Modeling the effects of focal adhesion placement during cell spreading

22 Pages·2017·2.5 MB·English

RESEARCHARTICLE
Center or periphery? Modeling the effects of
focal adhesion placement during cell
spreading
MagdalenaA.Stolarska1*,AravindR.Rammohan2*
1DepartmentofMathematics,UniversityofSt.Thomas,St.Paul,MN,UnitedStatesofAmerica,2Advanced
ModelingandAnalysis,CorningIncorporated,Corning,NY,UnitedStatesofAmerica
*[email protected](MAS);[email protected](ARR)
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Abstract
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Focaladhesionsareoftenobservedatthecell’speriphery.Weprovideanexplanationfor
thisobservationusingasystem-levelmathematicalmodelofacellinteractingwithatwo-
dimensionalsubstrate.Themodeldescribesthebiologicalcellasahypoelasticcontinuum
materialwhosebehavioriscoupledtoadeformable,linearelasticsubstrateviafocaladhe-
OPENACCESS
sionsthatarerepresentedbycollectionsoflinearelasticattachmentsbetweenthecelland
Citation:StolarskaMA,RammohanAR(2017)
thesubstrate.Theevolutionofthefocaladhesionsiscoupledtolocalintracellularstresses
Centerorperiphery?Modelingtheeffectsoffocal
whicharisefrommechanicalcell-substrateinteractions.Usingthismodelweshowthatthe
adhesionplacementduringcellspreading.PLoS
ONE12(2):e0171430.doi:10.1371/journal. cellhasatleastthreemechanismsthroughwhichitcancontrolitsintracellularstresses:
pone.0171430 focaladhesionposition,size,andattachmentstrength.Wealsoproposethatonereason
Editor:JungWeonLee,SeoulNationalUniversity whyfocaladhesionsaretypicallylocatedonthecellperipheryinsteadofitscenteris
CollegeofPharmacy,RepublicofKorea becauseperipheralfocaladhesionsallowthecelltobemoresensitivetochangesinthe
Received:August18,2016 microenvironment.Thisincreasedsensitivityiscausedbythefactthatperipherallylocated
focaladhesionsallowthecellstomodulateitsintracellularpropertiesoveramuchlarger
Accepted:January20,2017
portionofthecellarea.
Published:February3,2017
Copyright:©2017Stolarska,Rammohan.Thisis
anopenaccessarticledistributedundertheterms
oftheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,and
reproductioninanymedium,providedtheoriginal
Introduction
authorandsourcearecredited.
Cellbasedassaysareincreasinglybecominganimportantpartofdrugdevelopmentwherebio-
DataAvailabilityStatement:Allrawdatafilesfor
numericalsimulationsareavailablefromthe logicalcellsareplacedineitherfunctionalizedpetridishesormicroplatesofdifferentformats,
Zenododatabaseathttps://zenodo.org/record/ forexample96wellplates[1,2].Thekeytothesuccessofthesecellbasedassaysisthatthe
60425(DOI10.5281/zenodo.60425). functionalizedsurfacesallowthecellstobehaveassimilarlyaspossibletotheirnativeinvivo
Funding:Theauthorsreceivedsmallamountsof environments.Cellswhichbehavemostnaturallycanthenbeusedtoassesstheperformanceof
fundingfromCorning,Inc.andtheCenterfor candidatedrugmoleculesintheirabilitytoactivateordeactivatecertainbiologicalpathways.
AppliedMathematicsattheUniversityof Effectivedesignofthesefunctionalizedsurfacesrequiresafundamentalunderstandingof
St.Thomasforthiswork.Fundingwasnotpartofa
theinteractionbetweenacellandthesurface.Adherentcellsengagewiththeunderlyingsub-
grantreceivedbytheauthors.Rather,itwasused
strates(theextracelluarmatrix—ECM—invivo,orafunctionalizedsurfacewithpeptidecoat-
toprovidetravelfundssothattheauthorcould
attendconferencesandspendtimecollaborating ingsinvitro)throughtheformationoffocaladhesions(FAs).FAformationinvolvessensing
onthiswork.Thefundershadnoroleinstudy thechemicalandmechanicalmicroenvironmentonthesubstratebycellsurfacereceptors,
PLOSONE|DOI:10.1371/journal.pone.0171430 February3,2017 1/22
Centerorperiphery?Modelingfocaladhesionplacement
design,datacollectionandanalysis,decisionto whichformanycelltypesareintegrins,thatbindtospecificchemicalmotifsintheECM/func-
publish,orpreparationofthemanuscript. tionalizedsurface.Oncethisbindingoccurs,acomplexorchestrationofmanydifferentmole-
CompetingInterests:AravindRammohan’s culesoccursintracellularly,whichresultsintheaccumulationofoverseventyproteinsatthis
employmentatCorning,Inc.doesnotalterour cell-surfacejunction[3,4].Theseproteincomplexesarereferredtoasfocaladhesions,which
adherencetoPLOSONEpoliciesonsharingdata arebothchemicallyandmechanicallysensitive.Themechanicalsensitivityisreflectedin
andmaterials.
detailedexperimentalstudieswhichshowthatthesizeofthefocaladhesioncomplexesgrow
withthestiffnessoftheunderlyingsubstrates[5].Thechemicalsensitivityhasalsobeenwell
establishedwithdifferentchemicalmotifspatternedonthesurfaces[6].
InthelastdecadeFAshavebeencharacterizedextensivelyfromanexperimentalperspec-
tive[7,8].OfthedifferentaspectsofFAs,suchastheirgrowth,dynamics,andmechanosensi-
tivity,onewhichisoftenobservedbutisnotwellunderstoodisthelocationofFAsonthe
contactregionbetweenthecellandsubstrate.Whencellsareimagedonflatsurfacesitis
almostalwaysobservedthattheFAsarelocalizedatthecellperipheryandrarelyseeninthe
cellcenter[9–12].RecentexperimentalworkshavebeguntoelucidatetheeffectsofFAposi-
tioningoncellbehavior.Elinenietal.[10]haveshownthattheattachmentstrengthofcells
increaseswhenFAsarefoundintheperipheryversusinthecenter.Otherstudiesshowthat
increasesincellspreadarea,FAsize,andFAlocationcollectivelyleadtoanincreaseincell
tractionstresses[13].TheseresultsraisethequestionabouthowFApositionaffectsintracellu-
larmechanicsandthecellularresponsetoitsenvironment.
ExistingmodelslookatcertainspecificaspectsofFAs.Forexample,one-dimensionalmod-
elsofonlytheFAregionaimtounderstandhowFAsizesandpositionsrespondtoforces[14,
15].ModelsbyOlberdingetal.[16]andWalcottetal.[17]useathermodynamicapproachto
capturetheactiveandinactivestateoftheintegrinandlinkintegrinactivationtothegrowth
ofFAsbutdosousingaverysimplifiedrepresentationofthecell.Onedimensionalmodelsof
singlecellmotilityhaveelucidatedthebell-curveshapedrelationshipbetweenadhesive
strengthandcellvelocity[18].Othershavedevelopedtwodimensionalmodelsofcellsand
haveincorporatedFAsintothemodels.However,theaimofthesemodelsistobetterunder-
standtheevolutionofstressfibers,contractileactinbundles[19,20],thecomplexinterplayof
variouscellularcomponentsrequiredforcellmotility[21,22],ortheeffectsofFAmaturityon
spontaneouscellmigration[23].However,noneoftheseworksfocusontheeffectsofgrowth
andlocationofFAsoncellmovements.
Giventheseexistingmodels,inthispaperwesetouttodescribeatwo-dimensionalrepre-
sentationofaspreadingcellasamechanicalbodywiththedetailedbiochemistryassociated
withtheFA’schemicalandmechanicalresponsecapturedbyamathematicalmodelthathas
beensimplifiedsothatallkeyphysicalaspectsofcell-substrateattachmentsarecaptured.The
effectsoftheactinpolymerizationandactomyosincontraction,whichserveastheprimary
motilemachineryofthecellandarecontrolledbyahostofaccessoryproteins[24],arealso
modeledinanapproximatewaybyspecifyingthecell’srateofactivedeformation.Thesesim-
plifieddescriptionsofFAevolutionandcytoskeletaldeformation,inconjunctionwiththetwo
dimensionalmechanicaldescriptionofthecell,allowustodevelopadeterministicsystem
levelmodelofthecell.Furtherinthismodeldescriptionwearealsoabletoexplicitlyaccount
forthedeformationofthesubstrateandmodelitsmechanicalresponsetotheforcesexerted
onitbythecell.Thisallowsustocomparethemodelpredictedsubstratedeformationswith
experimentallydeterminedtractionforcemicroscopymeasurements,whichallowsustocali-
brateandvalidateoursimulations.Therefore,ouraiminthispaperistwo-fold.First,wewish
toillustratethatourmodelofthecell-substratesystemcanbeusedforavarietyofcelltypes
spreadingoverdifferentenvironments.Second,weusethisdescriptiontoinvestigatetheeffect
ofhavingFAsatthecellperipheryversusitscenter.SpecificallyweinvestigatetheeffectofFA
size,location,strengthofattachment,andsubstratestiffnessoncellularandsubstrateresponse,
PLOSONE|DOI:10.1371/journal.pone.0171430 February3,2017 2/22
Centerorperiphery?Modelingfocaladhesionplacement
whichisasteptowardsunderstandingwhyFAsaremoreoftenfoundonacell’speriphery
ratherthanitscenter.
Methods
Themodelofthecell-substratesystemconsistsoffourcomponents,asillustratedinFig1.To
modeltheaveragedresponseofthemixturecontainingcytoskeletalfilaments,organelles,cyto-
sol,andotherintracellularmacromoleculeswedescribethecellasahypoelasticmaterial[25]
(component(a)inFig1).Thereorganizationoftheactincytoskeletontodrivecellmovement
andgenerateintracellularstressesiscapturedinthemodelbyasinglerateofdeformationten-
sorfieldwhichdescribestheactivelocaldeformationsofthecell(component(b)).Thecellular
FAsandtheforcesthereinareincorporatedintothemodelasacollectionofdiscretelinear
springswhichattachthehypoelasticcelltothesubstrate(component(c)).Weassumethe
mechanicalresponseofthesubstratetobelinearlyelasticsinceitsdeformationsaregenerally
small(component(d)).ThemodeldescribingtheevolutionofFAsandallitsaccessorypro-
teinsiswrittensothatthepositionofFAsisdescribedbyasingle,stress-dependentscalar.Our
Fig1.Cell-substrateinteractionmodelisillustratedontheright.Thebiologicaldetailsofthe
mechanismsinvolvedincellspreadingarereducedtofourmajormodelcomponents,labeled(a)-(d)inthe
rightpaneltitled“Reducedmathematicalmodel”.Boththecellandthesubstratearetwodimensional.The
rheologyofthecellisdeterminedbylocalconcentrationsoffilaments,organelles,andadditional
macromolecules,anduponapplyinganexternalforceviamicroindentation,thecytoplasmicforcesundergo
relaxation[26].Weassumetheaveragematerialpropertiesofthecelltobehypoelastic(a),andweassume
thatthedeformablesubstrateislinearlyelastic(d).Actin-basedcellmovement[24,27]isdescribedbyan
activerateofdeformationtensor(b).TheinteractionbetweencellandsubstrateoccursviatheFAs,whichare
modeledascollectionsofdiscretelinearsprings(c).AdescriptionofthedynamicFAcomplexissimplifiedand
writtenasstressdependentevolutionofasinglescalarfieldrepresentingthevolumefractionoftheentireFA
complex[4].
doi:10.1371/journal.pone.0171430.g001
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Centerorperiphery?Modelingfocaladhesionplacement
aiminthisworkistoincludekeyqualitativeandquantitativeelementsofcellmovement,cell
adhesion,substrateresponse,andtheinteractionbetweentheseprocessesinarelativelysimple
manner,butonethatreflectstheessentialphysicalinteractionsrequiredtosimulateexperi-
mentallyobservablebehavior.
Cellmechanicalmodel
Wetreatthecellasatwo-dimensional,deformablecontinua.Thematerialresponseofthecell
isassumedtobegovernedbyahypoelasticmodel,whichisequivalenttolinearelasticityfor
smallstrains,whileforlargestrainsaccountsforflow-likeinelasticeffectsoflargedeformation.
Hypoelasticconstitutiveequationshavebeenshowntobethermodynamicallyconsistent[28]
andvalidforbiologicalmaterials[29,30]duetothefactthatbiologicalsystemsinvolvegrowth,
materialremodeling,andlargedeformations[31].Thefiniteelementanalysisofthesefeatures,
pursuedinthispaper,benefitsfromoccasionalremeshing(describedlater),whichrequires
associatedtransferofthefieldsinvolvedbetweenthecomputationalmeshes.Thisremeshing
processalsobecomessimplerinthecontextofhypoelasticmaterials.
Theequilibriumandconstitutiveequationsforthecellare,respectively,givenby
r(cid:1)σ (cid:0) k ðu (cid:0) u (cid:0) uÞdðx(cid:0) xÞ ¼ 0 ð1Þ
c a c a s c
(cid:18) (cid:19)
1
σr ¼C ðrv þrTv Þ(cid:0) DA ; ð2Þ
c c 2 c c
where
dσ
σr ¼ c(cid:0) ðrv Þσ (cid:0) σ ðrTv Þ: ð3Þ
c dt c c c c
TheCauchystresstensorwithinthecellisσ,andσr istheOldroydtimederivativetorender
c c
theconstitutiveequationframe-invariant.Therefore,Eq(2)combinedwithEq(3)describe
thematerialresponseofthecell,inwhichtheactiverateofdeformationisdenotedbytheten-
sorfieldDA.Theactiverateofdeformationisdiscussedinthenextsection.Thesecondterm
ontheleftofEq(1)representstheforcefromtheFAspringactingonthecell.Thedisplace-
mentsofthecellandthesubstrate,respectively,areu andu,andthevelocityofthecellisrep-
c s
resentedbyv ¼duc.ThecelldisplacementatthetimeaFAspringbindstothesubstrateis
c dt
representedbyu .Sincethedisplacementsofthesubstratearesmallrelativetothoseofthe
a
cell,wedonotexplicitlyaccountforitsdisplacementatthemomentofattachmentinthe
modelequations.TheFAspringstiffnessisk ,whichwesetto5×10−5dynes/μm[32].The
a
locationatwhichtheFAspringisconnectedtothecellisgivenbyx,andδ((cid:1))istheDirac
c
deltafunction.Thecellboundaryistractionfree.
TheconstantHooketensorrepresentingthematerialpropertiesofthecellisgivenbyC.
c
Sincethecellthicknessinthedirectionnormaltothesubstrateissmallcomparedtoitsradius,
soweassumethatthecellisinastateofplanestress,whichforisotropicmaterialsassumed
here,leadstotheHooketensorthathasthefollowingmatrixrepresentationinanyCartesian
coordinatesystem
2 3
1 n 0
c
E 6 7
C ¼ c 4n 1 0 5 ð4Þ
c 1(cid:0) n2 c
c
0 0 1(cid:0) n
c
whereE andν aretheYoung’smodulusandPoissonratioofthecell,respectively.
c c
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Centerorperiphery?Modelingfocaladhesionplacement
Activedeformation
InEq(2),DAistheactiverateofdeformationtensor,whichcharacterizesacell’slocalactive
rateofdeformationduetospreadingandcontractionandneedstobespecified.Weassume
thatthethetotalrateofdeformationtensor,D,canbeadditivelydecomposedintoastress-
relatedpassivepart,DP,andanactivepart,DA,suchthat
1
D¼ ðrv þrTv Þ¼DPþDA ð5Þ
2 c c
DAcaningeneraldependonthevariablesinthemodel,suchaslocalstressortheconcentra-
tionofanintracellularbiochemicalcomponent.Suchanadditivedecompositioniscoupledto
theassumptionthattheactivedeformationcomponentDAdescribesonlythelocaluncon-
strainedrateofactiveremodelingwhichisstressfree,andhypoelasticstressratesinthecell
arerelatedonlytothepassivecomponent,DP.InEq(2),DPiswrittenasD−DA.Decomposi-
tionssimilartothiswereoriginallyusedinmodelingofsmalldeformationsduetothermal
expansioninmaterialsandhavebeenusedinvariousmodelsinvolvinggrowthofbiological
tissues(seeforexample[33]).
Biologically,acell’smotilemachineryisdependentonthedynamicstructureoftheactin
network.Thefilamentousactinatthecellperipheryconsistsofabranchedactinnetworkthat
pushesoutthecell’smembraneviapolymerization,whiletheactinnetworkoftheinteriorof
spreadcellsconsistsofbundledcontractilestressfibersthatprovidetensionandstructureto
thecell[34,35].Incorporatingthesevariationsinactinstructurewouldresultinaspatially
andtemporallyinhomogeneousactiverateofdeformation.Asafirststep,inthispaperwe
assumethatthecellactivelyspreadsorcontractsataconstant,spatiallyuniform,andisotropic
rate,andthuswetakeDAtobe
" #
a 0
DA ¼ ð6Þ
0 a
Wesetthevalueα=0.00725min−1forspreading.ThisvalueisbasedonWakatsukietal.[36]
andischosensothatthediameterofacircularcellapproximatelydoublesoverthecourseof
twohours.Weestimatethecontractionratetobeα=−0.001min−1inordertoobtainexperi-
mentallyobservedcellshapes.Weassumethatthecellularmaterialthatisrequiredtoallow
thecelltospreadcomesfromthecellularregionswhichareoutsideofthetwo-dimensional
planeweconsiderinoursimulations.
Deformablesubstratemechanics
Thedeformationofthesubstrateisgovernedby
(cid:18) (cid:19)
C
r(cid:1) sðru þrTuÞ þk ðu (cid:0) u (cid:0) uÞdðx(cid:0) xÞ ¼ 0 ð7Þ
2 s s a c a s s
Here,C istheHooketensorforthesubstrate,andwithsuitablechoiceofvaluesforthe
s
Young’smodulusandPoissonratio,ithasthesameformasinEq(4).ThelocationoftheFA
springonthesubstrateisgivenbyx.
s
ThesystemEqs(1),(2)and(7)iscoupledthroughtheFAspringsthatexertforcesonboth
thecellandthesubstrate,asseeninEqs(1)and(7).Wesetthedisplacementattheedgesof
thesubstratetozero.
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Centerorperiphery?Modelingfocaladhesionplacement
Spatiotemporalevolutionoffocaladhesions
Themodelforcellularandsubstratedeformationdescribedabovecanbecoupledtoamodel
ofafullsignaltransductionnetworkdescribingmanyindividualintracellularbiochemicalpro-
cesses.Here,weemployasimplifiedapproachinwhichwemodeltheentireFAcomplex,con-
sistingofthecell-substratebindingproteins,suchasintegrins,andotheraccessoryproteins,as
asinglescalarfield,denotedbyϕ(x,t).Thisscalarvalueisdefinedateverypointonthecellular
computationaldomain,anditslocalvaluerepresentsthevolumefractionoftheFAcomplex.
ThemodelfortheevolutionoftheFAsemployedhereisbasedonearlierworkbyBesser
andSafran[15].WenotethatwechoosethismodelasabasisforourFAevolutioninpart
becauseofitssimplicity.Inourfutureworkweplanonincludingamoredetaileddescription
oftheFAcomponents.Intheirone-dimensionalmodel,BesserandSafranrepresenttheentire
FAcomplexbyasinglescalar-valuedfunctionofspaceandtime,andtheyassumethat
mechanosensorscontroltheevolutionofFAvolumefractionbychangingthecomplex’s
chemicalpotential.Intheirmodel,thechemicalpotentialdependsonchangesinconforma-
tionalenergyduetoadditionofcomplexunits,changesstrainenergyintheFAcomplexdue
tocompressionandtensionwithin,oronenergychangeduetomoleculardeformation.They
buildtheirmodelassumingthatcompressivestressesinthecell’scytoskeletonleadtoFAacti-
vationwhiletensilestressdeactivatestheFA.Accordingly,thefunctionϕisconstructedso
thatcompressivestressesincreaseϕ,andlargervaluesofϕimplyFAactivation.Besserand
Safrandescribetheevolutionofϕusing
(cid:20) (cid:18) (cid:18) (cid:19) (cid:19) (cid:21)
@(cid:30) (cid:15) b b @(cid:30) @2(cid:30)
¼C m (cid:0) b tanh DG(cid:0) f (cid:0) 1 (cid:0) gðf Þ þ(cid:15)(cid:30)(cid:0) c(cid:30)3þB ð8Þ
@t 1 b 2 2 2 a a @x @x2
ThechemicalpotentialofFAcomplexesinbulkisgivenbyμ .Thesecondtermontheright
b
handsidedescribestheeffectsoftheforceappliedtotheFAbythecytoskeleton,denotedf ,in
a
thechemicalpotential.Theremainingtermsontherighthandsidedescribehowthechemical
potentialisaffectedbyinteractionbetweentheFAcomplexesandbytheadditionofnewFA
complexestotheexistingFAs.TheenergeticcostforaconformationalchangeintheFAis
givenbyΔG,andthetermsC ,(cid:15) ,β,γ(f ),(cid:15),c,andBareparametersofthesystem.Whenone
1 b a
neglectstheFAcomplexinteractiontermsandreplacestheforcef withstressσ,@(cid:30)hasthe
a @t
formthatisgraphedinFig2.ThisfigureillustratesthatEq(8)capturestheactivationofFA
complexesbycompressivestresses(negativevaluesofσ).
Ithasbeenestablishedexperimentallythatincreasesinintracellularstressesarisingfrom
interactionwiththesubstrateandstressfibersincreaseFAsize[5,13].However,aspecific
functionaldependenceofFAgrowthratesonintracellularstresshasnotbeenestablished.
UsingEq(8),BesserandSafranprovideonemathematicaldescriptionofhowFAevolutionis
dependentonaconstantforcegeneratedbyactinstressfiberinaone-dimensionalframework.
WesimplifytheirmodelbyassumingthatintracellularstressesaffecttheFAchemicalpotential
andonlycompressivestressesaffectFAactivationwhiletensilestresseshavenoeffect.This
simplificationallowsustoextendthemodelforFAevolutiontoamorerealistictwo-dimen-
sionaldescriptionofthecellandtoexplicitlycoupleFAevolutiontocalculationsofintracellu-
larstresses.Asaresult,weareabletopredictthestress-dependentgrowthoftheFAsand,due
totheattachmentofthecelltothesubstrateviaFAsprings,accountfortheeffectoftheunder-
lyingsubstrate.
Theequationgoverningtheevolutionofϕinourmodelisgivenby
@(cid:30)ðx;tÞ
¼k tanhð(cid:0) k s(cid:22)ðx;tÞÞHð(cid:0) s(cid:22)ðx;tÞ ð9Þ
@t s 2
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Centerorperiphery?Modelingfocaladhesionplacement
(cid:2) (cid:0) (cid:0) (cid:1) (cid:1)(cid:3)
Fig2.Graphof@(cid:30)¼C m (cid:0) (cid:15)b tanh bDGþbs (cid:0) 1 .In[15],specificparametervaluesarenotprovided
@t 1 b 2 2 2
andadimensionlesssystemofequationsissolved.InthegraphweuseparametervaluesC =1,μ =0,(cid:15) =
1 b b
1,β=1,andΔG=0.Notethatinthefunctiongraphedweaddbs,whereasinEq(8)thetermbf issubtracted.
2 2 a
Thisisbecausecompressivestresses,whichareassumedin[15]toactivateFAformation,arenegative,and
f isdenotedtobeapositiveforceparameter.
a
doi:10.1371/journal.pone.0171430.g002
InEq(9),evolutionrateparametersaregivenbyk andk ,HistheHeavisidefunction.We
σ 2
definetheaveragebulkstressby
s ðx;tÞþs ðx;tÞ
s(cid:22)ðx;tÞ¼ xx yy ð10Þ
2
andnotethatthisscalarvalueisalsoreferredtoasthehydrostaticcomponentofthestressten-
sor,ofwhichσ andσ arecomponents.Eq(9)couplestheintracellularstresseswiththeevo-
xx yy
lutionoftheFA.ThisevolutionequationhasaverysimilarstructuretothatillustratedinFig2
butresultsinasmallernumberofparameterswhosevaluesmustbedetermined.Duetothe
factthattheexistingtheoreticaldevelopments(e.g.[15])offerlittleguidanceconcerningthe
parametersinEq(9),andbecausethereisnoquantitativeexperimentaldataavailablewhich
wouldallowustoformallydeterminevaluesofk andk ,weuseatrialanderrorapproach
σ 2
andobtainthefollowingparametervalues:k =360.0min−1andk =1.0kPa−1.Thesevalues
σ 2
ofk andk arechosensothatFAgrowthratesinresponsetointracellularstressesresultin
σ 2
totalsizesofadhesionareaobservableinlivecells(see,forexample,totalFAareasin[13]).In
addition,thesevalueallowfortractableandefficientsimulations.Suchanapproachissuffi-
cienttodemonstratethatthemainassumptionsofthemodelareviable.Whenquantitative
experimentalresultsbecomeavailable,moreformalapproaches(e.g.regressionmethods)may
beusedtodeterminek andk .
σ 2
FAsevolveaccordingtoEq(9),whichiscoupledtothefollowingrule
(cid:30)ðx;tÞ(cid:21)0:5 ! FA spring is placed at position x ð11Þ
(cid:30)ðx;tÞ<0:5 ! FA spring is removed from position x ð12Þ
ThetotalFAareaisthereforecalculatedbydeterminingtheareaoverwhichϕ(cid:21)0.5.
InadditiontomodelingFAturnoveraccordingtoEqs(9),(11)and(12),wealsoincorpo-
rateastretch-dependentFAruptureandamicrotubuledependentrupture.Incertain
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Centerorperiphery?Modelingfocaladhesionplacement
Table1.Parametervaluesusedinsimulations.
Parameter Description Value(s) Source
E CellYoung’smodulus 0.5–20kPa [39],[40]
c
E SubstrateYoung’smodulus 1.0–100kPa [39],[41],[40]
s
ν CellandsubstratePoissonratio 0.3 [42]
k FAspringconstant 5×10−5dynes/μm [32]
a
k FAevolutionparameter 360min−1 NewParameter
σ
k StressfeedbackofFAgrowthparameter 1.0dynes−1 NewParameter
2
doi:10.1371/journal.pone.0171430.t001
simulations,ifthestretchinanFAspringsurpassesacriticalvalue,thisattachmentbetween
thecellandthesubstratebreaksatthisspring.Inaddition,FAevolutionprocessesarealso
coordinatedbythemicrotubulenetwork[37,38].FAdisassemblybymicrotubulesisavery
complexprocessthatisnotwellunderstood.Possiblemechanismsincludemicrotubule-
inducedincreaseofcellcontractilitynearanFA,transportofendocytosedintegrinsviamicro-
tubulemotors,ormicrotubuletransportofECMdegradationmolecules(matrixmetallopro-
teases)totheFAleadingtothebreakdownofthecell-ECMattachments[38].Sinceithasbeen
establishedthatmicrotubulesbranchfromthenuclearenvelopetothecellperiphery,we
implementcontrolofFAsizeintothemodelbyassumingthatthesecytoskeletalstructuresdis-
assembleFAsfromthecellcenterouttothecelledge.WemodeltheeffectsofFAdisassembly
duetomicrotubulesbyspecifyingacriticalFAareatocellarearatio,andremoveanyFA
springsthatallowtheFAareatoexceedthiscriticalratio.Springsclosesttothecellcenterare
removedtokeeptheFAtocellarearatiobelowthecriticallevel.Thelocalvalueofϕisalso
loweredto0.2whenaspringisremoved,therebymodelingthedisassemblyoftheentirefocal
adhesioncomplex.
Anoteaboutnumericalimplementation
Wesolvethesystemofgoverningequationsforthecelldisplacement,substratedisplacement,
cellstresses,andFAvolumefraction(u,u,σ,andϕ,respectively)usingthefiniteelement
c s c
methodforspatialdiscretizationand,forthevariablesu,u,andσ,weuseabackwarddiffer-
c s c
enceschemefortimediscretization.TheevolutionofthevolumefractionofFA,ϕ,isdiscre-
tizedintimeusinganaforwarddifferencemethod.
Asbiologicalcellsmoveandspreadtheydeformsignificantly,andsomeelementsinthe
finiteelementmeshbecomesignificantlymisshapen.Thisinturnleadstoalackofconver-
genceoftheGauss-Seideliterativesolutionapproachusedinthesimulations.Toovercome
thisissue,weincorporateremeshinginthenumericalalgorithm.Remeshingisperformed
eitherafterafixednumberoftimestepsorwhenoneofthetriangularelementsfallsbelowa
criticalminimumheighttoarearatio.Sinceremeshingofthecomputationaldomainresultsin
anadditionalnumericalerror,tobeabletocompareresultsofdifferentsimulationsweforce
theremeshingtooccuratfixed,predeterminedtimesteps.Inthismanner,resultsthatare
beingcomparedhavebeencomputedonfiniteelementmeshesthatwereremeshedthesame
numberoftimes.Finally,wenotethatallprogrammingwasdoneinMatlab.Parametersused
insimulationsarelistedinTable1.
Results
Modelvalidation
Herewevalidateourmodelbycomparingthemodelpredictionswithexperimentalwork.
Tractionforcemicroscopy[43]allowstheexperimentaldeterminationoftractionsexertedby
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Fig3.Comparisonofsimulationstocellspreadinginaconfinedenvironment.Oursimulationsareshownandcorresponding
experimentalresultscanbefoundinFig3ofWangetal.[39](seecenterandrightmostpanelinmiddlerow.)Weillustratenumerically
computedsubstratedisplacementsandFAtractions.FAtractionmagnitudesareshownascontoursandtractionvectorsarealsoillustrated.
Wenotethatnumericalresultscomparewelltoexperimentsbothqualitativelyandquantitatively.CellandsubstrateYoung’smodulus:E =
c
0.5kPa,E =1kPa.FAspringsdonotrupture.
s
doi:10.1371/journal.pone.0171430.g003
acellonasubstrateduringmotilityorspreading.Wecompareournumericalresultstothe
tractionpatternsmeasuredforaspreadinghumanairwaysmoothmusclecell[39]andthe
tractionpatternsofacontractingcardiomyocyte[40].Theaimofthesecomparisonsistoillus-
tratethefidelityofthemodelinitsabilitytocorrectlycapturemechanicalresponsesfortwo
differentcelltypesonsubstrateswithdifferentstiffnessesandchemicalpatterningresultingin
differentadhesionpatternsandcellshapes.
Asdescribedin[39],smoothmusclecellswereplacedonto50-μmdiametercircularislands
madeupofpolyacrylamidegelwithacollagencoating.Cellswereplatedontheislandsand
culturedforseveralhours,duringwhichtimetheyspreadontotheisland.Wechoosethispar-
ticularexperimentalresultasamodelvalidationbecausethegeometryofthecellsspreading
onthecircularislandscanbemimickedverycloselyinournumericalsimulations.Smooth
musclecellsarerelativelysoftwithaYoung’smodulusofapproximately0.1to0.5kPa,andthe
polyacrylamidesubstrateusedin[39]hasaYoung’smodulusofapproximately1.3kPa.Inthe
correspondingsimulationswesettheYoung’smodulusto0.5kPaforthecelland1kPaforthe
substrate.RemainingparametervaluesforthisandothersimulationsarelistedinTable1.
Afterseveralhoursofspreading,thecelldiameterintheexperimentalresultsisapproximately
40μm,sowecomparetheexperimentalresultstoournumericalsimulationsat120minutes.
InFig3weshownumericalresultsforsmoothmusclecellspreading.Whentheseresults
arecomparedtoexperimentalresultsshowninFig3ofWangetal.[39],weobservemaximum
substratedisplacementsfrombothexperimentsandoursimulationsintheregionwherethe
cellisattachedtothesubstrate.Quantitatively,themaximumexperimentallydetermineddis-
placementsareapproximately3.5μmwhichareingoodagreementwiththemaximumcom-
puteddisplacementsof2.0μm.Further,bothexperimentsandthesimulationsshowaa
similarlocationofmaximumintractionforcesexertedbythecellsontothesurface.The
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maximumexperimentallydeterminedtractionsareapproximately500Pawhichareinreason-
ableagreementwiththemaximumcomputedtractionsof740Pa.
Thesecondexperimentalsystemthatwepicktovalidateourmodelisaratcardiomyocyte
platedonatwodimensionalsiliconesubstrateonwhichtheyarereportedtocontractwithan
averagefrequencyof38contractionsperminute[40].Inthisexperimentalwork,theeffectofa
rangeofsubstratestiffnessesvaryingbetween1–500kPa,specifically1,15,30,50,150and500
kPa,isexplored.Cardiomyocytesareknowntoexhibitarangeofstiffnesseswiththehealthy
cellhavingaYoung’smodulusintherangeof10–30kPaand,withagingandwithdifferent
diseasesoftheheart,cangoupto150kPa.Thesiliconsubstratemicrostructureconsistsofa
hexagonallatticeofmicrodotswithanedgelengthof2.5μm.Forthepurposeofanalyzing
theirresultstheauthorsshowthatthecellshapecanbeapproximatedbyatriangularregion
onthelattice.Inlinewiththis,wemodelthecelltohaveatriangularshapewithanedgelength
of30μm,whichiscomparabletotheedgelengthreportedin[40].Insimulationswecompare
themaximumtractionatthepeakofasinglecontraction.Here,wesetthecellYoung’smodu-
lusto20kPa.Sinceresultsin[40]correspondtoawiderangeofsubstrateYoung’smoduli,we
ransimulationswithfourdifferentsubstrateYoung’smoduli(E =15kPa,20kPa,25kPa,and
s
50kPa)andpresentresultscorrespondingtoasubstrateYoung’smodulusof15kPasincethe
predictedsubstratedisplacementcomparisonsarethemostaccurate.
AsillustratedinFig4,computedmaximumsubstratedisplacementsare0.06μmandmaxi-
mumexperimentallymeasuredsubstratedisplacementsare0.08μm.ThemaximumFAtrac-
tionmagnitudesfromthesimulationarearound0.2kPawhilethosefromtheexperimentsare
around0.08kPa,i.e.approximately2.5timeslargerfornumericallycomputedcardiomyocyte
contractionthanobservedinexperiments.Thisoverestimationofthecomputedtractionstress
couldbeduetoseveralfactors,suchasdifferencesinthecellYoung’smodulusorthefactthat
contractionsintheexperimentalcellsystemaretemporallyperiodicwhileinthesimulation
werunonesteadycontraction.Thesimilarpatternofdisplacementandtractiondistribution
indicatesthattheoverallfeaturesofthemodeldoreproduceexperimentalmeasurements.
Thefactthatthemodelpredictsthesubstratedisplacementsaccuratelyandcapturesthe
tractionmagnitudesreasonablywellfortwodifferentcellsystemsonverydifferentsubstrate
stiffnessesgivesusconfidencethatthemodelincorporatestheessentialphysicsthatdescribe
thedynamicmechanicalinteractionsbetweenanactivelymovingcellandadeformablesub-
strate.Inthenextsectionweusethismodelofmechanicalinteractionsbetweenaspreading
cellandsubstratetogeneratesomeinsightsonwhyfocaladhesionsaretypicallyobservedon
thecellperipheryandnotthecellcenter.Throughthisinvestigationwealsoseektoestablish
whataresomebehaviorsofthecellthiscurrentmodelisabletocaptureandwhataresome
shortcomingsofthemodelinitscurrentstate.Inthediscussionsectionweproposesome
mechanismsthatcouldpotentiallybeaddedtothemodeltoaddresssomeofthese
shortcomings.
EffectsofFApositioningoncellandsubstrateresponse
TogainabetterunderstandingofwhyexperimentallyobservedFAsaretypicallylocatedon
thecell’speripheryandnotitscenterweconstructtwomodelsystems.Bothmodelsystems
beginwithcircularcellswithradiusof10μmandeightFAswithanareaof3.14μm2each.
TheseFAsareplacedeitheratadistanceofR=4μmorR=7μmfromthecellcenter,asillus-
tratedinFig5.FAsoriginallyplacedatadistanceof7μmfromthecellcenterarereferredto
asperipheralFAsandthosethatareatadistanceof4μmfromthecenterarecalledcentral
FAs.Forthesetwomodelsystemsweinvestigatehowglobalresponseofthecellandthesub-
strateisaffectedbychangesinFAsize,FAstrength,andsubstrateYoung’smodulus.FAsizeis
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