|  Membrane-mediated interactions between arc-shaped particles strongly depend
on membrane curvature
---|---
|  Francesco Bonazzia and Thomas R. Weikla
|  Besides direct molecular interactions, proteins and nanoparticles embedded
in or adsorbed to membranes experience indirect interactions that are mediated
by the membranes. These membrane-mediated interactions arise from the membrane
curvature induced by the particles and can lead to assemblies of particles
that generate highly curved spherical or tubular membranes shapes, but have
mainly been quantified for planar or weakly curved membranes. In this article,
we systematically investigate the membrane-mediated interactions of arc-shaped
particles adsorbed to a variety of tubular and spherical membrane shapes with
coarse-grained modelling and simulations. We determine both the pairwise
interaction free energy, with includes entropic contributions due to
rotational entropy loss at close particle distances, and the pairwise
interaction energy without entropic components from particle distributions
observed in the simulations. For membrane shapes with small curvature, the
membrane-mediated interaction free energies of particle pairs exceed the
thermal energy $k_{B}T$ and can lead to particle ordering and aggregation. The
interactions strongly decrease with increasing curvature of the membrane shape
and are minimal for tubular shapes with membrane curvatures close to the
particle curvature.

##

††footnotetext: a Max Planck Institute of Colloids and Interfaces, Am
Mühlenberg 1, 14476 Potsdam Germany; email<EMAIL_ADDRESS>
## Introduction

The intricately curved shapes of biological membranes are generated by protein
assemblies 1, 2, 3, 4. BAR (BIN/Amphiphysin/Rvs) domain proteins, for example,
generate curvature by imposing their arc-like shape on membranes upon binding
5, 6, 7, 8 and can induce membrane tubules 9 covered by dense protein coats
10, 11, 12 or by less dense protein arrangements 13, 14, depending on the
protein type and concentration. While the assembly of proteins or particles in
solution is typically driven by direct molecular interactions 15, 16, 17, 18,
assemblies of membrane-associated proteins or membrane-adsorbed particles can
also result from indirect interactions that are mediated by the membrane 19,
20, 21, 22, 23. Simulations and numerical approaches with a variety of
different models 24, 25 indicate that such indirect interactions play a role
in the tubulation of membranes by assemblies of arc-shaped proteins and
particles 26, 27, 28, 29, 30, 31, 32, can lead the cooperative wrapping of
spherical 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43 and elongated particles
37, 38, and can result in assemblies of membrane-adsorbed Janus particles 44,
45, 46, 47, 48, 49, 50, elastic particles 51, and hinge-like particles 52, 53,
54. Membrane-mediated interactions and assembly of spherical particles 55, 56,
57, 58, 59, 60, 61 as well as the cooperative wrapping of spherical virus-like
particles 62 and rod-like particles 63 by membranes have also been observed in
experiments.

The membrane-mediated interactions result from a change of the membrane
curvature induced by the particles or proteins and, therefore, can be expected
to depend on the equilibrium curvature or shape of the membranes. However,
membrane-mediated pair interactions have mainly been quantified for model
proteins or particles that are adsorbed to or embedded in initially planar or
weakly curved membranes 64, 65, 66, 67, 68, 69, 70, 71, 44, 45, 72, 73, 74,
75. Recent exceptions are the pair interactions of spherical Janus particles
46, 48, which have been found to depend on whether the particles adsorb to the
inside or outside of the vesicles, and the pair interactions of spherical
particles adsorbed to vesicles with different sizes 76.

In this article, we determine the membrane-mediated pair interactions of arc-
shaped particles on tubular and spherical membrane vesicles from pair
distributions of the particles observed in Monte Carlo simulations. In our
coarse-grained model of membrane shaping, the membrane is described as a
triangulated elastic surface, and the particles as segmented arcs that induce
membrane curvature by binding to the membrane 31. In previous work, we found
that the vesicle morphologies induced by the particles are determined by the
arc angle and membrane coverage of the particles, and that membrane tubulation
is induced by the particles above a threshold coverage of roughly 50% 31, 32,
in agreement with experimental observations for amphiphysin N-BAR domains 14.
In this work, in contrast, the vesicle morphology is fixed by the volume-to-
area ratio of the vesicles, and the particle coverage in our simulations as
adjusted to relatively small values between 5% and 20% to identify the
particles’ pair interactions. We determine the interaction free energy of
particles bound to membrane tubules with different thicknesses and to
spherical membrane vesicles by comparing the pair distributions of the
particles obtained from simulations to ideal distributions of non-interacting
particles. These interaction free energies include entropic components from
the loss of rotational entropy of the arc-shaped particles at close distances,
besides the curvature-mediated interaction energies. In addition, we determine
the interaction energies of the particles by comparing the pair distributions
obtained from simulations to hard-cord distributions of ‘flattened’ particles
with the same shape, to quantify the curvature-mediated interaction of the
particles without entropic components. We find that both the interaction free
energies and the interaction energies strongly decrease with increasing
curvature of the membrane shape and are minimal for tubular shapes with
membrane curvatures close to the particle curvature.

## Results

Fig. 1: Exemplary simulation conformations for particles with arc angle
$60\degree$ at different area coverages $x$ and reduced volume $v$ of the
membrane. Fig. 2: Exemplary simulation conformations for particles with arc
angle $90\degree$ at different area coverages $x$ and reduced volume $v$ of
the membrane.

### Conformations of arc-shaped particles on tubular and spherical vesicles

Figs. 1 and 2 illustrate simulations conformations of closed tubular and
spherical membranes with bound arc-shaped particles. In our simulation model,
the membranes are dynamically triangulated surfaces composed of 2000 triangles
with a constrained total area $A$ (see Methods). The volume $V$ enclosed by
the membrane is constrained to different values in the simulations, which
results in the different membrane shapes shown in Figs. 1 and 2. The fixed
volume-to-area ratio of the membrane can be quantified by the reduced volume
$v=6\sqrt{\pi}V/A^{3/2}\leq 1$, which adopts its maximum value of 1 for an
ideal sphere. In our simulations of spherical membranes, we fix the reduced
volume to $v=0.95$ to allow for small variations and fluctuations around the
overall spherical shape. In our simulations of spherocylindrical, tubular
membranes, the reduced volume is fixed to $v=0.35$, $0.4$, $0.45$, $0.5$,
$0.55$, $0.6$, or $0.65$. The tubular membrane shapes are metastable, because
the stable membrane shape is an oblate ellipsoid for $0.592\lesssim v\lesssim
0.652$ and a stomatocyte for $0<v\lesssim 0.592$ 77, 78. The metastable
tubular shapes are protected against transformations into the stable membrane
shapes by an energy barrier that depends on the bending rigidity $\kappa$ 78.
To ensure a sufficiently large energy barrier, we use the relatively large
bending rigidity value $\kappa=30\,k_{B}T$ from the range of typical bending
rigidities of lipid membranes between about $10$ and $40\,k_{B}T$ 79, 80.

The arc-shaped particles of our model are composed of either 3 or 4 quadratic
segments with side length $a_{p}$ and an angle of $30\degree$ between adjacent
segments. The arc angle of the particles, i.e. the angle between the terminal
segments, thus is $60\degree$ for particles composed of 3 segments, and
$90\degree$ for particles composed of 4 segments. The particles bind to the
membrane with their inner, concave sides. A particle segment is bound to the
triangulated membrane of our model if its distance to the closest membrane
triangle is within a given range, and if the particle segment and membrane
triangle are nearly parallel with an angle that is smaller than the cutoff
angle $10\degree$ (see Methods for details). The relative area of the particle
segments and membrane triangles is chosen such that a particle segment can
only be bound to a single membrane triangle. The particles can bind to and
unbind from the membranes in the simulation. In each simulation, the area
fraction $x$ of the membrane covered by bound particles is kept at a constant
value between 5% and 20% by dynamically adjusting the adhesion energy $U$ per
particle segment. The total number of bound and unbound particles in our
simulations is 400.

The simulation conformations in Figs. 1 and 2 illustrate a tendency of the
particles to align side by side that increases with increasing reduced volume
$v$ of the membrane and with increasing arc angle of the particles. At the
reduced volume $v=0.95$, particles with arc angle 90°form linear aggregates on
the spherical membranes (see Fig. 2). This alignment and aggregation of the
particles is driven by membrane-mediated interactions, because the direct
particle-particle interactions are purely repulsive in our model (see
Methods).

### Interaction free energies

Fig. 3: (a) Two-dimensional interaction free energy of pairs of arc-shaped
particles with arc angle $60\degree$ for vesicle shapes with reduced volume
$v=0.5$, $0.65$, and $0.9$ and a particle coverage of 10%. (b) Two-dimensional
interaction free energy of pairs of arc-shaped particles with arc angle
$90\degree$ for vesicle shapes with reduced volume $v=0.5$, $0.6$, and $0.9$
and a particle coverage of 15%. (c,d) Minimum interaction free energy versus
membrane curvature for particles with arc angle $60\degree$ and $90\degree$ at
different particle coverages of the membrane. For tubular membrane shapes, the
membrane curvature here is calculated as the curvature for the tubular section
of an ideal spherocylinder with the same area $A$ as the membrane in our
simulations. For spherical shapes, the membrane curvature is calculated as the
curvature of an ideal sphere with the same area $A$ as the membrane in our
simulations. The dashed extrapolation lines to curvature 0 are guides for the
eye.

The membrane-mediated pair interactions of the particles can be quantified by
two-dimensional interaction free energies, which we obtain from the two-
dimensional pair distributions $P(x,y)$ of the particles observed in our
simulations. These free energies reflect the highly anisotropic interactions
of a particle along its side ($x$-direction) and along its tip ($y$-direction)
to a neighboring particle, and are two-dimensional generalizations of the one-
dimensional potential of mean force 81. The two-dimensional interaction free
energies are calculated as

$F(x,y)=-k_{B}T\ln\left[P(x,y)/P_{id}\right]$ (1)

where $P_{id}$ is the pair distribution of a non-interacting, ideal gas of
particles (see Methods for details). These interaction free energies include
entropic components from the reduction of the rotational entropy of the arc-
shaped particles at close distances, besides the curvature-mediated
interaction energies. For tubular vesicle shapes with reduced volume $v\leq
0.65$, we determine the pair distributions $P(x,y)$ in the central, tubular
membrane segments and exclude particle pairs at the spherical ends (see
Methods).

Fig. 4: Radial pair distributions $p(r)$ of membrane-bound particles obtained
from simulations with vesicle shapes for different reduced volumes $v$
(coloured data points), ideal distributions of non-interacting point particles
on the corresponding cylinders and spheres (grey dashed lines), and hardcore
distribution $p_{hc}(r)$ of ‘flattened’ particles with the same shape in a
plane (full grey line). The area coverages of the membrane-bound particles are
$x=0.05$ in (a) for particles with arc angle $60\degree$ and $x=0.09$ in (b)
for for particles with arc angle $60\degree$. For the tubular membrane shapes
with reduced volumes $v=0.4$, $0.5$, and $0.6$, the pair distributions $p(r)$
exhibit maxima when the particle distance $r$ is equal to the diameter $2R$.
The radial pair distributions of ideal gas on a cylinder here are calculated
as $p_{\rm ideal}(r)=8R\,f_{\rm
elliptic}[0.5\arccos[1-r^{2}/(2R^{2})],4R^{2}/r^{2}]$ where $f_{\rm elliptic}$
is the elliptic integral of the first kind.

Fig. 3(a) and (b) display the resulting two-dimensional interaction free
energies of the simulated particles with arc angle $60\degree$ and $90\degree$
at three different values of the reduced volume $v$ of the vesicles for
intermediate particle coverages of 10% and 15%, respectively. At the small
value $v=0.5$, the vesicles adopt elongated, thin tubular shapes (see Figs. 1
and 2). At this value of $v$, the two-dimensional free energy of particles
with arc angle $60\degree$ exhibits two minima with nearly equal depth of
about $-0.15$ $k_{B}T$ for pair conformations in which the particles are
oriented side-to-side and tip-to-tip, respectively (see Fig. 3(a)). At the
intermediate value $v=0.65$ at which the vesicles adopt thicker tubular
shapes, the depths of the minima increases to about $-0.45$ $k_{B}T$ for side-
to-side alignment of particles with arc angle $60\degree$, and to about $-0.3$
$k_{B}T$ for tip-to-tip alignment. At $v=0.95$, the vesicles adopt spherical
shapes, and the depth of the minimum for side-to-side alignment of the
particles further increases to nearly $-1.0$ $k_{B}T$, whereas tip-to-tip
alignment of the particles is no longer energetically favourable.

For particles with arc angle $90\degree$, the free-energy minima for pair
conformations in which the particles are oriented side-to-side are clearly
deeper compared to particles with arc angle $60\degree$, while free-energy
minima for tip-to-tip alignment of the particles no longer occur (see Fig.
3(b)). These deeper free-energy minima for side-to-side orientation of the
particles indicate clearly stronger curvature-mediated interactions of the
particles, because the rotational entropy loss in side-to-side conformations
is larger for the longer particles with arc angle $90\degree$, compared to
particles with arc angle $60\degree$. At the reduced vesicle volume $v=0.5$,
the free-energy minimum for side-to-side alignment of the particles has a
depth of about $-0.85$ $k_{B}T$. At the larger values $v=0.6$ and $v=0.95$,
several minima in side-to-side direction of the particles appear, which
reflects the tendency of the particles to form linear chains with side-to-side
orientation in the simulations at these values of $v$ (see Fig. 2). The
minimum at the smallest distance along the particle side reflects the
interaction free energy of nearest neighbours and has a depth of about $-1.6$
$k_{B}T$ for $v=0.6$ and $-3.4$ $k_{B}T$ for $v=0.95$.

Fig. 3(c) and (d) illustrate how the interaction free-energy minima for side-
to-side alignment of the particles depend on the curvature of the membrane at
the three different particle coverages considered in our simulations. For
tubular membrane shapes, the membrane curvature here is calculated as the
curvature for the tubular section of an ideal spherocylinder with the same
area $A$ as the membrane in our simulations. For spherical shapes, the
membrane curvature is calculated as the curvature of an ideal sphere with the
same area $A$ as the membrane in our simulations. For particles with arc angle
$60\degree$, the interaction is weakest at membrane curvatures of about
$0.45/a_{p}$, with a minimum interaction free energy of about $-0.15$
$k_{B}T$. These membrane curvature roughly corresponds to the induced
curvature of the particles, which can be estimated from the equation
$c=\theta/L$ for the curvature $c$ of a circular arc with angle $\theta$ and
arc length $L$. For the average induced angle of $52.5\degree$ between
membrane triangles bound to the terminal segments of a particle 31 and the
particle arc length $L\simeq 2a_{p}$ between the centers of these terminal
segments with side length $a_{p}$, we obtain $c\simeq 0.46/a_{p}$. The
interaction is strongest for spherical vesicles with a curvature of about
$0.15/a_{p}$, with a minimum interaction free energy of about $-1$ $k_{B}T$.
The dashed lines in Fig. 3(c) are extrapolations to planar membranes with
curvature $0$ and suggest pair interaction free-energy minima of about $-1.2$
$k_{B}T$ in planar membranes.

For particles with arc angle $90\degree$, the interaction is weakest at the
largest membrane curvature of about $0.53/a_{p}$ obtained for the thinnest
simulated tubules with reduced volume $v=0.35$ (see Fig. 3(d)). The
interaction free-energy minima at this largest curvature range from about
$-0.45$ to $-0.5$ $k_{B}T$, depending on the particle coverage $x$ of the
membranes. For particles with arc angle $90\degree$, the induced curvature is
about $c=\theta/L\simeq 0.48/a_{p}$ for an average induced angle $\theta\simeq
82.6\degree$ 31 and arc length $L\simeq 3a_{p}$ and, thus, slightly larger
than for particles with arc angle $60\degree$. The interaction is again
strongest for spherical vesicles with a curvature of about $0.15/a_{p}$, with
a minimum interaction free energy in the range from $-2.9$ to $-3.4$ $k_{B}T$.

### Side-to-side interaction energy

Fig. 5: (a) and (b) Interaction energy profiles of pairs of arc-shaped
particles with arc angle $60\degree$ at different values of the reduced volume
$v$ of vesicles and of the coverage $x$ of the particles. (c) Minima of the
interaction energy profiles versus membrane curvature for particles with arc
angle $60\degree$ at coverages of $x=0.05$, $0.1$, and $0.2$. The membrane
curvature here is calculated as in Fig. 3, and the dashed extrapolation lines
to curvature 0 are guides for the eye. (d) and (e) Interaction energy profiles
of pairs of arc-shaped particles with arc angle $90\degree$ at different
values of $v$ and $x$. (f) Minima of the interaction energy profiles versus
membrane curvature for particles with arc angle $90\degree$.

Besides interaction free energies of the particles, curvature-induced
interaction energies without entropic components from rotational entropy
losses can be obtained by comparing pair distributions $p$ from simulations to
hard-cord distributions $p_{hc}$ of ‘flattened’ particles with the same shape.
We focus now on one-dimensional radial distributions and on the curvature-
induced interaction energies for side-to-side aligned particle pairs, which
are reflected by energetic minima at small separations that the particles can
only achieve in side-to-side alignment. We calculate the one-dimensional,
distance-dependent interaction energy of the particles as

$E(r)=-k_{B}T\ln\left[\frac{p(r)}{p_{hc}(r)}\right]$ (2)

with hard-core distributions $p_{hc}(r)$ determined for ‘flattened’ particles
with an angle of $0$ between the quadratic segment of side length $a_{p}$,
rather than the angle of 30° of the curved, membrane-bound particles (see
Methods for details).

Fig. (4) illustrates the radial distributions $p(r)$ obtained from our
simulations at different reduced vesicle volumes $v$ (coloured data points) as
well as the hard-core distributions $p_{hc}(r)$ of the particles with arc
angle $60\degree$ and $90\degree$ (full grey lines), and compares these
distributions to the ideal distributions of non-interacting particles on
cylinders for $v\leq 0.6$ and spheres for $v\leq 0.95$ (dashed grey lines).
The distributions $p(r)$ and $p_{hc}$ are 0 at particle distances $r<a_{p}$
that are prevented by the hard-core interactions of the particles. At
distances $r>a_{p}$, the hard-core distributions $p_{hc}(r)$ gradually
increase and exhibit two kinks at distances $r$ at which the particles acquire
more rotational freedom. For particles composed of 3 segments, the first kink
of $p_{hc}(r)$ occurs at the distance $r=2a_{p}$ above which a particle that
is located at the side of another particle gains full rotational freedom. The
second kink occurs at the distance $r=3a_{p}$ above which a particle located
at the tip of another particle has full rotational freedom. For particles
composed of 4 segments, these kinks occur at corresponding, larger distances
$r$. The curvature-mediated interaction of the particles is reflected by
clearly larger values of the distributions $p(r)$ at close distances
$a_{p}<r<3a_{p}$ compared to the hard-core distributions $p_{hc}(r)$. For
particles with arc angle $60\degree$, the radial distributions $p(r)$ approach
the ideal distributions at larger distances $r>3a_{p}$. For particles with arc
angle $90\degree$, the tendency of the particles to form linear chains for
reduced vesicle volume $v\gtrsim 0.5$ (see Fig. (2)) leads to more global
deviations from ideal distributions, and to multiple maxima of $p(r)$ for
spherical vesicles with $v=0.95$.

Fig. (5) illustrates the interaction energy profiles $E(r)$ and minimum
interaction energies obtained from Eq. (2) for the distributions $p(r)$
obtained from our simulations and the numerically determined hard-core
distributions $p_{hc}(r)$ at the corresponding particle coverages $x$ (see
Methods for details). For particles with arc angle $60\degree$, the
interaction energy profiles $E(r)$ tend to values close to 0 at larger
distances $r>3a_{p}$ at which the distributions approach each other (see Figs.
5(a,b) and 4(a)). Our main aim are the minima of $E(r)$ at short distances $r$
that reflect the curvature-mediated interaction energy of the particles in
side-to-side alignment. These interaction energy minima strongly depend on the
membrane curvature (see Fig. 5(c)). As in Fig. 3(c), the interaction is
weakest at membrane curvatures of about $0.45/a_{p}$ with minimum values from
$-1.25$ to $-1.6$ $k_{B}T$, depending on the particle coverage $x$. The
interaction is strongest at the curvature of about $0.15/a_{p}$ of our
spherical vesicles with minimum values from $-2.0$ to $-2.2$ $k_{B}T$. The
minimum values of the interaction energies are roughly $1$ $k_{B}T$ lower than
the minima of the interaction free energies in Fig. (3(c)) at corresponding
curvatures. This difference of about $1$ $k_{B}T$ can be seen as the free-
energy contribution from the loss of rotational entropy of the particles in
the minimum-energy side-to-side pair conformations.

For particles with arc angle $90\degree$, the minima of the interaction energy
profiles $E(r)$ at short distances $r$ are clearly lower than for particles
with arc angle $60\degree$ (see Figs. 5(d) to (f)). For spherical vesicles
with $v=0.95$, the linear chains of particles observed in the simulations lead
to multiple minima of $E(r)$ at the larger particle coverages $x=0.15$ and
$0.20$, and to more global deviations from the hard-core distributions. The
global minimum values of the interaction energies are roughly $1$ to $2$
$k_{B}T$ lower than the minima of the interaction free energies in Fig. (3(c))
at corresponding curvatures (see Figs. 3(d) and 5(f)), which reflects a
slightly larger energy loss of rotational entropy for the longer particles
with arc angle $90\degree$, compared to particles with arc angle $60\degree$.

## Discussion and Conclusions

In this article, we have presented a general methodology to obtain the
membrane-mediated interaction free energies and interaction energies of
anisotropic curvature-inducing particles from distributions observed in
simulations at relatively low membrane coverages of the particles. In this
methodology, two-dimensional interaction free energies are calculated in
comparison to ideal distributions of non-interacting particles, as
generalization of the one-dimensional potential of mean force. The two-
dimensionality of the interaction free energy is necessary to correctly
capture the rotational entropy loss of the elongated particles at close
contact, and the low coverages are required to identify the pair interactions
of the particles, which should be independent of the membrane coverage of the
particles. For the membrane coverages $x$ between $5$% and $20$% in our
simulations, the minima of the interaction free energy depend only rather
weakly on $x$ (see Fig. 3(c) and (d)). Our main aim was to determine the
dependence of the particles’ interaction free energy on the overall curvature
to the membrane, which we adjusted by the reduced volume $v$ of membrane
vesicles, from values of $v=0.35$ for thin tubular vesicles to $v=0.95$ for
spherical vesicles (see Figs. 1 and 2). We found that the pair interaction is
smallest for tubular versicles with a curvature close to the particle
curvature, with free energy minima of around $-0.2$ $k_{B}T$ and $-0.5$
$k_{B}T$ for particles with arc angle $60\degree$ and $90\degree$,
respectively. The pair interaction is largest for spherical vesicles with free
energy minima of around $-1$ $k_{B}T$ and $-3$ $k_{B}T$ for the two types of
particles. We believe that the interaction free energies of our particles with
arc angle $60\degree$ are realistic for BAR domain proteins such as the
Arfaptin BAR domain and the Endophilin and Bin1 N-BAR domains, (i) because the
induced average angle $52.5^{\circ}$ of these particles 31 roughly corresponds
to the angle enclosed by these BAR domain proteins 82, and (ii) because the
electron tomography images of membrane tubules induced by Bin1 N-BAR domains
proteins show a rather loose protein arrangement with only short-ranged order,
similar to the tubular morphologies induced by the particles in our
simulations 31, 32. In general, the pair interaction free energy can be
expected to depend also on the coupling between the particles and the
membrane, which is determined by the particle-membrane interaction potential
in our model (see Methods).

Besides interaction free energies, we determined the interaction energies of
the particles without entropic component from the distributions observed in
our simulations and numerically determined hard-core distributions of
‘flattened’ particles with the same shape. The minima of the interaction
energy profiles are located at short particle distances at which the particles
are aligned side-by-side and exhibit the same dependence on membrane curvature
as the minima of the interaction free energies. However, the interaction
energy minima are about $-1$ $k_{B}T$ and between $-1$ to $-2$ $k_{B}T$ lower
compared to the free energy minima obtained for the particles with arc angle
$60\degree$ and $90\degree$, respectively. These differences between the
minima of the interaction energies and interaction free energies result from
the rotational entropy loss of the particles at close contact and can be seen
to quantify the free energy contribution of these entropy loss.

## Methods

### Model and simulations

We model the membrane as a dynamically triangulated, closed surface. The
membrane model is based on a standard discretization 83, 33 of the bending
energy ${\cal E}_{\text{be}}=2\kappa\oint M^{2}\,dS$ of closed fluid
membranes84 with bending rigidity $\kappa$ and local mean curvature $M$. Our
discretized membranes are composed of $n_{t}=2000$ triangles. In our Monte
Carlo (MC) simulations, the edges of the triangulated membrane are flipped to
ensure membrane fluidity (dynamic triangulation), and the vertices of the
triangulation are displaced to allow local changes of the membrane shape. The
edge lengths of the triangles are kept within an interval
$[a_{m},\sqrt{3}a_{m}]$ to limit triangle distortions. The vertex
displacements occur in MC steps in which a randomly selected vertex of the
triangulated membrane is translated along a random direction by a distance
that is randomly chosen from an interval between $0$ and $0.1a_{m}$. We impose
an overall tubular or spherical membrane shape by constraining both the
membrane area $A$ and the enclosed volume $V$ of the membrane via harmonic
constraining potentials. The volume-to-area ratio of the membrane therefore is
fixed in our simulations.

The discretized, arc-shaped particles of our model are linear chains of 3 or 4
identical planar quadratic segments with a side length $a_{p}$ and with an
angle of $30\degree$ between neighboring segments that share a quadratic
edge.31, 85 The arc angle of the particles, i.e. the angle between the first
and last segment, then adopts the values $60\degree$ and $90\degree$ for
particles composed of 3 and 4 segments, respectively. Each planar segment of a
particle interacts with the nearest triangle of the membrane via the particle-
membrane adhesion potential 31

$V_{\text{pm}}=-Uf_{r}(r)f_{\theta}(\theta)$ (3)

Here, $r$ is the distance between the center of the segment and the center of
the nearest triangle, $\theta$ is the angle between the normals of the
particle segment and this membrane triangle, and $U>0$ is the adhesion energy
per particle segment. The distance-dependent function $f_{r}$ is a square-well
function that adopts the values $f_{r}(r)=1$ for $0.25\,a_{m}<r<0.75\,a_{m}$
and $f_{r}(r)=0$ otherwise. The angle-dependent function $f_{\theta}$ is a
square-well function with values $f_{\theta}(\theta)=1$ for
$|\theta|<10\degree$ and $f_{\theta}(\theta)=0$ otherwise. By convention, the
normals of the membrane triangles are oriented outward from the enclosed
volume of the membrane, and the normals of the particle segments are oriented
away from the center of the particle arc. The particles thus bind with their
inward curved, concave surface to the membrane. The overlapping of particles
is prevented by a purely repulsive hard-core interaction that only allows
distances between the centres of the planar segments of different particles
that are larger than $a_{p}$. The hard-core area of a particle segment thus is
$\pi a_{p}^{2}/4$. We choose the value $a_{p}=1.5a_{m}$ for the side length of
the planar and quadratic particle segments. The particle segments then are
slightly larger than the membrane triangles with minimum side length $a_{m}$,
which ensures that different particle segments bind to different triangles.

The positions and orientations of the particles are varied by particle
translations and rotations in our simulations 31. In a particle translation, a
randomly selected particle is translated in random direction by a random
distance between $0$ and $a_{m}$. In a particle rotation, a randomly selected
particle is rotated around a rotation axis that passes through the central
point along the particle arc. For particles that consist of 3 segments, the
rotation axis runs through the center of the central segments. For particles
composed of 4 segments, the rotation axis runs through the center of the edge
that is shared by the two central segments. The rotation axis is oriented in a
random direction. The random rotations are implemented using quaternions 86,
87 with rotation angles between 0 and a maximum angle of about $2.3\degree$.
Our simulations thus consist of two types of MC steps for the membrane, vertex
translations and edge flips, and two types of MC steps for translating and
rotating the particles. The different types of MC steps occur with equal
probabilities for single membrane vertices, edges, or particles 31.

In each simulation, the area fraction $x$ of the membrane covered by bound
particles is kept at a constant value between 5% and 20% by dynamically
adjusting the adhesion energy $U$ per particle segment. The total number of
particles in the system is N = 400. The area of the membrane is constrained to
$A_{0}\simeq 0.677n_{t}a_{m}^{2}$. The strength of the harmonic constraining
potential is chosen such that the fluctuations of the membrane area are
limited to less than $1\%$. The simulations are run in a cubic box with
periodic boundary conditions and volume $V_{\rm box}\simeq 3\cdot
10^{5}a_{m}^{3}$. The simulation box volume is 64 times as large as the volume
of a perfect sphere with area $A_{0}$. After initial relaxation of the
simulations until a steady state regarding the number of bound particles is
reached, the simulation frames from which the particle distributions are
determined are extracted from the simulations at time intervals of 100 Monte
Carlo steps per vertex. For particles with arc angle $60\degree$, the total
number of analysed conformations for tubular vesicles with reduced volume
$v\leq 0.65$ is about $180\,000$ for the membrane coverage $x=5\%$, $145\,000$
for $x=10\%$, and $100\,000$ for $x=20\%$. For spherical vesicles with reduced
volume $v=0.95$, the total number of conformations is $150\,000$, $85\,000$,
and $61\,000$ at $x=5\%$, $10\%$, and $20\%$, respectively. For particles with
arc angle $90\degree$, the total number of analysed conformations for tubular
vesicles with reduced volume $v\leq 0.65$ is about $155\,000$ for the membrane
coverage $x=9\%$, $125\,000$ for $x=15\%$, and $90\,000$ for $x=20\%$. For
spherical vesicles with reduced volume $v=0.95$, the total number of
conformations is $150\,000$, $88\,000$, and $59\,000$ at $x=9\%$, $15\%$, and
$20\%$, respectively.

### Interaction free energy

We determined the two-dimensional interaction free energies of Fig. 3 from
two-dimensional pair distributions of the particles obtained from our
simulations. These two-dimensional pair distributions reflect the
distributions of neighbouring particles around a particle on the surface of
the spherical or tubular membranes. For spherical membrane shapes, we
determine the position of a neighbouring particle $j$ relative to a particle
$i$ based on an angle $\phi$ between two planes P and Q. Plane P runs through
the centres of the two terminal segments of particle $i$ and through the
center of mass of the spherical membrane and, thus, reflects the orientation
of particle $i$ on the membrane sphere. Plane Q runs through the centre of
particle $i$, the center of particle $j$, and the center of mass of the
spherical membrane. For particles with arc angle 60°, which are composed of
three segments, the particle center simply is the center of the central
particle segment. For particles with arc angle 90°, which are composed of four
segments, the particle center is the center of mass of the two central
particle segments. The value $\phi=0$ for the angle between planes P and Q
indicates that the center of particle $j$ is located at the tip of particle
$i$, and $\phi=\pi/2$ indicates that the center of particle $j$ is located at
the side of particle $i$. The two-dimensional particle distributions then are
calculated from the two coordinates $x=r\sin\phi$ and $y=r\cos\phi$ of all
particle pairs where $r$ is the distance of the particle centres. To determine
these distributions, we discretize the plane of the coordinates $x$ and $y$
into squares with side length $a_{p}/5$ and count the total number $M(x,y)$ of
particle pairs with coordinate values in the square centered at $(x,y)$ for
all conformations. The probability for finding a neighboring particle in the
square centered at $(x,y)$ can then be determined as
$P(x,y)=M(x,y)/(4N_{c}N_{b}(N_{b}-1))$ where $N_{c}$ is the number of
conformations and $N_{b}$ is the average number of bound particles in a
conformation. The factor 4 takes into account that the angle
$0\leq\phi\leq\pi/2$ is limited to one of the four angular quadrants because
of the particle symmetry, and $N_{b}(N_{b}-1)$ is the average number of
particle pairs per conformation that is considered in this calculation. For
ideal, non-interacting particles, the probability of finding a neighboring
particle in square $(x,y)$ is $P_{id}=A_{s}/A$ where $A_{s}=(a_{p}/5)^{2}$ is
the area of a discrete square and $A=A_{0}$ is the membrane area. The two-
dimensional interaction free energy then follows as

$F(x,y)=-k_{B}T\ln\left[\frac{P(x,y)}{P_{id}}\right]=-k_{B}T\ln\left[\frac{M(x,y)A}{4N_{c}N_{b}(N_{b}-1)A_{s}}\right]$
(4)

This two-dimensional interaction free energy takes into account whether a
neighboring particle is located at the side or at the tip of an arc-shaped
particle, and is a generalization of the one-dimensional potential of mean
force 81.

For tubular membrane conformations, we define a central, tubular membrane
segment of area $A=A_{0}/2$ based on the distance of the membrane triangles
from the center of mass of the membrane, and focus on particle pairs $i$ and
$j$ in which at least one particle is bound to the central membrane segment.
Pairs of particles that are both bound to a spherical membrane end are, thus,
excluded from the analysis. We next define the tubular axis of a conformation
based on a singular value decomposition of the positions of all particles
bound the membrane in this conformation, and project the particle centres on
an ideal cylinder with radius $r_{cyl}$ around this tubular axis. For each
particle pair, we determine the angle $\psi$ between the tubular axis and the
vector that connects the projected particle centres, and define two
coordinates as $x=r\cos\psi$ and $y=r\sin\psi$ where $r$ is the distance of
the (unprojected) particle centres. Because the particles are oriented on
average perpendicular to the tubular axis, the value $\psi=0$ corresponds to a
side-to-side orientation of a particle pair, and the value $\psi=\pi/2$
corresponds to a tip-to-tip orientation. We discretize the plane of the two
coordinates $x$ and $y$ as described above for spherical conformations, and
determine the two-dimensional free energy $F(x,y)$ from Eq. (4) after
rescaling the total number of particle pairs $M(x,y)$ by the factor
$\sqrt{1-y^{2}/(4r_{cyl}^{2})}$ to account for the cylindrical curvature in
$y$-direction. For tubular conformations, $N_{b}$ in Eq. (4) corresponds to
the average number of particles bound to the central tubular membrane
segments. In this calculation, the tubular radius $r_{cyl}$ of the particle
layer around the membrane is obtained from the maximum of the one-dimensional
distributions shown in Fig. 4.

### Interaction energy

The one-dimensional interaction energies of Fig. 5 were calculated based on
Eq. (2). To determine the rescaled distribution $p(r)$ of the membrane-bound
particles in this equation, we first determine the number of particle pairs
$M(r)$ with a distance of the particle centres within the shell $r\pm{\rm
d}r/2$ with width ${\rm d}r=a_{p}/10$. The rescaled distribution then follows
as $p(r)=M(r)/({\rm d}r\,N_{c}N_{b}(N_{b}-1)/A)$ where $N_{c}$ is the number
of simulation conformations and $N_{b}$ is the average number of bound
particles in the considered membrane area $A$. For tubular membrane
conformations, the area $A$ is the area $A_{0}/2$ of the central tubular
segment in which at least one of the bound particles in a pair needs to be
located (see above). For spherical membrane conformations, the area $A$ is the
total $A_{0}$ of the membrane.

Fig. 6: Hard-core distributions $p_{hc}(r)$ of particles with arc angle (a)
$60\degree$ and (b) $90\degree$, numerically determined for corresponding
‘flattened particles’ in a planar quadratic area $A$ at the indicated particle
coverages $x$. For determining the interaction energy (2) of particles on
spherical vesicles with reduced volume $v\leq 0.95$, the area $A$ for
calculating $p_{hc}(r)$ is taken to be the membrane area $A_{0}$, and the
coverage $x$ is taken as the corresponding coverage of the simulations. In the
case of tubular vesicles with $v\leq 0.65$, the area $A$ is the area $A_{0}/2$
of the central, tubular membrane segment used to calculate the distribution
$p(r)$ from simulation conformations. The dashed lines represent the ideal
distribution $p_{id}(r)=2\pi r$ of non-interacting particles.

The hard-core distributions $p_{hc}(r)$ in Eq. (2) are determined for
‘flattened’ particles with an angle of $0$ between the quadratic segment of
side length $a_{p}$, rather than the angle of $30\degree$ of the curved,
membrane-bound particles. To obtain these distribution, we first generate a
large number $N_{c}$ of non-overlapping conformations of $N_{b}$ ‘flattened’
particles in a quadratic area $A$. Here, $N_{b}$ is the average number of
membrane-bound particles and $A$ is the membrane in the corresponding membrane
simulation system, in which $p(r)$ has been determined. The hard-core
distributions $p_{hc}(r)$ then are calculated from the number $M_{hc}(r)$ of
particle pairs with a distance of the particle centres within the shell
$r\pm{\rm d}r/2$ as $p_{hc}(r)=M_{hc}(r)/({\rm d}r\,N_{c}N_{b}(N_{b}-1)/A)$.
We determine $M_{hc}(r)$ for standard periodic boundaries of the considered
quadratic area $A$, i.e. we take the distance $r$ as the minimum distance
among all periodic particle images. The hard-core distributions slightly
depend on the area coverage $x=N_{b}A_{p}/A$ of the particles where
$A_{p}=n_{s}a_{p}^{2}\pi/4$ is the hard-core area of a single particle, which
is composed of either $n_{s}=3$ or $4$ segments. The hard-core distributions
$p_{hc}(r)$ shown in Fig. 6 slightly increase with area coverage $x$ at close
distances $r<n_{s}a_{p}$ smaller than the particle length, because of an
increase of the pressure in the particle gas with $x$. This pressure pushes
the particles together, against their entropic repulsion at distances
$r<n_{s}a_{p}$. Because of the normalization implied by the definition of the
distributions, higher values of $p_{hc}(r)$ at small distances $r<n_{s}a_{p}$
lead to lower values at large distances. The hard-core distributions exhibit
two kinks at distances $r$ at which the particles acquire more rotational
freedom. For particles composed of 3 segments, the first kink of $p_{hc}(r)$
occurs at the distance $r=2a_{p}$ above which a particle that is located at
the side of another particle gains full rotational freedom. The second kink
occurs at the distance $r=3a_{p}$ above which a particle located at the tip of
another particle has full rotational freedom. For particles composed of 4
segments, these kinks occur at corresponding, larger distances $r$.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

Financial support from the Max Planck Society and from the Deutsche
Forschungsgemeinschaft (DFG) via the International Research Training Group
1524 “Self-Assembled Soft Matter Nano-Structures at Interfaces" is gratefully
acknowledged.

## References

  * Shibata _et al._ 2009 Y. Shibata, J. Hu, M. M. Kozlov and T. A. Rapoport, _Annu. Rev. Cell Dev. Biol._ , 2009, 25, 329–354.
  * Kozlov _et al._ 2014 M. M. Kozlov, F. Campelo, N. Liska, L. V. Chernomordik, S. J. Marrink and H. T. McMahon, _Curr. Opin. Cell Biol._ , 2014, 29, 53–60.
  * McMahon and Boucrot 2015 H. T. McMahon and E. Boucrot, _J. Cell Sci._ , 2015, 128, 1065–1070.
  * Baumgart _et al._ 2011 T. Baumgart, B. R. Capraro, C. Zhu, S. L. Das, S. R. Leone, P. S. Cremer, J. T. Groves and M. A. Johnson, _Annu. Rev. Phys. Chem._ , 2011, 62, 483–506.
  * Peter _et al._ 2004 B. J. Peter, H. M. Kent, I. G. Mills, Y. Vallis, P. J. G. Butler, P. R. Evans and H. T. McMahon, _Science_ , 2004, 303, 495–499.
  * Rao and Haucke 2011 Y. Rao and V. Haucke, _Cell. Mol. Life Sci._ , 2011, 68, 3983–3993.
  * Mim and Unger 2012 C. Mim and V. M. Unger, _Trends Biochem Sci._ , 2012, 37, 526–533.
  * Simunovic _et al._ 2019 M. Simunovic, E. Evergren, A. Callan-Jones and P. Bassereau, _Annu. Rev. Cell. Dev. Biol._ , 2019, 35, 111–129.
  * Takei _et al._ 1999 K. Takei, V. I. Slepnev, V. Haucke and P. De Camilli, _Nat. Cell Biol._ , 1999, 1, 33–39.
  * Frost _et al._ 2008 A. Frost, R. Perera, A. Roux, K. Spasov, O. Destaing, E. H. Egelman, P. De Camilli and V. M. Unger, _Cell_ , 2008, 132, 807–817.
  * Mim _et al._ 2012 C. Mim, H. Cui, J. A. Gawronski-Salerno, A. Frost, E. Lyman, G. A. Voth and V. M. Unger, _Cell_ , 2012, 149, 137–45.
  * Adam _et al._ 2015 J. Adam, N. Basnet and N. Mizuno, _Sci Rep_ , 2015, 5, 15452.
  * Daum _et al._ 2016 B. Daum, A. Auerswald, T. Gruber, G. Hause, J. Balbach, W. Kühlbrandt and A. Meister, _J. Struct. Biol._ , 2016, 194, 375–382.
  * Le Roux _et al._ 2021 A.-L. Le Roux, C. Tozzi, N. Walani, X. Quiroga, D. Zalvidea, X. Trepat, M. Staykova, M. Arroyo and P. Roca-Cusachs, _Nat. Commun._ , 2021, 12, 6550.
  * Shoemaker and Panchenko 2007 B. A. Shoemaker and A. R. Panchenko, _PLoS Comput Biol_ , 2007, 3, e42.
  * Marsh _et al._ 2013 J. A. Marsh, H. Hernández, Z. Hall, S. E. Ahnert, T. Perica, C. V. Robinson and S. A. Teichmann, _Cell_ , 2013, 153, 461–70.
  * Min _et al._ 2008 Y. Min, M. Akbulut, K. Kristiansen, Y. Golan and J. Israelachvili, _Nat. Mater._ , 2008, 7, 527–538.
  * Grzelczak _et al._ 2010 M. Grzelczak, J. Vermant, E. M. Furst and L. M. Liz-Marzán, _ACS Nano_ , 2010, 4, 3591–3605.
  * Weikl 2018 T. R. Weikl, _Annu. Rev. Phys. Chem._ , 2018, 69, 521–539.
  * Johannes _et al._ 2018 L. Johannes, W. Pezeshkian, J. H. Ipsen and J. C. Shillcock, _Trends Cell Biol._ , 2018, 28, 405–415.
  * Haselwandter and Phillips 2013 C. A. Haselwandter and R. Phillips, _Europhys. Lett._ , 2013, 101, 68002p1–68002p6.
  * Idema and Kraft 2019 T. Idema and D. J. Kraft, _Curr. Opin. Colloid Interface Sci._ , 2019, 40, 58–69.
  * Gao _et al._ 2021 J. Gao, R. Hou, L. Li and J. Hu, _Front Mol Biosci_ , 2021, 8, 811711\.
  * Hafner _et al._ 2019 A. E. Hafner, J. Krausser and A. Saric, _Curr. Opin. Struct. Biol._ , 2019, 58, 43–52.
  * Kumar _et al._ 2022 G. Kumar, S. C. Duggisetty and A. Srivastava, _J. Membr. Biol._ , 2022, 255, 757–777.
  * Simunovic _et al._ 2013 M. Simunovic, A. Srivastava and G. A. Voth, _Proc. Natl. Acad. Sci. USA_ , 2013, 110, 20396–20401.
  * Ramakrishnan _et al._ 2013 N. Ramakrishnan, P. B. S. Kumar and J. H. Ipsen, _Biophys. J._ , 2013, 104, 1018–1028.
  * Noguchi 2016 H. Noguchi, _Sci. Rep._ , 2016, 6, 20935.
  * Noguchi and Fournier 2017 H. Noguchi and J.-B. Fournier, _Soft Matter_ , 2017, 13, 4099–4111.
  * Olinger _et al._ 2016 A. D. Olinger, E. J. Spangler, P. B. S. Kumar and M. Laradji, _Faraday Discuss._ , 2016, 186, 265–275.
  * Bonazzi and Weikl 2019 F. Bonazzi and T. R. Weikl, _Biophys. J._ , 2019, 116, 1239–1247.
  * Gao _et al._ 2024 J. Gao, R. Hou, W. Hu, T. R. Weikl and J. Hu, _J. Phys. Chem. B_ , 2024, 128, 4735–4740.
  * Bahrami _et al._ 2012 A. H. Bahrami, R. Lipowsky and T. R. Weikl, _Phys. Rev. Lett._ , 2012, 109, 188102.
  * Saric and Cacciuto 2012 A. Saric and A. Cacciuto, _Phys. Rev. Lett._ , 2012, 109, 188101\.
  * Yue and Zhang 2012 T. Yue and X. Zhang, _ACS Nano_ , 2012, 6, 3196–3205.
  * Raatz _et al._ 2014 M. Raatz, R. Lipowsky and T. R. Weikl, _Soft Matter_ , 2014, 10, 3570–3577.
  * Raatz and Weikl 2017 M. Raatz and T. R. Weikl, _Adv. Mater. Interfaces_ , 2017, 4, 1600325\.
  * Xiong _et al._ 2017 K. Xiong, J. Zhao, D. Yang, Q. Cheng, J. Wang and H. Ji, _Soft Matter_ , 2017, 13, 4644–4652.
  * Tang _et al._ 2018 H. Tang, H. Ye, H. Zhang and Y. Zheng, _Nanotechnology_ , 2018, 29, 405102.
  * Yan _et al._ 2019 Z. Yan, Z. Wu, S. Li, X. Zhang, X. Yi and T. Yue, _Nanoscale_ , 2019, 11, 19751–19762.
  * Zuraw-Weston _et al._ 2019 S. Zuraw-Weston, D. A. Wood, I. K. Torres, Y. Lee, L.-S. Wang, Z. Jiang, G. R. Lazaro, S. Wang, A. A. Rodal, M. F. Hagan, V. M. Rotello and A. D. Dinsmore, _Nanoscale_ , 2019, 11, 18464–18474.
  * Spangler and Laradji 2021 E. J. Spangler and M. Laradji, _J. Chem. Phys._ , 2021, 154, 244902\.
  * Chen _et al._ 2021 T. Chen, Y. Zhang, X. Li, C. Li, T. Lu, S. Xiao and H. Liang, _J. Chem. Theory Comput._ , 2021, 17, 7850–7861.
  * Reynwar _et al._ 2007 B. J. Reynwar, G. Illya, V. A. Harmandaris, M. M. Müller, K. Kremer and M. Deserno, _Nature_ , 2007, 447, 461–464.
  * Reynwar and Deserno 2011 B. J. Reynwar and M. Deserno, _Soft Matter_ , 2011, 7, 8567–8575.
  * Bahrami and Weikl 2018 A. H. Bahrami and T. R. Weikl, _Nano Lett._ , 2018, 18, 1259–1263.
  * Bahrami and Bahrami 2019 A. Bahrami and A. H. Bahrami, _Nanotechnology_ , 2019, 30, 345101\.
  * Zhu _et al._ 2022 Y. Zhu, A. Sharma, E. J. Spangler and M. Laradji, _Soft Matter_ , 2022, 18, 4689–4698.
  * Zhu _et al._ 2023 Y. Zhu, A. Sharma, E. J. Spangler, J.-M. Y. Carrillo, P. B. S. Kumar and M. Laradji, _Soft Matter_ , 2023, 19, 2204–2213.
  * Sharma _et al._ 2024 A. Sharma, Y. Zhu, E. J. Spangler, T. B. Hoang and M. Laradji, _ACS Nano_ , 2024, 18, 12957–12969.
  * Midya _et al._ 2023 J. Midya, T. Auth and G. Gompper, _ACS Nano_ , 2023, 17, 1935–1945.
  * Li and Abel 2022 B. Li and S. M. Abel, _Soft Matter_ , 2022, 18, 2742–2749.
  * Li and Abel 2024 B. Li and S. M. Abel, _J. Chem. Phys._ , 2024, 160, year.
  * Nambiar _et al._ 2024 N. Nambiar, Z. A. Loyd and S. M. Abel, _J. Chem. Theory Comput._ , 2024, 20, 1732–1739.
  * Koltover _et al._ 1999 I. Koltover, J. Rädler and C. Safinya, _Phys. Rev. Lett._ , 1999, 82, 1991–1994.
  * van der Wel _et al._ 2016 C. van der Wel, A. Vahid, A. Saric, T. Idema, D. Heinrich and D. J. Kraft, _Sci. Rep._ , 2016, 6, 32825.
  * Sarfati and Dufresne 2016 R. Sarfati and E. R. Dufresne, _Phys Rev E_ , 2016, 94, 012604\.
  * van der Wel _et al._ 2017 C. van der Wel, D. Heinrich and D. J. Kraft, _Biophys J_ , 2017, 113, 1037–1046.
  * Lavagna _et al._ 2021 E. Lavagna, Z. P. Güven, D. Bochicchio, F. Olgiati, F. Stellacci and G. Rossi, _Nanoscale_ , 2021, 13, 16879–16884.
  * Azadbakht _et al._ 2024 A. Azadbakht, B. Meadowcroft, J. Májek, A. Šarić and D. J. Kraft, _Biophys. J._ , 2024, 123, 307–316.
  * Azadbakht _et al._ 2024 A. Azadbakht, T. R. Weikl and D. J. Kraft, _arXiv_ , 2024, 2403.05253.
  * Groza _et al._ 2024 R. Groza, K. V. Schmidt, P. M. Müller, P. Ronchi, C. Schlack-Leigers, U. Neu, D. Puchkov, R. Dimova, C. Matthaeus, J. Taraska, T. R. Weikl and H. Ewers, _Nat. Commun._ , 2024, 15, 2767.
  * van der Ham _et al._ 2024 S. van der Ham, J. Agudo-Canalejo and H. R. Vutukuri, _ACS Nano_ , 2024, 18, 10407–10416.
  * Goulian _et al._ 1993 M. Goulian, R. Bruinsma and P. Pincus, _Europhys. Lett._ , 1993, 22, 145–150.
  * Goulian _et al._ 1993 M. Goulian, R. Bruinsma and P. Pincus, _Europhys. Lett._ , 1993, 23, 155–155.
  * Park and Lubensky 1996 J. M. Park and T. C. Lubensky, _J. Phys. I_ , 1996, 6, 1217–1235.
  * Netz 1997 R. R. Netz, _J. Phys. I_ , 1997, 7, 833–852.
  * Weikl _et al._ 1998 T. R. Weikl, M. M. Kozlov and W. Helfrich, _Phys. Rev. E_ , 1998, 57, 6988–6995.
  * Kim _et al._ 1998 K. S. Kim, J. Neu and G. Oster, _Biophys. J._ , 1998, 75, 2274–2291.
  * Dommersnes _et al._ 1998 P. Dommersnes, J. Fournier and P. Galatola, _Europhys. Lett._ , 1998, 42, 233–238.
  * Dommersnes and Fournier 1999 P. G. Dommersnes and J. B. Fournier, _Eur. Phys. J. B_ , 1999, 12, 9–12.
  * Schweitzer and Kozlov 2015 Y. Schweitzer and M. M. Kozlov, _PLoS Comput. Biol._ , 2015, 11, e1004054.
  * Yolcu and Deserno 2012 C. Yolcu and M. Deserno, _Phys. Rev. E_ , 2012, 86, year.
  * Fournier and Galatola 2015 J.-B. Fournier and P. Galatola, _Eur. Phys. J. E_ , 2015, 38, 86\.
  * Kahraman _et al._ 2018 O. Kahraman, R. Langen and C. A. Haselwandter, _Sci. Rep._ , 2018, 8, 16383.
  * Vahid _et al._ 2017 A. Vahid, A. Šarić and T. Idema, _Soft Matter_ , 2017, 13, 4924–4930.
  * Seifert _et al._ 1991 U. S. Seifert, K. Berndl and R. Lipowsky, _Phys. Rev. A_ , 1991, 44, 1182–1202.
  * Bahrami and Hummer 2017 A. H. Bahrami and G. Hummer, _ACS Nano_ , 2017, 11, 9558–9565.
  * Dimova 2014 R. Dimova, _Adv. Colloid Interface Sci._ , 2014, 208, 225–234.
  * Nagle 2013 J. F. Nagle, _Faraday Discuss._ , 2013, 161, 11–29.
  * Chandler 1987 D. Chandler, _Introduction to modern statistical mechanics_ , Oxford University Press, 1987.
  * Qualmann _et al._ 2011 B. Qualmann, D. Koch and M. M. Kessels, _Embo J._ , 2011, 30, 3501–3515.
  * Jülicher 1996 F. Jülicher, _J. Phys. II_ , 1996, 6, 1797–1824.
  * Helfrich 1973 W. Helfrich, _Z. Naturforsch. C_ , 1973, 28, 693–703.
  * Bonazzi _et al._ 2021 F. Bonazzi, C. K. Hall and T. R. Weikl, _Soft Matter_ , 2021, 17, 268–275.
  * Frenkel and Smit 2002 D. Frenkel and B. Smit, _Understanding molecular simulation: from algorithms to applications_ , Academic Press, San Diego, 2nd edn., 2002, vol. 1.
  * Vesely 1982 F. J. Vesely, _J. Comput. Phys._ , 1982, 47, 291–296.