# Cell crawling on a compliant substrate:
a biphasic relation with linear friction

H. Chelly  A. Jahangiri  M. Mireux  J. Étienne jocelyn.etienne@univ-grenoble-
alpes.fr D.K. Dysthe<EMAIL_ADDRESS>C. Verdier claude.verdier@univ-
grenoble-alpes.fr P. Recho<EMAIL_ADDRESS>Université
Grenoble Alpes, Laboratoire Interdisciplinaire de Physique, CNRS, F-38000
Grenoble, France  University of Oslo, Department of Physics, Oslo, Norway

###### Abstract

A living cell actively generates traction forces on its environment with its
actin cytoskeleton. These forces deform the cell elastic substrate which, in
turn, affects the traction forces exerted by the cell and can consequently
modify the cell dynamics. By considering a cell constrained to move along a
one-dimensional thin track, we take advantage of the problem geometry to
explicitly derive the effective law that describes the non-local frictional
contact between the cell and the deformable substrate. We then couple such a
law with one of the simplest model of the active flow within the cell
cytoskeleton. This offers a paradigm that does not invoke any local non-linear
friction law to explain that the relation between the cell steady state
velocity and the substrate elasticity is non linear as experimentally
observed. Additionally, we present an experimental platform to test our
theoretical predictions. While our efforts are still not conclusive in this
respect as more cell types need to be investigated, our analysis of the
coupling between the substrate displacement and the actin flow leads to
friction coefficient estimates that are in-line with some previously reported
results.

###### keywords:

cell motility adhesion biphasic relation traction forces actin retrograde flow
elastic substrate

##  1 Introduction

The contact between a living cell and its substrate is mediated by focal
adhesion (FAs) linking the cell cytoskeleton and its actin meshwork to the
substrate. FA are constituted by a myriad of different proteins interacting
with each others in order to integrate extracellular chemical (e.g. cytokines)
and mechanical (e.g. substrate rigidity, external forces) cues (Geiger et al.,
2009). The cell then self-adjusts according to these cues by, for instance,
migrating from zones of lower substrate rigidity to zones of higher substrate
rigidity following a process called durotaxis (Lo et al., 2000). One of the
key components of this sensing machinery is the integrin, a transmembrane
protein which can bind actin filaments with substrate–coating proteins, such
as fibronectin. The integrin works as a bidirectional link as it is used as a
mechanical sensor of the substrate but also to transmit forces generated by
the active flow of the actin meshwork. In response, the binding and unbinding
of integrins creates a feedback on the actin flow. See Fig. 1.

Minimal agent-based models of sliding friction have been explored by
Srinivasan and Walcott (2009); Li et al. (2010); Sabass and Schwarz (2010);
Sens (2013) to investigate the interaction between an actin filament and
stochastic bonds, based on the Lacker-Peskin model (Lacker and Peskin, 1986).
The results show a biphasic (i.e. bell-shape) relationship between the cell
traction force and the actin velocity: at sufficiently small actin velocity,
the loading in the bonds increases slowly, which leads to a linear increase of
the traction force with the actin velocity. For higher actin velocity, the
loading rate in the bonds is high, therefore they break very rapidly and the
traction force decreases with the actin velocity. Using this microscopic
approach, Sens (2020) formulated a simple cell model consisting of one
stochastic adhesion module linked by a spring at both ends. This local non-
linear dependence of the traction force on the actin velocity has been used to
explain the well-established experimental observation that the global cell
velocity depends on the fibronectin ligand density in a biphasic way DiMilla
et al. (1991); Palecek et al. (1997). Similarly, starting from the general
trend that stationary cells generate large traction forces while showing a
slow actin flow and on the contrary motile cells generate small traction
forces with a fast retrograde flow, Barnhart et al. (2015) deduced that the
frictional slippage is weaker at the rear of a motile cell than in stationary
cells, and thus introduced a non-linear actin flow-dependent friction
coefficient to ensure a decreased cell-substrate coupling above a critical
actin flow velocity.

Nevertheless, the experimental observation (Schwarz and Gardel, 2012) that,
below a velocity threshold, the transmitted traction force evolves linearly
with the actin velocity, has led to disregard this complexity and consider the
simple viscous friction law $\bm{T}=\xi\bm{v}$, where $\bm{T}$ represents the
traction force exerted by the cell on the substrate, $\bm{v}$ is the actin
flow velocity and $\xi$ an effective viscous friction coefficient (see (Recho
et al., 2015) and references therein for a detailed discussion concerning this
reduction). In order to account for a non-uniform distribution of the adhesion
bonds under the cell (higher traction in the vicinity of the cell boundary), a
space dependent friction coefficient has been introduced in several studies
(Rubinstein et al., 2009; Mogilner and Verzi, 2003; Roux et al., 2016).

But the linear viscous friction model formulated above is only suited for
infinitely rigid substrates which do not deform under cell traction forces.
While this situation is standard _in vitro_ , substrates with a compliance
that matches the one of the cell are commonly encountered _in vivo_ and the
traction forces exerted by the cell can induce a motion of the substrate. In
order to generalize the model to the case of a deformable substrate, a simple
approach is to consider the difference between the actin velocity $\bm{v}$ and
the substrate velocity $\bm{v^{s}}$ in the friction law, such that
$\bm{T}=\xi(\bm{v}-\bm{v^{s}})$. Wong and Tang (2011) implemented this
friction law with a space-dependent friction coefficient to model the
interaction between a hyperelastic cell and a hyperelastic substrate. Hassan
et al. (2019) also used this law combined with a phase-field model for the
cell, where a migration directional bias was introduced in order to promote
cell migration in the direction of increasing substrate rigidity. More
recently, Zhang et al. (2020) used a similar approach with an actin
polymerization bias towards substrate tensile regions (resulting from external
forces or cell traction forces) to promote cell migration in these directions.
From another perspective which does not explicitly solve force balance within
the cell, Ziebert and Aranson (2013); Löber et al. (2014) coupled a phase-
field framework (Biben and Misbah, 2003), with reaction-diffusion adhesion
dynamics involving a substrate displacement-dependent detachment to ensure the
gripping (resp. slipping) below (resp. above) a substrate displacement
critical value. While Ziebert and Aranson (2013) used a single global spring
to model the elastic substrate in the fashion of Chan and Odde (2008), Löber
et al. (2014) improved this previous model by locally resolving the
viscoelastic incompressible substrate displacement. In these models a local
traction force exerted by a cell generates a non-local displacement of the
substrate under the cell and therefore induces a local reorganization of the
cell cytoskeleton which ultimately impacts its motility dynamics in a non
trivial way.

In the present paper, by extending one of the simplest model of the
cytoskeleton actin turnover driving cell crawling (Kruse et al., 2006;
Jülicher et al., 2007) to the case of a deformable elastic substrate, we show
that the account of the substrate displacement is sufficient to explain a
biphasic relation relating the cell velocity to the substrate stiffness
(Stroka and Aranda-Espinoza, 2009; Peyton and Putnam, 2005). Therefore it is
not necessary to invoke a non-linear dependence of the traction force on the
actin velocity as previously discussed or indeed a direct dependence of the
active force production machinery on the substrate stiffness (Dokukina and
Gracheva, 2010; Sarvestani, 2011) to reach this conclusion.

In order to understand the influence of the substrate rigidity on cell
motility from a purely mechanical standpoint, we need a relation linking the
flow of the cell cytoskeleton to substrate deformation. For simplicity we
assume a semi-infinite elastic substrate, implying that the thickness of the
extra-cellular matrix (ECM) substrate is large compared to the cell size as
opposed to the situation studied in (Nicolas and Safran, 2004) where the ECM
and adhesion plane form a thin film. The two opposed limits (thick or thin
ECM) have been further studied in (Nicolas and Safran, 2006) in the context of
the focal adhesion size regulation. In this semi-infinite case, the non-local
response of the compliant substrate to a local traction force within the small
deformation framework, is given by the Cerruti-Boussinesq solution in the
three dimensional case. However if the geometry of the contact is invariant in
one direction, a plane strain assumption can apply, leading to a Flamant
problem (Johnson, 1987). Both strategies have been employed to characterize
the contact between the cell and the substrate. The plane strain assumption
has been made by Qian et al. (2008) in order to simulate the detachment
process of a focal adhesion and estimate its lifetime, and by Lelidis and
Joanny (2013) to investigate the effect of substrate rigidity on cell motility
with uniformly distributed discrete focal adhesions. The plane strain
assumption represents a simplification valid only if the substrate deformation
orthogonal to the principal loading direction is negligible. Besides, the
substrate displacement found within this framework is not bounded at infinity,
thus an arbitrary length needs to be introduced, above which the displacement
vanishes. These issues do not arise when considering the three dimensional
problem. Traction Force Microscopy (TFM), the method to evaluate the cell
traction forces from a measured substrate displacement field, requires the
resolution of an inverse problem based on the Boussinesq-Cerruti solution
(Dembo et al., 1996; Ambrosi, 2006; Michel et al., 2013; Sabass et al., 2008).
In the present work we propose an approach to obtain a simple relationship
between the actin velocity and the substrate displacement under the
simplifying hypotheses of a one-dimensional cell crawling on a semi-infinite
incompressible elastic substrate. This model corresponds to the situation
where the cell is constrained to move on a thin adhesion track.

The paper is organized as follows. In sec. 2, we build the mathematical model
corresponding to our geometric assumptions by constructing the kernel
governing the response of the semi-infinite elastic substrate to the cell
traction force field. We operate under the hypotheses that the friction with
the substrate is linear, and that the substrate is a linear elastic medium.
Next, in sec. 3, we couple this model of the contact with one of the simplest
models of actin turnover-based cell motility to obtain a coupled system
relating the actin flow field, global cell velocity and substrate
displacement. We then study traveling waves (TW) solutions of this problem in
sec. 4 and give explicit solutions for some special cases of the actin
turnover dynamics as well as asymptotic solutions when the substrate is
infinitely hard (cell tractions negligible compared to the substrate
stiffness) or soft (cell tractions large compared to the substrate stiffness).
Then in sec. 5, we obtain numerically a biphasic behavior of the steady state
velocity as a function of either the substrate rigidity or the cell–substrate
friction coefficient. In sec. 6, we propose an experimental setup to validate
this theory, using T24 bladder cancer cells migrating on substrates of
different rigidities (5 kPa, 8 kPa and 28 kPa), confined along thin
fibronectin–coated tracks. The actin cytoskeleton flow velocity and the
substrate displacement are measured and projected along the cell major axis.
Although we were not able to obtain sufficiently motile cells to interrogate
our theoretical model in detail, we could estimate the friction coefficient
for static cells on substrates of different rigidities. The experimental setup
does contain some limitations to address in the future, including the lack of
accuracy of the actin acquisition and the need to build thinner cell tracks to
improve the applicability of the 1D hypothesis but the values obtained for the
friction coefficient are already consistent with the literature describing
cell crawling on a rigid substrate.

Figure 1: Composition of a focal adhesion (adapted from Elosegui-Artola et al.
(2018)). The force transmission from the cell to the substrate is ensured by
the transmembrane proteins, integrins, connecting the actin cytoskeleton to
protein receptors (e.g. fibronectin) on the substrate. The generated traction
force triggers a positive feedback loop, by inducing an increase of integrins
recruitement into the FA, which in return increases the traction. The actin
retrograde flow induced by polymerization and cell contractility, is slowed
down by the continuous de-/attachment of integrins to the substrate.

##  2 Model of the contact between the cell and the substrate

We consider a cell moving on a semi-infinite elastic substrate and restrict
our analysis to small deformations of the substrate, neglecting both physical
and geometrical non-linearities of the elastic problem. In particular, we
shall not differentiate the Eulerian and Lagrangian frames in our approach.
The orthonormal frame of reference $(\bm{e_{x}},\bm{e_{y}},\bm{e_{z}})$ is
chosen such that the substrate is semi-infinite opposite to the $\bm{e_{z}}$
direction and material points are labeled by the spatial coordinate
$\bm{r}=(x,y,z)$. We consider the case of a cell confined to move on a
straight track of width $\delta$ oriented along the $x-$direction (See Fig.
2). The cell length $L(t)$ is defined by $L(t)=l_{+}(t)-l_{-}(t)$, where
$l_{\pm}(t)$ denotes the cell fronts, and the geometric center is
$C(t)=(l_{+}(t)+l_{-}(t))/2$. Considering that the track constrains the cell
so that it is very thin compared to its length ($\delta\ll L$), we assume that
it only exerts traction forces tangentially to the substrate

$\bm{T}(\bm{r},t)=T_{x}(x,t)\,\bm{e_{x}}+T_{y}(x,t)\,\bm{e_{y}},$ (1)

where $x\in[l_{-}(t),l_{+}(t)]$. Taking advantage of the fact that the track
is thin compared to the cell length, we have supposed that the cell fronts are
straight and orthogonal to the track, and that the traction forces only depend
on the $x$-coordinate along the track.

Defining the Boussinesq-Green kernel for a semi-infinite incompressible
elastic medium of Young modulus $E_{s}$ (Landau and Lifshitz, 1959)

$\mathcal{G}(x,y,z)=\frac{3}{4(x^{2}+y^{2}+z^{2})^{3/2}\pi
E_{s}}\left(\begin{array}[]{ccc}2x^{2}+y^{2}+z^{2}&xy&xz\\\
xy&x^{2}+2y^{2}+z^{2}&yz\\\ xz&yz&x^{2}+y^{2}+2z^{2}\\\ \end{array}\right),$

we can express the displacement of the substrate $\bm{u}$ due to the traction
forces as

$\displaystyle\bm{u}(x,y,z,t)$
$\displaystyle=\int_{l_{-}(t)}^{l_{+}(t)}\int_{-\delta/2}^{\delta/2}\frac{3/(4\pi
E_{s})}{((x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2})^{3/2}}$
$\displaystyle\left(\begin{array}[]{c}T_{y}(x^{\prime},t)(x-x^{\prime})(y-y^{\prime})+T_{x}(x^{\prime},t)((2(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2})\\\
T_{x}(x^{\prime},t)(y-y^{\prime})(x-x^{\prime})+T_{y}(x^{\prime},t)((x-x^{\prime})^{2}+2(y-y^{\prime})^{2}+z^{2})\\\
(T_{x}(x^{\prime},t)(x-x^{\prime})+T_{y}(x^{\prime},t)(y-y^{\prime}))z\\\
\end{array}\right)\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}.$ (5)

Averaging the displacement over the $y$-direction leads to the following
simplification

$\displaystyle\bar{\bm{u}}(x,z,t)=\int_{l_{-}(t)}^{l_{+}(t)}\int_{-\delta/2}^{\delta/2}\int_{-\delta/2}^{\delta/2}\frac{3/(4\pi
E_{s})}{\delta((x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2})^{3/2}}\left(\begin{array}[]{c}T_{x}(x^{\prime},t)\left(2(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2}\right)\\\
T_{y}(x^{\prime},t)\left((x-x^{\prime})^{2}+2(y-y^{\prime})^{2}+z^{2}\right)\\\
T_{x}(x^{\prime},t)(x-x^{\prime})z\\\
\end{array}\right)\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\mathrm{d}y.$ (9)

Next, by taking the value of the displacement at $z=0$, we find

$\displaystyle\bm{u^{s}}(x,t)=\int_{l_{-}(t)}^{l_{+}(t)}\int_{-\delta/2}^{\delta/2}\int_{-\delta/2}^{\delta/2}\frac{3/(4\pi
E_{s})}{\delta((x-x^{\prime})^{2}+(y-y^{\prime})^{2})^{3/2}}\left(\begin{array}[]{c}T_{x}(x^{\prime},t)\left(2(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\right)\\\
T_{y}(x^{\prime},t)\left((x-x^{\prime})^{2}+2(y-y^{\prime})^{2}\right)\\\ 0\\\
\end{array}\right)\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\mathrm{d}y,$ (13)

which shows that the surface remains flat during the motion. As we shall only
investigate the force balance along the track, we project $\bm{u^{s}}(x,t)$ in
the $x$-direction and performing the integrals in $y$, we finally obtain

$u^{s}_{x}(x,t)=\int_{l_{-}(t)}^{l_{+}(t)}\phi\left(\frac{x^{\prime}-x}{\delta}\right)T_{x}(x^{\prime},t)\mathrm{d}x^{\prime},$
(14)

where

$\phi(x)=\frac{3\mathrm{log}\left(\frac{1+\sqrt{1+x^{2}}}{|x|}\right)}{2\pi
E_{s}}=\frac{3\text{arcsch}\left(|x|\right)}{{2\pi E_{s}}}.$ (15)

and arcsch denotes the inverse hyperbolic cosecant.

Figure 2:  Scheme of the model of a cell crawling on a semi-infinite
incompressible elastic substrate. The thin adhesion track (in red) of width
$\delta\ll L$ allows a one dimensional motion of the cell along the $x-$axis.
The cell (in green), extending from $l_{-}$ to $l_{+}$ exerts traction forces
$T_{x}$ on the substrate, inducing a displacement $u^{s}_{x}$ of the substrate
at the surface.

Note that, in the above formula, when $x\ll 1$, $\phi$ simplifies to the so-
called plane strain kernel $\phi_{0}(x)=-3\log(|x|/2)/(2\pi E_{s})$
(Timoshenko and Goodier, 1970) while it leads to the plane stress kernel
$\phi_{\infty}(x)=3/(2|x|\pi E_{s})$ in the opposite limit where $x\gg 1$
(Johnson, 1987), see Fig. 3. Although $\phi_{0}$ is singular at $x=0$, its
integral exists in the sense of Cauchy principal value. In contrast,
$\phi_{\infty}$ is singular and not integrable at $x=0$, which would lead to
an infinite displacement at $x=0$. On the other hand, while $\phi_{\infty}$
tends to zero at infinity, $\phi_{0}$ is unbounded at $x=\infty$, which led
Timoshenko and Goodier (1970) to introduce an arbitrary cut-off length
$x_{\infty}$ at which the displacement vanishes ($u^{s}_{x}(x_{\infty})=0$) to
regularize such situation. The kernel $\phi$ we obtain from the Boussinesq-
Cerruti solution within the thin track framework, encompasses the advantages
of both $\phi_{0}$ and $\phi_{\infty}$, namely the integrability at $x=0$ and
the vanishing substrate displacement at $x=\infty$, because $\phi$ behaves
like $\phi_{0}$ near the singularity $x=0$ and like $\phi_{\infty}$ far from
the origin.

Figure 3:  The non-local interaction kernel $\phi$ and the kernels $\phi_{0}$
and $\phi_{\infty}$ which represent the two limiting behaviors of plane strain
and plane stress, respectively. Parameter $E_{s}=1$.

We choose to model the traction force exerted by the cell using an effective
viscous friction coefficient:

$T_{x}(x,t)=\xi(v_{x}(x,t)-v_{x}^{s}(x,t)),$ (16)

where $\xi$ is an effective viscous friction coefficient that assumes a
sufficiently fast attachment/detachment of the integrins, $v_{x}$ is the actin
retrograde flow velocity and $v_{x}^{s}=\partial_{t}u^{s}_{x}(x,t)$ denotes
the velocity of the substrate. Such relation takes into account the cell’s
internal activity via the actin retrograde flow, while the influence of the
compliance of the substrate is encompassed in the substrate velocity.

In order to eliminate the time-dependence of the integral boundaries, we re-
map the problem using the change of variables $\bar{x}=2(x-C(t))/L(t)$ and
$\bar{t}=t$, which leads to the partial derivative relations:

$\partial_{x}(.)=\frac{2}{L}\partial_{\bar{x}}(.)\text{ and
}\partial_{t}(.)\overset{\text{def}}{=}\frac{D(.)}{D\bar{t}}=\partial_{\bar{t}}(.)-\frac{2}{L}\left(\dot{C}+\frac{\dot{L}}{2}\bar{x}\right)\partial_{\bar{x}}(.),$

where the superimposed dot denotes the time derivatives. Introducing the cell
aspect ratio $\epsilon(t)=2\delta/L(t)$ yields

$u^{s}_{x}(\bar{x},t)=\frac{L(t)}{2}\int_{-1}^{1}\phi\left(\frac{\bar{x}^{\prime}-\bar{x}}{\epsilon(t)}\right)T_{x}(\bar{x}^{\prime},t)\mathrm{d}\bar{x}^{\prime},$
(17)

and injecting (16) into (17) gives

$u^{s}_{x}(x,t)+\frac{\xi
L(t)}{2}\int_{-1}^{1}\phi\left(\frac{x^{\prime}-x}{\epsilon(t)}\right)\frac{Du^{s}_{x}(x^{\prime},t)}{Dt}\mathrm{d}x^{\prime}=\frac{\xi
L(t)}{2}\int_{-1}^{1}\phi\left(\frac{x^{\prime}-x}{\epsilon(t)}\right)v_{x}(x^{\prime},t)\mathrm{d}x^{\prime},$
(18)

where for simplicity of the notations and from now on, we shall use the same
symbol to denote the re-scaled variables, i.e. $\bar{x}:=x$ and $\bar{t}:=t$.

Assuming the actin retrograde flow velocity $v_{x}$ is known, (18) is a
singular integro-differential equation. In the case of an infinite domain of
integration, explicitly solving this type of equation is generally performed
by switching to the Fourier domain. In our case where the domain of
integration is a finite interval, the functional basis that diagonalizes the
symmetric operator $\phi$ is not explicit as in the case of the plane strain
kernel $\phi_{0}$ (Boyd, 2001; Canuto et al., 2012), making the expression of
$u^{s}_{x}$ as a function of $v_{x}$ non-transparent. Note that while the
integration of $x^{\prime}$ is performed over the finite segment $[-1,1]$, $x$
takes value on the whole real line such that the boundary conditions
associated to the convective time derivative $D/Dt$ are the canonical ones
($u^{s}_{x}$ and all its derivatives tend to zero at infinity).

##  3 Coupling with a simple model of protrusion based motility

One of the simplest theoretical models of cell motility that can be coupled
with the traction force transmission model (17) is based on the turnover of
actin pushing the cell boundaries (Kruse et al., 2006; Jülicher et al., 2007).
Modeling the cell moving along the one-dimensional track as a viscous gel with
a fixed length and uniform contractile properties, we write its constitutive
behavior as

$\sigma=\frac{2\eta}{L}\partial_{x}v_{x}+\chi c_{0},$ (19)

where, $x\in[-1,1]$ is the rescaled variable introduced above, $\sigma$ is the
axial stress in the cell actin meshwork, $\eta$ is the viscosity of the
meshwork and $\chi c_{0}\geq 0$ is the uniform contraction due to the
molecular motors (which we assume here to have a fixed concentration $c_{0}$).
Additionally, denoting $h$ the cell height and assuming a thin film
approximation $h\ll L$ (Roux et al., 2016), the force balance equation within
the cell reads,

$\frac{2h}{L}\partial_{x}\sigma=\xi\left(v_{x}-\frac{Du^{s}_{x}(x,t)}{Dt}\right).$
(20)

This equation is associated with stress boundary conditions at the cell moving
edges such that

$\sigma|_{\pm 1}=\sigma_{b},$

where $\sigma_{b}$ is an unknown residual stress representing the constraint
fixing the cell length and imposing that the two cell fronts move with the
same velocity $V(t)=\dot{l}_{-}=\dot{l}_{+}$. This boundary condition
typically emerges as a limit when an effective spring that connects the two
fronts has a stiffness tending to infinity while $L(t)$ tends to the rest
length of the spring. The unknown residual stress $\sigma_{b}$ results from
this double limit. See (Putelat et al., 2018) for more details. Taken together
with (19), this constraint on the stress leads to

$\partial_{x}v_{x}|_{-1}=\partial_{x}v_{x}|_{1}.$

Finally, the protrusion and retraction of the moving fronts are given by the
Stefan boundary conditions

$\dot{l}_{\pm}=v_{x}|_{\pm 1}+v_{\pm},$ (21)

where $v_{\pm}$ are the given polymerization and depolymerization velocities
at the leading edge ($l_{+}$) and trailing edge ($l_{-}$). See Recho and
Truskinovsky (2013) for details We introduce the quantities $\Delta
V=v_{+}-v_{-}$ representing the mismatch between polymerization and
depolymerization and $V_{m}=(v_{+}+v_{-})/2$ representing the average
turnover. More realistic albeit more complex models of actin protrusion and
retraction of the cell fronts can be found in Ambrosi and Zanzottera (2016);
Giverso and Preziosi (2018). A first mechanism at the origin of the motility
in the model described above is the accretion of actin at the leading edge
exactly compensated by the removal of actin at the trailing edge. This leads
to a propulsion at velocity $V_{m}$ which is independent of both the substrate
and the cell constitutive behavior. Added to this mechanism, the mismatch
between polymerization and depolymerization with a preserved length $L$ of the
cell implies a certain flow in the cell, exerting traction forces on the
substrate proportional to $\Delta V$. This second contribution depends on the
substrate stiffness as well as on the friction coefficient with the substrate
in a non-trivial manner that we seek to characterize.

We non-dimensionalize the time by $\eta/(\chi c_{0})$, the distance by $L/2$
and the stress by $\chi c_{0}$. Combining (18) with (19) and (20), and
denoting $u=u_{x}^{s}$ and $v=v_{x}-v_{0}$, where

$v_{0}(x)=-\Delta V\frac{\sinh(\alpha^{-1}x)}{2\sinh(\alpha^{-1})}$ (22)

is a Dirichlet lift function that accounts for the asymmetry in the
protrusion/retraction kinetics at the boundaries, we obtain the non-
dimensional problem

$v(x,t)=\frac{1}{\alpha}\int_{-1}^{1}G\left(\frac{x-x^{\prime}}{\alpha},\alpha\right)\frac{Du(x^{\prime},t)}{Dt}\mathrm{d}x^{\prime}\qquad\text{
and }\qquad
u(x,t)=\gamma\int_{-1}^{1}\Phi\left(\frac{x-x^{\prime}}{\epsilon}\right)\partial_{x^{\prime}x^{\prime}}[v(x^{\prime},t)+v_{0}(x^{\prime})]\mathrm{d}x^{\prime}.$
(23)

In (23), $v(x,t)$ is only defined for values of $x\in[-1,1]$ (i.e. within the
cell) while $u(x,t)$ is defined on the whole real line ($-\infty<x<\infty$).
The three non-dimensional parameters

$\epsilon=\frac{2\delta}{L}\text{, }\alpha=\sqrt{\frac{4\eta h}{\xi
L^{2}}}\text{ and }\gamma=\frac{3h\chi c_{0}}{\pi LE_{s}}.$

respectively represent the track aspect ratio, the ratio of the hydrodynamic
length and the cell length and the substrate compliance compared to the cell
contractile stress. For simplicity, we keep the same notations for the non-
dimensional (de)polymerization velocities $v_{\pm}$ and their dimensional
counterparts. The non-dimensional symmetric kernels representing the non-local
behavior of the active viscous cytoskeleton and the elastic substrate
respectively read:

$G\left(x,\alpha\right)=\frac{\cosh(\alpha^{-1}+x)}{2\sinh(\alpha^{-1})}-\text{H}(x)\sinh(x)\qquad\text{
and }\qquad\Phi(x)=\text{arcsch}\left(|x|\right),$

where H is the Heaviside function. Thus, $G$ corresponds to the resolvent of
the elliptic problem (Recho et al., 2015)

$-\alpha^{2}\partial_{xx}v+v=\frac{Du}{Dt},$

with periodic boundary conditions on $v$:

$\partial_{x}v|_{-1}=\partial_{x}v|_{1}\text{ and }v|_{-1}=v|_{1}.$ (24)

We remark that as $\dot{L}=0$ and $u$ is rescaled by $L/2$, the convective
derivative in (23) takes the simple form

$\frac{Du}{Dt}=\partial_{t}u-V\partial_{x}u.$

After $v$ and $u$ are obtained from (23), the dynamics of the moving fronts
can be found from (21) which leads to

$V(t)=V_{m}+\frac{v|_{-1}+v|_{1}}{2}$ (25)

We provide in Table 1 some rough estimates of the various rheological
coefficients entering the model. One should however bear in mind that these
coefficients can vary over several orders of magnitude depending on the
biological conditions and these values should therefore be taken with care.
The interest of the type of reduced model that we present rather lies in
capturing some physical effects with a minimal baggage than in describing cell
crawling at a quantitative level.

name | symbol | typical value
---|---|---
cytoskeleton viscosity | $\eta$ |  $100$ kPa s (Jülicher et al., 2007; Rubinstein et al., 2009)
contractility | $\chi c_{0}$ |  $1$ kPa (Jülicher et al., 2007; Rubinstein et al., 2009)
friction coefficient | $\xi$ |  $10$ kPa s $\mu\text{m}^{-1}$ (Kruse et al., 2006; Barnhart et al., 2011)
lamelipod height | $h$ |  $1$ $\mu$m (Kruse et al., 2006)
cell length | $L$ |  $50$ $\mu$m
(de)polymerization velocities | $v_{\pm}$ |  $0.2$ $\mu\text{m\,s}^{-1}$ (Larripa and Mogilner, 2006; Kruse et al., 2006)
substrate stiffness | $E_{s}$ |  $10$ kPa
track width | $\delta$ |  $5$ $\mu\text{m}$
characteristic length | $L/2$ |  $25$ $\mu$m
characteristic time | $\eta/(\chi c_{0})$ |  $100$ s
characteristic velocity | $\chi c_{0}L/(2\eta)$ |  $0.25$ $\mu\text{m}\,\text{s}^{-1}$
characteristic stress | $\chi c_{0}$ |  $1$ kPa
track slenderness | $\epsilon=2\delta/L$ | 0.2
hydrodynamic coefficient | $\alpha=\sqrt{4\eta h/(\xi L^{2})}$ | 0.1
substrate softness | $\gamma=3h\chi c_{0}/(\pi LE_{s})$ | 0.02
turnover asymmetry | $\Delta V=2\eta(v_{+}-v_{-})/(\chi c_{0}L)$ | 1
turnover average | $V_{m}=\eta(v_{+}+v_{-})/(\chi c_{0}L)$ | 1
Table 1: Rough estimates of the material coefficients, characteristic scales
and dimensionless parameters definitions. Some parameters are subjected to
several orders of magnitude variations.

##  4 Traveling wave solutions

We now seek for traveling wave solutions of (23)-(25) where $V$ is a constant
and $\partial_{t}u=0$, implying that $Du/Dt=-V\partial_{x}u$. In such a case,
(23) can be combined into the following single integral equation:

$\alpha^{2}f(x)-\gamma
V\int_{-1}^{1}\frac{f(x^{\prime})}{x^{\prime}-x}\mathrm{d}x^{\prime}-\int_{-1}^{1}R(x,x^{\prime})f(x^{\prime})\mathrm{d}x^{\prime}=-\frac{\Delta
V}{2}x+V-V_{m}$ (26)

where
$f(x)=\partial_{xx}(v(x)+v_{0}(x))=(v(x)+v_{0}(x)+V\partial_{x}u)/\alpha^{2}$
represents the substrate traction force and

$R(x,x^{\prime})=\gamma
V\underset{=R_{1}(x,x^{\prime})}{\underbrace{\frac{(x-x^{\prime})/\epsilon^{2}}{1+(x-x^{\prime})^{2}/\epsilon^{2}+\sqrt{1+(x-x^{\prime})^{2}/\epsilon^{2}}}}}+\underset{=R_{2}(x,x^{\prime})}{\underbrace{H(x-x^{\prime})(x-x^{\prime})+\frac{x+1}{2}x^{\prime}}}$

is a kernel that does not contain any singularity. The $R_{1}$ part is
reminiscent of the elastic interaction with the substrate (regular part) and
the $R_{2}$ contribution represents a second order antiderivative with
appropriate boundary conditions accounted for by the righthandside of (26).
The unknown crawling velocity $V$ in (26) is fixed by the global force balance
constraint:

$\int_{-1}^{1}f(x)\mathrm{d}x=0.$ (27)

#### Solutions for specific choices of the turnover dynamics

A simple solution of (26) can be found when $v_{+}=-v_{-}$ when the two fronts
are symmetrically polarized in opposite directions leading to $V_{m}=0$. In
this case, we can set $V=0$ and obtain $f(x)=f_{0}(x)=-\Delta
V\sinh(\alpha^{-1}x)/(2\alpha^{2}\sinh(\alpha^{-1}))$. This corresponds to
$v=0$ (i.e. $v_{x}(x)=v_{0}(x)$) and

$u(x)=\gamma
u_{0}(x)=\gamma\int_{-1}^{1}\Phi\left(\frac{x-x^{\prime}}{\epsilon}\right)\partial_{x^{\prime}x^{\prime}}v_{0}(x^{\prime})\mathrm{d}x^{\prime}.$

We show in Fig.4 the typical trend of such substrate displacement that is
induced by a traction force distribution with an even symmetry with respect to
the layer center.

Figure 4:  Displacement $u_{0}$ of the substrate induced by the actin velocity
field $v_{x}=v_{0}$ corresponding to a static configuration. The dashed lines
indicate the normalized cell fronts. Parameters are $\epsilon=0.2$ and $\Delta
V=1$.

The substrate displacement reaches its (anti-symmetric) maxima close to the
cell boundaries and sharply decays to zero outside of the cell as no traction
forces are imposed in this region. Another simple case is when $v_{+}=v_{-}$,
leading to $f=0$. Thus $v=0$, $u=0$ and $V=V_{m}$. This corresponds to the
trivial situation where the actin protrusion and retraction are happening at
the same rate at each cell edge, corresponding to a pure treadmilling movement
without any internal flow of the filamentous actin. As a result, there are no
traction forces and no substrate deformation. This specific situation is
rather unrealistic for most cell types as traction forces are known to be
applied on the substrate during motion.

More generally, we are interested in the dependence of $V$ on $\gamma$ for an
arbitrary choice of $V_{m}$ and $\Delta V$. For this, we start by analytically
analyzing the asymptotic behavior of $V$ when $\gamma\ll 1$.

#### Solutions in the limit of underformable substrate

When $\gamma=0$ (i.e. the substrate is undeformable), the solution of (26) is
again $f(x)=f_{0}(x)$ corresponding to $v=0$, $u=0$ and $V=V_{m}$. This is the
classical case investigated by Jülicher et al. (2007) in the case of an
infinitely stiff substrate and generalized by Recho and Truskinovsky (2013) in
the presence of external loading. We can further expand the solution of
(23)-(25) at first order when $\gamma\ll 1$

$v(x)\simeq\gamma v_{1}(x)\text{ and }V\simeq V_{m}+\gamma V_{1}.$

Note that the domain of accuracy of such approximation outside of the limit
$\gamma\ll 1$ should however be investigated numerically. The validity of such
approximation clearly also relies on the fact that $V_{1}$ remains finite,
such that $\gamma V_{1}$ is indeed small compared to $V_{m}$.

From (23), we obtain,

$v_{1}(x)=-\frac{V_{m}}{\alpha}\int_{-1}^{1}G\left(\frac{x-x^{\prime}}{\alpha},\alpha\right)\partial_{x^{\prime}}u_{0}(x^{\prime})\mathrm{d}x^{\prime}$

and thus,

$V_{1}=-\frac{V_{m}}{\alpha}\int_{-1}^{1}G\left(\frac{1-x^{\prime}}{\alpha},\alpha\right)\partial_{x^{\prime}}u_{0}(x^{\prime})\mathrm{d}x^{\prime},$

which after some standard manipulations takes the form:

$\frac{V_{1}}{V_{m}}=-\frac{\Delta
V\text{csch}^{2}\left(1/\alpha\right)}{4\alpha^{4}}\int_{0}^{2}\text{arcsch}\left(\frac{x}{\epsilon}\right)\left(\alpha\sinh\left(\frac{x}{\alpha}\right)+(x-2)\cosh\left(\frac{x}{\alpha}\right)\right)\mathrm{d}x.$
(28)

Figure 5: (a) Normalized value of the increase of velocity due to a small
deformability of the substrate as a function of the slippage coefficient of
the cell with respect to the substrate. (b) Asymptotic regimes for large and
small $\alpha$ as obtained by formulas (29). Parameter $\Delta V=1$.

We illustrate the behavior of $V_{1}$ as a function of the parameter $\alpha$
in Fig. 5 (a). As $V_{1}$ is always positive for a positive value of $\Delta
V$, the influence of a small substrate deformability is to increase the cell
velocity compared to its value $V_{m}$ on an undeformable substrate since the
flow induced by $\Delta V$ in this case promotes motion in the positive
direction. A more subtle effect is that the influence of the friction
coefficient on the additional speed due to the substrate small deformability
is not monotonic. In some parameter ranges, a more adhesive substrate can lead
to an increased velocity since traction forces stemming from the actin flow
are more effective for the propulsion. But the opposite effect is also true in
other cases since an increase of adhesion can also effectively increase the
friction force opposing to the motion. While this effect is lost in the limit
of an undeformable substrate, we demonstrate that it can readily be observed
for a slightly deformable substrate ($\gamma\ll 1$). We further illustrate
this property by analytically computing the dependence of $V_{1}$ in (28) when
$\alpha$ is either small or large to find

$\frac{V_{1}}{V_{m}}\underset{\alpha\rightarrow 0}{\sim}\frac{\Delta
V\epsilon}{4\alpha\sqrt{\epsilon^{2}+4}}\text{ and
}\frac{V_{1}}{V_{m}}\underset{\alpha\rightarrow\infty}{\sim}\frac{\Delta
V\epsilon\left(2\log\left(\frac{\sqrt{\epsilon^{2}+4}+2}{\epsilon}\right)+\epsilon-\sqrt{\epsilon^{2}+4}\right)}{4\alpha^{2}}.$
(29)

These asymptotic expressions, both decreasing with $\alpha$, are plotted on
Fig. 5 (b). The cross-over regime between them corresponds to a situation
where the velocity increases with $\alpha$, in a parameter range when the
corresponding hydrodynamic length is comparable to the cell length
($\alpha\sim 1$). Note that when $V_{1}$ blows up close to $\alpha=0$, the
first order expansion breaks down as the product $\gamma V_{1}$ is not
necessarily small any more. Thus, this behavior does not correspond to a
physically meaningful regime.  This type of non-linear dependence between the
cell velocity and the substrate adhesiveness was also found in a continuum
model representing an actin gel with turnover and linear friction with the
substrate by Callan-Jones and Voituriez (2013). However, the origin of the
effect is completely different as the authors consider an infinitely stiff
substrate with an internal active stress that depends on the actin density in
a non-linear fashion.

#### Solutions in the limit of highly deformable substrate

We now consider the opposite limit where $\gamma\gg 1$ and the substrate is
highly deformable. In this case, we seek a solution of (26) where $f\ll 1$
while $\tilde{f}=\gamma f$ remains finite. Thus, (26) reduces to

$-V\left(\int_{-1}^{1}\frac{\tilde{f}(x^{\prime})}{x^{\prime}-x}\mathrm{d}x^{\prime}+\int_{-1}^{1}R_{1}(x,x^{\prime})\tilde{f}(x^{\prime})\mathrm{d}x^{\prime}\right)=-\frac{\Delta
V}{2}x+V-V_{m}$ (30)

Observe first that this implies that results will be independent of $\alpha$
in this limit. Indeed, any finite value of the friction coefficient has the
same role in the transmission of traction forces because the substrate is
already highly deformable. This integral equation with a singular Cauchy
kernel is numerically solved following the approach of Karpenko (1966).

Figure 6:  Crawling velocity in the limit $\gamma\gg 1$ as a function of
$\epsilon$. (see (30)). The dashed line represents the asymptotic value
$V=V_{m}+\Delta V/4$. In inset, we show the traction forces exerted on the
substrate rescaled by $\gamma$ and the substrate displacement for some values
of $\epsilon$. The traction forces display a square root singularity at the
trailing edge $x=-1$. Parameters are $\Delta V=1$ and $V_{m}=1.5$.

The asymptotic value of $V$ is displayed on Fig. 6 as a function of the track
aspect ratio $\epsilon$. We also show in inset of Fig. 6 the spatial
dependence of the traction force and the substrate displacement in this limit
for several values of $\epsilon$. As expected from the combination of the two
limiting relations $\partial_{xx}(v+v_{0})\simeq 0$ and
$-\alpha^{2}\partial_{xx}v+v=-V\partial_{x}u$ (see (23)) with associated
boundary conditions (24), we numerically recover that the substrate
displacement on $[-1,1]$ assumes the quadratic form $u(x)=\Delta
V/(4V)x^{2}+(V_{m}/V-1)x+\text{Cst}$ where Cst is a constant that remains to
be set. For large track aspect ratios (which also correspond to a plane strain
situation), (30) can be further simplified to

$-V\int_{-1}^{1}\frac{\tilde{f}(x^{\prime})}{x^{\prime}-x}\mathrm{d}x^{\prime}=-\frac{\Delta
V}{2}x+V-V_{m},$ (31)

which solution can be expressed as (see Karpenko (1966))

$\tilde{f}(x)=\left(\tilde{f}_{0}P_{0}^{(1/2,-1/2)}(x)+\tilde{f}_{1}P_{1}^{(1/2,-1/2)}(x)\right)\sqrt{\frac{1-x}{1+x}}.$

In the above expression, $P_{0,1}^{(1/2,-1/2)}$ are the first two Jacobi
polynomials (of order zero and order one) with parameters $1/2$ and $-1/2$ and
their coefficients are given by

$\tilde{f}_{0}=V-V_{m}-\Delta V/4\text{ and }\tilde{f}_{1}=-\frac{\Delta
V}{2}.$

Given the integral condition (27) $\tilde{f}_{0}$ has to vanish, leading to
the asymptotic velocity when both $\gamma$ and $\epsilon$ are large
$V=V_{m}+\Delta V/4$.

Having clarified the two limits $\gamma\ll 1$ and $\gamma\gg 1$, which are
associated to two different values of the crawling velocity and traction
forces profiles, we turn to quantifying the dependence of the crawling on the
substrate softness $\gamma$ and the slip coefficient with the substrate
$\alpha$.

##  5 Biphasic relation of the cell velocity

Using again the approach developed in Karpenko (1966), we compute numerically
the solution of (26) for the realistic parameters reported in Table 1. The
unknown velocity in (26) is found by dichotomy starting from the value
$V=V_{m}$. We show on Fig. 7 and Fig. 8 the resulting dependence of the cell
velocity driven by the internal flow as a function of the substrate softness
and the slippage coefficient with the substrate.

Figure 7:  Dependence of the crawling velocity of the substrate softness. (a)
$V-V_{m}$ as a function of $\gamma$ displays a biphasic dependence in a
realistic range of $\gamma$ (see Table 1). For larger $\gamma$, the velocity
increases again to reach its asymptotic value given in Fig. 6. The dashed line
represents the slope of the curve for small $\gamma$ given by (28). Traction
forces exerted on the substrate (b) and substrate displacement (c) for several
values of $\gamma$. Parameters are $\epsilon=0.2$, $\alpha=0.1$, $\Delta V=1$
and $V_{m}=1.5$.

On the left panels of these figures, we also display the typical traction
force distributions exerted on the substrate and the displacement field at the
contact surface as a function of the spatial coordinate. The central symmetry
of these distributions is broken, which leads to a non zero flow-driven
crawling velocity. The traction forces self-adjust with the substrate
displacement and lead to biphasic behaviors of the crawling velocity as a
function of both the substrate rigidity and adhesion, as experimentally
observed in Palecek et al. (1997); Peyton and Putnam (2005).

Figure 8:  Dependence of the crawling velocity on the slip coefficient with
the substrate. (a) shows the biphasic regime as a function of $\alpha$.
Traction forces (b) and substrate displacements (c) are given for some
specific values of $\alpha$. Parameters are $\epsilon=0.2$, $\gamma=0.02$,
$\Delta V=1$ and $V_{m}=1.5$.

The crawling velocity cannot be directly related to the magnitude of the
traction force or the mechanical work performed on the substrate as the odd
component of $f$ does not contribute to motility. However, we do observe that
the $V(\gamma)$ curve is associated with a biphasic behavior of the work of
the traction forces $W=\int_{-1}^{1}f(x)u(x)\mathrm{d}x$ while their magnitude
$I=\int_{-1}^{1}|f(x)|\mathrm{d}x$ keeps decreasing as $\gamma$ increases.
This is consistent with Fig. 7 where we observe that the magnitude of traction
forces decreases while the one of the substrate displacement increases when
$\gamma$ varies between $0.01$ and $0.02$, which corresponds to the biphasic
region for $V$. We quantity this effect on Fig. 9, where we show the variation
of $I$ and $W$ as a function of $\gamma$.

Figure 9:  Evolution of the traction force magnitude
$I=\int_{-1}^{1}|f(x)|\mathrm{d}x$ and traction force work
$W=\int_{-1}^{1}f(x)u(x)\mathrm{d}x$ as a function of $\gamma$ at a fixed
$\alpha$. Parameters are $\epsilon=0.2$, $\alpha=0.1$, $\Delta V=1$ and
$V_{m}=1.5$.

A plausible explanation of the present biphasic regime of the velocity in
$\gamma$ is therefore that while the traction forces decrease with the
substrate stiffness, their work is maximal at a certain value of $\gamma$
leading to an optimal propulsion at least at a local level close to this
value.

But this explanation does not hold for the global biphasic behavior in
$\alpha$ shown in Fig. 8 where both the magnitude of the traction forces and
their mechanical work on the substrate keep decreasing as $\alpha$ increases.
In this case, the importance of the substrate stiffness is to regularize the
behavior of the velocity when $\alpha$ is small. Indeed, in the limit where
the substrate is very stiff, the velocity expansion blows up when $\alpha\ll
1$ as we show in Fig. 5, indicating that the expansion is no longer valid in
this regime. A finite $\gamma$ is sufficient to restore a finite value of the
speed when $\alpha\rightarrow 0$ (see Fig. 8) and regularizes the problem.
Physically, $\alpha\ll 1$ corresponds to a large friction coefficient where
outside of sharp boundary layers, $v_{x}=v_{x}^{s}$ such that the cytoskeleton
velocity is transmitted to the substrate. It is therefore expected that the
substrate deformability is playing an important role in fixing the velocity in
this regime. The other limit where $\alpha\gg 1$ corresponds to a vanishing
friction coefficient between the cell and the substrate and the two problems
therefore uncouple. We can thus expect that $V=V_{m}$ in this regime as the
substrate deformation stops contributing to the crawling motion. In between
these two regimes, Fig. 8 shows that there exists a friction coefficient which
maximizes the crawling velocity as traction forces are transmitted to the
substrate contributing to the propulsion while the coupling with the substrate
is minimally braking the motion.

Contrary to the biphasic regime with respect to the substrate stiffness, the
biphasic regime with respect to the substrate friction coefficient is not due
to the precise constitutive behavior of the substrate. To demonstrate this, we
consider in A an exponential kernel replacing $\Phi(x)$ by
$\Phi_{\text{exp}}(x)\sim\exp(-|x|)$. Such non-singular kernel mimics the non-
local feedback of the substrate on the cell motion but is of a shorter range
compared to $\Phi(x)$ as it implies that the characteristic lengthscale of the
displacement decay in response to a point force is $\sim\epsilon$ while it is
$\sim 1$ for the elastic kernel. In this case, while we recover a generic
biphasic regime for the $V(\alpha)$ curve, the $V(\gamma)$ curve is
monotonically increasing.

##  6 Experimental platform

We then aimed at constructing an in-house experimental platform to investigate
in details the influence of the substrate rigidity on the cell motion and
check the range of validity of our simple one-dimensional theoretical model.
To do so, it is necessary to produce a substrate with a suitable pattern and
appropriate stiffness. To answer this request, patterning methods and cell-
compatible hydrogels have been developed that are able to create protein
micropatterns on substrates with varied stiffness (Grevesse et al., 2014,
2013). Hydroxy-PAAm(polyacrylamide) are good candidates and can be
functionnalized with fibronectin. After cell seeding on such hydrogels, it is
possible to acquire microscopic images of the cells and track fluorescent
beads embedded within the substrate in order to study the relationship between
cell migration and substrate deformation. Added to this, we used T24
epithelial bladder cancer cells which have been transfected with actin–GFP
(Peschetola et al., 2013), so that the actin velocity can be measured. We can
therefore image at the same time the substrate beads and the actin meshwork as
shown in Fig.10 to estimate the substrate displacement and the actin velocity.
The necessary steps to follow are detailed in B.

Figure 10: Fluorescent cell (actin in cyan) migrating on a fibronectin track
(red) $w\sim 13\,\mu$m on a gel with embedded beads (dark blue).

In a real experimental context, the substrate is not semi-infinite but has a
finite thickness. We justify the use of such approximation in C

In order validate our 1D-model, since experimentally attainable tracks that
are not thin enough for our one-dimensional assumption (1) to be fully
adequate, we project the actin velocity and the substrate displacement on the
major axis of the cell, where the origin of the axis corresponds to the cell
center (see Fig. 11).

Figure 11:  (a) Actin displacement between two successive acquired time frames
($t_{2}$ and $t_{1}$ are separated by 5 s) and (b) its projection along the
track direction. (c) Beads displacements between a given time frame ($t_{1}$)
and a reference configuration where the cell is absent (after detachment)
($t_{0}$). The projection along the track axis is shown on (d). Our assumption
(1) that the traction forces only depends on the $x$-coordinate is not well
satisfied in our experiments and we will need to produce thinner tracks in the
future for a better applicability of our one-dimensional model.

For a reason that is not yet clear to us, our cells were poorly motile and we
therefore restricted our analysis to an acquisition sequence of 15 time frames
with a 5 s time lag between them where the cell remains almost static. In this
situation, we measured an incremental displacement of the substrate of the
order of the measurement uncertainties, corresponding to the resolution of the
measurement where $\partial_{t}u_{x}^{s}\ll v_{x}$. Therefore (18) becomes a
simple integral equation relating $u_{x}^{s}$ and $v_{x}$:

$u^{s}_{x}(x)=\frac{\xi
L}{2}\int_{-1}^{1}\phi\left(\frac{x^{\prime}-x}{\epsilon}\right)v_{x}(x^{\prime})\mathrm{d}x^{\prime},$
(32)

Due to the resolution of the measurements, a linear approximation of the actin
retrograde flow velocity is appropriate, as pushing the approximation to
higher order would essentially capture noise. Thus we write $v_{x}(x)\simeq
v_{1}x$, where the coefficient $v_{1}$ can be physically interpreted as the
slope of (22) at the origin. In order to reduce the effect of fluctuations, a
linear regression is performed at every time frame and then averaged over the
total number of frames:

$v_{x}(x)=\frac{1}{N}\sum_{n=1}^{N}v_{1}^{n}x,$

where $N$ is the total number of time frames in a sequence ($N=15$) and
$v_{1}^{n}x$ is the linear regression of the actin velocity for frame $n$.

(a) $E_{s}=5$ kPa

(b) $E_{s}=8$ kPa

(c) $E_{s}=28$ kPa

Figure 12: Estimation of the friction coefficient $\xi$ from the experimental
data for three different substrate rigidities (5 kPa, 8 kPa, 28 kPa). Over one
sequence (15 frames with a 5s timestep), the measured actin velocity is
represented using 40 blue boxplots and the measured substrate displacement
under the cell using 40 red boxplot. Each boxplot is computed considering all
the data points in 1/40 of the cell. The blue line represents the linear
regression of the measured actin velocity over the sequence. The red line is
the predicted substrate displacement obtained using our model (33), where the
value of $\xi$ is extracted by performing a least square minimization between
the predicted (red line) and measured (red boxplots) substrate displacements.

From (32), the substrate displacement for $x\in[-1,1]$ for the linearized
expression of $v_{x}$ reads,

$u^{s}_{x}(x)=\frac{3L\xi}{4\pi E_{s}}v_{1}x\mathcal{M}_{0}^{\epsilon}(x).$
(33)

where,

$\displaystyle\mathcal{M}^{\epsilon}_{0}(x)=$
$\displaystyle\epsilon\,\text{arcsinh}\left(\frac{1-x}{\epsilon}\right)+\epsilon\,\text{arcsinh}\left(\frac{1+x}{\epsilon}\right)$
$\displaystyle+$
$\displaystyle(1-x)\,\mathrm{log}\left(\frac{\epsilon+\sqrt{(1-x)^{2}+\epsilon^{2}}}{1-x}\right)+(1+x)\,\mathrm{log}\left(\frac{\epsilon+\sqrt{(1+x)^{2}+\epsilon^{2}}}{1+x}\right).$

We then estimate $\xi$ by minimizing the distance between the predicted
substrate displacement (33) for the measured actin retrograde flow velocity
and the substrate displacement from the experimental data. The results for
three different substrate rigidities ($E_{s}$ = 5 kPa, 8 kPa and 28 kPa) are
shown in Fig. 12. In these plots the actin velocity and substrate displacement
experimental data are represented using boxplots. We observe strong
fluctuations of the actin velocity for the 5 kPa and 28 kPa rigidities, while
the data points are much more concentrated in the 8 kPa case. This is due to
the fact, that in the cases of 5 kPa and 28 kPa, the cells where not elongated
enough (not 1D), thus at the same abscissae, we could measure two opposite
velocities in some cases. The actin retrograde flow was also difficult to
track because of the diffuse fluorescence.

While the friction coefficient seems to exhibit a slight dependency to the
substrate rigidity, ranging from $6.64\times 10^{9}\,\text{Pa.s.m}^{-1}$ to
$6.10\times 10^{10}\,\text{Pa.s.m}^{-1}$, more data are needed to extract a
clear tendency, as only one cell sequence was retained for each substrate
rigidity. Because $\partial_{t}u_{x}^{s}\ll v_{x}$, the viscous friction law
(16) reduces to $T_{x}(x,t)=\xi v(x,t)$, which corresponds to the friction law
on a rigid substrate. This gives us a comparison point to evaluate the
relevance of our model. Indeed, while obviously the substrate displacement
between soft and rigid substrates cannot be compared (being zero in the
latter), the friction law is identical in this specific case. Therefore to
validate our model the friction coefficient for a compliant substrate has to
be of the same order as the one for a rigid substrate, which we note
$\xi_{r}$. Extended work has already been done to characterize cell crawling
on rigid substrates by mean of a viscous friction law (Jülicher et al., 2007;
Rubinstein et al., 2009) and $\xi_{r}$ has been evaluated to be of the order
of $10^{9}-10^{10}\,\text{Pa.s.m}^{-1}$. With our model we successfully
recover a friction coefficient of the same order of magnitude as the one
obtained on a rigid substrate.

With this validated experimental setting, we plan to investigate in details
situations where the cell is strongly motile in the future.

##  7 Conclusion

Starting from a linear elastic semi-infinite substrate and a viscous friction
law linearly relating the cell traction forces and the relative velocity
between the actin and the substrate, we modeled the mechanical interaction
between a crawling cell and a compliant substrate for cells confined to move
on thin micropatterned fibronectin tracks. This model of the cell contact was
then coupled with one of the simplest model of cell propulsion based on actin
turnover. In such a model, the polymerization at the cell front and its
depolymerization at the cell back leads to two motility mechanisms. The
addition and removal of actin monomers induces a certain treadmilling velocity
that is independent of external conditions but also a retrograde flow of actin
from the cell front to its back exists and is dependent on the mechanical
coupling with the substrate. Because of such non-local coupling, we find that
the dependence of the cell velocity on either the substrate stiffness at a
given friction coefficient or the friction coefficient at a given substrate
stiffness contain a biphasic regime for a realistic choice of the model
parameters. This offers an interesting paradigm to complement other theories
directly invoking a local non-linear dependence of the friction force on the
actin flow to explain this global non-monotonicity. Experimentally, through
the simultaneous monitoring of the substrate displacement and the actin
retrograde flow velocity for static cells, we were able to extract the
effective friction coefficient that enters in our model for different
substrate rigidities by performing model-based data-fitting. We observed a
small variation of the friction coefficient with the substrate rigidity, but
the values globally lie in the same magnitude range. We find values of the
friction coefficient of the same order as the ones previously reported for
cells crawling on a rigid substrate suggesting that it is legitimate to
consider this parameter as fixed regardless of the substrate rigidity. In the
case of the 5 kPa and 28 kPa subtrate rigidities, the cells were not elongated
enough to consider them as one dimensional and the actin acquisition
resolution was poor. To address these issues it would be necessary to be able
to pattern thinner fibronectin tracks.

We expect our experimental method to be of greater interest for cell types and
substrate rigidities where the actin retrograde flow and the rate of
deformation of the substrate are comparable and the cells are more motile than
in the present work. Another perspective is to couple the present model of
non-local cell-to-substrate contact to the paradigmatic case of contraction-
driven cell motility (Recho et al., 2015) where we expect complex intermittent
gaits to appear from the coupling of the Keller-Segel instability to the non-
locality induced by the substrate elasticity.

##  8 Acknowledgements

We thank Alain Duperray for providing the transfected cells, and for related
discussions. We acknowledge support from a CNRS MOMENTUM grant and the ANR
(grant No. 12-BS09-020-01, TRANSMIG). C.V., J.E. and P.R. are members of the
LabeX Tec 21 (Investissements d’Avenir: grant agreement No. ANR-11-LABX-0030).

##  Appendix A Interaction kernel with the substrate with exponential decay

We formulate the new problem where we replace the expression of $\Phi$ in (23)
by $\Phi_{\text{exp}}(x)=A\text{e}^{-|x|}$ where we set the constant

$A(\epsilon)=\frac{e^{\frac{1}{\epsilon}}\left(\epsilon\log\left(\frac{\sqrt{\epsilon^{2}+1}+1}{\epsilon}\right)+\sinh^{-1}(\epsilon)\right)}{\left(e^{\frac{1}{\epsilon}}-1\right)\epsilon},$

such that
$\int_{-1}^{1}\Phi(x/\epsilon)\mathrm{d}x=\int_{-1}^{1}\Phi_{\text{exp}}(x/\epsilon)\mathrm{d}x$.
With such exponential kernel, system (23) can be written in differential form:

$\left\\{\begin{array}[]{c}-\alpha^{2}\partial_{xx}v_{x}+v_{x}=-V\partial_{x}u\\\
-\epsilon^{2}\partial_{xx}u+u=2\epsilon
A\gamma\partial_{xx}v_{x}\end{array}\right.\text{ with boundary conditions
}\left\\{\begin{array}[]{c}\partial_{x}v_{x}|_{-1}=\partial_{x}v_{x}|_{1}\text{
and }v_{x}|_{1}-v_{x}|_{-1}=-\Delta V\\\
\epsilon\partial_{x}u|_{-1}-u|_{-1}=0\text{ and
}\epsilon\partial_{x}u|_{1}+u|_{-1}=0\end{array}\right.$ (34)

and the unknown velocity is still fixed by the condition

$V=V_{m}+\frac{v_{x}|_{-1}+v_{x}|_{1}}{2}.$

System (34) is a fourth order boundary value problem with a free parameter $V$
that is set by the previous condition. Solving this problem using a
continuation method, we construct the $V(\gamma)$ and $V(\alpha)$ curves that
we superimpose with their analogue for the elastic kernel $\Phi$ on Fig. 13.

Figure 13:  Dependence of the crawling velocity on (a) the substrate softness
$\gamma$ and (b) the substrate slip coefficient $\alpha$ for both the
exponential and elastic interaction kernel with the substrate. Parameters are
$\epsilon=0.2$, $\Delta V=1$ and $V_{m}=1.5$.

We observe that while the global biphasic structure in $\alpha$ is still
present, the local one in $\gamma$ disappears.

##  Appendix B Experimental and post processing methods

$\blacktriangleright$ Synthesis of hydroxy-PAAm hydrogels

Figure 14: AcrylAmide (AAm, in black), N,N${}^{{}^{\prime}}$
methylene–bis–AcrylAmide (bis-AAm, in blue) and N-Hydroxy–EthylAcrylamide
(HEA, in red) were mixed together to form hydroxy-PAAm Grevesse et al. (2013).

In order to prepare hydroxy-PAAm gels to be coated with fibronectin for cell
adherence, it is necessary to use acrylamide (AAm) mixed with
N-hydroxyethylacrylamide (HEA) monomers. The latter contains HO groups that
can form a hydrophilic network (see Fig. 14). Then using bis–acrylamide as a
crosslinker and TEMED/APS for catalysis, rapid polymerization of acrylamid
monomers is obtained Grevesse et al. (2014); Abidine et al. (2018).

Here, we used the protocol explained in Abidine et al. (2018). We prepared
solutions by mixing acrylamide (30% weight per weight [w/w]), N-hydroxyethyl-
acrylamide (5.85% w/w), and N,N-methylene-bisacrylamide (2% w/w) in different
amounts (Sigma-Aldrich, St. Louis, MO). Three concentrations of bis–acrylamide
were used (0.1, 0.3, and 0.6 %), with the acrylamide (3.2%) and
N-hydroxyethyl-acrylamide (1.25%) contents remaining fixed in the final HEPES
solution (50 mM). Gels were 70 $\mu$m thick with an area 1.5 cm $\times$ 1.5
cm, and were prepared on a glass slide (pre–treated with
3–Aminopropyl–triethoxysilane, APTMS) glued at the bottom of a Petri dish.

The gel rigidity should be chosen as a compromise between the stiffness in
physiological conditions and the sufficiently large displacements to be
measured in soft gels (on the order of a few $\mu$m). Values of elastic
modulus ($E$) between 5kPa and 30 kPa were found to be adequate Peschetola et
al. (2013). The elastic Young moduli of the three hydrogels were measured
using an Atomic Force Microscope (JPK AFM, NanoWizard II, Berlin) in contact
mode, equipped with MLCT cantilevers (pyramidal tips, stiffness 0.01 N/m,
Bruker). Five different locations were selected and 5x5 elasticity maps were
performed at each location. The values were obtained using the classical
relationship $F=\frac{3}{4}\frac{E}{1-\nu^{2}}\,\text{tan}\theta\,\delta^{2}$,
where $F$ is the applied force, $\nu\sim 0.5$ is the Poisson ratio, $\theta$
the half–pyramid angle, and $\delta$ is the indentation. The resulting elastic
moduli were found to be 5 $\pm$ 1 kPa, 8 $\pm$ 1.5 kPa and 28 $\pm$ 3 kPa.
This is shown in Fig. 15 with hydrogel elastic moduli increasing with cross-
linker concentration as previously measured Abidine (2015).

Figure 15: The stiffness of hydroxy-PAAm hydrogels is increasing with the
amount of bis-AAm cross-linker Abidine (2015).

$\blacktriangleright$ Fibronectin micropatterning of hydroxy-PAAm gels

Fibronectin (FN) mediates a wide variety of cellular interactions with the ECM
and plays an essential role in cell adhesion, migration, growth, and
differentiation Pankov and Yamada (2002). Incubation of fibronectin straight
microtracks with a very narrow width (less than $15\,\mu$m) and far from each
other (more than $40\,\mu$m) on micro-fabricated hydroxy-PAAm hydrogels can be
used to simulate 1D migration of cells on soft substrates. Here we used a
fibronectin micropatterning process by designing the required
PDMS111Polydimethylsiloxane stamp Grevesse et al. (2013) as shown in Fig. 16.

Figure 16: Process of fibronectin micropatterning on hydrogel Grevesse et al.
(2013).

For the preparation of patterns on PDMS, we first made a silicon master using
optical lithography with a negative photoresist. Optical lithography is a
photographic method by which a light-sensitive polymer, named a photoresist,
is exposed and developed to form 3D features on the silicon substrate. A
negative photoresist is one whose UV exposed parts become cross-linked and
other parts remain soluble and can be washed away during development. One of
the commonly used epoxy–based negative photoresists is SU-8 GM1070, a
chemically amplified resist system with excellent sensitivity. The final
micropattern presents regions with photoresist and other parts are uncovered.
This coated micropattern is needed to shape the PDMS. The common steps for a
regular photolithography process are as follows: substrate preparation,
photoresist spin coating, prebaking, exposure, post-exposure baking,
development, and post-baking Mack (2016). In order to have the silicon master
with about $15\,\mu$m in height, we followed the SU8-Photoepoxy GM 1070
datasheet. The essential principle behind this photoresist operation is the
change in the solubility of the photoresist in the developer, upon exposure to
light. Here we used UV laser exposure with $10\,\mu$m laser beamwidth in order
to change the solubility of the photoresist and as a result we obtain designed
photoresist micro-patterns on top of the silicon wafer (Fig. 17). Then PDMS is
poured onto the substrate, baked at 60∘C for 2h, and is finally peeled off.

Figure 17: Process of the PDMS stamp fabrication

According to Fig. 17, by giving a designed pattern to the laser machine, it is
possible to make our micro–pattern directly on the photoresist. There are two
essential parameters, the percentage of exposure energy and the time of
exposure. The time of exposure comes from the velocity of the moving laser
beam. By testing different exposure energy percentages and velocities, we
found that 100 $\%$ and 25 mm/s give the best results for 15 $\mu$m height of
photoresist layer on the substrate.

In order to check the final shape of designed PDMS stamps with different
patterns (Fig. 18a), we cut very thin cross-section layers of PDMS and imaged
them under an optical microscope (Fig. 18b–e). Then we measured the width
($w$) and height ($h$) of the lines, as well as the distance between two lines
($D$) using FijiTM software. According to Fig. 18f, the results show that
$h\simeq 15\,\mu$m and $D\simeq 45\,\mu$m are almost similar for all PDMS with
different $w$. Also, Fig. 18g shows that the smallest $w$ that we could obtain
was $w\simeq 8\,\mu$m. Although for 1D migration it would be better to have
stamps with smaller $w$, the width of laser beam that was accessible
($10\,\mu$m) gave this limitation for us.

Figure 18: a) Schematic of a cross-section of PDMS stamp with $w$ (width of
lines), $h$ (height of lines), and $D$ (distance between two lines) b, c, d,
e) Images of PDMS cross-sections with different $w$ f) Distribution of
measured values $h\simeq 15\,\mu$m and $D\simeq 45\,\mu$m. g) Values of $w$:
8, 11, 12, and 14 $\mu$m for images b, c, d, and e respectively.

$\blacktriangleright$ Cell culture and seeding

First, cells were cultured at 37∘C and 5$\%$ CO2 atmosphere. Following growth,
they were detached using trypsin-EDTA, resuspended in complete culture medium
(RPMI + 10 $\%$ FBS + 1 $\%$ penicillin–streptomycin) and finally deposited on
the gel. A small drop ($50\,\mu$L) containing $\sim$ 2,000 cells was set onto
the micropatterned gel surface (1.5 cm $\times$ 1.5 cm) bound to the bottom of
a Petri dish, and left to adhere for 15 min. Then 2 mL of culture medium were
added into the Petri dish. It was then possible to observe them individually
under the microscope using the green channel (FITC) of the fluorescence
microscope.

$\blacktriangleright$ live cell microscopy

We used an Olympus IX83 inverted microscope equipped with a Hamamatsu Camera
(Orca G) to capture fluorescent images in green (FITC), blue (DAPI), and red
(TRITC). Thus, it is possible to capture fluorescence of the actin
cytoskeleton using the green channel (FITC), and the beads fluorescence within
the substrate via the blue channel (DAPI) of the microscope. TRITC was only
used in a few cases to check the track width, using rhodamin fibronectin
(Cytoskeleton, Inc.).

The microscope is equipped with the possibility to program the image
acquisition mode (CellSensTM). The loop we chose allowed to quickly capture
two images (green and blue). It is important to select the appropriate image
frequency, since we need to capture fast actin polymerization but we should
not expose the cells to too much light (phototoxicity). Our loop was a series
of 15 pairs of images (FITC then DAPI) taken every 5 seconds, then a 15–minute
pause was held before starting a new series of images. Thus actin fibers and
beads could be followed over short periods of time during cell migration for
about 30 minutes.

At the end of the experiment, we added trypsin to the Petri dish to detach the
cells from the substrate. After waiting for 15 min, the last beads image was
captured to obtain the initial beads position when the gel is in a rest state.
Finally, series of FITC and DAPI images could be treated using image
processing in MatlabTM and FijiTM software for tracking beads and PIV analyses
of actin motion.

$\blacktriangleright$ Image Processing

### Image acquisition

In this work, a code formerly written (Mireux (2019); Jahangiri (2020)) was
improved. The initial code is divided into four main parts. It reads 16-bit
images taken from the microscope and converts them into a data sheet. Next, it
finds the cell, produces masks at different times and calculates cell
contours. Then, it finds the beads and follows their movement in time.
Finally, it selects the beads under the cell and calculates the total
displacements of beads (DAPI) between the rest state and the present
configuration of the gel.

Due to the addition of trypsin to detach cells and also possible shifts of the
camera (thermal changes), it was necessary to rewrite the code in order to
process images. Here, the DAPI images (15 DAPI images taken every 5 seconds +
DAPI image taken after adding trypsin to the Petri dish) as blue color and
FITC images (15 FITC images taken every 5 seconds + blank image corresponding
to the absence of cell on the substrate) as green color were merged. Then we
used the StackReg plugin Thévenaz (2009) in the Fiji software to shift all
images in order to correct for any camera shift. The idea of this plugin is to
align or match a stack of images. When the plugin has finished, the current
slice works as a global anchor.

Then by separating the blue component from the green one, we could track beads
in time using the DAPI images (saved as blue color images) and achieve PIV
(Particle Image Velocity) measurements on FITC images (saved as green color
images).

### Tracking Beads

By using a previously written Matlab code (Mireux (2019)), we could find bead
positions at each time (for example DAPI images at $t_{0}$ and $t_{1}$ in Fig.
19). The particle tracker is a routine adapted for Matlab of a previous
program Crocker and Weeks (2011). This function reads the first DAPI image and
processes it interactively to figure out what settings are needed in order to
identify all the beads. It then repeats this process on all the images.
Finally, it links the beads coordinates to form trajectories.

By replacing the last image (obtained after adding trypsin, Fig. 19a at time
$t_{0}$) from the end of the treated DAPI stack to the first and running the
tracking function in Matlab, we obtain a list of trajectories including the
beads identities numbered from 1 to the total number of beads as well as their
positions ($x$,$y$) at each time $t$ (for example at time $t_{1}$, Fig. 19b ).
Using this data, it is possible to find displacements for each bead between
two times (Fig. 19c).

Figure 19: DAPI images. a) time $t_{0}$ (after adding trypsin), b) time
$t_{1}$; c) Beads displacement between $t_{0}$ and $t_{1}$

### Actin and PIV

We found Particle Image Velocimetry (PIV) to be a suitable technique to
estimate the actin velocity. In general, PIV is an optical technique of flow
visualization used to obtain instantaneous velocity measurements and related
properties in fluids. Basically, a pair of images is divided into smaller
areas named interrogation windows. The cross-correlation between these image
sub-regions measures the optical flow (incremental displacement or velocity of
the objects) between the two images. To improve resolution, higher PIV
resolution can be achieved by progressively decreasing the interrogation
window size Tseng (2014). The PIV analysis was conducted using the MATLAB tool
”PIVlab” Thielicke and Stamhuis (2014), typical outputs from the analysis are
shown Fig. 20.

Figure 20: Actin displacements between $t_{1}$ and $t_{2}=t_{1}+5$s (a); Cell
shape at $t_{1}$ (b) and $t_{2}$ (c)

The post-processing protocol is summarized in Fig. 21.

Figure 21: Post-processing method to extract the friction coefficient $\xi$
from the experimental data.

##  Appendix C Justification of the semi-infinite approximation for the
substrate

To justify the use of our semi-infinite approximation in our experimental
context, we set a simple homogeneous loading where $T_{x}(x,t)=T_{0}$ and
$x\in[-L/2,L/2]$ and compare the ensuing averaging displacement at the surface
and at a certain depth in the bulk. Thus with

$\bar{\bar{u}}_{x}(z)=\frac{3T_{0}}{4\pi\delta
LE_{s}}\int_{-L/2}^{L/2}\int_{-L/2}^{L/2}\int_{-\delta/2}^{\delta/2}\int_{-\delta/2}^{\delta/2}\frac{\left(2(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2}\right)}{((x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2})^{3/2}}\mathrm{d}x^{\prime}\mathrm{d}x\mathrm{d}y^{\prime}\mathrm{d}y,$

we wish to find $z$ such that $\bar{\bar{u}}_{x}(z)\ll\bar{\bar{u}}_{x}(0)$.
Assuming that both $L\gg\delta$ and $|z|\gg\delta$, we find that at leading
order,

$\bar{\bar{u}}_{x}(0)=\frac{3\delta T_{0}}{\pi
E_{s}}\tanh\left(\frac{2L}{\delta}\right)$

and ($z$ being negative)

$\bar{\bar{u}}_{x}(z)=-\frac{3L\delta T_{0}}{4\pi E_{s}z}.$

Thus, in order to have $|\bar{\bar{u}}_{x}(z)/\bar{\bar{u}}_{x}(0)|\ll 1$, we
typically need to impose that $|z|\gg L$. More precisely, in our case, the
cell is typically $L\sim 50$ $\mu$m and the substrate thickness is on the
order of 70 $\mu$m. Such value is roughly an order of magnitude larger than
the track width $\delta$. However,

$|\bar{\bar{u}}_{x}(z)/\bar{\bar{u}}_{x}(0)|\sim\frac{L}{4z}\left(\tanh\left(\frac{2L}{\delta}\right)\right)^{-1}\sim\frac{L}{4z}=\frac{50}{4\times
70}\sim 0.18$

is still a bit large and thicker substrates should be used to aim at a
quantitative agreement.

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