeldandtractioneldforanciallyconstructedexample.Thesimulationcon-sistsoftwopairsofpointtractionsourcesofdifferentmagnitudes,scaledtobesimilartothoseseeninrealcells.Notethatthetractionmaphasnon-zerotractionsonlyatthefoursourcepoints,whereasthedisplace-eldisnon-zerooverbroaderregions.(Becausethisisasimulationwithnoarticiallyaddednoise,therecoveryofthetractionmapwasidenticalbetweentheunconstrainedandconstrainedFTTCmethods.)Forthissimulation,themomentmatrices,orientationofprincipaltractions,andstrainenergyarelistedinTable3.Thisshowshowtheoff-diagonalelementsofarisewhenthelaboratorycoordinatesystemdoesnotcoincidewiththeprincipalaxesandhowwhensuitablyrotated(herebyfromthe-axis),themomentmatrixbecomesdiagonal().Thefactthatarebothnegativemeansthat,asexpectedfromFig.6,theprincipaltractionsarecontractile.Themag-nitudeofislargerthanthatof,correspondingtostrongercontractionalongthataxis.EffectofNoiseintheDisplacementDataWithrespecttoquestionsofresolutionandaccuracyinthepresenceofnoisydata,itisimportanttorecog-nizethatintheFTTCapproach,noisearisesonlyintheestimationofthedisplacementeldfromtheimagepairs;thecalculationofthetractioneldis,apartfromround-offerrorsinniteword-lengtharithmetic,anexactprocedure.Therearemanydifferentmethodsbywhichtoestimatethedisplacementeldfromimagepairs;theonewehavechosen(locatingpeaksinthecross-correlationfunctiononwindowedregionsofthetwoimages)isconvenientforourpurposes,butitisnotthefocusofthispaper.Wehave,however,character-izedtheeffectofdisplacementnoiseontherecoveredmomentmatrices,netcontractilemoment,andstrainenergy,asdescribedbelow.Weperformedsimulationswherethedisplacementeldconsistedofpurenoiseandexaminedthedeparturesoftherecoveredmomentsandstrainenergyfromzero.Weperformed100simulationsusingthesamegridandgelcharacteristicsasintherealimagesinFigs.2and3.ThedisplacementnoisewasGaussianwithameanofzeroandastandarddeviationof1m.Thetractionwasrecoveredbyusingboththeunconstrainedandcon-strainedFouriermethods,whereintheconstrainedcaseweusedthesamecelloutlineasinFig.3.Theresultsofthesesimulationsonthemomentmatrices,thenetcon-tractilemoment,andthestrainenergyareasfollows. Fig.6.Simulationofthedisplacementeld()associatedwith2pairsofpointtractionsources().Thisarticialexamplecorre-spondstoFigs.3and4fortherealcell.Thetractioneld()wasrecoveredwiththeuseofunconstrainedFTTC.ThetractionrecoveredwiththeuseofconstrainedFTTC(notshown),withaboundaryofanimaginarycellshownbythewhiteline,wasindis-tinguishablefromthetractioneldinTable3.MomentmatricescomputedbyunconstrainedFTTCfromarticialdata1.030.511.350 Netcontractilemoment1.6pNmOrientationofprincipaltractions32Totalstrainenergy0.11pJMomentmatricescomputedbyunconstrainedFTTCfromthetractioneldshowninFig.6cialexample).Con-strainedFTTCyieldedthesameresults.FOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 Theelementsofthemomentmatriceswerenotsignicantlydifferentfromzero.Thiswastobeexpected,be-causethetractionsarelinearfunctionsofthedisplace-mentsandtheexpectationvaluesofthenoiseinthedisplacementsarezero.Becausethisistrueofallele-mentsofthemomentmatrices,theeffectofthenoiseonthenetcontractilemomentalsoisnotsignicantlydif-ferentfromzero.Inanygivensinglerealization,however,itisimportanttoknowtheexpectedmagnitudeofthedeparturefromzero.Thestandarddeviationofthenetcontractilemomentarisingfrompurenoise,fromthesesimulations,is0.29and0.21pNmpermicrometerstan-darddeviationofdisplacementsintheunconstrainedandconstrainedcase,respectively.Theeffectofnoiseonstrainenergyisquitedifferent;unlikethetractions(andthereforethemoments),whicharelinearfunctionsofdisplacement,thestrainenergyisquadratic.Thisim-pliesthattheexpectationvalueofthestrainenergyduetopurenoiseisnon-zero.Inoursimulations,wefoundthattheenergyassociatedwithdisplacementnoiseis11.1and0.88pJpersquaremicrometerofdisplacementvarianceintheunconstrainedandconstrainedcase,re-spectively.Thissubstantialdifferenceinstrainenergyintheconstrainedandunconstrainedcases(roughlyafac-torof12)ispreciselywhatwasexpected,becausetheeldareaisroughly12timestheareaboundedbythecell(Fig.3),andthereisnostrainenergyinthegelconferredbysurfacetractionsexteriortothecellintheconstrainedcase.AdvantagesandDisadvantagesinUnconstrainedandConstrainedFTTCThereareanumberofadvantages()anddisadvan-tages()associatedwithunconstrainedandcon-strainedFTTC.UnconstrainedFTTC.)Alloftheobserveddataareused,includingthedisplacementsofbeadsexteriortotheperceivedcellboundary.Thefalloffindisplace-mentsexteriortothecellconstitutesadditionalinfor-mationregardingtheoveralltractioneld,especiallywhenthedistributionofbeadsissparsewithinthecellboundary.Thismayseemtobeanobviousadvantage;itisinlargemeasuretrue.However,thefalloffindisplacementsfromanyparticulartractionsourcepointislike1/,sotheinformationofdis-placementsexteriortothecelliscorrespondinglylow,andimportantinformationmayinfactnotbelost.)Thecellboundaryneednotbeidentied,and,assuch,noinvestigatorjudgmentisnecessarytoidentifythisboundary.Thisisanadvantageoverallcon-strainedmethodsinsofarasforcegenerationassoci-atedwithsmallatlamellipodia,orconnectivetissueelementsmaybemissedintheorig-inalmicrographimages.)AswithalldiscreteFourierproblemsimple-mentedoveranitespace,theinherentperiodicityintroducesartifactualtractionsattheboundaryoftheeldbecausethemeasureddisplacementsarenotstrictlyperiodic.Totheextentthattheseareremotefromthecell,theyposenodifculty.Moreover,suchanartifactisequivalenttoadipoleeldontheboundary,whichdoesnotstronglyinuencethecomputedcelltractions.Theseboundaryartifactsareeasytorecog-nizeandcanbesafelyignored.)Errorsintherecoveredtractionsexteriortotherealcellboundary,secondarytonoiseinthedisplace-eld,willhavezeromeanifthenoisehaszeromean.Thisfollowsfromthelinearityofrelationshipbetweentractionsanddisplacements.)Bycontrast,thestrainenergy(whichisqua-draticinthedisplacements)willbeartifactuallyhighduetothecontributionofnoiseinthetractionexteriortothecell.ConstrainedFTTC.)Ifthecellisexertingnon-zerotractionsonthesubstrateatlocationsexteriortotheperceivedcellboundary,theinteriortractionswillnecessarilybeinerror,becausetheymustcompensateforthesetractionsthatwereincorrectlyconstrainedtobezero.Thisresultsinartifactuallylargetractions,especiallyinthevicinityoftheimposedcellboundary.Thedangerhereisthatthelargetractionsattheperceivedcellboundarymaybeinterpretedmistakenlyasreectingrealtractionconcentrations.)ConstrainedFTTCrequiresaniterativeap-proach,andsocomputationalefciencyisnotguaran-teed.Inourexperience,however,wehavefoundthattheiterativeschemedescribedabovetypicallycon-vergesquitequickly(typically10iterations)sothatthisapproachisalsonotcomputationallyintensive.)Thenatureofthemixedboundaryconditionimpliesthatanassessmentofthenoiseontheeldisdifcult.Thisisbecausetheinducedlevelofnoisedependsontheboundaryofthecell.)ThestrainenergyinconstrainedFTTCisconbyconstruction,totheinterioroftheperceivedcellboundary,sothereisnocontaminationofthenetstrainenergyfromnoiseintheexteriordisplacementdata.ThereareseveraladvantagestoFTTCcommontoboththeconstrainedandunconstrainedimplementa-tions.Theseincludethefollowing:()ThemomentmatrixisanespeciallysimpleformulationinFourierspace.Noadditionalcalculationsareneeded.)TheuseoftheFFTimpliesthattheentireprob-lemoftractionmeasurementisnolongercomputation-allyintensive;largeimagepairscanbeanalyzedinsecondsorminutes.EffectofNoiseonTractionRecoveryintheExampleSmoothMuscleCellHerewedescribetheeffectofnoiseonouractualcomputationsofmomentsandstrainenergyassoci-atedwiththesmoothmusclecellshowninFigs.1Acomparisonoftherootmeansquaretractionsex-teriortothecellboundarywiththatassociatedwithpurenoise,asdescribedin,givesaroughestimateofthenoiseinthedisplacementeldandisaconservativeestimateinsofarasthereappeartobepatchesofnon-zerocorrelatedtractionsinFig.4,possiblysecondarytoothercellsexteriortotheFOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002 ofview.Inourcase,thisresultsinanestimateofmrootmeansquarenoiselevelinthedis-eld.Fromthepurenoisesimulationsquotedin,thisamountstoanuncertaintyinthenetcontractilemomentsof0.015and0.010pNmintheunconstrainedandconstrainedcases,respec-tively.ThesenumbersshouldbecomparedwiththoseinTables1and2,wherethenetcontractilemomentofthecellis3pNminmagnitudeandshowsthatnoiseisnegligibleinitscontributiontotheseestimates.Bycontrast,asremarkedabove,thestrainenergyisquadraticinthenoiselevel,sodisplacementnoiseof0.05mcorrespondstoroughly0.025and0.0025pJintheunconstrainedandconstrainedcases,respectively.Thedifferenceintheestimatedstrainenergiesfortherealcellintheunconstrainedandconstrainedmethods,fromTable1andTable2,isroughly0.1pJ,andweconcludethatnoisemayaccountforsomequarterofthisdifference.InspectionofFig.4,however,showsthatthisisalsotobeexpected,becausethepatchesofcorrelatedtractionswillcertainlycontributetothesedifferingestimates.RemarksontheDWMethodTheDWmethodspeciesbothtractions(exactly)anddisplacements(approximately)exteriortotheperceivedcellboundary.ThisparticularspecioftheproblemrequiresspecialtechniquesbecausetheseareapproximateCauchyconditions,forwhichtheellipticNavierequationsofelasticityingeneralhavenosolution.Thisisanill-conditionedproblemthatnecessitatessmoothingorregularizationtoob-tainstablesolutions.TheDWmethodutilizestheregularizationmethodintroducedbyTikhonovinthe1940s(summarizedinRefs.8and9)withthechoiceofsmoothingfunctionalandlevelofsmoothingintro-ducedbyPhillips(6)andTwomey(10).Inbrief,theresidualsofthedisplacementeldplusacertainamountoftheLnormofthegradientofthetractioneld(theregularizingfunctional)areminimized,andthedisplacementresidualsareexamined.Thelevelofsmoothingisthenvarieduntilthereisanappropriatelevelofvariationinthepredicteddis-placements,givenaprioriinformationaboutthenoiseinthedisplacementeld(6).Forproblemsthatarehighlyillconditioned,andwhenthereareunam-biguousaprioriconstraints,thiskindofapproachisoftenusefulandappropriate(2).Bycontrast,thedisplacementkernelintheBoussinesqsolutionde-cayslike,whichissufcientlyrapidthatinFTTCwedonotndunacceptablylargeoscillationsintherecoveredtractioneldthatwouldnecessitateaTi-khonov-typeapproach.RecommendationsandConclusionOnthebasisoftheresultspresented,thebeststrategyformeasuringtractioneldusingFTTCmaybesummarizedasfollows.Aninitialexamina-tionofthetractioneldrecoveredwiththeuseofunconstrainedFTTCwillrevealtheextentofsignif-icanttractionsexteriortotheperceivedcellbound-ary.Ifthesecanbedeterminedtoresultfromcon-tractilecellsexteriortotheeldofview,thetractionmapsobtainedwithconstrainedFTTCmaybemoreaccurate.Ontheotherhand,suchtractionsmightarisefromrealstructuralelementsthatarenotseeninphase-contrastmicroscopyandthatarepreservedinunconstrainedFTTC.Whichevermethodisused,thetractionmomentsareeasilycomputedinFourierspace,whereasthestrainenergyisbestcomputedwithconstrainedFTTC,integratingoverrealspace.Ifdesired,thenoiselevelinthedisplacementmaybeestimatedbyexaminationofitsFourierspec-trumathighfrequencies,wherethewhitenoiseismanifestasaconstantlevelofintensity.Inconclusion,Fouriertransformtractioncytometryisanewsolutiontotheproblemofmappingoftheeldbetweenacellanditssubstrate,giventheeldbetweentwomicrographimages.Thismethodhastheadvantagesofbeingexact,com-putationallyefcient,andnotsubjecttocertainarti-factsthatcanleadtomisleadingconclusions.Thisapproachalsoyieldssimplemeasuresofthenetcon-tractilemomentofthecell,thestrainenergyimpartedtothesubstratum,theorientationoftheprincipaltractions,andaquantitativeindexofcellpolarity.FTTCmayrepresentanewandimportanttoolforstudyingthemechanicalpropertiesandfunctionofadherentcells.WethankS.M.Mijailovichforperformingniteelementsimula-tionsoftestcasesearlyinthecourseofthiswork.WeespeciallythankE.J.MilletandN.Wangforhelpfuldiscussions.ThisworkwasstimulatedinlargemeasurebycollaborationwithN.Wangonhisstudiesofcellmechanics.WethankJ.Chenfortechnicalassis-tanceincellandgelpreparations.CellswerekindlysuppliedbyR.ThisworkwassupportedbyNationalHeart,Lung,andBloodInstituteGrantHL-P01BalabanNQ,SchwarzUS,RivelineD,GoichbergP,TzurG,SabanayI,MahaluD,SafranS,BershadskyA,AddadiL,andGeigerB.Forceandfocaladhesionassembly:acloserelationshipstudiedusingelasticmicropatternedsubstrates.NatCellBiol3:466472,2001.ButlerJP,ReedsJA,andDawsonSV.Estimatingsolutionsrstkindintegralequationswithnonnegativeconstraintsandoptimalsmoothing.SIAMJNumerAnal18:381397,1981.DemboMandWangYL.Stressesatthecell-to-substrateinterfaceduringlocomotionofBiophysJ76:23072316,1999.LandauLDandLifshitzEM.TheoryofElasticity(3rded.).Oxford,UK:Pergamon,1986.PelhamRJandWangYL.Highresolutiondetectionofme-chanicalforcesexertedbylocomotingbroblastsonthesub-MolBiolCell10:935945,1999.PhillipsDL.Atechniqueforthenumericalsolutionofcertainintegralequationsoftherstkind.JAssocComputMach9:8497,1962.D,MijailovichSM,TolicrrelykkeIM,ChenJ,andWangN.Cellprestress.II.Contributionofmicrotubules.AmJPhysiolCellPhysiol282:C617C624,2002.TikhonovAN.Solutionofincorrectlyformulatedproblemsandtheregularizationmeth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TikhonovAN.Regularizationofincorrectlyposedproblems.SovietMathDokl4:16241627,1963.TwomeyS.OnthenumericalsolutionofFredholmintegralequa-tionsoftherstkindbytheinversionofthelinearsystemproducedbyquadrature.JAssocComputMach10:97101,1963.WangN,TolicrrelykkeIM,ChenJ,MijailovichSM,ButlerJP,FredbergJJ,andStamenovicCellpre-stress.I.Rigidityandprestressarecloselyassociatedincontractileadherentcells.AmJPhysiolCellPhysiolC616,2002. FOURIERTRANSFORMTRACTIONCYTOMETRYAJP-CellPhysiolVOL282MARCH2002