Coordinated tractions increase the size of a collectively moving pack in a
cell monolayer

Aashrith Saraswathibhatla,1 Silke Henkes,2 Emmett E. Galles,1 Rastko
Sknepnek,3,4 Jacob Notbohm1

1Department of Engineering Physics, University of Wisconsin-Madison, Madison,
Wisconsin 53706, USA

2School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

3School of Science and Engineering, University of Dundee, Dundee DD1 4HN,
United Kingdom

4School of Life Sciences, University of Dundee, Dundee DD1 5EH, United Kingdom

Correspondence should be sent to<EMAIL_ADDRESS>
## Keywords

Collective cell migration, traction persistence, traction alignment, Self-
propelled Voronoi model

## Abstract

Cells in an epithelial monolayer coordinate motion with their neighbors giving
rise to collectively moving packs of sizes spanning multiple cell diameters.
The physical mechanism controlling the pack size, however, remains unclear. A
potential mechanism comes from assuming that cell–substrate traction forces
persist over some time scale: with large enough persistence time, collective
cell packs emerge. To test this hypothesis, we measured the velocity and net
traction of each cell. The data showed that in addition to having some
temporal persistence, tractions were spatially correlated, suggesting that
cells coordinate with their neighbors to apply tractions in the same
direction. Chemical inhibitors and activators of actomyosin contraction were
used to determine effects of altering the traction persistence and alignment.
Numerical simulations based on the self-propelled Voronoi model, augmented to
include both traction persistence and alignment and calibrated against the
experimental data, matched the experimentally measured pack size. The model
identified that if there were no alignment of traction between neighboring
cells, the size of the collective pack would be substantially smaller than
observed in the experiments. Hence, combining experiments and a simple
mechanical model, this study confirms the long-standing assumption of traction
persistence and adds the notion of traction alignment between neighbors.
Together, persistence and alignment are two factors controlling the size of a
collectively moving cell pack.

## 1\. Introduction

During tissue formation and repair, cells migrate in collective groups and
packs [1]. The characteristic size of a collective cell pack lies at an
intermediate point between extremes present in other collective systems—cell
motion is not fully random as it would be in systems driven by thermal
fluctuations [2], nor is the motion highly coordinated as in flocks of birds
[3] or schools of fish [4]. Instead, in collective cell swirls, packs, and
waves, the motion is correlated over length scales ranging from a few cells to
a couple dozen cells [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Hence, there
must be a mechanism for correlation of velocity between neighboring cells.

A plausible scenario, from predictions of prior models of epithelial cell
sheets, is that the finite time required for a cell to reorganize its force-
generating machinery leads to a temporal persistence to each cell’s motion.
The models, including both geometric (i.e., vertex) models [17, 18, 19, 20]
and simpler particle-based models [21, 22, 15], have shown that if the
persistence time is sufficiently large, collective packs can emerge [18, 15,
20]. While appealing for its simplicity, this explanation includes no explicit
mechanism for a cell to polarize and align its propulsive force with that of
its neighbors. Cell-induced substrate displacements are correlated over
multiple cell lengths [23], implying that there may exist some mechanism of
cell coordination. Several physical mechanisms have been proposed, including a
tendency of cells to migrate along the local orientation of the maximal
principal stress [24], a tendency to apply tractions that pull each cell
toward regions of empty space [25], force transmission across the cell layer
that causes collective stiffness sensing [26], collective cell migration in
response to electric fields [27], and alignment between neighboring cells
creating orientational nematic order [28, 29, 30]. Models of such coordination
have suggested that a coupling between the force on a cell and its traction
results in flock formation [21, 31]. In addition, cells can coordinate to
align stress fibers in the same direction [32, 33, 34], and there exist
molecular mechanisms that enable neighboring cells to coordinate front-back
polarization [9, 16]. These observations give circumstantial evidence that
cells may coordinate with their neighbors to apply tractions in the same
direction. Coordination of traction would then be a second mechanism—in
addition to temporal persistence—that controls the size of a collective cell
pack. Direct evidence, however, is lacking, because neither the temporal
persistence time nor a coordination of traction forces between neighbors has
been quantified experimentally at the single cell level.

In this letter, using traction force microscopy and cell tracking, we
developed a method to measure the net traction exerted by individual cells
over time. Using this new single-cell measure, we verified the long-standing
notion that cell tractions are persistent over time. Importantly, a spatial
autocorrelation of single cell traction demonstrated that the collective
motion is not controlled merely by a temporal persistence of traction; rather,
cells also coordinate to align directions of traction with those of their
neighbors. Traction persistence and alignment were investigated further, by
using chemical perturbations of actomyosin contraction and a self-propelled
Voronoi model that was calibrated against the experimental data and included
both traction persistence and alignment. Together, the experiments and model
reveal how the interplay between persistence and alignment affects the size of
a collectively moving cell pack.

## 2\. Methods

### 2.1. Cell Culture and Sample Preparation

Madin-Darby Canine Kidney (MDCK) type II cells were used in the experiments.
In all experiments except those with SiR-actin (described in the Results
section), the cells used were stably transfected with green fluorescent
protein (GFP) in the nucleus as described previously [11]. Cell monolayers
were micropatterned into 1.5 mm islands on polyacrylamide substrates of
Young’s modulus 6 kPa and thickness of 150 $\mu$m as described previously [35]
and detailed in the Supplemental Methods.

### 2.2. Microscopy

Time lapse microscopy was performed using an Eclipse Ti microscope (Nikon,
Melville, NY) with a 20$\times$ numerical aperture 0.5 or 40$\times$ numerical
aperture 1.15 objective (Nikon) and an Orca Flash 4.0 camera (Hamamatsu,
Bridgewater, NJ) running Elements Ar software (Nikon). For traction
experiments, after the imaging, cells were removed from the polyacrylamide
substrates by incubating in 0.05% trypsin for 20 min, and images of the
fluorescent particles were collected; these images provided a traction-free
reference state for computing cell-substrate tractions. Fixed imaging of
stress fibers used an A1R+ confocal microscope with a 40$\times$ NA 1.15
water-immersion objective with a step size of 0.5 $\mu$m using Elements Ar
software (all Nikon).

### 2.3. Single-Cell Velocity and Traction Measurements

Cell velocities were measured by using methods of particle tracking [36].
Images of GFP-labeled cell nuclei were segmented using StarDist [37]. At each
time point, every cell nucleus was mapped to the nearest cell nucleus at the
next time point using the knnsearch function in Matlab, allowing for cell
velocities and trajectories to be computed. Cell-induced displacements of the
fluorescent particles were measured using Fast Iterative Digital Image
Correlation [38] using $32\times 32$ pixel subsets centered on a grid with a
spacing of $8$ pixels ($2.6$ $\mu$m). Tractions were computed using
unconstrained Fourier transform traction microscopy [39] accounting for the
finite substrate thickness [40, 41]. To quantify the net traction applied by
each cell, a Voronoi tessellation was constructed using the center of each
cell nucleus, enabling the traction at each grid point to be mapped to each
cell. A vector sum of the traction within each cell was computed, giving the
net traction applied by each cell.

### 2.4. Numerical Model

Numerical simulations were performed with the Self-Propelled Voronoi (SPV)
model [18, 19] implemented in the SAMoS package [42]. In the SPV, a cell
monolayer is modeled as the Voronoi tiling of the plane with cell centroids
acting as Voronoi seeds. We recall that a Voronoi tessellation is a polygonal
tiling of the plane based on distances to a set of seed points. For each 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 smooth
deformations of the cells [19].

The state of cell $i$ is described by the position vector, $\mathbf{r}_{i}$ of
its centroid and the direction of self-propulsion (i.e., traction), described
by a unit-length vector $\mathbf{n}_{i}$. The traction vector can be written
in terms of the angle $\theta_{i}$ between $\mathbf{n}_{i}$ and the $x-$axis
of the laboratory reference frame as
$\mathbf{n}_{i}=\left(\cos\theta_{i},\sin\theta_{i}\right)$. The motion of
each cell is described by two equations, one for the cell’s position
$\mathbf{r}_{i}$ and one for $\theta_{i}$,

$\displaystyle\mu\dot{\mathbf{r}}_{i}$ $\displaystyle=$
$\displaystyle-\nabla_{\mathbf{r}_{i}}E+f_{0}\mathbf{n}_{i},$ (1)
$\displaystyle\dot{\theta}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{a}}\sum_{j}\sin\left(\theta_{j}-\theta_{i}\right)+\eta_{i},$
(2)

where $\mu$ is the friction coefficient that measures dissipation with the
environment, $f_{0}$ is the magnitude of the self-propulsion force in the
direction of the traction vector $\mathbf{n}_{i}$. Variable $\tau_{a}$
represents the characteristic time required for cell $i$ to align its traction
direction with that of its neighbors; smaller $\tau_{a}$ indicates greater
strength of alignment of neighboring tractions. The variable $\eta_{i}$
describes a Gaussian random process with $\langle\eta_{i}\rangle=0$ and
$\langle\eta_{i}\left(t\right)\eta_{j}\left(t^{\prime}\right)\rangle=\tau_{p}^{-1}\delta_{ij}\delta\left(t-t^{\prime}\right)$,
with $\tau_{p}$ being the characteristic persistence time of the direction of
cell traction, and $\langle\dots\rangle$ denoting an average with respect to
the distribution of the realizations $\eta_{i}$. 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. Finally,

$E=K_{A}\sum_{i}\left(A_{i}-A_{0}\right)^{2}+\Gamma_{P}\sum_{i}\left(P_{i}-P_{0}\right)^{2},$
(3)

is the energy functional of the vertex model [17]. $K_{A}$ is the area
modulus, i.e., the energy cost of the cell $i$ having area $A_{i}$ that is
different from the preferred area $A_{0}$. $\Gamma_{P}$ is the perimeter
modulus, i.e., the energy cost of cell $i$ having perimeter $P_{i}$ that is
not equal to the preferred perimeter $P_{0}$. We note that in Eq. (1), the
gradient is computed using the position of the cell centroid $\mathbf{r}_{i}$,
while the energy functional in Eq. (3) is typically written in terms of
positions of the vertices that are meeting points of three or more cell
junctions. If one restricts to the Voronoi tiling, as is the case in the SPV,
there is a one-to-one mapping between positions of cell centroids and
vertices, which makes it possible to directly compute the force on the cell
centroid from Eq. (1). The calculation is tedious but straightforward [19].
Cell velocities in the model are computed by taking the displacement at
different time points and dividing by time, which is consistent with the
method used to compute velocities in the experiments.

## 3\. Results

To begin, we seeded MDCK cells in confluent monolayers, imaged them over time,
and tracked the nuclei [37, 36] to measure each cell’s velocity (Fig. 1a). The
persistence of cell velocity was defined by the temporal autocorrelation,
$C(t^{\prime})=\langle\hat{\mathbf{v}}\left({t}\right)\cdot\hat{\mathbf{v}}\left({t}-{t}^{\prime}\right)\rangle$,
where $\hat{\mathbf{v}}({t})$ is the unit vector corresponding to the
direction of cell velocity at time ${t}$, and $\langle...\rangle$ indicates an
average over all cells in all pairs of time points that are $t^{\prime}$
apart. To quantify the persistence time, we used the time over which the
persistence decreased to a value of $0.2$, which corresponds to the time
required for the direction of cell velocity to change by 78∘. The persistence
time was $\approx 1.5$ h (Fig. 1b), consistent with prior research [15]. In
addition to persistence, the cell velocity fields showed packs of cells moving
in the same direction (Fig. 1a; Supplemental Video 1). The pack size was
quantified by a spatial autocorrelation of $\hat{\mathbf{v}}$, which showed
correlated cell velocities over distances of several cell diameters (Fig. 1d).
As with the persistence time, the pack size was quantified by the distance
over which the correlation dropped to $0.2$, which was $\approx 90$ $\mu$m
(Fig. 1d), setting a typical pack to have the size of $\approx 180$ $\mu$m
($\approx 8$ cell diameters), which matches the order of magnitude of previous
measurements of velocity correlations in confluent epithelial sheets [6, 10,
12, 13, 15].

Figure 1: Temporal persistence and spatial correlation of velocity and
traction. (a) Cell velocities (arrows) overlaid on an image of a cell
monolayer. Colors indicate the angle of the cell velocity (see color circle).
(b) Persistence, defined as the temporal autocorrelation, of single-cell
velocity and traction. In all figures, lines with error bars represent mean
values and standard deviations over at least four cell monolayers and 6 h of
imaging. (c) Single-cell tractions overlaid on a Voronoi tessellation of a
cell monolayer. Colors indicate the angle of traction (see color circle). (d)
Spatial autocorrelation of single-cell velocity and traction. In all figures,
vertical dashed lines represent the average cell diameter. (e) Representative
images of stress fibers reorienting over time. Solid lines indicate the
orientation of a pair of stress fibers at different time points. (f)
Representative images of a stress fiber disappearing over time (circled). (g)
Time required for stress fibers to reorient (circles) or disappear (stars).

To relate these temporal and spatial velocity correlations to the tractions at
the cell-substrate interface, we used traction force microscopy [38, 43, 39,
40, 41], which gave a grid of traction measurements between the cell monolayer
and the substrate spaced uniformly by $2.6$ $\mu$m. As cells are not uniformly
spaced, the traction data did not map directly to each cell. To resolve this
issue, we produced a Voronoi tessellation approximating the cell outlines
(Supplemental Fig. S1) [44], which enabled an approximate mapping between each
cell and its traction (Supplemental Fig. S2a). Then, the vector sum of
traction applied by each cell was computed, yielding the net traction produced
by that cell (Fig. 1c). To verify the accuracy of the net traction computed by
the Voronoi-based estimate of the cell outlines, we compared the Voronoi-based
net traction to the net traction computed from segmenting each cell, with
results showing close agreement (Supplemental Fig. S2b, c). Therefore, we used
the Voronoi-based estimate of the cell outlines for the remainder of our
study. The magnitude of net traction applied by each cell fluctuated in time
but rarely reached $0$ (Supplemental Fig. S2d). Importantly, the direction of
net traction coincided closely with the orientation of stress fibers in cells,
consistent with the notion that stress fibers generate tractions (Supplemental
Fig. S2e-f). The noise floor of this new measurement was quantified by
computing the net traction applied by isolated cells, which should be zero as
inertial forces are negligible [45, 46]. The average was $10$ Pa (Supplemental
Fig. S2g), which defines the experimental noise floor, and, importantly, was
nearly an order of magnitude smaller than the average net traction applied by
cells in a monolayer, $\approx 80$ Pa.

The net traction is not merely a high resolution measurement of tractions, but
rather a measurement of the total force applied by each cell to the substrate.
As such, it enables new investigation, for example, of the temporal
persistence and spatial correlation of each cell’s traction (Fig. 1b, d). The
traction persistence time, defined as the time over which the correlation
decreased to $0.2$, was found to be $\approx 1.5$–$2$ h (Fig. 1b, Supplemental
S2h). To investigate the cytoskeletal origins of the traction persistence
time, we performed live fluorescence imaging of F-actin labeled with SiR-actin
[47]. After observing the temporal evolution of stress fibers in multiple
cells, we classified our observations into two cases. In the first, the stress
fibers reoriented with time (Fig. 1e); in the second, the stress fibers
disappeared after some time (Fig. 1f). We then quantified the time required
for stress fibers to reorient or disappear, with times averaging $1.5\pm 0.5$
and $2.0\pm 0.7$ h, respectively (Fig. 1g). As these times match the
persistence times of traction, these data provide evidence that the
measurement of net traction is accurate and that its temporal dynamics are
physically meaningful. Turning now to the spatial correlation of traction, if
tractions applied by neighboring cells were fully uncorrelated, the
autocorrelation would decay to zero over a distance of approximately one cell
diameter ($\approx 22$ $\mu$m). By contrast, the averaged autocorrelation
decayed to zero over a distance of two cell diameters, indicating that cells
coordinate with their neighbors to apply traction in the same direction (Fig.
1d). Hence, cell tractions are correlated in both time and space.

We next sought to identify the relative contributions of traction persistence
and correlation on the size of a collectively moving cell pack. To begin, we
considered temporal persistence, as prior models have suggested that
increasing persistence increases the pack size [18, 15, 20]. In an effort to
alter actin dynamics, we used the F-actin stabilizer jasplakinolide (JSP)
[48], which increased traction persistence and decreased traction magnitude by
factors of $\approx 2$ (Supplemental Fig. S3a-c). Even though the JSP
increased traction persistence, it reduced the spatial correlation of velocity
(Fig. 2a), in contrast to our expectation. A potential resolution is that the
treatment also caused a statistically significant reduction in the spatial
correlation of traction (Fig. 2b, Supplemental Fig. S3d, e), suggesting that
the velocity correlation may be determined more so by the traction
correlation. Other chemical treatments produced similar results: using ML 141
to inhibit the front-back polarization molecule Cdc42 [49] and altering actin
polymerization with cytochalasin D [50] also increased the persistence time
but decreased the spatial correlation of traction. As with JSP, both ML 141
and cytochalasin D decreased the velocity correlation, though these treatments
also reduced the magnitude of traction produced (Supplemental Fig. S3).
Together, these data suggest that traction correlation may have a larger
effect than traction persistence on the size of a collectively moving cell
pack. An alternative explanation is that the treatments affected pack size by
altering the distance over which cell–cell forces propagate. To consider this
explanation, we quantified propagation of tensile stresses in the monolayer by
using monolayer stress microscopy [24, 51, 52] and computing an
autocorrelation. Results showed that the correlation length of tension was not
statistically affected by treatment with JSP (Supplemental Fig. S4), which
ruled out the explanation based on cell–cell forces, thus supporting the idea
that the spatial correlation of traction had the stronger effect on pack size.

Figure 2: Evidence for traction alignment spanning multiple cells. (a, b)
Spatial autocorrelation of cell tractions (a) and velocities (b) in
experiments and model. Values in the legend in panel a correspond to different
traction persistence and alignment times, $\tau_{p}$ and $\tau_{a}$, measured
in hours, and force $f_{0}$ measured in nN. (c, d) Representative images of
cell velocities (c) and net cell tractions (d) in the model, for
$\tau_{p}=1.3$ h, $\tau_{a}=2.5$ h and $f_{0}=53$ nN. Colors indicate the
angle of velocity or traction (see color circle). The scale bars correspond to
$100\>\mu$m.

A potential confounding factor in the experimental data is that the traction
magnitude changed in addition to the traction persistence and correlation
(Supplemental Fig. S3). Experiments alone cannot resolve this issue, because
the three variables (traction magnitude, persistence, and correlation) cannot
be controlled independently in the experiments. Therefore, we turned to the
SPV model [18, 19, 42]. Although the original implementation of the SPV model
[18] included only temporal persistence of traction, our experimental data
give evidence for both persistence and alignment. Therefore, we extended the
SPV model by adding a term in Eq. 2 for alignment of traction between
neighboring cells, similar to recent studies [19, 53]. The SPV model includes
sufficient cell-level detail to estimate, to order of magnitude, the
parameters of the experimental system [15]. Briefly, the estimates are as
follows. The unit of length was chosen to match the average cell diameter in
the experiments, $\ell=22$ $\mu$m, which set the the preferred cell area
$A_{0}=\pi\ell^{2}/4$ in the SPV model. Furthermore, $A_{0}$ can be combined
with the preferred cell perimeter $P_{0}$ into a single dimensionless
parameter, $p_{0}=P_{0}A_{0}^{-1/2}$, which is one factor controlling whether
the collective cell behavior in the model is liquid- or solid-like [54, 18].
In the SPV model simulations, we used an input value of $p_{0}=3.8$, which is
in the solid regime but near the border to a liquid. Adding self-propulsion
then raises the actual values of $PA^{-1/2}$ above the input value of $3.8$,
consistent with measured values of $PA^{-1/2}$ in MDCK cells [35] and with the
fluid-like migration observed in our experiments (Supplemental Video 1). We
used the stiffness parameters $K$ and $\Gamma$ estimated in ref. [15]. The
value of $f_{0}$ was estimated from the mean magnitude of net traction, $110$
Pa (Supplemental Fig. S3a), multiplied by the average cell area, $22\times 22$
$\mu$m2 to give 53 nN. The friction factor $\mu$ was tuned to achieve a match
between cell migration speeds in the experiments and model (average speed of
$18$ $\mu$m/h, Supplemental Fig. S5); a value of $\mu=265$ pN$\cdot$h/$\mu$m
gave the best match and is of the same order of magnitude as prior estimates
[30, 15]. Earlier references wrote the equations using variable
$v_{0}=f_{0}/\mu$ in place of $f_{0}$, with $v_{0}$ having units of velocity
[18, 19]. Here, we chose to use $f_{0}$ due to its connection to the single-
cell net traction measured in the experiments.

There remained two free parameters in the model, the persistence and alignment
time scales, $\tau_{p}$ and $\tau_{a}$, respectively. We used a value of
$\tau_{p}=1.3$ h to match the experimentally measured traction persistence
time (Fig. 1b). We then adjusted $\tau_{a}$ to achieve a good fit to the
traction correlations in the untreated control data, finding a good match for
$\tau_{a}=2.5$ h (Fig. 2b). Importantly, using the value $\tau_{a}=2.5$ h, the
model also matched closely to the velocity correlation data (Fig. 2a; see the
Supplemental Note for a comparison of experimental and simulated velocity
persistences). The matched value of $\tau_{a}=2.5$ h is slightly longer than
the $1.5$–$2$ h required for stress fibers to reorient in a cell (Fig. 1g),
which is reasonable given that the multicellular realignment of traction
associated with $\tau_{a}$ could not be faster than the time required for each
cell to reorganize its own stress fibers. Representative images of cell
velocity and traction in the model (Fig. 2c,d) show collective cell packs and
neighboring cells applying traction in the same direction, which is
reminiscent of the experimental data.

With the model parameters chosen to match experimental control conditions, we
then adjusted the parameters to mimic the experiments with JSP. To this end,
we increased the traction persistence time $\tau_{p}$ and decreased the
traction magnitude $f_{0}$ by factors of two, consistent with the experimental
data (Fig. S3a–c). The effect of this change on the velocity correlation was
small (Fig. 2a), suggesting that $\tau_{p}$ alone had a relatively small
effect on the velocity correlations. Moreover, these altered parameters
increased the spatial correlations in velocity and traction, opposite to the
experimentally observed effects of JSP (Fig. 2a, b). Matching the experimental
data required us to increase both $\tau_{p}$ and $\tau_{a}$, which indicates
that the multicellular traction alignment is essential to capture the
experimental data. We performed a more thorough theoretical study of the
combined effects of $\tau_{p}$ and $\tau_{a}$ by decreasing and increasing
$\tau_{p}$ by factors of two and subsequently varying $\tau_{a}$ through a
range of values (Supplemental Fig. S5). Consistent with prior models [18, 15,
20], increasing $\tau_{p}$ caused larger collective packs in the absence of
multicellular traction alignment (i.e., $\tau_{a}\to\infty$). Additionally,
for larger values of $\tau_{p}$, less traction alignment (i.e., larger
$\tau_{a}$) was required to match the experimental data (Supplemental Fig.
S5), indicating that increasing either single-cell persistence or
multicellular alignment increases the size of a collectively moving cell pack.
Importantly, for all values of $\tau_{p}$ studied, a finite $\tau_{a}$ was
required to match the simulations to the experimental data. In the absence of
traction alignment, the pack size was notably smaller than the experimental
values, typically by a factor of $\approx 4$ (Supplemental Fig. S5). Hence,
the model demonstrates that alignment of traction between neighboring cells
has a strong effect on the pack size.

Figure 3: Effect of stress fiber activity on cell-cell traction alignment and
collective pack size. (a) Image of F-actin in low serum conditions. (b)
Traction autocorrelation in experiments and model. (c) Image of F-actin in
CN03-treated cells. (d) Velocity autocorrelation in experiments and model. In
panels b and d, $\tau_{p}$ and $\tau_{a}$ are measured in hours; $f_{0}$ is
measured in nN.

The experiments and model indicate that cells coordinate their tractions with
their neighbors, which in turn affects the size of a collectively moving pack.
To explore this finding further, we wondered if it would be possible to turn
off the correlation of traction between neighbors, thereby reducing the pack
size. To this end, we decreased contractility by culturing the cells in low
(1%) serum for 24 h. Under these conditions, cells expressed few stress fibers
(Fig. 3a), and the traction autocorrelation dropped nearly to zero at one cell
diameter (Fig. 3b), suggesting essentially no coordination of traction between
neighbors. Importantly, even in low serum conditions, the temporal persistence
of traction stayed well above the imaging period of 10 min (Supplemental Fig.
S6b), indicating that the traction data remained above the noise floor of the
experiments. Hence, the loss of spatial correlation was not an artifact due to
noise but rather an indicator of reduced coordination between neighboring
cells. Addition of the Rho activator CN03 (3 $\mu$g/mL) reversed the effects
of low serum, causing more pronounced stress fibers that aligned between
neighboring cells (Fig. 3c) and increasing the traction autocorrelation such
that it decayed to zero over a distance of $\approx 2$ cell diameters (Fig.
3b). As expected, the velocity autocorrelation was smaller for serum-starved
compared to CN03-treated conditions (Fig. 3d). To confirm that the low serum
conditions essentially turned off the multicellular traction alignment, we
returned to the model. To mimic the low serum conditions, we switched off
traction alignment by setting $\tau_{a}\to\infty$, which resulted in a good
match with the experimental traction and velocity correlations. To match the
CN03-treated conditions, we noted that the CN03 increased net traction
magnitude and traction persistence by factors of approximately $4$ and $2$,
respectively (Supplemental Fig. S6). Therefore, in the simulations, we
respectively increased $f_{0}$ and $\tau_{p}$ by factors of $4$ and $2$ but
retained no multicellular traction alignment (i.e., $\tau_{a}\to\infty$).
Despite increasing the cell persistence time $\tau_{p}$, the simulations did
not reveal any significant change in velocity autocorrelation. Instead, a
match between the model and the CN03-treated experimental data required that
we use a finite traction alignment time, $\tau_{a}=5$ h (Fig. 3b, d). Hence,
the experiments and simulations confirm the multicellular traction alignment
and demonstrate that it results from alignment of stress fibers between
neighboring cells.

## 4\. Conclusions

Here, we have demonstrated that cells in a collective align their cell-
substrate tractions with those of their neighbors, which leads to complex
large-scale patterns in collective motion that are reminiscent of the rich
collective behaviors observed in flocks of birds or schools of fish. For birds
and fish, the collective alignment of propulsion results from visual cues. For
epithelial cells, the underlying causes of the alignment are not yet clear,
but it is likely that the alignment mechanism involves a combination of
biochemical signalling, e.g., via correlation of front-back polarity, which
can be coordinated between neighboring cells [9, 16], and mechanical
interaction. Numerical simulations of the SPV model revealed that the
collective motion resulting from the complex physics and biological signaling
can be captured quantitatively by a relatively simple model. Our results,
therefore, show that even without knowledge of the specific mechanisms for
traction alignment, the complex motion patterns in cell monolayers can be
explained within the framework of physics of active matter. This is in line
with the growing notion that mechanical cues play an equally important role as
chemical signaling in controlling collective cell behavior at tissue scales.
More broadly, by increasing the size of a collective cell pack, the
multicellular traction alignment tempers randomness in cell motion caused by
biological variability, which could be important for formation and repair of
tissues.

## Acknowledgments

We thank Christian Franck for use of the 40$\times$ microscope objective.
A.S., E.E.G., and J.N. were supported by National Science Foundation grant
number CMMI-1660703 and the University of Wisconsin-Madison Office of the Vice
Chancellor for Research and Graduate Education with funding from the Wisconsin
Alumni Research Foundation. R.S. acknowledges support by the UK BBSRC (Award
BB/N009789/1). S.H. acknowledges support by the UK BBSRC (grant number
BB/N009150/1-2).

## References

  * [1] P. Friedl, D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell Biol. 10 (7) (2009) 445–457.
  * [2] A. J. Liu, S. R. Nagel, Jamming is not just cool any more, Nature 396 (6706) (1998) 21–22.
  * [3] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. U.S.A. 105 (4) (2008) 1232–1237.
  * [4] A. J. Ward, D. J. Sumpter, I. D. Couzin, P. J. Hart, J. Krause, Quorum decision-making facilitates information transfer in fish shoals, Proc. Natl. Acad. Sci. U.S.A. 105 (19) (2008) 6948–6953.
  * [5] M. Poujade, E. Grasland-Mongrain, A. Hertzog, J. Jouanneau, P. Chavrier, B. Ladoux, A. Buguin, P. Silberzan, Collective migration of an epithelial monolayer in response to a model wound, Proceedings of the National Academy of Sciences 104 (41) (2007) 15988–15993.
  * [6] T. E. Angelini, E. Hannezo, X. Trepat, M. Marquez, J. J. Fredberg, D. A. Weitz, Glass-like dynamics of collective cell migration, Proc. Natl. Acad. Sci. U.S.A. 108 (12) (2011) 4714–4719.
  * [7] X. Serra-Picamal, V. Conte, R. Vincent, E. Anon, D. T. Tambe, E. Bazellieres, J. P. Butler, J. J. Fredberg, X. Trepat, Mechanical waves during tissue expansion, Nat Phys 8 (8) (2012) 628–634.
  * [8] M. Deforet, V. Hakim, H. Yevick, G. Duclos, P. Silberzan, Emergence of collective modes and tri-dimensional structures from epithelial confinement, Nat. Comm. 5 (2014) 3747.
  * [9] T. Das, K. Safferling, S. Rausch, N. Grabe, H. Boehm, J. P. Spatz, A molecular mechanotransduction pathway regulates collective migration of epithelial cells, Nat. Cell Biol. 17 (3) (2015) 276–287.
  * [10] S. Garcia, E. Hannezo, J. Elgeti, J.-F. Joanny, P. Silberzan, N. S. Gov, Physics of active jamming during collective cellular motion in a monolayer, Proc. Natl. Acad. Sci. U.S.A. 112 (50) (2015) 15314–15319.
  * [11] J. Notbohm, S. Banerjee, K. J. Utuje, B. Gweon, H. Jang, Y. Park, J. Shin, J. P. Butler, J. J. Fredberg, M. C. Marchetti, Cellular contraction and polarization drive collective cellular motion, Biophys. J. 110 (12) (2016) 2729–2738.
  * [12] A. J. Loza, S. Koride, G. V. Schimizzi, B. Li, S. X. Sun, G. D. Longmore, Cell density and actomyosin contractility control the organization of migrating collectives within an epithelium, Mol. Biol. Cell 27 (22) (2016) 3459–3470.
  * [13] C. Malinverno, S. Corallino, F. Giavazzi, M. Bergert, Q. Li, M. Leoni, A. Disanza, E. Frittoli, A. Oldani, E. Martini, et al., Endocytic reawakening of motility in jammed epithelia, Nat. Mater. 16 (5) (2017) 587–596.
  * [14] V. Petrolli, M. Le Goff, M. Tadrous, K. Martens, C. Allier, O. Mandula, L. Hervé, S. Henkes, R. Sknepnek, T. Boudou, et al., Confinement-induced transition between wavelike collective cell migration modes, Phys. Rev. Lett. 122 (16) (2019) 168101.
  * [15] S. Henkes, K. Kostanjevec, J. M. Collinson, R. Sknepnek, E. Bertin, Dense active matter model of motion patterns in confluent cell monolayers, Nat. Comm. 11 (2020) 1405.
  * [16] N. Hino, L. Rossetti, A. Marín-Llauradó, K. Aoki, X. Trepat, M. Matsuda, T. Hirashima, ERK-mediated mechanochemical waves direct collective cell polarization, Dev. Cell 53 (6) (2020) 646–660.
  * [17] R. Farhadifar, J.-C. Röper, B. Aigouy, S. Eaton, F. Jülicher, The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing, Curr. Biol. 17 (24) (2007) 2095–2104.
  * [18] D. Bi, X. Yang, M. C. Marchetti, M. L. Manning, Motility-driven glass and jamming transitions in biological tissues, Phys. Rev. X 6 (2) (2016) 021011.
  * [19] D. L. Barton, S. Henkes, C. J. Weijer, R. Sknepnek, Active vertex model for cell-resolution description of epithelial tissue mechanics, PLoS Comput. Biol. 13 (6) (2017) e1005569.
  * [20] J. A. Mitchel, A. Das, M. J. O’Sullivan, I. T. Stancil, S. J. DeCamp, S. Koehler, O. H. Ocaña, J. P. Butler, J. J. Fredberg, M. A. Nieto, D. Bi, J.-A. Park, In primary airway epithelial cells, the unjamming transition is distinct from the epithelial-to-mesenchymal transition, Nat. Comm. 11 (2020) 5053.
  * [21] B. Szabó, G. J. Szöllösi, B. Gönci, Z. Jurányi, D. Selmeczi, T. Vicsek, Phase transition in the collective migration of tissue cells: experiment and model., Phys. Rev. E 74 (6 Pt 1) (2006) 061908.
  * [22] D. Drasdo, S. Hoehme, M. Block, On the role of physics in the growth and pattern formation of multi-cellular systems: What can we learn from individual-cell based models?, J. Stat. Phys. 128 (1-2) (2007) 287–345.
  * [23] T. E. Angelini, E. Hannezo, X. Trepat, J. J. Fredberg, D. A. Weitz, Cell migration driven by cooperative substrate deformation patterns, Phys. Rev. Lett. 104 (16) (2010) 168104.
  * [24] D. T. Tambe, C. C. Hardin, T. E. Angelini, K. Rajendran, C. Y. Park, X. Serra-Picamal, E. H. Zhou, M. H. Zaman, J. P. Butler, D. A. Weitz, et al., Collective cell guidance by cooperative intercellular forces, Nat. Mater. 10 (6) (2011) 469–475.
  * [25] J. H. Kim, X. Serra-Picamal, D. T. Tambe, E. H. Zhou, C. Y. Park, M. Sadati, J.-A. Park, R. Krishnan, B. Gweon, E. Millet, et al., Propulsion and navigation within the advancing monolayer sheet, Nat. Mater. 12 (9) (2013) 856–863.
  * [26] R. Sunyer, V. Conte, J. Escribano, A. Elosegui-Artola, A. Labernadie, L. Valon, D. Navajas, J. M. García-Aznar, J. J. Muñoz, P. Roca-Cusachs, X. Trepat, Collective cell durotaxis emerges from long-range intercellular force transmission, Science 353 (6304) (2016) 1157–1161.
  * [27] D. J. Cohen, W. J. Nelson, M. M. Maharbiz, Galvanotactic control of collective cell migration in epithelial monolayers, Nat. Mater. 13 (4) (2014) 409–417.
  * [28] T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans, B. Ladoux, Topological defects in epithelia govern cell death and extrusion, Nature 544 (7649) (2017) 212–216.
  * [29] K. Kawaguchi, R. Kageyama, M. Sano, Topological defects control collective dynamics in neural progenitor cell cultures, Nature 545 (7654) (2017) 327–331.
  * [30] G. Duclos, C. Erlenkämper, J.-F. Joanny, P. Silberzan, Topological defects in confined populations of spindle-shaped cells, Nat. Phys. 13 (1) (2017) 58–62.
  * [31] F. Giavazzi, M. Paoluzzi, M. Macchi, D. Bi, G. Scita, M. L. Manning, R. Cerbino, M. C. Marchetti, Flocking transitions in confluent tissues, Soft matter 14 (18) (2018) 3471–3477.
  * [32] M. M. Nava, Y. A. Miroshnikova, L. C. Biggs, D. B. Whitefield, F. Metge, J. Boucas, H. Vihinen, E. Jokitalo, X. Li, J. M. G. Arcos, et al., Heterochromatin-driven nuclear softening protects the genome against mechanical stress-induced damage, Cell 181 (4) (2020) 800–817.
  * [33] J. M. López-Gay, H. Nunley, M. Spencer, F. Di Pietro, B. Guirao, F. Bosveld, O. Markova, I. Gaugue, S. Pelletier, D. K. Lubensky, et al., Apical stress fibers enable a scaling between cell mechanical response and area in epithelial tissue, Science 370 (6514).
  * [34] A. Popkova, O. J. Stone, L. Chen, X. Qin, C. Liu, J. Liu, K. Belguise, D. J. Montell, K. M. Hahn, M. Rauzi, et al., A Cdc42-mediated supracellular network drives polarized forces and drosophila egg chamber extension, Nat. Comm. 11 (2020) 1921.
  * [35] A. Saraswathibhatla, J. Notbohm, Tractions and stress fibers control cell shape and rearrangements in collective cell migration, Phys. Rev. X 10 (1) (2020) 011016\.
  * [36] J. C. Crocker, D. G. Grier, Methods of digital video microscopy for colloidal studies, J. Colloid Interface Sci. 179 (1) (1996) 298–310.
  * [37] U. Schmidt, M. Weigert, C. Broaddus, G. Myers, Cell detection with star-convex polygons, in: Med Image Comput Comput Assist Interv, Springer, 2018, p. 265.
  * [38] E. Bar-Kochba, J. Toyjanova, E. Andrews, K.-S. Kim, C. Franck, A fast iterative digital volume correlation algorithm for large deformations, Exp. Mech. 55 (1) (2015) 261–274.
  * [39] J. P. Butler, I. M. Tolic-Nørrelykke, B. Fabry, J. J. Fredberg, Traction fields, moments, and strain energy that cells exert on their surroundings, Am. J. Physiol. Cell Physiol. 282 (3) (2002) C595–C605.
  * [40] J. C. Del Alamo, R. Meili, B. Alonso-Latorre, J. Rodríguez-Rodríguez, A. Aliseda, R. A. Firtel, J. C. Lasheras, Spatio-temporal analysis of eukaryotic cell motility by improved force cytometry, Proc. Natl. Acad. Sci. U.S.A. 104 (33) (2007) 13343–13348.
  * [41] X. Trepat, M. R. Wasserman, T. E. Angelini, E. Millet, D. A. Weitz, J. P. Butler, J. J. Fredberg, Physical forces during collective cell migration, Nat. Phys. 5 (6) (2009) 426–430.
  * [42] R. Sknepnek, SAMoS: Soft Active Matter on Surfaces, https://github.com/sknepneklab/SAMoS (2013–2020).
  * [43] M. Dembo, Y.-L. Wang, Stresses at the cell-to-substrate interface during locomotion of fibroblasts, Biophys. J. 76 (4) (1999) 2307–2316.
  * [44] S. Kaliman, C. Jayachandran, F. Rehfeldt, A.-S. Smith, Limits of applicability of the voronoi tessellation determined by centers of cell nuclei to epithelium morphology, Front. Physiol. 7 (2016) 551.
  * [45] V. Maruthamuthu, B. Sabass, U. S. Schwarz, M. L. Gardel, Cell–ECM traction force modulates endogenous tension at cell–cell contacts, Proc. Natl. Acad. Sci. U.S.A. 108 (12) (2011) 4708–4713.
  * [46] H. Tanimoto, M. Sano, A simple force-motion relation for migrating cells revealed by multipole analysis of traction stress, Biophys. J. 106 (1) (2014) 16–25.
  * [47] G. Lukinavičius, L. Reymond, E. D’este, A. Masharina, F. Göttfert, H. Ta, A. Güther, M. Fournier, S. Rizzo, H. Waldmann, et al., Fluorogenic probes for live-cell imaging of the cytoskeleton, Nat. Methods 11 (7) (2014) 731–733.
  * [48] A. Holzinger, Jasplakinolide: an actin-specific reagent that promotes actin polymerization, in: Cytoskeleton Methods and Protocols, Springer, 2009, p. 71.
  * [49] J. Cau, A. Hall, Cdc42 controls the polarity of the actin and microtubule cytoskeletons through two distinct signal transduction pathways, J. Cell Sci. 118 (12) (2005) 2579–2587.
  * [50] J. F. Casella, M. D. Flanagan, S. Lin, Cytochalasin D inhibits actin polymerization and induces depolymerization of actin filaments formed during platelet shape change, Nature 293 (5830) (1981) 302–305.
  * [51] D. T. Tambe, U. Croutelle, X. Trepat, C. Y. Park, J. H. Kim, E. Millet, J. P. Butler, J. J. Fredberg, Monolayer stress microscopy: limitations, artifacts, and accuracy of recovered intercellular stresses, Plos one 8 (2) (2013) e55172.
  * [52] A. Saraswathibhatla, E. E. Galles, J. Notbohm, Spatiotemporal force and motion in collective cell migration, Sci Data 7 (1) (2020) 197.
  * [53] S.-Z. Lin, W.-Y. Zhang, D. Bi, B. Li, X.-Q. Feng, Energetics of mesoscale cell turbulence in two-dimensional monolayers, Commum. Phys. 4 (1) (2021) 21.
  * [54] D. Bi, J. Lopez, J. M. Schwarz, M. L. Manning, A density-independent rigidity transition in biological tissues, Nat. Phys. 11 (12) (2015) 1074–1079.