# Membrane shape deformation induced by curvature-inducing proteins consisting
of chiral crescent binding and intrinsically disordered domains

Hiroshi Noguchi<EMAIL_ADDRESS>Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan

###### Abstract

Curvature-inducing proteins containing a Bin/Amphiphysin/Rvs domain often have
intrinsically disordered domains. Recent experiments have shown that these
disordered chains enhance curvature sensing and generation. Here, we report on
the modification of protein–membrane interactions by disordered chains using
meshless membrane simulations. The protein and bound membrane are modeled
together as a chiral crescent protein rod with two excluded-volume chains. As
the chain length increases, the repulsion between them reduces the cluster
size of the proteins. It induces spindle-shaped vesicles and a transition
between arc-shaped and circular protein assemblies in a disk-shaped vesicle.
For flat membranes, an intermediate chain length induces many tubules owing to
the repulsion between the protein assemblies, whereas longer chains promote
perpendicular elongation of tubules. Moreover, protein rods with zero rod
curvature and sufficiently long chains stabilize the spherical buds. For
proteins with a negative rod curvature, an intermediate chain length induces a
rugged membrane with branched protein assemblies, whereas longer chains induce
the formation of tubules with periodic concave-ring structures.

##  I Introduction

In living cells, the shapes of biomembranes are changed dynamically depending
on their functions McMahon and Gallop (2005); Suetsugu  _et al._ (2014). For
example, spherical vesicles are formed via membrane budding during
endocytosis, exocytosis, and intracellular traffic. Curvature-inducing
proteins such as clathrin and coat protein complexes bend membranes into
spherical buds Johannes  _et al._ (2015); Brandizzi and Barlowe (2013); Hurley
_et al._ (2010); McMahon and Boucrot (2011); Kaksonen and Roux (2018). In
contrast, Bin/Amphiphysin/Rvs (BAR) superfamily proteins have a crescent
binding domain to bend the membrane along the domain axis; thus, their binding
generates cylindrical membrane tubes rather than spherical buds McMahon and
Gallop (2005); Suetsugu  _et al._ (2014); Johannes  _et al._ (2015); Itoh and
De Camilli (2006); Masuda and Mochizuki (2010); Baumgart  _et al._ (2011); Mim
and Unger (2012); Frost  _et al._ (2008); Adam  _et al._ (2015). Many BAR
proteins also have intrinsically disordered domains (from $50$ to $400$
residues Pietrosemoli  _et al._ (2013); Zeno  _et al._ (2019)). These domains
are unfolded in physiological conditions. They are often removed for in vitro
experiments to investigate the interactions between the BAR domains and
membranes. However, recent experiments Stachowiak  _et al._ (2012); Busch  _et
al._ (2015); Zeno  _et al._ (2018, 2019); Snead  _et al._ (2019) have shown
that such intrinsically disordered domains can significantly influence the
membrane shapes. They changed the length of disordered domains in BAR and
other proteins using gene manipulation and revealed that the disordered
domains induce curvature sensing and also the long disordered domains induce
the formation of small vesicles. Thus, it is important to understand the role
of disordered domains in membrane remodeling.

A disordered domain behaves like a linear polymer chain in a good solvent
Hofmann  _et al._ (2012), i.e., its mean radius of gyration scales like
$R_{\rm g}\sim{n_{\rm poly}}^{0.6}$, where $n_{\rm poly}$ is the number of
Kuhn segments Doi and Edwards (1986); Strobl (1997). Interactions between
anchored polymer chains and membranes have been intensively studied in theory
Lipowsky (1995); Hiergeist and Lipowsky (1996); Bickel  _et al._ (2001); Marsh
_et al._ (2003); Bickel and Marques (2006), simulations Auth and Gompper
(2003, 2005); Werner and Sommer (2010); Wu  _et al._ (2013), and experiments
Tsafrir  _et al._ (2001, 2003); Akiyoshi  _et al._ (2003); Nikolov  _et al._
(2007); Köth  _et al._ (2011). Polymer anchoring induces a positive
spontaneous curvature of the membrane and increases the bending rigidity.
These relations are analytically derived using Green’s function method for a
low polymer density (mushroom region) Lipowsky (1995); Hiergeist and Lipowsky
(1996); Bickel  _et al._ (2001) and scaling methods for a high density (brush
region) Lipowsky (1995); Hiergeist and Lipowsky (1996); Marsh  _et al._ (2003)
and are confirmed by Monte Carlo (MC) simulations Auth and Gompper (2003,
2005); Werner and Sommer (2010). In Ref. 18, an MC method similar to that
developed in Ref. 28 was used to analyze experimental data. Under phase-
separation conditions, the polymer anchoring can induce the division of a
large domain into smaller domains to gain a larger conformational entropy of
chains Wu  _et al._ (2013). The formation of caveolae-like dimple-shaped
membranes by the polymer anchoring was discussed in Ref. 37. In a poor
solvent, the polymer anchoring can induce a negative spontaneous curvature
Werner and Sommer (2010); Evans  _et al._ (2003). Experimentally, the
formation of spherical buds Tsafrir  _et al._ (2001, 2003); Nikolov  _et al._
(2007) and membrane tubes Tsafrir  _et al._ (2003); Akiyoshi  _et al._ (2003);
Köth  _et al._ (2011) has been observed. Thus, the effects of polymer
anchoring on membrane deformation are well understood. However, to the best of
our knowledge, the combination with the anisotropic bending of the BAR domains
has not been studied yet in theory and simulations.

The aim of this study is to clarify the collective effects of the anisotropic
curvature-inducing domains and anchored polymer chains. Previously, we have
simulated anisotropic curvature-inducing proteins using a meshless membrane
method Noguchi (2022, 2014, 2015, 2016a, 2016b); Noguchi and Fournier (2017);
Noguchi (2017, 2019a); Noguchi  _et al._ (2022). The assembly of crescent rods
induces tubulation from a flat membrane Noguchi (2016a, 2017), and the protein
chirality enhances this ability Noguchi (2019a). Moreover, vesicles are
deformed into disks and polyhedral shapes, since the edges are stabilized by
protein assemblies Noguchi (2014, 2015, 2016a). In this study, anchored chains
were added to the chiral protein rod model Noguchi (2019a) to investigate how
these shape deformations are changed by polymer anchoring to proteins.

The simulation model and method are described in Sec. II. The results are
presented and discussed in Sec. III. Deformations of the vesicles and flat
membranes are described in Secs. III.1 and III.2, respectively. Finally, a
summary is presented in Sec. IV.

##  II Simulation Model

A fluid membrane is represented by a self-assembled single-layer sheet of $N$
particles. The position and orientation vectors of the $i$-th particle are
${\bm{r}}_{i}$ and ${\bm{u}}_{i}$, respectively. The membrane particles
interact with each other via a potential
$U=U_{\rm{rep}}+U_{\rm{att}}+U_{\rm{bend}}+U_{\rm{tilt}}$. The potential
$U_{\rm{rep}}$ is an excluded-volume interaction with a diameter $\sigma$ for
all pairs of particles. The effective attractive potential $U_{\rm{att}}$
implicitly represents the interaction involving the solvent. The details of
the meshless membrane model and protein rods are described in Ref. 47 and
Refs. 39; 45, respectively. Except for intrinsically disordered domains, the
protein and membrane models are identical to those used in Ref. 45.

The bending and tilt potentials are given by $U_{\rm{bend}}/k_{\rm
B}T=(k_{\rm{bend}}/2)\sum_{i<j}({\bm{u}}_{i}-{\bm{u}}_{j}-C_{\rm{bd}}\hat{\bm{r}}_{i,j})^{2}w_{\rm{cv}}(r_{i,j})$
and $U_{\rm{tilt}}/k_{\rm
B}T=(k_{\rm{tilt}}/2)\sum_{i<j}[({\bm{u}}_{i}\cdot\hat{\bm{r}}_{i,j})^{2}+({\bm{u}}_{j}\cdot\hat{\bm{r}}_{i,j})^{2}]w_{\rm{cv}}(r_{i,j})$,
respectively, where ${\bm{r}}_{i,j}={\bm{r}}_{i}-{\bm{r}}_{j}$,
$r_{i,j}=|{\bm{r}}_{i,j}|$, $\hat{\bm{r}}_{i,j}={\bm{r}}_{i,j}/r_{i,j}$,
$w_{\rm{cv}}(r_{i,j})$ is a weight function, and $k_{\rm B}T$ is the thermal
energy. The spontaneous curvature $C_{0}$ of the membrane is given by
$C_{0}\sigma=C_{\rm{bd}}/2$. Shiba and Noguchi (2011) In this study, $C_{0}=0$
and $k_{\rm{bend}}=k_{\rm{tilt}}=10$ are used except for the membrane
particles belonging to the protein rods.

Using this parameter set, the bare membrane has a bending rigidity
$\kappa/k_{\rm B}T=15\pm 1$, area of the tensionless membrane per particle
$a_{0}/\sigma^{2}=1.2778\pm 0.0002$, area compression modulus
$K_{A}\sigma^{2}/k_{\rm B}T=83.1\pm 0.4$, edge line tension
$\Gamma\sigma/k_{\rm B}T=5.73\pm 0.04$ Noguchi (2014), and the Gaussian
modulus $\bar{\kappa}/\kappa=-0.9\pm 0.1$ Noguchi (2019b). These are typical
values for lipid membranes.

A BAR domain and the membrane underneath it are modeled together as a rod,
which is a linear chain of $n_{\rm{sg}}$ membrane particles Noguchi (2014)
(see the left snapshot in Fig. 1(b)). We use $n_{\rm{sg}}=10$, i.e., the rod
length $r_{\rm rod}=10\sigma$. The protein rods have spontaneous curvatures
$C_{\rm{rod}}$ along the rod axis and no spontaneous (side) curvatures
perpendicular to the rod axis. The protein-bound membrane is more rigid than
the bare membrane as $k_{\rm{bend}}=k_{\rm{tilt}}=80$. Since two identical
binding domains of BAR proteins form a dimer that has a two-fold rotational
symmetry and chiral, we model it as follows. Two particles are connected to
the last two particles next to the chain ends by a bond potential and located
laterally on opposite sides to represent the protein chirality Noguchi (2019a)
(see cyan particles in Fig. 1(b)). Two linear chains consisting of $n_{\rm
poly}$ particles are anchored to the third rod particles from both ends (see
green particles in Fig. 1(b)). These protein particles have excluded-volume
interactions with diameter $\sigma$ as the membrane particles. Neighboring
segments in the chains are connected by a harmonic potential $U_{\rm
bond}=(k_{\rm poly}/2)(r_{ij}-r_{\rm p0})^{2}$. In this study, $k_{\rm
poly}=10k_{\rm B}T$ and $r_{\rm p0}=1.2\sigma$ are used. These excluded-volume
chains are set on one side of the membrane (outside of a vesicle or upper side
of a flat membrane).

We primarily use a rod curvature of $C_{\rm rod}r_{\rm rod}=3$ that
corresponds to that of BAR-PH Zhu  _et al._ (2007). The length of the
excluded-volume chains are varied as $n_{\rm poly}=0$, $10$, $25$, $50$,
$100$, and $200$: the mean end-to-end distance of a free chain is $r_{\rm
end}\simeq r_{\rm p0}{n_{\rm poly}}^{0.6}=0$, $4.8\sigma$, $8.3\sigma$,
$13\sigma$, $19\sigma$, and $29\sigma$, respectively, where the exponent $0.6$
is for the excluded-volume chains in a good solvent Doi and Edwards (1986);
Strobl (1997). Since the Kuhn length of unfolded proteins is $\simeq 0.8$ nm
(approximately two residues) Hofmann  _et al._ (2012), $n_{\rm poly}=100$
corresponds to a disordered domain of $200$ residues. The excluded-volume
chains are attached to all protein rods unless otherwise specified. For
tubulation from a flat membrane, the case of the excluded-volume chains
attached only to 10% of protein rods is also examined. The total number of
membrane particles $N=25600$ is used for both vesicles and flat membranes. The
radius of the spherical vesicle is $50\sigma$. The protein density varies from
$\phi=n_{\rm{sg}}N_{\rm{rod}}/N=0.0375$ to $0.1$ for the vesicles and
$\phi=0.05$ and $0.2$ for flat membranes, respectively. Zero surface tension
is used for the flat membranes. To generate an initial conformation, straight
chains are anchored first and then the chain conformation is relaxed with or
without fixing the membrane position. Molecular dynamics with a Langevin
thermostat is employed Shiba and Noguchi (2011); Noguchi (2011). For the time
unit, $\tau=r_{\rm{rod}}^{2}/D$ is used, where $D$ is the diffusion
coefficient of the membrane particles in the tensionless membrane Noguchi
(2016b). Error bars are estimated from three and ten independent runs for
vesicles and tubulations, respectively.

Figure 1:  Snapshots of vesicles at $C_{\rm rod}r_{\rm rod}=3$ and $\phi=0.1$.
A protein is displayed as a crescent rod (red and yellow) connected with two
particles (cyan) on the laterally opposite sides and with two excluded-volume
chains (transparent green) consisting of $n_{\rm poly}$ particles, as shown in
the left panel in (b). The orientation vector ${\bf u}_{i}$ lies along the
direction from the yellow to red hemispheres. The cylinder width and
transparency of anchored chains are adjusted for each snapshot for clarity.
The membrane particles are displayed as transparent gray spheres. Spherical
vesicles at $C_{\rm rod}=0$ are used as initial states. (a) Time evolution for
the protein rods with no anchored chains ($n_{\rm poly}=0$). From left to
right, $t/\tau=0$, $30$, and $450$. The membrane particles are not displayed
in the right snapshot for clarity. (b) A vesicle shape at $t/\tau=800$ for
$n_{\rm poly}=50$. (c) A vesicle shape at $t/\tau=800$ for $n_{\rm poly}=200$.
In the left snapshot, the anchored chains are not displayed and the membrane
particles are displayed as small spheres for clarity.  Figure 2:  Time
development of (a) the asphericity $\alpha_{\rm sp}$, (b) aplanarity
$\alpha_{\rm pl}$, and (c) mean cluster size $\langle N_{\rm cl}\rangle$ at
$C_{\rm rod}r_{\rm rod}=3$ and $\phi=0.1$. The blue, green, and red curves in
(a),(b) correspond to those in Figs. 1(a), (b), and (c), respectively. (c) The
cluster size is averaged for three independent runs, and the error bars are
displayed at several data points. From top to bottom, $n_{\rm poly}=0$, $10$,
$25$, $50$, and $200$.

A protein is considered to belong to a cluster when the distance between the
centers of mass of the rod region and one of the rods in the cluster is less
than $r_{\rm{rod}}/2$. The mean size of the clusters is given by
$N_{\rm{cl}}=(\sum_{i_{\rm{cl}}=1}^{N_{\rm{rod}}}i_{\rm{cl}}^{2}n^{{\rm{cl}}}_{i})/N_{\rm{rod}}$,
where $n^{{\rm{cl}}}_{i}$ is the number of clusters of size $i_{\rm{cl}}$.

To quantify vesicle shapes, we calculate two shape parameters based on the
gyration tensor $a_{\alpha\beta}=(1/N)\sum_{i}(\alpha_{i}-\alpha_{\rm
G})(\beta_{i}-\beta_{\rm G})$ of membrane particles including the bound region
of the proteins (i.e., excluding anchored chains), where
$\alpha,\beta\in\\{x,y,z\\}$:

$\displaystyle\alpha_{\rm{sp}}$ $\displaystyle=$
$\displaystyle\frac{({\lambda_{1}}-{\lambda_{2}})^{2}+({\lambda_{2}}-{\lambda_{3}})^{2}+({\lambda_{3}}-{\lambda_{1}})^{2}}{2(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}},$
(1) $\displaystyle\alpha_{\rm{pl}}$ $\displaystyle=$
$\displaystyle\frac{9\lambda_{1}\lambda_{2}\lambda_{3}}{(\lambda_{1}+\lambda_{2}+\lambda_{3})(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{3}\lambda_{1})},$
(2)

where ${\lambda_{1}}\leq{\lambda_{2}}\leq{\lambda_{3}}$ are three eigenvalues
of the gyration tensor. Spherical, disk, and rod shapes are well distinguished
by $\alpha_{\rm{sp}}$ and $\alpha_{\rm{pl}}$. The asphericity,
$\alpha_{\rm{sp}}$, indicates the shape deformation from a spherical shape to
a thin-rod shape: Rudnick and Gaspari (1986) $\alpha_{\rm{sp}}=0$, $0.25$, and
$1$ for a sphere (${\lambda_{1}}={\lambda_{2}}={\lambda_{3}}$), thin disk
(${\lambda_{1}}=0,{\lambda_{2}}={\lambda_{3}}$), and thin rod
(${\lambda_{1}}={\lambda_{2}}=0$), respectively. A discocyte (red blood cell)
shape takes $\alpha_{\rm{sp}}\simeq 0.2$. Noguchi and Gompper (2005) The
aplanarity, $\alpha_{\rm{pl}}$, indicates the deviation from a plane: Noguchi
and Gompper (2006) $\alpha_{\rm{pl}}=0$ and $1$ for a flat membrane
(${\lambda_{1}}=0$) and sphere, respectively. In addition to vesicles Noguchi
and Gompper (2005); Noguchi  _et al._ (2015); Tamemoto and Noguchi (2020) and
polymer chains Noguchi and Yoshikawa (1998), these quantities were used to
characterize the cup-shaped membrane closing into a vesicle Noguchi (2019b)
and the arrangements of pores of the vesicle Noguchi (2016c).

Figure 3:  Transition between arc-shaped and circular assemblies of protein
rods at $C_{\rm rod}r_{\rm rod}=3$. (a),(b) Snapshots of (a) the arc-shaped
and (b) circular assemblies for $n_{\rm poly}=25$ and $100$ at $\phi=0.05$,
respectively. The membrane particles are displayed as small transparent
spheres for clarity. (c) Phase diagram. The circular (arc-shaped) assembly
exists in the region above the blue line (under the red line), when the
circular (arc-shaped) assembly is used as an initial conformation. These two
states coexist in the region between two lines.  Figure 4:  Sequential
snapshots of tubulation from a flat membrane at $C_{\rm rod}r_{\rm rod}=3$,
$\phi=0.2$, and $n_{\rm poly}=10$. From top left to bottom right, $t/\tau=0$,
$30$, $70$, $90$, $120$, and $465$.

##  III Results and Discussion

###  III.1 Vesicle Deformation

First, we consider the deformation of vesicles by the proteins (Figs. 1–3). At
$C_{0}=0$, the proteins are randomly distributed on the vesicle either with or
without anchored chains (see the left snapshot in Fig. 1(a)). As the curvature
is changed to $C_{0}r_{\rm rod}=3$ at $t=0$, the proteins self-assemble on the
vesicle from this random state (Figs. 1 and 2). Before discussing the effects
of the anchored chains, we describe the vesicle deformation in their absence
($n_{\rm poly}=0$). A parallel pair of proteins have an attraction in the
direction perpendicular to the protein axis via membrane-mediated interactions
Noguchi and Fournier (2017) so that the proteins form a side-by-side assembly
along one direction. At a sufficiently low protein density $\phi$, the protein
assembly exhibits an arc shape, leading to a disk-shaped vesicle Noguchi
(2014) (see Figs. 1(a)–(c) and Movie S1). At a higher density, the arc is
closed to a circular shape, and with increasing density the assembly elongates
to an elliptic shape Noguchi (2015, 2016a). At an even higher density,
polyhedral vesicles are formed Noguchi (2015). Note that the chirality gives
negligible effects on these straight assemblies Noguchi (2019a).

Anchored chains change the vesicle shape. As anchored chains become longer at
$\phi=0.1$, vesicles exhibit a twisted-arc shape at $n_{\rm poly}=50$ (Fig.
1(b)) and multiple arc-shaped protrusions at $n_{\rm poly}\gtrsim 100$ (Fig.
1(c) and Movie S2). Although this twisted-arc shape can be temporally formed
by proteins without chains at a higher density Noguchi (2016a), it quickly
transforms into a disk shape. Conversely, the twisted shape is maintained for
a long period (at least until $t/\tau=1000$) at $n_{\rm poly}=50$ owing to the
chain interactions. The anchored chains generate repulsion between neighboring
proteins as well as between protein assemblies. Moreover, they induce an
effective spontaneous curvature, such that the protein assembly is more curved
along the arc. These interactions suppress the fusion of protein assemblies
and stabilize the shape, as shown in Fig. 1(c). To investigate the assembly
dynamics in more details, the time development of the vesicle shapes and
protein cluster sizes is plotted in Fig. 2. The disk-shaped vesicle with no
anchored chains ($n_{\rm poly}=0$) has the asphericity $\alpha_{\rm sp}\simeq
0.2$ and aplanarity $\alpha_{\rm pl}\simeq 0.3$. The chain anchoring makes
vesicles more spherical shaped in the ellipsoidal approximation (Figs. 2(a)
and (b)). The initial stage of the protein assembly becomes slower as the
chain length increases from $n_{\rm poly}=0$ to $50$, and it is almost
constant at $n_{\rm poly}\geq 50$ (see Fig. 2(c)). This threshold length
$n_{\rm poly}=50$, the average distance ($\simeq 10\sigma$) between randomly
distributed protein rods is close to the end-to-end distance of free chains.
Hence, the initial assembly is slowed by the repulsive interaction between the
anchored chains, but this effect is saturated at a chain length equal to the
protein distance.

To clarify the interaction between neighboring protein rods in the assembly,
the closing transition of the arc-shaped assembly is investigated (Fig. 3).
When long chains are anchored to an arc-shaped assembly, the assembly
transforms into a circular shape, as shown in Fig. 3(b). Conversely, when the
chain length is reduced, the circular assembly opens into an arc shape, as
shown in Fig. 3(a). The thresholds of these changes are different, as shown in
Fig. 3(c), and the two states coexist at intermediate chain lengths.
Therefore, this is a discrete transition. The mean distance between
neighboring protein rods increases with increasing chain length. Even for low
density $\phi\simeq 0.04$, where less than a half of the circumference is
occupied by the protein rods, a circular assembly is formed at long chains
(Fig. 3(c)).

Figure 5:  Snapshots of tubulation from a flat membrane at $C_{\rm rod}r_{\rm
rod}=3$ and $\phi=0.2$ (a) $n_{\rm poly}=0$. (b) 10% of the protein rods have
anchored chains of $n_{\rm poly}=100$. (c) $n_{\rm poly}=25$. (d) $n_{\rm
poly}=100$. The anchored chains are not displayed in the left snapshots in (c)
and (d) for clarity. The right snapshots in (c) and (d) are from the top view.
Figure 6:  Time development of (a) mean cluster size $\langle
N_{\rm{cl}}\rangle$ and (b) vertical membrane span
$\langle{z_{\rm{mb}}}^{2}\rangle^{1/2}$ at $C_{\rm rod}r_{\rm rod}=3$ and
$\phi=0.2$. The solid lines represent the data of the chains anchoring to all
protein rods for $n_{\rm poly}=0$, $10$, $25$, $50$, and $100$. The dashed
lines represent the data of the chains anchoring to 10% of protein rods for
$n_{\rm poly}=100$.

Figure 7:  Tubulation from a flat membrane at $C_{\rm rod}r_{\rm rod}=3$ and
low density $\phi=0.05$. (a),(b) Snapshots for (a) $n_{\rm poly}=25$ and (b)
$n_{\rm poly}=100$ at $t/\tau=600$ and $800$, respectively. (c),(d) Time
development of (c) mean cluster size $\langle N_{\rm{cl}}\rangle$ and (d)
vertical membrane span $\langle{z_{\rm{mb}}}^{2}\rangle^{1/2}$. The solid
lines represent the data of the chains anchoring to all protein rods for
$n_{\rm poly}=0$, $25$, $50$, and $100$. The dashed lines represent the data
of the chains anchoring to 10% of protein rods for $n_{\rm poly}=100$.

###  III.2 Deformation of Flat Membrane

Next, we consider tubulation from a flat tensionless membrane (Figs. 4–7). At
high density ($\phi=0.2$), tubulation occurs for all values of $n_{\rm poly}$
at $C_{\rm rod}r_{\rm rod}=3$. For short chains ($n_{\rm poly}\leq 10$),
tubules are formed via bending of the protein assembly and fuse into a larger
tubule (see Fig. 4 and Movie S3 for $n_{\rm poly}=10$ and Fig. 5(a) for
$n_{\rm poly}=0$). The protein chirality promotes tubulation via helical
assembly on tubules Noguchi (2019a). With increasing chain length $n_{\rm
poly}$, the mean cluster size $\langle N_{\rm{cl}}\rangle$ and membrane
vertical span $\langle{z_{\rm{mb}}}^{2}\rangle^{1/2}$ decrease until $n_{\rm
poly}=25$ (Fig. 6). The vertical span of the membrane is calculated from the
membrane height variance as ${z_{\rm mb}}^{2}=\sum_{i}^{N}(z_{i}-z_{\rm
G})^{2}/N$, where $z_{\rm G}$ is the center of mass in the vertical ($z$)
direction. Interestingly, the vertical span increases from $n_{\rm poly}=25$
to $n_{\rm poly}=100$. At $n_{\rm poly}=25$, the short tubules are separated
by the repulsive interaction between anchored chains, where the distance
between tubules is greater than the end-to-end distance $r_{\rm end}$ (see the
right snapshot in Fig. 5(c)). A similar stabilization of small domains was
previously obtained in a phase-separated membrane, in which polymer chains
were anchored to one of the phases Wu  _et al._ (2013). At $n_{\rm poly}=100$,
$r_{\rm end}$ is greater than the tubule distance, so that the chains push the
tubules in the upper direction and also lead to narrower shapes to gain more
conformational entropy. Hence, vertically elongated tubules are obtained as
shown in Fig. 5(d) and Movie S4.

For the low density ($\phi=0.05$), tubulation is suppressed with increasing
chain length from $n_{\rm poly}=0$ to $n_{\rm poly}=100$ (see Fig. 7). Since
fewer protein assemblies are more separated, the dependency for $n_{\rm
poly}\leq 25$ at $\phi=0.2$ is seen even at $n_{\rm poly}=100$.

Until this point, the chains are anchored to all proteins. When the chains of
$n_{\rm poly}=100$ are anchored to 10% of the proteins, their effects are
negligibly small. For $\phi=0.2$, the difference from the unanchored proteins
is not recognized (compare the magenta and black dashed lines in Fig. 6, and
also compare the snapshots in Figs. 5(a) and (b)). For $\phi=0.05$, there are
small differences after tubules are formed (compare the magenta and black
dashed lines in Fig. 7). Thus, the interactions between anchored chains are
essential for the tubulation, whereas the effects of isolated chains are
marginal.

Next, we consider the formation of spherical buds. At $C_{\rm rod}=0$, the
flat membrane is stable on our simulation time scale ($\sim 1000\tau$, see
Fig. 8(a)). However, when a tubulated membrane is used as an initial state,
one to three spherical buds are formed at $n_{\rm poly}=50$ (see Fig. 8(b) and
Movie S5). Note that the bud shapes deviate form a perfect sphere. The
membrane is elongated in the vertical direction to gain the conformational
entropy of polymer chains, and the protein rods are orientated in the vertical
direction. For $n_{\rm poly}=100$, this elongation is enhanced. When the chain
length is reduced, one large bud remains at $n_{\rm poly}=25$, but all buds
disappear at $n_{\rm poly}=10$. Thus, the chain anchoring can induce buds,
although a high free-energy barrier exists for their formation.

Finally, we consider the case in which protein rods and anchored chains bend
the membrane in opposite directions, i.e., the chains are anchored to the
concave side of the rods ($C_{\rm rod}<0$). When these opposite bending
effects are compatible, proteins form branched assemblies on the membrane as
shown in Fig. 9(a). Thus, the membrane is rugged by these bumped assemblies.
This behavior agrees well with the experimental observation of vesicles
induced by a chimeric protein of I-BAR and disordered domains Snead  _et al._
(2019). They showed that the disordered domain suppressed inward membrane
invagination, and vesicles became a rugged shape. For longer chains ($n_{\rm
poly}=50$), the anchored chains have stronger bending effects; proteins are
distributed more randomly at the low density ($\phi=0.05$) and form shish-
kebab-shaped tubules, in which concave ring-shaped rod assemblies are formed
periodically, at the high density ($\phi=0.2$), as shown in Figs. 9(b)–(d).
The membrane vertical span $z_{\rm mb}$ increases accordingly (Fig. 9(e)). The
formation of this tubule is very slow and it continues to grow at
$t/\tau\simeq 1000$ (see Figs. 9(c),(e) and Movie S6). To examine longer
tubular structures, spherical buds are used as an initial state and the shish-
kebab-shaped tubules are obtained (see Fig. 9(d)). Small buds and tubules
disappear or merge into larger tubules, as shown in Movie S7. Although a
single ring assembly was previously obtained at the neck of a spherical bud
Noguchi (2017), the periodic ring structure was found here for the first time.
Since I-BAR proteins do not contain long disordered domains, this shish-kebab-
shaped tubule is not likely to exist in living cells but can be experimentally
constructed by designing chimeric proteins of I-BAR and longer disordered
domains.

Figure 8:  Snapshots of flat membranes at $C_{\rm rod}=0$, $\phi=0.2$, and
$n_{\rm poly}=50$. (a) Randomly distributed proteins in a flat membrane. (b)
Formation of spherical buds.

Figure 9:  Deformation of flat membranes at $C_{\rm rod}r_{\rm rod}=-3$.
Excluded-volume chains are anchored to the concave side of protein rods.
(a)–(d) Snapshots of membranes at (a) $(\phi,n_{\rm poly})=(0.05,25)$, (b)
$(\phi,n_{\rm poly})=(0.05,50)$, and (c),(d) $(\phi,n_{\rm poly})=(0.2,50)$.
Randomly mixed states at $C_{\rm rod}=0$ are used as initial states for
(a)–(c), while a tubulated state at $C_{\rm rod}r_{\rm rod}=3$ is used for
(d). (e) Time development of the vertical span of the membrane. Top to bottom:
$(\phi,n_{\rm poly})=(0.2,50)$, $(0.05,25)$, and $(0.05,50)$. The snapshots in
(a)–(c) correspond to the data at $t/\tau=800$.

##  IV Summary

We have studied the membrane deformation induced by the proteins consisting of
a chiral crescent domain and two anchored chains. We have shown that these two
types of domains can bend membranes both cooperatively and uncooperatively.
Anchored chains induce an effective spontaneous curvature of the membrane but
also generate repulsion between protein rods as well as between protein
assemblies. The latter can weaken or suppress the membrane deformation by the
protein rods, depending on the conditions. For vesicle deformation, long
chains induce multi-spindle vesicles and transition from arc-shaped protein
assembly to circular assembly. For tubulation from a flat membrane, small
tubules are formed by repulsion between the crowded chains of tubules.
However, long vertical tubules are formed by very long chains that are longer
than the mean distance between tubules. Thus, the chains suppress the lateral
assembly but promote vertical elongation. Spherical buds can be formed by
straight protein rods with sufficiently long anchored chains. When the rod and
chain domains bend membranes in opposite directions at similar amplitudes,
bumped network assemblies are formed on the membrane. In contrast, shish-
kebab-shaped tubules are formed when the long chains have a stronger bending
ability.

Our results suggest that the entropic (crowding) interactions between chains
are significant to induce membrane deformation. The highly packed brush states
of the chains induce large repulsion between protein assemblies and membrane
bending. In particular, brushed chains strongly push tubules away from the
membrane, when the membrane is completely covered by the chain crowds. Our
findings provide new insights into the mechanism of membrane shape regulation
by curvature-inducing proteins with disordered domains and the design of
curvature-inducing proteins.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number JP21K03481.

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