royalsocietypublishing.org/journal/rsif

Research

Cite this article: Lin S-Z, Bi D, Li B, Feng X-Q.2019 Dynamic instability and migration modes

of collective cells in channels. J. R. Soc.

Interface 16: 20190258.http://dx.doi.org/10.1098/rsif.2019.0258

Received: 8 April 2019

Accepted: 3 July 2019

Subject Category:Life Sciences–Physics interface

Subject Areas:biophysics

Keywords:collective cell migration, geometric

confinement, mode transition, active vertex

model

Authors for correspondence:Bo Li

e-mail: libome@tsinghua.edu.cn

Xi-Qiao Feng

e-mail: fengxq@tsinghua.edu.cn

Electronic supplementary material is available

online at https://dx.doi.org/10.6084/m9.

figshare.c.4576166.

© 2019 The Author(s) Published by the Royal Society. All rights reserved.

Dynamic instability and migration modesof collective cells in channels

Shao-Zhen Lin1, Dapeng Bi2, Bo Li1 and Xi-Qiao Feng1

1Institute of Biomechanics and Medical Engineering, Applied Mechanics Laboratory, Department of EngineeringMechanics, Tsinghua University, Beijing 100084, People’s Republic of China2Department of Physics, Northeastern University, Boston, MA 02115, USA

BL, 0000-0002-3792-2469; X-QF, 0000-0001-6894-7979

Migrating cells constantly experience geometrical confinements in vivo, asexemplified by cancer invasion and embryo development. In this paper,we investigate how intrinsic cellular properties and extrinsic channelconfinements jointly regulate the two-dimensional migratory dynamics ofcollective cells. We find that besides external confinement, active cell motilityand cell crowdedness also shape the migration modes of collective cells.Furthermore, the effects of active cell motility, cell crowdedness and confine-ment size on collective cell migration can be integrated into a unifieddimensionless parameter, defined as the cellular motility number (CMN),which mirrors the competition between active motile force and passiveelastic restoring force of cells. A low CMN favours laminar-like cell flows,while a high CMN destabilizes cell motions, resulting in a series of modetransitions from a laminar phase to an ordered vortex chain, and furtherto a mesoscale turbulent phase. These findings not only explain recentexperiments but also predict dynamic behaviours of cell collectives, suchas the existence of an ordered vortex chain mode and the mode selectionunder non-straight confinements, which are experimentally testable acrossdifferent epithelial cell lines.

1. IntroductionCollective cell migration is of essential significance in many physiological andpathological processes, for example, embryonic morphogenesis, cancer metasta-sis, epithelial homeostasis and regeneration [1–5]. Rich migration modes havebeen observed in two-dimensional multicellular sheets, including collectivedirected motions, dynamic swirling and robust rotation [6–8], which bear aresemblance to the collective dynamics of bacteria, insects and ants [9–11].The collective migration modes of cells are found to be dependent, to differentextents, on the density [4,12] and motility [13,14] of cells and intercellularadhesion [8,15]. In a confluent cell monolayer, for example, enhancing cellmotility may result in a solid-to-fluid transition [13,16], while disruptingintercellular junctions may yield randomly uncorrelated cell motions [8,15].

Besides the intrinsic properties of cells, extrinsic cues such as geometricconfinement [7,8,17–21], chemical factors [22] and electric field [23] also affecttheir dynamic behaviours. Migrating cells in vivo are often subject to geometricconfinements from the environment (extracellular matrix or other cells). Typicalexamples include the invasion of cancer cells into the porous peritumoralstroma [24] and the movement of border cells in Drosophila ovaries [25].In vitro experiments showed that a confluent Madin–Darby canine kidney(MDCK) cell monolayer confined to a circular patterned substrate may exhibiteither sustained global rotation or local swirling, depending on the size of con-finement [7,17–19]. Cells confined in a strip channel may exhibit eitherunidirectional cohort migration when the channel is narrow or local swirlingwhen the channel is wide (figure 1). These findings evidence the significantrole of geometric confinement in collective cell dynamics. To date, however,the biophysical mechanisms underpinning the rich dynamic migration modes

incr

easi

ng s

trip

wid

th

Figure 1. Experimental observation of migration mode transition of MDCK cell sheets on fibronectin strips with different width. Scale bars, 50 µm. Adapted from [8]with permission. (Online version in colour.)

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of collective cells under confinement remain elusive. Inthis work, we investigate the dynamic migration modesand their transition in coherent epithelial sheets confined ina straight channel. Our attention is focused on the regulatingroles of active cell motility, cell crowdedness and confinementsize. A new dimensionless parameter, referred to as the cellu-lar motility number (CMN), is defined to quantify theirroles in the emerging collective cell dynamics. It is foundthat a small CMN favours a laminar cell flow mode, and anincreased CMN will give rise to a self-sustained turbulentmode, reminiscent of the laminar-to-turbulent transition inpipe flow. In the mode-transition phase, a chain of stablevortex structures with alternately reversed handedness mayspontaneously emerge in the confined multicellular sheet.A scaling law between the cellular turbulence fraction andthe CMN is derived to elucidate the critical behaviour ofthe migration mode transition and to reveal the differencebetween the confined cellular motion and the classicalconfined flow of passive fluids.

2. Biophysical modelWe investigate collective cell migration in a coherentmonolayer by using an active vertex model. Cells in themonolayer are modelled as polygons (figure 2a). Due to theconfluency, they form an inter-connected polygonal tiling.The shapes and dynamics of cells are determined fromthe position of vertices, given by ri(t), which obeys theoverdamped equation of motion [26–29]

dridt

¼ 1gfUi þ

XJ[Ci

v0eJnJ

, ð2:1Þ

where g is the friction coefficient and fUi ¼ �@U=@ri denotesthe potential force acting on vertex i. nJ refers to the numberof connecting neighbours of cell J, and

PJ[Ci

computes a

summation over all neighbouring cells Ci of vertex i. v0 isthe self-propelled velocity accounting for the active motilityof cells, and eJ ¼ (cos uJ , sin uJ) is the polarity vector of cellJ. The evolution of polarity direction uJ satisfies [28]

duJdt

¼ mLA

nJ

XK[CJ

sin (u(vel)K � uJ)þ mCIL

nJ

XK[CJ

sin (aJ,K � uJ)

þ 1hJ(t), ð2:2Þ

where the first two terms on the right-hand side quantify theregulations of two competing kinds of intercellular socialinteractions: local alignment (LA) and contact inhibition oflocomotion (CIL), respectively. LA characterizes the tendencyof a migratory cell to follow its neighbours [26,30,31], resem-bling a Vicsek alignment interaction [32], whereas CILquantifies the repulsion effect between cells [33–36], mimick-ing the repolarization of cells upon contact. To date, themolecular mechanisms of LA and CIL have not been fullyunderstood [26,35]. Here, therefore, we omit the biochemicalsignalling details involved in these intercellular interactionsbut introduce two parameters, mLA and mCIL, to denote theintensities of LA and CIL, respectively. aJ,K ¼ arg (rJ � rK)represents the relative argument of cell J to its neighbour K,with rJ being the geometric centre of cell J. vK ¼ drK=dt isthe velocity of cell K, and u(vel)K ¼ arg (vK) is the correspondingargument.

PK[CJ

sums over all neighbouring cells CJ of cell J.The third term in equation (2.2) accounts for the effect ofnoise on cell re-orientation, where 1 denotes the intensity ofnoise and hJ(t) are independent unit-variance Gaussianwhite noises, satisfying hhJ(t)hK(t

0)i ¼ dJKd(t� t0), with dJKand d(t� t0) being the Kronecker delta and the Dirac deltafunction, respectively. It is emphasized that cell polarity canalso be affected by some other factors, e.g. cell memory[16,27,30,37] and intercellular mechanical interactions [38],which are either individual cell-based or passive. Here,we mainly account for the effects of two kinds of activecompeting intercellular social interactions (LA and CIL) on

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motion direction270°

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A B

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D

i j

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C

D

i¢j¢

inte

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iate

(a)

(b) (c)

Figure 2. Collective cell migration in a multicellular sheet under channel confinement. (a) Schematic of a coherent cell monolayer confined in a straight channel. Thelength and the width of the channel are Ls and Ws, respectively. Periodic boundary conditions are applied at the channel extremities x ¼ +Ls=2. At the interfacebetween cells and the channel, the cellular vertices are allowed to slide along the wall y ¼ +Ws=2. (b) Schematic of a T1 topological transition. When twoseparating cells (A and B) approach towards each other too close, they may squeeze out the neighbouring cells between them (C and D), breaking the original cell–cell interface (ij), and adhere to each other, forming new cell–cell interface (i0 j0). (c) Typical migratory modes and dynamic patterns of collective cells confined in achannel, including (I) unidirectional flow, (II) vortex chain and (III) mesoscale turbulence, emerge in the channels with different widths. The coloured arrows rep-resent the velocity vectors of cells, with the arrow length proportional to the velocity magnitude and the colour indicating its direction. The white solid triangles andsquares indicate the centres of right- and left-handed rotating swirls, respectively. (Online version in colour.)

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cell re-orientation. The roles of LA and CIL in regulatingcollective cell migration have been explored previously[26,28].

Considering cell area elasticity, cell contraction and inter-cellular interfacial tension, the potential energy of the systemis expressed as [28,39]

U ¼XJ

12Ka(AJ � A0)

2 þXJ

12KcL2J þ

X(i,j)

Llij, ð2:3Þ

where Ka denotes the cell area stiffness, A0 refers to thepreferred area, and AJ is the current area of cell J. Kc rep-resents the contractile modulus, and LJ is the perimeter ofcell J. L quantifies the interfacial tension between neighbour-ing cells, and lij is the length of common edge ij. The numberdensity of cells per unit area, rcell ¼ N=SJAJ , is defined toquantify cell crowdedness. The first term on the right-handside of equation (2.3) represents cell area elasticity arisingfrom the resistance to pulling/pushing from neighbouringcells, the second term describes active cell contraction of theactomyosin cortex, and the third term denotes the interfacialtension resulting from the competition between cell–celladhesion and cell cortical tension.

Equations (2.1)–(2.3) govern the collective dynamics in acoherent cell monolayer mediated by intercellular guidingcues. To facilitate simulations, these equations are normalizedin dimensionless forms via the length scale ‘ ¼ ffiffiffiffiffiffi

A0p � 30 mm

and the time scale t ¼ g=(KaA0) � 100 s, as estimated from

previous experimental measurements (see Material andmethods). Normalized key parameters include

~Kc ¼ Kc

KaA0, ~L ¼ L

KaA3=20

, ~mLA ¼ mLAt, ~mCIL ¼ mCILt,

~v0 ¼ v0tffiffiffiffiffiffiA0

p , ~rcell ¼ rcellA0, ~Ws ¼ WsffiffiffiffiffiffiA0

p , ~1 ¼ 1ffiffiffit

p:

9>>>=>>>;

ð2:4Þ

In this study, we focus on how active cell motility, cellcrowdedness and confinement size (i.e. channel width) coor-dinate to dictate collective cell dynamics under channelconfinement, and thus fix other parameters. Based on sys-tematic robustness tests, unless stated otherwise, we use thefollowing dimensionless parameter values in the simulations:~Kc ¼ 0:02, ~L ¼ 0:1, ~mLA ¼ 0:05 and ~mCIL ¼ 1:0 (Material andmethods). Notably, we examine the deterministic case ofcollective cell dynamics under channel confinement, andthus set the noise intensity ~1 ¼ 0. Including the regulatoryrole of noise is of interest and merits future work.

Throughout the paper, quantities decorated with tildestand for the dimensionless form of the correspondingparameters. In our simulations, periodic boundary conditionsare applied on the channel extremities. A uniform L isassumed across the whole monolayer, including all boundarycells. For cell vertices adhered to the wall, however, weintroduce an extra simulation constraint, where the boundary

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vertices may slide along the wall but are not allowed todetach from it (figure 2a). In this case, the adhesion betweencells and the wall will not affect collective cell migration, inas-much as the cell–wall adhesion energy remains unchanged.Besides, the boundary vertices of cells adhering to the wallmay have different coefficients of friction from the inner cellvertices due to the constraint of the wall. However, our simu-lations show that when the channel is wider than the size ofapproximately three cells, the coefficient of friction of bound-ary vertices has no significant influence on the collective cellmigration mode (see Material and methods). Therefore, weset the coefficient of friction of all cell vertices to be thesame in our simulations. The periodic length of the channelis set to be ~Ls ¼ 200, much larger than the spatial correlationlength measured in relevant experiments (approx. 10–20 cells[6,28]). During cell migration, T1 transitions may occur toaccomplish the exchange of cell neighbours (figure 2b). AT1 transition means that two originally separated butapproaching cells may push out their common neighbourssuch that they would get in contact [39]. In our simulations,a T1 transition is performed when the distance betweenthe two approaching cells becomes shorter than a thresholdDT1, taken as DT1 ¼ 2:5L=(KaA0) here [28], and the potentialenergy of the post-transition configuration is lower thanthat before the transition. Besides, the cells are randomlypolarized at the initial state, i.e. the polarity angles uJ aredistributed uniformly in [0, 2p), unless stated otherwise.

3. Results3.1. Collective migration modeWe first examine how the migration mode of collectivecells confined in a straight channel depends on the channelwidth ~Ws. Our simulations show that the cells in a narrowchannel will spontaneously organize into a unidirectionalcell flow mode along the channel with few cell rearrange-ments (figure 2c and electronic supplementary material,movie S1), resembling a solid flock [40]. In wide channels,however, a swirling pattern with frequent cell rearrange-ments emerges (figure 2c and electronic supplementarymaterial, movie S2), resembling a liquid flock [40]. Theseresults agree well with experimental observations (figure 1),demonstrating the feasibility of our theoretical model forstudying the dynamics of collective cells.

We quantify the mode transition of collective cellmigration by using the flocking order parameter,fv ¼ (1=N)

PJ exp [iu(vel)J ]

��� ���. fv approaches 1 when all cellscoordinate themselves to flow along the channel, and 0when cells move chaotically. Our simulations show thatwhen the channel is narrow, fv approaches 1 (figure 3a), indi-cating a highly ordered and unidirectional migration mode ofcells. When the channel width increases beyond a threshold~W

(cr)s � 12, fv sharply drops to zero (figure 3a), suggesting

a marked mode transition from directional flow to disorderedcellular migration. Intriguingly, right upon this phase tran-sition, a self-organized one-dimensional structure of vortexchain emerges, consisting of a chain of alternately left- andright-handed vortices with uniform spacing (figure 2c andelectronic supplementary material, movie S3). The emergingvortex chain is reminiscent of Rayleigh–Bénard spirals[41,42], but the underlying physical mechanisms are differ-ent. Notably, the vortex chain (or vortex lattice) pattern has

been observed in various motility assays including dense sus-pensions of bacteria [43] and spermatozoa [9], microtubules[44] and active nematics [45].

In the situation of ~Ws . ~W(cr)s , with the further increase

in the channel width, the one-dimensional vortex chainstructure will disappear and the cell motions will transitinto randomly distributed mesoscale turbulences (figure 2c).To quantify this mode transition, we detect the locations ofthe vortex cores (Material and methods), (xv, yv), and calcu-late their position variability, characterized by the standarddeviation s:d:(yv). With the increase in the channel width( ~Ws . ~W

(cr)s ), s:d:(yv) remains small (figure 3a), correspond-

ing to the vortex chain pattern. When the channel widthexceeds ~W

(cr2)s � 20, however, s:d:(yv) rises markedly with

~Ws (figure 3a), indicating the migration mode transitionfrom the vortex chain to randomly located turbulences.

The vortices in the vortex chain pattern and those inthe turbulent-like swirling pattern bear a characteristiclength scale (figure 2c). To quantify such characteristiclength scale of the emerging vortices, we calculate thespatial correlation function (SCF) of the velocity field,Cvv(r)¼ hdv(x) � dv(xþ r)ix,w=hdv(x) � dv(x)ix, where w ¼ arg (r)is the argument of the relative position vector r of cell–cellpairs, dv(x) is the fluctuation of velocity field, and r ¼ jrj isthe cell–cell distance. The SCF Cvv(r) quantifies how themotions of cell pairs at a separated distance r are spatiallycorrelated. The location of the minimum value of Cvv(r)corresponds to the characteristic size of swirls, js [6]. Ascan be seen from figure 3b, the swirl size js grows as thechannel width Ws increases in the vortex chain phase.When the channel width Ws . W (cr2)

s , js further increasesand finally approaches a plateau independent of Ws

(figure 3b). Therefore, the mesoscale cell turbulences possessan intrinsic characteristic length scale, which depends ontwo competing intercellular social interactions, LA andCIL [28].

3.2. Dynamic instability of unidirectional migrationThe observed mode transition of collective cell migrationinduced by the channel width is similar to that of passivepipe flow modulated by the pipe size. To further quantifythe mode transition of collective cell migration under channelconfinement and compare it with the traditional laminar-to-turbulent flow mode transition of passive fluids, weintroduce the transverse velocity q(x, t) ¼ hjvy(x, t)jix¼x,which corresponds to the average magnitude of the velocitycomponent perpendicular to the channel direction at eachchannel section x. Figure 3c illustrates the spatio-temporalevolutions of transverse velocity q(x, t) for the three typicalmodes of unidirectional flow, one-dimensional vortex chainand mesoscale turbulence. It is seen that the kymograph ofq(x, t) for the one-dimensional vortex chain pattern exhibitsperiodic oscillation along the x-direction, indicating thatthe vortex cores are highly stable and nearly stationary.For the fully developed turbulent mode, in contrast, thekymograph of q(x, t) is highly irregular, indicating itsintrinsic feature of instability. To characterize the turbulentdegree of collective cell migration, we define the indicatorfunction as [46]

I(x, t) ¼ 1, when q(x, t) . qthr;0, otherwise,

�ð3:1Þ

space (x)–100 1000

space (x)–100 1000

space (x)–100 1000

time

(t)

5000

2500

1.0 1.0

0.9

0.8

0.7

0.6

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0.4

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turb

ulen

ce f

ract

ion

c

0.4

0.2

0

0 10 20 30 40

qthr = 0.003

qthr = 0.006

qthr = 0.009

50 10–1 102 103 1041 10

0

0.02

0

0.01

unidirectional flow vortex chain mesoscale turbulence q (x, t)

unid

irec

tiona

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w

vort

ex c

hain

mesoscaleturbulence

1.0 1.0 2520151050 10 20

no. s

wir

ls

30 40 50

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0

–0.2

–0.4

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floc

king

ord

er p

aram

eter

fv

floc

king

ord

er p

aram

eter

fv

x s/A

01/2

00 10 20

channel width Ws/A01/2

channel width Ws/A01/2

cell–cell distance r/A01/2

SCF

Cvv

(r)

posi

tion

vari

abili

ty s

td(y

v)/A

01/2

30 40 50 0 20 40 60 80 100

15

12

9

6

3

0

std(yv)

fv

unid

irec

tiona

l flo

w

vort

ex c

hain

mesoscaleturbulence

space (channel axial)

time t/t

time

Ws = 5

50

unstable

stablemax

min

increasing channel width

~

~

~

Ws/A01/2

(a) (b)

(c)

(d) (e)

Figure 3. Collective cell dynamics regulated by channel width. (a) Phase diagram for the three migration modes of collective cells, characterized by the flockingorder parameter fv and the vortex core position variability std(yv) versus the channel width Ws. (b) Spatial correlation function (SCF) of the velocity field. Inset: theswirl size js versus the channel width Ws. (c) Kymographs of the transverse velocity q(x, t) for the unidirectional flow, vortex chain and mesoscale turbulencemodes. (d ) The cell turbulence fraction x versus the channel width Ws. (e) Temporal evolution of the flocking order parameter fv for the unidirectional flowof cells along the channel subjected to small perturbations on cell polarity (1P ¼ 0:1). Inset: kymographs of the transverse velocity q(x, t) for directed cellflow along the channel subjected to small perturbations on cell polarity. Parameter values: ~rcell ¼ 1:0 and ~v0 ¼ 0:1. (Online version in colour.)

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where qthr is a small threshold, taking the value of~qthr ¼ 0:006 here. Accordingly, the cell turbulence fraction iscalculated as x ¼ hI(x, t)ix, t, which quantifies the turbulentdegree of collective cell migration [46]. Figure 3d shows that xsharply increases around ~W

(cr)s , revealing a rapid migration

mode transition of collective cells from laminar flow to activeturbulence. To examine how this transition depends on thevalue of threshold qthr, we also vary qthr in the range of0:003 , ~qthr , 0:009 (figure 3d). While the overall shape of xversus ~Ws varies slightly for each qthr, we find that the criticalchannel width ~W

(cr)s � 12 is robust and does not depend on qthr.

To address the dynamic stability of unidirectional cellmigration under channel confinement, we simulate the tem-poral evolution of initially directed cell flow subjected to

small perturbations. More specifically, perturbations are intro-duced at the initial state through the polarity direction uJwhich distributes around zero with a small standarddeviation 1P. It is found that for cells in narrow channels,the flocking order parameter fv is insensitive to perturbations(figure 3e), indicating that the unidirectional flow is stable. Forcells in wide channels, however, small perturbations may leadto a big drop of the flocking order parameter fv, which mayeven become close to zero, suggesting the instability of thecell flow. The spatio-temporal evolutions of transverse velocityq(x, t) further illustrate the influence of confinement geometryon the dynamic stability of cell sheets (figure 3e). These resultsreveal that the migration mode transition of the confinedcollective cells results from the dynamic instability of cell flow.

1.0

0.8

0.6fl

ocki

ng o

rder

par

amet

er f

v

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ord

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ulan

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00.3 0.6 0.9

cell crowdedness rcellA0 cell motility v0 t/A01/2

1.2

mesoscaleturbulence

mesoscaleturbulence

1.5 0 0.05 0.10 0.15 0.20

fv

c

(a) (b)

Figure 4. Effects of cell density and motility on the migration mode of collective cells confined in channels. The flocking order parameter fv and the cell turbulencefraction x versus (a) cell density ~rcell and (b) cell motility ~v0. Parameter values: ~v0 ¼ 0:1 and ~W s ¼ 20 in (a); ~rcell ¼ 1:0 and ~W s ¼ 20 in (b). (Online version incolour.)

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3.3. Cellular motility numberSo far, we have examined the role of channel width incollective cell migration, reproducing reported experiments[8]. We next wonder how the intrinsic cellular propertiesincluding cell crowdedness and active motility of cells arecoupled with external confinement to shape the channel-confined motions of collective cells. Our simulations showthat both low crowdedness and weak motility of cellsfavour the ordered laminar flow (figure 4). Whereas increas-ing crowdedness rcell or enhancing active motility v0 of cellswill promote the emergence of vortex structures, and eventrigger a transition from the vortex chain pattern to thefully developed cell turbulence (figure 4). It reveals thatincreased cell crowdedness and motility tend to destabilizecell flow and disturb collective cell motions confined in thechannels. Interestingly, our simulations demonstrate thatincreasing cell crowdedness in a moderate range gives riseto turbulent cell motions, rather than jamming of the cellmonolayer (figure 4a). This suggests that the increase of cellcrowdedness could not be the major cause for the jammingof cell monolayers during maturation [47,48].

We have demonstrated that the migratory mode ofcollective cells confined in a channel can be regulated by avariety of variables. Rather than solely depending on thechannel width, it is also dictated by the crowdedness andactive motility of cells. More strikingly, the roles of channelwidth, cell crowdedness and cell motility in regulating themigration modes of collective cells under channel confine-ment are similar to the roles of pipe size, fluid density andflow velocity in determining the modes of passive pipeflow. This suggests that the two irrelevant systems bear some-what similarity in the emerging motion patterns. However,the Reynolds number of collective cell migration is ultralow[49,50] and the inertial effect can be ignored. Thus, it is inap-propriate to use the classic Reynolds number, which weightsthe influence of the inertial force and the viscous force, toquantify collective cell migration in channels. Instead, weintroduce a new dimensionless parameter for the confinedcell system

Z ¼ ~rcell~v0 ~Ws, ð3:2Þ

which encapsulates the effects of confinement size, cellactivity and crowdedness, and is referred to as the CMNhereafter. Interestingly, the CMN acts as a universal

parameter that unifies the behaviour of states with varyingconfinement size, cell activity and crowdedness. Our simu-lations show that the regulating roles of confinement size,cell activity and crowdedness in the dynamic instability andmigration mode transitions of collective cells under channelconfinement can be well captured by the proposed CMN(figure 5a). Clearly, unidirectional collective motions (laminarcell flow) take place at a small Z. Increasing Z would lead tothe emergence of a stable array of vortices in the channel,with the critical CMN Zcr � 1:2. The transition from thevortex chain pattern to the mesoscale cell turbulence occurswhen the CMN reaches another critical value Zcr2 � 2:0.

The reduced CMN z ¼ (Z� Zcr)=Zcr is used to describe thecritical behaviour of the laminar-to-turbulent transition. Wefind that near the critical state, the turbulent fraction x andthe reduced CMN j exhibit a scaling law of x � zb (figure 5b),with the exponent b � 1:71. Further, since the measurement ofcritical exponent b is notoriously error-prone, we examine howthe critical exponent b depends on the selected critical pointZcr. We vary Zcr within the interval [1.15, 1.25], which intui-tively covers the estimated transition point approximately1.2, and find that the measured critical exponent b decreaseswith the used critical point Zcr from 2.21 to 1.30 (figure 5b).It is known that during the laminar-to-turbulent transition oftraditional pipe flow, the turbulent fraction exhibits a similarscaling law with the reduced Reynolds number where the criti-cal exponent is b ¼ 0:58, which belongs to the universalityclass of the (2 + 1)-dimensional directed percolation(b ¼ 0:583) [51]. In spite of the phenomenological similaritybetween passive flows and active cell migration, these twosystems are distinct in physical nature.

3.4. Collective cell migration in a channel withvarying width

The concept of CMN can also be employed to characterizethe collective cell migration mode in a channel with complexgeometry. For illustration, we simulate the cellular motionsin a channel with varying width, Ws ¼ Ws(x), as shown infigure 6a. It is found that the cells will self-organize into het-erogeneous dynamic patterns consisting of differentmigration modes, resulting in the coexistence of differentphases. Active cell turbulence (i.e. liquid flock phase) takesplace in the wide regions at the two ends of the channel, and

1.0

0.5

01.0

0.5c

c

z = (Z–Zcr)/Zcr

b

fv

00 1 2

tran

sitio

n

Z

3

turbulent

laminar

varying Ws

varying v0

varying rcell

4

1

simulationfitting

slope = 1.71R2 = 0.9810–1

10–2

10–3

10–2 10–1

2.22.01.81.61.41.2

1.15 1.20 1.25

1

Zcr

(a) (b)

Figure 5. CMN controls the collective dynamics of cells confined in a straight channel. (a) The flocking order parameter fv and the cell turbulence fraction x versusthe CMN Z. (b) Scaling law of the cell turbulence fraction x versus the reduced CMN z near the laminar-to-turbulent transition. Inset: dependence of the measuredcritical exponent b on the evaluated critical point Zcr. (Online version in colour.)

space (x)–100 0 100

time

(t)

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q (x, t)0.02

1.0

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turbulent turbulent

fvc

(a)

(b) (c)

Figure 6. Collective cell migration in a channel with varying width. The channel has the minimum width at the middle, ~W s(~x ¼ 0) ¼ 6, and the maximum widthat the two ends, ~W s(~x ¼ +100) ¼ 30. (a) A typical velocity field. (b) Kymograph of the transverse velocity q(x, t). (c) Distribution of the height-averaged flockingorder parameter fv(x), the cell turbulence fraction x(x) and the CMN Z(x) along the channel. (Online version in colour.)

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it gradually transits into a laminar flow (i.e. solid flock phase)with the shrinking of the channel width (figure 6a,b). Wedivide the channel into several segments along its length,and the corresponding averaged flocking order parameterfv(x), averaged turbulent fraction x(x), and averaged CMNZ(x) are plotted in figure 6c. It is seen that the emerging hetero-geneous patterns can be well captured by the proposed CMN.

4. DiscussionThe CMN defined in equation (3.2) can be re-expressed interms of ~rcell, ~v0 and ~Ws as

Z ¼ rcellA0v0tffiffiffiffiffiffiA0

p WsffiffiffiffiffiffiA0

p ¼ rcellv0Wst

¼ rcellv0Wsg

KaA0¼ rcellWs(gv0)

KaA0

� motile forceelastic force

: ð4:1Þ

It is noted that the numerator rcellWs(gv0) represents theactive motile force of cells per unit length along the channel,and the denominator KaA0 scales to the elastic force. There-fore, the CMN actually accounts for the competitionbetween the active motile force (Fa) from the cell polarizationand the passive elastic restoring force (Fe) from the cytoskele-ton network. The balance of these two forces at the cellularlevel dictates the dynamic instability, mode selection and pat-tern formation at the tissue level. More specifically, for a smallCMN with Fa , Fe, the cells are hard to deform and passthrough their neighbouring cells, and thus the multicellularsheet behaves as a solid flock and shows a unidirectionalflow pattern. In the case of large CMN with Fa . Fe, how-ever, the cells have a high motility such that they canovercome the elastic restoring force, deform and navigatethrough their neighbouring cells. In this situation, the multi-cellular sheet will behave as a liquid flock and self-organizeinto distributed vortices.

Although the role of CMN in collective cell dynamicsunder channel confinement resembles that of the Reynolds

cancerization of normal cellssoft

1.0

0.8

0.6

floc

king

ord

er p

aram

eter

fv

cell

turb

ulen

ce f

ract

ion

c

0.2

cell area stiffness Ka/Ka0

0.4

0

1.0

0.8

0.6

0.2

0.4

0

101

stiff

unid

irec

tiona

l flo

w

vort

ex c

hain

mes

osca

le tu

rbul

ence

fv

c

Figure 7. Effect of cell stiffness on collective cell migration in channels. Theflocking order parameter fv and the cell turbulence fraction x versus the cellstiffness Ka. Here, Ka0 ¼ 50Kc=A0. (Online version in colour.)

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number in the motion transition of passive pipe flow, theunderlying physics is quite different: the former quantifiesthe ratio of the active motile force to the passive elastic restor-ing force, whereas the latter represents the ratio of the inertialforce to the viscous force. Interestingly, our results suggestthat confined multicellular systems of epithelial cell lines atvery different channel width and cell density could behavein the same way when the CMN is the same (figure 5).This theoretical prediction merits further experimentalvalidations.

In addition, equation (4.1) indicates that the cell areastiffness Ka plays an implicit but significant role in thedynamics of confined cell collectives. Stiffer cells favourlaminar cell flow, whereas softer cells prefer cell turbulence,as further verified by our simulations (figure 7). Recent exper-imental measurements showed that cells usually soften aftercancerization [52–54]. Thus, cancer cells would have alarger Z than normal cells and tend to exhibit more swirls.This finding may help explain the enhanced swirlingbehaviour observed in clinic papillary thyroid carcinoma [55].

The concept of CMN can also be employed to quantifycollective cell dynamics in other confined spaces. Doxzenet al. [18] experimentally observed that MDCK cells confinedin small circular domains can self-organize into a persistentrotation mode, which can be regarded as the laminar flowpattern in circular domains. By contrast, MDCK cellsconfined in large circular domains undergo dynamic swirlingpatterns [18], which can be regarded as turbulent flowpattern in circular domains. If we replace the channel widthWs in equation (4.1) with the diameter D of the circulardomain, such mode transition can be accounted for by theproposed CMN. In addition, previous experiments showedthat MCF10A cells confined in small domains tend toperform persistent rotation while cancerous MDA-MB-231cells prefer random and chaotic motion [18]. Equation (4.1)shows that among many factors that may influence cellmigration, the softening of cancerous cells could contributesignificantly to the emergence of these different motionpatterns. Therefore, the concept of CMN proposed in thispaper can be applied to more general cases of collective cellmigration in confined spaces. Generally, we can rewrite the

CMN as

Z ¼ rcellL(gv0)KaA0

, ð4:2Þ

where L is the characteristic size of confinement, for instance,the width of channels or the diameter of circular domains.It should be noted that for different confined spaces, thecritical CMN could be different, for instance, Zcr � 1:2 forchannels and Zcr � 2:0 for circular domains in terms of oursimulations.

It should be pointed out that although we have adoptedperiodic conditions at the two ends of the channel, ourmodel can be extended to investigate collective cell migrationin closed strip-like domains by modifying the boundaryconditions at the ends. Recent experiments demonstratedthat cells confined in rectangular domains may exhibit oscil-latory movements [56,57]. Our model can investigate theseoscillatory phenomena by properly modifying the boundaryconditions. We leave that to our future study.

Finally, it is worth emphasizing that collective cellmigration is also mediated by cell–cell social interactions[26,28], cell–cell adhesion [8,15] and cell–matrix adhesion[58]. We have performed robustness tests on the regulatingroles of cell–cell social interactions and cell–cell adhesion inconfined collective cell migration (see Material and methods).However, the present model has not taken the complex cell–matrix adhesion into account. Firstly, we have simplified eachcell as a polygon by ignoring such subcellular structures asfocal adhesions, which are non-uniformly distributed on thewhole cell–matrix interface and dynamically evolve duringcell migration [59,60]. Secondly, cell–matrix adhesion is sig-nificantly mediated by integrins, which repeatedly bind toand unbind from the matrix [59,60]. This binding–unbindingcycle depends on the mechanical forces applied on theintegrins, which are ordinarily heterogeneous and relatedto matrix stiffness [59,60]. Therefore, cell–matrix adhesioncannot be simply evaluated by the contact length betweencell edges and matrix. A refined collective cell migrationmodel merits future study to include the effect of thesemechanisms. Besides, it would be useful to weight therelative influence of cell autonomous properties and cell–environment interactions on collective cell migration.

In summary, we have investigated how the intrinsicallycellular properties and extrinsically confinement size coordi-nate to dictate the dynamic instability and migrationmode transition of collective cells in a monolayer under geo-metric confinement. The CMN, which quantifies the ratio ofthe active motile force to the passive elastic force, has inte-grated the effects of confinement size, cellular crowdedness,cell stiffness and active cell motility on the collective celldynamics. We show that the emergent collective migrationmodes, tissue-level dynamic patterns and their transitionsof collective cells in confined spaces can all be well capturedby the CMN. The dynamic instability dictated by the CMNsheds a promising perspective on large-scale collective celldynamics under geometric constraints. Our results suggesta potential strategy to speed up molecular or nutrient trans-port by lowering CMN, to enhance biogenic mixing at thetissue scales [11] and even to promote turbulence-controllablethrombopoiesis [61] by elevating CMN. In addition, the CMNopens a new avenue for exploring active morphodynamicsunder complex natural or in vitro confinements in both bio-logical and abiotic self-propelled systems.

0.200.150.100.050

2.0

1.5

1.0

0.5

0unidirectional flo

wmes

osca

le tu

rbul

ence

trans

ition

zone

boundary friction coefficient gw/g0.10 10–2 10210–1 1 100.080.060.040.020

0.10 1.0

0.8

0.6

0.4

0.2

0

0.05

0

–0.05

–0.10

mesoscale turbulence

unidirectional flow

transition zone

CIL

inte

nsity

mC

IL~

inte

rfac

ial t

ensi

on L~

LA intensity mLA~ contractile modulus Kc

~

Ws = 40, mesoscaleturbulence

~

Ws = 15, vortex chain~

Ws = 10, unidirectional flow~

floc

king

ord

er p

aram

eter

fv

(a) (b) (c)

Figure 8. Robustness test of the collective migration mode of cells confined in channels. (a) Regulations of LA and CIL. Parameters: ~K c ¼ 0:02, ~L ¼ 0:1,~rcell ¼ 1:0, ~v0 ¼ 0:1 and ~W s ¼ 30. (b) Regulations of cell contractile modulus and cell–cell interfacial tension. Parameters: ~mLA ¼ 0:05, ~mCIL ¼ 1:0,~rcell ¼ 1:0, ~v0 ¼ 0:1 and ~W s ¼ 30. (c) Effect of boundary friction coefficient gw. Parameters: ~K c ¼ 0:02, ~L ¼ 0:1, ~mLA ¼ 0:05, ~mCIL ¼ 1:0,~rcell ¼ 1:0 and ~v0 ¼ 0:1. (Online version in colour.)

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5. Material and methods5.1. Normalization schemeIn §2, we have given the controlling equations for the evolutionsof vertices’ positions, ri(t), and cell polarity directions, uJ(t). Sub-stituting equation (2.3) into equation (2.1), we obtain the detailedevolution equation for ri(t) as

dridt

¼�1g

XJ[Ci

12Ka(AJ �A0)[k� (r j2� r j1)]

� 1g

XJ[Ci

KcLJri� r j1jri� r j1jþ

ri� r j2jri� r j2j

� �� 1g

Xj[Vi

Lri� rjjri� rjjþ

XJ[Ci

v0eJnJ

,

ð5:1Þwhere k denotes a unit vector normal to the cell monolayer; j1 andj2 are the neighbouring vertices of vertex i in cell J; the summationP

j[Viis made over all neighbouring vertices Vi of vertex i.

To facilitate simulations, we normalize the controllingequations for the vertices’ positions ri (i.e. equation (5.1)) andthe cell polarity directions uJ (i.e. equation (2.2)) by introducingthe typical scales associated with multicellular system. Indetail, lengths are scaled by the reference cell size ‘ ¼ ffiffiffiffiffiffi

A0p

,time by the viscous dissipation time t ¼ g=(KaA0), and forcesby the elastic restoring force s ¼ Ka‘

3 ¼ KaA3=20 . Accordingly,

the normalized equations can be written as

d~rid~t

¼�XJ[Ci

12( ~AJ �1 )[k� (~r j2�~r j1)]

�XJ[Ci

~Kc~LJ~ri�~r j1j~ri�~r j1jþ

~ri�~r j2j~ri�~r j2j

� �

�Xj[Vi

~L~ri�~rjj~ri�~rjjþ

XJ[Ci

~v0eJnJ

ð5:2Þ

and

duJd~t

¼ ~mLA

nJ

XK[CJ

sin(uðvelÞK �uJ)þ ~mCIL

nJ

XK[CJ

sin(aJ,K�uJ)þ~1~hJ(~t): ð5:3Þ

The dimensionless parameters are defined as

~ri ¼ ri‘¼ riffiffiffiffiffiffi

A0p , ~t¼ t

t¼KaA0t

g, ~AJ ¼AJ

‘2¼AJ

A0, ~LJ ¼ LJ

‘¼ LJffiffiffiffiffiffi

A0p ,

~Kc ¼Kc‘

s¼ Kc

KaA0, ~L¼L

s¼ L

KaA3=20

, ~v0 ¼ v0t‘

¼ gv0KaA

3=20

,

~mLA ¼mLAt¼gmLA

KaA0, ~mCIL ¼mCILt¼

gmCIL

KaA0,

~1¼ 1ffiffiffit

p ¼ 1

ffiffiffiffiffiffiffiffiffiffiffig

KaA0

r, ~hJð~tÞ¼hJðtÞ

ffiffiffit

p ¼hJðtÞffiffiffiffiffiffiffiffiffiffiffig

KaA0

r:

9>>>>>>>>>>>=>>>>>>>>>>>;ð5:4Þ

5.2. Parameter values and robustness testsAccording to previously reported experimental measurements,the viscosity of biological tissues is of the order ofh � 104�105 N s m�2 [62], leading to an estimation of thefriction coefficient g � hh � 0:01�0:1 N s m�1, with h being theheight of cell monolayers taken as 1–10 µm. Take the cell’s arealstiffness for epithelial tissues as Ka � 105�107 N m�3 [63], andthe preferred area as A0 ¼ 900 mm2. The length scale, time scaleand force scale used for normalization are estimated as‘ ¼ ffiffiffiffiffiffi

A0p � 30 mm, t ¼ g=(KaA0) � 100 s and s ¼ Ka‘

3 � 10�8 Nin turn. The contraction force of cells is in the range of 1�10 nNfor epithelial cell sheets [64], which leads to a cell contractile mod-ulus Kc � 10�5�10�4 N m�1. The interfacial tension is of the orderL � 10�9�10�8 N, as estimated from experimental measurementson biological tissues [62]. Therefore, we have the estimationof the dimensionless cell contractile modulus ~Kc � 0:01�0:1and the dimensionless interfacial tension ~L � 0:01�0:1.Besides, the motile force actively driving cell migrationinvolves both actin polymerization of pseudopodia and cellu-lar pulling on the substrate via cell–matrix adhesion and is ofthe order 1 nN [65], which results in a self-propelled velocityv0 � 10�8�10�7 m s�1, with its dimensionless form of theorder ~v0 � 0:01�0:1.

To focus on the regulating roles of active cell motility, cellcrowdedness and geometric confinement size, we fix otherparameters, including LA intensity ~mLA, CIL intensity ~mCIL, cellcontractile modulus ~Kc and cell–cell interfacial tension ~L. Wehave performed robustness tests to extract reasonable parametervalues that lead to experimental observations. To address this,we simulate collective cell migration confined in wide channels( ~Ws ¼ 30), and examine the parameter ranges that reproducethe experimentally observed turbulent motion pattern ofcollective cells within wide channels [8] or without boundaryconstraints [6,47,48]. It is found that without CIL effect(~mCIL ¼ 0), the laminar-to-turbulent transition disappears(figure 8a). Given other parameters, only when the LA and CILintensities (~mLA, ~mCIL) are within the phase of mesoscaleturbulence (figure 8a) can we reproduce the experimentallyobserved turbulent motion pattern in wide channels. Forinstance, when ~mCIL ¼ 1:0, ~mLA should be small enough(approx. ~mLA , 0:08). In addition, cell contraction and cell–cellinterfacial tension (involving the effect of cell–cell adhesion)also affect collective cell migration in confined channels, asshown in figure 8b. We find that the cell contractile modulusshould be small enough such that the model reproduces theexperimentally observed turbulent motion pattern in wide chan-nels. Besides, strong cell–cell adhesion (i.e. small ~L) favoursturbulent cell motion, whereas weak cell–cell adhesion can leadto unidirectional flow pattern. Therefore, unless stated otherwise,

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10

we use the following dimensionless parameter values: ~Kc ¼ 0:02,~L ¼ 0:1, ~mLA ¼ 0:05 and ~mCIL ¼ 1:0.

Further, we perform simulations to investigate the role of thefriction coefficient (gw) of boundary vertices on the wall byvarying gw in a broad range (10�2 , gw=g , 102). It is foundthat provided the channel is not too narrow (larger than threecell size), gw has no significant influence on the collectivemigration mode of cells confined in channels (figure 8c),though the motion of individual cells adhered to the wall maybe speeded up (gw=g , 1) or slowed down (gw=g . 1). Previousexperiments of collective migration of MDCK cells adheringon fibronectin strips showed that there is no evident differencein motion speeds between boundary cells (those adhere tothe wall) and inner cells [8]. We thus set gw ¼ g in oursimulations.

R.Soc.Interface16:2019

5.3. Detection of vortex coresTo detect the vortex cores from the velocity field of cellularmotions, one can calculate the angle that the velocity field rotatesin a counterclockwise loop around each cell or each vertex

Dwv ¼ PiDw

ðiÞv , where Dw

ðiÞv ¼ w

ðiþ1Þv � w

ðiÞv , with w

ðiÞv being the

velocity direction of the ith point on the loop. If the examinedloop contains a vortex core, the rotation angle Dwv is equal to2p, regardless of whether the vortex is left- or right-handed [66].A core of source is excluded via defining a parameterc ¼ jFL=GLj with FL being the flux and GL the circulation alongthe loop L: if c , c0, we regard it as a vortex core; otherwise, itis a source core. Here, we set c0 ¼ 0:1 for analysis of all velocityfield. To examine whether the vortex is left- or right-handed,one can calculate the vorticity of the vortex core: it is positivefor a right-handed vortex while negative for a left-handed one.

Data accessibility. All data needed to evaluate the conclusions in thepaper are present in the paper. Additional data related to thispaper may be requested from the authors.

Authors’ contributions. S.-Z.L., B.L. and X.-Q.F. designed the project. S.-Z.L. performed the simulation and analysed the data. S.-Z.L andB.L. interpreted the simulation data. D.B., B.L. and X.-Q.F. discussedthe results. S.-Z.L., D.B., B.L. and X.-Q.F. wrote the manuscript. B.L.and X.-Q.F. supervised the project.

Competing interests. The authors declare that they have no competinginterests.

Funding. Supports from National Natural Science Foundation of China(grant nos. 11620101001, 11672161 and 11432008) are acknowledged.

0258

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