Shaping of membranes by arc-like particles
vorgelegt von
Master of Science
Francesco Bonazzi
von der Fakult¨at II - Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr.rer.nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Martin Schoen
Gutachterin: Prof. Dr. Sabine Klapp
Gutachter: Priv.-Doz. Dr. Thomas Weikl
Tag der wissenschaftlichen Aussprache: 14. Januar 2019
Berlin 2019
iii
iv
Abstract
This dissertation analyzes the interaction of biomembranes with arc-like particles, which
gives rise to complex morphologies of organelles inside cells. The shaping of mem-
branes is investigated with coarse-grained modeling and Monte Carlo simulations. In
the coarse-grained model, the membrane is described as a discretized elastic surface, and
the particles as segmented arcs.
Concave arc-like particles, i.e. particles interacting with the membrane on their con-
cave side, shape membranes by inducing positive curvature at their binding sites. The
membrane shape strongly depends on the overall angle and area concentration of the
adsorbed particles. Particles with sufficiently large angles can induce both tubules and
double-membrane disks, depending on their area concentrations. Double-membrane
disks are stabilized by particles that adsorb only to the highly curved edges of the
disks, while tubules are fully covered by particles.
If mixtures of both concave and convex particles are allowed to bind, morphologies
with regions of negative curvature appear on the membrane. A frequently observed
membrane morphology resembles a bulged ball, in which concave and convex particles
create bulges and invaginations on the membrane, respectively. For small particles and
low concentration of convex particles, U-shaped tubular and three-way junction mor-
phologies are observed, which are common in the peripheral endoplasmic reticulum.
Finally, this dissertation shows that the membrane-mediated interaction between
bound concave particles depends on the overall membrane curvature. The radial distribu-
tion function for bound particles is computed, and the membrane-mediated interaction
between particles is derived with the reversible work theorem. Membrane curvatures
that are compatible with the particle curvature have a weaker membrane-mediated in-
teraction, which indicates that the induction of curvature on the membrane plays an
important role for the membrane-mediated particle-particle interactions.
v
vi
Zusammenfassung
Diese Dissertation analysiert die Wechselwirkung von Biomembranen mit bogenf¨ormigen
Partikeln, die zu komplexen Morphologien von Organellen innerhalb einer Zelle f¨uhren.
Die Modellierung von Membranen wird mit vergr¨oberten Modellen und Monte-Carlo-
Simulationen untersucht. In dem vergr¨oberten wird die Membran als eine diskretisierte,
elastische Oberfl¨ache und die Partikel als segmentierte B¨ogen beschrieben.
Konkave bogenartige Teilchen, d. h. Teilchen, die mit der Membran ¨uber ihre
konkaven Seite wechselwirken, formen Membranen, indem sie eine positive Kr¨ummung an
ihren Bindungsstellen induzieren. Die Membranform h¨angt stark von dem Gesamtwinkel
und der Fl¨achenkonzentration der adsorbierten Teilchen ab. Teilchen mit ausreichend
großen Winkeln k¨onnen abh¨angig von ihren Fl¨achenkonzentrationen sowohl Tubuli als
auch diskoide Membranformen induzieren. Die diskoiden Formen werden durch Teilchen
stabilisiert, die nur an den stark gekr¨ummten R¨andern adsorbieren, w¨ahrend die Tubuli
vollst¨andig von Teilchen bedeckt sind.
Bei Mischungen von konkaven und konvexen Teilchen treten Morphologien auf, die
Bereiche mit negativer Kr¨ummung enthalten. Eine h¨aufig zu beobachtende Membran-
morphologie ¨ahnelt einer gew¨olbten Kugel, bei der konkave und konvexe Teilchen W¨olbun-
gen und Invaginationen auf der Membran erzeugen. F¨ur kleine Teilchen und geringe
Konzentrationen von konvexen Teilchen kommt es zu U-f¨ormigen tubul¨aren Morpholo-
gien und tubul¨aren Verzweigungen, die im peripheren Endoplasmatischen Retikulum
vorkommen.
Schließlich zeigt diese Dissertation, dass die membranvermittelte Wechselwirkung zwis-
chen gebundenen konkaven Teilchen von der globalen Membrankr¨ummung abh¨angt. Die
radiale Verteilungsfunktion f¨ur gebundene Teilchen wird berechnet, und die membran-
vermittelte Wechselwirkung zwischen Teilchen wird mit dem Reversible Work Theorem
abgeleitet. Membrankr¨ummungen, die mit der Kr¨ummung der Teilchen kompatibel sind,
weisen eine schw¨achere membranvermittelte Wechselwirkung auf, was darauf hindeutet,
dass die induzierte Membrankr¨ummung eine wichtige Rolle f¨ur die membranvermittelten
Teilchen-Teilchen-Wechselwirkungen spielt.
viii
Contents
Title i
Abstract v
Zusammenfassung vii
Contents ix
1. Introduction 1
1.1. Morphologies of biological membranes . . . . . . . . . . . . . . . . . . . . 1
1.2. Elastic energy of membranes . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Model and methods 7
2.1. Model ...................................... 7
2.1.1. Triangulated membranes . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2. Particles................................. 10
2.1.3. Particle-particle interaction . . . . . . . . . . . . . . . . . . . . . . 11
2.1.4. Membrane-particle potential . . . . . . . . . . . . . . . . . . . . . 11
2.2. Simulationmethods .............................. 13
2.2.1. Initial configurations for the membrane . . . . . . . . . . . . . . . 15
2.2.2. Starting configurations . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3. Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.4. Celllist ................................. 16
3. Shaping of membranes by concave particles 21
3.1. Angles induced by particles . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2. Membrane morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4. Shaping of membranes by mixtures of concave and convex particles 33
4.1. Membrane-mediated segregation and bulged balls . . . . . . . . . . . . . . 34
4.2. Converged morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3. Stability of three-way junctions . . . . . . . . . . . . . . . . . . . . . . . . 38
5. Membrane-mediated interactions between particles 49
5.1. Radial distribution functions for bound particles . . . . . . . . . . . . . . 51
ix
CONTENTS
A. Appendix 61
A.1. Initial membrane morphologies . . . . . . . . . . . . . . . . . . . . . . . . 61
A.2. Rotations with quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.3. Radii of tubules and disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A.4. Bending energy of ideal shapes . . . . . . . . . . . . . . . . . . . . . . . . 63
A.4.1.Tubule.................................. 63
A.4.2.Disk................................... 63
A.5. Radial distribution of an ideal gas on a cylinder . . . . . . . . . . . . . . . 65
Bibliography 67
x
1. Introduction
Biological membranes are fundamental structures around and inside cells. The formation
of membranes is regulated by the self-assembly of amphipathic lipids, which produces
lipid bilayers. The bilayers self-assemble as a consequence of the hydrophobic interaction
on amphipathic molecules: phospholipids are made of a hydrophilic head naturally at-
tracted to water, while their hydrophobic tail tries to avoid water contact. The resulting
self-assembly process allows the formation of a two-dimensional bilayer. The thickness
of lipid bilayers is in the order of magnitude of nanometers, while their surface extent is
usually in the order of micrometers.
Historically, it was a paper by Gorter and Grendel in 1925[19] to first hypothesize
that membranes are composed of lipid bilayers. They came to this conclusion after
observing a 2 to 1 ratio in the measurement of the surface areas with two different
methods. Their model failed to include other molecules, such as proteins which were
later included by Davson and Danielli, who had advanced doubts in 1935[10] on the
single bilayer hypothesis for cellular membranes. Their model proposed that membranes
are made of lipid bilayers with a double protein coating.
The modern understanding of biological membranes came only in 1972 when Singer
and Nicolson introduced the fluid-mosaic model[53]. They started from the observation
that proteins are able to move inside the membrane, evidence for a fluid state. This led
to the conclusion that, in addition to lipids, the membrane is composed of cholesterol and
proteins. Lipids are the principal component of the bilayer which, above their melting
temperature, are in a fluid state, whereas proteins can be peripheral or integral with
respect to the membrane. Integral membrane proteins contain parts, mostly hydrophobic
helices, that are embedded in the lipid bilayer. Peripheral proteins are bound to the
membrane, but are not embedded into the lipid bilayer.
1.1. Morphologies of biological membranes
Many biological membranes, especially inside cells, exhibit highly curved shapes[47, 26,
34], among which, as an example of interest, tubules and flattened sheets are recurrent
patterns. The endoplasmic reticulum (ER) is an organelle inside eukaryotic cells con-
taining both of these patterns in abundance[46]. Its inner part, known as the nuclear
envelope, surrounds the cellular nucleus while the peripheral ER stretches to the whole
cell. The ER is made of an interconnected network carved out of a single membrane,
which exhibits abundance of both tubular and sheet-like patterns. Tubules are sections
of membranes shaped into long cylindrical-like structures. They are most prominent in
the peripheral endoplasmic reticulum[58], where they can form complicated structures,
1
1. Introduction
such as densly packed tubular networks[34] as well as three-way junctions of tubules.
The sheets on the other hand consist of a membrane flattened into two parallel flat sur-
faces connected at their ends by a highly curved edge. Inside the ER, sheets are usually
located in the nuclear envelope[22], as helicoidal stack in the perinuclear region[14], and
as flattened cisternal structures close to the plasma[15]. The diameter of the tubules
and height of the sheets are both of the order of 50 nm in mammalian cells[42], which
has led to the hypothesis of a common formation mechanism.
Lipid membranes have a natural tendency to minimize their curvature, as highly bent
shapes are usually energetically unfavorable[62, 32]. Various mechanisms for the forma-
tion of highly curved membrane shapes are known, mostly involving interaction of the
membrane with proteins. Many protein types are responsible for the membane shap-
ing, for example BAR domains[54, 37], which are arc-like particles interacting with the
membrane on their concave side, and reticulons[57]. Reticulons are able to oligomerize
into arc-like scaffolds[48] and probably act similarly to BAR domains.
Sheets: It has been suggested that the strong curvature at the edge of the sheets is
induced by the interaction with assemblies of proteins of the reticulon class and other
similar ones[42]. Their structure has indeed interesting properties, such as hydropho-
bic hairpins, which could mold the membrane by shallow internalization into the lipid
bilayer, and a semicircular arclike shape, exerting pressure on parts of the membrane.
It is known that other proteins act as stabilizers between the two parallel surfaces of an
ER sheet by acting directly on them. Proteins like Climp63, p180, kineticin, and trans-
colon belong to this category[47]. In particular, experiments have shown that Climp63
causes a great abundance of sheets in the ER if overexpressed. On the other hand, the
absence of Climp63 appears not to prevent the formation of sheet-like structures, rather
causing the space separating the parallel surfaces of the sheets to shrink [60]. This leads
to the suggestion that such classes of proteins act merely as helpers to the construction
of the ER structures, but are not the unique builders.
2
1.1. Morphologies of biological membranes
Figure 1.1.1.: Illustration of the scaffolding hypothesis, in the upper image the membrane
has been shaped into a sheet of two parallel surfaces by an array of arc-like
particles. In the lower image the particles have shaped the membrane into
a tubule by surrounding its cylindrical side.
Tubules: Tubules occur frequently in cells, most notably in the peripheral ER.
Tubules are known to be induced by arc-shaped proteins[17, 23]. Reticulons allow for
the stabilization of tubules inside the ER and at the same time it is believed that they
play a vital role in the stabilization of the edges of flattened sheets in the ER[34, 14].
Another class of proteins often associated with stabilizing tubular structures are N-
BAR domains[37]. They are arc-shaped proteins with two domains containing amphi-
pathic termini. First evidence from electron microscopy [33, 1, 17] indicated that N-BAR
domains can form highly ordered helical coats around membrane tubules that are appar-
ently held together by direct protein-protein interactions. Recent tomographic imaging
of T-tubules[11], which are tubular structures found in skeletal and cardiac muscle cells,
showed that Bin1 N-BAR domains form rather loose and irregular arrangements, possi-
bly governed by weak or no direct interaction between neighboring proteins.
Apart from direct protein-protein interactions, the shaping by arc-shaped proteins
induced on the membrane has a cost in terms of bending energy. Therefore, it has been
suggested that energy minimization could be achieved by a close arrangement of such
particles [13, 25, 61, 41, 59]. This effect has been shown to result in a membrane-mediated
attractive force between proteins[41]. Other types of membrane-mediated interactions
3
1. Introduction
Figure 1.1.2.: Sub-tomogram of a T-tubule (blue) surrounded by Bin1 N-BAR domains
(yellow), reproduced from Daum-Meister 2016[11] with permission (CC
BY-NC-ND 4.0). It shows only short-range order, which suggests no direct
protein-protein interactions.
are caused by the reduction in thermal fluctuations on the membrane, which may result
in Casimir-like interactions between particles[52, 6, 20, 27].
3 way junctions: Three-way junctions are points of connection of three tubules
often found in the peripheral endoplasmic reticulum[40]. Tubular structures in the ER
are highly dynamic: they keep splitting and joining, ending up creating numerous 3-way
junctions. At the same time the tubules forming 3-way junctions may shift along one
another, possibly resulting in the destruction of the junction itself by separation of the
tubule or by fusion of junctions.
According to a recent hypothesis, the formation of 3-way junctions is made possible by
the Lunapark (Lnp1) protein[8, 9, 40]. Lnp1 is believed to act in concerted manner with
the Yop1[9]. Experimental evidence of variation of the concentration of Lnp1 has shown
that if Lnp1 is missing, the ER appears to collapse into a densely packed structure. On
the other hand, an overexpression of Lnp1 causes a large amount of polygonal structures
connected by 3-way junctions to appear[40].
Two transmembrane domains, the components of Lnp1, are likely shaped like a con-
cave wedge, suggesting the possibility of inducing negative curvature on the membrane
bound to Lnp1[8, 40], therefore acting in opposition to reticulons and BAR domains.
Furthermore, multiple Lnp1 molecules may possibly form larger structures contributing
to the formation and stabilization the 3-way junctions[8, 40].
1.2. Elastic energy of membranes
Lipid bilayers are self-assembled surface-like structures used as model systems for biolog-
ical membranes. The hydrophobic interaction together with the amphipathic behavior
makes phospholipids assemble in lipid bilayers. If the temperature is above the chain-
melting temperature, as it is the case of interest in this project, the lipid bilayer has a
two dimensional fluid nature.
There are two main types of membrane models: the molecular and the elastic ones.
The molecular models are atomistic and coarse-grained, they involve representations of
atoms in molecules or approximations thereof, with various degrees of discretization[16].
4
1.2. Elastic energy of membranes
(a)
(b)
(c)
Figure 1.2.1.: Examples of elastic surface deformations acting on a lipid bilayer. Di-
lation force (a) stretches (or compresses) the membrane uniformly in all
directions, while shearing (b) preserves the area. If the membrane is fluid,
there will be no resistence to shearing. Bending forces of the membrane
(c) act on the orthogonal direction to the surface.
These models are useful for simulations usually performed at small length and time
scales. The elastic models, on the other hand, treat the membrane as a fluid, elastic
sheet. They are suitable for larger membranes, as it is unfeasible to perform simulations
at molecular level of detail with the current available computational technology. The
most influential elastic model is due to Helfrich[21]. The triangulated membrane model
is a discretization of Helfich’s elastic membrane model.
The elastic models are justified by the fact that the lipid bilayer, once formed, acts
in a similar manner to an elastic surface. Under this assumption the membrane may be
handled with the classical elastic theory. In particular, the surface undergoes deforma-
tions due to bending, stretching and shearing, as illustrated in figure 1.2.1. The shearing
forces may be ignored, as the membrane is fluid and therefore any force resisting the
relocation of molecules in space is nonexistent. The hydrophobically-induced attraction
between phospholipidic tails is pretty strong, therefore any stretching force acting on
the membrane has minimal effects and fluid membranes are nearly incompressible[29].
This condition can be modeled by subjecting the area to the harmonic potential around
an equilibrium surface area
Ustretching ∝∆A2,(1.2.1)
where ∆A=A−¯
Arepresents the membrane area Aas displaced from the equilibrium
area ¯
A.
The bending modeling is more complex and requires the introduction of concepts from
differential geometry. Mathematical surfaces in space are regarded as two dimensional
differential manifolds embedded in the R3space. There are various ways to represent
them mathematically, for example with a map f:R2→R3from the local coordinates of
5
1. Introduction
the surface to the corresponding position in space (arguments about the existence of such
mapping, which usually holds, are neglected for simplicity). The curvature of a curve on
the surface passing through a point is the reciprocal of the radius of the osculating circle
tangent to that curve at that point. The principal curvatures C1and C2at a point of
the surface are the curvatures of two curves, orthogonal to each other, whose values are
the extrema of all curvatures of the curves passing through that point. A theorem by
Gauss guarantees the existence of the principal curvatures. Furthermore, the principal
curvatures are invariants as they do not depend on the chosen parametrization. The
principal curvatures are used to build the mean curvature
H=1
2(C1+C2) (1.2.2)
and the Gaussian curvature[45]:
K=C1C2(1.2.3)
The bending energy of the surface may be calculated as a surface integral[21]. As
the bending energy is a physical property and therefore has to be independent of the
parametrization, the bending energy can be expressed as a function of invariants only.
The integral may therefore be expanded up to the second Taylor term as
βEbending =Zh2κ(M−Msp)2+ ¯κKidA (1.2.4)
where Mand Kare the total and Gaussian curvatures, as previously defined. The
bending rigidity κis a constant that characterizes the rigidity of the membrane, in
biological membranes it is usually in the order of 10 −40kT [59]. The variable ¯κis
the modulus of the Gaussian curvature and is supposed to be in the same order of
magnitude as the bending rigidity κ≈¯κ[43], βis the inverted beta β=1
kBT, and, lastly,
the spontaneous curvature Hsp is used to quantify the tendency of the membrane to
naturally bend on one side, usually due to asymmetricity in the composition between
the two layers. Expression 1.2.4 gives a good approximation of the bending energy
of a real lipid bilayer, provided that its thickness is small compared to its radius of
curvature[62].
Only membranes of spherical topology are relevant to this research project, that is
membranes having a shape that can be continuously deformed into a sphere (membranes
without holes and without topological foldings such as in the Klein bottle). As a conse-
quence of Gauss-Bonnet theorem[28] the integral over the Gaussian curvature Kgives
a constant contribution to the total energy for fixed topology, and may therefore be
omitted. Furthermore, in this project the two layers of the membrane are assumed to
be symmetric, the consequence of dropping the spontaneous curvature term. As a last
reminder, equation 1.2.4 has a linear dependence between βand κ. These units can be
chosen so that βgets absorbed into κ. It is reasonable to operate on βunits (such that
kT = 1) and employ the value κ= 10. The energy expression may be rewritten as
Ebending =Z2κM2dA , (1.2.5)
6
2. Model and methods
In this chapter, a model for particles interacting with a triangulated membrane is pro-
posed. The triangulated membrane is a discretization suitable for computer simulations
of the continuous elastic model of equation 1.2.5. Particles interacting with the triangu-
lated surface are then introduced.
2.1. Model
2.1.1. Triangulated membranes
The integral expression 1.2.5 can be approximated in a standardized way by a summation
over a triangulated surface[5, 24], which can be used in computer simulations. The
mathematical surface is therefore replaced by a large polyhedron of triangular faces.
The number of vertices nV, of edges nEand of triangles nTare constant values and
obey the topological constraint
nV−nE+nT=χ= 2,
where χis known as Euler characteristic. The value χ= 2 is the condition for the
topology of the sphere, that is, a surface with neither holes nor foldings.
Geometrical properties of the membrane
Given a triangulation, geometrical quantities such as the surface area and the volume
in the enclosed space can be calculated from its geometrical elements. For instance, the
Figure 2.1.1.: Illustration of a lipid bilayer and a corresponding triangulated surface.
7
2. Model and methods
membrane surface area is the sum of the areas of all triangles on the triangulation. It is
given by
A=1
2X
i∈F|ei,1×ei,2|=1
2X
i∈F
li,1li,2sin θi,12 (2.1.1)
where the summation runs over all triangles. The vectors ei,1and ei,2are any two of
the three edges of triangle iand li,j is the length of ei,j.
In an analogous way, the volume of the space enclosed by the membrane is determined
by a summation of all signed tetrahedron volumes over all triangles
V=1
6X
i∈F
vi,1·(vi,2×vi,3),
where vi,j is the vector representing the position in space of the j-th vertex of triangle
i.
The reduced volume is then introduced as[44]
v= 6√πV
A3
2≤1,(2.1.2)
where Vis the volume of the space inside the membrane and Ais the surface area of
the membrane. The reduced volume is a dimensionless quantity which is proportional to
the volume of the space enclosed by the membrane if the surface area is kept constant.
The reduced volume is in the 0 < v ≤1 range depending on the shape of the membrane.
The sphere maximizes the value of the reduced volume to unity.
Ideally, the triangles of the triangulated membrane should be shaped as close as pos-
sible to equilateral triangles and not differ much in area. If all triangles were equilateral
triangles (which is almost always geometrically impossible to fit), the edge length would
be
aE=s4A
√3nF
(2.1.3)
The edges of triangles are subjected to a tether potential limiting their lengths to a
specific range. The minimal edge length is set to be 20% less than aE
am:= 4
5aE(2.1.4)
while the maximum edge length is √3am. Therefore the edges of the triangulated surface
are then subjected to a tether potential keeping their lengths in the l∈[am,√3am]
range[24, 5]. This choice allows the shapes and areas of the membrane triangles to cover
the majority of triangular shapes, with the exclusion of very thin ones. From now on,
amwill be used as the reference unit.
8
2.1. Model
Figure 2.1.2.: Illustration of two adjacent triangles of area Aiand Ajon a triangulated
surface. Their shared edge has length lij and their normal unit vectors ni
and njare shown. The dihedral angle φij between the planes identified by
the two triangles is related to the normals of the triangles by the relation
cos φij =ni·nj.
Bending energy of a triangulated membrane
The elastic energy of the membrane is calculated through an approximation on the
triangles of Helfrich’s integral for the bending energy, summed over the vertices[24]
Ebending = 2κ
nV
X
α=1
M2
α
Aα
,(2.1.5)
where Mαis the average contribution to the curvature of vertex α. Its expression is[24]
Mα=1
4X
(ij)
lijφij ,
where the summation ranges over all edges ij that share vertex α. The variable lij
denotes the length of the edge, while φij is the dihedral angle between the triangles i
and jadjacent to the edge. The area associated to the vertex is Aα=1
3PiAi, where
the summation, as before, runs over the three triangles that share the vertex α. The
bending rigidity is generically measured to be around κ≈10 −40kBTfor lipid bilayers,
where kBTis the thermal energy[59]. A reasonable value for the bending rigidity is
κ= 10 kBT[12]. The vertices of the membrane are allowed to fluctuate, while the edges
connecting them can be flipped and connected to other neighboring vertices, ensuring
membrane fluidity[18].
9
2. Model and methods
Dilations in triangulated membranes
Fluid membranes are nearly incompressible (cfr. equation 1.2.1). As a consequence, if
the amount of lipids does not vary, the total surface area of the membrane does not vary.
This is modeled by subjecting the membrane surface area to the harmonic potential of
equation 1.2.1, as
Ustretching =cB(A−A0)2(2.1.6)
Here, cBis the stretching constant, Ais the observed area of the membrane 2.1.1. A0is
a fixed area value kept at A0≈0.677nTa2
m, which is the area minimizing the stretching
energy. In this project the value of cBhas been calibrated to limit the fluctuations of
the area by around 1%. The suitable value of
cB= 2 ·105
1
βV
2
3
ref
(2.1.7)
has been used.
Volume potential
In certain cases, the membrane volume may be constrained to a certain value, which
causes the membrane to be shaped into tubular or spherical morphology[4]. The addi-
tional energy contribution is
EV=cV(v−v0)2,(2.1.8)
where vis the volume of the space enclosed by the membrane (see equation 2.1.2), v0is
the desired reduced volume and cV= 2·105is chosen so that the variation of the volume
is less than 1%. For simulations where the volume inside the membrane is allowed to
freely fluctuate, the value of cVis zero.
2.1.2. Particles
Membrane-shaping particles such as BAR-domains[54, 37] or reticulons[57] are arc-
shaped. Their shape allows them to exert bending force by lightly binding to the mem-
brane. Proteins or protein oligomers are modeled in this project as rigid bodies. These
particles are composed of a line of 3 to 7 contiguous quadratic segments of area a2
p, where
ap=3
2am.
Neighboring segments are curved 30◦with respect to each other, giving the particle an
arc-like shape. As the value of apis larger than the average edge length of a membrane
triangle, different segments bind to different triangles on the membrane. The angle
θdbetween adjacent segments is set to be 30◦. As these particles are rigid bodies, the
relative angles and distances between their segments will remain constants. Furthermore
two particles may not occupy the same space nor overlap nor penetrate the membrane.
10
2.1. Model
size 3 size 4 size 5 size 6 size 7
Figure 2.1.3.: Concave particles are finite sequences of 3, 4, 5, 6, 7 segments (orange).
The angle between the extremal segments are 60◦, 90, 120◦, 150◦, 180◦
respectively. Larger particles are expected to induce larger curvature on
the membrane.
Following this definition, all particles possess the same curvature, which can be estimated
by the radius of the cylindrical section comprising the centers of segments:
Rp=ap
2 tan θd
2(2.1.9)
2.1.3. Particle-particle interaction
A hard-core repulsion between particles is introduced to avoid overlapping. Specifically,
the hard-core repulsion is expressed by the interaction energy among segments with
centers at p1and p2as
Vpp =0 if |p1−p2|> ap
∞otherwise (2.1.10)
As a result of this interaction, the segments behave like hard-spheres of radius ap/2,
with the exception of the interaction with the membrane, which is differently defined.
2.1.4. Membrane-particle potential
Every segment of a particle interacts with the closest triangle on the membrane surface
with membrane-particle adhesion potential
V=−UX
i∈S
fr(ri)fθ(θi),(2.1.11)
where Sis the set of all particle segments, riis the distance between the center of segment
iand the center of the nearest triangle on the membrane, θiis the supplementary angle
to the angle between the normals of segment iand the nearest triangle. Two types of
particles are considered, depending on the direction of their normals. If the normals
of the particle segment point towards their curvature center, the particles are called
concave and induce positive curvature on the membrane, causing protrusions on it, see
11
2. Model and methods
figure 2.1.4(a). Convex particles, on the contrary, have their segment normals pointing
away from the center of curvature and induce negative curvature or invaginations on the
membrane, see figure 2.1.4(b). By convention, the normals of the membrane triangles
point outward from the enclosed volume. frand fθare properly chosen square-well
potentials:
fr(r) = 1 if r1< r < r2
0 otherwise (2.1.12)
fθ(θ) = 1 if |θ|< θc
0 otherwise (2.1.13)
The parameters r1,r2,θcregulate the interaction configurations of the particle to the
membrane. The values used in the simulations carried out in this project are r1=1
4am
and r2=3
4am. The angle threshold θc= 10◦is used, unless differently stated. The
variable Uis the adhesion energy for the particle-membrane interaction.
(a) (b)
Figure 2.1.4.: Illustration of two particle types and the mode of interaction with the
membrane. The displayed vectors, along with the distance between tri-
angle and square segment centers, determine the membrane-particle in-
teraction. The (orange) concave particles (a) induce positive membrane
curvature, as their normal vectors point towards the curvature center of
the particle. Particles interacting on their convex side (b) are colored blue,
they induce negative membrane curvature.
12
2.2. Simulation methods
Coverage area on the membrane
The overlap of particles is prevented by the repulsive forces defined in equation 2.1.10.
Every bound segment of a particle thus covers a surface area equal to
πa2
p
4(2.1.14)
on the membrane. Therefore the area fraction of the membrane covered by all bound
particles is given by
xcoverage =nBa2
pπ
4Am
(2.1.15)
where nBis the number of particle segments bound to the membrane triangles, Amis
the membrane area.
Anti-internalization algorithm
Particles are not allowed to enter the enclosed volume of the membrane. This can be
encoded in the simulation software by setting the condition that the vector pointing
from the membrane triangle to the particle segment must leave the membrane. Mathe-
matically, this can be defined as
(pi−fj)·nF
j>0
where piis the position of the segment i,fjis the position of the center of triangle jin
a neighborhood up to distance 3 −4amto the particle, and nF
jis the normal vector of
triangle j, pointing outward from the enclosed membrane volume.
2.2. Simulation methods
All simulations take place in a cubic box with periodic boundary conditions. The size
of the box is given by
3V
1
3
ref ≈50am.(2.2.1)
For each simulation, a single connected membrane is present in the box, typically with
a reservoir of many fluctuating particles. The particles can be divided into bound and
unbound states.
Metropolis Monte Carlo methods are employed in this project, whereby the evolution
of the system is generated by random processes. The time evolution in the simulations
relies on a Markov chain sampling of the Boltzmann distribution of energies through the
Metropolis-Hastings algorithm. This means that the system will undergo test random
state changes E−→ E′, with acceptance probability given by
P(E−→ E′) = min{exp(−β∆E),1},
13
2. Model and methods
Figure 2.2.1.: Illustration of the edge flipping transformation. A triangulation edge is
removed (left) and a new one is drawn (right) connecting the opposite
vertices of the adjacent triangles aand b. As a result, triangles cand dare
formed. This kind of transformation is needed to simulate the membrane
fluidity.
where Eis the energy of the current state, E′is the energy of the new state after a
random change and ∆E=E′−Eis the increase in energy. Notice that the random
change is always accepted in case of a decrease in energy. The simulation consists of
four different types of steps, randomly extracted each time with equal probability: vertex
translations, edge flipping, particle translation and particle rotation.
Translations Translations are possible changes in the position of particles and mem-
brane vertices. For vertex translations, a vertex is drawn randomly with all vertices
being equally likely to be chosen. The same uniformity of probability applies to edges
and particles. The maximum extent of translations and rotations is chosen such that
the acceptance ratio is close to 1
2. Given the maximum extent of translation tmax
Vfor the
vertex move, the translation vector is limited by the condition tx, ty, tz∈[−tmax
v, tmax
v],
with uniform probability in the interval. The same argument applies to particle trans-
lations, with maximum extent tmax
P. The values tmax
V=1
10 amand tmax
P=amare used in
this project.
Edge flipping Edge flipping is a move by which an edge Eis removed and the opposite
vertices of the triangles adjacent to edge Eare reconnected, as displayed in figure 2.2.1.
The edges to flip are chosen uniformly from all edges in the triangulated membrane.
This operation is required to assure that the membrane behaves like a two dimensional
fluid, which is the state of biological membranes in cells.
Rotations The rotation axis passes through the central point along the particle arc.
That is, the rotation axis passes through the middle point on the arc formed by the
segment centers. In particular, for particles with odd numbers of segments the rotation
axis passes through the center of the central segment. The maximum rotational angle
14
2.2. Simulation methods
is tuned to around 3 to 4 degrees. For further details on the implementation, confer to
A.2.
Averaging of simulation results The observed variables are measured at periodic time
intervals throughout the simulation run. These intervals contain on average 106ver-
tex moves for each vertex. The average values of the observables are taken to be the
arithmetic average of the last 10 measurements.
2.2.1. Initial configurations for the membrane
Every simulation is started with a single membrane of either nV= 1002 vertices, nE=
3000 edges and nT= 2000 triangles or nV= 2562 vertices, nE= edges and nT= 5120
triangles. The smaller membrane of nT= 2000 triangles has been shaped into three
different initial configurations (sphere, disk, spherocylinder, see figure 2.2.2a) while the
larger membrane has also been shaped into a three-way junction. The purpose of multiple
initial configurations for the same parameters is to identify stable and metastable states
and to investigate their convergences. In principle, sufficiently long simulations starting
with different initial membrane shapes should converge to the same final equilibrium
membrane shape if both simulations have the same parameters, for example the number
of total particles and interaction strength U. Actual simulations are not always long
enough for convergence to occur, especially with larger particles. In particular, the
configurations used in this project are:
Sphere The sphere can be described by the surface
x2+y2+z2=R2(2.2.2)
where Ris its radius. The reduced volume of the sphere is equal to one, which is the
largest value of reduced volumes among all possible shapes.
Ideal disk The ideal disk is composed of a low and thick cylinder and an outer semi-
torus on its curved side. The ideal disk is described by two parameters: R, the radius of
the cylinder, and r, the inner radius of the torus. The equations delimiting the disk are
|z|< r if x2+y2< R2
px2+y2−R2+z2=r2otherwise .(2.2.3)
The disk with nT= 2000 employed as starting configuration has a reduced volume of
0.5252. The radius of curvature of the disk edge is chosen to be the particle curvature
R=rpas defined in A.1.1, which is compatible with the curvature of the particles.
15
2. Model and methods
Ideal spherocylinder An ideal spherocylinder is a cylinder capped by two semi-spheres
on its flat sides. The spherocylinder is a cylinder of radius rand height hcapped by
two semispheres on its flat sides. The equations for the spherocylinder are
(x2+y2=r2if |z|< h
x2+y2+ (z−h)2otherwise .(2.2.4)
Once formed, the spherocylinder with nT= 2000 employed as starting configuration has
a reduced volume of 0.3754. The radius of the spherocylinder is chosen to be r=rp
from the definition A.1.1, compatible with the radius of curvature of the particles.
Three-way junction The three-way junction is a particular morphology connecting
three tubules at a single junction. The three-way junction used as starting shape is
created by shaping three semi-spherocylinders on a plane, whose axes are rotated by
120◦with respect to one another. This starting morphology has been created for a
membrane with nT= 5120 only, see figure 2.2.2b. For practical purposes the radius of
the three semi-spherocylinders constituting the junction is larger than rp.
2.2.2. Starting configurations
Starting particle configurations for the membrane in the disk and for the spherocylinder
shapes with bound particles have been used. In first place, simulations with frozen
membrane moves have been run, allowing enough particles to eventually bind to the
membrane. The resulting states are shown in figures 2.2.3(a) and 2.2.3(b). As soon as
enough particles are bound, the full simulation with the membrane dynamics is started
from that given state. For simulations starting with a spherical membrane shape, notice
that no particles were bound to the membrane at the start of the full simulation.
2.2.3. Convergence criteria
In order to determine whether a simulation run has entered a stable or a meta-stable
state, two variables are observed: the number of bound particles and the reduced volume.
The last 107MC steps per vertex of the simulation are divided into ten intervals of
106steps, and the averages of the number of bound particles and reduced volume are
calculated for these intervals. The simulations are considered converged if the standard
deviation of the averages of the ten intervals are below 0.2 for both the number of bound
particles and the reduced volume.
2.2.4. Cell list
In order to increase the speed of the simulation, it is necessary to make sure that in-
teractions between different objects are only calculated with neighboring objects, thus
avoiding spending computational time on the interaction energy between distant objects,
whose value is certainly null. The cell list is a useful data structure to quickly locate
16
2.2. Simulation methods
(a)
disk
sphere
tube
(b)
Figure 2.2.2.: Initial configurations for the membrane in the simulations performed in
this project, no bound particles are shown in this picture. The membrane
with nT= 2000 triangles has been shaped into three configurations (a):
the disk, the sphere and the spherocylinder. For the larger membrane
of nT= 5120 triangles an additional starting morphology (b) has been
introduced in simulations with convex particles: the three-way junction.
17
2. Model and methods
3 segments 4 segments 5 segments
6 segments 7 segments
(a)
3 segments 4 segments 5 segments 6 segments 7 segments
(b)
Figure 2.2.3.: Initial configurations for a membrane with nT= 2000 triangles. In 2.2.3(a)
the membrane has been shaped into a disk, while in 2.2.3(b) the shape is
a spherocylinder. The sizes of particles range from 3 to 7.
18
2.2. Simulation methods
neighboring particles: the space axes are divided in Nequal intervals, thus identifying
N3cubes. At this point, it is possible to create a list of N3elements, with each element
representing a box containing the indexes for the objects lying within the boundary of
that box.
Two such lists, indexing the triangles of the membrane and the segments of the par-
ticles, have been employed. Following the particle move, a search occurs to find all
neighboring triangles, with the purpose of verifying that no particle internalization into
the membrane occurs. After a vertex movement or an edge flip, all neighboring segments
are looked up, so as to perform the same interaction calculation.
19
2. Model and methods
20
3. Shaping of membranes by concave
particles
In this chapter, the model proposed in chapter 2 is applied to investigate the recurrent
morphological structures of flattened sheets and tubules induced by concave particles.
Concave particles can be interpreted as a model for the discretization of proteins form-
ing scaffolds around membranes. The investigation is carried out with a triangulated
membrane surrounded by concave particles. The simulations have been performed with
N= 200 or N= 400 concave particles of size 3 to 7. The membrane with nT= 2000
triangles is used in this chapter, and as starting configurations the sphere, the disk and
the spherocylinder, with bound particles have been used. During the simulations, shape
transitions may occur so that the converged membrane morphologies may be classified
as fluctuating spheres, tubules, disks. In the case of particles with 4 segments, a fourth
additional morphology has been observed, where the adsorbed particles are arranged in
rows dividing the membrane into flat regions. This configuration is referred to as faceted
membrane. In this chapter the constraint of equation 2.1.8 on the volume Vof the space
enclosed by the membrane is not enforced, or, equivalently, the constant of the volume
energy is set to cV= 0. Additionally the bending rigidity of the membrane is κ= 10kBT
for all simulations.
3.1. Angles induced by particles
Concave particles of 3 to 7 segments are illustrated in figure 3.1.1(a). These particles
induce curvature on the membrane by binding on their concave side, as schematically
visualized in figure 3.1.2. Each particle segment binds to only one membrane triangle,
with the range of angles for the triangle normals determined by equation 2.1.13. The
distribution of angles between the normals of the triangles bound to the first and last
particle segments may be used to quantify the induced curvature on the membrane.
The distributions for particles of size 3 to 7 and θc= 10◦are shown in Figure 3.1.1(b).
Similarly, figure 3.1.1(c) compares the same distributions for values θc= 3◦,5◦,10◦with
particles of size 3. The graph shows that larger particles are able to induce higher
curvatures on the membrane. Lower values for θcdecrease the distribution variance and
slightly increase the mean induced angles.
21
3. Shaping of membranes by concave particles
40o45o50o55o60o65o70o
0
0.05
0.10
0.15
40o60o80o100o120o140o160o180o
0
0.02
0.04
0.06
0.08
induced angle
induced angle
probability
probability
size 3 size 6
size 4 size 5
particle size 3
binding cutoff θc=10o
size 7
binding cutoff θc=10o
θc=5o
θc=3o
size 3 size 4 size 6 size 7size 5
(a)
(b)
(c)
Figure 3.1.1.: Distributions of angles induced by particles of various sizes. The angular
aperture induced by a bound particle is calculated as the angle between
the normal vectors of the two triangles of the membrane interacting with
the first and last square of said particle.
22
3.2. Membrane morphologies
Figure 3.1.2.: Illustration of the estimated angle induced by a particle (orange) on the
membrane (blue). The angle θis generally smaller than the angle formed
by the extremal segments of the particle.
3.2. Membrane morphologies
Figure 3.2.1 displays, for every particle size and value of θc, a plot of the converged
simulations along the membrane coverage and reduced volume. The points represent
simulations starting from different initial membrane shapes and parameters, like the
adhesion constant Uand number of concave particles N. The resulting morphologies
are classified as spherical, disk-like, tubular and faceted and the simulations are colored
accordingly. Simulations with the same parameters sometimes converge to different
shapes, which indicates metastabilities. Nevertheless, the plots show that the simulations
are located along curves, irrespective of stability or metastability. This implies that the
reduced volume may be considered as a function of the membrane coverage for a given
particle size and angle cutoff θc, therefore also the morphology is determined by the
membrane coverage.
Metastabilities
Simulations starting from different initial configurations do not always reproduce all
kinds of final states by varying adhesion energy and particle number. This indicates that
converged states may be metastable. In particular, for particles of size 3, 4, and 5 the
simulations starting from a spherical morphology are able to reproduce all final states,
while for larger particles, that is of sizes 6 and 7, no tubular or discoidal morphologies
emerge.
Phases and particle sizes
For particles of three segments, different simulations with angle cutoff θc= 3◦,5◦,10◦
have been performed. The equilibrium configurations of the membrane for particles of
three segments have been classified into two states, the fluctuating sphere and the tubule.
23
3. Shaping of membranes by concave particles
0 0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.0
reduced volume vreduced volume v
membrane coverage x membrane coverage x membrane coverage x
sphere
tube
disk
sphere
faceted
tube
diskfaceted
tube
tube
tube
tube
tube
sphere
disk disk
particle size 3 particle size 3 particle size 4
particle size 5 particle size 6 particle size 7
(a) (b) (c)
(d) (e) (f)
Figure 3.2.1.: These plots compare the membrane coverage and reduced volume for par-
ticles of specific size and θc. The colors represent different morpholo-
gies, with blue for spheres, red for tubules, orange for disks and green
for faceted. Grey points are either intermediate or strongly metastable
states. (a) displays particles of 3 segments with θc= 10◦while (b) dis-
plays the same particles with θc= 3◦(full circles) and θc= 5◦(open
circles). The remaining graphs (c), (d), (e), and (f) have θc= 10◦and
correspond to increasing particle sizes (4, 5, 6, 7). The gray points in (b)
and (c) are partly tubular, while the grey points in (e) are disks connected
to tubules. Finally, the grey points in (f) are partly formed tubules in a
strong metastable state.
24
3.2. Membrane morphologies
The tubular morphology appears at membrane coverage higher than 50% for θc= 10◦
or membrane coverage higher than 45% for θc= 3◦,5◦. Simulations with larger particles
have all been carried out only with θc= 10◦. The tubular morphology appears at high
membrane coverage for all particle sizes. Particles of four segments at low membrane
coverage display a region where the membrane still resembles a distorted sphere, with
particle lines and flattened regions, which is here denoted as faceted morphology. For
particles of 5, 6, 7 segments the disk-like configuration of the membrane appears as
an intermediate between tubular and spherical configurations for membrane coverages
lower than 45%, while tubular configurations appear at higher concentrations. Particles
of size 5 display a markedly more separated gap between the disk-like and tubular
regions. Tubular structures with particles of size 5 are indeed surrounded by three
lines of particles, which makes the transition to disk non-continuous. On the contrary,
for particles with 6 and 7 segments, the tubular morphologies have two lines of bound
particles, allowing a continuous transition between the disk-state and the tubule-state.
Figure 3.2.2 shows the membrane morphologies in the graph of membrane coverage
and mean induced angle by the particles. Small particles are able to bind at multiple
locations independently of one another, whereas binding events are infrequent for larger
particles. No binding events have been observed for particles of 7 segments if no other
particles are already bound to the membrane.
Tubular morphologies have been observed for every kind of particle size at high area
coverages. The radius of the tubule is related to the size of particles with smaller
particles inducing thicker tubes, as shown in figure 3.2.3 and plotted in 3.2.9. There is
an overall tendency of particles to be aligned side by side, which can also be observed in
the temporal evolution of the system, especially in simulations starting from a sphere.
Bound particles of size 3 do not display side-by-side alignments. On the contrary, larger
particles form lines of particles. Specifically, four lines for particles of size 4, three lines for
particles of size 5, and two lines for particles of size 6 and 7, see figure 3.2.4 for a schematic
illustration. As there are no direct particle-particle interactions in the model (except for
the hard-core repulsion), this ordering must be a result of membrane-mediated attractive
interactions[59]. In previous research on similar topics, two kinds of alignments have been
observed: the side-by-side one, as in this case, and the tip-to-tip one. The side-by-side
alignment is compatible with the energy minimization conditions[41] or for membranes
with high tensions[52]. The membranes in this project are tensionless. The tip-to-tip
alignment has been observed in molecular dynamics simulations of N-BAR domains[51,
50] and it is likely to be caused by direct interactions between proteins ensuing from
the molecular model. It is also relevant to note that multiple nucleation sites occur
with smaller particles while for larger particles the binding happens primarily through
membrane-mediation. The temporal evolution of the system starting from a sphere is
illustrated in the examples in figures 3.2.5(a), 3.2.5(b), 3.2.5(c), whereas figure 3.2.6
shows the transformation of a disk into a tubule.
The disk is made of two flattened quasi-parallel or slightly-swollen sheets connected
at their edges by a bent region, where the particles are bound. Disks have appeared
25
3. Shaping of membranes by concave particles
0 0.1 0.2 0.3 0.4 0.5 0.6
40o
60o
80o
100o
120o
140o
160o
180o
mean induced angle of particles
membrane coverage x of particles
sphere
faceted
tube
disk
Figure 3.2.2.: Morphology diagram showing the mean induced angle versus the mem-
brane coverage for the data points of figure 3.2.1. Open and solid circles
represent simulation results with θc= 3◦and θc= 10◦, respectively. The
mean induced angle is the mean of the values of the angular distribution
shown in figure 3.1.1. Colorings correspond to the morphology classifica-
tion of the converged simulations.
size 3
size 4
size 6
size 5
size 7
x = 0.54 x = 0.59 x = 0.64
x = 0.44 x = 0.53 x = 0.62
x = 0.46 x = 0.54 x = 0.62
x = 0.48 x = 0.52 x = 0.57
x = 0.46 x = 0.52 x = 0.57
Figure 3.2.3.: Illustrations of exemplary morphologies of tubules of particles of size 3 to
7 with different membrane coverages x.
26
3.2. Membrane morphologies
size 4 size 5 size 6 and 7
Figure 3.2.4.: Sectional scheme illustrating the lines arranged around tubules of particles
with 4 to 7 segments. Particles of size 4 form four lines around tubules,
while particles of size 5 the lines are three, and finally larger particles only
form two lines.
with particles of 5, 6, 7 segments, with an intermediate amount of membrane coverage
and a smaller thickness for larger particles, as illustrated in figure 3.2.7 and plotted in
3.2.9. These geometries are compatible with results from energy minimization[42]. It
must be emphasized that unlike tubules, where the whole membrane area is a potential
binding site, in disks only the edges have bound particles adsorbed to them. Therefore
the membrane coverage by particles for disks is expected to scale like the square root
√Aof the surface area A.
3 4 5 6 7
5
6
7
8
9
10
11
12
tube
disk
particle size
thickness [am]
Figure 3.2.9.: Plot of the estimated tubule radius, as well as the disk thickness Dtat
the center, measured in units of amand grouped by particle size. Both
quantities decrease in magnitude with the particle size. This graph also
shows that the ranges of both tubular radii and discoidal thicknesses are
larger for smaller particles.
27
3. Shaping of membranes by concave particles
1 5 1310 15
17 20 3525
0.01 0.02 0.05 0.1 0.4 0.7
(a)
10 20 30
1 2 74 8
0.04 0.05 0.09 0.2 0.4 0.6
(b)
1 643
13
2
15 24 30
35 40 43
(c)
Figure 3.2.5.: Temporal morphological evolution starting from a spherical configuration
and ending into a tubule. Units are 106Monte Carlo steps per membrane
vertex. Simulation with 400 particles. In 3.2.5(a) the particles have 3
segments and U= 18kBT, in 3.2.5(b) the particles have 4 segments and
U= 18kBT, in 3.2.5(c) the particles have 5 segments and U= 12kBT.
28
3.2. Membrane morphologies
048 12
15 22 25 35
Figure 3.2.6.: Temporal morphological evolution starting from a discoidal configuration
and ending into a tubule. Units are 106Monte Carlo steps per membrane
vertex. Simulation with 400 particles of size 5 and particle-membrane
binding constant U= 14kBT.
The computational results shown in this chapter can be compared with experimental
results. In particular, the angle induced on the membrane by particles of 3 segments
is compatible with the observed bendings caused by N-BAR domains such as Arfaptin,
endophilin, and amphiphysin[38, 31]. In our model, the particles of 3 segments loosely
surround the tubular morphologies, forming short-ranged arrangements only. This is
compatible with electron tomography observations of the Bin1 N-BAR domain[11]. As
follows, it is reasonable to hypothesize that Bin1 proteins lack of meaningful direct
protein-protein interaction in Bin1 proteins[11], as the model presented in this project
treats particle-particle interactions just as hard-core repulsions. The percentage of mem-
brane covered by particles is in agreement with fluorescence experiments. For example,
it has been determined that β2 centaurin covers around 37 ±9% of the membrane area,
while endophilin covers 44±27% of tubules pulled with optical tweezers from membrane
vesicles[49].
In previous computational models for the interaction of arc-shaped particles and bi-
ological membranes the particles were embedded in the membrane, as they could not
unbind. Tubular structures and disk-like sheets have already been observed with elastic
membrane models[2, 3, 39, 55, 35, 36]. Models treating curvature-inducing particles as
embedded elements of the membrane have been proposed. For example, a recent model
treats the particles as nematic objects on the vertices of a triangulated membrane[39, 55],
while another treats them as sequences of beads embedded in a membrane represented
by a two-dimensional sheet of beads[35, 36]. These models, having a fixed number of
particles on the membrane in a canonical ensemble, lead to aggregation areas of particles
where a strong membrane curvature is induced, while many regions with low particle
concentrations and weak membrane curvature persists.
29
3. Shaping of membranes by concave particles
size 6
size 5
size 7
top view
side view
top view
side view
top view
side view
x = 0.19 x = 0.26 x = 0.33
x = 0.29 x = 0.32 x = 0.35
x = 0.34 x = 0.35 x = 0.36
Figure 3.2.7.: Examples of system states in discoidal morphologies. The rows categorize
the particles of 5, 6, 7 segments while the variable xis the membrane
coverage ratio.
30
3.2. Membrane morphologies
0 5 8 21
(a)
02 8
19 27 32
36 42
(b)
Figure 3.2.8.: Formation of the disk morphology in a sytem with N= 400 particles. Time
is in 106Monte Carlo steps per vertex units. In 3.2.8(a) the particles have
5 segments and U= 10kBT. Two nucleation sites are visible, two lines
of particles are then formed, which end up merging forming the edge of
the disk. Starting from a tubule in 3.2.8(b), with particles of size 5 and
U= 8kBT.
31
3. Shaping of membranes by concave particles
32
4. Shaping of membranes by mixtures of
concave and convex particles
Mixtures of concave and convex particles can be expected to generate membrane mor-
phologies with regions of negative curvature. One of the motivations for the introduction
of a second type of particles, the is convex ones, is the suggestion that three-way junc-
tions are stabilized by such particles[9, 8, 40]. Three-way junctions are interconnections
of three membrane tubules occurring in the endoplasmic reticulum. They are mostly
present in the peripheral ER and are highly dynamic, continuously undergoing trans-
formations. The Lunapark protein (Lnp1) has been shown to stabilize the formation of
three-way junctions[9, 8, 40]. Experimental evidence suggests that this molecule is com-
posed of two transmembrane domains, possibly forming an arc-like shape, which may
resemble the BAR-domain or reticulon molecules that have been investigated in chapter
3, with the difference that the Lnp1 is expected to bind to the membrane on its convex
side, therefore inducing negative curvature or invaginations on the membrane[8]. The
three-way junction may be illustrated as the junction of three tubular legs converging
to an intersection point, see figure 4.0.1. At the junction, the three pairwise connections
between tubules include saddle-like regions of the surface, having directions with both
positive and negative curvatures. In this chapter, convex particles, which mimic the
proposed shape of Lnp1, are added to the system and the membrane morphologies are
investigated for different relative concentrations of particle types.
In our model, both concave or convex particles interact with the same adhesion energy
Uper particle segment. Therefore, the relative membrane coverage of the two types is
primarily regulated by the overall particle concentrations. The adhesion energy ranges
Figure 4.0.1.: Illustration of an ideal three-way junction connecting three tubules.
33
4. Shaping of membranes by mixtures of concave and convex particles
in the U= 7, . . . , 14 interval. In this chapter, a larger triangulated membrane has been
used, with nT= 5120 triangles. The larger size is motivated by a greater area required by
the three-way junction morphology. The angular threshold for equation 2.1.13 is always
kept at θc= 10◦in all simulations. Ensembles of mixtures of the two types of particles
are considered, with N1concave- and N2convex-interacting number of particles. The
concentration of convex-interacting particles relative to the total number of particles is
defined as
dconvex =N2
N1+N2
.
Concentrations of varying amounts of convex-interacting particles have been used, namely
dconvex = 0.0, 0.02, 0.05, 0.1, 0.2, 0.4, and 1.0. Particles of the same type have the same
number of segments, but the sizes of particles for the two different types may be differ-
ent. The combinations (s1, s2) of particle sizes used in the simulations are (3,3), (4,3),
(4,4), (5,3) and (5,5). The total number of particles in the system is held constant at
N1+N2=N= 400, which is enough to guarantee a reservoir of unbound particles. Low
concentrations of concave-interacting particles are the primary focus. In the simulations,
the initial morphologies for the membrane are the sphere, the tubule, and disks with the
addition of a three-way junction. The initial morphologies of tubule, disk, and three-way
junction are stabilized by particles that are already bound at the start of the simulation,
see section 2.2.1. In this chapter, the volume Vof the space enclosed by the membrane
is allowed to freely fluctuate. Therefore, the constant cVin equation 2.1.8 has the value
0. Additionally, the bending rigidity of the membrane is κ= 10kBTfor all simulations.
4.1. Membrane-mediated segregation and bulged balls
Particles of the same type tend to form lines through membrane-mediated interactions.
Segregation of particles by type is also observed, whereby side-by-side alignments of
concave particles form elongated bulges while side-by-side alignments of convex parti-
cles form invaginations resembling valleys. At high coverages of convex particles, the
membrane looks like a bulged ball, with many valleys of invaginations separated by lines
of bulges, see figure 4.1.1(a) and (b). In some cases, double lines of the same particle type
are also observed, see the first, third and fourth figures in 4.1.1(b). In this configura-
tion, the prevalent adsorbed particle type appears to form a connected region of aligned
particles on the membrane, while the other type of particles aggregate in disconnected
clusters.
Particles of 3 segments form moderate bulges on the membrane, and their side-by-
side alignments appear quite irregular, especially for the concave particles, see figure
4.1.1a. For particles with 4 or 5 segments, the protrusions are more pronounced and the
side-by-side alignments are straighter, see figures 4.1.1(b) and 4.1.1(c).
The dynamical formation of the bulged ball with particle segregation is straightforward
for all particle sizes. Figure 4.1.2(a) illustrates the temporal evolution of a sphere into a
bulged ball with particles of size 3 where the ratio of convex to total particles is dconvex =
0.2, resulting in a prevalence of concave particles covering the membrane. Concave
34
4.1. Membrane-mediated segregation and bulged balls
(a) 3-3
(b) 4-4
(c) 5-5
Figure 4.1.1.: Exemplary morphologies in which the membrane is covered by both con-
cave and convex particles. Membranes covered by a significant amount of
convex-interacting particles resemble bulged balls. Convex particles (blue)
induce invaginations while concave particles (orange) fill the hilly bulges.
The protrusions are less marked for smaller particle sizes. Group (a) shows
examples of morphologies of membranes covered by particles of 3 segments.
Group (b) shows membranes with particles of 4 segments and deeper val-
leys. Group (c) displays membranes with particles of size 5, where the
bulging is extremely marked and entire regions of the membrane are not
covered with particles at all. All figures display segregation of concave and
convex particles, arising from membrane-mediated interactions.
35
4. Shaping of membranes by mixtures of concave and convex particles
time=0.1 time=1 time=4 time=9 time=31 time=40
(a)
time=0.09 time=1 time=4 time=35
(b)
time=1 time=4 time=25 time=41
(c)
time=1 time=6 time=18 time=37
(d)
time=0.1 time=2 time=5
time=8 time=11 time=14 time=19 time=37
(e)
Figure 4.1.2.: Exemplary time sequences of simulations with morphological change to
bulged ball. Time is expressed in 106MC steps per vertex. The starting
morphology of the membrane for this simulation is a sphere. The adhesion
energy are U= 11 kBTfor (a), U= 12 kBTfor (b), U= 9 kBTfor (c),
U= 13 kBTfor (d), and U= 11 kBTfor (e). The concentrations of
convex particles are dconvex = 0.2, 0.4, 0.1, 0.05, 0.2 respectively. The
sizes of concave-convex particles are (3,3), (4,4), (5,5), (5,5), and (4,4)
respectively. (a), (b), (c), (d) start from an initial sphere whereas (e) starts
from a tubule.
36
4.2. Converged morphologies
particles form therefore a connected region isolating some clusters of disconnected convex
lines. A sample for the temporal evolution of particles of size 4 (both concave and
convex) is shown in figure 4.1.2(b), where the concentration of convex to total particles
is dconvex = 0.4. The larger size of the particles produces higher bulges, and the relatively
high coverage of convex particles creates isolated clusters of concave particle alignments.
Figures 4.1.2(c) and 4.1.2(d) illustrate the temporal evolution of a sphere into a bulged
ball with particles of 5 segments. In this case, the bulgings are extremely marked and
the membrane coverage is lower. The larger induced curvature, presumably, does not
allow alignments of particles of different types close to each other. Note that the bulged
ball morphology appears to be common at high concentrations of convex particles and
is probably independent of the initial shape. For example, figure 4.1.2(e) shows the
temporal evolution of a tubule into a bulged ball for particles of size 4 and dconvex = 0.2.
4.2. Converged morphologies
Mixtures of concave and convex particles with 3 segments induce various morphologies
depending on the relative coverage of convex and concave particles dconvex as well as
the total membrane coverage. Figure 4.2.1 shows a sequence of converged membrane
morphologies of simulations starting from a sphere with particles of size 3 ordered by
increasing reduced volume. The vesicles with low reduced volume (v < 0.55) exhibit
tubules, have a high membrane coverage ratio and are predominantly covered by concave
particles. Isolated clusters of convex particles are present. In some cases the tubules
are marked by sharp bends, which turn the membrane into a U shape. Small clusters
of convex particles are located on the inner side of the curve, suggesting that convex
particles are responsible for the production of U-shaped tubules. At intermediate values
of the reduced volume of the membrane v≈0.55 −0.90, the bulged ball is the primary
morphology. The membrane coverage is high and two different kinds of bulged balls are
distinguishable, depending on the relative concentration of concave and convex particles.
At higher values for the reduced volume, i.e. v > 0.90, lower membrane coverages are
observed. Both concave and convex particles appear to be loosely distributed around
the ball.
Figure 4.2.2 shows examples of membranes for simulations starting from a sphere
with particles of size 4 ordered by increasing values of their reduced volume. Mixtures
of particles with 4 segments at low reduced volume (v < 0.55) are predominantly cov-
ered by concave particles, they produce tubular morphologies or three-way junctions.
Convex particles along tubules may form isolated clusters of lines following the tubular
orientation. Three-way junctions are completely covered with concave particles except
for one of the three saddle regions at the junction, which presents a short line of con-
vex particles. It may be assumed that such a cluster acts as an anomaly generating a
third leg out of the tubule, thus inducing the morphology of a three-way junction. At
middle values for the reduced volume (0.55 < v < 0.8), the bulged ball occurs again,
the membrane coverage is still high but wth various concentrations of convex particles.
If the concentration of concave particles is not low, the bulged ball presents extremely
37
4. Shaping of membranes by mixtures of concave and convex particles
marked curvatures of valleys and outward bulgings. At high values for the reduced vol-
ume (v > 0.8), the membrane coverage by particles is very low and the membrane is in
a spherical morphology.
Figure 4.3.7 compares exemplary U-shaped membranes for particles of size (3,3), (4,3),
and (4,4). Figures 4.3.8, 4.3.9, and 4.3.10 illustrate examples of formation of U-shaped
membrane morphologies.
4.3. Stability of three-way junctions
Converged three-way junction morphologies can be observed in two kinds of simulations:
(1) simulations starting from an initial configuration other than the three-way junction
and (2) simulations starting from the three-way junction.
Formation of three-way junctions The formation of three-way junctions from a sphere
was observed only with particles of size 4. Figures 4.3.1 and 4.3.11 illustrate examples
of the formation of three-way junctions from a spherical morphology. A small cluster of
convex particles appears to be disrupting the formation of a tubular structure induced
by the concave particles. This allows a third leg to jut out of the sphere, which is
then extended by the adsorption of concave particles. The small line of convex particles
remains bound to one or two of the three saddle regions of the junction.
Simulations starting from three-way junctions In order to investigate the stability of
three-way junctions, further simulations have been run with the three-way junction as
initial shape for combinations of sizes of concave and convex particles (s1, s2) is (3,3),
(4,3), (4,4), and (5,3). Systems with both concave and convex particles of size 3 are
illustrated in figure 4.3.2. The three-way junction morphology has been preserved at
low coverage of convex particles and in some cases, surprisingly, even without adsorbed
convex particles. At high membrane coverages, a higher concentration of convex particles
among the adsorbed particles results in the structure breaking up and turning into an
elongated ball with clustered groups of convex particles. Some of the membranes have
turned into a U-shaped tubule, thus losing one of the three legs.
For simulations with concave particles of size 4 and convex particles of size 3, the
morphologies stabilized into a three-way junction have their saddle regions covered with
concave particles, unlike other cases, see figure 4.3.3. The convex particles tend to cluster
on the sides of the three-way junction. Double lines of concave particles aligned side-
by-side cover the three saddles of the junction, which are extensions of the four lines
surrounding the tubular morphologies with particles of size 4. For larger amounts of
convex particles, the three-way junction structure is disrupted and turns into a tubule
with invaginations.
When both concave and convex particles have size 4, the locations of the saddles of
the junction show a competition between coverings by double lines of concave particles
connecting two tubular legs, and short lines of side-by-side aligned convex particles, see
figure 4.3.4. Higher amounts of convex particles appear to disrupt the junction.
38
4.3. Stability of three-way junctions
Figure 4.2.1.: Converged morphologies with mixtures of particles of size 3 ordered by
the reduced volume of the enclosed space. All these samples have started
from an initial spherical membrane morphology. Ordinary and U-shaped
tubules are seen on the first line, at higher reduced volume the membrane
turns into a bulged ball and then into a sphere.
39
4. Shaping of membranes by mixtures of concave and convex particles
Figure 4.2.2.: Converged morphologies resulting from simulations with initial spherical
morphology with mixtures of particles of size 4 ordered by the reduced vol-
ume of the enclosed space. Tubules and three-way junctions are spotted at
low reduced volume, while the bulged balls appear at higher concentrations
of adsorbed convex particles.
40
4.3. Stability of three-way junctions
time=1 time=7 time=11 time=13 time=19 time=22
(a)
time=1 time=5 time=7 time=12 time=13 time=22
(b)
time=1 time=5 time=8 time=12 time=19 time=31
(c)
time=1 time=5 time=11 time=14 time=19 time=41
(d)
Figure 4.3.1.: Exemplary time sequences of morphological changes from spheres to three-
way junctions. Time is expressed in 106MC steps per vertex. The concen-
tration of convex-interacting particles is 0.02. Both the concave-interacting
particles and the convex-interacting particles have 4 segments. The start-
ing morphology of the membrane for this simulation is a sphere. The
adhesion energy in 4.3.1(a) is U= 10 kBT, in 4.3.1(b) is U= 11 kBT, in
4.3.1(c) is U= 12 kBTand in 4.3.1(d) is U= 14 kBT.
41
4. Shaping of membranes by mixtures of concave and convex particles
Figure 4.3.2.: Morphologies resulting from simulations starting as three-way junctions,
with particles of size 3. Low concentrations of convex particles appear
to stabilize the three-way junction structure, while higher concentrations
disrupt the structure into an elongated tubule.
Mixtures of particles with concave size 5 and convex size 3 always show the transition
from the three-way junction to either disks or faceted shapes with three lines of concave
particles, with convex particles loosely bound to isolated locations of the membrane, see
figure 4.3.5. It appears that three-way junctions are unstable states for these combination
of particle sizes.
Figure 4.3.6 compares examples of three-way junctions for combinations of particles
with (s1, s2) of (3,3), (4,3), and (4,4). The reduced volume for the membrane ranges
from 0.43 to 0.55.
42
4.3. Stability of three-way junctions
Figure 4.3.3.: Examples of simulations starting as three-way junctions with concave par-
ticles of size 4 and convex particles of size 3. Concave particles of size 4
are able to control the saddle-like regions of the junction, while convex
particles are loosely clustered on relatively flat regions.
43
4. Shaping of membranes by mixtures of concave and convex particles
Figure 4.3.4.: Examples of morphologies from simulations starting as three-way junctions
with particles of size 4. Convex particles tend to form short lines on one
saddle region of the junction. Higher amounts of adsorbed convex particles
disrupt the structure of the junction.
Figure 4.3.5.: Examples of two simulations starting as three-way junctions with concave
particles of size 5 and convex particles of size 3. The initial morphology
is not preserved, as the membrane is transformed into either a disk or a
folded disk, as shown in the pictures.
44
4.3. Stability of three-way junctions
(3,3)
v=0.5094 v=0.5323 v=0.5156 v=0.5537
(4,3)
v=0.4745 v=0.4795 v=0.4846 v=0.5017
(4,4)
v=0.4337 v=0.4502 v=0.4776 v=0.4806
Figure 4.3.6.: Examples of three-way junction morphologies for different particle sizes.
The reduced volume of the membranes is reported. Sizes for particle types
of (3,3), (4,3), and (4,4) are shown. Convex particles (blue) are located
near the junction while concave particles (orange) cover all sides.
45
4. Shaping of membranes by mixtures of concave and convex particles
(3,3)
v=0.5071 v=0.5135 v=0.5255 v=0.5722
(4,3)
v=0.4582 v=0.4991 v=0.5040 v=0.5101
(4,4)
v=0.4551 v=0.4646 v=0.4692 v=0.5722
Figure 4.3.7.: Examples of U-shaped tubule morphologies. The reduced volume of the
luminal space of the membranes is reported. Particle types of size (3,3),
(4,3), and (4,4) are shown. Convex particles (blue) are often located on
the saddles of tubular bends while concave particles (orange) cover the
whole membrane.
time=1 time=13 time=18 time=20
time=24 time=25 time=27 time=40
Figure 4.3.8.: Exemplary time sequence of a morphological change from sphere to U-
shaped tubule. Time is expressed in 106MC steps per vertex. In this
simulation, the adhesion energy is U= 13 kBT. The concentration of
convex-interacting particles is 0.02. Both the concave-interacting parti-
cles and the convex-interacting particles have 3 segments. The starting
morphology of the membrane for this simulation is a sphere.
46
4.3. Stability of three-way junctions
time=1 time=7 time=14 time=16
time=16 time=19 time=22 time=27
Figure 4.3.9.: Exemplary time sequence of a morphological change from sphere to U-
shaped tubule. Time is expressed in 106MC steps per vertex. In this
simulation, the adhesion energy is U= 9 kBT. The concentration of
convex-interacting particles is 0.02. Both the concave-interacting parti-
cles and the convex-interacting particles have 3 segments. The starting
morphology of the membrane for this simulation is a sphere.
time=1 time=5 time=14 time=19
time=25 time=31 time=38 time=39
Figure 4.3.10.: Exemplary time sequence of a morphological change from sphere to U-
shaped tubule. Time is expressed in 106MC steps per vertex. In this
simulation, the adhesion energy is U= 8 kBT. The concentration of
convex-interacting particles is 0.02. Both the concave-interacting parti-
cles and the convex-interacting particles have 4 segments. The starting
morphology of the membrane for this simulation is a sphere.
47
4. Shaping of membranes by mixtures of concave and convex particles
time=1 time=4 time=13 time=19
time=22 time=25 time=33 time=37
Figure 4.3.11.: Time sequence of a morphological change from sphere to three-way junc-
tion. Time is expressed in 106MC steps per vertex. In this simulation, the
adhesion energy is U= 9 kBT. The concentration of convex-interacting
particles is 0.02. Both the concave-interacting particles and the convex-
interacting particles have 4 segments. The starting morphology of the
membrane for this simulation is a sphere.
48
5. Membrane-mediated interactions
between particles
The adsorption of arc-like particles to the membrane leads to membrane-mediated in-
teractions between membrane-bound particles[59]. These indirect, membrane-mediated
interactions arise because the membrane curvature induced by the arc-like particles costs
bending energy[59]. This cost depends on the distance and relative orientation of the
particles and has been shown to result in a strong attraction between concave arc-like
particles if the particles are oriented side-by-side[41, 59].
In statistical mechanics, the free energy of the interaction between two particles in
a complex system may be deduced from the radial distribution function g(r), which
describes the density variation in number of particles with respect to the distance r
from a reference particle. The complete interaction energy between particles may be
computed from the radial distribution function g(r) of particle pairs and its link to the
particle-particle interaction energy w(r) provided by the reversible work theorem[7]
g(r) = exp −w(r)
kBT(5.0.1)
where ris the distance between two particles, kBis the Boltzmann constant, and Tis
the temperature of the system. The interaction between particles of 3 and 4 segments
adsorbed to tubules and spheres is studied and quantified in this chapter. The triangu-
lated membrane with nT= 2000 triangles has been employed. The simulation runs in a
box with periodic boundary conditions and volume Vbox ≈3·105a3
m. This is 64 times as
large as the volume of a perfect sphere, slightly larger than the size of the box described
in equation 2.2.1.
Previous investigations of the membrane-mediated interaction between particles bound
to a membrane have largely focused on planar morphologies[?]. Only recently, membrane-
mediated interactions between Janus particles adsorbed to curved morphologies have
been investigated[?]. Here, in order to investigate the membrane-mediated interaction
of concave particles bound to a curved membrane, a further constraint on the enclosed
volume of the membrane is introduced, which keeps the membrane morphology in a
stable or metastable tubular shape, even without the presence of curvature inducing
particles[4]. The membrane curvature can therefore be regulated by fixing its reduced
volume, with the addition of the contribution of equation 2.1.8 to the energy of the sys-
tem. The curvature induced by bound particles is expected to depend on the curvature
induced by the volume constraint, which in turn affects the membrane-mediated inter-
action. The values for the equilibrium reduced volume v0examined are v0= 0.35, 0.4,
0.45, 0.5, 0.55, 0.6, 0.65, 0.7 for the tubular morphology, and v0= 0.95 for the spherical
49
5. Membrane-mediated interactions between particles
morphology. It is important to note that the tubule is indeed a metastable morphology
for reduced volume constraints smaller than v0= 0.652, as the stable morphology is
either an oblate discocyte or an axisymmetric stomatocyte[4]. In order to prevent the
transition to other morphologies, a higher value for the bending rigidity, κ= 30kBT,
has been chosen[4]. The simulations for v0= 0.95 use the sphere as starting morphology
for the membrane, while all others use spherocylinder as described in section 2.2.1. The
total number of particles in the system is N= 400, their sizes are either 3 or 4 segments,
depending on the simulation. For each simulation, the membrane coverage ratio is kept
at either 5%, 10%, or 20% by dynamically varying the value of the adsorption energy
U. The adjustment occurs as
U−→ U−cU(NB−N0)
where NBis the average estimation of bound particles over the last 106MC steps,
whereas N0is the desired number of bound particles, determined by counting each par-
ticle segment as occupying an area a2
pπ/4 (see equation 2.1.14) on the membrane. Finally,
cU= 10−5has been chosen as proper value for the dynamics of U, so that the number
of adsorbed particles only varies by a few units from the desired number. The data for
the radial distribution function is only examined after the number of bound particles
has stabilized to a neighborhood of the equilibrium value N0. The radial distribution
function is then estimated to be
g(r) = ρ1(r)
ρ2(r)(5.0.2)
where ρ1(r) is the measured joint distribution function for finding a particle at distance
rfrom a particle at a properly chosen origin point on the tubule[7], whereas the function
ρ2(r) is the expected joint distribution for a free gas of particles with the same hard-core
shape. The distribution of a simple free gas is not correct as the particles in this project
are rigid bodies with spacial extension.
The function ρ1(r) is estimated as the distribution of the distances between all pairs
of particles for which at least one particle is central. Central particles are defined to be
particles bound to the central 50% area of the tubule. Therefore, in the case of tubular
structures, they exclude the particles bound to the spherical ends, that is, areas with a
different membrane curvature. The axis upon which the central particles are determined
is the first principal component of the covariance matrix of the spatial distribution of
membrane vertices. The pairs are collected at different points in time, at intervals of
100 MC steps per vertex for a period of around ≈107MC steps per vertex.
For symmetric particles on a planar surface it is expected to be ρ2(r) = 2πr for r
larger than the hardcore diameter. The fact that our particles are not symmetric makes
the composition of ρ2(r) more complicated. The value of function ρ2(r) for particles of 3
and 4 segments is estimated by running a simulation of randomly positioning flattened
particles on a plane with periodic boundary conditions. The flattened particles are
analogous to those defined in 2.1.2 with the exception of not being curved, as they lie on
a plane. The flattened particle is a line of 3 or 4 points at distance apin the sequence,
50
5.1. Radial distribution functions for bound particles
each having a hard-core radius of ap/2. A simulation is carried out to randomly position
the particles in space with random orientation. The relative position of the centers is
then measured as the estimation for ρ2(r). Figure 5.0.1 displays the estimation of the
ρ2(r) functions for particles of sizes 3 and 4. The curves have two non-smooth points,
corresponding to the distances at which two particles acquire more rotational freedom.
The curves in figure 5.0.1 are identical to ρ2(r) = 2πr for large values of r.
5.1. Radial distribution functions for bound particles
The radial distribution function shows that the correlation at short distance is the
strongest for membranes of reduced volume of v0= 0.95, whereas the minimal steepness
is for membranes with reduced volume of v0= 0.35 and v0= 0.4, see figures 5.1.1 and
5.1.2. Using equation 5.0.1 to derive the equivalent particle-particle interaction energy
with membrane mediation, the attraction of particles at low range (1.5amto 3 am) is
stronger for the spherical membrane morphology (reduced volume v0= 0.95) and for
larger tubular morphologies with reduced volume higher than v0= 0.4. The cases with
particles of size 4 and membrane coverage of 10% with reduced volume v0= 0.7, and
membrane coverage of 20% with reduced volumes v0= 0.65 and v0= 0.7 had a morpho-
logical transition of the membrane into a disk, and have therefore been excluded. The
condition v0= 0.327 is expected to create membranes with radius length closest to rp,
defined in equation A.1.1. Indeed, if the volume of the membrane is calculated under
the assumption that the membrane shape is a spherocylinder, one gets
4πr2
p+ 2πhrp=A, (5.1.1)
4
3πr3
p+πhr2
p=V. (5.1.2)
Solving equation 5.1.1 for hand substituting in equation 5.1.2, with the value A=
A0as defined in 2.1.6, gives approximately v0≈0.32689. As a consequence of the
definition of membrane-particle interaction in equation 2.1.11, the radius rpis the most
compatible curvature for a tubule to bind with the particle models used in this project.
The membrane-mediated interaction due to the bending energy cost is therefore expected
to be minimized for membranes of reduced volume v0= 0.327. It is no surprise that
both membranes of reduced volume v0= 0.35 and v0= 0.4 show the smallest attraction
at short distance between particles. The reduced volumes of v0= 0.45 up to v0= 0.7,
together with v0= 0.95 show an increasing stronger interaction energy at low ranges for
all membrane coverages and both particle sizes, see figures 5.1.3 and 5.1.4.
The total interaction between particles of size 4 on the membrane with reduced volume
v0= 0.95 shows a second minimum for membrane coverages of 10% and 20% for r=
3.5amto 4am, see figures 5.1.4(b) and 5.1.4(c). This corresponds to the effect of the
second coordination in the lines of side-by-side particle alignments, see figures 5.1.6(e)
and 5.1.6(f). In the range from r= 3.2amto 4.5amfor the membrane with reduced
volume V0= 0.95 and coverage of 5% the energy curve appears less steep, see figure
51
5. Membrane-mediated interactions between particles
a)
b)
Figure 5.0.1.: Distribution functions ρ2(r) for a free gas of particles of size 3 (a) and 4
(b), on a plane with toroidal boundary conditions. In both graphs there
are two non-smooth points on the curves. The first non-smooth point
corresponds to the distance of full rotational freedom for particles at the
side of a reference particle (3 amfor particles of three segments, 3√35
4am≈
4.437 amfor particles of four segments), while the second non-smooth point
corresponds to the distance of full rotational freedom for particles at the
top of a reference particle (4.5amfor particles of three segments, 6 amfor
particles of four segments). Notice that after the last non-smooth points
the curves are described by ρ2(r) = 2πr.
52
5.1. Radial distribution functions for bound particles
5.1.4(a), this can again be reconnected to the formation of lines of particles, see figure
5.1.6(d).
53
5. Membrane-mediated interactions between particles
a)
b)
c)
Figure 5.1.1.: Radial distribution functions for particles of size 3 bound to the membrane.
The rows correspond to membrane coverages of 0.05 (a), 0.1 (b), and
0.2 (c). The strongest correlation at low distance has been observed for
reduced volume v0= 0.95.
54
5.1. Radial distribution functions for bound particles
a)
b)
c)
Figure 5.1.2.: Radial distribution functions for particles of size 4 bound to the membrane.
The rows correspond to membrane coverages of 0.05 (a), 0.1 (b), and
0.2 (c). The strongest correlation at low distance has been observed for
reduced volume v0= 0.95.
55
5. Membrane-mediated interactions between particles
a)
b)
c)
Figure 5.1.3.: Membrane-mediated interaction energies for particles of size 3 bound to
the membrane. The rows correspond to membrane coverages of 0.05 (a),
0.1 (b), and 0.2 (c).
56
5.1. Radial distribution functions for bound particles
a)
b)
c)
Figure 5.1.4.: Membrane-mediated interaction energies for particles of size 4 bound to
the membrane. The rows correspond to membrane coverages of 0.05 (a),
0.1 (b), and 0.2 (c).
57
5. Membrane-mediated interactions between particles
a)
b)
c)
d)
e)
f)
g)
h)
Figure 5.1.5.: Comparison of tubular morphologies with different reduced volumes. The
membrane coverage is 10% and the particles have three segments. The
reduced volume is constrained to v0= 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65,
0.7 for (a,b,c,d,e,f,g,h), respectively.
58
5.1. Radial distribution functions for bound particles
a) b) c)
d) e) f)
Figure 5.1.6.: Exemplary morphologies with reduced volume constrained to v0= 0.95 for
particles of size 3 and 4 at different membrane coverages. The morphology
is spherical. The rows (a, b, c) and (d, e, f) display snapshots of the
membrane with bound particles of sizes 3 and 4, respectively. The columns
(a, d), (b, e), and (c, f) correspond to membrane coverages of 5%, 10%,
and 20%, respectively. Notice that (e) and (f) present lines of side-by-
side aligned particles, which is compatible with the second coordination
location visible in the potential in figures 5.1.3(d) and 5.1.3(f).
59
5. Membrane-mediated interactions between particles
60
A. Appendix
A.1. Initial membrane morphologies
In order to perform simulations with all different initial membrane configurations, the
membrane has been shaped into a disk and a spherocylinder by adding a squeezing po-
tential to the spherical configuration. That is, starting from the spherical configuration,
a MC simulation has been performed with vertex movements and edge flips, until the
desired shape has been reached. At this point, simulated annealing (i.e. a decrease of
temperature in the MC acceptance ratio) is performed to smoothen the surface. Given
the radius of curvature Rpof the particles, as defined in 2.1.9, and the membrane-
particle interaction potential described in equation 2.1.12, the ideal matching radius for
the membrane is determined to be
rp=Rp−r1+r2
2(A.1.1)
•For the disk, a radial potential around the z-axis is applied, which is minimized in
equations 2.2.3. The value of ris replaced by rp, and Ris calculated so that the
area of the disk is Aref.
•In order to shape the membrane into a spherocylinder, a semi-harmonic potential
is created for vertices in the triangulated surface depending on their distance to
the z-axis. The semi-harmonic potential is
c(rz−r)2Θ(rz−r)
where rzis the distance to the z-axis, cis a tunable constant and Θ is the Heaviside
step function.
In order to accelerate the process an additional pulling potential is created to
stretch the membrane along the z-axis until a reasonable length has been reached.
A.2. Rotations with quaternions
A computationally efficient implementation of rotations can be achieved with the repre-
sentation of rotations in quaternion algebra[16]. Quaternions are extensions of complex
numbers that have three different kinds of imaginary parts:
q=a+ib+jc+kd
61
A. Appendix
Under multiplication the imaginary symbols follow the rules:
ij =k jk =i ki =j
i2=−1j2=−1k2=−1.
Unit quaternions, that is quaternions whose components observe a2+b2+c2+d2= 1
may be employed to represent rotations, by acting on vectors by
p′=q·p·q−1,
where pis in the form pxi+pyj+pzk. The correspondence between a unit quaternion
and the axis-angle (r,θ) representation of a rotation is given by:
q= cos θ
2+rsin θ
2,(A.2.1)
where r=rxi+ryj+rzkis the axis of rotation and θis the rotation angle.
Small random rotations can therefore be performed by picking a random unit quater-
nion qεin the neighborhood of 1, such that the Euclidean distance to unity is less than
ε. Given a random uniformly distributed unit quaternion q, a suitable neighborhood of
unity can be found by[16]
qε=1 + εq
k1 + εqk,(A.2.2)
where εis the rotational magnitude tuning parameter.
In order to generate random unit quaternions, an algorithm originally developed by
Marsaglia has been used[56, 30]. The procedure is as follows: generate numbers x1and
x2independent and uniform in the interval [−1,1], such that S1=x2
1+x2
2<1. The
same procedure is repeated to generate numbers x3and x4independent and uniform
in the interval [−1,1], such that S2=x2
3+x2
4<1. After calculating c=q1−S1
S2, the
random unit quaternion will be q= (x1, x2, c ·x3, c ·x4). The distribution of rotation
axes is uniform (because of symmetry). The distribution of angles on the other hand
is non-uniform and slanted towards θmax. For a given ε, the maximum angle is θmax =
2 arccos √1−ε2. The value ε= 0.02 has been employed throughout this project.
A.3. Radii of tubules and disks
Triangulated membranes of tubular shapes can be characterized by their radius. Sim-
ilarly, disk-like shapes vary in thickness. These quantities can be extracted from the
triangulated structure of the model membrane. The thickness of disks can be defined as
the distance between the two sheets at the center of mass of the disk. For an ideal disk,
the two sheets are parallel and their distance is equal to 2Rp, where Rpis the curvature
radius of the particles. In reality, the results of the simulations show that disks tend to
bulge slightly and the size of particles influences the thickness.
A geometrical method has been devised to estimate these quantities. Namely, for
every vertex of the membrane i, a vertex j(i) opposite to iwith respect to the center
62
A.4. Bending energy of ideal shapes
of mass is associated. Formally, by calling pithe position with respect to the center
of mass of vertex iof the membrane, the vertex j(i) is the one such that pi·pj(i)
|pi||pj(i)|is
minimal, which means as close to negative unity as possible. The vertex j(i) will be
located close to the antipodal position on the membrane to the vertex i. By calling
the set of such pairs X={i, j(i)}i∈vertices, the estimate for the tubular radius and the
discoidal thickness is given by the expression min {|pi−pj|}i,j∈X, that is, the minimum
of the distances of vertex pairs in set X.
RpDt
Figure A.3.1.: Schematic view of the disk thickness at its center.
A.4. Bending energy of ideal shapes
The bending energy of membranes given by equations 1.2.5 can be solved for ideal shapes
of tubules and disks as parametrized in section 2.2.1 .
A.4.1. Tubule
The ideal tubule is modeled as a spherocylinder, which is a cylinder (of height hand
radius r) capped with two semispheres of radius r. The computed bending energy is
Et=πκ 8 + h
r.
This equation can be expressed as a function of rby inverting it with the area equation
At= 4πr2+ 2πhr
therefore
Et= 6πκ +32/33
pπ
2κ
r2
A.4.2. Disk
The ideal disk is a cylinder of height 2rand radius R, whose curved surface is capped
by an outer semi-torus parametrized by rand R. The bending energy is given only by
the curved region of the surface. A parametrization of the outer semi-torus with angles
θand φis introduced, where θ∈(−π
2,π
2) spans the section of radius r, while φ∈(0,2π)
revolves around the rotational symmetry axis of the disk.
63
A. Appendix
Bending energy
The equation 1.2.5 is solveable for the disk. The area element for the torus is
dA =r(R+rcos θ)dθ dφ
and the mean curvature is given by
H=1
21
r+cos θ
R+rcos θ.
At this point, equation 1.2.5 is expressed as
Ed=1
2κZ2π
0Zπ
2
−π
21
r+cos θ
R+rcos θ2
r(R+rcos θ)dφ dθ,
which results in
Ed= 4πκ
R24 tanh−1R−r
√(r−R)(r+R)+ 5 tanh−1q2r
r+R−1
rp(r−R)(r+R)+ 2
.
Area of the disk
The area of the disk can be calculated by integrating or by using Pappus’ theorem,
resulting in
Ad= 4πr2+ 2π2rR + 2πR2.
By inverting the area equation, under the assumption of constant membrane area, it is
possible to express the bending energy as a function of ronly
Ed=√2π2/3κA−π4/3r2(X+Y)
rqπ2/3rπ2/3A−(π2−6) r−62/3
+ 8πκ,
where
X= 4 tanh−1 A−3
√π(2 + π)r
p2π4/3Ar −2π2/3(π2−6) r2−2 62/3!,
Y= 5 tanh−1 s−43
√πr
(π−2) 3
√πr −A−1!,
and A=pπ2/3(π2−8) r2+ 2 62/3is a repeated subexpression.
64
A.5. Radial distribution of an ideal gas on a cylinder
A.5. Radial distribution of an ideal gas on a cylinder
In this section the radial distribution function for a free gas on an infinite cylinder is
computed. Under the assumption of constant surface density ρfor particles of a free
gas, the expected density at distance Rfrom a point on the cylinder can be described
as the area of the cylinder inside a sphere of radius Rhaving its center on the cylinder.
The cylinder is represented by the implicit parametrization
x2+y2=r2,(A.5.1)
while the sphere with center on the edge of the cylinder at (r, 0,0) and radius Ris given
by
(x−r)2+y2+z2=R2.(A.5.2)
In what follows, the intersection curve between the cylinder (A.5.1) and the sphere
(A.5.2) is parametrized with polar coordinates in the xy-plane. The surface area of the
cylinder that lies inside the sphere is calculated by solving A.5.2 for z, for any sphere of
size R, giving
x=rcos φ
y=rsin φ
z=pR2+ 2r2(cos φ−1)
.(A.5.3)
The area element for the integration on a cylindrical surface is dA =r dz dφ. At this
point, by calling B=pR2+ 2r2(cos φ−1), one derives
ZA
−AZB
−B
r dz dφ, (A.5.4)
where Bis the integration boundary along the zaxis, while Ais the maximum value of
φfor a given value of R. This integral can be divided in two parts, the case in which
the sphere does not envelope the entire cylindrical section (i.e. R < 2r) and the case in
which it does (i.e. R > 2r). For the first case, the value A= arccos(R2−2r2
2r) is replaced
in equation A.5.4. Solving the integral, the result is
8·r·R·E1
2arccos 2r2−R2
2r2|4r2
R2,(A.5.5)
where E(a, b) is the incomplete elliptic integral of the second kind, given by
E(a|b) = Za
0
√1−bsin x dx.
For the second case, the angular integration occurs around the whole cylinder, therefore
A=π. Setting up the limiting cases
65
A. Appendix
Zπ
−πZB
−B
r dz dφ,
which results in
8rpR2−4r2E−4r2
R2−4r2,(A.5.6)
where E(b) is the complete elliptic integral of the second kind
E(b) = E π
2|b=Zπ
2
0
√1−bsin x dx.
In order to get the radial distribution function, the formulae A.5.5 and A.5.6 are derived
by R, resulting in
8·r·F1
2arccos 1−R2
2r2|4r2
R2if R < 2r
8·r·R·K−4r2
R2−4r2
√R2−4r2if R > 2r
,
where F(a, b) is the incomplete elliptic integral of the first kind
F(a|b) = Za
0
1
p1−bsin2xdx
and K(a) is the complete elliptic integral of the first kind
K(b) = F π
2|b.
66
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