Title:Self-organizationofstresspatternsdrives
statetransitionsinactincortices
Authors:TzerHanTan1+,MayaMalikGarbi2+,EnasAbu-Shah2,3+#,JunangLi1,Abhinav
Sharma4,6,FredC.MacKintosh4,KinneretKeren2,3,5*,ChristophF.Schmidt6,7*,Nikta
Fakhri1,6*
*
Correspondingauthors
+
#
Theseauthorscontributedequally
Currentaddress:KennedyInstituteofRheumatology,UniversityofOxford,Oxford,UK
Affiliations:
1
DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA,USA
2
DepartmentofPhysics,Technion–IsraelInstituteofTechnology,Haifa,Israel
3
RussellBerrieNanotechnologyInstitute,Technion–IsraelInstituteofTechnology,
Haifa,Israel
4
DepartmentofPhysicsandAstronomy,VrijeUniversiteit,Amsterdam,TheNetherlands
5
NetworkBiologyResearchLaboratories,Technion–IsraelInstituteofTechnology,
Haifa,Israel
6
ThirdInstituteofPhysics–Biophysics,GeorgAugustUniversity,Göttingen,Germany
7
GermanCenterforCardiovascularResearch(DZHK),Göttingen,Germany
1
Abstract
Biologicalfunctionsrelyonorderedstructuresandintricatelycontrolledcollective
dynamics.Incontrasttosystemsinthermodynamicequilibrium,orderistypically
establishedandsustainedinstationarystatesbycontinuousdissipationofenergy.
Non-equilibriumdynamicsisanecessaryconditiontomakethesystemshighly
susceptibletosignalsthatcausetransitionsbetweendifferentstates.Howcellular
processesself-organizeunderthisgeneralprincipleisnotfullyunderstood.Here,we
findthatmodelactomyosincortices,inthepresenceofrapidturnover,displaydistinct
steadystates,eachdistinguishedbycharacteristicorderanddynamicsasafunctionof
networkconnectivity.Thedifferentstatesarisefromasubtleinteractionbetween
mechanicalpercolationoftheactinnetworkandmyosin-generatedstresses.
Remarkably,myosinmotorsgenerateactinarchitectures,whichinturn,forcethe
emergenceoforderedstresspatterns.Reminiscentofsecondorderphasetransitions,
theemergenceoforderisaccompaniedbyacriticalregimecharacterizedbystrongly
enhancedstrainfluctuations.Thestrikingdynamicsinthecriticalregimewere
revealedusingfluorescentsingle-walledcarbonnanotubesasnovelprobesofcortical
dynamics.
OneSentenceSummary
Actincorticesdisplaytransitionsbetweensteadystateswithdistinctstructuresand
stresspatternsymmetries.
2
MainText
Phasesofmatteraretypicallycharacterizedbysymmetryorcollectiveorderofits
components.Changesinthisorderorsymmetryoftencorrespondtophasetransitions
inclassicalequilibriumstatisticalphysics(1).Dynamicfunctionsoflivingsystemsalso
requirespecificformsandcollectiveorder.However,incontrasttoequilibriumsystems,
organismsbuildorderedstructuresthroughthedissipationofmetabolicenergy(2-4).A
generalizationofequilibriumconceptscanprovideinsightintohoworderemergesin
thesenon-equilibriumsystems.
Dissipativecellularstructureshavetoberobustandformstablesteadystates
lastingfromsecondstothelifetimeoftheorganism.Atthesametime,theyhavetobe
highlyadaptiveandreorganizeinresponsetointernalorexternalsignals.Aclassical
caseofcollectivedynamicswithhighsensitivitytocontrolparametersisobservedin
systemsnearacriticalpointorsecond-orderphasetransition.Suchtransitionsare
characterizedbythecontinuousdevelopmentoforder,whichdistinguishestwophases
(1).Atsuchatransition,thesusceptibilitytocertaincontrolparametersdiverges,and
thesystemdisplayslargefluctuationswithextendedspatialandtemporalcorrelations.
Criticalityisemergingasanimportantfunctionalfeatureinbiologicalsystems,with
regulatorymechanismspositioningsystemsnearcriticality(5).
Aprominentexampleofanadaptivedissipativestructureisthecellcortex.The
cortexisaquasi-2Dnetworkofdynamicallycrosslinkedactinfilamentsthat
continuouslypolymerizeanddepolymerize(6).Thismicronthicknetworkisanchoredto
thecellmembraneandisinternallyactivatedbymyosinmotorproteins.Thecortex
providesmechanicalintegrityandrigiditytocellsoverextendedtimescales,whileits
componentsturnoverwithinminutes(7,8).Inaddition,thecortexundergoesdramatic
reorganizationsduringcellularprocessessuchasdivision,generationofpolarityand
motility(9-11).Theseprocessesoftenentailtransitionsfromhomogeneousto
inhomogeneousstatesvianetworkcontraction(12,13).Howdoesthecortexachieve
theseeminglyimpossiblefeatofswitchingbetweenstabilityandlarge-scale
reorganization?
3
Cortexstructureanddynamicsarecontrolledbyasubtleinterplaybetweenactin
turnover,networkconnectivity,aswellasstrainsandstressesgeneratedbynon-muscle
myosinmotors(14-16).Characteristictimescalesareontheorderofsecondsforall
theseprocesses(14,17).Experimentalaccesstotheactivemechanicsofcorticeshas
beenseverelylimited.TechniquessuchasAFMindentation(18)orpipetteaspiration
(19)onlyprovideglobal,coarse-grainedmeasurements,notcapturingspatial
inhomogeneity,anisotropyandtime-dependence(14).Microrheology(19)could,in
principle,providelocalinformation,butitsuseinthecortexhasbeenlimitedbythelack
ofsuitableprobes;micron-sizedcolloidalbeadsaretoolargetopenetratethethin
cortexwithoutperturbingit,whilenanometer-sizedparticles,suchasquantumdots,are
toosmalltoremaininthenetworkforlongenoughtoprobeit.
Here,weusefluorescentsingle-walledcarbonnanotubes(SWNTs)asprobesto
demonstratethatasubtleinteractionbetweenmechanicalpercolationofthenetwork
andpropagationofmyosin-generatedstressesleadstostructurallyanddynamically
distinctsteadystatesinmodelactincortices.Transitionsbetweenthesestatesare
accompaniedbystrikingchangesinthesymmetryofstresspatterns,includingacritical
regimecharacterizedbystronglyenhancedstrainfluctuations.
Weformeddynamiccortices,invitro,byencapsulatingXenopuseggextractin
water-in-oilemulsiondroplets(Fig.1)(20).LocalizationofActAproteintothewater-oil
interfaceactivatedArp2/3-mediatednucleationofbranchedactinnetworks(Fig.1B),
generatinghomogeneous,~1µmthick,corticalactinnetworks.Thesenetworks
exhibitedcontinuousactinturnoverwithatypicaltimescaleof~1min(Fig.S1)aswell
asmyosin-drivennetworkdynamics,inastablecell-likegeometry.Dropletswere
flattenedandconfinedbetweenhydrophobiccoverslipsforobservation.Thedroplets
were~100µmindiameterandhadafixedheightof30µm.Thecorticalactinnetworks
wereimagedbyconfocalmicroscopy(Fig.1A).Toresolvenetworkdynamicsathigher
resolutions,weusednear-IRfluorescentSWNTsasnovel“stealth”probes(Fig.1B).
SWNTsare~1nmindiameterand~100-300nminlength(17).WefoundthatSWNTs
easilypenetratethethincortexwithoutperturbingthenetworkand,duetotheirlarge
4
aspectratio(21),getentrappedforlongenoughtimestoreportonnetworkdynamics.
Duetotheirextremephotostability,theSWNTscouldbetrackedoverabroadwindow
oftimescales(millisecondstohours).
Indropletsofbareextract,weobservedhomogeneouscortices(Fig.1A)withno
discernablenetworkmovement.Sincemyosinscanonlydrivelarge-scalemovement
whenthenetworkissufficientlycrosslinked,weincreasednetworkconnectivityby
addinganactincrosslinker,α-actinin.Increasingcrosslinkerconcentrationinthissystem
leadstobreakingofsymmetryandformationofapolarcap(20).Wefoundthat,asa
functionofcrosslinkerconcentration,wecoulddrivethesystemintothreedramatically
differentdynamicsteadystates,referredtoaslow,intermediateandhighconnectivity
(Fig.2).Inallstates,continuousactinturnoverwasmaintained(FigS1).
Atlowconnectivity(0-1.5µMaddedα-actinin),corticesremained
homogeneous,andSWNTsfluctuatedrandomly(Fig.2A,MovieS1).Thehomogenous
actindistributionreflectsadynamicsteadystate,withcontinuousactinpolymerization
catalyzedbytheActAatthesurfaceanddepolymerizationtothebulk(Fig.S1).Acontrol
experimentwithmyosinimmuno-depletedextracts(Fig.S2)showedthatthespatially
uncorrelatedandrandomprobemotionswerelargelymotordriven.
Atintermediateconnectivity(2.0-2.5µMaddedα-actinin),coherentlymoving
clustersofSWNTsappearedwithdiametersupto10µm.Strikingly,theclustersmoved
overlongdistanceswithinthecortex,withoutgeneratinglarge-scaleactindensity
inhomogeneities(Fig.2B,MovieS2).Theclustersmovedinavorticalmanner,withhigh
directionalpersistence,attypicalspeedsof~10µm/min.Thismotionwaslikelydriven
bycontractileforcesactingbetweenclusters.Occasionalrapidchangesofcluster
velocities(MovieS3)canbeexplainedbytheruptureoftenuousbridges,contractedby
fewmyosinminifilaments.Notethatthemechanismofforcegenerationisdifferent
fromthatofactiveswimmers(22),suchasfishinaschool,whereparticlespropel
themselvesbyexertingaforceagainstanembeddingmediumorasurface.The
dynamicsinourmodelcorticesarealsodifferentfromtheobservedphenomenain
5
highlyconcentratedmotoractivatedsolutionsofactinfilaments(23)andmicrotubules
(24),whereorderisimposedthroughstericrepulsionbetweenlongfilaments.
Athighconnectivity(3.0-4.0µMaddedα-actinin),corticesphaseseparatedand
formedalargecap,whichextendedoverasubstantialfractionofthedropletsurface,
butremainedessentially2-dimensional.Smallclusterscontinuouslynucleatedand
flowedradiallytowardsthecap.Thiscontractionwasvisibleinboththeactinandthe
SWNTfluorescencechannels(Fig.2C,MovieS4andS5).Thevelocityoftheconverging
clusterswas~10µm/minintheperipheryanddecreasedtowardsthecap(MovieS5).
Thecapsformedatrandompositions,butonceformed,remainedstableforatleast2
hours.Theratioofintensitiesbetweenthecapregionandperipheralaccretionzones
alsoremainedconstantovertime(Fig.S3).Thisshowsthatthesystemisatsteadystate,
maintainedbythecontinuousdisassemblyofactinfromthecap.
Astrikingfeatureofthesenon-equilibriumsteadystatesisthattheyalldisplay
distinctpatternsofmotionwhileonlythehighconnectivitystateexhibitsan
inhomogeneousdensitypattern.Two-dimensionalcorrelationanalysisofSWNT
velocitiesshowsarapidbutcontinuousgrowthofcorrelationlengthatintermediate
connectivity(Fig.S4).Increasingcorrelationlengthreflectsincreasingconnectivityinthe
actincortex,andtheshapeofthecurve(Fig.S4)indicatescooperativityandis
reminiscentofconnectivitypercolation(25).Breakingupofthenetworkintoindividual
clustersiscommonlyobservedasaconsequenceofcrosslinking,especiallyin
combinationwithcontractilemotors(26,27).Incontrasttoprevioussystems,inour
dissipativesteady-statesystem,theclustersmaintaintheirsizebutaredynamic,with
theirlifetimeexceedingthatoftheircomponents,i.e.theydisplaycollectivestructural
memory.Themostprominentfeatureweobserveisthelongdistancemovementofthe
clustersdrivenbymyosinmotors,demonstrating,inaddition,acollectivemotion
memory.
Atintermediateconnectivity,patternsofmotionsfluctuatestrongly(MovieS2
andFig.2B),analogoustofluctuationsobservednearacriticalpointofanequilibrium
system.Toquantifyfluctuations,weextractchangesofvelocityinfixedpositionsinthe
6
cortex.Wecalculatedthenormalizedcoarse-grainedvelocityfluctuation
autocorrelationintime(Fig.3A), C! Δ (τ ) = ΔV̂α ( t ) ⋅ ΔV̂α ( t + τ )
αT
.Velocityfluctuation,


ΔV ,isthedifferencebetweenthevelocityinagivencoarse-grainedbox Vα ( t ) andthe

time-averagedvelocityinthesamebox Vα T (SeeSI).Figure3Ashowsthevelocity
fluctuationautocorrelationintimefora5x5gridatthedifferentcrosslinker
concentrations.Localaveragevelocitiesareclosetozeroatlowandintermediate
connectivitybutnon-zeroandconvergingtothecapathighconnectivity.Fluctuations
havesmallamplitudesandarerapidatlowconnectivity,butgainamplitudeand
persistenceatintermediateconnectivity,tobecomeofsmallamplitudeagainathigh
connectivity.Weconstructedascalarorderparametertoquantifythepersistenceof
localvelocities.Theorderparameterwascalculatedastheangularcorrelationof
velocitiesinthecoarse-grainedboxes, cosφ
50 s
,averagedovertimeintervalsof50s,
longerthanthecorrelationtimeofamyosinmini-filament(~5s(17))(Fig.3B).The
orderparametershowsasteepincreaseatintermediateconnectivity,analogoustoa
secondorderequilibriumphasetransition.Theincreasingorderinprobeparticle
velocitiesreflectsspatialorderingofthelocalstresses,ratherthanliquid-crystalline
nematicstructureofthefilamentsinthecortex.Fromtherelativenumbersof
homogeneousandphase-separatedcorticesatagivencrosslinkerconcentration,
inferredfromimagesofrhodamine-labeledactin,weconstructedanactin-densitybased
structuralorderparameter(Figs.3CandS5).Thisorderparameterconfirmsaglobal
phaseseparationathighconnectivity.
Nearcriticalpoints,systemsexhibitextendedcorrelationsoverlength-scales
thatsubstantiallyexceedmolecularscales.Toextracttheaveragespatialextentof
correlatedvelocityfluctuationsdrivenbythefluctuatingarrangementsofstress
generatingmotors,weusedthedeviationsfromtheoveralllocalvelocityaverages.The
normalizedvelocityfluctuationcorrelationfunctioninspace(seeSI)(Fig.3D)was
( ) (
"
"
" "
calculatedas C! Δ ( R ) = Δv̂i ( ri ,t ) ⋅ Δv̂ j rj ,t δ ri − rj − R
)

ijT
.Velocityfluctuation, Δvi ,is

thedifferencebetweenthevelocityofanindividualparticle vi andtheoverallaverage
7

velocityinthecoarse-grainedbox α containingparticle i , Vα
T
(SeeSI).Atlow
connectivity,spatialcorrelationsofvelocityfluctuationsaresmall.Athighconnectivity,
deviationsfromthelocalaveragevelocity,whichiszerointhecapregionandnon-zero
intheperiphery,arealsosmall,sothatcorrelationsbecomeburiedinthenoise.At
intermediatecrosslinkdensitiesormarginalconnectivity,however, C Δ ( R ) showsa
characteristicfluctuationcorrelationlengthof~15µmsmaller,thanbutapproaching
thesystemsize(Fig.3E).
Anotherhallmarkofequilibriumcriticalpointsisdivergingsusceptibility,i.e.a
divergingresponsetoexternaldrivingsuchasanappliedforcefield(1).Inthenonequilibriumsteadystatesofthecortices,drivingisnotexternal,butcausedbymyosin
motorscreatinganinternalstressfield.Wedefinesusceptibilityastheamountof
changesinpatternsofmotioninresponsetosmallchangesinthestressfield,i.e.the
additionorremovalofafewmyosinmotors.Analogoustothermalfluctuationsin
equilibrium,inourmodelsystem,myosinon/offkineticsprovidesstochasticforce
fluctuations.Thuswecaninfersusceptibilityfromthestrainfluctuationsdrivenbythese
forcefluctuations.Toobtainameasureofsusceptibilityfromtheobservedpatternsof
motion,wecalculatedthevarianceofvelocityfluctuationcorrelationsatafixed
distancelargerthanthetypicallengthofasingleactinfilament(>1µm).Figure3F
showsthatthisvarianceexhibitsabroadpeakatintermediateconnectivity.
Interestingly,anumericalsimulationofanisotropicallystretched2Dnetworkoflinear
springsalsoshowscorrelatedstrainfluctuations,i.e.movingclusters,nearconnectivity
percolation(Fig.S6,seeSI).
Thethreestatesdifferinthesymmetryofthepatternsofmotionoftheprobe
particles.Toquantifyaveragesymmetryandorderoftherespectivestates,we
calculateddirectionalcorrelationfunctionsfortheentiresetsofdata(Fig.4).This
functionwasconstructedbycalculatingthevelocity-velocitydirectionalcorrelationfor
eachmovingprobe,withallthesurroundingprobes,asafunctionofdistanceandangle
withrespecttoitsowndirectionofmotion,andthenaveragingoverallmovingparticles
inallmovies(28)(seeSI).
8
Atlowconnectivity,patternsarefullysymmetric,andnocorrelationisobserved
(Figs.4AandS7).Atintermediateconnectivity,thevectorfieldofprobevelocities
beginstoshowcurlbutnodivergence(Figs.4BandS7).Theinnerregionofpositive
correlationinFig.4Breportsaverageclustersize,consistentwithclustersize
determinedfromdisplacementcorrelations(Fig.S8).Thebreakingofrotational
symmetryisevidentfromtheextendedpositivecorrelationsaround0°and180°,i.e.a
finiteprobabilitythatparticlestrailingandleadingagivenparticlemoveinthesame
direction.Suchabreakingofrotationalsymmetryischaracteristicfornematicliquid
crystals.Notethatinthiscase,orderingofvelocitiesimpliesapolarnematic(22,29).In
oursystem,orderisnotgeneratedbyalignmentofactinfilamentsthroughentropy
maximization,asistypicallythecaseforequilibriumnematics.Rather,thealigned
velocityvectorslikelyreflectunderlyingorderingofcontractilemyosinminifilaments
thatcanbeconceptualizedasforcedipoles.Athighconnectivity,thevelocityvector
fieldaroundthecapshowsdivergence,ratherthancurl,i.e.itreflectsbroken
translationalsymmetry(Figs.4CandS7).Thedirectionalcorrelationfunctionclearly
showsfirst,anextendedcorrelationlengthorcapsize,andsecond,anasymmetry
between0°and180°whichisasignatureofaconvergingflowfield(Fig.S9).This
convergingvelocityfieldlikelyreflectsaradialstresspattern,i.e.aradialorientationof
myosinforcedipoles.Suchglobalcentrosymmetricpolarorderisnotobservedintypical
liquidcrystals.Thecrucialdifferenceinoursystemisthelackofconservationof
particlesbecauseofcontinuousactindepolymerizationinthecapandrecyclingof
monomerstothebulk.
Figure4(D-F)presentsaconceptualmodelfortheobservednon-equilibrium
steadystates.Thegenerationoforderandthebreakingofsymmetryinthecortexare
drivenbytheinterplayofnetworkconnectivityandactivestresses.Increasing
connectivitydrivesthenetworktowardsmechanicalpercolation,whichactslikeaclutch
allowingmyosinmotorstoexertincreasinglylong-rangeforces.Thisleadstolong-range
transportinthecortexandstructuralrearrangements,whichinturn,feedsbackonthe
arrangementoftheforce-generatingmyosinmotors.Theobservednematicand
9
centrosymmetricpolarordersinprobevelocitiesthusreflecttheorderinthe
orientationofthestressgenerators(myosinminifilaments).Atlowconnectivity(Fig.
4D),myosinwillengageactin,butwillnotdrivecorrelatedmotionsoverdistances
longerthanfilamentlength.Atintermediateconnectivity(Fig.4E),theeffectiverange
ofstresspropagationexpands.Thecombinationofcontractilityandcrosslinkingcreates
relativelyrigidislandsoflimitedsize,wherethedensityisnotdifferentenoughtobe
visiblebyactinfluorescence,butthemobilityofprobeparticleswithinclustersis
stronglysuppressed.Myosin-activatedcontractilebridgesmovetheclustersinrandom
andchangingdirections.Notethatcompetingmotorscanalsoacceleratebridge
rupture,creatinganegativefeedbackmechanismforclustergrowth(MovieS3).We
speculatethatinourcase,inthepresenceofconstantturnover,clustersize(rc)is
primarilylimitedbythecompetitionbetweenaccretionattheboundary(∝rc)and
depolymerizationwithinthecluster(∝rc2).Athighconnectivity(Fig.4F),atransition
occurswhenthelargestclusterreachessystemsizesandconcentratesactinfilaments,
motors,andcrosslinkers,suchthataradialgradientislikelytoemerge.Thiscreatesnet
polymerizationintheperipheryandnetdepolymerizationinthecap,setsupasteady
flowofsmallclusterstowardsthecap,andinturnorientsthemotordipolesradiallyin
theperiphery.Thismodelpredictsasystem-sizedependenceofthesecondtransition.
Theobserveddynamicshighlightsomeoftheuniquepropertiesof“active
matter”:activestressesandmechanicalstructurescancoupleincomplexandsubtle
wayssuchthatdrasticallydifferentdynamicpatternscanresultfromshiftsinthe
balancebetweencompetingmoleculartimescales.Theintermediatecaseofmarginal
connectivityresemblesacriticalstate,albeitinanon-equilibriumsituation,andshows
maximalsusceptibilitytointernalstressvariations.Thiscriticalstateextendsovera
relativelybroadcrosslinkerconcentrationrange,whichisafeatureofarobuststate.
Thisrobustnessisahallmarkofself-organizedcriticality(26,30).Invivodynamic
transitionssuchascorticalsymmetrybreakinginC.elegans(9,10),thetransitionintoa
contractileringinoocytes(31),membranestirring(32)ortheformationof
immunologicalsynapses(33)bearstrikingresemblancetoourobservations.This
10
suggeststhatthesamenon-equilibriummechanismsareatworkinvivo.Boththe
connectivityofthenetworkandtheactivityofmyosinmotorscanbeactivelytunedby
cellsinresponsetovarioussignalssuchasCa2+ions(15,34).WithSWNTsasideal
probes,itnowremainstotacklethemuchmorechallengingproblemofmapping
corticaldynamicsinlivingcells.
11
Figure1.Dynamicmodelactincortices
Quasi-2DactinnetworksweregeneratedbyencapsulatingXenopuseggextractin
water-in-oilemulsiondropletsandconfiningactinpolymerizationtotheinterface.(A)
Equatorialcrosssectionofflatteneddroplets.SimultaneousconfocalimagingofbodipyconjugatedActA(green,left)andrhodamine-labeledactin(magenta,right).Amphiphilic
bodipy-ActAlocalizestothewater-oilinterfaceandcatalyzestheformationofaquasi
2DdynamicactinnetworkbylocalactivationofArp2/3.(B)Top:Schematicofthe
experiment.Corticaldynamicsaretrackedneartheflatbottomsurfaceofthedroplet.
Thereiscontinuousturnoverbetweenthethinpolymericactinlayerandactin
monomersinthebulk(arrows).Bottom:Zoomedinschematicoftheessential
componentsofthecortex.IRfluorescentSWNTsareinsertedasprobesofcortex
dynamics.
12
[α-actinin] level
A
B
C
Time(s)
200
0
20μm
Figure2.Signaturesofdistinctsteadystates.
Networkpercolationcanbeinducedbyincreasingcrosslinkersdensity(α-actinin).Three
distinctsteadystatescanbeidentifiedbasedonqualitativedifferencesincollective
networkdynamicsimagedinthebottomcorticallayerinthedroplets.(A)Low
crosslinking(0-1.5µMaddedα-actinin,imagesshown:1.5µM)(B)Intermediate
crosslinking(2-2.5µMaddedα-actinin,imagesshown:2.5µM).(C)Highcrosslinking(34µMaddedα-actinin,imagesshown:3µM).(A-C)Topleft:Fluorescenceimageof
insertedSWNTprobes.Topright:Confocalimageoftheactindistribution.Bottom:
IndividualSWNTsweretrackedinmoviesof200slongwith2000frames(MoviesS1-
S4).TracksofindividualSWNTscolor-codedforprogressionintime.(Scalebar=20µm)
13
1
B
0.6
0.4
Low
Int.
High
0.2
0
40
Lag time, τ (s)
60
1.5 μM
2.0 μM
3.0 μM
0
0
0
80
20
D
~
Velocity fluctuation
correlation C∆(R)
1
20
Correlation length (μm)
0
10
20
30
Distance, R (μm)
40
50
16
12
1
2
3
[α-actinin] (μM)
0.8
0.6
Low
Int.
High
0.2
0
2.0
E
C
0.4
4
1.6
1
2
3
[α-actinin] (μM)
4
F
1.2
Low
Int.
High
8
0.8
Low
Int.
High
0.4
4
0
Fraction with Polar Cap
Velocity fluctuation
correlation C∆(τ)
~
1.5 μM
2.0 μM
3.0 μM
C∆(R≈5 μm) x10-4
0.8
A
<cos(φ)>
1
0
1
2
3
[α-actinin] (μM)
4
1
2
3
[α-actinin] (μM)
4
Figure3.Collectivedynamicsandcriticalfluctuations.
Steadystatesdifferinthedegreeoforderandcorrelationofprobeparticlevelocities.
(A)Normalizedvelocityfluctuationautocorrelationisplottedasafunctionoflagtime
fordifferentα-actininconcentrationsevaluatedina5x5coarsegrainedgridfor200s
movies,averagedoverallgridpoints.(B)Orderparametercalculatedfromtheaveraged
velocitydirectionalpersistenceasafunctionofα-actininconcentration.(C)Anactin
density-basedstructuralorderparametershowingthefractionofphase-separated
corticesasafunctionofcrosslinkerconcentration(0-4µMofaddedα-actinin),inferred
fromimagesofactinfluorescence.(D)Thenormalizedvelocity-velocityfluctuation
cross-correlationatzerolagtimeisplottedasafunctionofprobedistancesfordifferent
α-actininconcentrations(symbols).Dataarefittedwithsingleexponentials(lines).(E)
Characteristiccorrelationlengthsfromfitsin(D)asafunctionofα-actinin
concentrations.Errorbars:95%confidenceinterval.(F)Susceptibilityofclustermotion
tostressfluctuationsquantifiedfromthevarianceoftheamplitudeofthevelocityvelocityfluctuationcorrelationtakenatR=5µm,plottedasafunctionofα-actinin
concentration.Errorbarscorrespondtostandarddeviations.
14
Figure4.Orderandsymmetriesinthestresspatterns.
Increasingcrosslinkersdensityleadstoincreasingorderandbreakingofsymmetriesin
thearrangementofstressgenerators.(A-C)Directionalvelocity-velocitycorrelation
mapsthedegreeoforderandshowsbreakingofsymmetryforincreasingα-actinin
concentrations.Foreachconcentration,thecolorsportraytheaveragealignmentof
pairsofprobevelocities,asafunctionofthedistanceandanglebetweenthem(Ainset)
Atlowcrosslinking(1µM;A)thereisnoappreciablecorrelation.Atintermediate
connectivity(2.5µM;B),themapshowsenhancedcorrelations,distributed
anisotropically,primarilyalongthefront-backaxis,signifyingtheemergenceofnematic
polarorder,andthebreakingofrotationalsymmetry.Athighconnectivity(3µM;C),the
convergingnetworkflowleadstoanasymmetrybetweentopandbottom,indicating
theappearanceofcentrosymmetricpolarorderandthebreakingoftranslational
symmetry.(D-F)Schematicillustrationoftheconceptualmodelforcortexdynamicsasa
functionofconnectivity.Atlowconnectivity(D),thenetworkisnotmechanically
connected,somyosincannotgeneratelong-rangeforces,andthenetworkfluctuates
randomly.Atintermediateconnectivity(E),thenetworkreachesthemechanical
percolationthreshold.Myosin-generatedstressescanpropagateoverlargerdistances,
leadingtotheformationofrigidclusters,whichmoveinavorticalmanner.Theinterplay
betweenstressesandnetworkrearrangementleadtolocalnematicorderingofthe
force-generatingmyosinminifilaments.Athighconnectivity(F),thenetworkself
15
organizesintoasinglelargecapwithpersistentconvergingflowofclustersintothecap,
andpolarorderingofthemyosinforcedipoles.
16
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Acknowledgements
WethankAlexanderSolon,JacquesProst,SriramRamaswamy,MehranKardarandAlex
Mogilnerfordiscussion.ThisresearchwassupportedbyaHumanFrontierScience
ProgramFellowship(N.F.),J.HandE.V.WadeFundAward(N.F.),theClusterof
ExcellenceandDFGResearchCenterNanoscaleMicroscopyandMolecularPhysiologyof
theBrain(CNMPB)(C.F.S.),theEuropeanResearchCouncilAdvancedGrantPF7ERC2013-AdG,Project340528(C.F.S),theDeutscheForschungsgemeinschaft(DFG)
CollaborativeResearchCenterSFB937(ProjectA2)(C.F.S.)andagrantfromtheIsrael
ScienceFoundation(K.K.).
18