# Modelling cellular spreading and emergence of motility in the presence of
curved membrane proteins and active cytoskeleton forces

Raj Kumar Sadhu1, Samo Penič2, Aleš Iglič 2,3 and Nir S. Gov1 1 Department of
Chemical and Biological Physics, Weizmann Institute, Rehovot, Israel  2
Faculty of Electrical Engineering, University of Ljubljana, Ljubljana,
Slovenia  3 Faculty of Medicine, University of Ljubljana, Ljubljana, Slovenia

###### Abstract

Eukaryotic cells adhere to extracellular matrix during the normal development
of the organism, forming static adhesion as well as during cell motility. We
study this process by considering a simplified coarse-grained model of a
vesicle that has uniform adhesion energy with a flat substrate, mobile curved
membrane proteins and active forces. We find that a high concentration of
curved proteins alone increases the spreading of the vesicle, by the self-
organization of the curved proteins at the high curvature vesicle-substrate
contact line, thereby reducing the bending energy penalty at the vesicle rim.
This is most significant in the regime of low bare vesicle-substrate adhesion.
When these curved proteins induce protrusive forces, representing the actin
cytoskeleton, we find efficient spreading, in the form of sheet-like
lamellipodia. Finally, the same mechanism of spreading is found to include a
minimal set of ingredients needed to give rise to motile phenotypes.

##  I Introduction

The adhesion of cells to an external substrate is an essential process
allowing cells to form cohesive tissues, migrate and proliferate
geiger2009environmental . The stages of cellular spreading over an adhesive
surface have been studied experimentally Sheetz2004PRL ; cavalcanti2007cell ;
cuvelier2007universal ; Sheetz2008 ; Sheetz2011 ; Sheetz2014 ;
schaufler2016selective ; Nils2017 , and involve an initial stage of non-
specific and weak adhesion, followed usually by spreading that is driven by
the formation of thin sheet-like lamellipodia. These structures form when
actin polymerization is recruited to the leading edge of the lamellipodia
Oakes2018 . The actin provides both a protrusive force that pushes the
membrane outwards and traction forces that enhance the growth of adhesion
complexes. When actin polymerization is inhibited, cells exhibit very weak
spreading, small adhered area Sheetz2008 , and strongly retract (if the drug
is delivered after normal spreading bar1999pearling ).

While this complex process has been explored from its biological aspects, a
more basic physics understanding of the cell spreading process is lacking
schwarz2013physics . There are several theoretical treatments of the cell
spreading process, with various levels of coarse-graining and detail, starting
from the simplest dynamical-scaling model cuvelier2007universal . Some models
focus on the role of actin and actin-adhesion coupling during the spreading
and adhesion, but do not describe the membrane shape dynamics in detail
Sheetz2010 ; nisenholz2014active ; gong2018matching ; dedenon2019model . Other
models describe in detail the cell shape and the stress-fibers that span the
adhered cell loosli2010cytoskeleton ; fang2020active , with the higher realism
obtained at a price of much higher model complexity.

The simpler process of vesicle adhesion, which has been explored using in-
vitro systematic experiments gruhn2007novel ; reister2008dynamics ;
streicher2009integrin ; maan2018adhesion ; bibissidis2020 , is amenable to
theoretical physics description bibissidis2020 . A large number of theoretical
studies treated the coarse-grained adhesion of a vesicle with uniform adhesion
lipowski1990 ; lipowsky1991 ; lipowski2005 , while other studies have explored
the molecular-scale adhesion dynamics boulbitch2001kinetics ; smith2008force ;
sengupta2018adhesion , which include ligand binding/unbinding as well as
diffusion and aggregation of ligands on the membrane-substrate interface
(through direct and membrane-induced interactions farago2010fluctuation ;
fenz2017membrane ).

We aim here to help bridge the gap between our understanding of vesicle
adhesion and the more complex process of active cellular spreading. We do this
by exploring a simple, coarse-grained theoretical model of vesicle adhesion
which contains two ingredients: (i) we add a fixed density of curved membrane
proteins, and (ii) exert active protrusive forces at the locations of the
membrane proteins. Both of these components are motivated by experimental
properties of cells: cell membranes contain a plethora of curved membrane
proteins zimmerberg2006proteins ; suetsugu2014dynamic , and many of these
curved proteins (or membrane-bound protein complexes) are involved in the
recruitment of actin polymerization activity to the membrane scita2008irsp53 ;
kuhn2015structure . These two ingredients have been recently shown
theoretically miha2019  and experimentally begemann2019mechanochemical ;
graziano2019cell  to be sufficient to induce the formation of sheet-like
lamellipodia protrusions in cells. We therefore set out to explore
theoretically the role of these ingredients during active spreading of cells.

##  II Model

We use here the same theoretical model of miha2019 , adapted to include
membrane-substrate adhesion. We consider a three-dimensional vesicle that is
described by a surface of $N$ vertices, each connected to its neighbours with
bonds of length $l$, to form closed, dynamically triangulated, self-avoiding
network, with the topology of a sphere, as shown in Fig. 1. An adhesive
surface is placed near the vesicle, parallel to x-y plane and at position
$z=z_{ad}$ (see Fig. 1). The total energy of the vesicle is the sum of four
contributions, (1) the local bending energy due to its curvature, (2) the
energy due to binding between neighboring proteins (direct interaction
energy), (3) the energy due to the active cytoskeleton force and (4) the
adhesive energy due to the attractive interaction between the vesicle and the
substrate.

Note that the term ”curved membrane proteins” stands for any complex of such
proteins and lipids (such as in nanodomains) in general, that has a
spontaneous curvature, and can induce local polymerization of the cortical
actin cytoskeleton.

The bending energy can be mathematically expressed using the Helfrich
expression helfrich1973  as,

$W_{b}=\frac{\kappa}{2}\int_{A}(C_{1}+C_{2}-C_{0})^{2}dA$

where, $C_{1}$ and $C_{2}$ are principle curvatures, $C_{0}$ is the
spontaneous curvature at any position of the vesicle, and $\kappa$ is the
bending rigidity. The bending energy is properly discretized following the
refs. gompper1996 ; ramakrishnan2011 ; samo2015 . We model the spontaneous
curvature as discrete entities that is occupied by a vertex. The spontaneous
curvature of a vertex that is occupied by curved proteins is taken to have
some non-zero value $C_{0}=c_{0}$, and zero otherwise. In our model, we
consider a positive $c_{0}>0$, i.e. convex spontaneous curvature. Note that we
describe here isotropic curved proteins or isotropic curved nanodomains
kralj_iglic1996 ; fovsnarivc2006influence .

The energy due to the binding between proteins is expressed as,

$W_{d}=-w\sum_{i<j}{\cal H}(r_{0}-r_{ij})$

where, ${\cal H}$ is the Heaviside step function, having a value of unity if
the argument is positive, otherwise vanishes, $r_{ij}$ is the displacement
between proteins and $r_{0}$ is the range of attraction, beyond which the
attractive force becomes zero and $w$ is a positive constant (throughout the
paper we use: $w=1~{}k_{B}T$). In our model, we choose $r_{0}$ to be such that
only the proteins in neighbouring vertices can bind with each other.

The actin cytoskeleton that is recruited by the curved proteins exerts an
outward force, which therefore gives the following energy contribution,

$W_{F}=-F\sum_{i}\hat{n_{i}}.\overrightarrow{x_{i}}$

where, $F$ is the magnitude of the force, $\hat{n_{i}}$ be the outward normal
of the vertex that contains a protein and $\overrightarrow{x_{i}}$ is the
position vector of the protein. Due to the thermal fluctuations of the vesicle
and the diffusion of the proteins, this force changes its direction with time.
Thus, this is equivalent to a force exerted due to a time-varying external
potential, which therefore drives the system out of equilibrium.

Finally, the vesicle can adhere on the adhesive surface, due to which it has
the energy contribution,

$W_{A}=-\int_{A}V(z)dA$

where, $V(z)$ is the interaction potential between the adhesive surface and
the vesicle. In our model, we choose the interaction potential $V(z)$ to be a
step potential, such that, $V(z)=E_{ad}$ (a constant, termed as adhesion
strength) for $z_{ad}\leq z(i)\leq(z_{ad}+\Delta z)$, and zero otherwise;
where $z_{ad}$ is the z-coordinate of the adhesive surface, $\Delta z$ is the
width of potential energy and $z(i)$ is the z-coordinate of $i$-th vertex (see
Fig. 1 (b)). The adhesive surface is considered to be a rigid object, such
that a vertex can not penetrate it.

Thus, the total energy of the system can be written as,

$W=W_{b}+W_{d}+W_{F}+W_{A}$ (1)

Figure 1: A schematic representation of our model. (a) A three-dimensional
vesicle is placed on the adhesive surface, having uniform adhesion interaction
with the vesicle throughout. The position of the adhesive surface is at
$z=z_{ad}$ and parallel to $x-y$ plan. (b) The range of adhesive interaction
is within a distance of $\Delta z$ above the adhesive surface. The total
adhesion energy will be $E_{ad}$ times the number of vertices within this
interaction range, where $E_{ad}$ is the adhesion strength, defined as the
adhesion energy per adhered vertex.

The simulation details are given in the appendix A. Throughout these
simulations we do not conserve the vesicle volume, which is appropriate for
cells that are observed to change their volume significantly during spreading
and adhesion guo2017cell ; xie2018controlling . One can however add to the
model the effects of an internal osmotic pressure that inflates the vesicle
miha2019 .

Note that during the process of vesicle adhesion and spreading, there are
hydrodynamic processes that we do not include in our model, such as fluid flow
within the vesicle, between the vesicle and the substrate, and of the fluid
membrane, i.e. visco-elastic properties of the system were neglected (see for
example bernard2000strong ). These omissions mean that the dynamics that we
extract from the simulations, in MC time steps, may not be simply mapped to a
real time-scale.

##  III Results

In order to validate our simulation method, we first compared the steady-state
shapes of adhered protein-free vesicles to those previously obtained using
detailed numerical solutions raval2020shape ; bibissidis2020 . The very good
agreement between the two different methods (Fig. A-1 of appendix B) verifies
the accuracy of our simulations. The equilibrium vesicle shapes minimize the
energy, striking a balance between the adhesion energy that drives the
spreading and the bending energy that resists the deformation of the membrane.

Next, we explored the effects of the curved proteins and the active forces
that they recruit.

###  III.1 Spreading of vesicles with passive curved proteins

We start by exploring the effects of passive curved proteins, without active
cytoskeletal forces. Since, there is no active force acting on the system, the
system reaches an equilibrium configuration after evolving it for sufficiently
long times. We aim to understand here, how the presence of the curved proteins
affects the vesicle shape, as well as the demixing of the curved proteins.

Figure 2: Typical steady state configurations of the vesicles and cluster
distribution in the $E_{ad}-\rho$ plane for the passive case (F=0). The
background color is showing the fraction of adhered area of the vesicle on the
adhesive surface. The blue part in the vesicle denotes the protein free
regions and the red color denotes the curved proteins. In the inset, we show
the cluster distribution of proteins near each snapshot; x-axis is the size of
the cluster and the y-axis is the corresponding frequency of having the
particular cluster in the ensemble. The y-axis is shown in the log-scale. We
show the snapshots for $E_{ad}$=0.75, 1.5, 2.5 (in units of $k_{B}T$ ) and
$\rho$=3.45 %, 10.36 %, 17.27 % and 24.18 %. Other parameters are: Total
number of vertices, $N=1447$, $\kappa=20~{}k_{B}T$ and $w=1~{}k_{B}T$. The
width of the potential, $\Delta z$ is taken to be $l_{min}$, and the
spontaneous curvature of the curved proteins is taken to be
$c_{0}=1~{}l^{-1}_{min}$.

In Fig. 2, we plot snapshots of typical equilibrium adhered vesicle shape and
the protein cluster-size distribution for different values of adhesion
strength ($E_{ad}$) and the number density of proteins ($\rho=N_{c}/N$). The
background color is showing the adhered area fraction (AAF, $A_{ad}/A$), where
$A_{ad}$ is the area adhered on the adhesive surface and $A$ is the total area
of the vesicle (so that the maximal possible value is $A_{ad}/A\simeq 0.5$ for
very small volume).

For small $\rho$ and small $E_{ad}$, the vesicle shape is quasi-spherical,
similar to adhered protein-free vesicles. The proteins are weakly clustered
and there are small patches of proteins around the whole vesicle. As $\rho$
increases keeping $E_{ad}$ small, the vesicle spreads, becomes flatter (since
the volume is not fixed). This spreading is driven by the aggregation of the
curved proteins at the high-curvature region kralj_iglic1996 ; markin1981
where the adhered vesicle contacts the substrate (”contact line”, Fig.3 (a)).
Due to their large convex spontaneous curvature, the aggregation of the
proteins in this region lowers significantly the bending energy of the
vesicle, facilitates stronger bending and larger spreading on the substrate
compared to the protein-free vesicle (Fig.3 (b)).

Note that demixing and phase-separation of membrane proteins on an adhered
vesicle were considered theoretically rouhiparkouhi2013adhesion , however the
curvature-based demixing aimon2014membrane  that we discuss here was not
previously treated during membrane adhesion. A simplified analytic model (see
appendix C) allows us to qualitatively recover the same trends found in the
simulations. We find that the aggregation of curved proteins along the high
curvature rim of the vesicle leads to a monotonous increase in the AAF and
decrease of the bending energy with increasing protein density (for low
$\rho$, Fig. A-4).

As $\rho$ increases, the vesicle flattens and surplus proteins that have no
more space along the contact line form necklace-like protein clusters around
the vesicle (Fig. 2). The necklace-like structures are formed because of the
isotropic membrane proteins used in our simulation miha2019 ; Luka2017 . The
AAF increases with $\rho$ as long as the shape of the vesicle remains quasi-
spherical or pancake-like. However, for large $\rho$, the shape of the vesicle
changes and the surplus curved proteins lead to budding all over the membrane.
These buds (isolated and necklace-like), deform the membrane away from the
flat shape, and give rise to a decrease of the AAF with increasing $\rho$
(Fig. 2).

As $E_{ad}$ is increased, the vesicle is more spread, and the natural
curvature along the contact line increases. This has the effect of aggregating
the curved proteins more strongly and the flattening effect of the curved
proteins sets in at lower densities (Fig. 2). Similarly to low $E_{ad}$, the
AAF decreases for large $\rho$, and the peak in the AAF shifts to smaller
values of $\rho$. As $E_{ad}$ increases the proteins form a large cluster at
the contact line, where almost all the proteins are clustered in a single
cluster.

To quantify the effect of the curved proteins on the vesicle spreading, we
measure the ratio of AAF between vesicles with highly curved passive proteins
and protein-free vesicles (Fig.3 (c)). We find that this ratio is larger for
small $E_{ad}$, where the protein-free vesicles are very weakly adhered. As
expected from Fig. 2, the enhancement of spreading due to the curved proteins
has a maximum as function of $\rho$, with the peak shifted to lower values of
$\rho$ as $E_{ad}$ increases. Note that for the passive case ($F=0$), a
protein-free vesicle is similar to a vesicle with flat proteins ($c_{0}=0$).
For details, please see appendix D.

Figure 3: Curved proteins facilitate spreading at low adhesion energy. (a)
Protein distribution along the angle $\theta$ for the passive case with
$E_{ad}=0.50~{}k_{B}T$ and various $\rho$ along with the snapshots. The angle
($\theta$) is defined as the angle between the line parallel to the x-y plane,
passing through the COM of vesicle and the line joining the surface of the
vesicle and its COM, as shown in figure. P($\theta$) is the probability that
there is a protein at the angle $\theta$, averaged over the azimuthal
direction, such that $\theta$ varies from $-\pi/2$ to $\pi/2$. (b) Bending
energy as a function of AAF with and without proteins. The blue circles are
for protein-free vesicle and the red triangles are for passive case. We also
show the snapshots for each case, in the minimum and the maximum adhered
state. For protein-free vesicle, we vary $E_{ad}$ from $0.50~{}k_{B}T$ to
$4.0~{}k_{B}T$, while for passive vesicle, we fix $E_{ad}$ to $0.50~{}k_{B}T$
and vary $\rho$ from $0.70~{}\%$ to $27.64~{}\%$. (c) Ratio of AAF between
vesicles with highly curved passive proteins (Fig.2) and protein-free
vesicles. The ratio is close approaches unity for either $\rho\rightarrow 0$
or for $E_{ad}\rightarrow\infty$.

###  III.2 Spreading of vesicles with active curved proteins

Next, we study the active system, with active cytoskeletal forces acting
outward on the proteins. To highlight the effects of the active force, we
start with a large value of $F=4~{}k_{B}T/l_{min}$ (Fig. 4). Despite the
presence of the active force, we find that for sufficiently large times most
systems do reach a well-defined steady state, which allows us to extract
average quantities, such as the AAF and the cluster distribution. When the
density $\rho$ is small, the shapes of the adhered vesicle are quite different
compared to the passive case (Fig. 2). However, for large $\rho$, the shapes
are quite similar to the passive case.

Figure 4: Typical steady-state configuration of vesicle shape and cluster-size
distribution with active forces. Similar to Fig. 2 for an active protrusive
force $F=4~{}k_{B}T/l_{min}$. The green dashed line-circles denotes the
transition to a pancake-like shape. The red dashed vertical line denotes the
density for the pancake transition for a free vesicle (without adhesion,
miha2019 ). We estimate these transition lines by measuring the mean cluster
size, which shows a sharp jump at the critical $\rho$ (Fig. 5(c)). Snapshots
are shown for $E_{ad}$=0.75, 1.5, 2.5 (in units of $k_{B}T$) and
$\rho=3.45~{}\%,10.36~{}\%,17.27~{}\%$ and $24.18~{}\%$. All the other
parameters are the same as in Fig. 2.

For small value of $\rho$, unlike the passive case (Fig. 2), the shape of the
vesicle is highly non-rotationally-symmetric (Fig. 4). The transition into
this class of shapes, for $\rho$ to the left of the vertical dashed green line
(around $5.25~{}\%$), is very sharp and was also observed for free vesicles
with active curved proteins (denoted by the vertical dashed red line,
$\rho\sim 7.26~{}\%$) miha2019 . At a low number of proteins, there are simply
not enough proteins to complete a circular aggregate around the rim of the
adhered vesicle. Instead, the proteins settle into two opposing arc-like
aggregates, which exert opposing forces on the vesicle, that therefore assumes
a stretched tube-like shape.

In Fig. 5(a) we demonstrate the dynamics of the spreading in the low $\rho$
regime. We find that often the proteins form three or more aggregates that
drive the spreading, but they coarsen over time to form the stable two-arc
shapes. These two-arc shapes are mostly non-motile, but due to asymmetry
between the sizes of the arc-like protein aggregates on either side, there can
be a net force that leads to sliding of these vesicles in the direction of the
end that has the larger aggregate (see Fig.9(c)).

For $\rho$ values to the right of the red dashed line in Fig. 4 there is a
transition to pancake-like shapes, that correspond to very efficient spreading
and high AAF. For even larger protein density, there is not enough space on
the outer rim of the vesicle to accommodate all the proteins, so the pancake-
like shape does not remain stable. Small bud like structure appear around the
vesicle and the AAF decreases, similar to the passive case at high densities
(Fig. 2).

The transition between the two-arc and pancake-like shapes for different
values of $\rho$ and $E_{ad}$ is shown in more detail in Fig. 5(b). As
$E_{ad}\rightarrow 0$, the transition density increases and approaches its
value for a free vesicle. The pancake transition is quantified by the mean
cluster size ($\langle N_{cl}\rangle$), which we plot as function of $\rho$
(Fig. 5(c)). The quantity $\langle N_{cl}\rangle$ exhibits a sharp jump near
the pancake transition, where one cluster contains almost all the proteins
(phase separation).

Figure 5: Transition to pancake-like, highly-spread shape, for active curved
proteins ($F=4~{}k_{B}T/l_{min}$). (a) Snapshots of the vesicle for small
density, at different instant of time. Here, we show that there are different
possible metastable states in the small density regime. We use here
$\rho=3.45~{}\%$, $E_{ad}=0.75,1.5,2.5$ (in units of $k_{B}T$). (b)
Configurations of vesicle for different values of $E_{ad}$ and $\rho$. The
green line with circles denotes the density for a given $E_{ad}$ at which the
pancake-like shape is obtained. (c) Mean cluster size of proteins, $\langle
N_{cl}\rangle$, as a function of $\rho$ for different values of $E_{ad}$. The
sharp increase in the value of $\langle N_{cl}\rangle$ shows the discontinuous
transition to pancake-like shape. We also show the snapshots of vesicle for
$E_{ad}=2.5~{}k_{B}T$ for different densities to show how the jump in the
value of $\langle N_{cl}\rangle$ gives rise to the pancake-like shape
transition.

Next, we study the AAF as a function of $F$, for different values of $E_{ad}$
and a fixed density $\rho=10.36~{}\%$ (Fig. 6 (a)). For large $E_{ad}$ the
force increases the AAF smoothly, as the vesicles are already spread even in
the absence of active forces. For small $E_{ad}$, the AAF shows a large
increase with $F$, including an abrupt jump for $E_{ad}=0.25~{}k_{B}T$ at
$F\sim 0.8~{}k_{B}T/l_{min}$. This jump corresponds to crossing the pancake
transition line for these parameters. In Fig. 7 we plot a more complete phase-
diagram, of the steady-state shapes of the vesicle for low adhesion
($E_{ad}=0.5~{}k_{B}T$), as function of the proteins density and active force.
The active force is seen to shift the transition to the pancake (or the two-
arc) shape, to lower values of the density, compared with the passive system
(at $F=0$). This shift to lower densities due to the force is also seen for
the initiation of protein aggregation (appendix E, Fig.A-8).

Figure 6: Variation of adhered fraction due to the active force ($F$). (a)
Fraction of adhered area with $F$ for different values of $E_{ad}$ for a fixed
$\rho$ ($=10.36~{}\%$). For small $E_{ad}$, the fraction $A_{ad}/A$ increases
with $F$ significantly, however, as $E_{ad}$ increases, the value of
$A_{ad}/A$ does not vary much. We also show the shape of the vesicle for
$E_{ad}=0.25~{}k_{B}T$. (b) The ratio of adhered area of active vesicle with
$c_{0}=1~{}l^{-1}_{min}$ to the passive vesicle with $c_{0}=0$. The ratio is
maximal for small $E_{ad}$ over a wide range of $\rho$. (c) The ratio of
adhered area of active to passive vesicle, with spontaneous curvature
$c_{0}=1~{}l^{-1}_{min}$. We note that the maximum increase in the adhered
area of the active vesicle over passive one is in the small $\rho$ and small
$E_{ad}$ region.

The role of the active force in increasing the steady-state AAF is emphasized
in Fig. 6 (b), where we plot its ratio with the AAF of the protein-free
vesicle. We find that the largest increase in AAF due to the active curved
proteins is for low $E_{ad}$. Compared to the vesicle containing curved
passive proteins, this enhanced spreading is extended to lower values of
$\rho$ due to the pancake transition (compare to Fig. 3(c)). This is
emphasized in Fig. 6 (c) where we plot the ratio of the AAF of the active and
passive curved protein systems. The largest contribution of the active force
is for low $E_{ad}$ and $\rho$, where the passive proteins do not form strong
aggregation at the contact line and are ineffective in driving spreading,
while the added active forces drive the pancake transition and strongly
enhanced spreading.

We note that for large $E_{ad}$ and large $\rho$, the AAF is also increased
due to activity, compared to the passive curved proteins (Fig. 6 (c)). This is
the region where the passive proteins form large necklace-like structures
which decrease the adhered area (Fig. 2), while the active forces tend to
destabilize them and therefore increase the AAF.

The active forces exerted at the locations of the curved proteins may give
rise to a non-zero net force. While the planar component of this force simply
pushes the vesicle on the substrate, the vertical component (along
$z$-direction) can affect the adhesion. Since the curved proteins prefer the
free (dorsal) side of the vesicle over the perfectly flat basal side, this
force tends to overall push the vesicle away from the substrate. In the regime
of very low $E_{ad}$ and large $\rho$, the active forces exerted by the
proteins can lead to lowering the AAF by partially detaching the vesicle. In
Fig. A-9 of appendix F, we show the behavior of vesicles where we apply an
external force that balances the total vertical component of the active
forces. For a living cell, this condition corresponds to assuming that actin
filaments that are pushing the top membrane upwards exert an equal and
opposite force on the bottom membrane. We see that except at the lowest
$E_{ad}$, there is no qualitative difference, compared to the previous results
(Fig.4).

The importance of coupling the force to curvature, is demonstrated by
simulating the adhesion of a vesicle with flat active proteins (zero
spontaneous curvature, $c_{0}=0$). As in previous studies miha2019 ;
graziano2019cell , we find the formation of long protrusions, that are highly
dynamic (Fig. A-10 of appendix G). Due to the adhesion, the long protrusions
are found to often grow along the substrate. However, when they point upwards,
they lead to partial detachment of the vesicle. Clearly, active forces that
are not coupled to curvature do not contribute to effective spreading and
adhesion.

When comparing our results with experimental observations of the shapes of
adhered cells, we begin by noting that cells undergo a much diminished
spreading (or strong retraction) when actin polymerization is inhibited
(adhered area decreases by factor of $\sim 4$ Sheetz2008 ). This suggests that
the bare adhesion of the cell to external substrates is typically low, so that
in terms of our model cells are usually in the regime of low $E_{ad}$. In this
regime, we demonstrate that self-organization of the actin polymerization
recruited by curved membrane proteins can increase the adhered area by factors
that are similar to those observed experimentally (Fig.6b,c).

However, the actin polymerization in the cell does more than just provide a
protrusive force, as we assumed in our model. The actin retrograde flow
produces shearing forces that triggers the growth of integrin-based adhesion
complexes cavalcanti2007cell ; schaufler2016selective . This suggests that the
activity of actin polymerization also effectively increases $E_{ad}$ for the
cell, compared to the actin-inhibited cell. Similarly, increased adhesion
strength ($E_{ad}$) allows for stronger mechanical coupling between the actin
filaments and the substrate, inducing a larger effective protrusive force $F$
chan2008traction ; gong2018matching . These effects mean that when comparing
our model to cell shapes, the effective actin protrusive force $F$ and the
effective value of $E_{ad}$ are not independent of each other.

Many adhered cells are found not to be circularly spread, but have a distinct
spindle-like shape with usually two oppositely formed lamellipodia
protrusions. This typical shape appears naturally in our model when the
density of the curved proteins is below the pancake transition value, and the
adhered vesicle assumes the elongated two-arc shape (Figs. 4, 5). Note that
since the critical density for the pancake transition increases for decreasing
$E_{ad}$ (Fig.5b), we expect that cells can transform from the pancake to the
elongated two-arc shape with decreasing adhesion strength. This is indeed
observed in experiments cavalcanti2007cell ; schaufler2016selective ;
singh2020cell . The morphology of two oppositely oriented lamelipodia (similar
to our two-arc shapes) was observed to stretch cells, and is sometimes
utilized to drive cell division kee2012mechanosensory ; flemming2020cortical .

Figure 7: Typical steady-state configurations of vesicle shape and cluster-
size distribution for different $F$ and $\rho$, for a given small adhesion
strength: $E_{ad}=0.50~{}k_{B}T$. The snapshots are shown for $F$=0.50, 1.50,
2.50 and 3.50 (in units of $k_{B}T/l_{min}$), and
$\rho=3.45~{}\%,10.36~{}\%,17.27~{}\%$ and $24.18~{}\%$. The yellow dotted
lines denotes the transition to a pancake-like shape. For large $F$, the
transition to a pancake shape from a two-arc shape is very sharp, and the
transition line is estimated by measuring the mean cluster, as explained in
Fig. 5. For small $F$, the transition is not very sharp. Here, in order to
identify the transition line, we measure the largest cluster, and we
approximate the shape to be a pancake when the largest cluster is at least
$60~{}\%$ of the total number of proteins (see Fig. A-12 of appendix I). The
green dotted line represents the transition from a quasi-spherical to a two-
arc type shape. The slope of this transition line diverges as $\rho\rightarrow
0$. This transition is also estimated by measuring the largest cluster and the
threshold value of the largest cluster is taken to be $30~{}\%$ in this case,
as there are two separate clusters in the two-arc shape. All the other
parameters are the same as in Fig. 2.

###  III.3 Spreading dynamics

We compare our results for the spreading dynamics of a protein-free vesicle, a
vesicle with passive-curved and active-curved proteins respectively, in Fig.
8(a-c). The vesicles with proteins are shown in the interesting regime of low
$E_{ad}$. We note that the active vesicle spreading is much noisier than the
passive spreading. This is because in the active case the vesicle may
transiently get locally de-adhered from the substrate, which gives rise to
large variations in the measurement of the AAF (Fig. 8(c)). In Fig. A-11 of
appendix H, we plot the cross-sectional shapes, side-views and three-
dimensional shapes, of the spreading vesicles as function of time, for all
three cases. Clearly, the passive systems are observed to spread more
isotropically, compared to the active system. The anisotropic spreading of the
active vesicles is quantified in Fig. 8 (f), where we see a sharp reduction in
the circularity of the adhered region during the initial stages of spreading.

Next, we plot the increase of the adhered radius ($R_{ad}$) as function of
time, which is defined as $\sqrt{A_{ad}/\pi}$, where $A_{ad}$ is the adhered
area (Fig. 8(d-e)). We find that for the passive systems the adhered radius
($R_{ad}$) grows with time as $\sim t^{\beta}$, where the exponent $\beta$ is
different for the different cases. Although this plot is given in MC time-
steps, and does not include the hydrodynamic effects of the the membrane flow
and the fluid flow within and around the vesicle, the calculated dynamics of
the passive vesicles resemble the experimental observations for spreading
artificial vesicles streicher2009integrin .

For the active case (Fig. 8e), we averaged over those cases which adhered
smoothly without significant events of de-adhesion, in order to get a less
noisy curve. For this case, we find that there is a slower growth in the
beginning (Fig. 8d), followed by a faster growth regime. The exponent of the
faster growth stage depends on $E_{ad}$ (Fig. 8 (e)). The slower initial
growth is due to the low circularity of the active vesicle, with the protein
aggregates spread over the vesicle and pushing the membrane upwards and in
uncoordinated manner. Once the proteins form the circular aggregate along the
contact-line, they induce a very efficient and rapid spreading, which
corresponds to the fast growth phase.

The calculated spreading dynamics for an active vesicle resemble several
aspects experimentally observed in spreading living cells Sheetz2004PRL ;
dubin2004nanometer ; Sheetz2014 : (i) The active vesicles exhibit an initially
accelerating radial growth, followed by a growth with almost constant velocity
(Fig. 8 (c,d)). These features are observed in living cells, and do not appear
for our passive vesicles. (ii) The initiation of the rapid spreading phase
takes longer to appear for cells on substrates of lower adhesiveness, as we
also find (Fig. 8 (e)). (iii) Compared to the passive system, the active
vesicle initially grows more slowly (Fig. 8 (d)), similar to experimental
observations cuvelier2007universal .

Figure 8: Dynamics of the spreading process. (a) Variation of adhered area
with time for a protein free vesicle. Here, we use $E_{ad}=5.0~{}k_{B}T$. (b)
Variation of adhered area with time for a passive vesicle with
$\rho=13.82~{}\%$ and $E_{ad}=0.50~{}k_{B}T$. (c) Variation of adhered area
with time for an active vesicle with $F=4~{}k_{B}T/l_{min}$, $\rho=10.36~{}\%$
and $E_{ad}=0.50~{}k_{B}T$. We note that the protein free vesicle adhere much
faster than the other cases. The spreading of passive vesicle is very smooth,
while the active vesicle spreading is very noisy. Here, the unit of time (t)
is $2\times 10^{3}$ MC steps for (a)-(c). (d) Variation of adhered radius
($R_{ad}$) with time, for protein-free, passive and active vesicles. The blue
circle is for protein-free vesicle, the red triangles are for passive vesicle
and the green boxes are for active vesicle. We note that the growth of a
protein free vesicle and passive vesicle is uniform in the beginning, and then
saturates, however, for active case, there are two phases of growth in the
small time regime before $R_{ad}$ saturates. For protein free vesicle,
$R_{ad}(t)\sim t^{0.152}$, while for a passive vesicle, $R_{ad}(t)\sim
t^{0.183}$. Unlike the protein-free and passive vesicles, the active vesicle
exhibits two growth phases: a slower initial growth of $R_{ad}(t)\sim
t^{0.063}$ followed by a faster growth $R_{ad}(t)\sim t^{0.267}$. Here, we use
$E_{ad}=5.0~{}k_{B}T$ for protein-free vesicle, and $\rho=10.36~{}\%$ and
$E_{ad}=0.50~{}k_{B}T$ for passive and active vesicles. For active vesicle, we
use $F=4~{}k_{B}T/l_{min}$. (e) Variation of $R_{ad}$ with time for an active
vesicle with various values of $E_{ad}$: from bottom to top, star symbols are
for $E_{ad}=0.25~{}k_{B}T$, boxes are for $E_{ad}=0.50$, circles are for
$E_{ad}=1.0k_{B}T$ and triangles are for $E_{ad}=2.0~{}k_{B}T$. Note that the
fast growth phase takes longer to appear for lower $E_{ad}$, while the growth
exponent of the fast growth phase increases with $E_{ad}$. We vary $E_{ad}$
from $0.25~{}k_{B}T$ to $2.0~{}k_{B}T$, that gives rise to variation in the
exponent from $0.51$ to $0.20$. Here we use $\rho=10.36~{}\%$ and
$F=4~{}k_{B}T/l_{min}$. The unit of time (t) is $2\times 10^{4}$ MC steps for
(d)-(e). (f) The circularity of the vesicle shows non-monotonic variation with
time. Here, we use $F=4~{}k_{B}T/l_{min}$, $E_{ad}=0.50~{}k_{B}T$ and
$\rho=10.36~{}\%$.

Other features of cell spreading are also manifested in our spreading active
vesicles: Similar to the case of spreading cells Sheetz2010 , we find that the
circularity of the spreading active vesicles decreases sharply during the
beginning of the spreading process, and recovers slowly afterwards.
Furthermore, the active forces often give rise to the transient upwards
detachment of the leading edge in our simulations (Fig.8 (c)), resembling the
ruffles observed at the leading edge of spreading cells Sheetz2011 ; Nils2017
; begemann2019mechanochemical ; simeonov2019high .

Note that since we do not conserve the vesicle volume, we find that the volume
strongly decreases as the vesicles spread. We discuss this in more detail in
the appendix J (Fig. A-13).

###  III.4 Motile vesicles at low protein densities

In the regime of low (curved and active) protein density, where there are not
enough proteins to form a circular aggregate around the cell rim (pancake
shape), we find that the vesicles can form motile shapes. By ”motile” we mean
that the active proteins form a single large aggregate on one side of the
vesicle, that results in an unbalanced force that pushes the vesicle along the
adhesive substrate. Such motile crescent shapes are shown for example in Figs.
9 (a,c).

In Fig. 9a we show the regime of active force and adhesion energy that gives
rise to the motile crescent shapes, at very low protein density
($\rho=3.45~{}\%$). We find that the regime where the crescent shapes appear
coexists with two-arc shapes, and the two can transiently convert into each
other (see for example Fig. 9(c)). When moving, the crescent shape remains
persistent by maintaining a sharp leading edge (Fig. 9(c), inset), due to the
active force concentrated in a single cluster, while the rear region is less
curved due to minimization of bending energy and area conservation.

In Fig. 9(b) we plot the average velocity of the crescent shapes, calculated
from the simulations, divided by an effective friction coefficient that is
assumed to be linear in $E_{ad}$ (the vesicle speed before this scaling is
shown in Fig. A-14 of appendix K). Note that also the two-arc shapes are
weakly motile, as the protein aggregates at each end of the cell are not
identical in size and there is a small residual force (Fig. 9(d)). The
crescent shapes exhibited persistent motility with a well defined velocity in
the high force regime, while for weak active forces they exhibited diffusive
motility (see Fig. A-15 of appendix L).

The region of crescent shapes in Fig. 9(a) is bounded by two transition lines.
The lower line denotes the line below which the proteins do not form large
clusters. In this regime the proteins form disordered small clusters, and the
vesicle remains approximately hemispherical. Above this line, the proteins
form one or two large aggregates, thereby enabling the formation of the
crescent or two-arc shapes. Above the transition line the large adhesion
energy or strong active forces induce the sufficiently high curvature at the
vesicle contact line, which concentrates the proteins and drives the formation
of large aggregates. This is similar to the pancake (and two-arc) transition
line denoted in Fig. 7, where increasing $E_{ad}$ corresponds to higher $\rho$
at the contact-line region. Note that the direct protein-protein interaction
strength $w$ plays a minor role in this transition, which can occur even for
$w=0~{}k_{B}T$ (Fig. A-16(a) of appendix M).

The upper transition line that bounds the crescent shapes regime in Fig. 9(a)
can be estimated by comparing the bending and adhesion energies of a two-arc
shape versus the crescent shape. Compared to the crescent shape, the two-arc
shape has a lower adhesion energy, since the elongated cylindrical part is
more weakly adhered as it is devoid of curved proteins. It is also more
strongly curved compared to the circular shape of the crescent vesicle. On the
other hand, the work done by the active forces that elongate the two-arc
vesicle counts as a negative energy contribution. The net difference between
the two classes of shapes can be approximately written as

$\Delta F=-NFL+2\pi RL\frac{\kappa}{2R^{2}}+E_{ad}\Delta A$ (2)

where $R$,$L$ are the radius and length of the cylindrical segment of the two-
arc shape, $N$ is the number of proteins that pull the membrane at the two
ends of the cell and $\Delta A$ is the difference in adhered area between the
two-arc and crescent shape. We first minimize with respect to the radius $R$,
taking the total area of the cylindrical segment to be conserved $A_{cyl}=2\pi
RL$. We find that $R=2\pi\kappa/(NF)$. We substitute this value in Eq. 2, and
calculate the critical adhesion energy at which $\Delta F=0$

$\Delta F=0\rightarrow E_{ad}=\frac{A_{cyl}}{2\kappa\Delta
A}\left(\frac{NF}{2\pi}\right)^{2}$ (3)

We assumed here for simplicity that $\Delta A$ is constant along this
transition line, which is approximately obeyed by the simulation results (Fig.
A-17, appendix N). The relation in Eq. 3 appears to capture correctly the
essence of the transition between the two shapes, as shown in Fig. 9a. We fit
the Eq. 3 with simulation data points, and obtain the relation $E_{ad}\simeq
2.04~{}F^{2}$, that matches well with the simulation. We also estimate this
prefactor of $F^{2}$ by calculation $\Delta A$ and $A_{cyl}$ from simulation,
and obtain a value $\simeq 1.96$ close to the value obtained by fitting the
curve (see A-17 for details).

When comparing the motile shapes that our model produces, they bear similarity
to the shapes of different motile cells mogilner2009shape . Similar to
experimental observations dimilla1991mathematical ; klank2017biphasic ;
liu2020cell , we predict a maximum of the migration speed as function of the
adhesion strength (Fig. 9b). These results indicate that the coupling of
curved proteins that recruit the actin polymerization, and adhesion, can self-
organize into a spontaneously motile shape. However, these ingredients give
rise to rather delicate and transient motility, which can either disappear
spontaneously (Fig. 9(c)) or when confronted with an external perturbation
(Fig. A-18, appendix O). Within the wider context of active-matter systems,
our motile vesicles can be compared to recent works that have shown similar
symmetry breaking that is driven by self-organization of active elements
abaurrea2019vesicles .

Figure 9: Crescent-shaped (motile) vesicle, its speed and stability. (a)
Probability for a vesicle to be found in a crescent-shaped state (background
color), in the $E_{ad}-F$ plane for small $\rho=3.45~{}\%$. In the region
where there is non-zero probability of obtaining a crescent-shaped vesicle,
the snapshots of a crescent-shaped vesicles are shown, otherwise a typical
snapshot is shown. The yellow solid line is the analytical prediction
separating the two-arc phase to a crescent-phase (Eq.3). Here, we fit the
numerical data and obtain a good fit at $E_{ad}\simeq 2.04~{}F^{2}$. We also
numerically calculated the prefactor and it turns out to be $\simeq 1.96$,
close to the value obtained by fitting the curve (see Fig. A-17 for details).
The yellow dots are the simulation point separating the two-arc shapes from
crescent-shaped vesicle: For a given $E_{ad}$, these dots represent the value
of $F$, above which we do not find crescent shaped vesicles. The green dashed
line separates the region of vesicles with disordered small protein clusters
(below) and the regime of clustered proteins (either two-arc or crescent
shape). Here, we estimate this line by measuring the largest cluster, and if
the largest cluster is less than $30\%$ of the total proteins, we regard it to
be in the disordered state of small protein clusters. The snapshots are shown
for $E_{ad}=$ 0.50, 1.5, 2 and 3 (in units of $k_{B}T$) and $F=$ 0.50, 2 and 4
(in units of $k_{B}T/l_{min}$). (b) Speed of the crescent-shaped vesicle,
scaled by $E_{ad}$. We calculate the speed as the displacement of COM per MC
step (divided by $E_{ad}$), and then normalize all the values by the maximum
speed found for the range of parameters shown here, where the maximum value of
the speed is $0.164~{}l_{min}$ per MC step. For large $E_{ad}$ but small $F$,
the crescent vesicles exhibit diffusive behaviour (Fig.A-15). (c) Spontaneous
transition from crescent-shape to two-arc shape, demonstrating the transient
nature of the motile shapes. In the inset, we show the cross-sectional view of
a motile crescent-shaped vesicles. The red dot is showing the location of the
high protein density. (d) Examples of asymmetric two-arc shapes, which exhibit
weak residual motility. Below each shape, we also mention the ratio of
smallest to largest cluster. For both (c,d) we use $E_{ad}=3.0~{}k_{B}T$,
$\rho=3.45~{}\%$ and $F=4~{}k_{B}T/l_{min}$.

##  IV Discussions

We have shown here how interacting curved membrane proteins, passive and
active, affect the process of vesicle spreading and adhesion. We find that
large density of passive curved proteins can greatly enhance the adhesion of
vesicles on low adhesion substrates. Coupling the curved proteins with active
protrusive forces extends this enhancement to lower densities of curved
proteins. By spontaneously self-organizing curved proteins at the cell-
substrate contact line, the active forces drive a shape transition into a flat
geometry with high adhered area and robust spreading. At very low densities of
curved proteins the protrusive activity can stabilize either spindle-like
elongated cells, or motile crescent shapes.

Our simplified model does not contain all the complexities of a real cell,
which strongly affect its final adhered shape. One such component, the network
of stress-fibers, is known to determine the cell shape in many cell types
schwarz2013physics . In addition, the cytoskeleton and internal organs (such
as the nucleus) hinder the shape changes of the cell, and exert volume
constraints. Future extensions of our model can include additional components
of the cell adhesion process. For example, we could add non-uniform adhesion
that is activated closer in proximity to the curved proteins, to describe the
activation of adhesion by actin retrograde flow bershadsky2003adhesion ;
geiger2009environmental ; gardel2010mechanical ; Ibata2020nascent .
Nevertheless, our model describes many features of spreading cells, allowing
to relate the observed cell spreading dynamics and the cell shape to the
parameters of the model.

Observations in living cells emphasize the central role played by actin
polymerization during cell spreading and adhesion. These observations suggest
that in living cells the membrane density of highly curved proteins is
relatively low, and cells are not likely to be in the regime where a high
density of curved proteins alone drives the spreading and adhesion (Fig. 2).
Loading the membrane with a large density of such curved proteins may be
problematic for the cell, and limit its ability to dynamically control and
modify its spreading and adhesion strength. Our model demonstrates that by
having a low bare adhesion, and low density of curved proteins, the cell can
achieve robust and dynamic adhesion by activating the protrusive force of
actin polymerization, in a highly localized and self-organized pattern. The
spontaneous aggregation of the curved proteins along the cell-substrate
contact line, driven by the actin-induced forces (and attractive direct
interactions between the proteins), provides a highly controllable mechanism
for cell spreading and adhesion.

In addition to non-motile steady-state shapes of adhered vesicles, we found
that in the low $\rho$ regime the vesicles may form a polarized, crescent
shape, that is motile (Fig.9). This motile vesicle resembles the shapes of
motile cells, that depend on adhesion barnhart2017adhesion , and demonstrates
that the combination of curved proteins that recruit the actin polymerization,
and adhesion, provide a minimal set of ingredients needed for motility.
However, in order to make the polarization that drives the motility robust and
persistent (as opposed to transient), cells have evolved additional
biochemical feedbacks of various types ridley2003cell ; maiuri2015actin ;
rappel2017mechanisms . Our model does not contain many components that play
important roles in cell motility, such as contractility, and more realistic
treatment of the actin-adhesion coupling, such as catch and slip-bond
dynamics. Our results however highlight that curvature-force coupling, with
adhesion, provide the basic coarse-grained components that can self-organize
to spontaneously break the symmetry and form a motile system.

##  V Acknowledgements

We thank Orion Weiner, Benjamin Geiger, Ronen Zaidel-Bar, Robert Insall, Sam
Safran, Jeel Raval and Wojciech Gozdz for useful discussions. N.S.G.
acknowledges that this work is made possible through the historic generosity
of the Perlman family. N.S.G. is the incumbent of the Lee and William
Abramowitz Professorial Chair of Biophysics and this research was supported by
the Israel Science Foundation (Grant No.1459/17). A.I. and S.P. acknowledge
the support from Slovenian Research Agency (ARRS) through program No. P2-0232
and the funding from the European Union’s Horizon 2020 - Research and
Innovation Framework Programme under grant agreement No. 801338 (VES4US
project).

##  Appendix A Simulation details

The time evolution of the vesicle in our MC simulations consists of miha2019 :
(1) vertex movement, and (2) bond flip. In the vertex movement, a vertex is
randomly chosen and attempt to move by a random length and direction within a
sphere of radius $s$ drawn around the vertex. In the bond flip movement, a
single bond is chosen, which is a common side of two neighbouring triangles.
The bond connecting the two vertices in diagonal direction is cut and
reestablished between the other two, previously unconnected vertices. In order
to satisfy self avoidance, the ratio of maximum and minimum bond length, i.e.,
$l_{max}/l_{min}=1.7$ and the maximum possible displacement of a vertex in a
given attempt is taken to be $s=0.15$ in units of $l_{min}$.

We use Metropolis algorithm to update our system. Any movement that increases
the energy of the system by an amount $\Delta E$ occurs with rate $exp(-\Delta
E/k_{B}T)$, otherwise if the movement decreases the system energy, it occurs
with rate unity. We let the system evolve according the above rule and wait
till the system reaches steady state. All the average quantities are measured
after the system reaches steady state.

In the simulations presented in this paper we use the following model
parameters: Total number of vertices, $N=1447$, the bending rigidity
$\kappa=20~{}k_{B}T$, the protein-protein attraction strength $w=1~{}k_{B}T$.
The width of the potential, $\Delta z$ is taken to be $l_{min}$. Among all the
$N$ vertices, $N_{c}$ of them are occupied by curved membrane proteins. The
spontaneous curvature at all $N_{c}$ vertices are taken to be
$c_{0}=1~{}l^{-1}_{min}$, unless stated otherwise. We chose this set of
parameters to be in the interesting regime where the curved proteins form
aggregates, and exhibit a force-driven phase-separation into a pancake-like
shape miha2019 .

##  Appendix B Comparison of simulated and detailed numerical solutions of the
adhered vesicle shape

In this section, we compare the results of our MC simulations for the shapes
of adhered protein-free vesicles, with detailed numerical solutions that
appeared recently in Ref. raval2020shape ; bibissidis2020 . In the detailed
numerical solutions, the parameters used are $\tilde{w}$, the scaled adhesion
strength and $v$, the reduced volume. The parameter $\tilde{w}$ is defined as,
$\tilde{w}=E_{ad}R_{s}^{2}/\kappa$, where $E_{ad}$ is the adhesion energy per
unit adhered area and $R_{s}$ is the radius of a spherical vesicle with same
volume as the original. Since, in our model we define $E_{ad}$ as the adhesion
energy per vertex, we properly scaled it before comparison. In raval2020shape
; bibissidis2020 , the reduced volume $v$ is fixed, however, in our model, we
can not fix $v$ before the adhesion and spreading dynamics. In order to access
different values of $v$ for the same $\tilde{w}$, we use the osmotic pressure
difference $p=p_{inside}-p_{outside}$, that adds one more energy term $-pV$ to
Eq. 1, where $V$ is the total volume of the vesicle miha2019 . In Fig. A-1, we
show the comparisons of the shapes of the vesicle from our simulation and the
detailed numerical solution (as given in Fig. 5 of ref. bibissidis2020 ). We
note that the shape is comparable to our MC simulations, and the agreement is
very reasonable, thereby validating the MC approach.

Figure A-1: Comparison of the results of MC simulation with detailed numerical
solutions raval2020shape ; bibissidis2020 . The red circles are for simulation
results and the blue boxes are for detailed numerical solution. (a)
$\tilde{w}=6.4$, $v=0.545$, (b) $\tilde{w}=6.4$, $v=0.75$, (c)
$\tilde{w}=6.4$, $v=0.85$ and (d) $\tilde{w}=6.4$, $v=0.95$. For detailed
numerical solution, the data is extracted from Fig. 5 of ref. bibissidis2020
using the ‘digitize image’ tool from ‘OriginLab’ software.

##  Appendix C Analytical model

We now present an analytic calculation of the adhered shape of the vesicle, in
the presence of curved proteins and active forces. This calculation correctly
describes the qualitative features that we found in the simulations, but fails
quantitatively. The main failure is the use of protein-protein interactions
that are good for the dilute limit (Eq.A-3), and do not capture the large
density increase at the contact line as in the simulations. In regimes where
the protein-protein interactions are not playing an important role, for
example when the active force is large, we find quantitative agreement
(Figs.A-5,A-6 below).

We assume the average shape of the vesicle consists of three parts: (1) The
base area, which is having a circular shape, with radius $R_{b}$, (2) the
annulus part curving around the circular base, which is the part of a torus,
with radius of the tube $R_{t}$, and (3) the spherical cap, which is a part of
sphere, with radius $R_{c}$ (see Fig. A-2). $\theta$ is the angle between the
vertical line passing through the centre of the sphere (OQ), and the line
joining the centre of sphere and the contact point of sphere and the torus
(OP), as shown in Fig. A-2. We neglect any thermal fluctuation in the shape of
the vesicle. Total area of the vesicle, $A=A_{b}+A_{t}+A_{c}\simeq\pi
R_{b}^{2}+2\pi(1-cos\theta)R_{c}^{2}+2\pi R_{t}R_{b}(\pi-\theta)$, where
$A_{b}$ denotes the area of the base, $A_{t}$ is the area of the annulus part
(torus) and $A_{c}$ is the area of the spherical cap. Total area $A$ is taken
to be constant lipowski2005 .

Figure A-2: Schematic representation of the analytical model. $R_{c}$ is the
radius of the spherical cap, $R_{t}$ be the radius of the annulus part
(torus), $R_{b}$ be the radius of the circular disk at the base. $\theta$ is
the angle between the line perpendicular to the plane ‘OQ’ and the line
joining the center of sphere ‘O’ and the point ‘P’, where the spherical cap
and torus section meets.

Let $\rho$ be the density of the spontaneous curvature $c_{0}$, $\rho_{c}$ is
the density on cap, and $\rho_{t}$ is the density on the annulus, such that,
$\rho A=\rho_{t}A_{t}+\rho_{c}Ac$. Here, we assume that bottom part of the
vesicle does not contain proteins, which is also seen in the simulations. Now,
the adhesion energy is given by,

$W_{A}=-E_{ad}\pi R_{b}^{2},$ (A-1)

the bending energy is given by,

$W_{b}=\frac{1}{2}\kappa
A_{t}(\frac{1}{R_{b}}+\frac{1}{R_{t}}-c_{0}\rho_{t})^{2}+\frac{1}{2}\kappa
A_{c}(\frac{2}{R_{c}}-c_{0}\rho_{c})^{2},$ (A-2)

the protein-protein nearest neighbour attraction energy is of the form,

$W_{d}=-\frac{w}{2}(A_{c}\rho_{c}^{2}+A_{t}\rho_{t}^{2}),$ (A-3)

the energy due to active force,

$W_{F}=-FR_{t}\rho_{c}A_{c}-FR_{b}\rho_{t}A_{t},$ (A-4)

and finally, the entropy,

$S=-A_{c}\\{\rho_{c}ln(\rho_{c})+(1-\rho_{c})ln(1-\rho_{c})\\}-A_{t}\\{\rho_{t}ln(\rho_{t})+(1-\rho_{t})ln(1-\rho_{t})\\},$
(A-5)

where, $E_{ad}$ is the adhesion energy per unit adhered area and $\kappa$ is
the bending rigidity. Note that the entropy is only due to the thermal
fluctuation of proteins and any entropy due to the thermal fluctuation in the
vesicle shape is not considered here. In the calculation of entropy, we assume
that only the spherical cap of the vesicle and the annulus part (torus part)
contain proteins while the base of the vesicle does not contain proteins,
which is also seen in simulations. Since, the slope of the surface changes
continuously along the angle $\theta$, the slope at the contact point of torus
and spherical cap should be the same. This gives us a constrains,
$R_{c}sin\theta=R_{b}+R_{t}sin\theta$. The free energy is given by,

$F=W_{b}+W_{A}+W_{d}+W_{F}-TS$ (A-6)

We assume the total area $A$ to be a constant. We then minimize the free
energy (Eq. A-6), in the ($R_{b},\rho_{t},\theta$) plane, and express other
parameters in terms of these three parameters. More explicitly, we solve the
equations, $\frac{\partial F}{\partial R_{b}}=0$, $\frac{\partial
F}{\partial\rho_{t}}=0$, and $\frac{\partial F}{\partial\theta}=0$ with the
constraint $\frac{\partial^{2}F}{\partial R_{b}^{2}}>0$ and
$\frac{\partial^{2}F}{\partial\rho_{t}^{2}}>0$,
$\frac{\partial^{2}F}{\partial\theta^{2}}>0$. Among several solutions, we
consider the physical one.

In the analytical model, we define $E_{ad}$ as the adhesion energy per unit of
adhered area (having dimension of energy/length2), while, in the simulation,
we define it as the adhesion energy per adhered number of vertex (with
dimension of energy). In order to compare the simulation and analytical
results, we properly scale $E_{ad}$ such that the definition becomes
consistent in both the cases. The value of the parameters used here are:
$A=2200~{}l_{min}^{2}$, which is approximately the average area of a
unstretched vesicle in our simulation, $c_{0}=1.0~{}l^{-1}_{min}$,
$\kappa=20~{}k_{B}T$, $w=1.0~{}k_{B}T$. The value of $k_{B}T$ is taken to be
unity.

For passive case ($F=0$), we compare our analytical results for the cross-
sectional shape of the vesicles with different $\rho$ in Fig. A-3. We note
that the effect of increasing $\rho$ is not very strong in the analytical
model. We show the comparison of adhered fraction, density of proteins in the
curved regions ($\rho_{t}$) and the different energies in Fig. A-4. We note
that the fraction of adhered area increases with $E_{ad}$ similar to our
simulations. With increasing $E_{ad}$ the vesicle becomes more and more flat
and tends to the value $1/2$ for large enough $E_{ad}$ (Fig. A-4(a)). The
analytical prediction is however lower than the simulation results. We also
measure the density of curved proteins in the annulus part, $\rho_{t}$. As
$E_{ad}$ increases, $\rho_{t}$ also increases and tends to unity for
simulations, however, the analytical prediction is very low (Fig. A-4(b)).
This indicates that for large $E_{ad}$ most of the proteins are aggregated in
the curved annulus region. In our simulation, we also note that for large
$E_{ad}$, we do have a pancake-like phase, where all the proteins are
clustered in the curved region. We also measure the fraction $A_{ad}/A$ as a
function of $\rho$ for given $E_{ad}$ (Fig. A-4(c)). We note that similar to
the simulation results, our analytical model also shows non-monotonic
variation in $A_{ad}/A$ with $\rho$, however, here also, the quantitative
comparison is not very well. Finally, we also show the variation of $\rho_{t}$
with $\rho$, which also show an increase similar to simulation results (Fig.
A-4(d)).

We also compare the different energies in the lower panel of Fig. A-4 for
simulation and analytical cases. We note that the energy values are quite
different but the maximum difference is the protein-protein interaction energy
(Fig. A-4(g)), due to which, increasing $\rho$ is not very sensitive in our
analytical model. Thus, we conclude that because of the simplicity of the
analytical model, the quantitative comparison is not so good, however, it
could describe the qualitative features of our system very well. Next, we
compare the results for the active case.

Figure A-3: Comparison of the shape of vesicle for simulation results and
analytical results for passive case with $E_{ad}=0.50~{}k_{B}T/l_{min}^{2}$
and various $\rho$. Red circles are for simulation results and blue line is
for analytical results. Figure A-4: Results for analytical model for passive
case ($F=0$) and comparison with simulation. Red circles are for simulation
results and blue line is for analytical results. (a) Fraction of adhered area
with adhesion strength $E_{ad}$ for $\rho=10.36\%$. The adhered area increases
and tends to $1/2$ for large $E_{ad}$. (b) The density of curved proteins in
the annulus part of the vesicle ($\rho_{t}$) as a function of $E_{ad}$ for
$\rho=10.36\%$. It shows that the density $\rho_{t}$ increases and tends to
unity for large $E_{ad}$. Here also, the analytical prediction is smaller that
the simulation results. (c) Fraction of adhered area with the protein density
$\rho$ for a given $E_{ad}$. We note that the analytical prediction also shows
non-monotonic variation as seen in the simulation. (d) $\rho_{t}$ with $\rho$
for given $E_{ad}$. (e) The adhesion energy ($W_{A}$)) as a function of
$\rho$. (f) The bending energy ($W_{b}$) as a function of $\rho$. (g) The
protein-protein interaction energy ($W_{d}$) as a function of $\rho$. (h) The
total energy ($W$) as a function of $\rho$. For (c) to (h), we use
$E_{ad}=0.50~{}k_{B}T/l_{min}^{2}$. For analytical results, we use here
$A=2200~{}l^{2}_{min}$, which is the average area of a unstretched vesicle in
our simulation, $c_{0}=1~{}l^{-1}_{min}$, $\kappa=20~{}k_{B}T$,
$w=1.0~{}k_{B}T$. For analytical results, $k_{B}T$ is taken to be unity. For
MC simulations, the other parameters are same as Fig. 2.

In the active case, we compare the cross-sectional shape of the vesicle for a
given $\rho$ and different values of $F$ in Fig. A-5. We note that the effect
of increasing $F$ is very effective in analytical model. We also compare other
results in Fig. A-6. We note that the adhered fraction is very close to our MC
simulations in the large $F$ region (Fig. A-6(a)). In this case also, the
energy values are not very comparable (see Fig. A-6(c-g)).

Figure A-5: Comparison of the shape of vesicle for simulation results and
analytical results for active case with $\rho=10.36~{}\%$,
$E_{ad}=0.40~{}k_{B}T/l_{min}^{2}$ and various $F$. Red circles are for
simulation results and blue line is for analytical results. Figure A-6:
Results for analytical model for active case and comparison with simulation.
Red circles are for simulation results and blue line is for analytical
results. (a) Fraction of adhered area as a function of $F$. (b) The density of
curved proteins in the annulus part of the vesicle ($\rho_{t}$) as a function
of $F$. (c) The adhesion energy ($W_{A}$)) as a function of $F$. (d) The
bending energy ($W_{b}$) as a function of $F$. (e) The protein-protein
interaction energy ($W_{d}$) as a function of $F$. (f) The active energy due
to cytoskeleton forces ($W_{F}$) as a function of $F$ (g) The total energy
($W$) as a function of $F$. Here we use $\rho=10.36~{}\%$ and
$E_{ad}=0.40~{}k_{B}T/l_{min}^{2}$. Other parameters are same as Fig. A-4

##  Appendix D Passive vesicle with proteins having zero spontaneous curvature

In Fig.A-7 we show our results for a passive vesicle with proteins having zero
spontaneous curvature, i.e., $c_{0}=0$. This serves as a verification of our
calculation: since the proteins and the membrane are identical, we indeed find
that the adhered area remains constant with the density of proteins, only
depending on $E_{ad}$.

Figure A-7: Microstate of vesicle and cluster distributions for passive case
with flat proteins ($c_{0}=0$). The background color is showing the fraction
of adhered area. For the snapshots, we use
$\rho=3.45~{}\%,10.36~{}\%,17.27~{}\%,24.18~{}\%$ and $E_{ad}=0.75,1.5,2.5$
(in units of $k_{B}T$). The other parameters are the same as Fig. 2.

##  Appendix E Linear stability analysis for budding transition and comparison
with the contour for $<N_{cl}>=2$

In miha2019 , an expression for the critical temperature ($T^{c}$) is derived
using linear stability analysis (Eq. 6 of miha2019 ), below which the proteins
will start forming aggregates (buds) Gladnikoff2009 . We use this expression
and obtain the critical force $F^{c}$ as a function of $\rho$ and other
variables as, $F^{c}=\frac{12w}{l^{2}_{min}c_{0}}\frac{1}{(1-1/\rho
R_{0})}\Big{(}\frac{k_{B}T}{12w\rho(1-\rho)}-1\Big{)}^{2}$. Here, $1/R_{0}$ is
the mean curvature at the site of the curved membrane protein. In the limit
$R_{0}\rightarrow\infty$, the expression for $F^{c}$ will turn out to be,

$F^{c}(R_{0}\rightarrow\infty)\rightarrow\frac{12w}{l^{2}_{min}c_{0}}\Big{(}\frac{k_{B}T}{12w\rho(1-\rho)}-1\Big{)}^{2}$
(A-7)

We use the above equation to calculate the $F^{c}$ for our case, and compare
this line with the contour, where mean cluster $<N_{cl}>=2$. Since, the
budding forms along the contact line with the adhesive substrate, where the
protein density is higher ($\rho_{t}$, say) than the average $\rho$ (see for
example Fig.3a and appendix C), we use in Eq.A-7 the value of $\rho_{t}$
obtained from the simulations to calculate the critical force $F^{c}$, and
plot it as a function of the average $\rho$ (see Fig. A-8). We note that the
line is almost vertical, and in the large $F$ regime, it is in between the
pancake transition (see Fig.7) and the line $<N_{cl}>=2$. Qualitatively, the
shape of this analytic line, and its dependence on the density at the contact
line, describe the transition of the proteins into small aggregates. The
pancake transition at larger values of $\rho$ has qualitatively the same
shape, but of course can not be captured by the linear stability analysis.

Figure A-8: The critical force $F^{c}$ obtained by linear stability analysis
(Eq.A-7) using the protein density at the contact line, and is compared with
the line where $<N_{cl}>=2$. The other transition lines are as in Fig.7.

##  Appendix F Simulations with balanced total vertical force

The active force may in general act in a direction, that tries to de-adhere
the vesicle from the adhesive surface. In order to cancel this effect, we
apply an external force in the vertical direction (along $z$-direction), when
the net vertical force acts upward (that tends to de-adhere the vesicle). We
note that even after applying the external force, see Fig. A-9, there is no
qualitative change in the results. The benefit of applying this external force
is that we could now explore much smaller $E_{ad}$ which could not be explored
before, due to the de-adhesion of vesicle from the substrate.

Figure A-9: Microstates and cluster distribution of vesicle for active case
with balanced total vertical force. The external force is applied to all the
vesicle nodes, such that it balances the net vertical force when is acts in
the upward direction (i.e., when it acts against the adhesion process). We
show the snapshots and cluster distribution of the vesicle with
$E_{ad}=0.20,0.75,1.5$ (in units of $k_{B}T$) and $\rho$=3.45 %, 10.36 %,
17.27 % and 24.18 %. The green dashed line denotes the transition to a
pancake-like shape. The red dashed vertical line denotes the density for the
pancake transition for a free vesicle (without adhesion, miha2019 ). Compared
to the calculation without applying a balancing external force (Fig.4), we do
not observe any qualitative change except for the fact that in this case, we
could explore much smaller $E_{ad}$ values. Other parameters are the same as
in Fig. 4.

##  Appendix G Active vesicle with zero spontaneous curvature proteins

In this section, we show our results for the active case with proteins having
zero spontaneous curvature. The shape of the vesicle is very dynamic in this
case and changes with time. We show few snapshots of the vesicle in Fig.
A-10(a). The shapes shown in this figure should not be assumed to be a steady
state shape. We also show the dynamic nature of the vesicle in Fig. A-10(b),
where we show the snapshot for a given density ($\rho=10.36~{}\%$) at
different instant of time, for few values of $E_{ad}$. In Fig. A-10 (c), we
plot the fraction of adhered area with time, and also the snapshots near the
plot. This plot also explain the dynamic nature of the vesicle.

Figure A-10: Results for active case ($F=4~{}k_{B}T/l_{min}$) with proteins
having zero spontaneous curvature. (a) Microstate of vesicle and cluster
distributions for different values of $\rho$ and $E_{ad}$. We show the
snapshots for $\rho=3.45~{}\%,6.91~{}\%,10.36~{}\%,13.82~{}\%,17.27~{}\%$ and
$E_{ad}=3,4,5$ (in units of $k_{B}T$). (b) The dynamic nature of the vesicle
for a given density $10.36~{}\%$ and $E_{ad}=3,4,5,6$ (in units of $k_{B}T$).
(c) Variation of adhered fraction with time along with snapshots. We use here
$\rho=10.36~{}\%$ and $E_{ad}=0.50~{}k_{B}T$. The other parameters are the
same as in Fig.4.

##  Appendix H Spreading dynamics and time-dependent shapes of vesicles

Here, we show our results for the dynamics of spreading of the vesicles, by
plotting the cross-sectional shapes and snapshots with time (Fig.A-11). We
study three different cases: A protein free vesicle (Fig.A-11 (a)), a vesicle
with passive proteins (Fig.A-11 (b)) and a vesicle with active proteins
(Fig.A-11 (c)). For the protein-free vesicle, we chose a large adhesion energy
$E_{ad}=5.0~{}k_{B}T$ such that the vesicle could spread significantly. For
the passive vesicle, we chose a small adhesion energy $E_{ad}=0.50~{}k_{B}T$
and $\rho=13.82~{}\%$, in the regime where the curved proteins dominate the
spreading dynamics. For the active case, we use $F=4~{}k_{B}T/l_{min}$,
$E_{ad}=0.50~{}k_{B}T$ and a comparably smaller density of proteins
$\rho=10.36~{}\%$, where the active forces dominate the spreading. For each of
the three cases, we plot (i) the cross-section of adhered area, (ii) The side
view of the vesicle, and (iii) The three-dimensional view of the vesicle.

Figure A-11: Spreading dynamics. (a) Spreading of a protein free vesicle.
Here, we choose $E_{ad}=5~{}k_{B}T$. (i) The bottom view of the adhered area
with time, showing that the vesicle adhere quickly. (ii) The side view of the
vesicle, showing how the cross-sectional area decreases with the adhesion of
vesicles. (iii) Three-dimensional view of the vesicle with time. (b) Spreading
of passive vesicle with $\rho=13.82~{}\%$ and $E_{ad}=0.50~{}k_{B}T$. Panels
(i), (ii) and (iii) are the same as above. The adhered area grows uniformly
from the beginning. (c) Spreading of an active vesicle, with
$\rho=10.36~{}\%$, $E_{ad}=0.50~{}k_{B}T$ and $F=4~{}k_{B}T/l_{min}$. In the
beginning, the growth of adhered area is slow, then accelerates and finally
the vesicle takes the shape of a pancake. The unit of time (t) is taken as
$2\times 10^{3}$ MC steps.

##  Appendix I Quantifying pancake transition by measuring the fraction of
largest cluster

In Fig.A-12 we show our results for the fraction of largest cluster as
function of $\rho$ for active case with different values of $F$ and a small
value of $E_{ad}=0.50~{}k_{B}T$. We note that for large $F$, there is a sharp
jump in the value of the fraction at the pancake transition (dashed vertical
line in Fig. A-12 (a)), however, for small $F$, the fraction does not show any
big jump. In order to quantify the pancake transition for small $F$, we choose
a threshold value of the fraction, above which we assume the shape to be a
pancake. By monitoring the data of Fig. A-12, we choose the threshold value to
be $0.60$, above which the shape looks like a pancake.

Figure A-12: Fraction of largest cluster with $\rho$ for various values of $F$
and $E_{ad}=0.50~{}k_{B}T$. (a) $F=2.0~{}k_{B}T/l_{min}$, (b)
$F=1.0~{}k_{B}T/l_{min}$, and (c) $F=0.50~{}k_{B}T/l_{min}$. The vertical
dashed line of (a) is showing the pancake transition, which is a sharp
transition. For (b) or (c), the pancake transition is not very sharp, and the
shape is assumed to be a pancake when the size of largest cluster is at least
$60~{}\%$ of the total number of proteins.

##  Appendix J Adhered area-volume relation

As the cells spread, their volume is observed to decrease guo2017cell ;
xie2018controlling . We find similar dynamics in our model simulations. The
steady-state relation between the vesicle volume and adhered area, is also
similar to the experimental observations, although we have explored a smaller
range of values (see Fig. A-13). Note that in cells there are internal
organelles, such as the large nucleus, that limit the decrease in volume and
which our model does not describe. In addition, osmotic pressure, and ion
fluxes also contribute to the volume control in cells adar2020active .

Figure A-13: Area-Volume relation. (a) The variation of area and volume with
time for an active vesicle. The red triangles are for the volume and the blue
circles are for the area. We use here, $\rho=10.36~{}\%$,
$E_{ad}=0.50~{}k_{B}T$ and $F=4~{}k_{B}T/l_{min}$. (b) The area-volume
relation in the steady state for active case with $E_{ad}=0.50~{}k_{B}T$ and
two different values of $\rho$. The hollow magenta circles are for
$\rho=4.84~{}\%$ and the green solid circles are for $\rho=10.36~{}\%$. We
also show the snapshot of vesicles for both the $\rho$, for their highest
area. Here, we vary $F$ in order to access different area and volume in the
steady state. We note that for larger $\rho$ (which corresponds to the pancake
shape), the system can access larger area and smaller volume in comparison to
lower $\rho$ (which corresponds to the two-arc elongated shapes).

##  Appendix K Speed of the crescent-shaped vesicle in the $E_{ad}-F$ plane

In Fig.A-14 we show the speed of the crescent-shaped vesicle in the $E_{ad}-F$
plane, without scaling by $E_{ad}$ (as is shown in Fig.9b). We normalize the
speed by the maximum value, where the maximum value of the speed is
$0.328~{}l_{min}$ per MC steps. We note that, the speed is maximum for large
$E_{ad}$ and large $F$ region.

Figure A-14: Speed of the crescent-shaped vesicle in the $E_{ad}-F$ plane.
Here, we calculate the speed as the displacement of COM per MC step and then
normalize all the values by the maximum speed, where the maximum value of the
speed is $0.328~{}l_{min}$ per MC step. For large $E_{ad}$ but small $F$, the
vesicle speed is zero but it might show diffusive behaviour as well. The speed
is maximum for large $E_{ad}$ and large $F$ region.

##  Appendix L Mean square displacement of the crescent-shaped vesicle

In Fig.A-15 we show the mean square displacement of the COM of the crescent-
shaped vesicle with time, for $E_{ad}=2.0~{}k_{B}T$, $\rho=3.45~{}\%$ and
various values of $F$. We note that for large $F$, there is a finite speed
which increases with $F$, however, for small $F$, the displacement grows
diffusively.

Figure A-15: Mean square displacement (MSD) of the COM of vesicle. (a)
$F=0.25~{}k_{B}T/l_{min}$, (b) $F=0.50~{}k_{B}T/l_{min}$, (c)
$F=2.0~{}k_{B}T/l_{min}$, and (d) $F=4.0~{}k_{B}T/l_{min}$. We note that for
small $F$, MSD shows diffusive behaviour while for large $F$, the vesicle has
a finite speed. The y-axis is in the unit of $l_{min}$, and the typical size
of a vesicle (radius of a pancake shape, say) is $\sim 20~{}l_{min}$. Here, we
use $E_{ad}=2.0~{}k_{B}T$, and $\rho=3.45~{}\%$.

##  Appendix M Effect of varying the protein-protein interaction strength $w$

Throughout the paper we kept the value of the protein-protein interaction
strength ($w$) to be fixed at $w=1~{}k_{B}T$. In Fig.A-16(a), we show that
even if we take $w=0~{}k_{B}T$, the qualitative nature of the shapes of
vesicles do not change. We still obtain the two-arc shapes or the crescent-
like shapes. However, increasing $w$ to a value such that $w\gg F\cdot
l_{min}$ changes qualitatively the shapes of the vesicle, as shown for
$w=10~{}k_{B}T$ in Fig.A-16)b. The bud-like protein clusters are now solid-
like, and do not easily deform or break-up, and do not merge over time.

Figure A-16: Snapshots of vesicle with various values of protein-protein
interaction strength $w$. (a) For $w=0~{}k_{B}T$, we obtain the two-arc and
crescent-like shapes as found for the case of $w=1~{}k_{B}T$ (Fig.9). For (i),
we use $E_{ad}=2.0~{}k_{B}T$, $F=4.0~{}k_{B}T/l_{min}$ and $\rho=3.45~{}\%$,
and for (ii), we use $E_{ad}=3.0~{}k_{B}T$, $F=2.0~{}k_{B}T/l_{min}$ and
$\rho=3.45~{}\%$. (b) For very large $w=10~{}k_{B}T$, the vesicle forms small
isolated clusters, that do not break-up or merge after long time. Here, for
(i), we use $E_{ad}=2.0~{}k_{B}T$, $F=3.0~{}k_{B}T/l_{min}$ and
$\rho=3.45~{}\%$, and for (ii), we use $E_{ad}=2.0~{}k_{B}T$,
$F=4.0~{}k_{B}T/l_{min}$ and $\rho=3.45~{}\%$.

##  Appendix N Difference in the adhered area of a crescent-shaped vesicle and
a two-arc shaped vesicle across the crescent to two-arc transition line
(Fig.9(a))

The analytical calculation of the transition line (Fig. 9(a)) separating a
crescent shape and two-arc shape (Eq.3) depends on the value of $\Delta A$,
which is difference in adhered area of the crescent-shaped vesicle and the
two-arc shaped vesicle across the transition line. Here, we plot the value of
$\Delta A$ across this transition line as a function of $E_{ad}$ as extracted
from the simulations (Fig. A-17(a)). The value of $\Delta A$ does not show any
monotonic variation with $E_{ad}$, and is approximately constant along the
transition line (as was assumed in Eq.3). The average value of $\Delta A$ is
$\simeq 144.32$. The analytical calculation also contains the area of the
cylindrical part of the vesicle, $A_{cyl}$. We also measure this area, and
note that this value is also roughly a constant along the transition line
(Fig. A-17(b)). The average value of this area turns out to be $A_{cyl}\simeq
1083.52$. We assume the number of proteins at each end of the cell to be
$N=25$, half of the total number of proteins. Since, the unit of $E_{ad}$ is
different in simulations and analytical calculation, we properly scale it to
make it consistent for both the cases. Using all these values, we obtain the
prefactor of $F^{2}$ in Eq.3 as $\simeq 1.96$ This value if very close to the
value of $2.04$ obtained by fitting the simulation points along the transition
line (Fig.9a) with the equation.

Figure A-17: (a) The difference in the adhered area of a crescent-shaped
vesicle and a two-arc shaped vesicle across the crescent to two-arc transition
line, $\Delta A$ (in units of $l^{2}_{min}$), as a function of $E_{ad}$. We
note that the value of $\Delta A$ does not show any monotonic variation with
$E_{ad}$, and is approximately constant along the transition line (as was
assumed in Eq.3). (b) The area of cylindrical part in a two-arc shape
$A_{cyl}$ (in units of $l^{2}_{min}$), as a function of $E_{ad}$. This value
also seems to be roughly a constant along the transition line.

##  Appendix O Motile vesicle growing against an immobile barrier

In Fig.A-18 we allow a motile crescent shape to hit a rigid non-movable
barrier, placed perpendicular to the direction of motion of the crescent. We
take a non-adhesive as well as an adhesive barrier. In both the cases, the
crescent shape finally breaks into two-arc shape.

Figure A-18: Motile vesicle growing against an immobile barrier. (a) We plot
the snapshots of vesicle with time for a non-adhesive barrier. (b) The
snapshots of the vesicle with time for an adhesive barrier. (c) The side view
of the vesicle for the adhesive case. We note that in both the adhesive and
non-adhesive cases, the crescent-shaped vesicle breaks into two-arc shape.
Here, we use $\rho=3.45~{}\%$, $F=4~{}k_{B}T/l_{min}$ and
$E_{ad}=3.0~{}k_{B}T$. For the adhesive barrier also, we use
$E_{ad}=3.0~{}k_{B}T$.

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