Coordinated tractions increase the size of a collectively moving pack in a cell monolayer
Aashrith Saraswathibhatla, Silke Henkes, Emmett E. Galles, Rastko Sknepnek, Jacob Notbohm
CCoordinated tractions control the size of a collectively moving pack in a cell monolayer
Aashrith Saraswathibhatla, Silke Henkes, Emmett E. Galles, Rastko Sknepnek,
3, 4 and Jacob Notbohm ∗ Department of Engineering Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom School of Science and Engineering, University of Dundee, Dundee DD1 4HN, United Kingdom School of Life Sciences, University of Dundee, Dundee DD1 5EH, United Kingdom
Cells in an epithelial monolayer migrate in collective packs of sizes spanning multiple cell diam-eters, but the physical mechanisms controlling the pack size remain unclear. Here, we measuredcell velocity and traction with single-cell resolution, enabling us to identify both a persistence timeof traction and a correlation of traction between neighboring cells. A self-propelled Voronoi modelmatched the experiments only when it accounted for alignment of traction between neighboringcells. Hence, traction alignment, in addition to persistence, affects the pack size in collective cellmigration.
During tissue formation and repair, cells migrate in col-lective groups and packs [1]. The characteristic size of acollective cell pack lies at an intermediate point betweenextremes present in other collective systems—cell motionis not fully random as it would be in systems driven bythermal fluctuations [2], nor is the motion highly coordi-nated as in flocks of birds [3] or schools of fish [4]. Instead,in collective cell swirls, packs, and waves, the motion iscorrelated over length scales ranging from a few to a cou-ple dozen cells [5–16]. Hence, there must be a mechanismfor correlation of velocity between neighboring cells.A plausible scenario is that the finite time requiredfor a cell to reorganize its force-generating machineryleads to a temporal persistence to each cell’s motion.Computational models of epithelial cell sheets, includingboth geometric (i.e., vertex) models [17–20], and sim-pler particle-based models [15, 21, 22] show that if thepersistence time is sufficiently large, collective packs canemerge [15, 18, 20]. While appealing for its simplicity,this explanation includes no explicit mechanism for a cellto polarize and align its propulsive force with that of itsneighbors, as can occur through chemomechanical signal-ing [9].Cell-induced substrate displacements are correlatedover multiple cell lengths [23], implying that there mayexist some mechanism of cell coordination. Several phys-ical mechanisms have been proposed, including a ten-dency of cells to migrate along the local orientation ofthe maximal principal stress [24] and a tendency to ap-ply tractions that pull each cell toward regions of emptyspace [25]. Cells can also align with their neighborsto create orientational nematic order [26–28]. Modelsof such coordination have suggested that a coupling be-tween the force on a cell and its traction results in flockformation [21, 29]. In addition, cells can coordinate toalign stress fibers in the same direction [30–32], and thereexist molecular mechanisms that enable neighboring cellsto coordinate front-back polarization [9, 16]. These ob-servations give circumstantial evidence that cells may co-ordinate with their neighbors to apply propulsive trac-tions in the same direction. Coordination of traction would then be a second mechanism—in addition to tem-poral persistence—that controls the size of a collectivecell pack. Direct evidence, however, is lacking, becauseneither the temporal persistence time nor a coordinationof propulsive forces between neighbors has been quanti-fied experimentally at the single cell level.In this letter, using traction force microscopy and celltracking, we developed a method to measure the net trac-tion exerted by individual cells over time. Using this newsingle-cell measure, we verified the long-standing notionthat cell tractions are persistent over time. Importantly,a spatial autocorrelation of single cell traction demon-strated that the collective motion is not controlled merelyby a temporal persistence of traction; rather, cells alsocoordinate to align directions of traction with those oftheir neighbors. Traction persistence and alignment wereinvestigated further using chemical perturbations of ac-tomyosin contraction and a self-propelled Voronoi modelthat was calibrated against the experimental data andincluded both traction persistence and alignment. To-gether, the experiments and model reveal how the inter-play between persistence and alignment affects the sizeof a collectively moving cell pack.We seeded Madin-Darby Canine Kidney (MDCK) cellsin confluent monolayers and imaged them over time. Thecells expressed green fluorescent protein in their nuclei,enabling us to track each nucleus [33, 34] and to mea-sure each cell’s velocity (Fig. 1a). The persistence ofcell velocity was defined by the temporal autocorrela-tion, C ( t ) = h ˆ v ( t ) · ˆ v ( t − t ) i , where ˆ v ( t ) is the unitvector corresponding to the direction of cell velocity attime t , and h ... i indicates an average over all cells in allpairs of time points that are t apart. To quantify thepersistence time, we used the time over which the per-sistence decreased to a value of 0 .
2, which is comparableto statistical noise. The persistence time was ≈ . v , which showed correlated cell velocities a r X i v : . [ phy s i c s . b i o - ph ] F e b T i m e ( h ) a bc def g
100 µm20 µm P e r s i s t en c e S pa t i a l au t o c o rr e l a t i on -0.200.20.40.60.81-0.200.20.40.60.81 VelocityTraction r ′ (µm) FIG. 1. Temporal persistence and spatial correlation of ve-locity and traction. (a) Cell velocities (arrows) overlaid onan image of a cell monolayer. Colors indicate the angle ofthe cell velocity (see color circle). (b) Persistence, definedas the temporal autocorrelation, of single-cell velocity andtraction. In all figures, lines with error bars represent meanvalues and standard deviations over at least four cell mono-layers and 6 h of imaging. (c) Single-cell tractions overlaid ona Voronoi tessellation of a cell monolayer. Colors indicate theangle of traction (see color circle). (d) Spatial autocorrela-tion of single-cell velocity and traction. In all figures, verticaldashed lines represent the average cell diameter. (e) Repre-sentative images of stress fibers reorienting over time. Solidlines indicate the orientation of a pair of stress fibers at dif-ferent time points. (f) Representative images of a stress fiberdisappearing over time (circled). (g) Time required for stressfibers to reorient (circles) or disappear (stars). over distances of several cell diameters (Fig. 1d). Aswith the persistence time, the pack size was quantified bythe distance over which the correlation dropped to 0 . ≈ µ m (Fig. 1d; S3e [35]), setting a typicalpack to have the size of ≈ µ m ( ≈ . µ m. As cells are not uniformly spaced, the traction data didnot map directly to each cell. To resolve this issue, weproduced a Voronoi tessellation approximating the celloutlines (Fig. S1 [35]) [41], which enabled an approxi-mate mapping between each cell and its traction (Fig.S2a [35]). Then, the vector sum of traction applied byeach cell was computed, yielding the net traction pro-duced by that cell (Fig. 1c). The magnitude of net trac-tion applied by each cell fluctuated in time but rarelyreached 0 (Fig. S2b, [35]). Importantly, the directionof net traction coincided closely with the orientation ofstress fibers in cells, consistent with the notion that stressfibers generate tractions (Fig. S2c-d [35]). The noise floorof this new measurement was quantified by computingthe net traction applied by isolated cells, which should bezero as inertial forces are negligible [42, 43]. The averagewas 10 Pa (Fig. S2e [35]), which defines the experimen-tal noise floor, and, importantly, was nearly an order ofmagnitude smaller than the average net traction appliedby cells in a monolayer, ≈
80 Pa.Our single-cell traction measurement enabled us tocompute the temporal persistence and spatial correlationof traction (Fig. 1b, d). The traction persistence time,defined as the time over which the correlation decreasedto 0 .
2, was found to be ≈ . . ± . . ± . ≈ µ m).By contrast, the averaged autocorrelation decayed to zeroover a distance of two cell diameters, indicating that cellscoordinate with their neighbors to apply traction in thesame direction (Fig. 1d). Hence, cell tractions are corre-lated in both time and space.We next sought to identify the relative contributions oftraction persistence and correlation on the size of a col-lectively moving cell pack. To begin, we considered tem-poral persistence, as prior models have suggested that in-creasing persistence increases the pack size [15, 18, 20]. Inan effort to alter actin dynamics, we used the F-actin sta-bilizer jasplakinolide (JSP) [45], which increased tractionpersistence and decreased traction magnitude by factorsof ≈
100 120 r ′ (µm) T r a c t i on au t o c o rr e l a t i on V e l o c i t y au t o c o rr e l a t i on r ′ (µm) a cb d Ctrl
JSP τ p = ; τ a = 2.5 ; f = 53 τ p = ; τ a = 10 ; f = 26 τ p = ; τ a = 2.5 ; f = 26 FIG. 2. Evidence for traction alignment spanning multiplecells. (a, b) Spatial autocorrelation of cell tractions (a) andvelocities (b) in experiments and model. Values in the legendin panel a correspond to different traction persistence andalignment times, τ p and τ a , measured in hours, and force f measured in nN. (c, d) Representative images of cell velocities(c) and net cell tractions (d) in the model, for τ p = 1 . τ a = 2 . f = 53 nN. The scale bars correspond to100 µ m. traction persistence, it reduced the spatial correlation ofvelocity (Fig. 2a), in contrast to our expectation. A po-tential resolution is that the treatment also reduced thespatial correlation of traction (Fig. 2b), suggesting thatthe velocity correlation may be determined more so bythe traction correlation. Other chemical treatments pro-duced similar conclusions: using ML 141 to inhibit thefront-back polarization molecule Cdc42 [46] and alter-ing actin polymerization with cytochalasin D [47] alsoincreased the persistence time of traction but decreasedthe velocity correlation, though these treatments also re-duced the magnitude of traction produced (Fig. S3 [35]).Together, these data indicate that traction persistenceand correlation together affect the size of a collectivelymoving cell pack, though in a coupled, complex way.To establish more clearly the relative contributionsof traction persistence and correlation on pack size, weused the Self-propelled Voronoi (SPV) model [18, 19, 48],which treats each cell as a polygon constructed by aVoronoi tessellation of the plane with cell centers usedas seed points. The energy of the i th cell is a quadraticfunction of the deviations of its area A i and perimeter P i from target values A and P , i.e., E i = K A ( A i − A ) +Γ P ( P i − P ) . K A and Γ P , respectively, are area andperimeter moduli and are assumed to have the same valuefor all cells. A and P , also assumed to have the samevalue for all cells, can be combined into a single dimen-sionless parameter, p = P A − / , which is one factorcontrolling whether the collective cell behavior in themodel is liquid- or solid-like [18, 49]. Here, we used an input value of p = 3 .
8, which is in the solid regime butnear the border to a liquid. Adding self-propulsion thenraises the actual values of
P A − / above the input valueof 3 .
8, consistent with observations in MDCK cells [50].In the plane, the direction of self-propulsion (i.e., trac-tion) is specified by vector n i = (cos θ i , sin θ i ), where θ i isthe angle between vector n i and the x -axis of the labora-tory reference frame. The motion of each cell is describedby two equations, one for the cell’s position r i and onefor θ i , µ ˙ r i = −∇ i E + f n i , ˙ θ i = τ − a X j sin ( θ j − θ i )+ η i , (1)where µ is a friction coefficient, and f is the magnitude oftraction that produces the self-propulsion force in direc-tion n i [35]. Earlier references used variable v = f /µ inplace of f , with v having units of velocity [18, 19]; here,we chose to use f due to its connection to the single-cellnet traction measured in the experiments. Consistentwith our experimental observation of both traction per-sistence and correlation, the time evolution of θ i containsboth angular noise and polar spatial alignment. The vari-able η i describes a random process with h η i i = 0 and h η i ( t ) η j ( t ) i = τ − p δ ij δ ( t − t ), with τ p being the char-acteristic persistence time of the direction of cell traction.The inverse polar alignment strength τ a is the character-istic time required for cell i to align its traction directionwith that of its neighbors; smaller τ a indicates greaterstrength of alignment of neighboring tractions. The j -sum is over all neighbors of cell i within a cutoff distance r cut , set to be equal to the average cell diameter.The SPV model includes sufficient cell-level detail tomake order-of-magnitude estimates of parameters of theactual experimental system [15]. The unit of length usedin the model sets A / , and it was chosen to match theaverage cell diameter in the experiments, 22 µ m. We usedthe stiffness parameters K and Γ estimated in ref. [15].The value of f was estimated from the mean magnitudeof net traction, 110 Pa (Fig. S3a [35]), multiplied by theaverage cell area, 22 × µ m to give 53 nN. The frictionfactor µ was tuned to achieve a match between cell migra-tion speeds in the experiments and model (average speedof 18 µ m/h, Fig. S4 [35]); a value of µ = 265 pN · h /µ m(corresponding to v = 200 µ m/h) gave the best matchand is of the same order of magnitude as prior estimates[15, 28]. Representative images of cell velocity and trac-tion in the model (Fig. 2c,d) show collective cell packsand neighboring cells applying traction in the same di-rection, which is reminiscent of the experimental data.The remaining free parameters in the model are thepersistence and alignment time scales, τ p and τ a , respec-tively. We began with τ p = 1 . τ a = 2 .
5h (Fig. 2a, b; see Supplemental Note [35] for a compari-son of experimental and simulated velocity persistences).The matched value of τ a = 2 . . τ a could not befaster than the time required for each cell to reorganize itsown stress fibers. In an effort to mimic the experimentswith JSP, we increased the traction persistence time τ p and decreased the traction magnitude f by factors oftwo. The effect of this change on the velocity correla-tion was small (Fig. 2a), suggesting that τ p alone had arelatively small effect on the velocity correlations. Addi-tionally, this change increased the correlations in velocityand traction, opposite to the experimentally observed ef-fect of JSP (Fig. 2a, b). Matching the experimental datarequired increasing both τ p and τ a , indicating that themulticellular traction alignment is essential to capturethe experimental data. A more thorough study on thecombined effects of τ p and τ a was performed by decreas-ing and increasing τ p by factors of two and subsequentlyvarying τ a through a range of values (Fig. S4 [35]). Con-sistent with prior models [15, 18, 20], increasing τ p causedlarger collective packs in the absence of multicellular trac-tion alignment (i.e., τ a → ∞ ), and for larger values of τ p ,less traction alignment (i.e., larger τ a ) was required tomatch the experimental data (Fig. S4 [35]), indicatingthat increasing either single-cell persistence or multicel-lular alignment increases the size of a collectively movingcell pack. Importantly, for all values of τ p studied, a finite τ a was required to match experimental data, indicatingthat cells align their tractions with those of their neigh-bors.The experiments and model indicate that cells coor-dinate their propulsive tractions with their neighbors,which in turn affects the size of a collectively movingpack. To explore this finding further, we wondered ifit would be possible to turn off the correlation of trac-tion between neighbors, thereby reducing the pack size.To this end, we decreased contractility by culturing thecells in low (1%) serum for 24 h. Under these condi-tions, cells expressed few stress fibers (Fig. 3a), and thetraction autocorrelation dropped nearly to zero at onecell diameter (Fig. 3b), suggesting essentially no coor-dination of traction between neighbors. Addition of theRho activator CN03 (3 µ g/mL) reversed this effect, caus-ing more pronounced stress fibers that aligned betweenneighboring cells (Fig. 3c) and increasing the traction au-tocorrelation such that it decayed to zero over a distanceof ≈ T r a c t i on au t o c o rr e l a t i on V e l o c i t y au t o c o rr e l a t i on
100 120 r ′ (µm) r ′ (µm) a bc d
20 µm
Low serum
CN03 τ p = 1.3; τ a = ∞ ; f = 14 τ p = 2.6; τ a = ∞ ; f = 53 τ p = 2.6; τ a = 5; f = 53 FIG. 3. Effect of stress fiber activity on cell-cell tractionalignment and collective pack size. (a) Image of F-actin inlow serum conditions. (b) Traction autocorrelation in experi-ments and model. (c) Image of F-actin in CN03-treated cells.(d) Velocity autocorrelation in experiments and model. Inpanels b and d, τ p and τ a are measured in hours; f is mea-sured in nN. off (i.e., τ a → ∞ ). To match the CN03-treated con-ditions, we noted that the CN03 increased net tractionmagnitude and traction persistence by factors of approx-imately 4 and 2, respectively (Fig. S5 [35]). Therefore,we increased f and τ p by factors of 4 and 2 but retainedno multicellular traction alignment (i.e., τ a → ∞ ). De-spite increasing the cell persistence time τ p , there was nosignificant change in velocity autocorrelation. Instead, amatch to the CN03-treated data required that we use a fi-nite traction alignment time, τ a = 5 h (Fig. 3b,d). Hence,this experiment confirms the multicellular traction align-ment and demonstrates that it results from alignment ofstress fibers between neighboring cells.Here, we have demonstrated that cells in a collectivealign their propulsive forces with those of their neigh-bors. In flocks of birds or schools of fish, visual cues pro-vide guidance for neighbors to align their propulsion. Al-though the specific mechanisms by which cells in epithe-lial sheets align their tractions remains unclear, it is likelythat the alignment mechanism involves a combinationof biochemical signalling, e.g., via correlation of front-back polarity, which can be coordinated between neigh-boring cells [9, 16], and mechanical interaction. 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Phys. , 1074 (2015). [50] A. Saraswathibhatla and J. Notbohm, Phys. Rev. X ,011016 (2020). upplemental Material: Coordinated tractions control the size of a collectivelymoving pack in a cell monolayer Aashrith Saraswathibhatla, Silke Henkes, Emmett E. Galles, Rastko Sknepnek,
3, 4 and Jacob Notbohm ∗ Department of Engineering Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom School of Science and Engineering, University of Dundee, Dundee DD1 4HN, United Kingdom School of Life Sciences, University of Dundee, Dundee DD1 5EH, United Kingdom
I. SUPPLEMENTAL FIGURES )05001000 V o r ono i c e ll a r ea ( µ m ) a bc d
100 µm
FIG. S1. Tracking each cell and its outline in a cell monolayer. (a) Cellular trajectories built by tracking each cell’snucleus which is labeled with green fluorescent protein. (b) Voronoi tessellation built using the nuclei centroids. (c)Overlay of the Voronoi tessellation on a phase contrast image of the corresponding cell monolayer. (d) Scatter plotcomparing cell areas from the Voronoi tessellation and the actual cell areas. ∗ Correspondence should be sent to: [email protected] a r X i v : . [ phy s i c s . b i o - ph ] F e b abb M agn i t ude o f N e t T r a c t i on ( P a ) e M agn i t ude o f N e t T r a c t i on ( P a ) Isolated cells Cells inmonolayer c
50 µm θ (°)010203040 N u m be r o f c e ll s θ d t ′ . ( h ) f FIG. S2. Non-zero net traction of cells in a monolayer. (a) Overlay of Voronoi tessellation on traction color map.(b) Magnitude of vector sum of traction (i.e., net traction) applied by five representative cells (colors) over time.(c) Images of F-actin labeled with phalloidin. Tractions were measured simultaneously (see materials and methodsbelow); the arrows show the net traction applied by each cell. (d) Histogram of angle θ between the orientation ofstress fibers and direction of traction for different cells. The inset defines angle θ where stress fibers are shown bythe black lines and traction directions are shown by the cyan arrows. (e) Magnitude of net traction for isolated cellsand cells in a monolayer. Each dot represents a different cell. As isolated cells produce zero net traction, the datafor isolated cells quantifies the noise floor of the measurement. The magnitude of net traction applied by cells in amonlayer is statistically larger than that applied by isolated cells ( p < . t . , thetime at which the temporal persistence of traction reaches a value of 0.2, for ≈
600 cells. -0.200.20.40.60.81 T r a c t i on pe r s i s t en c e t ′ (h) r ′ (µm) V e l o c i t y au t o c o rr e l a t i on b cd ea r ′ . ( µ m ) C t r l M L J S P C y t o D t ′ . ( h ) C t r l M L J S P C y t o D M agn i t ude o f N e t T r a c t i on ( P a ) C t r l M L J S P C y t o D CtrlJSPML141CytoD020406080100120
FIG. S3. Perturbing traction persistence. (a) Magnitude of net traction after treating cell monolayers with Jas-plakinolide (JSP), ML 141, or cytochalasin D (Cyto D). All treatments are statistically different from control (Ctrl)( p < .
01, ANOVA with Tukey correction for multiple comparisons). (b) Average temporal persistence of tractionin response to the different treatments. (c) Temporal persistence time of traction. All treatments are statisticallydifferent from control ( p < .
01, ANOVA with Tukey correction). (d) Spatial autocorrelation of velocity in responseto the different treatments. (e) Correlation length of velocity. All treatments are statistically different from control( p < .
01, ANOVA with Tukey correction). In panels a, c, and e, each dot represents an average from a different cellmonolayer. -0.200.20.40.60.81 T r a c t i on au t o c o rr e l a t i on r ′ (µm) T r a c t i on au t o c o rr e l a t i on r ′ (µm) r ′ (µm) V e l o c i t y au t o c o rr e l a t i on r ′ (µm) V e l o c i t y au t o c o rr e l a t i on ∞ V e l o c i t y pe r s i s t en c e t ′ (h) V e l o c i t y pe r s i s t en c e t ′ (h) ∞ τ p = . h τ p = . h -0.200.20.40.60.81 T r a c t i on au t o c o rr e l a t i on r ′ (µm) r ′ (µm) V e l o c i t y au t o c o rr e l a t i on ∞ V e l o c i t y pe r s i s t en c e t ′ (h) τ p = . h P r obab ili t y P r obab ili t y P r obab ili t y Speed (µm/min)
Speed (µm/min)
Speed (µm/min) FIG. S4. Combined effects of persistence and alignment times, τ p and τ a . Histograms of cell speed and graphs oftraction autocorrelation, velocity autocorrelation, and velocity persistence are shown for τ p = 2 .
6, 1 .
3, and 0 . τ a in units of h. For all plots, f = 53 nN. Low serumCN03 τ p = 1.3; τ a = ∞ ; f = 14 τ p = 2.6; τ a = 5; f = 53 Low serumCN03-0.200.20.40.60.81 V e l o c i t y pe r s i s t en c e -0.200.20.40.60.81 T r a c t i on pe r s i s t en c e t ′ (h) t ′ (h) a b c M agn i t ude o f N e t T r a c t i on ( P a ) Lowserum CN03
FIG. S5. Effects of Rho-activator CN03. (a, b) CN03 treatment increased traction magnitude (a, p < . τ p and τ a haveunits of h; f has units of nN. II. SUPPLEMENTAL VIDEO CAPTION
Video 1. Time lapse video of cell velocities (arrows) overlaid on phase contrast images of the cell monolayer. Colorsindicate the angle of the cell velocity with respect to the horizontal axis (see color circle in Fig. 1). The size of thefield of view is 360 × µ m. III. SUPPLEMENTAL NOTE—PERSISTENCE OF VELOCITY
Here, we compare the temporal persistence of velocity between experiments and model. A temporal autocorrelationof cell velocities showed that increasing traction alignment (i.e., decreasing τ a ) caused a greater temporal correlation,indicating more persistent collective motion (Fig. S4). Compared to the aucorrelations in traction and velocity, thepersistence of velocity did not agree as well with the experimental data—at short time scales it decayed faster inthe model than in the experiments for studied values of τ a . Additionally, for long times and small values of τ a , thevelocity persistence in the model was larger than the experiments. These differences suggest that there are additionalfactors in the experiments that are not considered in the model. One such factor is likely to be that in MDCK cellswaves of collective cell motion oscillate over space with time periods of a few hours [ ? ? ? ], which would reduce thepersistence on these time scales. These oscillations were not included in the model, which would explain the largerpersistence at long times in the model than in the experiments. IV. MATERIALS AND METHODSA. Cell culture
Madin-Darby Canine Kidney type II cells were used in the experiments. In all experiments except those withSiR-actin, the cells used were stably transfected with or without green fluorescent protein (GFP) in the nucleus asdescribed previously [ ? ]. The cells were maintained in low-glucose Dulbecco’s modified Eagle’s medium (12320-032;Life Technologies, Carlsbad, CA) with 10% fetal bovine serum (Corning, NY) and 1% G418 (Corning) in an incubatorat 37 ◦ C and 5% CO2. The cells were passaged every 2 or 3 days until passage 20. For experiments that treated cellswith cytochalasin D, jasplakinolide, and ML-141, and the corresponding control, cell medium was replaced withmedium containing 2% fetal bovine serum 8-12 h before the start of the experiment. For experiments referred to aslow serum, medium was replaced with medium containing 1% serum 24 h before the start of the experiment
B. Polyacrylamide substrates and micropatterning
Cell monolayers were micropatterned on polyacrylamide substrates as described previously [ ? ]. Briefly, we fabri-cated polyacrylamide gels with Young’s modulus of 6 kPa and thickness of 150 µ m with fluorescent particles located atthe top. Next, polydimethysiloxane masks with holes (1.5 mm diameter) were placed on the gels before functionalizingwith sulfo-SANPAH and collagen I, thereby constraining the collagen to circular patterns on the gels. 500 µ L of cellsolution of concentration 0.5 million cells/mL was pipetted onto the masks and incubated at 37 ◦ C for 2 h. The maskswere removed, and confluent cell monolayers were formed within 10-12 h.
C. Microscopy
Time lapse microscopy was performed using an Eclipse Ti microscope (Nikon, Melville, NY) with a 20 × numericalaperture 0.5 objective (Nikon) and an Orca Flash 4.0 camera (Hamamatsu, Bridgewater, NJ) running Elements Arsoftware (Nikon). Live imaging of stress fibers and the experiments that measured tractions and imaged stress fiberssimultaneously used the same microscope and a 40 × numerical aperature 1.15 objective (Nikon). Fixed imaging ofstress fibers used an A1R+ confocal microscope with a 40 × NA 1.15 water-immersion objective with a step size of0.5 µ m using Elements Ar software (all Nikon).For time lapse imaging, fluorescent and phase contrast images were captured every 10 min for 8-10 hr. During timelapse imaging, the cells were maintained at 37 ◦ C and 5% CO2 using a H301 stage top incubator with UNO controller(Okolab USA Inc, San Bruno, CA). After the imaging, cells were removed from the polyacrylamide substrates byincubating in 0.05% trypsin for 20 min, and images of the fluorescent particles were collected; these images provideda traction-free reference state for computing cell-substrate tractions.
D. Single cell tracking
Images of GFP-labeled cell nuclei were segmented using StarDist, a plugin in ImageJ [ ? ]. For computing cellvelocities, we tracked the nucleus of each cell over time from time lapse imaging performed every 10 min. The centersof cell nuclei were identified from segmented images at each time point. For each time point, every cell nucleus wasmapped to the nearest cell nucleus at the next time point using the knnsearch function in Matlab, allowing for celltrajectories and velocities to be computed. As the cell speed was typically no more than 0.5 µ m/min and imaging wasperformed every 10 min, the average cell displacement was < µ m, which is much less than the average cell diameterof 22 µ m. Hence, our method avoided mismatches between time frames. In the event of cell division, only one of thedaughter cells was tracked. E. Individual cell traction analysis
Cell-induced displacements of the fluorescent particles were measured using Fast Iterative Digital Image Correlation[ ? ] using 32 ×
32 pixel subsets centered on a grid with a spacing of 8 pixels (2 . µ m). Tractions were computedusing unconstrained Fourier transform traction microscopy [ ? ] accounting for the finite substrate thickness [ ? ? ].Next, a Voronoi tessellation was constructed using the center of each cell nucleus and the traction at each grid pointwas mapped to each cell. Finally, a vector sum of the traction within each cell was computed, giving the net tractionapplied by each cell.In the experiments that measured cell tractions and stained F-actin simultaneously, fluorescent particles in the sub-strate were imaged before seeding the cells, which provided the stress-free reference state for traction force microscopy.After seeding the cells, we simultaneously imaged fluorescent particles in the deformed state of substrate with cellson it, fixed the cells, and stained for F-actin as described below. F. Chemical treatments
Chemical treatments were cytochalasin D (Sigma-Aldrich), ML-141 (Sigma-Aldrich), Jasplakinolide (Sigma-Aldrich), and CN03 (Cytoskeleton, Inc, Denver, CO). Stock solutions of cytochalasin D, ML-141, and CN03 wereprepared at 2 mM, 2 mM, and 0.1 g/L, respectively, all dissolved in dimethylsulfoxide except CN03 in water. Thestock solutions were diluted in phosphate buffered saline to obtain desired concentrations for the experiments.
G. Stress fiber imaging
For live imaging of stress fibers, the cells were seeded on a thin sheet of polyacrylamide ( < µ m in thickness)and treated with 0.2 µ M SiR-actin (Cytoskeleton, Inc). For fixed imaging of stress fibers, the cells were rinsed twicewith PBS and fixed with 4% paraformaldehyde in PBS for 20 min. The cells were washed with Tris-buffered salinetwice for 5 min each and then incubated in 0.1% Triton X-100 for 5 min at room temperature, and treated with 3-5units/mL Phallodin Dylight 594 (Life Technologies catalog no. 21836).
H. Numerical simulations model
Here we briefly summarize the model used for numerical simulations and show that in the absence of alignmentinteractions there are no spatial correlations between tractions of neighboring cells.
1. Self-propelled Voronoi (SPV) model
Numerical simulations were performed with the Self-Propelled Voronoi (SPV) model [ ? ? ] implemented in theSAMoS package [ ? ] and extended to include alignment interaction between tractions of neighboring cells. In theSPV, a cell monolayer is modeled as the Voronoi tiling of the plane with cell centroids acting as Voronoi seeds. Werecall that a Voronoi tessellation is a polygonal tiling of the plane based on distances to a set of seed points. Foreach seed there is a corresponding polygon that contains the region of the plane closer to that seed than to any other.The assumption of Voroni tiling is essential to ensure that smooth motion of cell centroids corresponds to smoothdeformations of the cells [ ? ].The state of cell i is described by the position vector, r i of its centroid and the direction of the traction, describedby a unit-length vector n i . The traction vector can be written in terms of the angle θ i between n i and the x − axis ofthe laboratory reference frame as n i = (cos θ i , sin θ i ). The equations of motion are then µ ˙ r i = −∇ r i E + f n i , (S1)˙ θ i = 1 τ a X j n.n. i sin ( θ j − θ i ) + η i , (S2)where µ is the friction coefficient that measures dissipation with the environment, f is the magnitude of the self-propulsion force in the direction of the traction vector n i , τ a is the time scale it takes two neighboring traction vectorsto align with each other, and η i is a random variable with h η i i = 0 and h η i ( t ) η j ( t ) i = τ − p δ ij δ ( t − t ), where h . . . i is the ensemble average, and τ p is the characteristic persistence time of the direction of cell traction. Finally, E = K A X i ( A i − A ) + Γ P X i ( P i − P ) , (S3)is the energy functional of the Vertex Model [ ? ]. K A is the area modulus, i.e., the energy cost of the cell i havingarea A i that is different from the preferred area A . Γ P is the perimeter modulus, i.e., the energy cost of cell i havingperimeter P i that is not equal to the preferred perimeter P . We note that in Eq. (S1), the gradient is computedusing the position of the cell centroid r i , while the energy functional in Eq. (S3) is typically written in in terms ofpositions of the vertices that are meeting points of three or more cell junctions. If one restricts to the Voronoi tiling,as it is the case in the SPV, there is a one to one mapping between positions of cell centroids and vertices, whichmakes it possible to directly compute the force on the cell centroid from Eq. (S1). The calculation is tedious butstraightforward [ ? ]. Cell velocities in the model are computed by taking the displacement at different time pointsand dividing by time, which is consistent with the method used to compute velocities in the experiments.
2. Traction alignment without alignment interaction
In order to find the spatial and temporal correlation between traction vectors, we need to calculate h n ( r , t ) · n ( r , t ) i .We find, h n ( r , t ) · n ( r , t ) i = h n x ( r , t ) n x ( r , t ) i + h n y ( r , t ) n y ( r , t ) i = h cos θ ( r , t ) cos θ ( r , t ) i + h sin θ ( r , t ) sin θ ( r , t ) i = h cos ( θ ( r , t ) − θ ( r , t )) i (S4)where in going from the second to the third line we used cos ( θ ( r , t ) − θ ( r , t )) = cos ( θ ( r , t )) cos ( θ ( r , t )) +sin ( θ ( r , t )) sin ( θ ( r , t )). In the absence of alignment between neighboring cells, the first term on the right-hand-sidein Eq. (S2) vanishes (i.e., τ a → ∞ ) and the equation of motion for the orientation of the traction vector n i decouplesfrom its neighbors. Therefore, for τ a → ∞ , ˙ θ i = η i . (S5)In this case, it is straightforward to show that D ( θ i ( t ) − θ j ( t )) E = 2 τ p ( t + t − ψ ( t, t )) δ ij , (S6)where ψ ( x, y ) is the minimum function that is equal to the smaller of the two of its arguments. We note that t + t − ψ ( t, t ) = | t − t | . (S7)If we recall that cos( θ ) = Re (cid:0) e iθ (cid:1) , and use the cumulant expansion, (cid:10) e iθ (cid:11) = e i h θ i− ( h θ i −h θ i ), with h θ i = 0, Eqs.(S4) and (S6) give h n ( r , t ) · n ( r , t ) i = e − | t − t | /τ p δ ( r − r ) ..