Cooperation of dual modes of cell motility promotes epithelial stress
relaxation to accelerate wound healing

Michael F. Staddon1,2, Dapeng Bi3, A. Pasha Tabatabai4,5, Visar Ajeti4,5,
Michael P. Murrell4,5,6, Shiladitya Banerjee1,2*,

1 Department of Physics and Astronomy, University College London, London, UK

2 Institute for the Physics of Living Systems, University College London,
London, UK

3 Department of Physics, Northeastern University, Boston, USA

4 Department of Biomedical Engineering, Yale University, New Haven, USA

5 Systems Biology Institute, Yale University, New Haven, USA

6 Department of Physics, Yale University, New Haven, USA

* Correspondence<EMAIL_ADDRESS>
## Abstract

Collective cell migration in cohesive units is vital for tissue morphogenesis,
wound repair, and immune response. While the fundamental driving forces for
collective cell motion stem from contractile and protrusive activities of
individual cells, it remains unknown how their balance is optimized to
maintain tissue cohesiveness and the fluidity for motion. Here we present a
cell-based computational model for collective cell migration during wound
healing that incorporates mechanochemical coupling of cell motion and adhesion
kinetics with stochastic transformation of active motility forces. We show
that a balance of protrusive motility and actomyosin contractility is
optimized for accelerating the rate of wound repair, which is robust to
variations in cell and substrate mechanical properties. This balance underlies
rapid collective cell motion during wound healing, resulting from a tradeoff
between tension mediated collective cell guidance and active stress relaxation
in the tissue.

## Author summary

Many developmental processes involve collective cell motion, driven by
migratory behaviours of individual cells and their interactions with the
extracellular environment. An outstanding question is how cells regulate their
internal driving forces to maintain tissue cohesiveness while promoting the
requisite fluidity for collective motion. Progress has been limited by the
lack of an integrative framework that couples cellular physical behavior with
stochastic biochemical dynamics underlying cell motion and adhesion. Here we
develop a cell-based computational model for collective cell migration during
epithelial wound repair that integrates tissue mechanics with active cell
motility, cell-substrate adhesions, and actomyosin dynamics. Using this model
we show that an optimum balance of protrusive cell crawling and actomyosin
contractility drives rapid directed motion of cohesive cell groups, robust to
variations in cell and substrate physical properties. We further show that
disparate modes of individual cell migration can cooperate to accelerate
collective cell migration by fluidizing confluent tissues.

## Introduction

Collective cell migration is central to tissue morphogenesis, wound repair and
cancer metastasis [1]. During tissue repair after wounding [2], or during
closure of epithelial gaps [3, 4], collective cell migration enables the
regeneration of a functional tissue. Gap closure is usually mediated by two
distinct mechanisms for collective cell movement [5, 6, 7]. First, cells both
proximal and distal to the gap can crawl by Arp2/3 driven forward
lamellipodial protrusions [6, 7, 8]. Secondly, cells around the gap can
collectively assemble a supracellular actomyosin cable, known as a purse-
string, which closes tissue voids via active contractile forces [6, 9]. It
remains poorly understood how these two modes of collective cell movement,
driven by the assembly of distinct actin network architectures, are regulated
in diverse biophysical conditions.

Many experimental studies have provided key insights into the physical forces
driving collective cell migration [9, 8, 10, 11, 12, 13, 7]. Recent in vitro
wound healing experiments have shown that closure of large wounds is initiated
by cell crawling, followed by the assembly of purse string that dominates
closure at smaller wound sizes [12, 13]. Purse-string acts like a cable under
contractile tension, pulling in the wound edge at a speed proportional to its
local curvature [14]. By contrast, crawling driven closure occurs at a
constant speed, regardless of wound morphology [7]. However, it remains
unknown how the mechanochemical properties of individual cells and their
interactions with the extracellular matrix regulate crawling and purse-string
based collective cell motion. While experiments are limited in the extent to
which mechanical effects are separated from biochemical processes, theoretical
and computational models can decouple these variables precisely.

Extensive theoretical work has been done to model collective cell migration
during tissue morphogenesis and repair [15, 16, 17, 18, 19, 20, 21]. However,
existing models do not explain how individual cells adapt their migratory
machineries and interactions with neighboring cells to move collectively like
a viscous fluid while maintaining tissue cohesion. Continuum models of tissues
[22] as viscoelastic fluids [16, 13] or solids [15, 23, 17, 14] have been
successful in describing collective flow and traction force patterns observed
experimentally. However, such macroscopic models cannot capture cellular scale
dynamics, and therefore unsuited for connecting individual cell properties to
collective cell dynamics. On the other hand, cell-based computational models,
including the Cellular Potts Model [24, 25], Vertex Model [26, 27], phase-
field [28] or particle-based models [20, 29, 30] explicitly account for
dynamic mechanical properties of individual cells and their physical
interactions. However, these models have not yet been developed to integrate
the mechanics of cell motion with cell-substrate adhesions and intracellular
cytoskeletal dynamics. It remains poorly understood how migrating cells sense
changes in their physical environment and translate those cues into
biomechanical activities in order to facilitate collective motion. This is
particularly important for epithelial wound healing, where wound edge cells
actively remodel their cytoskeletal machineries and the resulting modes of
motility in response to changes in wound size, shapes and substrate properties
[31, 14, 12].

To overcome these limitations, we propose an integrative modeling framework
that incorporates the mechano-chemical coupling of cell motion and adhesion
with stochastic transformation between protrusive and contractile cell
behaviors. In contrast to previous cell-based models of wound healing [32, 18,
31], our approach explicitly accounts for the spatiotemporal regulation of
protrusive and contractile activities, cell-matrix interactions, adhesion
turnover, and cell polarity. Using this model, we ask: How do migrating cells
sense changes in their physical environment? How do cells regulate their modes
of motilities to optimize the speed of collective motion? What roles do tissue
mechanical properties play in stress propagation and relaxation during wound
repair? In particular, we find that an optimum mixture of protrusive and
contractile cell activities at the wound edge accelerates the rate of wound
healing under diverse conditions. The optimum mixed mode of migration is
robust to changes in substrate rigidity, wound shape, intercellular adhesions
and cortical tension. A unique insight offered by our study is that a mixture
of protrusive and contractile activities promotes faster wound repair by
optimizing the tradeoff between collective cell guidance and local stress
relaxation. Finally, we propose a fundamental mechanism by which tissues can
locally fluidize to drive rapid collective cell motion while maintaining their
overall mechanical integrity.

## Cell-based mechanochemical model

Our model consists of several computational components that simulate: (1)
mechanical interactions between cells, (2) biochemical dynamics (protrusions,
adhesions), and (3) transitions between distinct cell motility modes.
Mechanical interactions between cells are simulated using the vertex model for
epithelial mechanics [26, 18, 27, 21, 33, 34, 35], where the geometry of each
cell is defined by a two-dimensional polygon, with mechanical energy given by:

$E_{i}=K(A_{i}-A_{0})^{2}+\Gamma P_{i}^{2}+\gamma P_{i}\;.$ (1)

The first term in (1) represents the energy cost for cell compressibility,
where $A_{i}$ is the area of cell $i$, $A_{0}$ is the preferred cell area, and
$K$ is the elastic constant. The second term, $\Gamma P_{i}^{2}$, is the
energy due to contractile forces in the actomyosin cortex. The last term in
(1) represents the interfacial tension between cells, which is the difference
between cortical tension and the cell-cell adhesion energy per unit length.
The elastic substrate is modeled as a triangular mesh of harmonic springs
(Methods). Focal adhesion complexes are modeled as stiff springs that anchor
the cell vertices to the substrate mesh, with attachment and detachment rates
given by $k_{\text{on}}$ and $k_{\text{off}}$, respectively (see Methods). The
net mechanical force acting on the cell vertex $\alpha$ is given by
$\mathbf{F}^{\alpha}=-\partial E_{\text{tot}}/\partial\mathbf{x}^{\alpha}$,
where $E_{\text{tot}}=\sum_{i=1}^{n}E_{i}+E_{\text{adh}}$ is the total
mechanical energy of the cells and the cell-substrate adhesions.

In addition to mechanical forces (Fig 1A), cells within the bulk tissue
actively move with a self-propulsion velocity $v_{0}\hat{\mathbf{p}}_{i}$ (Fig
1B), where $\hat{\mathbf{p}}_{i}$ defines the random polarity vector for cell
motion, and $v_{0}$ is the self-propulsion speed. Cells at the wound leading
edge initiate motion by crawling towards the wound center [12, 13], with a
force $\mathbf{f}_{p}$ (Fig 1A). At each time step, crawling cell fronts can
transition to a purse-string at a constant rate $k_{\text{p}}$. This leads to
an increased line tension on the wound edge due to actomyosin contractility
(Fig 1A) (see Methods). Assuming over-damped dynamics, cell vertex $\alpha$ at
the wound edge moves as:

$\mu\frac{d\mathbf{x}^{\alpha}}{dt}=\mathbf{F}^{\alpha}+\mathbf{f}_{p}^{\alpha}\;,$
(2)

where $\mu$ is the friction coefficient. Cell vertices in the bulk of the
tissue move according to following equation of motion

$\mu\frac{d\mathbf{x}^{\alpha}}{dt}=\mathbf{F}^{\alpha}+\frac{1}{n^{\alpha}}\sum_{i\in\alpha}\mu
v_{0}\hat{\mathbf{p}}_{i}\;,$ (3)

where the last term is the averaged self-propulsion force over $n_{\alpha}$
neighboring cells sharing the vertex $\alpha$ (Fig 1B). We estimate the model
parameters from available experimental data (Methods, Table 1).

Fig 1: Collective migration during wound healing driven by a mixture of
crawling and purse-string based cell motility. A: Model schematic, showing
physical forces and model elements. B: Illustration of self-propulsion force.
The central vertex for a cell inside the tissue has a resultant force (red
arrow) equal to the average force from its adjacent cells (blue arrows). The
polarity vector (blue arrow) for a wound edge cell bisects the angle between
the lines from the cell centroid to the boundary vertices. C: Tissue
configuration during wound closure with $k_{\text{p}}=4$ hr-1, at $t=15$ min
(left), $t=33$ min (middle), $t=64$ min (right). Arrows indicate traction
forces. D-F: Kymograph of the (D) radial component of cell velocity field,
$v_{r}$, (E) magnitude of the azimuthal velocity, $|v_{\theta}|$, and (F)
radial traction stress, $T_{r}$, for the mixed modality of closure
corresponding to (C). G: Log-linear plot for wound area vs time for crawling
($k_{\text{p}}=0$), purse-string ($k_{\text{p}}=1000$ hr-1), and mixed
($k_{\text{p}}=4$ hr-1) modes of closure. Inset: Time evolution of the
percentage of wound perimeter covered by purse-string. See Table 1 for the
full list of default model parameters.

## Results

### Cooperation of distinct modes of cell migration during wound repair

To elucidate the mechanisms of collective cell motion during wound repair, we
simulated healing of a circular wound for a mixed modality of closure:
$k_{p}=4$ hr-1. Initially, cells close the wound by crawling (Fig 1C), but
over time they switch to the purse-string mode, resulting in rapid contraction
of cell edges lining the wound periphery (Fig 1C, S1 Video.). To quantify the
spatiotemporal patterns of collective cell motion, we calculated spatially
averaged radial and azimuthal velocities as a function of the radial distance
from the wound center at each time point (Fig 1D-E). Initially, both radial
and azimuthal velocities are highest around the wound edge and decay with
distance inside the monolayer. As crawling cells pull on the substrate, the
resultant traction forces point radially outwards and away from the wound (Fig
1C,F). Halfway through the closure process, the purse-string fully assembles
(Fig 1G-inset) and the traction forces switch to pointing radially inwards
(Fig 1F), in quantitative agreement with experimental data [31].  Consistent
with experiments, tangential traction stresses are comparable in magnitude
with the radial components of the traction stress (S2 Fig:B). Our model
reproduces the experimental observation that focal adhesions are oriented
towards the wound center for crawling cells [31, 36] (S3 Fig:A,C). By
contrast, purse-string adhesions have a higher probability of orienting
tangentially at the leading edge than crawling cells (S3 Fig:B,D,E). As
closure proceeds, the band of high radial velocities around the wound narrows
(Fig 1D), while the azimuthal velocity narrows and decreases around the wound
(Fig 1E). This results in more coordinated inward motion of the cells.

Increasing $k_{p}$ from $0$ (crawling only) to $1000$ hr-1, monotonically
increases the proportion of wound perimeter covered by the purse-string over
time (Fig 1G-inset). For non-zero values of $k_{p}$, wound area shrinks in a
biphasic manner: an initial slow exponential decay, followed by fast
exponential decay, consistent with experimental data [36]. In contrast to the
mixed mode of closure (Fig 1C,G), the traction forces for crawling mediated
closure are always directed radially outwards (S2 Fig:C), because crawling
cells pull on the substrate. While further inside the monolayer the traction
forces point radially inwards as the rear end of crawling cells retract via
cortical contraction. In purse-string mediated closure, the wound shape
remains circular throughout (S2 Video.), in contrast to the ruffling
morphology observed for crawling cell fronts (S2 Video.). Traction forces
point into the gap, and increases in magnitude as the wound size gets smaller
(S2 Fig:D). For a fixed set of parameters, we find that a balance of purse-
string and crawling mediated closure results in faster wound healing (Fig 1G).
To determine how the relative proportion of purse-string and lamellipodia is
optimized for rapid collective motion, we turned to examine how the purse-
string assembly rate ($k_{p}$) regulates wound closure time for varying
physical properties of the cells, the underlying substrate, wound size and
shape.

### Mixture of crawling and purse-string based motilities accelerates wound
closure independent of substrate rigidity

Since the speed of cell crawling and the magnitude of traction forces are
sensitive to substrate rigidity [37, 38], we first investigated the role of
substrate stiffness on wound closure time. To this end, we varied the
substrate Young’s modulus, $E_{s}$, and the purse-string assembly rate,
$k_{\text{p}}$, for fixed physical properties of the tissue and the wound. We
find that wound closure time increases with $E_{s}$ for higher values of
$k_{\text{p}}$, but remained insensitive for crawling mediated closure (Fig
2A). Strikingly, there exists an optimum value of $k_{\text{p}}$
(corresponding to mixed modality) for any value of $E_{s}$, which results in
minimum closure time (Fig 2A). For fixed $k_{\text{p}}$, strain energy
transmitted to substrate decreases monotonically with increasing stiffness for
$E_{s}>0.5$kPa (Fig 2B) (see Methods for calculation details). For all values
of $E_{s}$ and $k_{p}$, faster wound closure coincides with higher strain
energy transmitted to the substrate, signifying a positive correlation between
energy cost and the speed of wound healing.

Fig 2: Mechanosensitivity of wound closure. A-B: Dependence of closure time
(A) and mean strain energy (B) for different values of substrate stiffness
($E_{s}$) and $k_{\text{p}}$. Starred cells indicate the fastest wound closure
for a given $E_{s}$ with varying $k_{p}$ (PS: $k_{\text{p}}=1000$ hr-1; C:
$k_{p}=0$). Each data point corresponds to the average of 5 simulations. C:
Pearson’s correlation coefficient, $r$, between closure time and substrate
stiffness, for crawling ($k_{\text{p}}=0$), purse-string ($k_{\text{p}}=1000$
hr-1), and mixed ($k_{\text{p}}=4$ hr-1) modes of closure. The asterisk
represents coefficients significantly different from zero (p$<0.05$); ** means
p$<0.01$, and *** means p$<0.001$. For each mode, n = 25. D-E: Schematic of
purse-string driven (D) and crawling mediated (E) cell motility on an elastic
substrate. Blue arrows represent reaction force from the substrate, green
(red) arrows represent purse-string (protrusion) driving forces, and black
arrows represent cortical contractile forces. F: Effective friction from the
substrate on the leading edge of the wound for crawling ($k_{\text{p}}=0$),
purse-string ($k_{\text{p}}=1000$ hr-1), and mixed ($k_{\text{p}}=4$ hr-1)
modes of migration. Error bars represent standard deviation (n = 5).

Our results agree with experimental findings that wound closure time is not
sensitive to changes in substrate stiffness for moderate to high rigidities
[31, 36]. On very soft substrates ($<500$ Pa), our model predictions are
inconsistent with experiments by Anon et al [7], who showed that crawling-
based migration fails to close wounds on very soft gels ($\sim 100$ kPa), as
lamellipodia do not form. This may be captured by implementing additional
biochemical feedback mechanisms between protrusive activity and substrate
stiffness, beyond the scope of our mechanical model.

As $E_{s}$ is increased, purse-string driven motion slows down. To quantify
the dependence of closure time on stiffness, we calculated the Pearson’s
correlation coefficient between wound closure time and substrate stiffness for
different modes of wound closure (Fig 2C). We find that purse-string based
motility slows down with increasing stiffness, with a positive correlation
coefficient significantly different from zero (p-value $<0.05$). In contrast,
crawling based motility and have the least significant correlation coefficient
(p-value $>0.05$).

The sensitivity of purse-string driven motility to substrate rigidity (Fig
2A-C) can be explained by a mechanical force balance argument (Figs. 2D-E).
Purse-string driven contractile forces drag the border cells into the gap, in
competition with cortical tension retracting the rear cell edges. This results
in a large net resistive force from the deforming elastic substrate (Fig 2D).
By contrast, crawling cells pull the substrate backwards at the wound edge and
contractile forces pull the substrate forward at the cell rear (Fig 2E). This
dipole-like traction pattern results in a net assistive force from the
substrate, pointing towards direction of cell crawling. During mixed mode of
migration, a combination of net assistive and resistive forces should
therefore lead to the least sensitivity to substrate stiffness.

To test this hypothesis, we computed the net radial traction force, $F_{r}$,
on the substrate under the first row of cells at the wound edge. We then
calculate the time-averaged ratio between the radial force and the radial
velocity, $v_{r}$, of the wound edge, to obtain an effective friction
coefficient: $\mu_{\text{eff}}=\langle F_{r}/v_{r}\rangle$ (Fig 2F). We find
that $\mu_{\text{eff}}$ monotonically increases in magnitude with increasing
substrate stiffness (for all modes of migration), consistent with previous
theoretical predictions [39]. For all values of substrate stiffness, purse-
string motion leads to the highest positive $\mu_{\text{eff}}$, suggesting
high resistance and sensitivity to substrate rigidity. Crawling driven
motility leads to negative $\mu_{\text{eff}}$, indicative of assistive motion.
By contrast, the mixed mode of migration leads to the lowest magnitude of
$\mu_{\text{eff}}$, i.e. least drag from the substrate.

Rigidity sensing by different modes of collective migration is expected to be
strongly coupled to focal adhesion kinetics. While we have assumed constant
rates of binding and unbinding of cell-substrate adhesions, experiments have
demonstrated that integrin-ligand pairs form catch bonds [40], such that
$k_{\text{off}}$ decreases under low forces and increases under larger forces.
To test the if the mechanosensitivity of cell-substrate adhesion bonds impact
our results, we implemented a catch bond model for adhesions, assuming a
single bound state and two unbinding pathways [41] (see Methods). As a result,
the crawling mode of closure is now more sensitive to changes in substrate
stiffness, with closure time increasing with stiffness, before decreasing at
higher stiffnesses due to increased adhesion lifetime (S4 Fig). Purse-string
driven closure shows an increase in sensitivity compared to the default case,
while the mixed mode of closure is least sensitive to changes in substrate
stiffness. However, the mixed mode of migration is always the fastest,
irrespective of force sensitivity of the adhesions.

Aside from mechanosensitivity of different modes of wound closure, the driving
force for closure is expected to be strongly dependent on the relative
proportion of purse-string and crawling cells. Since actomyosin purse-string
is a cable under tension, the driving force for closure is proportional to the
wound curvature. As a result, purse-string driven closure is expected to be
sensitive to the wound geometry [14, 12]. By contrast, crawling driven closure
has been found to reduce wound area at a constant speed [7]. Therefore, we
sought to investigate how the coaction of purse-string and crawling based
motilities modulate collective motion for varying wound morphologies.

### Wound geometry regulates the optimum modality of collective motion

For circular wounds of varying radii we recapitulate the experimental result
that closure time increases with wound radius (Fig 3A) [7]. However, the
optimum purse-string assembly rate ($k_{\text{p}}$) for fastest closure
decreases with wound radius, such that closure time is highly sensitive to
$k_{\text{p}}$ for larger wounds. This is because purse-string driven forces
are higher near the end of closure, and that purse-string force is low in the
beginning of closure of a large wound. For larger wound radii, an optimum
mixture of purse-string and protrusive cell crawling leads to fastest closure.
We find that the average strain energy on the substrate increases
monotonically with wound radius for $k_{p}$ (Fig 3B), but is more sensitive to
wound size for purely crawling mediated migration ($k_{p}=0$).

Fig 3: Wound geometry regulates the optimum modality of collective migration.
A: Closure time for different values of wound radius and $k_{\text{p}}$. B:
Mean strain energy vs wound radius for different values of $k_{p}$. C: Closure
time for different values of wound aspect ratio and $k_{\text{p}}$. D:
Substrate strain energy as a function of wound aspect ratio and
$k_{\text{p}}$. E: Wound closure time for concave wound shapes for crawling
($k_{\text{p}}=0$), purse-string ($k_{\text{p}}=1000$ hr-1), mixed
($k_{\text{p}}=4$ hr-1) and curvature-sensing modes of closure. Error bars
represent standard error of mean ($n=5$). F: Velocity against curvature during
simulations of the right most shape in (E) for crawling, purse-string and
mixed modes of closure. G: Evolution of wound morphology during closure by (i)
pure crawling, (ii) pure purse-string, (iii) a combination of crawling and
purse-string, and (iv) curvature-dependent formation of purse-string. Colors
progressively change from black to red with increasing time.

Next we simulated elliptical shaped wounds of fixed area but varying aspect
ratios. We find that regardless of the migratory mode, closure time decreases
with increasing aspect ratio (Fig 3C). In addition, there exists an optimum
value of $k_{\text{p}}$ for a given aspect ratio that leads to minimal closure
time. Thus, a mixed mode of closure is always the fastest, but isn’t much
faster than crawling mediated closure for high aspect ratio wounds. This is
because crawling cells advance at a constant speed perpendicular to the wound
edge. Therefore only the short axis distance must be crossed for the wound to
close (S4 Video.) (S5 Fig:A). For purse-string driven closure, the high
curvature ends of elliptical wounds move rapidly inwards, leading to faster
closure than circular shapes (S5 Fig:C). At all values of aspect ratio, strain
energy is inversely proportional to closure time (Fig 3D).

Since purse-string behaves as a contractile cable, then for wounds with
concave morphologies (positive curvatures), cells should be pulled away from
the wound by the purse-string tension. To investigate this we simulated
concave wound shapes as in ref. [14]. For varying degrees of concavity (with
fixed area), we observed that a mixed mode of closure leads to fastest wound
closure (Fig 3E). To quantify the relationship between wound healing speed and
curvature, we measured the local velocity and curvature at the wound
perimeter. We find that the purse-string velocity is proportional to the
curvature, crawling velocity is curvature-independent, while a mixture of
crawling and purse-string leads to faster collective motion, with velocity
decreasing with curvature (Fig 3F, S5 Video.). These findings quantitatively
agree with experimental data [14].

Previous studies suggest the possibility that purse-string and lamellipodia-
based migration during wound healing can be geometrically coupled [31, 14],
such that the formation of protrusive borders may be directly coupled to the
assembly of purse-string cables on neighboring wound edges with opposite
curvatures. Such a mechanism is not captured by a purely stochastic transition
between protrusive and contractile activities. To this end, we implemented a
model of curvature sensing motility of the wound leading edge, where the
switching between crawling and purse-string mechanisms is regulated by the
local curvature of the wound (S7 Fig). Based on this model, if the curvature
of a cell’s leading edge is larger than a threshold curvature, it contracts
via purse-string. Otherwise, the cell moves via protrusive crawling (Methods,
S7 Fig). We applied this model to wounds with non-uniform curvatures as in Fig
3E. Consequently, the convex regions move forward by crawling, whereas
contractile purse-string cables assemble in the concave regions. We find that
for all three concave shapes in Fig 3E, the curvature sensing mechanism closes
the wound at least as fast as in the mixed case with stochastic switching of
motility modes (Fig 3E-G). We note that the curvature-sensing mechanism may
not be applicable to the closure of undamaged epithelial gaps where purse-
string cables do not form [7].

### Optimum balance of protrusive and contractile cell activities promotes
rapid wound healing via active stress relaxation

Our cell-based model predicts many differences in collective cell motility
driven by contractile and protrusive activities (Fig 1-3). In particular,
purse-string tension rounds the wound edge and leads to solid-like, radial
deformation of the tissue (Fig 3F-inset). By contrast, crawling cells ruffle
the wound leading edge (Fig 3G, S3 Video., S5 Video.), suggestive of lack of
guided motion. To quantify differences in tissue deformation and their
relationship to collective motion, we measured the angle ($\theta$) between
cell center velocity and the unit vector pointing towards the wound center
(Fig 4A). In purse-string driven closure ($k_{\text{p}}=1000$ hr-1), the angle
distribution shows a single peak at $\theta=0$, corresponding to radially
inward deformation (Fig 4B). By contrast, crawling cells ($k_{\text{p}}=0$)
have a wider distribution of angles, with secondary peaks at $\theta=\pm\pi$
(Fig 4B), representing outward motion from cell neighbor exchanges (Fig 4A).
To quantify the distributions, we define collective cell guidance,
$\mathcal{G}$, as the probability that a cell moves towards the wound center:
$\mathcal{G}$=$\int_{-\pi/2}^{\pi/2}P(\theta)d\theta$, which monotonically
increases with increasing $k_{p}$ (Fig 4C).

Fig 4: Tension mediated cell guidance and active stress relaxation promotes
rapid wound healing. A: Definition of the angle $\theta$ between the cell
center velocity and the radial vector to wound center. Right: Representative
cell velocity fields for crawling and purse-string modes of closure. B:
Probability density distribution for $\theta$ for crawling ($k_{\text{p}}=0$),
purse-string ($k_{\text{p}}=1000$ hr-1), and mixed ($k_{\text{p}}=4$ hr-1)
modes of closure. C: Guidance parameter for different values of shape
parameter $p_{0}$ and $k_{p}$. D: Rate of T1 transitions in the wounded tissue
for varying $p_{0}$ and $k_{p}$. E: Closure time for different values of
$p_{0}$ and $k_{\text{p}}$. Starred cells indicate the fastest wound closure
for a given $p_{0}$.

Since tissue deformation properties depend on cortical tension, cell
contractility, and cell-cell adhesions [42, 43, 44, 34], we investigate how
cellular mechanical properties regulate collective guidance ($\mathcal{G}$).
We can rewrite the mechanical energy of cells (Eq (1)) as:

$E_{i}=K(A_{i}-A_{0})^{2}+\Gamma(P_{i}-P_{0})^{2}\;,$ (4)

where $P_{0}=-\gamma/2\Gamma$ is the preferred cell perimeter. The non-
dimensional shape parameter $p_{0}=P_{0}/\sqrt{A_{0}}$ controls cell shape
anisotropy and the emergent rigidity of confluent tissues [45]. Increasing
$p_{0}$ reduces cortical tension relative to cell-cell adhesions, which
softens the tissue. It has been shown that confluent tissues behave like a
jammed solid for $p_{0}<3.81$, whereas it exhibits fluid-like behaviour for
$p_{0}>3.81$ [45]. Activity in the form of cell motility, division, or death
can fluidize tissues further by lowering the critical $p_{0}$ for rigidity
transition [46, 47, 48]. In our model, activity arises from self propulsion
($v_{0}$) (S1 Fig:B, S8 Fig), and cell crawling whose relative strength is
regulated by $k_{p}$. We find that increasing $p_{0}$ decreases $\mathcal{G}$,
regardless of $k_{p}$ (Fig 4C). The decrease in $\mathcal{G}$ with increasing
$p_{0}$ arises from an increased rate of cellular neighbor exchanges (T1
transitions) that locally fluidizes the tissue (Fig 4D). Surprisingly, for a
fixed $p_{0}$, T1 rates in the wounded tissue is highest for intermediate
values of $k_{p}$, resulting in minimum closure time (Fig 4E). With higher
$p_{0}$, cells have a higher preferred perimeter, such that both contractile
and protrusive motilities experience lower mechanical resistance from tension
in the border cells (S6 Video.) (S9 Fig). This enables a faster reduction in
wound area as compared to rigid tissues with lower $p_{0}$ (Fig 4E).

These findings elucidate the mechanical basis for rapid collective migration
via a mixture of protrusive and contractile cell activities. Purse-string
driven tension maximizes collective cell guidance and leads to the lowest
frequency of tissue rearrangements, such that cell movements are impeded by
mechanical resistance from the surrounding tissue. By contrast, purely
crawling motion exhibits the lowest collective guidance due to randomized
protrusions of individual cells at the wound leading edge. We find that an
optimum mixture of crawling and purse-string leads to intermediate collective
guidance, while maximizing the frequency of local tissue rearrangements (Fig
4D). This mechanism of active fluidization enables tissues to locally relax
their mechanical stress, promoting rapid wound healing. When intercalations
are disabled in the model, tissue mechanical energy increases due to increase
in cell elongation around the wound (S10 Fig). This results in cell jamming
and slowing down of wound closure. Therefore, cell intercalations, promoted by
a mixture of contractile and protrusive forces, lead to efficient wound
closure by minimizing both tissue mechanical energy and wound closure time.
Recent experiments, however, suggest that cells may not necessarily try to
minimize energy or closure time during wound healing [36]. But rather, they
tend to coordinate the assembly of diverse actin architectures to conserve the
amount of mechanical work done per unit time.

## Discussion

Our cell-based computational model quantitatively captures a wide range of
experimental trends including the patterns of collective cell motion and
traction stress organization for crawling and purse-string mediated wound
closure (Fig 1). We reproduced the experimentally observed size-dependence of
wound closure times, the curvature dependence of purse-string velocity, and
independence of cell crawl speeds to variations in wound morphology. We
predict that increasing aspect ratio of the wound speeds up closure as
crawling cells can rapidly cross the short axis of the wound, whereas purse-
string cables can generate rapid movements on regions of high curvature (Fig
3).

Robust to variations in substrate and tissue mechanical properties, we find
that an optimum proportion of protrusive and contractile motilities
accelerates wound closure. While purse-string driven motion slows down on
stiffer gels due to an increased resistance from drag on the substrate,
crawling driven migration is largely independent of substrate stiffness (Fig
2). We find that a mixed mode of collective migration is more efficient
regardless of substrate stiffness. Robust to parameter variations, an increase
in closure speed is associated with an increase in the strain energy
transmitted to the underlying substrate (S11 Fig). As a result, migrating
cells actively dissipate more mechanical energy to their environment in order
to speed up collective motion.

A source of active stress dissipation comes from cellular neighbor exchanges
that locally fluidize the tissue, resulting in faster wound closure (Fig 4).
These T1 transitions have previously been observed in vivo, during wound
closure in Drosophila embryo epidermis [49]. T1 transitions are also observed
in our in vitro laser-ablation experiments on MDCK monolayers, where the
number of cells at the wound edge decreases over time via wound edge
intercalations (S12 Fig). In our model, the mechanism of active fluidization
via intercalation is promoted by a mixture of protrusive and contractile
activities of wound edge cells, and reduced contractility or increased cell-
cell adhesion in the bulk of the tissue. The ability to actively remodel an
elastic tissue, coupled with tension-driven collective cell guidance,
constitute the two key mechanisms for rapid directed motion in adherent
environments. While the stress relaxation mechanism in our model comes only
from cell neighbor exchanges, other dissipative mechanisms can also be
triggered by mechanical forces including cell shape fluctuations [50], cell
division [51] or cell death [4]. In these cases, our prediction will remain
very similar, with the rate of cell movement into free space augmented by the
sum of relaxation rates of various dissipation modes [46]. A future challenge
is to identify the molecular pathways that activate distinct stress relaxation
modes during tissue development and regeneration.

## Methods

### Cell-substrate interactions

We model the substrate as a triangular mesh of springs with a spring constant
$k_{s}$. The Young’s modulus of the substrate is given by
$E_{s}=2k_{s}/\sqrt{3}h_{s}$, where $h_{s}$ is the substrate thickness, and
the Poisson’s ratio for a triangular mesh is $\nu=1/3$.

Since focal adhesions and cellular traction forces typically localize at the
cell periphery [44], we implement adhesions at the cell boundaries. We model
the focal adhesion complexes as stiff springs with stiffness $k_{f}$, which
connect the cell vertices with the substrate mesh. Bound focal adhesions can
detach stochastically with a rate $k_{\text{off}}$, whereas unbound cell
vertices can attach to the nearest node of the substrate mesh with a rate
$k_{\text{on}}$. The resultant force on the cell vertex is,

${\bf f}_{\text{adh}}^{\alpha}=-\frac{\partial
E_{\text{adh}}^{\alpha}}{\partial\mathbf{x}^{\alpha}}\;,$ (5)
$E_{\text{adh}}^{\alpha}=\sigma_{\alpha}\frac{k_{f}}{2}(|\mathbf{x}^{\alpha}-\mathbf{r}^{\alpha}|-|\mathbf{x}_{0}^{\alpha}-\mathbf{r}_{0}^{\alpha}|)^{2}\;,$
(6)

where $E_{\text{adh}}^{\alpha}$, is the adhesion energy, $\sigma_{\alpha}$ is
the state variable for cell-substrate attachment (0: detached; 1: attached),
$\mathbf{r}^{\alpha}$ is the position of the substrate mesh connected to ${\bf
x}^{\alpha}$, and $\mathbf{x}_{0}^{\alpha}$ and ${\bf r}_{0}^{\alpha}$ are the
initial positions of the cell and the substrate vertices at the time of
adhesion formation.

### Active cell motility

Each cell carries a unit polarity vector, $\hat{{\bf p}}_{i}$, which
represents the front/rear polarization of a motile cell [52]. The polarity
vector is an internal state variable of cell that specifies the preferred
orientation of cell motion, not their actual direction of motion. Cells in the
bulk of the tissue, i.e. not on the wound edge, move due to self-propulsion
[47]. The polarity of a bulk cell $i$ is defined by a unit vector with angle
$\theta_{i}$ that undergoes rotational diffusion:

$\partial_{t}\theta_{i}=\eta_{i}(t),\
\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=2D_{r}\delta_{ij}\delta(t-t^{\prime})\;,$
(7)

where $D_{r}$ is the rotational diffusion constant, and $\eta_{i}(t)$ is a
Gaussian white noise with mean $0$ and variance $2D_{r}$. The self-propulsion
of cell $i$ results in a force on the vertex $\alpha$ as:
$\frac{1}{n_{\alpha}}\sum_{\alpha\in i}\mu v_{0}\hat{\mathbf{p}}_{i}$, where
$v_{0}$ is the self-propulsion speed, and the sum is over all neighboring
cells to vertex $\alpha$ (S1 Fig:B).

Here, we have neglected alignment interactions between cell polarity vectors
in the bulk of the tissue, which can drive coherent swirling motion of cell
collectives [53]. Without such polarity alignment rules, cell velocity vectors
remain correlated over $\sim 5$ cell diameters due to mechanical interactions
(S13 Fig), somewhat less than the correlation lengths measured in experiments
in the absence of a wound [54].

To model lamellipodia based crawling, we allow cell vertices at the wound edge
to protrude in the direction of polarity before attaching to the substrate (S1
Fig). This pushes the cell front outwards, while cortical tension pulls the
rear of the cell forwards. The polarity vector of cells along the wound points
into the gap, and is determined by the mid-point of the wound edges. The
direction of protrusion is given by the unit vector
$\hat{\mathbf{v}}_{\alpha}^{i}$ of wound cell $i$, which makes half the angle
between the two lines joining the centroid of cell $i$ to the vertices on the
wound that neighbour other cells, i.e. are on the boundary of internal and
external edges (S1 Fig:C). This ensures contact inhibition of locomotion [30],
preventing collision of two neighbouring cells. For a cell $i$ neighbouring
the wound, the crawling force on vertex $\alpha$ on the wound edge is given
by: ${\bf
f}_{\text{p}}^{\alpha}=f_{p}(1-\sigma_{\alpha})\hat{\mathbf{v}}^{\alpha}_{i}$,
where $f_{p}$ is the protrusion force magnitude. For simplicity we have
assumed that $f_{p}$ is independent cell-substrate adhesions. However,
protrusive activity remains strongly correlated to focal adhesion kinetics,
since the frequency of the protrusions is slaved by the rate of focal adhesion
binding and unbinding. As a consequence of this feedback, increasing the duty
ratio of adhesions leads to slower crawl speeds and increased closure time (S4
Fig:D).

### Curvature sensing model for purse-string formation

Here we describe the model where the switching between crawling and purse-
string modes is dependent on the local geometry of the wound leading edge. At
each time step in the simulation, cells at the wound edge makes a decision to
switch its motility phenotype based on the local curvature of the wound edge.
We calculate the curvature of a wound edge cell as the inverse of the radius
of a circle inscribed to that cell edge. Curvature is defined as positive if
the wound is convex (e.g. a circle), and negative otherwise. If the curvature
is above a threshold value, then the cell switches to a purse-string mode. If
the curvature is below the threshold value, then the cell moves by crawling.
As a result, cells typically start by crawling and switch to the purse-string
mode as the wound shrinks in size, consistent with experimental findings [31].

To determine the optimum value of the threshold curvature, we varied the
threshold curvature for switching to a purse-string mode, and computed the
resultant wound closure time for a given initial wound shape. The optimum
threshold curvature is given by the curvature value that minimizes wound
closure time, as shown in S7 Fig:B.

### Catch bond model for cell-substrate adhesions

We implemented a catch-bond model for cell-substrate adhesions, where the
detachment rate of the adhesion bonds, $k_{\text{off}}$ is a function of the
bond tension, $f$, as given below:

$k_{\text{off}}(f)=k_{0}e^{-f/f_{0}}+k_{1}e^{-f/f_{1}}\;.$ (8)

The functional form for the detachment rate is taken from a catch bond model
for integrin-ligand bonds that assumes a single bound state and two unbinding
pathways [41]. The parameters $k_{0}$, $k_{1}$, $f_{0}$, and $f_{1}$ have
previously been estimated for single integrin ligand bonds [55]. Based on that
estimate, we calibrate these parameters for the coarse-grained adhesion bonds
in our simulations that represent several ligand-integrin pairs. We used
parameter values of $k_{0}=25$ hr-1, $k_{1}=0.006$ hr-1, $f_{0}=3.125$ $\mu$N,
and $f_{1}=0.6944$ $\mu$N, which results in the default unbinding rate at zero
force, and showed high sensitivity to substrate stiffness in the range 1-16
kPa.

### Model Implementation

The vertex model is implemented using Surface Evolver [56]. We generate a
wound by removing any cells that lie totally or partially within the wounded
area. Edges surrounding the wound are then moved to the target wound shape. We
then relax the energy of the remaining cells without adhesions so that all
vertices on the wound lie on the target wound perimeter and system is at an
energy minimum. To initiate gap closure, cells around the wound are set to
crawling mode. We then execute the following steps (S1 Fig) until wound
closure:

  * •

Update adhesion states for cell vertices. Adherent vertices attempt to unbind
with a rate $k_{\text{off}}$ at each time step. Detached vertices attempt to
attach to the nearest node of the substrate mesh with a rate $k_{\text{on}}$.
For cell edges at the wound border, attachment occurs via protrusion into the
nearest substrate vertex.

  * •

Refine cell edges by subdividing edges longer than a maximum length, and
merging edges shorter than a minimum length. This ensures an even distribution
of adhesions, and allows the cells to assume curved shapes.

  * •

Perform neighbour exchanges, also known as T1 transitions, when a cell edge
shrinks below the threshold length, $L^{*}$, such that it lowers the total
mechanical energy. Once an edge goes below the threshold length $L^{*}$, then
that edge is replaced by a perpendicular contact of the same length.

  * •

Update modes of cell movement. Cells at the wound edge switch from crawling to
purse string modes at a rate $k_{\text{p}}$. In the purse-string mode, cells
can no longer crawl but instead carry a higher line tension around their
wounded edge, $\gamma_{\text{ps}}$, modelling contractility of the actomyosin
cable. Once cells are in the purse-string mode they remain so until wound
closure or when the cell edge length shrinks to zero.

  * •

Move the cell vertices according to the overdamped equation of motion (Eq. (2)
or Eq. (3)). Individual nodes of the substrate spring mesh move at a velocity
proportional to the net force resulting from focal adhesions and the gradient
of mechanical energy of the spring mesh.

### Model parameters

Table 1 lists the parameters used in our simulations. The number of cells was
chosen to be large enough to avoid finite size effects and displacement on the
outer row of the cells. To confirm this, we ran wound healing simulations
using different numbers of cells. As the number of cells increases from 50,
closure time increases and then quickly plateaus after cell count reaches 100
(S14 Fig:A). We use a default value of 150 cells, but increase the cell number
(in the range 150-250) while running simulations for wounds with larger sizes
(Fig 3A). Substrate node density was chosen to be small enough so that a cell
vertex is always close to a node in the substrate spring mesh, allowing focal
adhesions to form with a relatively short length. As shown in S14 Fig:B, we
find little dependence of closure time on node density, and use 0.6 $\mu$m-2
as the default value.

The preferred area of the cell, $A_{0}$, is chosen to be approximately the
same as the average area of MDCK cells in wound healing assays [31, 13]. The
preferred perimeter $P_{0}$ is chosen so that the cell shape index,
$p_{0}=P_{0}/\sqrt{A_{0}}$ is close to the value for a regular hexagon,
enabling us to study the effects of cell shape anisotropy on wound healing
speed. The substrate stiffness was chosen as a typical value for gels used in
in vivo wound healing assays [31]; the Poisson’s ratio of $1/3$ for the
substrate is a consequence of using a triangular mesh of linear springs. The
Young’s modulus of the substrate defines the force scale in the simulations.
The wound radius was chosen to be in the range 5-30 $\mu$m, similar to those
in experimental studies [31, 13, 12].

Purse-string tension was estimated by taking the product of the force
generated by a single myosin motor, $3$ pN [57], with the typical number of
myosin motors in a contractile ring of length $15$ $\mu$m and thickness $1$
$\mu$m, $10^{5}$ [58], which gives a tension of $300$ nN. Next, we fit
parameters for cell area and perimeter elasticities, $K$ and $\Gamma$,
adhesion binding and unbinding rates, $k_{\text{off}}$ and $k_{\text{on}}$.
Together, these parameters determine the overall tissue motility and the
magnitude of traction force generation. Thus we fit them simulataneously to
the experimental data for typical closure speed and traction force magnitudes
generated during closure [31, 13, 12]. In addition, we examine the
spatiotemporal pattern of traction forces generated during closure. For
example, traction stresses are normally localized around the wound but are not
evenly distributed around the perimeter. Low adhesion time leads to smooth
closure but little traction force while higher adhesion binding times lead to
an even distribution of traction around the wound but the closure dynamics are
less smooth.

We estimate the protrusion force, $f_{p}$, by comparing to single cell
crawling speeds of $~{}15$ $\mu$m hr-1 [7]. To this end, we simulated a single
crawling cell with a fixed polarity vector, and calibrated $f_{p}$ to the
value that resulted in a crawl speed of $~{}15$ $\mu$m hr-1. Internal motility
speed was set to a similar value as cell crawling speeds. Dependence of wound
closure time for variations in $f_{p}$ and $\gamma_{\text{ps}}$ are shown in
S2 Fig:F. Whereas, the dependence of closure time on internal motility,
$v_{0}$ is shown in S8 Fig:A,C. The range of purse-string assembly rates were
chosen so that the minimum value, $k_{p}=0$ yields pure crawling, the maximum,
$k_{p}=1000$ hr-1, yields 100% purse-string coverage, and intermediate values
produce a combination of purse-string and crawling.

Table 1:  Default Parameter Values Parameter | Default Value
---|---
Cell |
Area elastic modulus, $K$ |  $0.2$ nN $\mu$m-3
Preferred area, $A_{0}$ |  $100\ \mu$m2
Preferred perimeter, $P_{0}$ |  $36$ $\mu$m
Contractile tension, $\Gamma$ |  $20$ nN $\mu$m-1
T1 threshold edge length, $L^{*}$ |  $1$ $\mu$m
Protrusion force, $f_{p}$ |  $2$ $\mu$N
Internal motility, $v_{0}$ |  $10$ $\mu$m hr-1
Rotational diffusion, $D_{r}$ |  $5$ hr-1
Substrate |
Node density in the spring mesh | 0.6 $\mu$m-2
Young’s modulus, $E_{s}$ |  $4$ kPa
Poisson’s ratio, $\nu$ | 1/3
Friction, $\mu$ |  $7.2\times 10^{4}$ nN $\mu$m-1s-1
Thickness, $h_{s}$ |  $5$ $\mu$m
Adhesion stiffness, $k_{f}$ |  $4\times 10^{5}$ nN $\mu$m-1
Adhesion unbinding rate, $k_{\text{off}}$ |  $25$ hr-1
Adhesion binding rate, $k_{\text{on}}$ |  $500$ hr-1
Wound |
Radius  |  $15$ $\mu$m
Aspect ratio  | 1
Purse-string line tension, $\gamma_{\text{ps}}$ |  $300$ nN
Purse-string transition rate, $k_{\text{p}}$ |  $4$ hr-1
Other |
Simulation timestep  |  $3.6$ s
Cell count  | 150

### Traction stress computation

We record displacements of the substrate mesh, ${\bf u}={\bf r}-{\bf r}_{0}$,
at each timestep during the simulation. These vectors are then interpolated to
a square grid, from which strain is evaluated using the finite difference
discretization of:
$\epsilon_{kl}=\frac{1}{2}(\partial_{k}u_{l}+\partial_{l}u_{k})$, where $k$
and $l$ are in-plane spatial coordinates. The resultant stress is:

$\sigma_{kl}=\frac{E_{s}\nu}{(1+\nu)(1-2\nu)}\delta_{kl}\epsilon_{mm}+\frac{E_{s}}{(1+\nu)}\epsilon_{kl}\;.$
(9)

The traction stress is calculated using $T_{k}=h_{s}\partial_{l}\sigma_{kl}$.
The computed traction force vectors in the square grid are in excellent
agreement with forces directly inferred from spring displacements in the
triangular mesh (S15 Fig). The strain energy density is given by
$U=\frac{1}{2}\epsilon_{kl}\sigma_{kl}$. For each simulation we calculate the
mean strain energy as total strain energy transmitted to the substrate
averaged over simulation time, $T$:

$\langle\text{SE}\rangle=\frac{1}{T}\int_{0}^{T}dt\int_{A}dA\
h_{s}U(x,y,t)\;.$ (10)

### In vitro wound healing experiments

Madin-Darby Canine Kidney (MDCK.2) cells (CRL-2936${}^{\text{TM}}$; ATCC,
Manassas, VA) were cultured in Eagle’s Minimum Essential Medium (ATCC)
containing 10% fetal bovine serum (GIBCO Life Technologies) and 1%
penicillin/streptomycin at 370C and 5% CO2 in a humidified incubator. MDCK.2
cells are stably transfected with a plasmid construct encoding for
FTRActinEGFP (a gift from Sergey Plotnikov, University of Toronto).

Polyacrylamide gels are polymerized onto a glass coverslip at a ratio of
12%:0.086% polyacrylamide:bis-acrylamide to create a gel with an elastic
modulus of 12.2 kPa [59]. After polymerization is complete, the polyacrylamide
gels are reacted with 2mg/mL Sulfo-SANPAH (Thermo Fisher Scientific) and
incubated with 1mg/mL Type 1 rat tail collagen (Corning, high concentration)
for 2 hours in the dark [60]. Excess collagen is removed by rinsing with 1X
Phosphate-buffered saline.

Confluent cell monolayers were grown on a polyacrylamide gel substrate with an
elastic modulus of 12.2 kPa. Wounds were formed by laser ablation of a single
cell using a 435 nm wavelength laser (Andor Technology, Belfast, Northern
Ireland). Cell death causes monolayer retraction for $\sim$20 min after which
the wounds close.

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## Supporting information

#### S1 Fig

Computational pipeline in the wound healing assay. A: From left to right: 1)
Update adhesion states for cell vertices. Adherent vertices attempt to unbind
with a rate $k_{\text{off}}$, and unbound vertices attempt to bind to the
nearest substrate mesh with a rate $k_{\text{on}}$. 2) Update cell modes from
crawling (red) to purse-string (green) with a probability $k_{\text{p}}\Delta
t$. 3) Protrude cell edges in crawling mode (red arrows) and contract cell
edges on purse-string mode (green arrows). 4) Minimize mechanical energy to
move the cell vertices down their mechanical energy gradient (black arrows).
B: Illustration of self-propulsion force on a vertex in the bulk. The central
vertex has a resultant force (red arrow) equal to the average force from its
adjacent cells (blue arrows). C: The polarity vector (blue arrow) for a cell
around the wounds bisects the angle between the lines from the cell centroid
to the boundary vertices (dashed lines).

#### S2 Fig

Forces driving wound closure. A: Traction stress distribution around a closing
wound with $k_{\text{p}}=4$ hr-1, at $t=5$ min (left), $t=30$ min (middle),
$t=60$ min (right). B: Kymograph of tangential traction stress for the mixed
mode of closure ($k_{\text{p}}=4$ hr-1). C: Kymographs of radial and
tangential traction stress for the crawling ($k_{\text{p}}=0$ hr-1) mode of
closure. D: Kymographs of radial and tangential traction stress for the purse-
string ($k_{\text{p}}=1000$ hr-1) mode of closure. E: Total strain energy
transmitted vs time for crawling, purse-string, and mixed modes of closure. F:
Closure time as a function of purse-string tension (green) and protrusion
force (red) for a mixed mode of closure ($k_{\text{p}}=4$ hr-1). Error bars
represent standard error of mean.

#### S3 Fig

Orientation of cell-substrate adhesions in leading edge cells. Histograms of
the local angle between cell-substrate bonds and the radial vector to the
wound center, in (A) crawling and (B) purse-string cells at the leading edge.
(C) Representative image of crawling cells with focal adhesions oriented
normal to the wound edge. (D) A purse-string edge flanked between two crawling
edges have its focal adhesions parallel to the wound edge. (E) Purse-string
only wounds have a majority of adhesions oriented normal to the wound edge,
due to normal driving forces arising from contractile tension in the purse-
string. Green segments represent purse-string edges, while red segments are
crawling cells.

#### S4 Fig

Effect of cell-substrate adhesion kinetics on wound closure time. (A) Mean
adhesion lifetime, $k_{\text{off}}^{-1}$, vs applied force for a catch-bond
model (blue) and constant $k_{\text{off}}$ (yellow). (B) Substrate stiffness
dependence of wound closure time for a catch-bond model of cell-substrate
adhesions, for crawling (red), purse-string (green) and mixed ($k_{p}=4$ hr-1,
black) modes of closure. (C) Wound closure time vs substrate stiffness for
constant $k_{\text{off}}$. Each data point represent average over 6
simulations. Error bars show standard deviation. (D) Closure time vs duty
ratio of focal adhesion bonds, $k_{\text{on}}/(k_{\text{off}}+k_{\text{on}})$,
for crawling, purse-string and mixed modes of wound closure. Duty ratio is
varied by changing the detachment rate, $k_{\text{off}}$, for a fixed
$k_{\text{on}}$.

#### S5 Fig

Shape dependent dynamics of wound closure. Wound morphologies for (A) crawling
($k_{\text{p}}=0$ hr-1), (B) mixed ($k_{\text{p}}=4$ hr-1), and (C) purse-
string ($k_{\text{p}}=1000$ hr-1) modes of closure, at $t=6$ min (left),
$t=18$ min (middle), $t=30$ min (right). The initial aspect ratio of the wound
is $4$.

#### S6 Fig

Dependence of traction stress on cell, substrate and wound properties.
Temporal mean of spatially averaged traction stress during wound closure for
different values of $k_{p}$ and (A) substrate stiffness, (B) wound radius, (C)
wound aspect ratio, and (D) shape parameter $p_{0}$.

#### S7 Fig

Model for curvature dependent purse-string formation. (A) Schematic showing
purse-string and crawling edges for a wound with non-uniform curvature. Purse-
string (PS; green) forms on leading edges with curvature $\kappa>\kappa^{*}$,
where $\kappa^{*}$ is a threshold curvature. Cells prefer to crawl (C; red) if
$\kappa<\kappa^{*}$. (B) Wound closure time vs $\kappa^{*}$ for the concave
shaped wound in (A). The optimum threshold curvature is chosen to be the one
that minimizes wound closure time. Dashed lines indicate wound closure times
for pure crawling (red), pure purse-string (red) and stochastic mixed (black)
modes of closure.

#### S8 Fig

Internal motility accelerates the rate of wound closure. (A) Closure time, and
(B) mean strain energy for different values of internal motility $v_{0}$ and
$k_{p}$. Starred cells indicate the fastest wound closure for a given $v_{0}$
with varying purse-string assembly rates. (C) Closure time, and (D) average
strain energy for different values of shape parameter $p_{0}$ and internal
motility $v_{0}$, for a mixed mode of closure ($k_{\text{p}}=4$ hr-1).

#### S9 Fig

Tissue morphology prior to wound closure for different values of shape
parameter $p_{0}$, from 3.0 (solid-like tisue) to 4.0 (fluid-like tissue).

#### S10 Fig

Intercalations reduce tissue mechanical energy. (A) Total tissue mechanical
energy vs time, with intercalations enabled and disabled during wound closure.
(B) Mean cell shape parameter vs time. Shaded regions represent one standard
deviation. With intercalations disabled, cells elongate and have more
variability in shape. (C-D) Simulation image showing tissue morphology before
closure with intercalations (C), and in a jammed state without intercalations
(D). Cells are much more elongated when intercalations are disabled. (E) Total
substrate strain energy, and (F) total focal adhesion strain energy, over time
with intercalations enabled and disabled.

#### S11 Fig

Faster closure leads to higher strain energy transmitted to the substrate.
Mean strain energy vs closure time. Each data point represents a different
simulation. The color corresponds to the parameter that was being varied in
that simulation.

#### S12 Fig

Intercalation events can occur during wound healing. (A) Time-lapse images of
fluorescent F-actin within MDCK cells closing a wound and (B) the drawn
outlines of cells initially at the leading edge. Cells at the leading edge at
each time point are numbered in red, whereas cells excluded from the leading
edge during closure are numbered in cyan. (C) The probability distribution of
fractional cell loss for $N=18$ wounds where the average number of cells
initially at the leading edge is $9\pm 2$. $N=8$ wounds exhibit a loss of
cells at the leading edge during closure. Within this subset, the average
percentage of cells lost is $0.23\pm 0.14$. (C-inset) The closure timescale,
$\tau$, calculated from $A(t)=A(0)e^{-t/\tau}$, where $A(t)$ is the area of
the wound at time $t$, vs the fractional cell loss at the leading edge.

#### S13 Fig

Velocity-velocity correlation of cells during wound closure. Figure shows
velocity-velocity correlation function, $C_{vv}(r)=\langle{\bf v}(0).{\bf
v}(r)\rangle/\langle{\bf v}(0)^{2}\rangle$, where $r$ is the distance between
two cell center velocity vectors, ${\bf v}$. $C_{vv}(r)$ is shown at different
time points (indicated by color) for (A) crawling, (B) purse-string, and (C)
mixed modes of wound closure. Velocity vectors of cells on opposite sides of
the wound are anti-correlated.

#### S14 Fig

Dependence of wound closure time on cell count and substrate node density.
Closure time vs (A) cell count, (B) density of nodes in the substrate spring
mesh, for a wound of fixed initial size.

#### S15 Fig

Comparison of traction force computation methods. Figure shows traction force
vectors using two different methods computed during wound closure at $t=30$
min (left column), $t=36$ min (middle column), and $t=42$ min (right column).
(A) Traction force vectors computed using the continuum elasticity equation
(9). (B) Continuum model based forces in (A) interpolated on the substrate
triangular mesh. (C) Traction forces directly computed from displacements in
the substrate spring mesh. (D) Error map showing the difference of traction
force vectors in (B) and (C). Lengths of arrows are proportional to the
magnitude of the traction force, and the scale is consistent between images.

#### S1 Video.

Wound healing driven by a mixture of crawling and purse-string.

#### S2 Video.

Wound healing driven by pure purse-string

#### S3 Video.

Wound healing driven by pure cell crawling.

#### S4 Video.

Wound closure simulations for a circular and an elliptical wound.

#### S5 Video.

Wound closure simulations for a concave wound shape.

#### S6 Video.

Effect of tissue fluidity on wound closure. Left: $p_{0}=2.6$, Right:
$p_{0}=4.6$.