Mechanics of motility initiation and motility arrest in crawling cells

1.8 MB·English

by  Pierre Recho

Mechanics of motility initiation and motility arrest in crawling cells
P.Rechoa,b,T.Putelatc,L.Truskinovskya
aLMS,CNRS-UMR7649,EcolePolytechnique,RoutedeSaclay,91128Palaiseau,France
bPhysicochimieCurie,CNRS-UMR168,InstitutCurie,CentredeRecherche,26rued’UlmF-75248ParisCedex05,France
cDEM,Queen’sSchoolofEngineering,UniversityofBristol,Bristol,BS81TR,UK
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0 Abstract
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Motility initiation in crawling cells requires transformation of a symmetric state into a polarized state. In contrast,
l
u motility arrest is associated with re-symmetrization of the internal configuration of a cell. Experiments on kera-
J tocytes suggest that polarization is triggered by the increased contractility of motor proteins but the conditions of

2 re-symmetrizationremainunknown. Inthispaperweshowthatifadhesionwiththeextra-cellularsubstrateissuffi-
2
cientlylow,theprogressiveintensificationofmotor-inducedcontractionmayberesponsibleforbothtransitions:from

static (symmetric) to motile (polarized) at a lower contractility threshold and from motile (polarized) back to static
]
h (symmetric)atahighercontractilitythreshold. Ourmodeloflamellipodialcellmotilityisbasedona1Dprojectionof
p thecomplexintra-cellulardynamicsonthedirectionoflocomotion. Intheinterestofanalyticaltransparencywealso
-
o neglectactiveprotrusionandviewadhesionaspassive. Despitetheunavoidableoversimplificationsassociatedwith
i theseassumptions,themodelreproducesquantitativelythemotilityinitiationpatterninfishkeratocytesandrevealsa
b
crucialroleplayedincellmotilitybythenonlocalfeedbackbetweenthemechanicsandthetransportofactiveagents.
.
s Apredictionofthemodelthatacrawlingcellcanstopandre-symmetrizewhencontractilityincreasessufficientlyfar
c
i beyondthemotilityinitiationthresholdstillawaitsexperimentalverification.
s
y
h
p
[
1. Introduction

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v Theabilityofcellstoself-propelisessentialformanybiologicalprocesses. Intheearlylifeofanembryo,stem
5 cellsmovetoformtissuesandorgans. Duringtheimmuneresponse,leukocytesmigratethroughcapillariestoattack
8 infections. Wound healing requires the motion of epithelial cells. While the biochemistry of such motility is rather
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wellunderstood,theunderlyingmechanicsofactivecontinuummediaisstillinthestageofdevelopment(Bray,2000;
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Mogilner,2009;CarlssonandSept,2008;JoannyandProst,2011;AdlerandGivli,2013;ZiebertandAranson,2013;
0
. GiomiandDeSimone,2014;Rechoetal.,2014;CoxandSmith,2014).
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Ataverygenerallevel,acellcanbeviewedasanelastic‘bag’whoseinteriorisseparatedfromtheexteriorbya
0
5 bi-layerlipidmembrane. Themembraneisattachedfrominsidetoathincortex-anactivemuscle-typelayermain-
1 tainingthecell’sshape. Theinteriorisfilledwithapassivemedium, thecytosol, whereallessentialcellorganelles
: are immersed. The active machinery inside the cytosol ensuring self-propulsion is contained in the cytoskeleton: a
v
i perpetuallyrenewednetworkofactinfilamentscross-linkedbymyosinmotorsthatcaninflictcontractilestresses.The
X
cytoskeletoncanbemechanicallylinkedtothecellexteriorthroughadhesionproteins(Albertsetal.,2002).
r The elementary mechanisms responsible for the steady crawling of keratocytes (flattened cells with fibroblastic
a
functions) have been identified (Abercrombie, 1980; Bell, 1984; Stossel, 1993; Bellairs, 2000). The advance starts
withprotrusiondrivenbyactivepolymerizationoftheactinnetworkinthefrontalareaofthecell(thelamellipodium)
withasimultaneousformationofadhesionclustersattheadvancingedge. Aftertheadhesionoftheprotrudingpart
ofthecellissecured,thecytoskeletoncontractsthankstotheactivityofmyosinmotors. Thiscontractionleadstothe
Emailaddress:[email protected](L.Truskinovsky)
PreprintsubmittedtoJournaloftheMechanicsandPhysicsofSolids July23,2015
detachmentattherearandlesseningoftheactinnetworkthroughde-polymerization. Alltheseactivephenomenaare
drivenbyATPhydrolysisandarehighlysynchronizedwhichallowsthecelltomovewithastableshapeandrelatively
constantvelocity(Barnhartetal.,2011).
Theinitiationofsuchmotilityrequiresapolarizationofthecellwhichisaprocessthatdiscriminatestheleading
edge from the trailing edge. The implied symmetry breaking turns a symmetric stationary configuration of a cell
into a polar motile configuration. While both contraction and protrusion contribute to steady state cell migration,
contractionappearstobethedominatingmechanismofpolarization: ithasbeenshownexperimentallythatmotility
initiationinkeratocytesmaybetriggeredbyraisingthecontractilityofmyosin(Verkhovskyetal.,1999;Csucsetal.,
2007; Lombardi et al., 2007; Yam et al., 2007; Vicente-Manzanares et al., 2009; Poincloux et al., 2011). It is also
knownthatcellsmayselfpropelbycontractiononly(Kelleretal.,2002).
Inphysicalterms,thecontraction-drivenpolarization/motilityisperformedby‘pullers’(contractileagents)while
‘pushers’(protrusiveagents)remainlargelydisabled. Somenumericalmodelssuggestthattherelativeroleof‘push-
ers’and‘pullers’incellularmotilitymaybetightlylinkedtothetasktobeperformed(SimhaandRamaswamy,2002;
SaintillanandShelley,2012)andeventothenatureofthecargo(RechoandTruskinovsky,2013). However,itisstill
notfullyunderstoodwhyincaseofkeratocytesthemotilityinitiationisprimarilycontraction-driven. Incontrastto
motilityinitiation,thereciprocalprocessofmotilityarrest isassociatedwithre-symmetrizationandsuchsymmetry
recoveryisatypicalprecursorofcelldivision(Stewartetal.,2011;Lancasteretal.,2013;LancasterandBaum,2014).
Itisnotyetclearwhetherre-symmetrizationisalsopredominantlycontraction-drivenandifyes,whetheritrequires
contractility reduction or contractility increase beyond the motility initiation threshold. It is, however, known that
somecellscanswitchfromstatictomotilestateasaresultofadecreaseinthelevelofcontractility(Liuetal.,2010;
Huretal.,2011).
A large variety of modeling approaches targeting cell motility mechanisms can be found in the literature, see
thereviewsbyRafelskiandTheriot(2004);CarlssonandSept(2008);Mogilner(2009);Wangetal.(2012). Insome
models,theactinnetworkisviewedasahighlyviscousactivefluidmovingthroughacytoplasmbygeneratinginternal
contractile stresses (Alt and Dembo, 1999; Oliver et al., 2005; Herant and Dembo, 2010; Kimpton et al., 2014). In
other models, the cytoskeleton is represented by an active gel whose polar nature is modeled in the framework of
the theory of liquid crystals (Kruse et al., 2005; Joanny et al., 2007; Julicher et al., 2007; Joanny and Prost, 2011;
Callan-Jones and Julicher, 2011). The active gel theory approach, which we basically follow in this study without
anexplicitreferencetolocalorientationalorder,wasparticularlysuccessfulinreproducingrings,asters,vorticesand
someothersubcellularstructuresobservedinvivo(DoubrovinskiandKruse,2007;SankararamanandRamaswamy,
2009;DoubrovinskiandKruse,2010;Duetal.,2012). Atsufficientlyfasttimescales, thecytoskeletoncanbealso
modeledasanactivesolid withahighlynonlinearscale-freerheology(BroederszandMacKintosh,2014;Pritchard
etal.,2014).
Variousspecificsub-elementsofthemotilitymechanismhavebeensubjectedtocarefulmechanicalstudy.Thus,it
wasshownthatinsomecasestheplasmicmembranewithanattachedcortexcanbeviewedasapassiveelasticsurface
andmodeledbyphasefieldmethodsallowingonetogosmoothlythroughtopologicaltransitions(Wangetal.,2012;
GiomiandDeSimone,2014). Inothercases,themembranemayalsoplayanactiverole,forinstance,anasymmetric
distributionofchannelsonthesurfaceofthemembranecanberesponsibleforaparticularmechanismofcellmotility
relying on variation of osmotic pressure (Stroka et al., 2014). While most models assume that the cell membrane
interacts with the exterior of the cell through passive viscous forces, active dynamics of adhesion complexes has
recently become an area of intense research driven in part by the finding of a complex dependence of the crawling
velocity on the adhesive properties of the environment (DiMilla et al., 1991; Novak et al., 2004; Deshpande et al.,
2008; Gao et al., 2011; Lin et al., 2008; Ronan et al., 2014; Lin, 2010; Ziebert and Aranson, 2013). The account
of other relevant factors, including realistic geometry, G-actin transport, Rac/Rho-regulation, etc., have led to the
development of rather comprehensive models that can already serve as powerful predictive tools (Rubinstein et al.,
2009;Wolgemuthetal.,2011;Tjhungetal.,2012;GiomiandDeSimone,2014;Barnhartetal.,2015).
The more focused problem of finding the detailed mechanism of motility initiation, is most commonly ad-
dressedintheframeworkoftheoriesemphasizingpolymerization-drivenprotrusion(MogilnerandEdelstein-Keshet,
2002; Dawes et al., 2006; Bernheim-Groswasser et al., 2005; Schreiber et al., 2010; Campas et al., 2012; Hodge
and Papadopoulos, 2012). With such emphasis on ‘pushers’, spontaneous polarization was studied by Kozlov and
Mogilner(2007);Callan-Jonesetal.(2008);Johnetal.(2008);Hawkinsetal.(2009);HawkinsandVoituriez(2010);
DoubrovinskiandKruse(2011);Blanch-MercaderandCasademunt(2013).InBanerjeeandMarchetti(2011);Ziebert
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etal.(2012)andZiebertandAranson(2013),polarizationwasinterpretedasaresultofaninhomogeneityofadhesive
interactions. YetanothergroupofauthorshavesuccessfullyarguedthatcellpolaritymaybeinducedbyaTuring-type
instability(Morietal.,2008;Altschuleretal.,2008;Vanderleietal.,2011;JilkineandEdelstein-Keshet,2011). Such
a variety of modeling approaches is a manifestation of the fact that very different mechanisms of motility initiation
areengagedincellsofdifferenttypes.
Theobservationthatcontractionmaybetheleadingfactorbehindthepolarizationofkeratocyteshasbeenbroadly
discussedintheliterature. Itwasrealizedthatactivecontractioncreatesanasymmetry-amplifyingpositivefeedback
becauseitcausesactinflowwhichinturncarriestheregulatorsofcontraction(Kruseetal.,2003;Ahmadietal.,2006;
Salbreuxetal.,2009;Rechoetal.,2013;Barnhartetal.,2015). Inconstrainedconditionssuchpositivefeedbackgen-
eratespeaksintheconcentrationofstressactivators(myosinmotors)(Boisetal.,2011;Howardetal.,2011)andthis
patterningmechanismwasusedtomodelpolarizationinducedbyangularcortexflow(Hawkinsetal.,2009,2011).
CloselyrelatedheuristicmodelsoftheKeller-Segeltype(Perthame,2008)describingsymmetrybreakingandlocal-
izationwereindependentlyproposedbyKruseandJu¨licher(2003);Calvezetal.(2010).Inallthesemodels,however,
the effect of contraction (pullers) was obscured by the account of other mechanisms, in particular, polymerization
induced protrusion (pushers), and the focus was on generation of internal flow and the resulting pattern formation,
ratherthanontheproblemofensuringsteadytranslocationofacell.
Thisshortcomingwasovercomeinmorerecentmodelsofcontraction-inducedpolarizationrelyingonsplayinsta-
bilityinanactivegel(Tjhungetal.,2012;GiomiandDeSimone,2014;Tjhungetal.,2015).Inthesemodel,however,
‘pushers’werenottheonlyplayers,inparticular,polarizationwasinducedbyalocalphasetransitionfromnon-polar
topolargel. InCallan-JonesandVoituriez(2013),themotilityinitiationwasattributedtoacontraction-inducedin-
stability in a poro-elastic active gel permeated by a solvent. Here again the non-contractile active mechanism was
involvedaswellandthereforethedomineeringroleofcontractioncouldnotbemadeexplicit.
Thegoalofthepresentpaperistofocusonthespecialroleofbarecontractioninsymmetrybreakingprocesses
by studying a minimalistic, analytically transparent model of motility initiation in a segment of an active gel. Fol-
lowingpreviouswork,weexploittheKeller-Segelmechanism,butnowinafreeboundarysetting,andshowthatthe
underlyingsymmetrybreakinginstabilityisfundamentallysimilartoanuphilldiffusionoftheCahn-Hilliardtype. In
contrasttomostpreviousstudies,ourcontractiondriventranslocationofacelliscausedexclusivelybytheinternal
flowgeneratedbymolecularmotors(pullers)andnootheractiveagentsareinvolved. Each‘puller’contributestothe
stressfieldandsimultaneouslyundergoesbiasedrandommotionresultinginanuphilldiffusionalongthecorrespond-
ingstressgradient. Inotherwordsour‘pullers’(activecross-linkers)usethecontinuumenvironment(passiveactin
network)asamediumthroughwhichtheyinteractandself-organize.
Weemphasizethatthecontractionmechanismofpolarizationandmotility(Rechoetal.,2013,2014)isconcep-
tually very close to chemotaxis, however, instead of chemical gradients, the localization and motility is ultimately
drivenbytheself-inducedmechanicalgradients. Morespecifically,thepullerspropelthepassivemediumbyinflict-
ingcontractionwhichcreatesanautocatalyticeffectsincethepullersarethemselvesadvectedbythismedium(Mayer
etal.,2010). Theinevitablebuildupofmechanicalgradientsintheseconditionsislimitedbydiffusionwhichresists
therunawayandprovidesthenegativefeedback. Afterthesymmetryofthestaticconfigurationisbrokeninthecon-
ditionswheremattercancirculate,theresultantcontraction-drivenflowensurestheperpetualrenewalofthenetwork
andthenfrictionalinteractionwiththeenvironmentallowsforthesteadytranslocationofthecellbody.
The next natural question is how such steady translocations can be halted. For instance, if motility initiation is
contraction-driven, canmotilityarrestbealsocontractiondrivenandwhatasteadilymovingcellcandoinorderto
stopandsymmetrize? Severalcomputationalmodelsprovidedanindicationthatmotilityinitiationandmotilityarrest
mayberelatedtoare-entrantbehaviorofthesamebranchofmotileregimes(KruseandJu¨licher,2003;Tjhungetal.,
2012; Recho et al., 2013; Giomi and DeSimone, 2014). To make the link between motility initiation and motility
arrestmoretransparentwestudyinthispaperananalyticallytractableproblemwhichcapturesthecomplexityofthe
underlyingphysicalphenomena. Whilemostoftheelementsoftheproposedmodel(Rechoetal.,2013,2014)have
been anticipated by some comprehensive computational approaches (e.g. Rubinstein et al., 2009), it was previously
notapparentthattheinitiationofmotility,steadytranslocationandthearrestofmotilitycanbeallcapturedinsucha
minimalsetting.
Our model of lamellipodial cell motility is based on a 1D projection of the complex intra-cellular dynamics
on the direction of locomotion. In the interest of analytical transparency, we decouple the dynamics of actin and
myosinbyassuminginfinitecompressibilityofthecross-linkedactinnetwork(Julicheretal.,2007;Rubinsteinetal.,
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2009). To ensure that the crawling cell maintains its size, we introduce a simplified cortex/osmolarity mediated
quasi-elastic interaction between the front and the back of the self-propelling fragment (Banerjee and Marchetti,
2012; Barnhart et al., 2010; Du et al., 2012; Loosley and Tang, 2012); a comparison of such mean field elasticity
withmoreconventionalbulkelasticitymodelscanbefoundinRechoandTruskinovsky(2013). Weremarkthatthe
couplingbetweenthefrontandtherearofthefragmentmayalsohaveanactiveoriginresultingfromdifferentratesof
polymerizationanddepolymerizationattheextremitiesofthelamellipodium(RechoandTruskinovsky,2013;E´tienne
etal.,2014). Inotherrespectsweneglectactiveprotrusion(pushers). Wealsoviewadhesionasfullypassive.
Despite the unavoidable oversimplifications associated with these assumptions, we show that our model repro-
ducesquantitativelythemotilityinitiationpatterninfishkeratocytesandrevealsacrucialroleplayedincellmotility
bythenonlocalfeedbackbetweenthemechanicsandthetransportofactiveagents. Italsoprovidescompellingevi-
dencethatboth,theinitiationofmotilityanditsarrest,maybefullycontrolledbytheaveragecontractilityofmotor
proteins.
Moreprecisely, weshowthattheincreaseofcontractilitybeyondawelldefinedthresholdleadstoabifurcation
fromastaticsymmetricsolutionofthegoverningsystemofequations(ofKeller-Segeltype)toanasymmetrictrav-
eling wave (TW) solution corresponding to steadily moving cells. While several TW regimes may be available at
the same value of parameters, we show that stable TW solutions localize motors at the trailing edge of the cell in
agreement with observations (Verkhovsky et al., 1999; Csucs et al., 2007; Lombardi et al., 2007; Yam et al., 2007;
Vicente-Manzanaresetal.,2009;Poinclouxetal.,2011). Moreover,weshowthatifadhesionwiththeextra-cellular
substrate is sufficiently low, the increase of motor-induced contraction may induce transition from the steady state
TWsolutionbacktoastaticsolution. Thisre-symmetrizationtransition,leadingtothemotilityarrest,canbedirectly
associatedwiththebehaviorofkeratocytespriortocelldivisionandourmodelshowsthatsuchre-entrantbehavior
canbeensuredby‘pullers’withoutanyengagementofeitheractiveprotrusionorliquidcrystalelasticity.
Thepaperisorganizedasfollows. InSection2,wepresentadiscrete“modelofamodel”whichconveysthemain
ideas of our approach in the simplest form. In Section 3, we develop a continuum analogue of the discrete model,
study its mathematical structure and pose the problem of finding the whole set of TW solutions incorporating both
staticandmotileregimes. InSection4,allstaticsolutionsoftheTWproblemarefoundanalytically. InSection5,we
studythefinestructureofmultiplebifurcationsproducingmotilesolutionsfromthestaticonesandidentifyparametric
regimeswhenthesebifurcationsbecomere-entrant. InSection6,weinvestigatenumericallytheinitialvalueproblem
whichallowsustoqualifysomeofthemotileTWsolutionsasattractors. Thereconstructionthebackgroundturnover
of actin, which takes place in our model without active protrusion at the leading edge, is discussed in Section 7.
In Section 8, we demonstrate that our model can quantitatively match the experiments carried on keratocytes. The
lastSectionhighlightsourmainconclusionsandmentionssomeoftheunsolvedproblems;threeappendicescontain
materialoftechnicalnature.
Some of the results of this paper have been previously announced in two pre-publications (Recho et al., 2013,
2014) but without any details. In addition to providing a necessary background for Recho et al. (2013, 2014), here
we develop a new discrete model, investigate the nonlocal nature of the coupling between mechanics and diffusion
ofactiveagents, giveathoroughanalysisofthestaticregimes, studythebifurcationpointsbyusingtheLyapunov-
Schmidt reduction technique, investigate the non-steady problem numerically, generalize the model to account for
nonlineardependenceofcontractilestressonmotorconcentrationandprovideadetailedquantitativecomparisonof
themodelwithexperiment.
2. Thediscretemodel
Ourpointofdepartureisaconceptualdiscretemodelelucidatingthemechanismofcontraction-drivencrawling
and emphasizing the role of symmetry breaking in achieving the state of steady self propulsion. This ”model of a
model”allowsustoclarifytheroleofdifferentcomponentsofthecontraction-dominatedmotilitymachineryandlink
theproposedmechanismwiththepreviousworkonoptimizationofthecrawlingstrokeirrespectiveoftheunderlying
microscopicprocesses(e.g.DeSimoneandTatone,2012;Nosellietal.,2013). Itdoesnot,however,addressdirectly
themainissuesofmotilityinitiationandmotilityarrestthatrequiremoreelaborateconstructions.
Recallthatincrawlingcells,the‘motorpart’containingcontractingcytoskeleton(lamellipodium),isathinactive
layer located close to the leading edge of the cell, see Fig. 1. We assume that all mechanical action originates in
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lamellipodiumandthatfromthemechanicalviewpointtherestofthecell,includingthenucleus,canbeinterpretedas
cargo.Themaintaskwillbetodevelopamodeloffreelymovinglamellipodiumwhichweschematizeasasegmentof
activegelinviscouscontactwitharigidbackground.Theactinnetworkinsidethegeliscontractedbymyosinmotors
whichleadstoaninternalflowopposedbytheviscousinteractionwiththebackground. Theunidirectionalmotionin
alayeradjacenttothebackgroundthatultimatelypropelsthecellisaresultoftheasymmetryofcontraction.
Figure1: Conceptualdiscretemodelofthemotilitymechanisminacrawlingkeratocytecell.
Atoymodelelucidatingthispointinvolvesthreerigidblocksofsizel placedinafrictionalcontactwitharigid
b
support, characterized by the viscous drag coefficient ξ. The neighboring blocks are connected by active pullers
(forcedipoles)exertingcontractileforces. Theessentiallongrangeinteractionsrepresentingglobalvolumeconstraint
(due to passive elastic structures and osmotic effects, see Section 3) are modeled by a linear spring with stiffness
k connecting the first and the last block. To regularize the problem we place in parallel with contractile elements
additionaldashpotscharacterizedbytheviscositycoefficientη. Intheabsenceofinertia,wecanthenwritetheforce
balanceequationsintheform
−l ξx˙ +kx3−x1−L0 +χ −ηx˙1−x˙2 =0
−l ξbx˙ 1−χ +χL0 −ηx˙2−1x˙1 −ηx˙l2b−x˙3 =0 (1)
−bl ξx˙2 −k1x3−x12−L0 −χlb −ηx˙3−x˙lb2 =0,
b 3 L0 2 lb
where x (t),x (t),x (t)arethecurrentpositionsoftheblocksand L isthereferencelengthofalinearspring. This
1 2 3 0
springdescribesthemembrane-cortex‘bag’aroundthelamellipodiumallowingtheinhomogeneouscontractiontobe
transformedintoaprotrudingforce. Weassumethatpolarizationhasalreadytakenplaceandthereforethecontractile
forcedipolesχ ≥ 0andχ ≥ 0actingbetweenthetwopairsofblocksarenotthesameχ (cid:44) χ . Thepolarization
1 2 1 2
itselfrequiresadditionalconstructsandwillbeaddressedlater.
System(1)canberewrittenasthreedecoupledequationsforthelengthofouractivesegmentL(t)= x (t)−x (t),
3 1
itsgeometriccenterG(t)=(x (t)+x (t))/2andthepositionofacentralblockx (t)representingtheinternalflow:
3 1 2
−l ξ(1+l2/l2)L˙ =χ +χ +2k(L/L −1)
b 0 b 1 2 0
2l ξ(1+3l2/l2)G˙ =χ −χ (2)
b 0 b 1 2
−l ξ(1+3l2/l2)x˙ =χ −χ
b 0 b 2 1 2
(cid:112)
where l = η/ξ is the hydrodynamic length scale which will ultimately play the role of a regularizing parameter.
0
Thefirstequationshowsthatthelengthisconvergingtoasteadystatevalue:
L = L (cid:2)1−(χ +χ )/(2k)(cid:3).
∞ 0 1 2
Noticethatinordertoavoidthecollapseofthelayerduetocontraction,itisnecessarytoensurethatthespringhas
sufficientlylargestiffnessk>(χ +χ )/2.WealsoobservethatindependentlyofthevalueoftheevolvinglengthL(t),
1 2
thevelocityofthegeometricalcenterofourtrainofblocksV isalwaysthesame
χ −χ
V =G˙ = 1 2 . (3)
2l ξ(1+3l2/l2)
b 0 b
Onecanseethatthesystemcanmoveasawholeonlyifχ (cid:44) χ ,whichemphasizesthecrucialroleformotilityof
1 2
thepolarizationandtheensuinginhomogeneityofcontraction.
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Weobservethatthemiddleblockmovesinthedirectionoppositetothemotionofthecenterofthesystemwith
a constant velocity x˙ = −2V. Therefore, it takes a finite time ∼ L /(3V) for the central block to collide with the
2 ∞
Figure2: (a)Schematicrepresentationofthemotionofindividualparticles(blocks)formingthemotorpartofacrawlerinasteadystateregime
(threeparticlecase). Trajectoriesinspacetimecoordinatesoftheparticlesx1 (magenta,OBCEF), x2 (green,ABDEG)andx3 (red,ACDFG);
dashedlinesshowthejumppartsofthecrawlingcycle. Continuousflowshavetoovercomefrictionwhilethejumpsareassumedtobefriction
free.(b)Aclosedloopconstitutingonefullstrokeintheparameterspace(x2−x1,x3−x2).Thetimeofonefullstroke(AtoG)isTs=L∞/Vand
thedistancetraveledbythecrawlerperstrokeisVTs=L∞.
block at the rear. Beyond this time, system (1) formally collapses and additional assumptions are needed to extend
the dynamics beyond the collision point. The origin of the problem is our focus on the layer adjacent to the rigid
backgroundandtheneglectoftheglobalflowofactin.
To make the model more adequate we have to take into consideration that while the flow of F-actin (polymer-
izedorfilamentousactin)iscontinuousalongthecontactsurface,thecytoskeletalnetworkdisintegratesintoG-actin
(unpolymerizedmonomers)atthetrailingedgeandreintegratesfromtheavailableG-actinattheleadingedge. The
polymerizationinduceddepletionofG-actinattheleadingedgeiscompensatedbythediffusivecounter-flowofactin
monomersfromthebackofthecelltoitsfront. Thiscounter-flowcannotbedescribeddirectlyinthe1Dsetting.
Itcanbemodeled,however,inanindirectwaybymassandmomentumpreservingperiodicboundaryconditions
allowing F-actin to disappear at the rear and reappear in the front. This situation is rather typical for continuum
mechanicswhereunresolvedspatialscalesareoftenreplacedbybalance-law-preservingjump/singularityconditions
asinthecaseofshockwaves,cracktipsandboundarylayers.
Morespecifically,toaccountforglobalcirculation(turnover)ofthecytoskeletoninaonedimensionalsetting,we
assumethatthereisasingularsourceofactinatthefrontofthecellthatiscompensatedbytheequivalentsingular
masssinkofactinattherearofthecell. Thisassumptionallowsustoclosethetreadmillingcycle,eventhoughthe
details of the discontinuous part of the cycle, involving both the polymerization reaction and the diffusive transport
ofmonomers,arenotexplicitlyresolvedinthemodel. WeessentiallypostulatethatthereisapoolofG-actinwhich
is replenished as fast as it is depleted and that the resulting reverse flow of actin is synchronous with the direct
flow. Under these assumptions the reverse flow is viewed as passive and and is assumed to be driven exclusively
by inhomogeneous contraction. In particular, we neglect active propulsion on free boundaries due to growth and
lesseningofthenetwork.
Wedescribetheseprocessesinourtoymodelbyassumingthepossibilityofcreationanddestructionoftheblocks.
Ourgoalistoaccountforthefactthatactinpolymerizesattheleadingofthecell(whereblocksareassembled)and
depolymerizes at the trailing edge of the cell (where blocks are disassembled). We offer two interpretations of the
underlyingcontinuousprocessintermsofdiscreteblocksemphasizingdifferentsidesofsuchpassivetreadmilling.
Inafirstinterpretation,weassumethatasaresultofeachcollisionablockattherearisinstantaneouslyremoved
fromthechainandatthesametimeanidenticalblockisaddedatthefront. Inotherwords, each(equilibrium)de-
polymerizationeventattherearismatchedbyan(equilibrium)polymerizationeventatthefront. Hereweimplicitly
refertotheexistenceofastationarygradientofchemicalpotentialofactinmonomersandofalargepoolofmonomers
ready to be added to the network at the front as soon as one of them is released at the rear. The ’instantaneous’
reappearanceofthedisappearingblocksshouldbeunderstoodasameantomodeltheoverallcontinuityoftheflow.
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The structure of the resulting stroke in the t,x plane and in the x − x ,x − x plane is shown in Fig. 2. One
2 1 3 2
can see that each block maintains its identity through the whole cycle and that its trajectory involves a succession
ofcontinuoussegmentsdescribedby(1)thatareinterruptedbyinstantaneousfrictionlessjumpsfromthereartothe
front. Notice that in this interpretation the blocks can change order and the condition x < x < x is not always
1 2 3
satisfied. Forinstance,whentheblocksx andx collideatpointB,theblockx disappearsattheback(pointB)and
1 2 1
reappearsatthefront(pointC)aheadoftheblockx . Thisjumpmimicsthefrictionlesspartofthetreadmillingcycle.
3
Similarly, when when the block x collides with the block x at point D, the latter reappears at the point E ahead
3 2
of the block x . This interpretation is attractive because it allows one to trace the trajectories of the blocks through
1
subsequenttreadmillingcycles. Itis,however,abitmisleadingbecauseinrealitytheblockthatdisappearsattheback
andtheblockwhichinstantaneouslyreappearsatthefrontaredefinitelynotthesameeventhoughtheyareidentical.
According to a second interpretation, illustrated in Fig. 3, the middle block is the only one to undergo cycling
motion. As a result, the ordering x < x < x is always preserved and the distances between the first two blocks
1 2 3
l = x −x andthelasttwoblocksl = x −x canbeonlypositive. Wecanalternativelysaythatnowthenotations
1 2 1 2 3 2
x ,x ,x indicate positions only and can refer to different blocks in different times. In this interpretation, when the
1 2 3
middleblockshitstherearone,itisthemiddleblockthatgetsrecycledtothefrontwhiletherearonekeepsmoving
continuously.
Figure3: Schematicrepresentationofthecontinuous(α,β)andthejump(β,α)partofthecrawlingstroke. Thezeroarealoopinl1,l2 plane
illustratingthestrokeisshownin(a).Theloopisnotsymmetricbecausecontinuousflowhavetoovercomefrictionwhilethejumpsareassumed
tobefrictionfree. The‘tankthread’analogyin(b)isnotfullyadequatebecausethe‘departing’blocksatpointβ,thatenterthepoolofactin
monomers,andthearrivingblocksatpointα,thataresimultaneouslytakenfromthesamepool,arenotthesame.
In coordinates (l ,l ) the cycle collapses on a single line, which is traveled continuously in one direction and
1 2
discontinuously in the other direction, see Fig. 3(a). Notice that the internal parameters l (t) and l (t) undergo a
1 2
periodic sequence of extensions and contractions which resemble the mechanism propelling the swimming sheet
(Taylor,1951)anditscrawlinganalogue(DeSimoneandTatone,2012). Themaindifferenceisthatinourcasethe
propulsionisachievedbecauseoftheasymmetryoffrictionforcesactinginthedifferentphasesofthestroke. More
specifically, we assume that during the continuous phase of the cycle the blocks move with friction (polymerized
filamentsexperienceeffectivedragtransmittedbyfocalcontacts),whileduringthediscontinuousphasethedissipation
(associated with reaction and diffusion) can be neglected. The situation is remotely analogous to that of a rotating
‘tank tread’, see Fig. 3(b), even though in reality the disappearing block and the appearing block are not the same.
This interpretation is closer to the physical picture where the points of the membrane (cortex) represented by two
side blocks move with a constant speed ensuring that the cell maintains its length. We reiterate that both discrete
interpretationsareschematicandwillbebackedlaterinthepaperbyanappropriatecontinuummodeling.
Sincetheobtainedexpressionforvelocity(3)remainsfiniteinthelimitl /l →0itappearsthatthedashpotsplay
0 b
aredundantroleinthismodelandcanbedropped. Toillustratetheroleofthedashpotswenowconsiderthecaseof
N coupledblocks. Then,theforcebalanceforthecentralblocks j∈[2,N−1]reads
−l ξx˙ −χ +χ −ηx˙j−x˙j−1 −ηx˙j−x˙j+1 =0.
b j j−1 j l l
b b
7
Thissystemofequationscanbewritteninthematrixform,
Tx˙ =b, (4)
wherewedenotedbyx˙ theunknownvectorx˙ ,...,x˙ . Thetri-diagonalmatrix
1 N
T= −(20001+ ll0b22) −(2.100.+. llb022) .0101.. −(2.001.+. llb022) −(20010+ lb2) 
l2
0
describestheviscouscouplingandfrictionalinteractionwiththebackgroundwhilethevector
 
b= ξllb02  χ−Nχ−χ11N+−χ−12σσ−−...00χχ−−2Nξ−ξlllblb10202xx˙˙1N 
with σ = −k(x − x −L )/L carries the information about the active forcing, the mean field type elasticity and
0 N 1 0 0
the boundary layer effects. To find the solution x˙, we need to invert the matrix T and then solve a system of two
coupledlinearequationsx˙ =(Rb) andx˙ =(Rb) whereR=T−1. ThecomponentsofthematrixRcanbefound
1 1 N N
explicitly(Meurant,1992)
cosh((N+1− j−i)λ)−cosh((N+1−|j−i|)λ)
R = ,
i,j 2sinh(λ)sinh((N+1)λ)
whereλ=arccosh(1+l2/(2l2)).Knowingthe‘velocityfield’,wecannowcomputethesteadystatevalueofthelength
b 0
L∞ = L01− (cid:80)(cid:80)NjN=−1−11ccoosshh((λλ((jj−−NN//22))))χkj.
j=1
From this formula we see again that a finite stiffness is necessary to prevent the collapse of the system under the
actionofcontractilestresses: assumingforinstancethatχ =χweobtainthelowboundfortheadmissibleelasticity
i
modulusk>χ.
ThesteadyvelocityV =(x˙ +x˙ )/2ofthegeometricalcenterofthesystemcanbealsocomputedexplicitly
N 1
l (cid:80)N−1sinh(λ(j−N/2))χ
V =− b j=1 j.
2ηsinh(λN/2)
WhenN iseven,bydenotingM = N/2,wecanrewritethisexpressionintheform
V =−lb(cid:80)Mj=−11sinh(jλ)(χM+j−χM−j).
2ηsinh(λM)
8
fromwhereitisclearthat(asinthecaseofthreeblocks)thesymmetryofthevectorχwithrespecttothecentermust
bebrokenforthesystemtobeabletoself-propel.
Ifwenowformallydropthedashpotsbyassumingthatl = 0weobtainsimilarexpressionsforthevelocityand
0
forthesteadystatelengthasinthethreeblock(N =3)case
χ −χ (cid:18) χ +χ (cid:19)
V = N−1 1,L = L 1− 1 N−1 . (5)
2ξl ∞ 0 2k
b
Thereasonbehindthissimilarityisthat,inthislimit,the‘flow’fullylocalizesinthetwoboundaryelements,theonly
onespresentinthecase N = 3. Moreprecisely,thesolutionofthediscreteproblemdependssingularlyontheratio
l2/l2 and becomes progressively more concentrated around the boundary elements as l2/l2 → ∞. Such localization
b 0 b 0
presentsacertainanalyticalproblemifweconsiderthecontinuumlimitwhen N → ∞andl → 0while Nl → L,
b b
whereListhecontinuumlengthoftheself-propellingsegment. Indeed,inthislimitthesizeofboundarylayerstends
tozeroandthediscretesolutionconvergestoadistribution.Theviscosity,introducingalengthscalel ,isthusneeded
0
topreservetheregularityofsolutionsinthecontinuumlimit.
Observealsothatthelimitsl →0(droppingdashpots)andl →0(continuumapproximation)donotcommute.
0 b
Forinstance,ifwechoosein(5)themotordistributionwithallχ =0exceptforoneχ =χ∗ >0weobtainV =0for
i 2
anyvalueofl ,inparticular,whenl →0westillhaveV →0. Ifinsteadwefirstperformthecontinuumlimitwhile
b b
keepingl finiteweobtain
0 L∞ = L01− (cid:82)0L∞co2skhl[(sxin−hL[L∞/2/()2/ll0])]χ(x)dx (6)
0 ∞ 0
and
(cid:82)L∞sinh[(x−L /2)/l ]χ(x)dx
V =− 0 ∞ 0 . (7)
2ηsinh[L /(2l )]
∞ 0
Ifwenowtakeadistributionofmotorsχ(x) = χ∗δ whereδ istheDiracmassat x = 0, whichcanbeviewedasa
0 0
continuumanalogofthediscretedistributionconsideredabove,weobtainthatV =χ∗/(2l2ξ).Theninthelimitl →0
0 0
weobtainthatV → ∞whichisinconflictwithourpreviousassertionthatV = 0,obtainedwhentheorderoflimits
wasreversed. Assumenowthatl ∼ N−1 andhencel2/l2 ∼ 1/(ηN2). Onecanseethatthecrossoverscalingη ∼ N−2
b b 0
separatesthetwonon-commutinglimitingregimes. Forl2/l2 →∞(whichisadimensionlessversionofη(cid:28) N−2)the
b 0
internalflowlocalizesintheboundarylayerswhosethicknessdisappearswhenη→0;whenwedroppedthedashpots
inthethreeelementmodelwecouldnotdetectthislocalizationbecausethetwoboundarylinksweretheonlyones
presentinthesystem. Intheotherlimitl2/l2 → 0(dimensionlessversionofη (cid:29) N−2)theviscositydominatesthe
b 0
dynamicsandtheinternalflowbecomesuniform.
In the next sections the formulas (6) and (7) will be obtained directly from the continuum model. We will also
see more clearly how the introduction of the viscosity-related internal length scale and the associated nonlocality
regularizesthecontinuummodelwhichotherwisehasonlysingularsolutions.
3. Thecontinuummodel
Wemodelthelamellipodiumasaonedimensionalcontinuumlayerinfrictionalcontactwitharigidbackground,
seeFig.4. Assumingthatthedragisviscousandneglectinginertiawecanwritetheforcebalanceintheform
∂ σ=ξv, (8)
x
whereσ(x,t)istheaxialstressandv(x,t)isthevelocityofthecytoskeleton(actinnetwork). Eq. (8)isthecontinuous
analogofthesystem(4)inthediscreteproblem.
Asinthediscretemodel,wedenotedbyξthecoefficientofviscousdrag. Suchrepresentationofactiveadhesion
is usual in the context of cell motility (Rubinstein et al., 2009; Larripa and Mogilner, 2006; Julicher et al., 2007;
Shaoetal.,2010;DoubrovinskiandKruse,2011;Hawkinsetal.,2011). Itimpliesthatthetime-averagedshearstress
generated by constantly engaging and disengaging focal adhesions is proportional to the velocity of the retrograde
flow,seeTawadaandSekimoto(1991)foramicroscopicjustification. Thereisevidence(bothexperimental(Gardel
9
Figure4: Schematicrepresentationofacontinuummodelsimulatinglamellopodialcontraction-drivencrawling.
etal.,2008,2010;Mogilner,2009;Boisetal.,2011;SchwarzandGardel,2012)andtheoretical(DiMillaetal.,1991;
Mietal.,2007))thatthisassumptiondescribesthebehavioroffocaladhesionsaccuratelyonlywhentheretrograde
flowissufficientlyslow. Thebehaviorofadhesionstrengthinthebroaderrangeofvelocitiesisbiphasicandsincewe
neglect this effect, we potentially misrepresent sufficiently fast dynamics. Observe though that for both keratocytes
andPtK1cellstherateoflamellaractinretrogradeflowvariesfrom5to30nm.s−1 inusualexperimentalconditions
(Schwarz and Gardel, 2012) and in this range a direct proportionality relationship between traction stress and actin
retrogradeflowhasbeenconfirmedexperimentally(Gardeletal.,2008;Fournieretal.,2010;Barnhartetal.,2011).
Thecharacteristicvelocityoftheflowinourproblemis20nm.s−1 whichfallswellintotheaforementionedinterval
wherethebiphasicbehaviorcanbeneglected.
FollowingKruseetal.(2006);Julicheretal.(2007);Boisetal.(2011)andHowardetal.(2011),weassumethat
thecytoskeletonisaviscousgelwithactivepre-stress. Weneglectthebulkelasticstressesthatrelaxoveratimescale
of1−10s(Rubinsteinetal.,2009;Wottawahetal.,2005;Koleetal.,2005;Panorchanetal.,2006;Mofrad,2009;
RechoandTruskinovsky,2013)whichismuchshorterthancharacteristictimescaleofmotilityexperiments(hours).
Wecanthendescribetheconstitutivebehaviorofthegelintheform
σ=η∂ v+χc, (9)
x
where η is the bulk viscosity, c(x,t) is the mass concentration of motors and χ > 0 is a contractile pre-stress (per
motor) representing internal activity. The constitutive relation (9) generalizes the parallel bundling of dashpots and
contractileunitsinthediscretemodel. Theimportantnewelementisthatthestrengthofthecontractileelementsmay
nowvaryinbothspaceandtime.
Inthediscretemodeltheconcentrationofmotorscwasagivenasafunctionofx. Toobtainamoreselfconsistent
descriptionweassumethatthefunctionc(x,t)satisfiesaconvection-diffusionequation(Rubinsteinetal.,2009;Bois
etal.,2011;Barnhartetal.,2011;Wolgemuthetal.,2011;Hawkinsetal.,2011)
∂c+∂ (cv)= D∂ c, (10)
t x xx
whereDisthediffusioncoefficient.Behind(10)istheassumptionthatmyosinmotors,activelycross-linkingtheactin
network,areadvectedbythenetworkflowandcanalsodiffusewhichaccountsforthermalfluctuations.
To justify this model, consider a simple mixture model with two species representing attached and detached
motors. Theattachedmotorsareadvectingwiththevelocityofactinfilamentsandcandetach. Thedetachedmotors
arefreelydiffusing,andcanalsoattach. Supposethattheattachment-detachmentprocesscanbedescribedbyafirst
orderkineticequation. Thenthesystemofequationsgoverningtheevolutionoftheconcentrationsofattachedc and
a
detachedc motorscanbewrittenas:
d
∂tca+∂x(cav)=koncd−koffca
∂tcd−D˜∂xxcd =koffca−koncd
wherekon andkoff arethechemicalratesofattachmentanddetachmentandD˜ isthediffusioncoefficientofdetached
motors in the cytosol. Now suppose that the attachment-detachment process is chemically equilibrated and hence
10