# The role of vimentin-nuclear interactions in persistent cell motility
through confined spaces

Sarthak Gupta1, Alison E. Patteson1, J. M. Schwarz1,2 1 Physics Department and
BioInspired Institute, Syracuse University, Syracuse, NY USA,
2 Indian Creek Farm, Ithaca, NY USA

###### Abstract

The ability of cells to move through small spaces depends on the mechanical
properties of the cellular cytoskeleton and on nuclear deformability. In
mammalian cells, the cytoskeleton is composed of three interacting, semi-
flexible polymer networks: actin, microtubules, and intermediate filaments
(IF). Recent experiments of mouse embryonic fibroblasts with and without
vimentin have shown that the IF vimentin plays a role in confined cell
motility. Here, we develop a minimal model of a cell moving through a
microchannel that incorporates explicit effects of actin and vimentin and
implicit effects of microtubules. Specifically, the model consists of a cell
with an actomyosin cortex and a deformable cell nucleus and mechanical
linkages between the two. By decreasing the amount of vimentin, we find that
the cell speed increases for vimentin-null cells compared to cells with
vimentin. The loss of vimentin increases nuclear deformation and alters
nuclear positioning in the cell. Assuming nuclear positioning is a read-out
for cell polarity, we propose a new polarity mechanism which couples cell
directional motion with cytoskeletal strength and nuclear positioning and
captures the abnormally persistent motion of vimentin-null cells, as observed
in experiments. The enhanced persistence indicates that the vimentin-null
cells are more controlled by the confinement and so less autonomous, relying
more heavily on external cues than their wild-type counterparts. Our modeling
results present a quantitative interpretation for recent experiments and have
implications for understanding the role of vimentin in the epithelial-
mesenchymal transition.

00footnotetext: ∗ E-mail<EMAIL_ADDRESS>∗∗ E-mail:
<EMAIL_ADDRESS>∗∗∗ E-mail<EMAIL_ADDRESS>
##  I Introduction

Cell migration is a fundamental process that contributes to building and
maintaining tissue. To be able to migrate, the cell cytoskeleton, which is
comprised of three dynamic polymeric systems: F-actin, microtubules, and
intermediate filaments (IFs), generates forces. While actin and microtubules
are more studied cytoskeletal filaments, intermediate filaments (IFs) also
play a role in a range of cell and tissue functions Leduc and Etienne-
Manneville (2015); Danielsson  _et al._ (2018); Pekny and Lane (2007).
Vimentin is an IF protein whose expression correlates with in vivo cell
motility Helfand  _et al._ (2011); Mendez  _et al._ (2010) behaviors involved
in wound healing Eckes  _et al._ (2000); Rogel  _et al._ (2011) and cancer
metastasis Kidd  _et al._ (2013); Satelli and Li (2011), and, yet, its role in
three-dimensional cell migration is poorly understood.

In vivo, cells move through a confining tissue environment made out of other
cells and extracellular matrix (see. Figs 1(a) and 1(b)) Friedl and Bröcker
(2000). Emerging experimental studies show that the structure and dynamics of
the cytoskeleton in three-dimensional motility differ from those for cells on
surfaces Friedl and Bröcker (2000); Petrie and Yamada (2012). In confined
settings, actin tends to accumulate at the cell cortex Balzer  _et al._
(2012a) and microtubules align with the direction of the confining track Doyle
_et al._ (2009). Moreover, in highly confining environments, the nucleus, one
of the stiffest organelles in the cell, can be an inhibitor of cell migration
due to steric hindrance Friedl  _et al._ (2011). Earlier cell motility models
in such environments, therefore, focus on the nucleus Reversat  _et al._
(2020); Le Berre  _et al._ (2013); Hawkins  _et al._ (2009), in contrast with
cell motility models for surfaces Danuser  _et al._ (2013). Recent studies
have shown that in three-dimensional settings the centrosome displays an
increased probability to be near the rear of the cell Zhang and Wang (2017).
When cells change direction in a narrow track, the centrosome repositions,
moving from one side of the cell to the other, to repolarize the cell by
developing a new trailing edge of the cell and setting up the polar direction
of the cell. Interestingly, the nucleus appears to be decoupled from this
phenomenon in that the removal of the cell nucleus does not alter the
centrosome repositioning in most cells Zhang and Wang (2017). Though for
strong confinement, the cell nucleus will presumably play a more dominant role
in the repositioning of the centrosome as it deforms.

Figure 1: Modeling confined cell motility: (a) Cells move through confining
environments due to the extracellular matrix (in gold) and other cells. (b)
Schematic of an individual cell with a cytoskeletal network containing actin,
microtubules, and intermediate filaments, a nucleus, and a centrosome
connected to the nucleus via the protein dynein. (c) A simulation model for
the cell: The cortex is made up of blue monomers connected with springs, the
nucleus is also made up of red monomers with springs connecting them. The bulk
cytoskeletal network is simplified and modeled as springs connecting the
nucleus and the cortex. The cell and nucleus contain cytoplasmic and
nucleoplasmic material, each of which is modeled as area springs.

Given the roles of actin, microtubules, and centrosome positioning in confined
cell motility, one is naturally led to ask about the role of vimentin, which
couples to both actin and microtubules and forms a cage around the cell
nucleus Costigliola  _et al._ (2017). Recent experiments Patteson  _et al._
(2019a); Battaglia  _et al._ (2018) highlight new roles of vimentin in
mediating cell speed and cell persistence in three-dimensional confining
environments. Specifically, when comparing the motion of wild-type mouse
embryonic fibroblasts (mEFs) with their vimentin-null counterparts moving
through three-dimensional micro-fluidic channels, the loss of vimentin
increases cell motility and increases rates of nuclear damage in the form of
nuclear envelope rupture and DNA breaks. Moreover, unlike in unconfined
motility, the loss of vimentin increases the spontaneous persistent migration
of cells through 3D micro-channels and 3D collagen matrices van Bodegraven and
Etienne-Manneville (2020).

Inspired by these experiments, we seek to quantitatively understand the role
of vimentin in confined cell motility via a computational model. To do so, we
develop a minimal, yet detailed, cell motility model incorporating actin,
microtubules, and vimentin. The starting point of the model is the notion that
vimentin plays a distinct role in mediating forces between the actomyosin
cortex and the nucleus Patteson  _et al._ (2020a, b). Our model, therefore,
contains both an actomyosin cortex and a nucleus, whose interaction via a set
of linker springs is strengthened by the presence of vimentin. With this level
of detail, a new mechanism for cell polarity for motility in which vimentin
plays an important role naturally emerges. As the cell moves through the model
micro-channel, we will quantitatively show how vimentin modulates cell speed,
nuclear shape, dynamics, and cell persistence. In addition to quantitative
comparison with experiments, we pose new insights for the role of vimentin in
confined cell motility more generally. In particular, we address the
upregulation of vimentin typically found in the epithelial-mesenchymal
transition Mendez  _et al._ (2010) in terms of how vimentin affects a cell’s
interaction with its environment.

##  II Model

###  II.1 The players

Cellular cytoskeleton: The cellular cytoskeleton is composed of actin,
microtubules, and intermediate filaments. The persistence length of actin
filaments is smaller than microtubules, but larger than intermediate filaments
Janmey  _et al._ (1991). Myosin motors exert forces on actin filaments to
reconfigure them. Many actin filaments and myosin motors reside in proximity
to the cell membrane to form the actomyosin cortex. The actomyosin cortex is
an important piece of the cell motility machinery as its reconfiguring drives
cell motion. Microtubules typically originate from the microtubule-organizing
center, or centrosome, and have a crucial role in cell polarity as they as
required to generate traction forces Bouchet and Akhmanova (2017).
Microtubules also have a role in controlling cell shape as they typically span
the entire cell and are the stiffest cytoskeletal filaments Janmey  _et al._
(1991). Vimentin filaments exist as a cage or mesh structure around the
nucleus and are also present in the cytoskeleton in fibrous form as can be
seen in Fig. 1(b) Costigliola  _et al._ (2017); Patteson  _et al._ (2020a).
Studies show that vimentin provides structural integrity to the cell Patteson
_et al._ (2019b); Guo  _et al._ (2013).

The various types of cytoskeletal filaments do not act independently of each
other. In fact, actin, microtubules, and intermediate filaments are rather
interconnected with each other Seetharaman and Etienne-Manneville (2020). For
instance, vimentin and actin directly interact with each other via the tail
domain of vimentin Esue  _et al._ (2006). Moreover, plectin is a major
crosslinker among all three types of filaments Svitkina  _et al._ (1998).

Cell nucleus: The nucleus is typically the largest, and stiffest, organelle in
the cell, yet it is still deformable. Recent studies showed a cell under a
highly confined environment, and thereby, the nucleus is so squeezed such that
DNA becomes damaged Patteson  _et al._ (2019b). Cells cannot move through a
particular confining geometry if the nucleus cannot do so Friedl  _et al._
(2011). Therefore, the nucleus is also an important player in confined cell
motility. There also exists a nuclear envelope, or nuclear cortex, consisting
of inner nuclear lamins and outer vimentin.

The interconnected cytoskeletal networks and the nucleus are also connected to
each other. For instance, LINC(Linker of nucleoskeleton and cytoskeleton)
complexes made of nesprins and SUN (Sad1 and UNC84) proteins act as connecting
bridges between nucleoskeleton, containing lamins, and the actin network in
cytoplasm Crisp  _et al._ (2006). Microtubules are also joined to the nucleus
via kinesin-1 which talks to nesprin-4 Roux  _et al._ (2009). Intermediate
filaments are connected via plectin which connects to nesprin-3, which is
joined with nucleus Wilhelmsen  _et al._ (2005). Disruption of these nucleus-
cytoskeleton links also leads to impairment of 3D cell migration Khatau  _et
al._ (2012); Harada  _et al._ (2014).

Centrosome: Mechanisms for driving cell polarity, or the direction in which a
cell moves, is also crucial for understanding cell motility. Recent
experiments in two-dimensional motility suggest that removing the centrosome
induces microtubules to grow symmetrically in all directions and, as a result,
the cell forms lamellae in many directions and, therefore, loses its
polarization Wakida  _et al._ (2010). In a 3D setting, the centrosome is
typically found to be posterior of the cell, and microtubules are oriented in
the direction of motion (for certain cell types) Doyle  _et al._ (2009).
Therefore, the position of the centrosome sets the polarity of the cell by
defining the tail/back of the cell during migration since the change in cell
direction results only after the centrosome moves to the new posterior side of
the cell Ueda  _et al._ (1997).

###  II.2 Cell as a spring network

Given the complexity of such interactions of the above players, we took a
reductionist approach and simplified this highly tangled picture to a two-
dimensional network of harmonic springs with an outer ring of springs
representing the cellular cortex and an inner ring of springs representing the
nuclear envelope, and harmonic springs connecting the cellular cortex with the
nuclear cortex (see Fig. 1(c)). Our model is a two-dimensional cross-section
of the three-dimensional system. As we can see from the schematic, each cell
cortex monomer is joined to each nuclear cortex monomer via a linker spring.
Each spring type, the cellular cortex, the nuclear cortex, and the interacting
bulk linker spring, has its own stiffness with the respective potential
energies $V_{cc}=\frac{K_{cc}}{2}(r_{cc}-r_{cc,o})^{2}$,
$V_{nc}=\frac{K_{nc}}{2}(r_{nc}-r_{nc,o})^{2}$,
$V_{l}=\frac{K_{l}}{2}(r_{l}-r_{l,o})^{2}$, respectively, with $r_{cc}$
denoting the distance between the centers of neighboring cell cortex monomers,
for example, and $r_{cc,o}$ represents the rest length of the spring. We also
include potentials in the form of two area springs,
$V_{cell}=\frac{K_{cell}^{area}}{2}(A_{c}-A_{c,o})^{2}$ and
$V_{nuc}=\frac{K_{nuc}^{area}}{2}(A_{n}-A_{n,o})^{2}$, where $A_{c}$ and
$A_{n}$ denote the areas of the cell and nucleus, respectively. The two area
springs prevent the cell and the nucleus from collapsing as the cell becomes
increasingly more confined.

How do we explore the role of vimentin in such a mechanical model,
particularly when considering wild-type fibroblasts versus their vimentin-null
counterparts? Since vimentin-null cells are softer than wild-type cells Guo
_et al._ (2013) and exhibit more DNA damage Patteson  _et al._ (2019b), to
capture both cell lines we change the stiffness of the cytoplasmic/linker
springs $\text{K}_{\text{l}}$ and nucleus area springs
$\text{K}_{\text{nuc}}$. Since removing vimentin does not significantly affect
the cortical stiffness Guo  _et al._ (2013), we do not alter the cortical
spring constant among two cell lines.

The cell also interacts with its confining walls which are considered to be
adhesive due to fibronectin or collagen I or some other kind of protein that
is usually coated inside the channels. Specifically, the micro-channel is
modeled as two lines of wall monomers fixed in place that interact with the
cell cortex monomers via adhesion. In previous studies, adhesion to the
surface is modeled as catch and slip bonds Bangasser  _et al._ (2013); Lopez
_et al._ (2014), where the cell forms a bond to the substrate, and then as it
moves those connections peel off. We, therefore, model the adhesion
interaction with the Weeks-Chandler-Anderson potential, or

$V_{cell-
wall}=\begin{cases}4\epsilon_{Ad}\left[\left(\frac{\sigma_{cw}}{r}\right)^{12}-\left(\frac{\sigma_{cw}}{r}\right)^{6}\right]+\epsilon&r\leqslant
1.2\sigma_{cw}\\\ 0&r>1.2\sigma_{cw},\end{cases}$ (1)

where $\epsilon$ quantifies the adhesion strength and the minimum of the
potential is located at $2^{1/6}\sigma_{cw}$. While the potential is purely
repulsive in the short-range, there is also an attractive component. Cell
experiences no adhesion until the distance between cortex and wall is less
than or equal to $1.2\sigma_{cw}$. After which, cell experiences an attractive
force till the potential minima $2^{1/6}\sigma_{cw}=1.122\sigma_{cw}$. The
difference between cutoff distance and potential minima $0.077\sigma_{cw}$ or
$0.15507\,\mu m$ or $155.07\,nm$, thus represents a typical focal adhesion
size Changede and Sheetz (2017); Evans and Calderwood (2007). We did not vary
the adhesion strength between the two cell lines.

###  II.3 Polymerization and adhesion forces

As the cell moves, actin is polymerized at the leading edge of the cell to
translate the cell in a particular direction with microtubules setting the
direction Meiring  _et al._ (2020). We model actin polymerization via an
active force, $\mathbf{F}_{a}$. The active force is present for half cortex of
the cell, the leading edge half, and has a magnitude $F_{a}$. The direction of
$\mathbf{F}_{a}$, which is set by microtubules emanating from the centrosome,
is initially chosen to be towards the opening of the micro-channel, which
determines the leading edge half—its polarization direction. There are small
fluctuations in the direction of $\mathbf{F}_{a}$ as it moves through the
micro-channel.

Polarity-mechanism: If one allows for large fluctuations in the direction of
$\mathbf{F}_{a}$, the probability is enhanced of the cell turning around in
the channel. Cell length is also found to be correlated with the time required
for cells to change the direction, which is the time the centrosome takes to
move to the other side of the cell to define a new tail Zhang and Wang (2017).
Therefore, we posit that the more the centrosome is located away from the cell
center of a crawling cell, the more biased cell migration becomes. Thus, the
centrosome is essential for the preservation of polarized cell morphology.

While there is no explicit centrosome, nor microtubules, in our model, the
centrosome is also connected to the nuclear outer membrane by a protein emerin
Salpingidou  _et al._ (2007). Given the strong coupling between the nucleus
and the centrosome, we effectively include a centrosome and a direction of
cell polarity as determined by the direction of microtubule polymerization. To
do so, we define $\mathbf{d}$ representing the difference between the center
of mass of the cell and the center of mass of the nucleus. Its angle is
measured from the positive $x$-axis. To be specific, if the cell moves
parallel to the channel away from the designated entrance, $\theta=90^{o}$; if
the cell performs a “180” to head directly back towards the designated
entrance, $\theta=180^{o}$. For incorporating the centrosome role in motility,
we define the following empirical equation:

$\Delta\theta=\frac{\pi}{2}\left(1-\frac{|\mathbf{d}|}{R_{nuc}}\right),$ (2)

where $R_{nuc}$ denotes the rest radius of the nucleus. With this empirical
equation, we define the bounds on the direction a cell can take. We take the
nuclear cortex-cellular cortex (NC-CC) axis as our reference axis and for the
upper bound and lower bound we can add and subtract $\Delta\theta$ from this
axis. See Fig. 4(a). Then, we choose a random angle under these bounds and
that represents the direction of the nuclear, or centrosome, axis given the
strong coupling between the two, which then drives the direction of migration.

We found that the negative $\Delta\theta$ is highly unlikely in this model as
the purely repulsive LJ interaction ensures that the distance between the
center of the cell and the center of the nucleus does not increase more than
the radius of the nucleus. For instance, when the nucleus gets displaced off-
center, the maximum it can go is to the cortex wall. Also, even if
$\Delta\theta$ is negative, the upper and lower bound of available angles
would simply flip. Thus, it would not change the range of available angles for
choosing the next direction. In our current analysis, we ignore the ensembles
where $\Delta\theta$ is negative.

What is the timescale for changing $\Delta\theta$? We assume that vimentin
acts as a template for microtubules in the cell on a time scale of 10-20
minutes since (1) vimentin has a slow turnover rate than microtubules and (2)
microtubules in vimentin-null cells show less orientation than in wild-type
cells Gan  _et al._ (2016). Therefore, this polarity mechanism repeats itself
every $15$ minutes as, on average, this is the time after which vimentin
restructures Gan  _et al._ (2016); Yoon  _et al._ (1998). For simplicity, we
assume that the repolarization time in the vimentin-null cell is the same as
wild-type, as it is not known if microtubule turn-over time changes between
the two or not. We leave the investigation of this assumption for future work.

Additionally, we ensure that all cells move toward the channel initially so
that they will be in contact with the mouth of the channel. Then the polarity
mechanism kicks in and the cell chooses a new direction. We have used the
initial radius of the nucleus as a normalization constant in Eq. 3. Note that
there is no memory of the previously taken direction. The NC-CC axis is,
again, taken as a reference after each $15$ min. and depending on the distance
of the nucleus from the cell cortex, $\Delta\theta$ is calculated, and
eventually, a new angle is again chosen randomly from within the bounds. Note
that the cell does not exactly follow the NC-CC axis for cell migration due to
the intracellular dynamics and fluctuations in the cell.

In addition to adhering to the wall, it has been observed that in confined
environment experiments, actin bundles start forming at the edges of the cell
where it is interacting with the walls Paul  _et al._ (2016). We are proposing
that due to this interaction with the wall, the cell generates forces to
enhance motility in the direction in which it is polarized. Therefore, any
cell cortex monomer with some proximity of a wall monomer exerts an additional
force, $\mathbf{F}_{w}$ in the direction of the leading edge. For avoiding
overlap (volume exclusion), we apply $\mathbf{F}_{rep}$ which is derived by
WCA potential with a cut-off at minima($2^{1/6}\sigma$), which makes this
force purely repulsive in short-range and zero for a greater distance than
minima. $\mathbf{F}_{rep}$ is applied to all particles except particles
connected via two-body harmonic spring.

Figure 2:  Model quantitatively agrees with cell speeds from micro-channel
experiments: (a) Time series of a wild-type mEF cell moving in a microchannel
(b) Same as (a) but images are from a computational model. (c) Average cell
speed as a function of channel width for wild-type and vimentin-null cells for
the computational model. Prior experimental results are also plotted for
comparison. Note that the experimental results indicate the average maximum
cell speed, while the computational results indicate the average speed.

###  II.4 Dynamics

Now that we have detailed the forces involved, here is the equation of motion
for each cell monomer of type $i$ at position $\mathbf{r}_{i}$:

$\dot{\mathbf{r}}_{i}=\mu_{i}(\mathbf{F}_{a}+\mathbf{F}_{w}+\mathbf{F}^{c})+\sqrt{2D_{i}}\mathbf{\xi}_{i}(t),$
(3)

where $\mathbf{F}_{a}$ is an active force representing the actin forces at the
front end of the cell in the direction of motion and $\mathbf{F}_{w}$ denotes
the force generated at the wall in the direction of the leading edge. Finally,
$\mathbf{F}^{c}$ represents the conservative forces in the system, or
$\mathbf{F}^{c}=\mathbf{F}_{cc}+\mathbf{F}_{nc}+\mathbf{F}_{l}+\mathbf{F}_{cell}+\mathbf{F}_{nuc}+\mathbf{F}_{cell-
wall}+\mathbf{F}_{rep})$, which are the two-body springs, the area springs,
both modeling the mechanics of the cell and the adhesion force between the
cell and the confinement (wall) and purely repulsive volume-exclusion force.
For the parameters varied in the simulations, see Table 1.

Regarding parameters, the diffusion constant of both the cell and nuclear
membranes are rooted in biologically relevant time scales such that membrane
displacements are much less than the global motion of the cell and nucleus,
respectively. The diffusion constant of the cell membrane, $D_{cc}$, is
estimated from the actin bundle motion just below the cell membrane Shirai
_et al._ (2017). For estimating the nuclear membrane diffusion constant,
$D_{nc}$, we look at the fluctuations of the cell nucleus Chu  _et al._
(2017). The values $D_{cc}$ and $D_{nc}$ are close, but not the same
($D_{cc}>D_{nc}$ ) to account for the fact that these two objects reside in
slightly different environments. See Table 1. As for the mobilities, prior
work has implemented cell cortex mobilities an order of magnitude lower Lopez
_et al._ (2014). There is some range of applicability of these estimates as
they are indeed estimates. A different mobility can be absorbed into changing
the time scale and the strength of the noise (see Eq. 2).

We use simulation units defined as unit simulation length equal to
$\mu\text{m}$, unit simulation time is equal to $sec$ and unit simulation
force is $nN$. The respective diameter of the monomers are
$\sigma_{cc}=\sigma_{cw}=2\,\mu m$, $\sigma_{nc}=1\,\mu m$, all neighbouring
springs in actomyosin cortex and nuclear cortex is $2\,\mu m$ and $1\,\mu m$
respectively and linker spring length is $5.73\,\mu m$. Relevant length scale
for converting kPA to nN/$\mu m$ are cortex diameter($20\,\mu m$), nucleus
diameter($10\,\mu m$), and linker spring length for cortex spring, nucleus
spring, and linker spring strength respectively. Our simulations used $36$
monomers in both cortices, $74$ monomers to simulate each side of the straight
part of the wall, and $7$ monomers for each side of the slanted channel entry
and exit. To iterate Eq. 2, we use the Euler-Maruyama method.

Parameters | Wild-type | Vimentin-null | Refs.
---|---|---|---
$K_{cc}$ | $100\,nN/\mu m$ | $100\,nN/\mu m$ | Guo  _et al._ (2013)
$K_{nc}$ | $1000\,nN/\mu m$ | $1000\,nN/\mu m$ | Caille  _et al._ (2002); Vahabikashi  _et al._ (2019)
$K_{l}$ | $10\,nN/\mu m$ | $1\,nN/\mu m$ | Guo  _et al._ (2013); Vahabikashi  _et al._ (2019)
$K_{cell}^{area}$ | $0.01\,nN/\mu m^{3}$ | $0.01\,nN/\mu m^{3}$ | -
$K_{nuc}^{area}$ | $0.05\,nN/\mu m^{3}$ | $0.005\,nN/\mu m^{3}$ | -
$F_{a}$ | $3\,nN$ | $3\,nN$ | Prentice-Mott  _et al._ (2013)
$F_{w}$ | $3\,nN$ | $3\,nN$ | -
$\epsilon_{ad}$ | $1\,nN\,\mu m$ | $1\,nN\,\mu m$ | -
$\epsilon_{rep}$ | $1\,nN\,\mu m$ | $1\,nN\,\mu m$ | -
$D_{cc}$ | $0.02\,\mu m^{2}/s$ | $0.02\,\mu m^{2}/s$ | Shirai  _et al._ (2017)
$D_{nc}$ | $0.04\,\mu m^{2}/s$ | $0.04\,\mu m^{2}/s$ | Chu  _et al._ (2017)
$\mu_{cc}$ | $0.01\,\mu m/nN\cdot s$ | $0.01\,\mu m/nN\cdot s$ | -
$\mu_{nc}$ | $0.02\,\mu m/nN\cdot s$ | $0.02\,\mu m/nN\cdot s$ | -
Table 1: Table of parameters used, unless otherwise specified. Figure 3:
Vimentin affects nuclear shape and position. (a) Images of a wild-type mEF
cell and a vimentin-null mEF cell moving in a microchannel and their
computational counterparts. In addition, the vector difference between the
cell center of mass and the nucleus center of mass, $\vec{d}$, is labeled on
each cell image. (b) The circularity of the cell nucleus as a function of
confinement for both cell types. (c) The position of the center of mass of the
cell with respect to the center of mass of the nucleus as a function of
channel width for both cell types.

###  II.5 Model summary

We model the cell as containing an outer actomyosin cortex, represented by
monomers connected by springs, a nuclear cortex, also represented by monomers
connected by springs, and bulk cytoskeletal filaments—including vimentin,
represented as linker springs connecting the two inner and outer cortices. See
Fig. 1(c). There is an energetic cost to changing the area of both cortices as
well to capture the incompressibility of the cell and the nucleus. The cell
also interacts with each wall of the microchannel composed of fixed monomers
via an adhesive potential. To study the motility of the cell through the
microchannel, there is an active force, $\bf{F}_{a}$ on the half of the acto-
myosin cortex monomers in a direction initially towards the opening of the
microchannel and whose direction is determined by the microtubule-centrosome
system. Should any actomyosin cortex monomer come within some proximity of the
wall, there is an additional force, $\bf{F}_{w}$, exerts on an actomyosin
cortex monomer in the direction of the leading edge. The direction of the
leading edge is chosen uniformly at random within some range of angle. We
propose a form for the range of possible leading edges that depends on the
magnitude of the difference between the center of the mass of the cell and the
center of mass of the nucleus, with the position of the nucleus serving as a
readout for the position of the centrosome. The forces in the model are
updated using over-damped dynamics. For a detailed explanation motivating each
force in the model, please see the Model section above. See Figs. 2a and 2b
for snapshots of the experiments and the simulations of a cell moving through
a microchannel. Two supplementary movies generated with Visual Molecular
Dynamics (VMD) Humphrey  _et al._ (1996) have also been provided.

###  II.6 Analysis

We measure the average speed of the center of mass of the cell while it is in
the channel. This cell speed average is then averaged over approximately 1000
realizations for each channel width. Error bars represent the standard
deviation of the mean of approximately $10^{3}$ ensemble runs. We also measure
the persistence in the motility as defined as the ratio of the path length of
the center of mass of the cell divided by the length of the channel. Flux is
defined as the fraction of cells that exit the channel on the side different
from the entry side. The circularity of the nucleus $C$ is defined as
$C=\frac{4\pi A}{P^{2}}$, where $A$ is the area of the cell and $P$ is its
perimeter. When $C=1$, the nucleus is a circle. The time-averaged energy of
the cell is calculated by taking the ensemble average of energy contributions
from the conservative potentials. The initial cell energy for the conservative
potentials is zero.

##  III Results

###  III.1 Cell speed is non-monotonic with confinement and affected by
vimentin

Strikingly, we observe for both cell types that as the channel width
decreases, the average cell speed is non-monotonic (Fig. 2c). This trend has
also been observed in experiments Charras and Sahai (2014); Irimia and Toner
(2009); Jacobelli  _et al._ (2010); Balzer  _et al._ (2012b). How does such a
trend emerge? As the channels become narrower, the cell’s cortex increases its
contact with the wall. This increase in contact generates more driving force
to increase cell speed. However, this trend is competing with the
deformability of the cell. As the channel width becomes even narrower, the
linker springs become more deformed (more compressed) and so these springs act
to increase the effective adhesion to the wall given that unbinding to the
wall is driven by a distance threshold. This increased adhesion time leads to
a slower cell speed. Thus, the non-monotonic trend emerges from these
competing factors, contact with the wall increasing the driving force and
increasing the adhesion time as the cells become increasingly more deformed.
We can modulate this competition by either increasing the driving force or the
adhesion strength. If one increases the driving force, either by increasing
the active force or the wall force, not only will the maximum speed increase,
the peak will broaden towards smaller channel widths (Fig. S1). If one
decreases the adhesion strength, a similar effect occurs (Fig. S2).

How is such a trend modified in vimentin-null cells? Since the presence of
vimentin makes cells stiffer Eckes  _et al._ (1998); Wang and Stamenovic
(2002); Guo  _et al._ (2013); Patteson  _et al._ (2019a), the vimentin-null
cell line is described by a decreased linker spring strength and a decreased
nucleus area spring strength, with the latter capturing the lack of a vimentin
cage around the nucleus. We indeed observe a similar non-monotonic trend, but
with a larger average cell speed as compared to wild-type cells, at least for
narrower channel widths. Moreover, the maximum average cell speed occurs at a
narrow channel width. Again, as confinement increases, the bulk cytoskeleton
in both cell lines also starts deforming (compressing). However, the wild-type
cell provides more resistance as it is a stiffer cell, and, thus, pushes
against the walls more to effectively act a stronger adhesion to the wall.
This effective adhesion to the wall is weaker for the vimentin-null cell line
because linker spring strength is weaker. This effect results in both a larger
maximum average cell speed and the driving force out-competing the adhesion
for a larger range of change of channel widths. Note that our results also
depend on the nucleus area spring strength. Decreasing the nucleus area spring
strength also decreases the effective adhesion to the wall, as the anchoring
of the springs to the nucleus is less stiff and so also enhances the average
cell speed (Fig. S3).

Now let us compare the computational model with the experiments. Our average
cell speed is in reasonable quantitative agreement with the experimental cell
speed measurements for both cell types and for two different channel widths
(Fig. 2(c)) with wild-type cells moving more slowly than vimentin-null cells
in the microchannels channels Patteson  _et al._ (2019a). This outcome is in
stark contrast with two-dimensional cell motility studies Helfand  _et al._
(2011). The experiments demonstrate that as confinement increases, vimentin-
deficient cell’s average speed also increases, whereas wild-type cell speed
remains largely unchanged Patteson  _et al._ (2019a). We find similar behavior
with the non-monotonic trend in cell speed as a function of channel width
weaker in the wild-type case. This non-monotonicity also agrees with the
experimental observations of other cell lines migrating in the channels
Charras and Sahai (2014). For vimentin-null cells, we predict that for
channels less than a 10-micron width that the cell speed will begin to
decrease. Hints of this prediction are evident in transwell experiments
Patteson  _et al._ (2019b).

###  III.2 Vimentin’s dual role of stress transmitter and nuclear protector

Figure 4: Nucleus-centrosome-based polarity mechanism. (a) Schematic of the
polarity mechanism. (b) Time series of a cell moving into the channel and
changing direction. For experimental methods, please see Patteson  _et al._
(2019a) (c) Computational example of a cell changing direction in the channel.
(d) Pers istence as a function of channel width for both cell types.

Since the linker spring strength mediates the transmission of forces between
the actin cortex and the nuclear cortex, one, therefore, may anticipate the
shape of the nucleus to depend on the linker spring strength. For instance, we
expect the nucleus shape to be less correlated with the cell cortex shape as
the linker spring strength becomes weaker. Prior experiments indeed indicate
that the shape of the nucleus is affected by vimentin as cells move in
confinement Lowery  _et al._ (2015). To quantify the effects of the linker
spring strength on nuclear shape, we calculate nuclear circularity C, which is
defined as $\frac{4\pi A}{P^{2}}$, where $A$ denotes the cross-sectional area
of the nucleus and $P$ its cross-sectional perimeter (Figs. 3(a) and (b)).
When $\text{C}=1$, the nuclear cross-section is a circle. As the channel width
decreases, nuclear circularity decreases for both cell types, however, the
nuclei of the vimentin-null cells are typically more deformed than their wild-
type counterparts. The greater decrease in $C$ for the wild-type cells with
increasing confinement is due to the larger stresses mediated by the bulk
cytoskeleton, thereby potentially motivating the need for a mesh-like vimentin
cage around the nucleus to help mediate them.

Interestingly, experiments indicate that the nucleus in vimentin-null cells
has more wrinkles than wild-type and has less effective volume Lele  _et al._
(2018). As discussed in the modeling section, in addition to modifying the
linker spring strength to go from one cell type to the other, we also modify
the area spring constant for the nucleus, with the latter accounting for the
mesh-like vimentin cage surrounding the nucleus. When we increase the area
spring constant for the nucleus, as well as increase the linker spring
stiffness, we find that even though the cytoskeleton still transfers forces
from outside to the nucleus with increasing confinement, the nucleus resists
such deformations with the circularity tracking very similarly for the smaller
channel widths between the two cell types. To see the further effects of
linker spring constant and nuclear area spring constant on nucleus shape,
please refer to Fig. S10.

The nuclear envelope is typically viewed as dominated by the nuclear laminas
McGregor  _et al._ (2016). Here we suggest that vimentin around the nucleus
also has a role in stabilizing the shape of the nucleus. Thus, we can conclude
that vimentin protects the nucleus from deformations as well as mediates
stress transmission between the two cortices, the actin one and the nuclear
one Maniotis  _et al._ (1997); Neelam  _et al._ (2015). Now we have
quantitative modeling results to substantiate this concept.

Since nuclear positioning is important for cell migration, we next compute the
center of mass of the nucleus, the center of mass of the cell, and determined
the distance between the two, defined as the magnitude of $\vec{d}$ with the
origin of $\vec{d}$ at the center of mass of the nucleus as shown in Figs.
3(a) and 3(c). This metric estimates the location of the nucleus with respect
to the rest of the cell. When $|\vec{d}|\approx 1\mu m$, the nucleus is closer
to the center of the cell as compared to the vimentin-null case. For larger
$|\vec{d}|$, the nucleus moves toward the rear of the cell or away from the
leading edge of the cell. We find that the nuclei in the vimentin-null cells
are typically positioned toward the rear of the cell, while the nuclei for the
wild-type cells are closer to the center of the cell (Fig. 3(c)). As the
channel width narrows, this difference becomes even starker.

How does this trend of nuclear positioning emerge? For the more malleable
linker springs, springs in the leading half of the cell more readily attach to
the wall creating a typically flatter leading edge for the part of the cell
cortex not attached to the walls. The linker springs in the leading half of
the cell associated with the leading edge are, therefore, more extended (see
Fig. 3(a)). From an energetic point of view, the extra tension (stretching) in
the leading half of the cell is compensated for by the springs in the rear
half of the cells configuring to be close to their rest length. This argument
also tells us that the nucleus is being pulled by the leading half of the cell
as it moves through the channel. We confirm this by calculating the forces on
the nucleus due to the cortex (Fig. S4). Our results hopefully prompt
additional experiments focusing on nuclear positioning for confined cell
motility.

Figure 5: Energy barriers in confined cell motility. (a) The average energy of
each cell due to the conservative forces as a function of time for different
channel widths. (b) The time-averaged energy as a function of channel width
for both cell types. (c) The magnitude of the numerical derivative of the
time-averaged energy, which is proportional to the stress on the cell due to
the channel, as a function of channel width for both cell types. (d) The flux
of each cell type as a function of the channel width. (e) A schematic of the
internal organization of more polarized/persistent cells versus less
polarized/persistent ones with the centrosome denoted by the white circle.

###  III.3 Loss of vimentin increases cell persistence in micro-channels

Based on the observed changes in nuclear positioning, we propose a novel
polarity mechanism based on the position of the nucleus, which is a readout
for the position of the centrosome in the cell. The centrosome plays an
important role in cell polarity and is also connected to the nucleus via
various crosslinkers and proteins Luxton and Gundersen (2011). We assume that
such connections between the centrosome and nucleus remain strong as the cells
move through the channel and so the position of the nucleus tracks the
position of the centrosome. We posit in confinement that as the magnitude of
$\vec{\text{d}}$ increases, the cell becomes more polarized with the possible
range of angles of microtubule polymerization scaling as
$|\Delta\theta|=\frac{\pi}{2}\left(1-\frac{\left|\vec{d}\right|}{R_{Nuc}}\right)$,
where $R_{Nuc}$ is the rest radius of the nucleus and $\pm\Delta\theta$ is
defined clockwise/counter-clockwise from the reference angle of $\vec{d}$
respectively (Fig. 4(a)). There are two reasons for this scaling. First, as
the nucleus is more displaced, only the longest microtubules polymerize
towards the leading edge of the cell. The lateral confinement and rear end of
the cell disrupt microtubule polymerization in the remaining directions. This
spatial arrangement biases microtubule polymerization parallel to the
confining walls Lagomarsino  _et al._ (2007); Pinot  _et al._ (2009). See
Supplementary Fig. S5 for the distribution of angles possible for the leading
edge of the cell (with respect to the $x$-axis) and the distribution of angles
chosen by the cell.

In Figs. 4(b) and (c), we show the time series for a wild-type cell from the
earlier micro-channel experiment Patteson  _et al._ (2019a) and compare it
with our computational model. As in the experiments, we find that the wild-
type cells are more likely to change direction in the channel. To quantify
this, we measure the contour length of the trajectory normalized by the length
of the micro-channel such that a persistence of unity occurs when the cell
does not change direction in the channel. We find that the vimentin-null cell
line is more persistent in the channels as compared to the wild-type for all
channel widths but more notably different for the smaller channel widths (Fig.
4(d)). We have also studied the persistence as a function of $k_{L}$,
$k_{Nuc}^{Area}$, $F_{a}$, $F_{w}$, and $\epsilon_{Ad}$. See Supplementary
Figs. S6, S7, S8. Given the more asymmetric nuclear positioning in the
vimentin-null, a stronger $F_{a}$ will enhance the tensioning in the leading
half of the cell and, thus, decrease $\Delta\theta$ to enhance the
polarization, for example.

In the wild-type cells, we note that persistence is initially increasing with
increased confinement (Fig. 4(d)). We also see there is a decreased
persistence towards the tighter channels. This behavior tracks the non-
monotonic behavior observed in the magnitude of $\vec{\text{d}}$ as a function
of the channel width. To explain this non-monotonic trend, we turn to the
delicate balance between the stiffness of the cell and strain due to channel
width. In intermediate channel widths, the wild-type spends less time, thus,
has less chance of turning to the other side. The more time wild-type cells
spend in the channels, the greater the chance of moving back towards the
entrance or getting stuck in the channel. In wider channels, cells do not have
enough wall contact but in very narrow channels, stiffness of the cells kicks
in and so the cells spend more time in the channel as they travel more slowly
in terms of speed.

Indeed, our model recapitulates the experiments. More precisely, experiments
found that vimentin-null mEFs were more persistent than wild-type mEFs. For
example, in $10\mu m$ channels about half of the wild-type cells did not cross
the channel, whereas most of the vimentin-null cells passed to the other side
Patteson  _et al._ (2019a). These results are somewhat surprising as the cells
behave opposite on 2D substrates. In 2D, vimentin-deficient cells form
lamellipodia in all directions thereby preventing them from polarizing, which
is not as likely to occur with their wild-type counterparts.

###  III.4 Energy barriers in confined migration

It has been observed that cells tend to migrate in the direction of least
confinement to minimize energetic costs Zanotelli  _et al._ (2019). To test
the notion of confinement as an energetic barrier in cell motility, we compute
the time-averaged energy due to conservative potentials while the cell is in
the channel. See Fig. 5(a). We repeat this measurement for the different
channel widths. See Fig. 5(b). For the two cell lines, the average energy
increases more for the wild-type cells than for the vimentin-null cells as the
confinement increases, as anticipated.

The energy increases as the channel width decreases and so translates to an
increasing energy barrier that the cells must overcome to enter and move in
the microchannel. In Fig. S9, for smaller channel widths, we demonstrate that
channel width correlates linearly with cell strain. In other words, the
channel width is a placeholder for cell strain in the direction perpendicular
to the walls. Since the derivative of the energy with respect to strain, or
channel width, relates to the compressive stress of the cell, the compressive
cellular stress increases faster than linear with decreasing channel width
below some critical channel width (see Fig. 5c). In other words, the cells
exhibit compression stiffening as they enter the channel. Compression
stiffening, a nonlinear rheological property in which a material’s moduli
increase with increasing uniaxial compressive strain, has recently been
discovered in static cells Gandikota  _et al._ (2020). Here, we observe a
dynamic version, if you will. The vimentin-null are more deformable and so
their compression stiffen is less dramatic and the onset occurs at a slightly
higher strain (see Fig. 5c).

What are the implications for compression stiffening? We observe a drop in
flux for the wild-type cells as the channel width decreases. Since flux is a
measure of the fraction of those cells whose center of mass enters the channel
and ultimately emerges out of the other side of the microchannel, the drop in
flux indicates that a large fraction of the wild-type cells enter the
microchannel but then ultimately turn around in the microchannel to come out
the side they entered. This non-linear response of the cells suggests that the
cell pushes against the walls in a non-linear manner to effectively enhance
the adhesion to the wall even more so than a linear material. For the wild-
type cells, the non-linear response is more pronounced. Enhanced adhesion
translates to slower speeds and, hence, more time spent in the microchannel
and so there is a much higher likelihood that the wild-type cells will change
direction in the microchannel to go back to the entrance of the microchannel,
given the polarization mechanism at play. As discussed in the previous
subsection, the persistence of the wild-type cells also decreases a
consequence, again, for the smaller channel widths.

In addition to the energetic barrier, there is also a time scale for entering
the channel that translates into a rate for attempting to hop over the energy
barrier. This attempt rate depends on the polarization of the cell. Prior to
entering the channel, the cell effectively sees a two-dimensional surface. In
this case, the magnitude of $\vec{d}$ is smaller for both cell types, so that
we anticipate more changes in cell direction for both cell types. Since we
initialize the cells to move towards the channel opening, we do not explore
the attempt rate here and leave it for future work.

##  IV Discussion

To focus on vimentin’s role in confined cell motility, we use a computational
model that captures explicitly the roles of actin and vimentin and implicitly
the roles of microtubules as well as the cell nucleus. We investigate the
potential dual role of vimentin: the first being the mechanical protector of
the nucleus and the second being a stress regulator between the cell cortex
and the cell nucleus. For the first role, we modify the stiffness of the
nucleus, and for the second, we modify the stiffness of the mechanical
connections, or linker springs, between the inner and outer cortices. For the
wild-type cells, we find a nonmonotonic dependence of cell speed with channel
width. As the channel width narrows, the cell’s cortex increases its contact
with the wall, which, in turn, generates more driving force to increase cell
speed. Yet, this trend competes with the bulk deformability of the cell via
the linker springs and nucleus to increase the effective adhesion as the
channel width decreases, leading to a slower cell speed. For the vimentin-null
cells, we observe a similar non-monotonic trend, but with a larger cell speed.
Moreover, the maximum cell speed occurs at a narrow channel width, as compared
to wild-type cells, given the enhanced bulk deformability of the cells. Thus,
for nondeformable confinement with simple geometry, vimentin-null cell speed
is typically faster, which is seemingly contrary to the notion that during the
epithelial-to-mesenchymal-transition (EMT), epithelial cells, which are more
stationary, typically upregulate vimentin to be able to move more efficiently
as mesenchymal cells Leggett  _et al._ (2021); Lamouille  _et al._ (2014);
Eriksson  _et al._ (2009).

And yet, there is another ingredient to cell motility beyond the speed, which
is cell direction, or cell polarity. Our results demonstrate that vimentin-
null cells are more polarized in confining microchannels. This trend emerges
because the nucleus is typically located more to the rear of the cell in the
vimentin-null case, which biases the orientation of the longer microtubules,
thereby determining the direction of the leading edge. The enhanced
polarization indicates that the vimentin-null cells are more subjugated to the
confinement since their own internal polarization mechanism that depends on
cross-talk of the centrosome with the nucleus and other cytoskeletal
connections is diminished. In other words, the vimentin-null cells rely more
heavily on external cues, at least in this stiff microchannel environment, and
so are less autonomous. See Fig. 5(e). Finally, since energetic costs are
known to be a predictor of the migration path in confined cell motility, we
find a higher nonlinear energy barrier for wild-type cells entering more
confined channels as compared to the vimentin-null cells, which, again, is
seemingly contrary to the notion of the upregulation of vimentin enhancing
cell motility. Moreover, restructuring of the cytoskeleton occurs on longer
migration timescales to potentially alter the energy barrier.

What do our findings tell us about the interaction between a cell and its
microenvironment more generally? In the absence of vimentin, the cells become
more deformable and so more mechanically sensitive to their microenvironment.
The downregulation of vimentin helps the cell travel more effectively from one
place to another in a confined, straight channel. However, the real tissue
environment, with its interstitial spaces, is more complex. Therefore, for a
cell to upregulate vimentin in order to enhance motility translates to a cell
enhancing its own internal polarization mechanism to effectively search the
microenvironment for a minimal energy barrier by being able to more readily
able to change direction. In other words, with the nucleus less displaced from
the center of the cell, the cell is more capable of altering its own direction
to be more autonomous and get itself out of potential “dead ends” in the
extracellular environment. So cells upregulate vimentin despite the potential
increase in a confinement energy barrier. They also have developed coping
mechanisms, such as Arp2/3 branched actin around the cell nucleus helping it
to squeeze through small pores Thiam  _et al._ (2016) to be able to move
efficiently even with the upregulation of vimentin. Such effects are not
currently accounted for in the model. Additionally, with the upregulation of
vimentin, there is more mechanical cross-talk between the two cortices to
perhaps increase the role of the nucleus itself in regulating cell mechanics.
Recent work suggests that compressed nuclei release calcium into the cytoplasm
to help reconfigure the cytoskeleton Lomakin  _et al._ (2020). Therefore, our
theoretical findings provide a much richer interpretation of how vimentin
affects cell migration with the combination of stress coupler between inner
and outer parts of the cell, nuclear protector, and now polarity regulator.

How does our model compare with other models of confined cell motility? Active
gel models of the cells predict that cells push their way through confined
spaces much like a climber chimneying off a wall Hawkins  _et al._ (2009).
Such a model does not explicitly include a nucleus nor a centrosome such that
cell polarity is an input. Another model based on the molecular-clutch
mechanism explores glial cells moving through microchannels Prahl  _et al._
(2020). Cell polarity is, again, an input. A third model with the complexity
of an actomyosin cortex and a nucleus and couplings in between demonstrates
that there is a nonmonotonic relationship between cell speed and matrix
stiffness in two-dimensions Pathak (2018). On the other hand, with our model,
we have not incorporated actin retrograde flow as the cell is modeled as a
collection of two-body and area springs. Therefore, we do not account for such
effects as actin retrograde flow. Enhanced actin retrograde flow potentially
leads to faster turnover of the actin and so potentially a larger actin
polymerization force as well as a faster time scale for focal adhesion
disassembly that we do not account for in this model.

Cell confinement in this model is explored by constraining the beads/nodes to
stay within a channel geometry. The cell direction of motion is randomly
chosen every so many minutes. While cell speed is studied, cell persistence is
not. Here, cell polarization emerges from an intra-cellular detail rooted in
the position of the centrosome assumed to be in close proximity with the
nucleus. There are indeed additional models, notably, Refs. Brückner  _et al._
(2020); Camley and Rappel (2014). Our model walks the fine line between being
minimal and yet detailed enough to quantify the new functionality of vimentin.
In terms of comparison with two-dimensional cell motility models Danuser  _et
al._ (2013), we anticipate that the polarizability of the cell changes from
our new mechanism to one that depends on the fluctuations in $d$, as opposed
to the average. Moreover, we anticipate the fluctuations in $d$ being larger
in the vimentin-null case, which corresponds to more possible directions in
the cells.

While we have focused on comparison with mEFs, we expect our findings to
generalize to other mesenchymal cell types. Specifically, another cell type
would presumably translate to a different set of parameters. Since our
findings are robust for a range of several parameters, we expect our
conclusions to generalize to other cell types. Given the paradigm of the EMT,
we aim to test our predictions on cell types that do undergo an EMT transition
while also accounting for the typical downregulation of keratin. Since we have
focused here on mesenchymal cells, it would be interesting to generalize our
model to include lamins in non-mesenchymal cells and explore the role of
vimentin in other motility modes of migration Petrie and Yamada (2015). It
would also be interesting to alter the geometry of the microchannel as well as
study multiple cells moving in confinement to determine the robustness of our
findings. Extending our polarization mechanism to include multicellular
interactions will shed more light on the phenomenon of contact inhibition of
locomotion in which motile cells stop moving or change direction upon contact
with another cell.

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##  V Acknowledgements

The authors acknowledge helpful feedback from an anonymous reviewer. JMS
acknowledges NSF-DMR-1832002 and an Isaac Newton Award from the DoD. JMS and
AEP acknowledge a Syracuse University CUSE grant and a Syracuse BioInspired
grant. SG acknowledges a graduate fellowship from Syracuse University.

##  VI Supplementary Figures and Movies

Supplementary Movie 1: Simulation of wild-type MEFs each moving through a
channel of different widths from a range of 8 to 24 microns in increments of 2
microns.
Supplementary Movie 2: Simulation of vimentin-null MEFs each moving through a
channel of different widths from a range of 8 to 24 microns in increments of 2
microns.

Figure S1:  Cell speed as a function of channel width for different actin
forces at the leading edge and actin forces at the wall.’ In (a) and (b), the
magnitude of the actin force at the leading edge $F_{a}$ is fixed and the
magnitude of the actin force at the wall $F_{w}$ is varied for each cell type
and as a function of channel width. In (c) and (d), the magnitude of the actin
force at the leading edge is now varied. Figure S2:  Cell speed as a function
of channel width for different adhesion strengths. (a) Vimentin-null cell (b)
Wild-type cell. Figure S3: Cell Speed as a function of channel width for
different linker and nuclear area spring strengths: (a)-(c) Varying nuclear
area spring strength $K_{nuc}^{area}$ for different linker spring strengths,
$K_{L}$. (d)-(f) Varying linker spring strengths, $K_{L}$, for different
nuclear area spring strengths $K_{nuc}^{area}$. Figure S4: Forces on the
nucleus due to the actomyosin cortex: The net force in the y-direction on the
top (upper) half of the nucleus due to the leading half of the actomyosin
cortex and the net force in the y-direction on the bottom (lower) half of the
nucleus due to the rear half of the actomyosin cortex as a function of channel
width for each cell type.  Figure S5: Angular information.(a) Probability
distribution function of the angle between x-axis and the Nucleus-axis for
both cell types. (b) Probability distribution function of the angle choosen by
the cell for a $10\,\mu\text{m}$ channel width. We are measuring all angles
with respect to the $x$-axis with the nucleus center as the origin and
$90^{\circ}$ as the direction towards exit from other side and $270^{\circ}$
is the direction towards the designated entrance of the channel. This figure
shows multiple peaks, but the highest ones for both cell lines are around
$90^{\circ}$ (exit direction or the other end of the channel) with smaller
peaks at $270^{\circ}$ as well, which represent the cells turning around
towards channel’s entry. We observe that for vimentin-null cells, the peak
around $90^{\circ}$ is higher than wild-type, which implies that the vimentin-
null cells are more persistent (not turning around in the channel) than the
wild-type cells. Figure S6: Persistence as a function of channel width for
different linker and nuclear area spring strengths: (a)-(c) Varying nuclear
area spring strength $K_{nuc}^{area}$ for different linker spring strengths,
$K_{L}$. (d)-(f) Varying linker spring strengths, $K_{L}$, for different
nuclear area spring strengths $K_{nuc}^{area}$. Figure S7:  Persistence as a
function of channel width for different actin forces at the leading edge and
actin forces at the wall.’ In (a) and (b), the magnitude of the actin force at
the leading edge $F_{a}$ is fixed and the magnitude of the actin force at the
wall $F_{w}$ is varied for each cell type and as a function of channel width.
In (c) and (d), the magnitude of the actin force at the leading edge is now
varied. Figure S8:  Persistence as a function of channel width for different
adhesion strengths. (a) Vimentin-null cell (b) Wild-type cell. Figure S9:
Strain on cell. Compressive strain on the cell versus channel width for both
cell types.  Figure S10: Nucleus circularity as a function of channel width
for different linker and nuclear area spring strengths: (a)-(c) Varying
nuclear area spring strength $K_{nuc}^{area}$ for different linker spring
strengths, $K_{L}$. (d)-(f) Varying linker spring strengths, $K_{L}$, for
different nuclear area spring strengths $K_{nuc}^{area}$.