Monfared
et al
. eLife 2023;12:e82435. DOI: https://doi.org/10.7554/eLife.82435
1 of 20
Mechanical basis and topological routes
to cell elimination
Siavash Monfared
1
*, Guruswami Ravichandran
1
, José Andrade
1
,
Amin Doostmohammadi
2
*
1
Division of Engineering and Applied Science, California Institute of Technology,
Pasadena, United States;
2
Niels Bohr Institute, University of Copenhagen,
Copenhagen, Denmark
Abstract
Cell layers eliminate unwanted cells through the extrusion process, which underlines
healthy versus flawed tissue behaviors. Although several biochemical pathways have been identified,
the underlying mechanical basis including the forces involved in cellular extrusion remains largely
unexplored. Utilizing a phase-
field model of a three-
dimensional cell layer, we study the interplay of
cell extrusion with cell–cell and cell–substrate interactions in a flat monolayer. Independent tuning of
cell–cell versus cell–substrate adhesion forces reveals that extrusion events can be distinctly linked
to defects in nematic and hexatic orders associated with cellular arrangements. Specifically, we
show that by increasing relative cell–cell adhesion forces the cell monolayer can switch between the
collective tendency towards fivefold, hexatic, disclinations relative to half-
integer, nematic, defects
for extruding a cell. We unify our findings by accessing three-
dimensional mechanical stress fields to
show that an extrusion event acts as a mechanism to relieve localized stress concentration.
Editor's evaluation
In this work, Monfared et al. construct a valuable three-
dimensional phase-
field model for cell mono-
layers and use this to investigate the relationship between single-
cell extrusion events and topo-
logical defects in cellular arrangement. The extension of existing 2D phase field models to three
dimensions is an important contribution of this paper, which will be of general interest to the theo-
retical modelling of epithelial monolayers. Here the model is used to study the importance of cell-
cell and cell-
substrate interaction in extrusion from cell monolayers, which will be of practical interest
to biologists and physicists working on this process. This paper presents convincing evidence that
extrusion events are distinctly linked to defects in nematic and hexatic orders in the cell monolayer.
Introduction
The ability of cells to self-
organize and collectively migrate drives numerous physiological processes
including morphogenesis (
Chiou and Collins, 2018
;
Vafa and Mahadevan, 2022
), epithelial–mesen-
chymal transition (
Barriga et al., 2018
), wound healing (
Brugués et al., 2014
), and tumor progression
(
De Pascalis and Etienne-
Manneville, 2017
). Advanced experimental techniques have linked this
ability to mechanical interactions between cells (
Maskarinec et al., 2009
;
Ladoux, 2009
;
Ladoux
and Mège, 2017
). Specifically, cells actively coordinate their movements through mechanosensitive
adhesion complexes at the cell–substrate interface and cell–cell junctions. Moreover, cell–cell and
cell–substrate adhesions seem to be coupled (
Balasubramaniam et al., 2021
), further complicating
the interplay of mechanics with biochemistry.
While advances in experimental techniques are followed by more nuanced theoretical and compu-
tational developments, a majority of current approaches to simulate multicellular layers are limited
RESEARCh ARTICLE
*For correspondence:
monfared@caltech.edu (SM);
doostmohammadi@nbi.ku.dk
(AD)
Competing interest:
The authors
declare that no competing
interests exist.
Funding:
See page 10
Preprinted:
18 August 2021
Received:
03 August 2022
Accepted:
22 March 2023
Published:
18 April 2023
Reviewing Editor:
Pierre Sens,
Institut Curie, CNRS UMR168,
France
Copyright Monfared
et al
. This
article is distributed under the
terms of the Creative Commons
Attribution License, which
permits unrestricted use and
redistribution provided that the
original author and source are
credited.
Research article
Physics of Living Systems
Monfared
et al
. eLife 2023;12:e82435. DOI: https://doi.org/10.7554/eLife.82435
2 of 20
to two-
dimensional systems, hindering in-
depth
exploration of intrinsically three-
dimensional
nature of the distinct forces that govern cell–cell
and cell–substrate interactions. Furthermore,
some of the most fundamental processes in cell
biology such as cell extrusion – responsible for
tissue integrity – are inherently three-
dimensional.
Thus, studying the underlying mechanisms neces-
sitates access to both in-
plane and out-
of-
plane
forces in the cell layers.
Cell extrusion refers to the process of removal
of excess cells to prevent accumulation of unnec-
essary or pathological cells (
Rosenblatt et al.,
2001
). This process can get initiated through
apoptotic signaling (
Rosenblatt et al., 2001
),
oncogenic transformation (
Hogan et al., 2009
),
and overcrowding of cells (
Marinari et al., 2012
;
Eisenhoffer et al., 2012
;
Levayer et al., 2016
)
or induced by replication stress (
Dwivedi et al.,
2021
). Most importantly, cell extrusion plays an
important role in developmental (
Toyama et al.,
2008
), homeostatic (
Eisenhoffer et al., 2012
;
Le
et al., 2021
), and pathological processes (
Slattum
and Rosenblatt, 2014
), including cancer metas-
tasis. However, the underlying mechanisms that
facilitate cell extrusion are still unclear.
The similarities between cellular systems and liquid crystals, studied both theoretically and experi-
mentally, featuring both nematic order (
Saw et al., 2017
;
Kawaguchi et al., 2017
;
Duclos et al., 2018
;
Blanch-
Mercader et al., 2018
;
Tan et al., 2020
;
Zhang et al., 2021
) and hexatic order (
Classen et al.,
2005
;
Sugimura and Ishihara, 2013
;
Pasupalak et al., 2020
;
Maitra et al., 2020
;
Hoffmann et al.,
2022
) with the two phases potentially coexisting (
Armengol-
Collado et al., 2022
) and interacting
provide a fresh perspective for understanding cellular processes. The fivefold disclinations in hexatic
arrangement of cells are numerically shown to favor overlaps between the cells in two-
dimensions
(
Loewe et al., 2020
), potentially contributing to the cell extrusion in three-
dimensions. In this vein,
it is shown that a net positive charge associated with hexatic disclinations can be associated with the
maximum curvature of dome-
like structures in model organoids and in epithelial cell layers (
Rozman
et al., 2020
;
Rozman et al., 2021
;
Hoffmann et al., 2022
). Moreover, in cellular monolayers, comet-
and trefoil-
shaped half-
integer topological defects, corresponding to +1/2 and -1/2 charges, respec-
tively, are prevalent (
Doostmohammadi et al., 2015
;
Doostmohammadi et al., 2016
). These are
singular points in cellular alignment that mark the breakdown of orientational order (
de Genne and
Prost, 1998
). Recent experiments on epithelial monolayers found a strong correlation between extru-
sion events and the position of a subset of +1/2 defects in addition to a relatively weaker correlation
with -1/2 defects (
Saw et al., 2017
). These recently introduced purely mechanical routes to cell extru-
sion have opened the door to new questions on the nature of forces that are involved in eliminating
cells from the monolayer and challenge the purely biological consensus that an extruding cell sends a
signal to its neighbor that activates its elimination process (
Rosenblatt et al., 2001
). Nevertheless, it is
not clear whether these different mechanisms are related, and whether, depending on the mechanical
features of the cells, the cell layers actively switch between different routes to eliminate the unwanted
cells. Since all the existing studies so far have only focused on effective two-
dimensional models of
the cell layers, fundamental questions about the three-
dimensional phenomenon of cell extrusion and
its connection to the interplay between cell-
generated forces at the interface between cells and the
substrate, with multicellular force transmission across the cell layer, remain unanswered.
In this article, we explore three-
dimensional collective cell migration in cellular monolayers. Based
on large-
scale simulations, we examine (i) the underlying mechanisms responsible for cell extru-
sion, including any correlations with ±1/2 topological defects and fivefold disclinations, and (ii) the
Figure 1.
Cell extrusion in a 3D representation of a
confluent cell layer. (
a
) A representative simulation
snapshot (cell–substrate adhesion
ω
cw
=0.0025
and
relative cell–cell adhesion
Ω
=
ω
cc
/
ω
cw
=0.4
) of a
three-
dimensional cell monolayer. Two cells are visibly
extruding. (
b
) A cross-
section (dotted yellow line
A−A
′
) of the cell monolayer highlighting the two
extruding cells via the normalized out-
of-
plane velocity
(
̃
v
z
=
(
⃗
v
·
⃗
e
z
)
/
v
max
z
), where
v
max
z
is the maximum value
of the
v
z
component of the velocity field
⃗
v
in the shown
cross- section.
Research article
Physics of Living Systems
Monfared
et al
. eLife 2023;12:e82435. DOI: https://doi.org/10.7554/eLife.82435
3 of 20
interplay of cell–cell and cell–substrate adhesion with extrusion events in cellular systems. Moreover,
by mapping the full three-
dimensional mechanical stress field across the entire monolayer, we identify
localized stress concentration as the unifying factor that governs distinct topological routes to cell
extrusion.
Results and discussion
Topological routes to cell extrusion: Nematic and hexatic disclinations
In the absence of self-
propulsion forces, the initial configuration tends to equilibrate into a hexagonal
lattice (see
Appendix 1—figure 11
in Appendix 1 for an example). As we introduce self-
propulsion
forces associated with front-
rear cell polarity (see ‘Materials and methods’ for polarization dynamics),
the system is pushed away from its equilibrium hexagonal configuration, resulting in defects mani-
fested as fivefold and sevenfold disclinations, as shown in Figure 2b.
Figure 1a
shows a simula-
tion snapshot with two extrusion events taking place. An extrusion event is detected if the vertical
displacement of a cell, relative to other cells in the monolayer, exceeds
R
0
/2
, where
R
0
is the initial
cell radius.
Figure 1b
displays the out-
of-
plane normalized velocity profile,
⃗
̃
v
z
=
(
⃗
v
(
⃗
x
)
·
⃗
e
z
)
/
v
max
z
where
v
max
z
is the maximum value of the velocity component in
⃗
e
z
direction in the displayed cross-
section of
the monolayer, clearly marking the extruding cells as they get expelled from the monolayer and lose
contact with the substrate.
Figure 2.
Nematic and hexatic disclinations govern cell extrusion. A representative analysis corresponding to the
configuration shown in
Figure 1a
and projected into
xy
−
plane (
z
=0
, i.e., the basal side). (
a
) A coarse- grained
director field with coarse-
graining length of one cell size
ℓ
dir.
=
R
0
and +1/2 (filled circles with the line indicating
orientation) and -1/2 (three connected lines with threefold symmetry) nematic defects. (
b
) Number of neighbors
z
for each cell, including fivefold and sevenfold disclinations mapped into the monolayer. The symbol + denotes the
center of mass for two extruding cells. (
c, d
) Probability densities of the normalized minimum distance between
extruding cells and the nearest ±1/2 defect,
̃
d
min
=
d
min
/
R
0
, for varying cell–cell to cell–substrate adhesion ratios
Ω
for (
c
) -1/2 and (
d
) +1/2 topological defects (inset: distribution mean
m
=
⟨
̃
d
+1/2
min
⟩
vs.
Ω
). (
e
) The probability
density of average coordination number
̄
z
for an extruding cell during
̃
t
=
(
t
/
τ
0
)
∈
[
̃
t
e
−
2.5,
̃
t
e
+0.3125]
, where
̃
t
e
denotes extrusion time,
τ
0
=
ξ
R
0
/
α
and for varying cell–substrate to cell–cell adhesion ratios
Ω
. The data in
(
c
–
e
) corresponds to four different realizations.
Research article
Physics of Living Systems
Monfared
et al
. eLife 2023;12:e82435. DOI: https://doi.org/10.7554/eLife.82435
4 of 20
In order to probe the possible mechanical
routes to cell extrusion, we begin by characterizing
topological defects in cell orientation field and
disclinations in cellular arrangements. To this end,
we first map the orientation field of the cells from
the 2D projected cell shape profile on
xy
−
plane
(
z
=0
, i.e., the basal side) and identify topological
defects as the singularities in the orientation field.
The results (example snapshot in
Figure 2a
) show
the continuous emergence of half-
integer (±1/2),
nematic, topological defects that spontaneously
nucleate in pairs and follow chaotic trajectories
before annihilation (see
Appendix 1—figure 9
in
Appendix 1 for energy spectra characterization).
It is noteworthy that unlike previous studies of
active nematic behavior in 2D cell layers (
Mueller
et al., 2019
;
Wenzel and Voigt, 2021
), the
nematic defects here emerge in the absence of
any active dipolar stress or subcellular fields as
the only active driving in these simulations is the
polar force that the cells generate. Therefore,
although the cells are endowed with polarity
in terms of their self-
propulsion, the emergent
symmetry in terms of their orientational align-
ment is nematic, which is in line with experimental
observations in cell monolayers (
Saw et al., 2017
;
Blanch-
Mercader et al., 2018
), discrete models
of self-
propelled rods (
Bär et al., 2020
;
Meacock
et al., 2021
), and recently proposed continuum
model of polar active matter (
Amiri et al., 2022
).
Remarkably, in accordance with experimental
observations (
Saw et al., 2017
), we find that the
extrusion events can be correlated with the posi-
tion of both +1/2 comet-
shaped and -1/2 trefoil-
shaped topological defects. To quantify this,
Figure 2c and d
display the probability density of
the normalized minimum distance
̃
d
±
1/2
min
=
d
±
1/2
min
/
R
0
between an extruding cell and ±1/2 topological
defects in the interval
̃
t
∈
[
̃
t
e
−
5.625,
̃
t
e
+0.625]
,
where
̃
t
=
t
/
τ
0
is the normalized time,
τ
0
=
ξ
R
0
/
α
,
ξ
corresponds to cell–substrate friction,
α
denotes the strength of polarity force, and
̃
t
e
is the (normal-
ized) extrusion time. This temporal window is chosen based on the first moment of a defect’s lifetime
distribution (see
Appendix 1—figure 5
in Appendix 1). The data in
Figure 2c and d
is based on four
distinct realizations and for varying cell–substrate to cell–cell adhesion ratios,
Ω
=
ω
cc
/
ω
cw
. For both
defect types, the probability density peaks in the vicinity of the eliminated cell (
≈
1.5
R
0
), at a much
smaller distance relative to a typical distance between two defects (see
Appendix 1—figure 7
in
Appendix 1), and falls off to nearly zero for
d
±
1/2
min
5
R
0
(
=40
)
. Furthermore, laser ablation experiments
have established that an induced extrusion event does not favor the nucleation of a pair of nematic
defects (
Saw et al., 2017
).
In a hypothesis-
testing approach, we check whether these peaks in the minimum distance repre-
sent a correlation between extrusion events and nematic defects. To this end, we set out to falsify the
hypothesis that the extrusion events are uncorrelated with the nematic defects. We utilize a Poisson
point process to randomly generate positions for extrusion events and quantify the minimum distance
between each event and the nearest half-
integer nematic defect. For each simulation, we generate
five different realizations for the extrusion events using a Poisson point process with the intensity set
Figure 3.
Topological, rather than geometrical, route
to cell extrusion. (
a
) Probability density functions for
normalized minimum distance between an extrusion
event and a +1/2 defect,
̃
d
+1/2
min
, based on simulation
results and randomly generated through a Poisson
point process and for
Ω
=0.4
. (
b
) Comparison
of Lewis’s linear and quadratic relations with our
simulations.
̄
A
z
is the average area for cells with
z
neighbors and
̄
A
is the average area of all cells. The
color bar indicates simulation time step, and the data
correspond to the case
ω
cw
=0.0025
and
Ω
=0.4
.