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Broad-tailed force distributions and velocityordering in a heterogeneous membrane model

for collective cell migration

Tripti Bameta, Dipjyoti Das, Sumantra Sarkar, Dibyendu

Das and Mandar M. Inamdar

EPL, 99 (2012) 18004

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EPL, 99 (2012) 18004 www.epljournal.org

doi: 10.1209/0295-5075/99/18004

Broad-tailed force distributions and velocity ordering in a

heterogeneous membrane model for collective cell migration

Tripti Bameta1, Dipjyoti Das1, Sumantra Sarkar1, Dibyendu Das1(a) and Mandar M. Inamdar2(b)

1Department of Physics, Indian Institute of Technology - Bombay, Powai, Mumbai-400 076, India2Department of Civil Engineering, Indian Institute of Technology - Bombay, Powai, Mumbai-400 076, India

received 4 April 2012; accepted in final form 14 June 2012published online 10 July 2012

PACS 87.18.Hf – Spatiotemporal pattern formation in cellular populationsPACS 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion

Abstract – Correlated velocity patterns and associated large length-scale transmission of tractionforces have been observed in collective live cell migration as a response to a “wound”. We arguethat a simple physical model of a force-driven heterogeneous elastic membrane sliding over aviscous substrate can qualitatively explain a few experimentally observed facts: i) the growth ofvelocity ordering which spreads from the wound boundary to the interior; ii) the exponential tailsof the traction force distributions; and iii) the swirling pattern of velocities in the interior of thetissue.

Copyright c© EPLA, 2012

Introduction. – The phenomenon of collective cellmigration arises in biological processes of morphogenesis,wound healing, as well as cancer growth, and is an activetopic of current research interest [1–7]. To understand thebasic features in collective cell migration as a responseto wound healing, two-dimensional monolayer patches ofMadin-Darby canine kidney (MDCK) cells on deformablesubstrates have been studied in different experiments[4,6–8].For a physical scientist, there are many interesting

aspects that these experiments reveal. The spatiallyheterogeneous swarming and swirling velocity patternsexhibited by the cells, studied by particle image velocime-try [6,7], are reminiscent of similar pattern formationin active nematics and driven granular matter [9]. Astime passes, a zone of velocity order starting from thewound boundary invades the interior of the MDCKtissue [7], reminding one of phase ordering kinetics [10].On the other hand, another set of experiments [4,8]have shown that the local traction forces exerted by theMDCK cells on the substrate have large fluctuations—the distribution of the forces being non-Gaussian withdistinct broad exponential tails, akin to force distri-butions in static granular piles [11]. Yet, the MDCKcells forming the tissue are held to each other and tothe substrate by a network of Cadherin and Integrin

(a)E-mail: dibyendu@phy.iitb.ac.in(b)E-mail: minamdar@iitb.ac.in

proteins, respectively [1,12], and they self-generate activeforces due to internal Actin and Myosin dynamics. Thus,at the microscopic level, they show no resemblance tomechanically driven loose granular rods or discs. Needlessto say, it is quite a challenge to model every observedfeature of the MDCK tissue system, as seen in differentsets of experiments. In this paper, we propose a simplestatistical-mechanical model for the system, in the spiritof toy models in biophyics [13]. We show that the modelcan simultaneously give a qualitative explanation of thegrowth of velocity ordering with time, and of the largeforce fluctuations.It is well known that the behaviour of large cell

collectives [14] is qualitatively distinct from that of asingle cell [15]. Attempts have been made to model cellassemblies incorporating signal transduction and cell-cellsignaling in Dictyostelium discoideum [16]. Collectivecell migration studies incorporating cell division has beendone [17–21], but the experiments that we are concernedwith [4,6,7] have noted, that over the relevant time-scales,the growth of cell number via cell division is not expectedto play a role in the features of interest in this paper. Thegeometrical instabilities such as fingering and tip splittingof the wound boundary in experiments [6,7] have beentheoretically modelled using ideas of interface growthkinetics [2,22]. The velocity patterns in the interior of thecell sheet [6,7] have recently been studied by a mechanicalmodel, where the cells with a local orientation fieldcollectively behave in a viscoelastic fashion [3]. We show

18004-p1

Tripti Bameta et al.

below that a much simpler model compared to these cannevertheless capture two key features of the MDCK tissuesystem, namely, the velocity ordering invasion, and thelarge force fluctuations. In particular, the novel aspect ofour model is to show the crucial role that heterogeneityplays in determining the large force fluctuations.

Model and parameters. – We model the cell sheet asan elastic membrane, although some earlier studies havetreated cell monolayers as viscoelastic [3]. The motiva-tion for this choice comes from the experimental find-ing [7] that the average distance between MDCK cellsdoes not change for many hours. It was clearly concludedthat the movements of the cells are correlated showingvery limited rearrangement, making the monolayer locallymore elastic than viscous [7]. Similarly, a recent exper-imental study of internal stresses in cell monolayers [5]have also assumed the latter to be elastic. In our model,the cells are represented as discrete points in continuousspace, and the cell-cell cadherin connections are repre-sented by simple harmonic springs of effective stiffness κ0.The elastic membrane is considered heterogeneous withκ0 drawn from a probability distribution function (p.d.f)P (κ0). Since there is inherent randomness in the numberof intercellular cadherin connections [23], and the possi-bility of any such connection to be in multiple distinctstructural states with different mechanical properties [24],our assumption of heterogeneity seems reasonable. Next,it is expected that the connections of the cells with thesubstrate via the integrin proteins will break and remakeas the tissue advances [12]. For a single cell, it has beentheoretically demonstrated that this complicated processof cell-substrate interaction can be effectively replaced bya linear viscous drag [25]. We extend the latter at thetissue level and assume that the substrate exerts a localdrag force −c0Vi where Vi is velocity of the i-th cell andc0 is the drag constant. The average position of the “woundboundary” (see fig. 1) defines the Y -direction, and thedirection orthogonal to it will be called X. For simulat-ing, we take a N ×N tilted square grid of points (fig. 1),but while thinking of a continuum limit, we will assumeN →∞ such that effectively the wound boundary will bevery far from the center of the tissue (as in actual exper-iments [4,7]). Finally, we supply the “live thrust forces”(originating from the cytoskeletal acto-myosin activityin the cells) by hand in two alternate ways: i) Cell-iwith position Ri = (Xi, Yi) is given a space and timedependent random force Fi =Fave,i+ηi, with Fave,i =−F0 exp(−ni/ξ)x. Here ni is the row number, from theboundary, of the i-th cell. The boundary row is numbered0, and ξ is a length scale. The components of ηi ≡(ηX , ηY )i are Gaussian white noise with zero mean and〈ηα,i(t1)ηβ,j(t2)〉= 2σ exp(−ni/ξ)δi,jδα,βδ(t1− t2) with αand β taking values X,Y . We will refer to this as partic-ipatory model (PM) (see fig. 1). Thus, in the PM model,the magnitude of the noise on force, just like the meanforce, decays with increasing distance from the boundary.ii) Only the cells at the wound boundary row are given a

X

Y

Drag

Fig. 1: (Color online) A schematic picture (top view) of thedeformed discretized membrane modeled as a tilted squarelattice with undeformed spring length a0 of unity. The woundboundary cells move towards −X. The cells are denoted bythe circles and the active forces are denoted by arrows (sizeproportional the force magnitude) on them. The varying springstiffnesses are denoted by multiple colors. The overall averagesubstrate resistance is denoted by “Drag”.

space and time dependent random force with an averageforce Fave,i =−F0δni,0x added to Gaussian white noise ηi,with zero mean and 〈ηα,i(t1)ηβ,j(t2)〉= 2σδi,jδα,βδ(t1−t2) (α≡ {X,Y }, β ≡ {X,Y }). We will refer to this as theleader-driven model (LDM).The motivation for comparing PM vs. LDM comes from

the discussions in ref. [26] following the experiment ofref. [4]. The mathematical equation used to simulate thesystem is

c0dRidt=∑

j

κij0 (|Rj −Ri| − a0)eij +Fi. (1)

The inertial term has been dropped as the system isclearly overdamped. The index j in the sum in eq. (1)goes over nearest neighbours of i and κij0 is the randomstiffness constant of the spring connecting i and j. Theunit vector eij = (Rj −Ri)/|Rj −Ri|, and a0 is the unde-formed length of the spring.Here we report numerical results with parameter valuesc0 = 10, N = 128, F0 = 1, σ= 0.2, a0 = 1, and P (κ

ij0 ) is a

uniform box distribution between 0.25 to 0.75. We willshow later that these choices of parameters lead to reason-able correspondence to experiments. In the presence ofthe active applied force Fi, velocities are obtained directlyusing eq. (1), and positions from numerically integratingVi =dRi/dt. The simulation results for figs. 2, 3, and 4,are obtained by providing the cells with zero initial veloc-ities, and initial random displacements (from the equi-librium positions) with components uniformly distributedover [−δ, δ]. This may be expected to be a generic initialcondition for the cell collective. Periodic boundary condi-tions are assumed along Y .

Results. – In this section we present the three impor-tant results of our model.

18004-p2

Broad-tailed force distributions in heterogeneous membrane model

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

(a) (b)

(c) (d)

t1/2

t1/2

(h1/2

)

Φ Φ

LDMPM

0

0.2

0.4

0.6

0.8

1

1 3 5

10-5

10-4

10-3

10-2

10-1

0 20 40 60n

-−Vx

LDM

PM

t

X

φn-0.2 0.2 0.6 1

Fig. 2: (Color online) Velocity ordering in the cell sheet usingδ= 0.025. (a) Velocity of the cell points as a function of rownumber for two different times (100 and 1000). The circles andsquares represent simulation results for PM (ξ = 5) and LDM,respectively, and the fitting lines correspond to the theoreticalmodel in eq. (2). (b) The order parameter φn at spatial rowposition x of the cell at different times t. The x-range is 0–120and t-range is 0–1000. The velocity ordering front is clearlyvisible. (c) The order parameter Φ for the complete tissue, asa function of time t, with a fit of t1/2 and t for the LDM andPM, respectively. (d) Time dependence of experimental orderparameter in [7] fitted with t and t1/2.

100

101

102

103

104

-0.25 1 2

P(T

x)

(a) (b)

(c) (d)

Tx

-1 0 1

100

101

102

103

104

P(T

y)

Ty

100

101

102

103

104

0 0.5 1

P(T

x)

Tx

-0.25 0 0.25

100

101

102

103

104

P(T

y)

Ty

Fig. 3: (Color online) Distributions P (TX) and P (TY ) oftraction forces TX and TY (in units of 10

−3) for LDM, att= 300 and δ= 0.025. Gaussian fits are shown for uniform κij0in (a) and (b). For random κij0 , deviation from Gaussianity,and resulting asymptotic exponential tails are shown in (c) and(d). The data is for layers 18 (filled symbols) and 19 (emptysymbols).

Invasion of velocity ordering. We first proceed toshow how under the action of active forces the cellvelocities acquire a bulk ordering. As the active forces arepreferentially oriented along −x (for both the LDM andPM), the velocities of the cells, starting with the ones closeto the boundary and followed by the ones in the bulk,gradually orient themselves towards −x. This velocityordering is shown in fig. 2(a) —with increasing time, forboth the LDM and PM, mean row velocity −VX of deeperlayers (with larger n) rise in magnitude. Thus, an order-disorder boundary moves towards larger n. Interestingly,

100

101

102

103

-2 0 2 5

P(T

x)

(a) (b)

(c) (d)

Tx

-3 0 3

100

101

102

103

P(T

y)

Ty

100

101

102

103

0.5 3 6

P(T

x)

Tx

-2 0 2

100

101

102

103

P(T

y)

Ty

Fig. 4: (Color online) Distributions P (TX) and P (TY ) oftraction forces TX and TY (in units of 10

−3) for LDM, att= 300, for a distorted lattice with random bond lengths.The uniform stiffness constant k0 = 1. Gaussian fits are shownfor ǫ= 0.15 in (a) and (b). For ǫ= 0.45, deviation fromGaussianity, and resulting asymptotic exponential tails areshown in (c) and (d). The data is for layers 18 (filled symbols)and 19 (empty symbols).

there is a quantitative difference between the two models—the shape of the −VX(n) curve is Gaussian for LDM andhas an exponential tail for PM. A local order parameterfor every row n can be defined as φn =−VX,i/|Vi| (withall velocities −VX,i < 10−14 set to zero to avoid spuriouscontributions). In fig. 2(b), φn is plotted (with largermagnitude corresponding to lighter color) as a functionof space X and time t for LDM —the movement of theorder-disorder boundary with increasing t is clearly seen.A similar plot was observed experimentally in [7]. Next,a global order parameter for the whole system can bedefined as Φ= 〈φn〉n. In fig. 2(c), Φ is shown to increaseas ∼ t1/2 for LDM, and ∼ t for PM. To compare withthe experiments, we have plotted the experimental datafor Φ from [7] in fig. 2(d); two curve-fits of ∼ t1/2 and∼ t are put against the data, showing that both theseforms work reasonably well. Thus, we have demonstratednumerically that our models have similar velocity orderingas seen experimentally in MDCK tissues [7] during woundhealing. We will now proceed to understand analyticallywhich ingredients of our model are essential for the abovephenomenon, and in particular the reason for quantitativedifferences between LDM and PM.It is interesting to note that the growth of VX(t)

above can be understood analytically from an analogous1-dimensional problem. We can solve the one-dimensionalproblem of a pulled, non-disordered, elastic chain:

c0∂u

∂t= κ0

∂2u

∂x2+F (x, t). (2)

The above equation is a simple 1-D, linearized, contin-uum version of eq. (1). Here u(x, t) is the displacementof any cell, x(∈ [0, L]) is the continuum space variablecorresponding to the row number ni, and the non-randomforce is F =−F0 exp(−x/ξ), with ∂u/∂x= 0|x=0 (for PM),

18004-p3

Tripti Bameta et al.

and F = 0, with ∂u/∂x= F0/κ0|x=0 (for LDM). The initialcondition is taken as u(x, 0) = 0 both for PM and LDM.Equation (2) can be solved for velocity v(x, t) = ∂u/∂t inthe limit of large system size (L→∞) with the bound-ary condition u(∞, t) = 0. This gives unique analyticalsolutions:

PM: v(x, t)− vCM = −F02c0e(τ−x)

(

1+Erf

[

x− 2τ2√τ

]

+e2x Erfc

[

x+2τ

2√τ

])

, (3)

LDM: v(x, t)− vCM =−F0√c0πκ0

e−c0x2/4κ0t

√t

. (4)

In eq. (3) the symbols x= x/ξ and τ = κ0t/c0ξ2 are scaled

dimensionless space and time, respectively, and Erf andErfc refer to Error and Complementary Error functions[27], respectively. The profile of the ordered velocity as afunction of x in the 1-D LDM model (eq. (4)) is clearlyGaussian, while in the PM model it is modified (eq. (3))to have an exponential profile for large x. The agreementin mathematical forms between the 1-D analytical resultand the 2-D simulation result in fig. 2(a) shows thatthe velocity ordering phenomenon is not particularlydependent on stiffness randomness or noise, and its essenceis captured even in a 1-D problem. As can be seen fromfig. (2)c, the PM model gives a growth law Φ∼ t for atransient period, the reason for which may be understoodfrom eq. (3). For large x, ln(−v(x))∼ τ − x+const, toleading order, while for small x, ln(−v(x))∼ τ − x2f(τ)+const (see PM in fig. 2(a)), where f(τ) is a function of τ .Thus, there is an short distance quadratic profile, followedby a long distance linear profile in x. Spatial expanse of thequadratic profile keeps increasing, and beyond a certaintime the growth law in PM will become ∼ t1/2 just likeLDM.To see if a drive from the boundary cell layer without

any participation from the cells in the bulk can invokeother experimentally observed phenomena, henceforth wefocus on LDM. While this simple approach provides usefulinsights, since the actual migrating cell sheets have thrustsprovided also by bulk cells, and possible mechanical feed-backs [4,5], the LDM may not be quantitatively reliable.

Traction force fluctuations. A second interestingresult of our model is that the traction force fluctuationsare unusually large as in the experiments in ref. [4]. Inour model, the local effective traction force on any cellis Ti =Fi− c0V (see eq. (1)). In LDM, Fi = 0 (for anynon-boundary cell) and so T’s are proportional to thelocal velocities V. The probability distributions of thecomponents TX and TY for the LDM are shown in fig. 3.For homogeneous membrane (constant κij0 ), the distri-butions are clearly Gaussian (figs. 3(a), (b)). To makesure that the latter is not a trivial consequence ofthe Gaussian distributed random thrust forces in theboundary layer, we checked the traction distributionswhen the boundary forces were drawn from a i) box,

and ii) an exponential distribution. In both of thesedistinct cases, we found (data not shown) that thetraction force components TX and TY are Gaussiandistributed. Thus, there is no doubt that central limittheorem (CLT) is valid and due to it, the local forces inthe bulk (being sum of random neighbouring forces) turnout to be normally distributed. On the other hand, for aheterogeneous membrane (random κij0 ) the distributionsdevelop exponential tails (figs. 3(c), (d)), indicating adeparture from the CLT. The shape of the curves ofP (TX) (with mean at TX = 0) and P (TY ) (with meanat TY = 0) have qualitative resemblance to experiments—in particular, the widths decrease with the increase ofthe distance from the wound.Breakdown of CLT in [4] is a priori quite intriguing.

It was speculated in ref. [4], that a q-model [11] likemechanism may be at play. Recognizing that the tissueis heterogeneous, here we are specifically suggesting thatan effective q-model like mechanism may arise due tounequal stress propagation and accumulation mediated bythe random cadherin connections. Since the possibility ofmanipulating the strengths of cadherin connections hasbeen experimentally demonstrated [5], our result is opento experimental test.The membrane can also be made heterogeneous in

another way by introducing variable bond lengths of thecell-cell connections, while keeping the spring stiffnesshomogeneous. This naturally leads to deviation fromregular lattice symmetry considered so far. The bondlengths aij0 are made disordered by imparting new randomequilibrium positions to the cells. To do so, the cellsare shifted from the regular lattice by ǫx (along-X) andǫy (along-Y ), where ǫx and ǫy are drawn from uniformdistribution over [−ǫ, ǫ]. As can be seen from eq. (1), themagnitude and direction of the spring force is independentof the bond length (to the first order). Thus for “small”lattice distortion the distributions for TX and TY areexpected to be the same as that of the uniform non-distorted lattice. This is seen in our simulations for achoice of ǫ= 0.15 —the traction distributions are indeedGaussian (see figs. 4(a) and (b)). Contrary to this, with“large” random lattice distortion, one would expect fromeq. (1) random harmonic forces, leading to a departurefrom the latter result. We indeed see this when we makeǫ large (say 0.45) —the traction distributions developexponential tails as shown in figs. 4(c) and (d). Thus,in two types of membrane heterogeneity —random springstiffnesses and random bond lengths— we have found thatnon-Gaussian traction force distributions arise.

Swirling patterns in the bulk. We now turn ourattention to velocity patterns which develop in the bulkof the system. The cells in the confined region of theexperimental setup in refs. [6,7] are expected to beunder internal stress due to the confinement from theboundary before the cell sheet is allowed to expand. Inorder to mimic this internal stress, we provide genericrandom initial positions to our cell lattice. The elastic

18004-p4

Broad-tailed force distributions in heterogeneous membrane model

0

0.04

0.08

0 5 10 15 20

Cvv(− n

)

−n

Fig. 5: The velocity-velocity correlation function Cvv againstscaled distance n along Y , at t= 500 (•) and t= 1000 (◦)for LDM using δ= 0.45. Inset: velocities of cells in the bulkregion (dimension 8× 8) are shown (at t= 1000) —swirls canbe clearly seen.

relaxation of this “pre-strained” sheet help excite spatialmodes through the harmonic couplings. The smallerwavelength modes damp out faster, leaving large wave-length velocity swirls at late times (inset of fig. 5 forLDM). To precisely quantify the correlations in thesepatterns, we show the velocity-velocity correlationfunction Cvv(n) = 〈v(n0).v(n0+ n)〉/〈v2〉 in fig. 5; heren= y/

√2a0 is the scaled distance in units of average

inter-cellular spacing along Y . The average 〈· · · 〉 is doneover ensembles as well as cell locations n0, belonging toa strip in the bulk region where 〈v(n0)〉= 0. We notethat Cvv(n) shows change of sign beyond some layers (asseen also in experiments [28]) reflecting the bending ofvelocity field over space. The correlation range increaseswith the time of observation as expected —it is ≈ 8–10cell layers which is roughly similar as distances seen inexperiments [7,28]. The correlations observed in ref. [28]are certainly influenced by substrate deformability andcell birth. But the fact that such correlations are alsoobserved otherwise [6,7] indicates that they may not bethe only factors governing the swirling patterns. Althoughour model cannot make these distinctions, we note thatit elucidates the role of inherent elasticity and pre-strainof the cell sheet in producing such patterns.

Numerical estimates of length, time and force scales.

We have shown so far that a boundary layer driven hetero-geneous elastic sheet can produce qualitatively manyexperimental observations in collectively migrating epithe-lial cells. We now make numerical estimates of variousquantities to find out whether our results are quantita-tively meaningful in comparison with the experiments.As shown in fig. 1, the bond length a0 is unity. In

real units it may be taken as 20μm (the average cell-cell separation [7]). To estimate the unit of time t0, westart from figs. 2(c) and (d). For the LDM in 2(c), the

slope of the line in units of t−1/20 is 0.0139, while the

experimental data in 2(d) has a slope 0.139 h−1/2. Equat-ing the two, gives t0 ≈ 0.01 h = 36 s. This tells us thatthe non-dimensional velocity in fig. 2(a) in the bound-ary layer (for LDM) ∼ 10−2 corresponds to 10−2a0/t0 ≈20μm/h. This is in the ballpark of the velocities quoted

in experiments [7]. The curves in fig. 2(a) are for timest= 1h (100 in simulation) and 10 h (1000 in simulation).To make contact of forces in fig. 3 with experiments,we choose to first obtain the bond stiffness κ0 in realunits. The properties of a continuum cell sheet may bechosen as those appearing in the supplementary materialof [5]: Young’s modulus E = 10 kPa, Poisson’s ratio ν =0.5, and sheet thickness of h= 5μm. We first consider acell sheet of dimension a0× a0×h, on which homogeneoustension 1Pa is applied. The relative change in surface areaΔA/A= 2(1− ν)/E = 10−4. On the other hand, applyingequivalent edge force of F = 1Pa× a0×h= 10−10 N on aa0× a0 square, whose sides are made of springs of stiff-ness κ0, we get ΔA/A= F/a0κ0. Equating the two rela-tive area changes, we get κ0 = 0.05N/m. But we have usedan average stiffness value of 0.5 in dimensionless unitsin our simulation. This gives the actual force unit to bef0 = 0.05× a0/0.5N= 2× 10−6N. In dimensionless unitsour forces in fig. 3 for the 19th layer are ∼ 0.5× 10−3,which is equivalent to 10−9N. This implies a traction of10−9/a20 = 2.5Pa, which is one order of magnitude lowerthan that reported in [4]. The dimensionless time 300reported in fig. 3 translates to ≈ 3 h. Using a0, t0 andf0, we see that the dimensionless value of c0 = 10 used inour simulation, is equivalent to 10× f0t0/a0 ≈ 36N s/min real units. This can be converted into drag coefficientζ = c0/a

20 = 25pN h/μm

3, which is an order of magnitudelower than reported in [3]. In fig. 5, at our simulationtime t= 1000 (equivalently ≈10 h) the length scale asso-ciated with the first minimum of the correlation curve is5× a0 = 100μm. In [28] at around 10 h, a similar reportedlength scale is 200μm (or ≈10 cell layers), which is higherthan our result only by a factor of two. Thus we see that,although our model is simple, it can make reasonably closecontact to experiment even quantitatively.One serious drawback is that our boundary tractions are

very large, ∼103 times compared to the bulk —in reality[4] forces do not diminish so fast. It would be interestingto modify LDM in the future by incorporating cellularthrusts from the bulk, to see if the description becomesmore realistic in a quantitative sense.

Discussion and conclusion. – Recent experimentson collective migration of MDCK cells have raised severalinteresting puzzles, which in turn have spurred varioustheoretical modeling attempts. Before we summarize themain results of this paper, we would like to situate ourwork with respect to the contributions made by theearlier theoretical models. The model of ref. [3] treats thecell sheet as a viscoelastic medium supplemented with adirector field to describe the local cellular orientations.This model obtains the dependence of the velocity ofthe wound boundary on the viscoelastic parameters ofthe cell sheet, and also shows complex correlated velocitypatterns in the bulk. Another model [2] concentratesspecifically on the dynamics of the boundary of the cellsheet. By introducing a competition between the curvaturedependent driving force, and the elastic and viscous

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resistance of the cell sheet, the fingering instability as seenin experiment [7] is reproduced by this model. In contrastto these models, we treat the cell sheet as an elasticmembrane, and hope to capture some of the phenomenaat early times. This is motivated by a direct experimentalobservation [7], and supported by treatment of cell sheetas an elastic material in another set of experiments [5]. Wenote that the phenomena we address in this paper, namely,the growth of bulk velocity order parameter [7], andtraction force distributions [4] have not been addressed inthe aforementioned publications [2,3]. At the same time,the LDM cannot produce pronounced fingering due tolack of flowy behaviour, and absence of coupling betweenaverage edge thrusts and boundary surface geometry.A very recent paper [20], which studies the effect of cellproliferation and migration leading to contact inhibition,introduces a simple one-dimensional model, where the cellsplastically spread in presence of cellular thrust forces fromthe boundary (similar to LDM in our paper). Nevertheless,since this model is one-dimensional, quite naturally, itcannot capture the two-dimensional phenomenology.In this paper we have identified few minimal mechani-

cal ingredients —heterogeneous elastic membrane, fluid-viscous drag, and the active drive of cells from theboundary to mechanically pull the system— which canexplain three aspects of collective cell migration: a) macro-scopic velocity ordering; b) breakdown of CLT for tractionforce fluctuations; and c) velocity correlations associatedwith swirls. Perhaps the most interesting result is, thatwithout resorting to any ad hoc physical or biologi-cal mechanisms, membrane heterogeneity can naturallyinduce non-Gaussian tails in the distribution of tractionforces. The mechanism that we propose here is very remi-niscent of the q-model for static granular assemblies [11].At the same time, we would like to point out that there aresignificant differences of our model from the q-model. Wehave a mobile network of cells, velocity-dependent dissipa-tive forces, and tensile force transmissions, as opposed tothe static transmission of compressive forces in the granu-lar assemblies. These differences may invite further analyt-ical exploration of our current model in the future.The three results in the paper show a close qualita-

tive resemblance to the experiments. Even the quantitativeestimates seem reasonable, albeit with a major drawbackthat the values of traction forces and velocities dimin-ish much faster than the experimental values. This canbe attributed to the fact that our model does not pumpenergy in the interior of the cell layer through activecellular thrusts. Recent experiments [5] hint that cellu-lar polarizations and cellular active forces are possiblytied to mechanical stress cues from surrounding cells. Thisdemands our model to go beyond being purely mechanical,by including a coupled dynamics (a cross-talk) betweenactive cellular forces and mechanical harmonic forces.While we shall explore these in the future, we concludeby noting that this work sets a benchmark by showing theachievements and limitations of a rather simple mechani-cal model for collective cell migration.

∗ ∗ ∗

Dipjyoti Das thanks CSIR, India (JRF AwardNo. 09/087(0572)/2009-EMR-I). MMI thanks Prof. P.Banerji for helpful discussions.

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