# Lipid membrane-mediated attraction between curvature inducing objects

Casper van der Wel  Biological and Soft Matter Physics, Huygens-Kamerlingh
Onnes Laboratory, Leiden University, PO Box 9504, 2300 RA Leiden, The
Netherlands  Afshin Vahid  Department of Bionanoscience, Kavli Institute of
Nanoscience, Delft University of Technology, Van der Maasweg 9, 2629 HZ Delft,
The Netherlands  Anđela Šarić  Department of Physics and Astronomy, Institute
for the Physics of Living Systems, University College London, Gower Street,
London, WC1E 6BT  Timon Idema  Department of Bionanoscience, Kavli Institute
of Nanoscience, Delft University of Technology, Van der Maasweg 9, 2629 HZ
Delft, The Netherlands  Doris Heinrich  Biological and Soft Matter Physics,
Huygens-Kamerlingh Onnes Laboratory, Leiden University, PO Box 9504, 2300 RA
Leiden, The Netherlands  Fraunhofer Institute for Silicate Research,
Neunerplatz 2, 97082 Würzburg, Germany  Daniela J. Kraft  Biological and Soft
Matter Physics, Huygens-Kamerlingh Onnes Laboratory, Leiden University, PO Box
9504, 2300 RA Leiden, The Netherlands<EMAIL_ADDRESS>
###### Abstract

The interplay of membrane proteins is vital for many biological processes,
such as cellular transport, cell division, and signal transduction between
nerve cells. Theoretical considerations have led to the idea that the membrane
itself mediates protein self-organization in these processes through
minimization of membrane curvature energy. Here, we present a combined
experimental and numerical study in which we quantify these interactions
directly for the first time. In our experimental model system we control the
deformation of a lipid membrane by adhering colloidal particles. Using
confocal microscopy, we establish that these membrane deformations cause an
attractive interaction force leading to reversible binding. The attraction
extends over 2.5 times the particle diameter and has a strength of three times
the thermal energy (). Coarse-grained Monte-Carlo simulations of the system
are in excellent agreement with the experimental results and prove that the
measured interaction is independent of length scale. Our combined experimental
and numerical results reveal membrane curvature as a common physical origin
for interactions between any membrane-deforming objects, from nanometre-sized
proteins to micrometre-sized particles.

## Introduction

Interactions between membrane proteins are of key importance for the survival
of cells as they are involved in many dynamical processes. The organization of
membrane proteins into complexes and their effect on membrane shape enables
for instance intracellular transport, cell division, cell migration, and
signal transduction [1]. Understanding the underlying principles of protein
organization is therefore crucial to unravel processes such as cell-cell
signalling in the brain [2] or disease mechanisms like membrane-associated
protein aggregation in Parkinson’s disease [3].

Besides specific protein-protein interactions and interactions with the
cytoskeleton, protein organization in membranes is thought to be driven by a
universal interaction force arising from membrane deformations. Theoretical
models [4, 5, 6, 7, 8] and simulations [9, 10, 11] predict that by deforming
the membrane locally, membrane proteins can self-assemble into complex
structures such as lines, rings, and ordered packings [10, 12, 13, 14].
Observations in living cells [3, 15] support the existence of such membrane-
mediated interactions, but have yet to provide conclusive experimental proof
of their common physical origin: separation of contributions arising from
specific protein-protein interactions and interactions with the cytoskeleton
is extremely challenging.

Further experimental indications for a universal membrane-mediated interaction
stem from simplified model systems: phase-separated membrane domains are known
to repel each other [16] while colloidal particles have been observed to
irreversibly stick together when attached to lipid vesicles [17, 18]. However,
the hypothesized connection between curvature and interaction force has not
been quantified to date: even the sign of the force is still under debate.

Existing model systems for studying surface-mediated interactions are
typically based on deformations of liquid-liquid or liquid-air interfaces [19,
20, 21]. In these systems, interactions are governed by surface tension, while
in lipid vesicles elastic _surface bending_ is expected to be the dominant
factor. In addition, lipid vesicles are bilayers of molecules that cannot
exchange molecules with the surrounding medium, which makes them profoundly
different from other liquid interfaces. The experimental quantification of
interface-mediated interactions in lipid membranes thus requires a clean and
dedicated model system.

Figure 1: Experimental model system for studying membrane-mediated
interactions. (a) Three-dimensional confocal image of a typical Giant
Unilamellar Vesicle (GUV, in magenta) with attached colloidal particles (in
green). Supplementary Video S1 contains the corresponding movie. (b) Schematic
of the avidin-biotin linkage between membrane and particle. By varying the
avidin concentration on the particles we control the adhesion strength.
Polyethylene glycol (PEG) suppresses electrostatic interactions between
membrane and particles, as well as non-specific adhesion between particles.

In this article, we describe such a specialized model system consisting of
membrane-adhering colloidal particles on Giant Unilamellar Vesicles (GUVs). We
characterize for the first time the effect of a single adhesive colloidal
particle on the local membrane shape using confocal microscopy. We find that
the particle is either fully wrapped by the membrane or not wrapped at all,
depending on the adhesion strength. Next, we measure the interaction potential
for particles in these two states and we find that only wrapped particles show
a reversible attraction, which implies that the attraction is purely caused by
the membrane deformation. Monte Carlo simulations of the bending-mediated
interaction between wrapped particles result in an interaction potential that
quantitatively agrees with the experimental result. Since these simulations do
not contain any absolute length scale, we conclude that the measured
attraction caused by lipid membrane deformations is scale-independent. Our
combined model system and simulations therefore quantitatively describe the
interactions of any membrane-deforming object, ranging from nanometre-sized
proteins to micrometre-sized colloidal particles.

## Results

### Particle-induced membrane deformation

As a dedicated model system for membrane-deforming proteins we use micrometre-
sized colloidal particles (polystyrene, $0.98\pm
0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter) adhered to single-
component Giant Unilamellar Vesicles (GUVs, diameters ranging from
$5100\text{\,}\mathrm{\SIUnitSymbolMicro m}$), allowing us to study membrane-
mediated interactions with confocal microscopy (see Figure 1). The GUVs
consists of DOPC lipids, which is above its melting point at room temperature,
ensuring a single-phase liquid membrane. The connection between membrane and
particle is realized by coating the particles with varying amounts of avidin,
a protein that binds strongly and specifically to biotin [22], which we attach
to the membrane through a functionalised lipid. The concentration of avidin
linkers on the particle surface allows us to effectively tune the adhesion
strength of the particle to the membrane.

By choosing different fluorescent markers for the particles and lipid
membranes, we are able to visualize the effect of a single particle on a lipid
membrane (see Figure 2). We find that particles exist in either a completely
wrapped state or a completely non-wrapped state: partial wrapping is only
observed as a transient situation. Non-wrapped particles are located on the
outside of the vesicle without deforming the membrane (Figure 2a-c), while
wrapped particles are protruding into the interior of the vesicle (Figure
2d-f). Co-localization of the membrane fluorescence with the particle
fluorescence further corroborates the fully wrapped state.

Figure 2: The effect of particle linker density on the membrane wrapping
state. (a) Fluorescence signal of a non-wrapped particle (green) and a
membrane (magenta). The separate fluorescence signals of the membrane and
particle are displayed in (b) and (c), respectively. In (d)-(f) the wrapped
state is displayed analogously. The scale bar is
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (g) Fraction of wrapped particles
as a function of linker density on floppy membranes (blue triangles, membrane
tension $\sigma<$10\text{\,}\mathrm{n}\mathrm{N}\mathrm{/}\mathrm{m}$$) and
tense membranes (red circles,
$\sigma>$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{N}\mathrm{/}\mathrm{m}$$).
The solid line is the analytic model at $\sigma=0$ derived from Equation 1 via
the Boltzmann factor. Horizontal error bars show the spread (one standard
deviation) in linker density. Figure 3: Interactions between membrane-attached
particles. (a)-(c) Confocal images of non-wrapped particles that stick
irreversibly together via lipid structures. The membrane fluorescence from the
image in (b) is shown separately in (c). (d)-(e) Interaction energy $u(s)$ as
a function of geodesic particle separation distance $s$ for (d) two non-
wrapped particles and (e) two wrapped particles. For non-wrapped particles (d)
there is no significant interaction on both tense
($\sigma>$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{N}\mathrm{/}\mathrm{m}$$)
and floppy ($\sigma<$10\text{\,}\mathrm{n}\mathrm{N}\mathrm{/}\mathrm{m}$$)
membranes in a vesicle diameter range of D${}_{\text{v}}$ =
$16.840\text{\,}\mathrm{\SIUnitSymbolMicro m}$. For wrapped particles (e) the
interaction potential shows a long-ranged attraction. The data for tense
vesicles is obtained from particle trajectories on a single tense membrane
with D${}_{\text{v}}$ = $36\text{\,}\mathrm{\SIUnitSymbolMicro m}$, while the
interaction energy for floppy vesicles is obtained from an average transition
probability matrix of particle trajectories on 3 floppy membranes with
D${}_{\text{v}}$ = $1440\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Every
measurement point is based on $201400\text{\,}\mathrm{i}$ndependent pair
measurements. Error bars denote one standard deviation. Scale bars are
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

This two-state behaviour is in agreement with theoretical predictions [23].
Using a similar approach, we express the total energy of particle wrapping
derived from the Canham-Helfrich energy functional [24]:

$E=\bigg{(}\frac{2\kappa}{R^{2}}+\sigma-u_{ad}\bigg{)}A.$ (1)

Here, $\kappa$ denotes the membrane bending rigidity, $R$ the particle radius,
$\sigma$ the membrane tension, $A$ the contact area, and $u_{ad}$ the adhesion
energy per unit area. This equation states that the energy is minimized by
either minimizing or maximizing the contact area, depending only on the sign
of the prefactor between brackets. The value of this prefactor in turn depends
on the tunable parameters membrane tension and adhesion energy.

We vary the membrane tension $\sigma$ by adjusting the salt concentration in
the vesicle exterior. For this we discern two extreme situations: _tense_
vesicles with a (non-fluctuating) spherical shape
($\sigma>$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{N}\mathrm{/}\mathrm{m}$$)
and _floppy_ vesicles that exhibit clear fluctuations around a spherical shape
($\sigma<$10\text{\,}\mathrm{n}\mathrm{N}\mathrm{/}\mathrm{m}$$). The values
of the surface tension have been derived from the spectral analysis of the
fluctuating vesicle contour according to [25].

In order to vary the adhesion energy $u_{ad}$, we coat particles with
different amounts of linker protein avidin. We measure the distribution of
linker densities in a fluorescence assay (see Method section) and relate this
to the fraction of particles that are wrapped by floppy membranes (see Figure
2g and S1). We find that wrapping occurs above a critical linker density of
$513\pm 77\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}$. On tense membranes
we never observe wrapping of particles.

Our DOPC membranes have membrane bending rigidity
$\kappa=$21\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$$ [26]. In the case of
floppy membranes ($\sigma\ll 2\kappa R^{-2}$) Equation 1 yields a
corresponding adhesion energy per unit area of
$168\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}\mathrm{/}\mathrm{{}^{2}}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.
Comparing this adhesion energy to the literature value of the binding energy
per avidin-biotin bond ($17\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$ [22]),
we conclude that effectively only 2 % of the surface linkers are binding. This
is probably due to the presence of the polymer between biotin and lipids: the
bulky polymer may reduce the binding energy per linker, prevent access to some
avidin binding sites, and cause an additional non-specific steric repulsion
because of overlap with polymers on the particles.

Note that while wrapping requires floppy membranes, it is irreversible and
affects the membrane tension: an initially floppy membrane gradually increases
its surface tension upon wrapping particles, due to the effective removal of
membrane surface area. In this way, a tense membrane with wrapped particles
can be obtained as well.

The observed “all or nothing” wrapping behaviour provides a means to control
local membrane deformations through easily accessible experimental parameters.
We will use this experimental control in the next section to investigate the
forces between local membrane deformations.

### Membrane shape mediated interactions

When two wrapped particles approach within a distance of several particle
diameters, we observe a reversible, long-ranged attraction between them (see
Supplementary Video S2). Excitingly, this interaction is absent for particles
that are adhered to but not wrapped by the membrane (see Supplementary Video
S3). This implies that the interaction observed between wrapped particles is
purely caused by the local deformation arising from particle wrapping.

To be able to single out the membrane-mediated force, we exclude all other
relevant forces on the particles. Firstly, electrostatic interactions are
screened up to a Debye-Hückel screening length of
$1\text{\,}\mathrm{n}\mathrm{m}$ by adding $50\text{\,}\mathrm{m}\mathrm{M}$
of salt to the vesicle exterior. Secondly, gravity is compensated with
buoyancy by carefully increasing the water density with heavy water
(D${}_{\text{2}}$O). Thirdly, wall interactions are suppressed by coating the
coverglass with polymer [27]. Finally, we ensure that particles do not attract
via Van der Waals forces by grafting a high density of poly(ethylene) glycol
(PEG) to the particle surface, which acts as a steric stabilizer [28]. We
confirm with particle tracking in three-dimensional confocal images that the
particles indeed do not interact or sediment (see Supplementary Fig. S2
online).

In order to quantify the membrane shape mediated interaction, we track the
membrane-adhered particles using confocal microscopy at a frame rate of
$2957\text{\,}\mathrm{H}\mathrm{z}$. We are able to extract the 3D particle
coordinates from these 2D image sequences by simultaneously tracking the
vesicle, to which the particles are confined (see Method). The pair
interaction energy is inferred from direct measurement of the transition
probability matrix $P_{ij}$, describing the probability for particles to move
from separation distance $s_{i}$ to separation distance $s_{j}$ [29]. Here,
the distance $s$ is the geodesic distance between the points where the
particles connect to the membrane. From $P_{ij}$ a stationary probability
distribution for $s$ is obtained. This is equal to the equilibrium
distribution, assuming that the hydrodynamic drag forces on the particle do
not depend on their separation. From the Boltzmann distribution we then
determine the energy of two interacting particles, $u(s)$.

Using this method, we infer the pair interaction energy $u(s)$ between wrapped
particles and between non-wrapped particles. Clearly, non-wrapped particles do
not interact, while wrapped particles show a long-ranged attraction (Figure
3d-e). The shape of the interaction potential for wrapped particles does not
depend on the membrane tension, although the interaction strength is lower on
tense membranes. We find that the interaction strength for floppy membranes is
and that the attraction extends over a range of
$2.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, which is equivalent to 2.5
particle diameters. As the interaction energy is larger than
k${}_{\text{B}}$T, this attraction can be observed by eye from the relative
movement of membrane-wrapped particles in Supplementary Video S2. The
interaction force is only present for particles that deform the membrane:
therefore we conclude that the reason for the interaction is the membrane
deformation only and that the membrane mediates this force.

In earlier work by Koltover et al. [17] and Ramos et al. [18], an attraction
between membrane-bound particles was observed that lead to irreversible
aggregation. Strikingly, in our experiments the interaction potential does not
feature this short-ranged permanent binding, but a long-ranged reversible
attraction. We explain this as follows: similar to the referenced work [17,
18], our initial experiments also contained particle aggregates on the
membrane (see Figure 3a-c and supplementary Fig. S3 online). However, confocal
microscopy revealed that these aggregates are mediated by small
($<$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) lipid vesicles always present
in GUV solutions [30]. As their membrane composition is equal to that of the
GUVs, they contain biotin linkers as well and thereby irreversibly bind to the
adhered particles. This “bridging” process gives rise to a short ranged
irreversible attraction. Previously, these lipid structures could not be
identified because they are invisible in bright field microscopy due to their
small size, while they are detected easily in the confocal microscopy
experiments presented here. We deliberately remove the small lipid structures
in our experiments by filtration (see Methods section) enabling us to single
out the membrane-mediated interaction.

We note again that lipid membranes are profoundly different from liquid/air
and liquid/liquid interfaces where surface bending is negligible and surface
tension effects such as capillary forces dominate. In the absence of gravity,
two ideal spheres bound to a liquid/air or liquid/liquid interface do not
interact, because on these types of surfaces, a sphere will adjust its height
until it accommodates the wetting angle and therefore does not induce any
surface deformation[31, 32]. Capillary forces due to interface deformations
are only observed for particles with an anisotropy in their shape or roughness
of the particle-interface contact line[31, 20, 19, 32]. In the case of lipid
membranes, however, this consideration is only valid in the limit of high
membrane tension, as hypothesized recently by Sarfati and Dufresne[33]. For
our low tension membranes, curvature energy is the most significant energy
contribution, as will be corroborated further in the next section.

Analytic approximations for a membrane-bending mediated interaction in the
weakly curved limit predict a fluctuation mediated attraction as well as a
bending mediated repulsion [4, 6], at least in the case of isotropic
deformations. Our attractive interaction, however, cannot be caused by
fluctuations since such an attraction is negligible compared to
k${}_{\text{B}}$T at this length scale [34]. The repulsion due to membrane
curvature should thus be dominant, but clearly cannot explain the attraction
we observe. Therefore, we conclude that the deformations induced by the
wrapped colloidal particles cannot be described by linearised theory.

With non-linear field theory it is possible to calculate the interaction force
from the exact membrane shape, even in the highly curved limit [7]. Without
this information, however, it is not even clear whether the particles repel or
attract. In the limit of asymptotically flat membranes, Reynwar et al. [35]
computed the interaction energy explicitly by numerically solving the membrane
shape equation. They found an attraction with a well depth of the interaction
potential on the same order of magnitude as in our experiments, , albeit with
an additional energy barrier at longer ranges. To more closely resemble our
experimental system, we therefore performed computer simulations on spherical
membranes with fixed area.

### Simulations of membrane mediated interactions

To investigate the origin of the observed membrane mediated interaction, we
simulate the interaction between two particles adhered to a spherical fluid
membrane. Our approach is based on earlier work by Šarić and Cacciuto [10, 36,
13] and is explained in detail in the method section. In short, we describe
the vesicle using a dynamically triangulated network consisting of 5882
vertices. Between the vertices we apply hard-core repulsion such that the
minimum edge length of the network is $l$. The fluid nature of the membrane is
taken into account by allowing the edges of this network to flip. The vesicle
itself, in equilibrium, forms a sphere of diameter $D_{v}=50l$. We introduce
two colloidal particles with diameter $D_{p}=8l$, chosen such that the
$D_{p}/D_{v}$ ratio is similar to the experimental value. Having set the
volume and surface area of the vesicle to the target values, we apply an
adhesion potential between the attractive part of the particles (which in our
system is about $90\%$ of the particles’ total area) and the vertices in the
vesicle to let the membrane wrap around the particles. We use a Monte Carlo
annealing algorithm to identify the equilibrium shape of the membrane for
different positions of the particles. Note that there is no absolute length
scale involved in these simulations.

Figure 4: Computer simulation of the interaction between two membrane-wrapped
particles. (a) The numerical results are displayed together with the
experimental results, which are rescaled by the bending rigidity
$\kappa=$20.9\pm 2.5\text{\,}\mathrm{k}_{\mathrm{B}}\mathrm{T}$$ [26] and the
particle diameter $\mathrm{D_{p}}=$0.98\pm
0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. (b)-(c) show cross-sections of
the simulations of two particles (green) adhered to a coarse-grained membrane
(magenta), at separation $s=$3.7\text{\,}\mathrm{D}_{\mathrm{p}}$$ (b) and
$1.5\text{\,}\mathrm{D}_{\mathrm{p}}$ (c). The membrane vertices are denoted
by small spheres with diameter $l$.

As shown in Figure S4, once the particles are wrapped by the membrane, the
adhesion energy and degree of wrapping remain constant, but the curvature
energy depends on the separation between the particles. The excess energy of
the membrane for different separations $s$ is shown in Figure 4, together with
the experimental data. We find that the curvature energy of the membrane
favours attraction between the particles for distances $1.5D_{p}\lesssim
s\lesssim 2.2D_{p}$, in excellent agreement with the experimental results. For
larger distances the energy of the vesicle is barely affected by a change of
the separation between the particles. The minimum distance is set by the
resolution of our coarse grained description of the membrane: at $1.5D_{p}$ we
can be sure to always have two layers of vertices between the particles.
Because of this limitation our simulations cannot capture the short-range
effects observed in the experiment as well as in the higher resolution
simulations of Reynwar et al. [9, 35]. In contrast to earlier work, however,
our simulations do take the overall curved shape of the vesicle into account,
as well as the fact that it is a closed surface with fixed area and enclosed
volume. For these conditions, we find that there is no long-range repulsion
between the particles, in contrast to some earlier numerical predictions on
asymptotically flat membranes [35]. The observed attraction is entirely due to
a decrease in the bending energy of the membrane upon approach of the
particles and is quantitatively agreeing with our experimental results on
vesicles.

## Conclusion

We established an experimental system of particles adhering to Giant
Unilamellar Vesicles that enables quantitative measurement of interactions
mediated by lipid membrane curvature. For individual membrane-adhering
particles, we showed that there are only two states of membrane deformation: a
non-wrapped and a fully wrapped state. This “all-or-nothing” behaviour is
controlled by the particle-membrane adhesion strength and membrane tension,
which agrees with a simple model based on bending, tension and adhesion
energies.

The two-state particle wrapping allows us to selectively measure the effect of
local membrane deformations on the pair interaction. For two membrane-wrapped
particles, we observed a reversible attraction of three times the thermal
energy over a distance of several microns. As the interaction is absent for
non-wrapped particles, we conclude that it is mediated by the lipid membrane
and originates solely from the particle-induced membrane deformation.

To further probe the underlying physical cause of this attraction, we used a
coarse-grained numerical model and Monte Carlo method from which we determined
the interaction energy between two wrapped particles. Apart from the geometry
and bending modulus of the membrane, which we respectively set to the
experimental and literature values, the model requires no adjustable
parameters. The energy profile we obtained from our simulations is in
excellent quantitative agreement with the experiments. In particular, we find
a long-ranged attraction between the wrapped particles, which is entirely due
to a decrease in the bending energy of the membrane as the two particles
approach each other.

As the observed interaction is determined by bending energy only, there is no
absolute length scale involved in the simulations. This implies that the
membrane curvature mediated force equally applies to the described colloidal
particles as well as to membrane proteins such as proteins containing a BAR
domain[15]. In fact, our experimental measurements quantitatively model
protein interactions on closed membranes, as long as the Helfrich energy
description holds. More local membrane deformations induced by for instance
transmembrane proteins[1] might also deform the membrane in a similar fashion,
but on a length scale comparable to the membrane thickness. In this case the
membrane cannot be described anymore by a two-dimensional surface, and
consequently other effects such as membrane thickness modulations[37] could
lead to interactions. On the other hand, aggregates of transmembrane proteins
may again act as larger membrane-deforming objects that are described by our
model.

To more closely mimic biological systems, lipid membranes with multiple
components and colloidal particles with anisotropic shapes or adhesion patches
may be employed in the future. Multi-component membranes may locally phase-
separate and affect the measured interaction potential through an intricate
process: the object-induced curvature may influence the local membrane
composition[38] and thus spatially modulate the elastic constants of the
membrane. A preference of the particle for a particular phase may further
induce a local phase-separation that adds an additional force driven by the
line tension of the phase boundaries[39]. Besides multi-component membranes,
colloidal particles with anisotropic shapes or site-specific adhesion patches
may be used to mimic the complicated deformation profiles of proteins. These
additional complexities will enable quantitative modeling of the interaction
profile between membrane proteins of various geometries and thus further
improve our understanding of cellular processes that involve membrane-shaping
proteins.

## Methods

### Chemicals

Styrene, itaconic acid, 4,4’-Azobis(4-cyanovaleric acid) (ACVA),
1,3,5,7-tetramethyl-8-phenyl-4,4-difluoroboradiazaindacene (BODIPY),
methoxypoly(ethylene) glycol amine (mPEG-NH${}_{\text{2}}$, M${}_{\text{w}}$ =
5000), N-hydroxysulfosuccinimide sodium salt (Sulfo-NHS), sodium phosphate,
D-glucose, methanol, ethanol, acetic acid, ammonium hydroxide 28-30%w/w
(NH${}_{\text{4}}$OH), Hellmanex III, Pluronic F-127, deuterium oxide 70%
(D${}_{\text{2}}$O), 3-(trimethoxysilyl)propyl methacrylate (TPM), and Bovine
Serum Albumin (BSA) were purchased from Sigma-Aldrich; sodium chloride, sodium
azide, hydrogen peroxide 35%w/w (H${}_{\text{2}}$O${}_{\text{2}}$),
acrylamide, N,N,N’,N’-tetramethylethylenediamine (TEMED), and ammonium
persulfate (APS) from Acros Organics; 1-ethyl-3-(3-dimethylaminopropyl)
carbodiimide (EDC) from Carl Roth; NeutrAvidin from Thermo Scientific; DNA
oligonucleotides (biotin-5’-TTTAATATTA-3’-Cy3) from Integrated DNA
Technologies; $\Delta$9-cis 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC),
1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(lissaminerhodamine B
sulfonyl) (DOPE-rhodamine), and 1,2-dioleoyl-sn-
glycero-3-phosphoethanolamine-N-[biotinyl(polyethylene glycol)-2000] (DOPE-
PEG-biotin) from Avanti Polar Lipids. Unless stated otherwise, chemicals were
used as received. Deionized water is used with
$18.2\text{\,}\mathrm{M}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$
resistivity, obtained using a Millipore Filtration System (Milli-Q Gradient
A10).

### Probe particles

Polystyrene particles are synthesized from styrene, itaconic acid, ACVA, and
BODIPY in water using a surfactant-free radical polymerization described in
[40], resulting in monodisperse spheres with a diameter of $0.98\pm
0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (see supplementary Fig. S4 online
for a scanning electron microscopy image). Resulting particles are coated with
NeutrAvidin and mPEG-NH${}_{\text{2}}$ using an a protocol adjusted from [41].
All subsequent reactions are done at $4\text{\,}\mathrm{\SIUnitSymbolCelsius}$
to slow down NHS hydrolysis. $1\text{\,}\mathrm{m}\mathrm{L}$ 20%w/w particles
are mixed with $80\text{\,}\mathrm{\SIUnitSymbolMicro mol}$ EDC and
$25\text{\,}\mathrm{\SIUnitSymbolMicro mol}$ Sulfo-NHS in
$10\text{\,}\mathrm{m}\mathrm{L}$ water at pH = 5.3 and stirred for
$30\text{\,}\mathrm{min}$. The pH of the resulting NHS-activated particles is
brought to 8.6 using $0.2\text{\,}\mathrm{M}$ NaOH.
$750\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of the 2%w/w activated particles
is then mixed with $0.550\text{\,}\mathrm{\SIUnitSymbolMicro g}$ NeutrAvidin.
After $30\text{\,}\mathrm{min}$, $4\text{\,}\mathrm{m}\mathrm{g}$ mPEG-
NH${}_{\text{2}}$ is added and the reaction proceeds for
$40\text{\,}\mathrm{h}$. Then the pH is brought to 12 with
$1\text{\,}\mathrm{M}$ NaOH, the particles are ultrasonicated for
$5\text{\,}\mathrm{min}$ and then washed 1 time with $0.01\text{\,}\mathrm{M}$
HCl and 3 times with water. Finally, sodium azide is added to a concentration
of $3\text{\,}\mathrm{m}\mathrm{M}$ to prevent bacterial growth.

### Biotin binding sites assay

In order to quantify the number of biotin binding sites (the ‘linker density’)
on each particle, we measure fluorescence of biotin- and dye-functionalised
DNA strands. DNA strands have the advantage that they are well soluble in
water, so that there is no non-specific adhesion to the particle surface.
$10\text{\,}\mathrm{\SIUnitSymbolMicro L}$ $6\text{\,}\mathrm{\micro M}$ DNA
(in water) is mixed with $10\text{\,}\mathrm{\SIUnitSymbolMicro L}$ 0.5%w/w
particles in a total volume of $310\text{\,}\mathrm{\SIUnitSymbolMicro L}$
$50\text{\,}\mathrm{m}\mathrm{M}$ PBS buffer with 0.5%w/w Pluronic F-127. The
mixture is heated to $55\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$30\text{\,}\mathrm{min}$ and washed 3 times with water. The sample is diluted
10 times in a PBS buffer inserted into an untreated rectangular glass
capillary, which immobilizes the particles. The fluorescence intensity is
quantified using fluorescence microscopy (Nikon Intensilight) with
reproducible settings and using a reference value obtained from commercial
particles with a known amount of avidin (Spherotech PC-S-1.0) we can obtain a
distribution of avidin linkers per particle.

### GUV preparation

Vesicles are prepared using a standard electroswelling technique [42]. A lipid
mixture of 97.5%w/w DOPC, 2%w/w DOPE-PEG-biotin, and 0.5%w/w DOPE-rhodamine is
used, ensuring a liquid bilayer at room temperature.
2x$20\text{\,}\mathrm{\SIUnitSymbolMicro g}$ of the lipids in chloroform are
dried on two $2525\text{\,}\mathrm{m}\mathrm{m}$ ITO-coated glass slides
($1525\text{\,}\mathrm{\SIUnitSymbolOhm}$, Sigma-Aldrich), placed in
$1.8\text{\,}\mathrm{m}\mathrm{L}$ of a solution with
$100\text{\,}\mathrm{m}\mathrm{M}$ glucose and
$0.3\text{\,}\mathrm{m}\mathrm{M}$ sodium azide in 49:51 (mass)
D${}_{\text{2}}$O:H${}_{\text{2}}$O. The cell is subjected to
$1.1\text{\,}\mathrm{V}$ (rms) at $10\text{\,}\mathrm{H}\mathrm{z}$ for
$2\text{\,}\mathrm{h}$, with the first $2\text{\,}\mathrm{min}$ a linear
increase from $0\text{\,}\mathrm{V}$. GUVs are stored in a BSA-coated glass
vial at room temperature. In order to remove small lipid structures [30],
$100\text{\,}\mathrm{\SIUnitSymbolMicro L}$ GUV solution is pipetted on a
Whatmann $5.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pore size
cellulosenitrate filter and slowly flushed with $5.0\text{\,}\mathrm{mL}$ of
glucose solution. $100\text{\,}\mathrm{\SIUnitSymbolMicro L}$ purified GUVs
are harvested from the filter and used the same day. All handling is done with
care not to mechanically shock the solution.

### Coverglass treatment

We employed a polymerization of acrylamide onto TPM-coated glasses [27], as
follows: coverglasses are cleaned for $30\text{\,}\mathrm{min}$ in a 2%v/v
Hellmanex solution, rinsed 3 times with water, immersed in 5:1:1
H${}_{\text{2}}$O:NH${}_{\text{4}}$OH:H${}_{\text{2}}$O${}_{\text{2}}$ for
$30\text{\,}\mathrm{min}$ at $70\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
rinsed 3 times with water, and 2 times with ethanol. TPM functionalisation is
done by immersing $15\text{\,}\mathrm{min}$ in ethanol with 1%v/v acetic acid
and 0.5%v/v TPM, rinsing 3 times with ethanol and incubating for
$1\text{\,}\mathrm{h}$ at $80\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
Polymerization is done in a 2%w/w solution of acrylamide (evacuated in vacuum
for $30\text{\,}\mathrm{min}$ to remove oxygen), with 0.035%v/w TEMED and
0.070%w/w APS for $2\text{\,}\mathrm{h}$. Resulting coverglasses were kept
inside the polymerization solution at
$4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ until use. Directly before use, a
coverglass is rinsed with water and blow-dried with nitrogen.

### Density matching

A density-matched PBS stock buffer of
$200\text{\,}\mathrm{m}\mathrm{O}\mathrm{s}\mathrm{m}$ is prepared containing
$10.0\text{\,}\mathrm{m}\mathrm{M}$ sodium phosphate,
$82.0\text{\,}\mathrm{m}\mathrm{M}$ sodium chloride, and
$3.0\text{\,}\mathrm{m}\mathrm{M}$ sodium azide. Density-matching with the
probe particles was achieved by gradually adding D${}_{\text{2}}$O until no
sedimentation or creaming occurred at $10\,000\text{\,}\mathrm{g}$ for
$1\text{\,}\mathrm{h}$. The mass ratio D${}_{\text{2}}$O:H${}_{\text{2}}$O for
water is roughly 51:49; for the buffer it is roughly 45:55 because the solutes
increase the density. Using ratios between the stock buffer and density-
matched water, buffers at different osmolarities are obtained. Density
matching is confirmed for each mixture separately.

### Sample preparation

Samples are prepared on polyacrylamide-coated coverglasses and density matched
with D${}_{\text{2}}$O. $2\text{\,}\mathrm{\SIUnitSymbolMicro L}$ 2%w/w
particles, $4\text{\,}\mathrm{\SIUnitSymbolMicro L}$
$150\text{\,}\mathrm{m}\mathrm{O}\mathrm{s}\mathrm{m}$ PBS buffer, and
$20\text{\,}\mathrm{\SIUnitSymbolMicro L}$ filtered GUVs are incubated for
$10\text{\,}\mathrm{min}$ in a plastic microtube. Then
$10\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of this mixture is slowly
distributed into the sample holder with $50\text{\,}\mathrm{\SIUnitSymbolMicro
L}$ $100\text{\,}\mathrm{m}\mathrm{O}\mathrm{s}\mathrm{m}$ PBS buffer already
inside. The sample holder consists of a Teflon ring clamped on a pretreated
coverglass. For tense GUVs, the sample holder is closed with vacuum grease and
a second coverglass; for floppy GUVs, the sample holder is kept open to air
for $30\text{\,}\mathrm{min}$ so that evaporation leads to an increase in
osmotic pressure, and consequentially a decrease of membrane tension. All
experiments were performed at a room temperature of
$1922\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

### Imaging

Imaging is done with an inverted Nikon TiE microscope equipped with a Nikon
A1R confocal scanhead with both galvano and resonant scanning mirrors. High-
speed trajectory imaging is achieved with a horizontal resonant mirror
scanning lines at $15\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$; single-particle
close-ups are done with the galvano mirrors. We use a 60x water immersion
objective (NA = 1.2) to reduce axial aberration due to index of refraction
mismatch. The excitation laser is passed through a quarter wave plate to
mitigate polarization effects of bilayer-attached dye molecules. Excitation
(at $488\text{\,}\mathrm{n}\mathrm{m}$ and $561\text{\,}\mathrm{n}\mathrm{m}$)
and detection are performed simultaneously (for trajectory imaging) or
sequentially (for close-ups) using a dichroic mirror splitting the emission
signal onto $500550\text{\,}\mathrm{n}\mathrm{m}$ and
$565625\text{\,}\mathrm{n}\mathrm{m}$ filters. The sample is mounted on an MCL
NanoDrive stage to enable fast Z stack acquisition.

### Image analysis

The raw images of the interaction measurements are high speed
($2957\text{\,}\mathrm{H}\mathrm{z}$) confocal images containing two separate
colours, being the vesicle fluorescence and the particle fluorescence. Images
are convolved with a Gaussian kernel with an rms width of 1 pixel to reduce
random noise.

The particle fluorescence signals are tracked using a widely employed centre-
of-mass based particle tracking algorithm in a Python implementation [43],
which is available online [44]. All particle trajectories are checked manually
for missing coordinates and corrected if necessary. Because the used centre-
of-mass refinement technique systematically finds coordinates of overlapping
features too close together, we refine overlapping signals additionally by
least-squares fitting to a sum of Gaussians.

The three-dimensional coordinates of the particles relative to the vesicle
($x_{rel}$, $y_{rel}$, $z_{rel}$) are determined as follows:
$x_{rel}=x-x_{c}$; $y_{rel}=y-y_{c}$;
$z_{rel}^{2}=(R+h)^{2}-x_{rel}^{2}-y_{rel}^{2}$. The vesicle radius $R$ is
obtained in a separate three-dimensional confocal measurement; the particle-
vesicle distance $h$ is known from the wrapping state of the particle; the
vesicle centre ($x_{c}$, $y_{c}$) is measured simultaneously with the particle
tracking ($x$, $y$) from the high-speed 2D confocal images.

For the vesicle tracking, we interpolate the image on lines that are drawn
outwards from a rough estimate of the vesicle centre. The maximum value on
each of these lines provides an estimate of the vesicle perimeter. Around each
maximum, a fit region of 5 pixels is defined for further refinement: linear
regression on the discrete derivative provides the position of the vesicle
perimeter with sub-pixel resolution. Finally, we perform a least-squares fit
to a circle (for two dimensions) or ellipsoid (for three dimensions) to obtain
the refined vesicle centre and radius. The here described algorithm is
available online [45].

### Modelling Details

The curvature energy of a biological membrane is described by the Helfrich
energy functional[24] as:

$\displaystyle u_{\mathrm{Curv}}=\frac{\kappa}{2}\int_{A}(2H)^{2}dA,$ (2)

where $H$ is the mean curvature of the membrane, which is defined as the
divergence of the surface normal vector,
$H=-\frac{1}{2}\nabla\cdot\mathbf{n}$. We model the vesicle by a network of
vertices with the minimum length of $l$ that are connected in a triangular
network. The curvature energy of our discretised membrane is given by:

$\displaystyle
u_{\mathrm{Curv}}=\sqrt{3}\kappa\sum_{<ij>}1-\textbf{n}_{i}\cdot\textbf{n}_{j},$
(3)

where $\textbf{n}_{i}$ and $\textbf{n}_{j}$ are the normal vectors to any pair
of adjacent triangles $i$ and $j$, respectively. The summation runs over all
pairs of such triangles. To simulate the fluidity of the membrane, we change
the connectivity of the network: we cut and reattach connections between the
four vertices of any two neighbouring triangles. The surface area $A$ and
volume $V$ of the vesicle, are maintained by constraints
$u_{A}=K_{A}(A-A_{t})^{2}/A_{t}$ and $u_{V}=K_{V}(V-V_{t})^{2}/V_{t}$ with
$K_{A}=10^{3}k_{B}T/l^{2}$ and $K_{V}=4\times 10^{3}k_{B}T/D_{p}l^{2}$, where
$k_{B}T$, $D_{p}$, $A_{t}$ and $V_{t}$ are the thermal energy, the diameter of
the particles, the target surface area and the target volume of the vesicle,
respectively. In each simulation we set the target values of surface area and
volume of the vesicle with diameter $D_{v}=50l$ as $A_{t}=1.05A_{0}$ and
$V_{t}=V_{0}$, respectively. These parameters cause the final volume and
surface area to deviate less than $0.01\%$ from the target values. To let the
vertices of the membrane wrap around the particles, we introduce an attraction
potential between them:

$\displaystyle u_{\mathrm{Ad}}=\begin{cases}-\varepsilon(l_{m}/r)^{6}&\mbox{if
}\theta\leq\theta_{\mathrm{Wr}},\\\ 0&\mbox{ otherwise},\end{cases}$ (4)

where $\varepsilon$ is the particles’ adhesion energy and $r$ is the centre to
centre distance between particles and vertices. $\theta$ is the angle between
the vector normal to the active area of the particles and the vectors that
connect the particles to vertices (see supplementary Fig. S5 online). The
maximum angle $\theta_{\mathrm{Wr}}$ is defined to control the area that is
forced to be wrapped by the membrane, preventing very sharp membrane bends.
$l_{m}=(l+D_{p})/2$ is the shortest distance between particles and vertices,
where the diameter of the particles is set to $D_{p}=8l$. We set a cut-off
radius for the attraction potential at $1.2l$ to make sure that other than
forming a layer of membrane on the surface of the particles, it has no extra
effects. The total energy $u_{\mathrm{T}}$ of the system is the sum of the
curvature energy (Eq. 3) and the adhesion energy (Eq. 4).

To analyse the equilibrium shape of the membrane, we implement the Monte Carlo
simulated annealing in order to minimize the total energy of the system. For
our Monte Carlo simulations, we use the Metropolis algorithm to move vertices
and particles, and flip the edges of the membrane triangulation, in order to
change the configuration of the system (shape of the membrane). The
temperature of the system is also slowly decreased so that we suppress the
fluctuation of the membrane and identify the minimum-energy configuration.

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## Acknowledgements

This work was supported by the Netherlands Organisation for Scientific
Research (NWO/OCW), as part of the Frontiers of Nanoscience program and VENI
grant 680-47-431. We thank Jeroen Appel and Wim Pomp for advice on the
protocol design and Marcel Winter and Ruben Verweij for experimental support.

## Author contributions statement

D.J.K., D.H., and T.I. designed the project; C.v.d.W. carried out experiments
and data analysis supervised by D.J.K. and D.H..; A.V. performed simulations
supervised by T.I. and A.Š.; all authors contributed to writing and revising
the manuscript.

## Additional information

Competing financial interests The authors declare no competing financial
interests.

## Supplementary material for “Lipid membrane-mediated attraction between
curvature inducing objects”

Author list: Casper van der Wel, Afshin Vahid, Anđela Šarić, Timon Idema,
Doris Heinrich, and Daniela J. Kraft

## Supplementary Videos

Videos are separately available online. Here, still images of the videos are
shown together with their captions.

Supplementary Video S1: Three-dimensional confocal image sequence of a Giant
Unilamellar Vesicle with attached colloidal particles. GUVs are shown in
magenta and particles in green. On the left, a three-dimensional rendering is
shown. On the right, the particle tracking is shown in an overlay on three
maximum intensity projections (centre: xy, bottom: xz, right: yz). The scale
bar denotes the pixel size of the xy projection. Z axes are compressed because
the physical size of one voxel is larger in the z-dimension than in the xy
dimensions. Label “-1” denotes the vesicle centre. Particles are
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter. Supplementary Video S2:
Three wrapped particles on a tense vesicle. From the particle trajectories, it
is clear that the particles attract each other. The video shows an image
sequence of confocal slices of a spherical vesicle (in magenta) with colloidal
particles (in green), displayed in real time. By using information from the
particle-vesicle distance and the vesicle radius, the full three-dimensional
coordinates of the particles can be reconstructed. This is shown on the right
in a three-dimensional rendering. Particles are
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter. Supplementary Video S3:
Non-wrapped particles adhered to a vesicle. The particles do not interact with
each other. The video shows a confocal image sequence of the top part of a
spherical vesicle (in magenta) with colloidal particles (in green), displayed
in real time.

## Supplementary Figures

Supplementary Figure S1: Distribution of particle linker densities for four
samples with different amounts of linker protein avidin. The linker amount
noted in the legend is the amount added to $15\text{\,}\mathrm{m}\mathrm{g}$
particles during synthesis, see Method section. In order to relate the
fraction of wrapped particles to these distributions, the right tail of each
distribution is shaded up to the measured fraction of wrapped particles. From
this, we estimate the critical linker density to be $513\pm
77\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}$. Note that we observed no
wrapping for the $20\text{\,}\mathrm{\SIUnitSymbolMicro g}$ sample (yellow).
Supplementary Figure S2: Three-dimensional radial distribution function and
sedimentation profile of particles suspended at a volume fraction of 0.0011 in
a $50\text{\,}\mathrm{m}\mathrm{M}$ density matched PBS solution. (a) The
radial distribution g(r) shows no interaction between particles. The red line
indicates particle contact. The sharp peak at a distance of 1 diameter is due
to the presence of a few dimers originating from the particle synthesis. (b)
The density profile shows that there is no gradient in concentration due to
gravity. Supplementary Figure S3: Particle aggregates mediated by small lipid
structures. In (a)-(c), permanent particle aggregates are shown (in green)
that are mediated by lipid structures, that are visible by their fluorescence
(in magenta). In (d)-(e) different fluorescence channels from the same
permanent dimer is shown, the bright spot of membrane fluorescence in (e) is
the lipid structure that causes the irreversible binding. Scalebars are
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Supplementary Figure S4: Scanning
Electron Microscopy image of the $0.98\text{\,}\mathrm{\SIUnitSymbolMicro m}$
polystyrene colloidal particles used in this work. Images are obtained with an
FEI nanoSEM 200 at $15\text{\,}\mathrm{k}\mathrm{V}$. From the two-dimensional
crystallization, it is clear that the size polydispersity is low
($0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The inset shows the smooth
surface of the particles. Supplementary Figure S5: Wrapping of particles by
the membrane and the resulting total membrane energy in our numerical model.
(a) Wrapping happens through adhesion of membrane vertices to colloid
particles, due to a strong adhesion potential (Equation 4). We can specify an
inactive region at the top of the colloid, preventing the membrane from making
very sharp turns (with very high bending energies); in the given example,
$\theta_{\mathrm{Wr}}=11\pi/12$. (b) Curvature, adhesion, and total energy of
the system, with zero set at the value of two wrapped particles located at
opposite poles of the vesicle. After the wrapping process, the adhesion energy
($u_{\mathrm{Ad}}$) does not change significantly and therefore the curvature
energy ($u_{\mathrm{Curv}}$) determines the behaviour of the particles.