Phys. Biol. 12 (2015) 035002 doi:10.1088/1478-3975/12/3/035002

PAPER

Model for adhesion clutch explains biphasic relationship betweenactin flow and traction at the cell leading edge

ErinMCraig1, Jonathan Stricker2,Margaret Gardel2 andAlexMogilner3,4

1 CentralWashingtonUniversity, Department of Physics, 400 E.UniversityWay, Ellensburg,WA98926-7422,USA2 Institute for Biophysical Dynamics, James Frank Institute, andDepartment of Physics, University of Chicago, Chicago, IL 60637,USA3 Courant Institute andDepartment of Biology, NewYorkUniversity, 251Mercer Street, NewYork, NY 10012,USA4 Correspondence: Courant Institute, NewYorkUniversity, NewYork, NY 10012,USA

E-mail:mogilner@cims.nyu.edu

Keywords: cell adhesion, actin flow, traction force

AbstractCellmotility relies on the continuous reorganization of a dynamic actin–myosin–adhesion network atthe leading edge of the cell, in order to generate protrusion at the leading edge and traction betweenthe cell and its external environment.We analyze experimentallymeasured spatial distributions ofactinflow, traction force,myosin density, and adhesion density in control and pharmacologicallyperturbed epithelial cells in order to develop amechanicalmodel of the actin–adhesion–myosin self-organization at the leading edge. Amodel inwhich the F-actin network is treated as a viscous gel, andadhesion clutch engagement is strengthened bymyosin but weakened by actinflow, can explain themeasuredmolecular distributions and correctly predict the spatial distributions of the actin flow andtraction stress.We test themodel by comparing its predictions withmeasurements of the actinflowand traction stress in cells with fast and slow actin polymerization rates. Themodel predicts how thelocation of the lamellipodium–lamellumboundary depends on the actin viscosity and adhesionstrength. Themodel further predicts that the location of the lamellipodium–lamellumboundary isnot very sensitive to the level ofmyosin contraction.

1. Introduction

Cell migration results from a cycle of protrusion,adhesion, and contraction. The cell leading edge isprotruded by the polymerizing front of a broad andflat network of F-actin filaments called the lamellipo-dium [1]. Near the leading edge, the polymerization ofthe lamellipodium only partially translates into pro-trusion, because the network also undergoes retro-grade motion toward the cell center. The retrogradeflow of actin is slower in the lamellum (actomyosinnetwork that begins 2–4 μm away from the leadingedge).While the faster retrogradeflownear the leadingedge is driven mainly by growing actin filamentspushing against the membrane [2], the slower flow inthe lamellum is caused by myosin II-generated con-traction of the actin network [2] (figure 1). In additionto driving the actin network backward, actomyosincontraction pulls the cell rear forward and is alsoinvolved in maturation of adhesion sites [3, 4]. Theadhesion sites are initiated in a myosin force-

independent manner within the lamellipodia as verydynamic nascent adhesions [5]. The nascent adhesionsbecome partly stabilized, forming dot-like focal com-plexes, and then grow, elongate, and become furtherstabilized producing elongated mature focal adhe-sions [6].

Adhesions connect the actin network to the extra-cellular matrix (ECM) [7] and transmit stresses gener-ated in the actin network by the actin pushing againstthe membrane and by the myosin contraction to theECM. These traction stresses generally orient away fromthe leading edge, in the same direction as the flow ofactin, allowing the polymerizing actin network to exertprotrusive force on the leading edge membrane [8, 9].The mechanical role of the adhesions is often likened toa molecular clutch, which slows actin retrograde flowand allows the actin polymerization to contribute toleading edge protrusion [10–12]. Thus, mechanicalproperties of the adhesions are crucial for cellmotility.

While the mechanics of single adhesive moleculesis being actively investigated [13], experimental

RECEIVED

7November 2014

REVISED

3April 2015

ACCEPTED FOR PUBLICATION

7April 2015

PUBLISHED

13May 2015

© 2015 IOPPublishing Ltd

understanding of adhesion complex mechanics is verypoor. So far, the only, relatively crude, way to estimatethe adhesive strength is to measure simultaneously theactin flow and traction stress [14, 15], and to interpretthe ratio of the stress and flow speed as an effectiveadhesive viscous drag. To fill this gap in our knowl-edge, recent modeling studies [16–19] borrowed ideasfrom the theories of molecular friction [20] and simu-lated adhesions as sticky springs dynamically bindingand unbinding to the deformable surfaces that thesesprings connect. Even these simple models revealed agreat wealth of mechanics of the dynamic adhesion,including, among other possibilities, biphasic andstick–slip force–velocity relations, so that the adhesivestrength can be great at slow actin flow rate and weakat faster rates. These models focused largely on thedynamics and mechanics of individual adhesions. Onthe other hand, a number of other models consideredcoupling of adhesion-generated forces to active con-tractile forces generated by myosin and passive resis-tance of the lamellipodial actin network todeformations [21–23] assuming constant uniformadhesion strength across the cell. Two recent model-ing studies [24, 25] went further: in Welf et al [24], abalance of four sources of stress—originating fromcontraction, membrane ‘recoil’, adhesion clutch andretrograde flow—was considered. Notably, the adhe-sion force had viscous character—it was proportionalto the product of the density of nascent adhesions andthe retrograde flow rate, and the number of engagedadhesions was a decreasing functions of the forceapplied to them. In addition, positive feedbacks fromprotrusion to adhesion, and from tension on theclutch adhesions to myosin, were introduced. In

another study [25], adhesions were modeled as elasticsprings between the actin network of the cell and thedeformable substrate, which disengage at a criticalforce. The focus of these two studies was not on thespatial self-organization of the adhesions, actin flow,myosin and traction force. Complex and stochastictemporal leading edge dynamics was investigated in[24], while myosin and stresses and flow in actin net-work were not modeled in [25], besides, the very spe-cial case of wide and fast keratocyte’s lamellipodiumwas addressed in [25].

A recent paper by Shemesh et al [26] consideredthe interactions of the F-actin network and force-sen-sitive adhesion dynamics in a coordinated, spatiallyexplicit way for the first time. In [26], the actin net-work ismodeled as an elastic gel. Adhesions detach at arate that increases with local force, grow if the force isabove a certain force threshold and shrink if the forceis below another force threshold. These dynamics leadto separation between the leading edge and a band ofadhesions, because adhesions too close to the frontexperience so much local force that the gel momenta-rily detaches from the adhesion, switching from asticking mode (bound to adhesion) to a slip mode. Inthe slip mode, the adhesions do not apply traction onthe gel, but they re-stick at some rate. The adhesionsnear the leading edge experience more time in the slipmode, but the average traction forces over time causethem to grow slowly. At a greater distance, adhesionsgrow faster because they share the applied force, whichallows them to grow but rarely induces the stick–sliptransition. At yet greater distance, adhesions arescreened from the elastic forces and decay. Thus, themodel predicted that the band of the stable adhesions

Figure 1. Schematic ofmodel. The viscous actin network undergoes the retrograde flowpushed by themembrane tension and pulled bythemyosin-generated stress balanced by the traction stress. Bothmyosin and adhesion units engage/disengage with/from the actinnetwork, the latter with the rates affected by theflow rates andmyosin forces.

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Phys. Biol. 12 (2015) 035002 EMCraig et al

develops at some distance from the leading edge, andthe authors of [26] proposed, following experimentalstudy [27], that this band demarcates the lamellipo-dium–lamellumboundary.

Simultaneous measurements of the traction stressand actin flow rate in Ptk1 epithelial cells [14] revealedthat this adhesion band at the lamellipodium–lamel-lum boundary also is slightly distal to a peak in thetraction stress, which is smaller both at the front, in thelamellpodium, and at the rear, in the lamellum(figure 2). On the other hand, the actin flow rate

decreases away from the leading edge (figure 2),revealing a biphasic spatial relationship between F-actin retrograde flow speed and traction stress. Tounderstand this complex spatial organization of theactin, myosin, and adhesions underlying the adhesionclutch and the resultant regions of inverse and directcorrelation between actin flow and traction, we buildon the ideas of [24, 26] and develop a continuousmechanicalmodel for coupled actin flow,myosin con-tractility, adhesion dynamics and traction stress.

Such a model is necessary for the following rea-sons: first, [24–26] did not focus on the spatial

Figure 2. Sample experimental data fromPtk1 cells. (a) Left: actin flow vector field overlaid on paxillin intensity image for a control Ptk1cell. Right: traction vector field overlaid on same image. (b) Actinflow speed and tractionmagnitude as a function of distance from theleading edge for a sample control cell. (c) Fluorescent images of paxillin intensity (left) andmyosin intensity (right) for sample controlcells. (d) Intensity as a function of distance from the leading edge for the sample cells in (c). Bars, 10microns.

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Phys. Biol. 12 (2015) 035002 EMCraig et al

distribution of the actin flow and traction stress. Sec-ond, [26] assumed that the actin network is mechani-cally elastic (studies [24, 25] did not explicitly modelthe spatial distribution of stress in the actin network).However, at characteristic values of the lamellipodialwidth and actin flow speed in slow motile epithelialcells, it would take on the time scale of minutes for anactin filament to traverse the lamellipodial width,while actin filaments turn over many times on thistime scale [28]. This turnover makes the elastic defor-mations dissipate and the actin network becomesmechanically viscous [29, 30]. (We also demonstratein the appendix that amodel based on elastic deforma-tions predicts an actin flow distribution qualitativelydifferent from that observed.) Third, [26] did not con-sidermyosin contraction in their work, which was jus-tified by the idea that myosin does not influence thelamellipodial dynamics. Here we explicitly considermyosin dynamics; its weak (but not negligible) role inthe lamellipodium is one of the predictions of themodel. Finally, the continuous and deterministiccharacter of our model complements the discrete,computational and stochastic nature of the study [26],providing further insight into how key mechanicalparameters determine lamellipodial–lamellar geo-metry, forces andmovements.

We demonstrate that a model in which adhesionclutch engagement is strengthened by myosin forcebut weakened by actin flow (or traction force) quan-titatively explains the experimentally observedbiphasic traction force distribution and decreasingactin flow (figure 2) [14], and also qualitativelyreproduces measured distributions of myosin andadhesion density (figure 2). The model further pre-dicts that, for cells with slow uniform retrograde flowspeed across the lamella, the traction peak is locatedat the leading edge and the lamellar actin flow speed isrelatively insensitive to myosin-generated stress. Weconfirm these predictions by measuring actin flowand traction stress in U2OS epithelial cells, whichexhibit relatively slow uniform flow in contrast to thesteeply decaying retrograde flow speed distributionobserved in Ptk1 cells.We treated the U2OS cells withvarying concentrations of the Rho-kinase inhibitorY-27632 compound, which leads to reduced myosinforces, and we observe relatively unchanged actinflow speed in the lamella over a broad range of totalmyosin force [31]. The insensitivity of lamellar actinflow to myosin-generated stress in U2OS cells isexplained in our model by the myosin-dependentadhesion strengthening: greater myosin pullingincreases the adhesion strength, and so the flow(which is the ratio of the former and the latter) doesnot change. We also discuss the predicted depen-dence of the lamellipodial width onmechanical para-meters of the cell.

2.Model

Figure 1 illustrates the physics of our model of theleading edge adhesion clutch system. Because therelevant forces and flows are directed radially inward,we consider a one-dimensional (1D) description ofthe system, where the variable x denotes the inwarddistance from the cell leading edge. We treat the F-actin network as an isotropic gel [32], and we assumethat internal network stresses are viscous. There isevidence in biophysical literature [30, 31] that on timescales longer than a few seconds, the actin gel, evencross-linked heavily, is mostly viscous. This is becauseactin-binding proteins attach to and detach from actinfilaments on the second scale, and over a few secondsmost of the elastic bending energy dissipates. Thecharacteristic rate of the actin flow is ten nm persecond, and over a few seconds, the actin displacementis less than one-hundred nm. This is not only smallerthan the scale of severalmicrons over which character-istic features of actin flow and traction are observed,but it is even smaller than the usualfilament size. Thus,the actin network has to be modeled as a viscous orviscoelastic fluid (with flowing viscous and elasticelements in series). Also, in the appendix, we demon-strate that the assumption of an elastic nature of theactin network predicts an actin flow profile qualita-tively different from that observed.

2.1.Model equationsThere are three forces that are balanced throughoutthe actin network [33]: passive deformation force,traction force between the cell and the ECM andmyosin contractile force (figure 1). The balance ofthese forces (more appropriately, balance of linearmomentum) that assumes the viscous nature of theactin gel has the form:

⏟μ

ζ

∂∂

= −

= = ∂∂

V

xT x F x

T x N V F x fM

x

( ) ( ) ,

( ) , ( ) . (1)

2

2

viscous forcetraction force

myo

contractile force

adh myo

Here, μ is the actin gel’s viscosity coefficient and V(x)is the actin retrograde flow speed. The spatial deriva-tive of the velocity is the rate of strain. Assuming thatthe actin network is a Newtonian fluid, μ V xd /d is theviscous stress, and the force is proportional to itsderivative. The same assumptions were used before inmany modeling papers including [21, 22]. In theappendix, we consider the cases of elastic and viscoe-lastic rheology of the actin network.T(x) is the tractionforce between the adhesions and the ECM, andFmyo(x) is themyosin contractile force within the actinnetwork. The second and thirdmodel assumptions arereflected in the expressions for the traction andmyosinforces in equation (1). Namely, we assume, followingprevious models [19, 21, 22, 24], that the traction

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Phys. Biol. 12 (2015) 035002 EMCraig et al

between the F-actin network and the ECM has aviscous character and thus is proportional to the actinflow speed. At relevant slow rates of the actin flow,representation of adhesions with sticky springs leadsto this model [18, 19]. Furthermore, experiments inwhich a single molecule is dragged along a filament ora surface demonstrate effectively viscous proteinfriction [34]. The parameter ζ represents an effectiveviscous drag per adhesion unit, andNadh(x) is the localdensity of the adhesion units. The expression for Fmyo

is based on the assumption that near the leading edgemost ofmyosin is distributed throughout the isotropicactin gel, rather than concentrated in the stress fibersconnecting mature adhesions and the actin network.We further assume that the myosin exerts isotropicstress, fM, proportional to the local myosin densityM. Effectively, f is the force per myosin motor. Then,the contractile force is proportional to the localgradient of the density of myosin in the actin net-work [22, 35].

Note that the membrane tension in our model isgenerated by the force of actin polymerization: pre-sumably, the rate of actin flow at the leading edge isless than the free polymerization rate, which meansthat the membrane is tensed and loads the growingactin filaments slowing them down. The membrane

tension (equal to μ− ∣∂∂ = )V

x x 0 therefore balances the

viscous stress at the very leading edge, because bothmyosin and traction forces are negligible at x= 0. As aresult, the implicit assumption of thismodel is that themembrane tension is equal to μ− ∂ ∂V x/ (0). As wehave the measured value of ∂ ∂V x/ (0), and have esti-mates of the actin viscosity from the literature (seebelow), we can estimate the membrane tension (seediscussion). Finally, if the myosin force is very smallthrougout the lamellipodium, which the model pre-dicts (see below), the traction force has to be equal tothe integral of the traction stress across the lamellipo-dium,which provides another testable prediction.

Based on experimental observations, we choosethe following two boundary conditions forequation (1): (1) =V V(0) :0 constant leading edgeactin flow speed. We focus on the characteristic casefor slowmotile epithelial cells in which the net protru-sion is much slower than the retrograde flow at thefront, and so parameter V0 is the polymerization rate.(2) Another boundary condition is based on the obser-vation (figure 2) that the actin flow speed becomesuniform in the lamellum. This likely means that thetraction andmyosin forces equilibrate in the lamellumand viscous deformation of the actin gel disappears.Thus, we use the condition ∂ ∂ =V x L/ ( ) 0 at distance

μ=L 8 m from the leading edge—farther than theobserved lamellipodium–lamellum boundary. In fact,the model then predicts that the flow speed becomesconstant closer to the leading edge than L. Note thatthis boundary condition indirectly predicts that themembrane tension is determined by the integral of thetraction stressminus the integral of themyosin force.

Tomodel the myosin distribution, we assume thatmyosin in the cytoplasm attaches to the actin networkwith rate mon and detaches from it with rate moff, andthat attached myosin flows inward with actin, suchthat the density of attachedmyosin,M(x), is given by:

∂∂

= − − ∂∂

M

tm m M

xVM( ). (2)on off

Based on fluorescent measurements of myosin densityin Ptk1 cells (figure 2 and [14]), we choose theboundary condition =M (0) 0. In the appendix, wediscuss mass continuity equation for actin anddemonstrate that the actin dynamics does not expli-citly affect the dynamics of flow and myosin andadhesion density distributions.

Similarly, the local density of adhesion units,Nadh(x), is determined by the rates of assembly (kon)and disassembly (koff). How forces and actin flowaffect growth of adhesions is still not entirely clear, butboth of these factors do influence the adhesiondynamics [13]. Here we test the hypothesis that theadhesion clutch is strengthened by an indirect effect oflocalmyosin force butweakened by fast flowof actin:

β

∂∂

= −

= + =

( )

( )

N

tk F k V N k

k F k kV

V

( ) ,

1 , exp*

. (3)

adhon myo off adh on

on0

myo off off0

⎛⎝⎜

⎞⎠⎟

Here, kon0 is the adhesion assembly rate in the absence

of myosin, and koff0 is the disassembly rate in the

absence of actin flow. Based on fluorescent imaging ofpaxillin as an indicator of adhesion density in Ptk1cells (figure 2 and [14]), we choose the boundarycondition =N (0) 0.adh The parameter β characterizesthe effect of adhesion strengthening, and V* is acharacteristic actin flow speed above which adhesionsrapidly detach and disassemble.

The data in [36] show that the adhesion complexesnear the leading edge are first attached to the actin net-work and slip relative to the substrate, and then at acritical stress attach to the substrate and slip relative tothe actin network.However, thesemovements are on avery short, sub-micron spatial scale, and so we do notinclude them in the model. Adhesions are multi-com-ponent protein complexes, whose content and prop-erties depend onmany factors and on the stage of theirdynamics [5, 36, 37]. However, the detailed morpho-logical cycle of adhesion maturation and turnover isbeyond the scope of the current study, as there is littleunderstanding of respective mechanical changes. Wefocus simply on the density of the generalized adhe-sion units and its influence on the forces.

Two key assumptions of the model, which we willtest by comparison to experimental data, are intro-duced in equation (3): (1) the adhesion disassemblyrate, koff, increases with actin flow speed. This hypo-thetical property is sometimes referred to as ‘stick–slip’ behavior [19, 26, 38]. A qualitative microscopicexplanation for this mechanism is that at fast actin

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Phys. Biol. 12 (2015) 035002 EMCraig et al

flow speeds, the bonds between actin and the ECM arequickly stretched to their breaking point, whereas atslower sliding speeds weak or broken bonds have timeto reform. The stick–slip mechanism for adhesionweakening may explain the experimentally observedbiphasic relationship between actin flow and tractionstrength. Note also that assuming that the rate ofdetachment is a function of the retrograde flow velo-city is mathematically equivalent to assuming it to be afunction of the traction force. Indeed, the tractionstress is ζ=T N V ,adh so the force per adhesion com-plex is equal to ζ=T N V/ adh because local stressimposed on the adhesions in any small area is dis-tributed, on the average, equally among them. In fact,the functional form for the detachment dependence

on the force, = =( ) ( )k k V V k f fexp / * exp /*

,off off0

off0

where ζ ζ= =f V f V,* * is the characteristic force of

breaking an adhesion, is equivalent to the Bellequation used often to describe the force-dependentbreaking of molecular bonds [24]. (2) The growth rateof adhesions, kon, increases with a factor proportionalto the local myosin density gradient. It was suggestedin the past that force application at adhesion sites helpsstabilize new adhesive contacts [37, 39] and promotesadhesion maturation and growth [6, 38–40]. How-ever, recently it was shown that absolute values of theintracellular forces do not correlate directly with theadhesion growth rate [31]. Rather, this study sug-gested that adhesion growth depends sensitively onstructure and dynamics of the actin network whichadhesions engage with [31]. Besides, while the tractionforce is applied to adhesions directly, the local myosinforce is generated by myosin clusters contracting actinfilaments that are part of the actin lamellipodial net-work. Some of these filaments interact with adhesionmolecules, and so the adhesion complexes are unlikelyto feel the local myosin force directly. What we pro-pose is that myosin density gradient could inducestructural changes in the actin network leading togrowth of actin templates favorable for adhesionstrengthening. In effect, there is not a direct effect ofthemyosin-generated force on the adhesion assembly,but rather an indirect effect of this force on the actinnetwork dynamics and structure, which in turn affectsthe adhesion assembly.

We also examine the possibility of the assumptionthat the adhesion assembly rate is proportional not tothe gradient of the myosin density, but to the myosindensity itself. Consequences of this assumption arereported in supplemental figure 7 and show that ifadhesion growth is proportional to the myosin den-sity, the predicted flow and traction distributions aredifferent from those observed experimentally. On theother hand, if adhesion growth is proportional to themyosin density gradient (as in equation (3)), themodel reproduces all experimental observations.

The model equations can be expressed in the fol-lowing simplified dimensionless form in steady state:

λ ∂∂

= −V

xT x F x

˜

˜˜( ) ˜ ( ), (5)

2

2 myo

= ∂∂

F x FM

x˜ (˜)

˜

˜, (6)myo

γ= − ∂∂ ( )M x

xVM˜ (˜) 1

˜˜ ˜ , (7)

= + ∂∂

−( )N x bFM

xV v˜ (˜) 1

˜

˜exp ˜

* , (8)adh

⎛⎝⎜

⎞⎠⎟

= + ∂∂

−( )T x bFM

xV v V˜(˜) 1

˜

˜exp ˜

*˜ , (9)

⎛⎝⎜

⎞⎠⎟

where = ≡ =V V V V x V˜ ( 0) ˜ ,0 = ≡M M M M˜ ˜ ,m

m0on

off

= ≡N N N N˜ ˜ ,k

kadh 0 adh adhon0

off0 = ≡ ζ

T T T T˜ ˜,k V

k0on0

0

off0 and

μ= ≡x L x x˜ (1 m) ˜.0 The steady-state distributionsare defined by the five dimensionless free parameters:

λ ≡ μ( ),T

V

L

1

0

0

02 ≡ ( )F ,

T

fm

L m

1

0

on

0 offγ ≡ ,

V

m L0

off 0β≡b T ,0

and ≡v* .*V

V0Note that λ is the measure of actin

viscosity,F is the measure of myosin strength, γ is themeasure of the myosin rate of detachment from theactin network, b is the measure of myosin forceinfluence on the adhesion strengthening, and v* is themeasure of howmuch smaller is the flow rate at whichadhesions are weakened by the flow than the polymer-ization rate at the front.

2.2. Characteristicmodel parametersIn the definition of parameter γ, the ratio L V/0 0 is thetime over which the actin flow carries myosin mole-cules across a small part of the lamellipodium. Theonly way the model reproduces the experimentalobservations is if myosin molecules do not detachbefore the flow moves them to the lamellum’sboundary; otherwise, myosin would be distributedmuch more evenly between the lamellipodium andlamellum than observed. Thus, we assume that the rateof myosin detachment is orders of magnitude smallerthan the ratio L V/ .0 0 Good agreement with the data isachieved if γ ~ 1000. Note that comparison of themyosin distribution model with the data in keratocytecells earlier led to a similar estimate [22]. To explainthe experimental relationship between flow speed andtraction, the actin flow speed at which the flow nolonger weakens adhesions must be somewhat slowerbut on the same order of magnitude as the character-istic flow speed V .0 Therefore, the parameter v*is lessthan 1, but not by an order ofmagnitude.

In the simulations, we use v* ~ 0.3. Note that inthe case of U2OS cells when parameter V0 is muchsmaller, parameter v* is much greater (supplementalfigure 1). Parameter b has to be significantly greaterthan 1 for adhesions to be sensitive to the indirecteffect of the myosin force, so in the simulations, weuse b ~ 10. Parameter F is the measure of the ratio ofthe maximal myosin stress to the characteristic trac-tion stress. The characteristic traction stress and myo-sin force are of the same order of magnitude, because

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Phys. Biol. 12 (2015) 035002 EMCraig et al

the myosin stress and membrane tension are two fac-tors contributing to the traction, and both are eitherlikely to be of the similar order of magnitude or themembrane tension ismuch less than themyosin force.Maximal myosin stress, however, is likely muchgreater than the resultingmyosin force because of rela-tively smooth myosin distribution in space leading topulling from neighboring myosin clusters largely can-celing each other and generating smaller force. Thus,parameter b is likely significantly greater than 1. Com-parison of the data and simulations show that theycompare well if F ~ 10. Finally, the order of magni-tude of the parameter λ can actually be estimated fromthe published data. From [14], μ= −T 20 pN m ,0

2

T L V/ ~ 100 02

03 pN s μm−1. On the other hand, the

reported actin network viscosity is of the order of×2 103 pN s μm−2 [29]. Parameter μ can be obtained

from the measured viscosity by multiplying the latterby the lamellipodial thickness of the order of amicron.Thus, parameter λ ~ 2.This value actually gives a verygood qualitative agreement with the data for PtK1cells. For U2OS cells, we have to use much greatervalues (supplemental figure 1).

We explore the dependence of the model on eachof these parameters in figure 4 and supplementalfigures 2–4. We constrain the model parameters bydetermining values that can simultaneously fit theactin flow and traction distributions for cells underseveral different experimental perturbations. We notethat there is a significant amount of cell–cell variationin the experimental data, and our goal is to illustratehow the hypothetical clutch mechanism produces thequalitative spatial organization observed experimen-tally, rather than to obtain precise numerical values foreach of these parameters.

We obtained numerical solutions to modelequations using MATLAB (The MathWorks, Natick,MA) codes that were based on finite difference numer-ical methods. We solved the partial differentialequations of the model using an explicit steppingmethod with the first-order upwind scheme for theadvection terms. The spatial domain was divided into40 equal segments, and the time steps were set suffi-ciently small to guarantee numerical stability. Compu-tational time was on the order of several minutes on apersonal computer.

2.3. Experimental data used to test themodelThe experimental data from Ptk1 epithelial cells infigures 2–3 were originally published in [14], and areincluded here for the purpose of illustration and directcomparison with the predictions of our theoreticalmodel. Traction and actin flow data in humanosteosarcoma (U2OS) epithelial cells (supplementalfigure 1) used to test the model predictions wereoriginally published in [31].

3. Results

3.1. Bimodalflow–traction relationship arises fromcompetingmechanisms of adhesion dynamicsregulation:flow-dependent detachment andmyosin-dependent attachmentAs first reported in [14], correlation between actindynamics and traction in Ptk1 epithelial cells reveals aninverse relationship between actin flow speed andtraction magnitude in the lamellipodium and a directcorrelation in the lamellum (figure 2). These cellsexhibit rapid actin flow at the leading edge(∼20 nm s−1) that decays across the lamellipodia, andrelatively uniform slowflow (∼2 nm s−1) in the lamella.The corresponding traction stress magnitude increaseswith distance across the lamellipodium and decreaseswith distance across the lamellum. Interestingly, thelocation of the traction peak depends in a non-trivialway on the strength of myosin contractility (figure 3;[14]): the peak is closer to the leading edge inblebbistatin-treated cells (where myosin contractilityhas been inhibited) than in control cells. However inCARho-treated cells, which exhibit increased myosincontractility, the locationof the tractionpeak shifts backtoward the leading edge relative to control cells. Theactin flow speed corresponding to maximal traction isrelatively similar under each of these conditions [14].

Our model for the adhesion clutch reproduces thebimodal relationship between actin flow speed andtraction magnitude observed in Ptk1 cells (figure 3).The spatial distributions for individual cells can bereproduced by tuning the myosin strength parameterF to correspond to experimental changes in myosincontractility (increasing F for blebbistatin, control,and CARho-treated cells, respectively), and using alower effective viscosity λ to fit blebbistatin data(figure 3). Note that it is reasonable to expect that actinnetworks with reduced myosin contractility are lessstiff [41]. Other system parameters were not variedfrom one cell to the next. Our model also reproducesthe experimental observations that adhesion andmyo-sin densities (as identified by microscopic tracking offluorescent markers [14]) are relatively small in thelamellipodia, and dramatically increase near thelamellipodia/lamella boundary (figures 2 and 3).

The bimodal flow–traction relationship observedin Ptk1 cells can be explained by considering the inter-play between velocity-dependent adhesion detach-ment and myosin-dependent adhesion strengthening(equation (3)). Near the leading edge, actin poly-merization against the membrane produces a fast ret-rograde flow speed, resulting in a low adhesiondensity. For a very rigid actin network (high λ), retro-grade flow is uniform throughout the network. On theother hand, for a deformable viscous network (lowerλ), the retrograde flow speed decays with distancefrom the leading edge (as observed in Ptk1 cells,figures 2 and 3). As retrograde flow decays, adhesionsdetach less frequently, according to the stick-slip

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Phys. Biol. 12 (2015) 035002 EMCraig et al

mechanism in equation (3), leading to an increase inadhesion density. For the region near the leading edgewhere >V v˜

*, the adhesion density increases morerapidly with distance than the flow speed decays, sothat the traction ( =T N V˜ ˜ ˜)adh increases with distance.This effect is responsible for the observed trend nearthe leading edge for traction to increase as retrogradeflowdecreases.

In the lamella region further from the leadingedge, the trend for traction to decrease with distancecan be explained by the force-dependent adhesionstrengthening mechanism described in equation (3).Myosin attaches to the actin network and is swept backwith actin retrograde flow, producing a steep increasein attachedmyosin density where the flow speed levelsoff (figure 3). The peak inmyosin force at this locationcreates a corresponding peak in adhesion density.Beyond this peak, traction decreases with distance dueto the decrease in adhesion density, which becomessmaller due to the decreasing myosin force beyond thepeak, as well as a slight continued decay in retrogradeflow speed (figure 3).

3.2.Myosin contractility regulates traction peaklocation bymodulating adhesion distributionOur model also reproduces the experimental observa-tion (comparing blebbistatin, control, and CARho-

treated cells) that increasing myosin contractilityinitially shifts the traction peak to higher x values, butfurther increasing myosin contractility shifts the peakback toward the leading edge (figures 3 and 4). This isbecause, in the absence of myosin force (as inblebbistatin-treated cells), the traction distribution is

given by = −( )T V v V˜ exp ˜*

˜ (equation (9)), which is

maximal when =V v˜*. In this case, the balance of

internal viscous forces and external traction determinethe location of the peak (supplemental figure 2). In thepresence of myosin forces (as in the control andCARho-treated cells), the traction distribution alsodepends on the location of the myosin force peak(equation (9)). Asmyosin strength F initially increasesfrom zero, the location of the traction peak shifts in the+x-direction toward the location where the myosingradient is highest (in the region where the flow speedlevels off). Further increasing F shifts the traction peakback toward the leading edge, because higher adhesiondensity causes the flow speed to fall off more steeply(figure 4).

3.3. Traction is highest at leading edge for cells withslowuniformflowAs discussed above, the bimodal traction–flow rela-tionship observed in Ptk1 epithelial cells results

Figure 3.Model fits to sample experimental Ptk1 data. (a)–(c) Actin flow speed (left) and traction (middle) and predictedmyosindensity (whiskers) and adhesion density (squares) (right) for a blebbistatin-treated cell (a), a control cell (b), and aCA-Rho treatedcell (c).

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Phys. Biol. 12 (2015) 035002 EMCraig et al

from the interplay between velocity-dependent adhe-sion detachment and myosin-dependent adhesionstrengthening.However, in cells with very slow leadingedge polymerization speed and corresponding retro-grade flow speed ( ≪V V* ),0 adhesion disassemblyis insensitive to actin flow speed and adhesiondensity is no longer kept small near the leading edge.In this case, the predicted traction distribution is

= + ∂∂( )T bF V˜ 1 ˜M

x

˜

˜(equation (9)), and we predict

that traction will decay with distance from the leadingedge if the myosin force contribution is relativelysmall, and that the traction distribution willmirror themyosin force distribution for increasing myosinstrength (supplemental figures 1(d) and (e)). For cellswith slow spatially uniform actin flow ( =V x( )

≪V V* ),0 the balance of forces on the actin network

becomes =F T˜ ˜myo (equation (5)). The steady-state

myosin density predicted by equation (7) is

= −γ−( )M̃ 1 exp ,x

V

˜˜ and the resulting myosin con-

tractile force is =γ−( )F F˜ exp ,x

Vmyo˜˜ and the traction is

predicted to decay with distance from the leading edgefor all values ofmyosin strength.

This prediction is consistent with our measurementsof actin flow and traction in U2OS epithelial cells(supplemental figures 1(d) and (e)). These cells exhi-bit relatively uniform flow (∼2–4 nm s−1) across thelamellipodia/lamella, with the leading edge speed onlyslightly higher than the speed of flow in the lamella.The corresponding traction profile in these cells ishighest near the cell membrane and decays with dis-tance from the leading edge.

3.4. Actinflow speed insensitive tomyosin strengthin regions of uniformflowA second prediction we can make for cells with slowuniform flow is the dependence of this flow speed onthemagnitude of themyosin force. If myosin turnoveris slow compared to the rate of transport across thelamellipodia (γ ≫ 1 ), then the myosin force andcorresponding traction in the lamella are approxi-mately uniform, = ≈T F F˜ ˜ .myo In this case, thesteady-state traction distribution (equation (9)) yields:

=+

VF

bF˜

1. (10)

An increase in the myosin force driving actin flow iscanceled out by increased traction due to myosin-

Figure 4.Role ofmyosin-dependent and actin-flow-dependent adhesion dynamics. Unless otherwise stated, parameters are: λ= 3, b= 15,v* = 0.3, and γ= 1000. (a)–(c) Fullmodel withmyosin-dependent adhesion strengthening and flow-dependent adhesionweakening.(a) and (b) Actin flow speed and traction as a function of distance from the leading edge. (c) Position of traction peak inmm(blackdots) and actin flow speed (in units of v*) corresponding to traction peak (blue exes) as a function of F. (d)–(f) Same as (a)–(c), exceptwithoutmyosin-dependent adhesion strengthening (by setting b= 0). (g)–(h) Actinflow and traction as a function of distance fromthe leading edge, in the absence of actin-flow-dependent adhesionweakening (by setting v*≫1). (i)Balance of forces on actin network(traction (black),myosin force (blue), internal viscous force (red)), for F= 10, b= 15, v* = 0.3.

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Phys. Biol. 12 (2015) 035002 EMCraig et al

dependent adhesion strengthening. For saturatinglevels ofmyosin strength, we predict that the actin flowspeed will be insensitive tomyosin strength. Assumingthat the total traction force between the cell and thesurface is proportional to the local myosin stress( ∝F F ),tot we can re-express equation (10) as a Hill-like relationship between total force and lamellar flowspeed,VLM :

=+

V VF

F F. (11)LM

kmax

tot

tot

⎛⎝⎜

⎞⎠⎟

We test this prediction by treating U2OS cells withvarious concentrations of the Rho-kinase inhibitor Y-27632 compound [31], which interferes with acto-myosin-based contractility, allowing us to tune themagnitude of myosin contractile force. In agreementwith the model, despite significant changes in themeasured total contractile force magnitude, the actinflow speed in the lamella is relatively unchanged for allbut the highest concentrations of Y-27632 compoundtested (supplemental figure 1).

4.Discussion

We have shown that mechanical feedback between F-actin flow and focal adhesion dynamics allows theleading edge of motile cells to self-organize intodistinct regions of inverse and direct correlationbetween actin flow and sub-cellular traction, asobserved in epithelial cells [14]. These characteristicdistributions arise from a balance of viscous stressin the actin network, acto-myosin contraction, mem-brane tension and adhesive traction. This ability toself-organize into functionally distinct spatial regionsallows the cell to undergo rapid actin polymerizationnear the leading edge while adhering firmly to thesurface near the lamellipodia/lamella boundary.Althoughwe consider several plausible actin–adhesionfeedback mechanisms (appendix), the model thatreproduces experimental distributions of actin flowand traction is one in which the adhesion ‘clutch’engagement is strengthened by an effect of myosincontractility on actin templates favorable for adhesionassembly and weakened by actin retrograde flow. Thespatial distributions predicted by our model resultfrom a balance of internal forces in the actin network(myosin contraction and internal viscosity) and exter-nal traction forces exerted on the network. The modelpredicts that viscous deformation is the major factorgenerating the traction stress near the front, whilemyosin force is the main contributor to the tractionstress farther than a few microns from the leadingedge. While there is substantial cell–cell variation, asexpected, we find that our model reliably predictscharacteristic features of the data including thelocation of the traction peak, dependence of tractionmagnitude on myosin strength, and the spatial dis-tribution of the actin retrograde flow.

Our model also reproduces qualitative features ofcells that have very different leading edge actin flowand traction profiles from the Ptk1 epithelial cells dis-cussed above. We looked at simultaneous actin flowand traction stress measurements in human osteo-sarcoma (U2OS) epithelial cells and in contrast to thePtk1 cells, and these cells exhibit slow relatively uni-form retrograde flow across the lamellipodia/lamellaand traction that is highest at the leading edge. Theslower more uniform flow in these cells may be theresult of physiological differences such as a more rigidactin network, slower actin polymerization dueto lower available G-actin pool, or differences in lead-ing edge membrane tension. In the framework ofour model, cells with slow uniform flow are effectivelyutilizing only one of our hypothetical adhesion regula-tionmechanisms (myosin-dependent clutch strength-ening), but are in a physical regime where they do notexperience the other hypothetical mechanism (‘stick–slip’, or flow-dependent clutch weakening). The myo-sin-dependent mechanism alone reproduces keyobservations in U2OS cells: (1) traction is highestat the leading edge; and (2) lamellar flow speed isrelatively insensitive to myosin strength. The insensi-tivity of actin flow speed to perturbations in myosincontractility may allow the cell to maintain robustprotrusive activity in the presence of fluctuatingmole-cular concentrations.

Our model makes a very simple prediction for thelamellipodial width. Our simulations show that thiswidth is not sensitive tomyosin strength becausemyo-sin is largely'swept away’ from the leading edge andstarts affecting the flow and traction only close to thelamellum’s boundary where the flow already slowsdown. Near the leading edge, the viscous force largely

balances the traction: μ∂∂

T~ ¯.V

x

2

2The boundary condi-

tions at the lamellipodium–lamellum boundary at dis-tance X from the leading edge are

≈ ≈∂∂

V X X( ) 0, ( ) 0.V

xSolving this equation for the

distance-dependent actin flow velocity yields

− +μ μ

V V x x~ (0) .TV T2 ¯ (0) ¯

22 The condition V ~

− +μ μ

V X X0 ~ (0) TV T2 ¯ (0) ¯

22 gives μX ~ .V

T

2 (0)¯

Furthermore, the traction stress can be estimated as

ζT V¯ ~ ˜ (0) where ζ ζ ( )k k˜ ~ /on0

off0 is the character-

istic adhesion strength. Thus, we arrive at the very

simple result: μζ

X ~ ,˜ and we predict that the lamel-

lipodial width is proportional to the square root ofthe actin viscosity, and inversely proportional to thesquare root of the average adhesion strength at theleading edge. This is the spatial scale that seems to befundamental for cell mechanics: the effect of a forceapplied to the actin network at a point dissipates overthis distance [22, 42]. At realistic values of parameters,the characteristic lamellipodial width is predicted to beon the order of a few microns, as observed.

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Phys. Biol. 12 (2015) 035002 EMCraig et al

Importantly, this scale is largely insensitive to para-meters other than the viscosity and adhesion strength.

Another interesting prediction of our model isthat, because the myosin force in the lamellipodium issmall, the membrane tension roughly balances theaverage traction stress multiplied by the lamellipodialwidth, which in turn is approximately equal to theactin viscosity multiplied by the flow rate gradient.The traction stress, lamellipodial width and the flowrate gradient were measured in [14], which allows usto predict a viscosity value of μ∼ 103 pN s μm−2 Thisagrees with the viscosity measured in [29], and themembrane tension magnitude (∼60 pN μm−1), whichis also of the same order of magnitude normally mea-sured in slow-moving cells [30, 43].

Our model is, of course, ignoring many complex-ities of the cell motile machinery. We did not considerpotentially very complex dynamic feedbacks betweenactin, adhesion andmyosin [44]. There aremore com-plex dynamics of adhesion growth and turnover thatdepend not only on force but on signaling [45, 46] aswell as spatial-temporal history [47]. Behind thesteady-state distributions of the cytoskeletal elements,flows and forces at the cell leading edge, there aremany non-steady processes on various spatial-tem-poral scales [48].We also did not investigate the effectsof complex kinetics of lamellipodial and lamellumactin networks and respective actin binding proteins[49, 50]. We did not consider the discrete character ofthe actin networks on the micron scale or the aniso-tropic nature of the contractile network. All these fac-tors, however, have to be included only after morecoarse-grained models of the type that we proposehere are sufficiently tested.

Acknowledgments

We thank Jun Allard and Kun-Chun Lee for usefuldiscussions. The study was funded by the NationalInstitute of Health GM068952 to A M and BurroughsWelcome Career Award and Packard Fellowship to MLG.

AppendixA. Characterizing each adhesionregulationmechanism separately

In the main text, we consider a model for adhesiondynamics regulation that relies on two mechanisms:(1) flow-dependent adhesion disassembly; and (2)myosin-dependent adhesion assembly. Here we dis-cuss each of thesemechanisms separately.

A.1.Myosin-dependent adhesion assembly(supplemental figure 1)

Here we consider a variation of the model in the maintext in which the assembly rate for adhesions is given

by the myosin-dependent expression in equation (3),but the disassembly rate is kept constant instead of theflow-dependent expression in equation (3) (i.e., we set

≫v* 1).In the absence of flow-dependent adhesion dis-

assembly, the steady-state adhesion density distribu-tion mirrors the myosin force distribution( = +N bF˜ 1 ˜ ).adh myo For increasing values of myosinstrength F, the adhesion density and correspondingtractionmagnitude increase. This causes the actin flowspeed to decay more steeply with distance, whichresults in the myosin force peak location moving clo-ser to the leading edge with increasing F. Becausemyo-sin is swept away from the leading edge by actin flow,the adhesion density near the leading edge is uniformand the resulting traction is directly proportional tothe decaying actin flow speed.

Thismechanism alone creates a spike in traction ata location that is sensitive to myosin strength, and canreproduce lamellar flow insensitivity to myosinstrength for high F, similar to experimental observa-tions. However, the traction and actin flow near theleading edge are directly proportional, in contrast tothe experimental observation of inverse flow–tractionnear the leading edge.

A.2. Actin flow-dependent adhesiondisassembly (supplemental figures 2–3;and figures 4 (d)–(f) in themain text)

Here we consider a variation of the model in the maintext where the disassembly rate for adhesions is givenby the flow-dependent expression in equation (3), butthe assembly rate is kept constant instead of themyosin-dependent expression in equation (3) (i.e., weset b = 0).

In this case, the steady-state traction distribution is

given by = −( )T V v V˜ exp ˜*

˜ (equation (9)), which is

maximal for =V v˜* independent of other system

parameters. The location of the traction peak dependson the internal viscosity of the actin network, λ, whichdetermines the decay rate of actin flow speed as a func-tion of distance from the leading edge (supplementalfigure 2).

The myosin distribution is determined by theparameter γ, which characterizes the time scalefor myosin to stay attached to the actin network (sup-plemental figure 3). For low γ, such that myosin turn-over is fast compared to the timescale for actinretrograde flow across the lamellipodia, the myosindistribution is spatially uniform. For higher γ, suchthat myosin stays attached long enough to be trans-ported by actin across the length of the leading edge,the myosin distribution is small near the leading edgeand increases rapidly as the actin flow speed levels off,in agreement with fluorescent images of myosindistributions.

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Phys. Biol. 12 (2015) 035002 EMCraig et al

Although this variation of the model can producethe experimentally observed bimodal relationshipbetween actin flow and traction, there are someexperimentally observed features it fails to reproduce:(1) the location of the traction peak increases mono-tonically with myosin strength (see figure 4(f) in themain text), in contrast to experimental results com-paring control, blebbistatin-treated, and CARho-trea-ted cells (figure 3 in the main text). In theseexperiments, the traction peak location initiallyincreases with increasing myosin strength, but movesback toward the leading edge for higher myosinstrength. (2) Thismechanism alone cannot explain theinsensitivity of lamellar actin flow speed to myosininhibition observed in U2OS cells (supplementalfigure 1).

Appendix B.Model dependence onstrength ofmyosin-dependent assemblyparameter b (supplemental figure 4)

Because the adhesion density simply depends on theproduct bF, the model dependence on b is similar tothe dependence on F (see figure 4 in the main text),with the following exception: for increasing F, themyosin contractile driving force increases along withthe adhesion density, so the corresponding flow speedis independent of F at saturating levels (supplementalfigure 4(b)). For increasing b, the adhesion densityincreases with no corresponding increase in drivingforce, so the actin flow speed at the traction peakapproaches zero for high b.

AppendixC.Model variations andalternativemodels (supplementalfigures 5–7)

C.1. Uniformmyosin force distribution(supplemental figure 5)

If we assume that myosin contractile force is spatiallyuniform, many of the qualitative features predicted bythe model in the main text are reproduced. Inparticular, the bimodal relationship between actinflow and traction distributions is not dependent on theparticular myosin force distributions predicted by thedynamic equation for myosin (equation (2)). How-ever, for uniform myosin force, the location of thetraction peak is relatively insensitive to myosin forcestrength F and depends primarily on actin networkviscosity λ. Given this limitation, this form of themodel cannot simultaneously reproduce the actinflowand traction profiles measured experimentally. If λ ischosen high enough to produce a traction peak at theexperimentally observed distance from the leading

edge, the corresponding actin flow speed falls off lesssteeply than experimentally observed.

C.2. Step functionmyosin distribution(supplemental figure 6)

If we treat the myosin force as a step function (zeronear the leading edge and constant beyond a thresholddistance), adhesion density is very small near theleading edge due to a lack of myosin-dependentassembly in combination with rapid disassemblyresulting from fast flow. For this reason, the tractionpeak spikes at the step-function boundary and thenrapidly decays with increasing distance due to con-tinuing decay in flow speed. As in the case of uniformmyosin force distribution discussed above, this versionof the model does not reproduce experimental flowand traction distributions as well as the fullmodel withmyosin force calculated based on the predictedmyosindistribution (equation (7)).

C.3.Myosin force directly proportional tomyosin density (supplemental figure 7)

Here, we consider a variation of the model in whichmyosin force is directly proportional to myosindensity, instead of proportional to the gradient of themyosin density as in the main text (equation (1)). Inthis case, adhesion density is close to zero near theleading edge, and increases in proportion to myosindensity away from the leading edge. The resultingtraction increases with distance from the leading edge,rather than having a distinct peak as observed in Ptk1cells and predicted by themodel in themain text.

AppendixD.Note about actin density

Unlike equations (2) and (3) that deal with conserva-ble densities of myosin and adhesions, respectively,equation (1) is the force balance equation. The velocitythat this equation describes is the velocity of the actinnetwork, and the mass conservation law has to besatisfied for actin. The equation for actin density A has

the form: = − −∂∂

∂∂( )r z A z A VA( ) ( ) ( ).A

t x1 2 Here

z A( )1 is the non-dimensionless term describing theactin assembly rate, which can be dependent on theactin density, and z A( )2 is the non-dimensionlessterm describing the actin disassembly rate, which alsocan be dependent on the actin density. Dimensionalparameter r is the dimensional characteristic rate ofactin turnover. The usual rates of actin turnover are0.1 s−1. Introducing the time scale equal to r1/ andspatial scale equal to the lamellipodial length L ,0 wecan re-scale and non-dimensionalize the actin density

equation as: ε= − −∂∂

∂∂( )z A z A vA( ) ( ) ( ).A

t x˜ 1 2 ˜Here

ε= = = =( )t rt x x L V rL v V V˜ ; ˜ / ; / ; / .0 0 0 0 Small

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Phys. Biol. 12 (2015) 035002 EMCraig et al

parameter ε μ μ×− −( ) ( )~ 0.02 m s / 0.1 s 2 m ~ 0.11 1

therefore the transport term can be neglected, and

approximately, ≈ −∂∂ ( )z A z A( ) ( ) .A

t̃ 1 2 Thus, assum-

ing there is a unique stable steady state for the actinturnover, the model predicts a constant in space actindensity. Figure 1(E) of [14] shows actin fluorescentsignal in the lamellipodium, and other than increasingpatchiness away from the front, the average actindensity is roughly constant, in agreement with themodel. On the relevant time scale of 10 s, the actinnetwork displacement is but 0.1 μm, orders of magni-tude less than the lamellipodial width. Therefore, theactin flow is likely irrelevant for the actin density,which is determined by the balance of assembly anddisassembly. Therefore, in the model, we did notinclude possible assumptions about (a)myosin viscos-ity being proportional to actin density, (b) attachmentrate for adhesions and/or myosin being proportionalto actin density.

Appendix E. Assumption that actinnetwork is elastic leads to predicteddistribution of actin flowqualitativelydifferent from the observed spatialflowprofile

Let us assume that the lamellipodial actin network iselastic and consider two material points at time

=t 0 —one at the very leading edge, with coordinate=x 0, another a small distance away from the front,

with coordinate δ=x . We investigate the case whenthe retrograde actin flow is steady and its velocitydistribution is given by the function V x( ) where thevelocity is positive and directed to the rear, away fromthe leading edge. Over a small time increment, amaterial point displacement at =x s would be

=s V s td ( )d , then, ∫ = t ,x t s

V s0

( ) d

( )where x t( ) is the

coordinate of the first material point (with coordinate=x 0 at time =t 0 ). Similarly, for the second

material point (with coordinate δ=x at time =t 0 ),

∫ =δ

t ,y t s

V s

( ) d

( )where y t( ) is the coordinate of

this second material point. Then, ∫ =x t s

V s0

( ) d

( )

∫ ∫+δ

δs

V s

x t s

V s0

d

( )

( ) d

( )and ∫ ∫= +

δ δ

y t s

V s

x t s

V s

( ) d

( )

( ) d

( )

∫ .x t

y t s

V s( )

( ) d

( )Comparing these two expressions and

considering that ∫ ∫= =δ

t ,x t s

V s

y t s

V s0

( ) d

( )

( ) d

( )we find:

∫ ∫=δ

.x t

y t s

V s

s

V s( )

( ) d

( ) 0

d

( )In the limit of very small values

of δ, we have: ≈ δ−,

y x

V x V( ) (0)or δ− ≈y x .V x

V

( )

(0)At the

very leading edge, the actin network is compressed bythe membrane tension. Let us define the width of thenetwork between the two considered material pointswithout this compression as Δ. Then, we can definethe strain ε of the network between the two consideredmaterial as ε = = −Δ

ΔδΔ

− −1.

y x V x

V

( ) ( )

(0)If Y is the

Young modulus of the network (which we willconsider constant for simplicity, as well as rescaled forthe 1D problem), and Fmemb is the membrane tension(similarly rescaled for the 1D problem), then the strain

at the leading edge is: =Δ δΔ− .

F

Ymemb Therefore,

the elastic stress is equal to σ ε= =x Y x( ) ( )

− −( )Y F Y .V x

Vmemb( )

(0)On the other hand, the spatial

derivative of this stress (in 1D problem) is equal to thedifference between the traction force and the myosin

force: = −σ T x F x( ) ( ).x

d

d myo Thus, =−( )Y F

V

V

x(0)

d

dmemb

−T x F x( ) ( ).myo The constant in the bracket,−( ),

Y F

V (0)memb is positive because in the framework of the

linear elasticity theory that we use, the force at theboundary has to be significantly smaller than theYoungmodulus. In the case of the blebbistatin-treatedcase, ≈F x( ) 0,myo and the traction force, >T x( ) 0(this force is directed to the leading edge, compressing

the network). Therefore, > 0:V

x

d

dthe retrograde flow

velocity should increase toward the rear, contradictingthe experimental measurements showing that the flowrate decreases toward the rear (figure 2). The explana-tion of the theoretical result is very simple: at the veryfront, the network is compressed maximally by themembrane tension balanced by the distributed trac-tion force. As we go away from the leading edge to therear, smaller and smaller part of the total tractioncompresses the network. This is similarly physically tothe elastic rod standing vertically in the gravitationfield: at the floor, the material is compressed maxi-mally by the total gravitational force balanced by thecounter-force from the floor. As we go higher, the rodmaterial is compressed less and less and finally the topof the rod is uncompressed at all.We conclude that theelastic rheology of the actin network is incompatiblewith the observation that the retrograde velocitydecreases toward the rear.

It is similarly easy to demonstrate that if the net-work is viscoelastic (Kelvin model with spring anddashpot in parallel), then the prediction is the same,and this is not a viable approximation in the sense thatthe model prediction would not agree with the data.However, if the network is described by the Maxwellviscoelastic model (with spring and dashpot in series),then equation (1) in the main text has to be modified

as follows [22]: μ α σ− +∂∂

∂∂

(1 )x

V

x ve⎡⎣ ⎤⎦ = −T x( )

σ τ μα+ + =σ σ∂∂

∂∂

∂∂

F x V( ), ,t x

V

xmyo veve ve⎡⎣ ⎤⎦ where para-

meter α determines a non-Newtonian fraction of visc-osity, σve is the viscoelastic part of the stress tensor(scalar in the 1D case), and τ is the relaxation time. Inthe case of the steady actin flow, μ α−(1 )

σ+ = − +σT x F x( ) ( ),V

x x

d

d

d

d myo ve2

2

ve τ μα=σV .

x

V

x

d

d

d

dve

Let us consider the case when elastic deformationsrelax fast compared to the characteristic time of flowacross the lamellipodial domain: τ < < L V/ (0).0

Then, using the rescaled relaxation timeτ τ= V L˜ (0)/ 0 as the small parameter, we can expand

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Phys. Biol. 12 (2015) 035002 EMCraig et al

the velocity and stress into the perturbation series andretain only linear terms with respect to the small para-meter: τ σ σ τσ= + = +V v v˜ , ˜ .0 1 ve 0 1 In the zeroth

order, σ μα μ= = −T x F x, ( ) ( ),v

x

v

x0d

d

d

d myo0

20

2and in

the first order: μ α− + =σ(1 ) 0

v

x x

d

d

d

d

21

2

1 and

σ μα+ =σv .

x

v

x0d

d 1d

d0 1 From the last equation,

σ μα μα= − v .v

x

v

x1d

d 0d

d

12

0

2Substituting this expression

into the previous equation, we obtain:

α= ( )v .v

x x

v

x

d

d

d

d 0d

d

21

2

20

2 From the observations, we can

approximate the flow rate as decreasing exponentiallyon the spatial scale L: = −v V x L(0) exp( / ).0 Then,

α= − +( ) cexp ,v

x

V

l

x

l

d

d

(0) 212

2 and we finally obtain

the approximate formula for the flow rate distribu-

tion: = + − −τα τα( ) ( )V V (0) exp expV

l

x

l

V

l

(0)

2

(0)

2

2 2

−( )x

l

2 .

The result is intuitive: the elastic and non-New-tonian part of the stress in the actin network causesmodest increase of the flow rate everywhere. Simplyspeaking, elastic resistance to deformation tends todiminish the spatial gradient of the flow rate. How-ever, qualitatively, the velocity still decreases awayfrom the leading edge, and its spatial profile is qualita-tively the same as in the viscous Newtonian case. Thus,the viscoelastic Maxwell model is a possible descrip-tion of the actin network rheology.

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Supplemental figure 1: U2OS data and model fit for lamella flow speed as a function of force. (A) Actin flow speed in the lamella as a function of Y‐compound concentration. (B) Total measured traction force as a function of Y‐compound concentration. (C) Lamellar flow speed as a function of total force (black dots), fit to model prediction (eq. 11, Vmax=6.6nm/s, Fk=23nN). (D) Model fit (black line) to experimental data (blue dots) for actin flow speed in a U2OS cell treated with 2mM Y‐compound. Parameters:  =50, =1000, v*=100, b=1.5, F=300 (E) Model fit to traction data for same cell as in (d). 

Supplemental figure 2: Model dependence on . Unless otherwise stated, parameters are: F=b=0, v*=0.3, =1000.  (A) Position of traction peak (black dots) and actin flow speed (in units of v*) corresponding to traction peak (blue exes) as a function of . (B‐E) Actin flow, traction, myosin density, and adhesion density as a function of distance from the leading  edge. Black dashed line in (b) corresponds to  v=v*.

Supplemental figure 3: Model dependence on . Unless otherwise stated, parameters are: F=b=0, v*=3, =1000.  (We set F=0, in order to illustrate the dependence of the myosin distribution on for the same flow profile in each case, rather than for changing flow profiles that would occur for F>0). (A) Actin flow speed (black) and traction (red) as a function of distance from the leading edge. (B) Myosin distribution for different values of . (C) Myosin gradient for same values of  as in (B). 

Supplemental figure 4: Model dependence on b . Unless otherwise stated, parameters are: =3, F=1, v*=0.3 and =1000. (A) Position of traction peak in mm (black dots) and actin flow speed (in units of v*) corresponding to traction peak (blue exes) as a function of b. (B‐F) Actin flow, traction, myosin density, myosin force, and adhesion density as a function of distance from the leading edge. Black dashed line in (B) corresponds to v=v*. (G‐I) Balance of forces on actin network (traction (black), myosin force (blue), internal viscous force (red)), for myosin strength b=10 (G), b=40 (H), and b=100 (I).

Supplemental figure 5: Variation of model with spatially uniform myosin force (Fmyo= F). Unless otherwise stated, parameters are: =3, v*=0.3, b=15 and =1000. (A‐D) Actin flow, traction, adhesion density, and myosin density as a function of distance from the leading edge . Black dashed line in (b) corresponds to v=v*. (E‐F) Balance of forces on actin network (traction (black), myosin force (blue), internal viscous force (red)), for F=0.2 (E) and F=0.4 (F).

Supplemental figure 6: Variation of model with uniform myosin force in the lamella (represented by a step function: Fmyo= F for x>5m, Fmyo=0 for x<5m). Unless otherwise stated, parameters are: =3, v*=0.3, b=15 and =1000. (A‐D) Actin flow, traction, adhesion density, and myosin density as a function of distance from the leading edge . Black dashed line in (B) corresponds to v=v*. (E‐F) Balance of forces on actin network (traction (black), myosin force (blue), internal viscous force (red)), for F=0.2 (E) and F=0.4 (F).

Supplemental figure 7: Variation of model with myosin force proportional to myosin density (Fmyo=FM). Unless otherwise stated, parameters are: =3, v*=0.3, b=15 and =1000. (A‐D) Actin flow, traction, adhesion density, and myosin density as a function of distance from the leading edge. Black dashed line in (b) corresponds to v=v*. (E‐F) Balance of forces on actin network (traction (black), myosin force (blue), internal viscous force (red)), for F=2 (E) and F=4 (F).