Membrane morphologies induced by arc-shaped scaffolds are determined by arc angle and coverage
MMembrane Morphologies Induced by Arc-Shaped Scaffolds areDetermined by Arc Angle and Coverage
Francesco Bonazzi and Thomas R. Weikl
Max Planck Institute of Colloids and Interfaces, Department of Theory and Bio-Systems, Science Park Golm, 14424 Potsdam, Germany
ABSTRACT The intricate shapes of biological membranes such as tubules and membrane stacks are induced by proteins. Inthis article, we systematically investigate the membrane shapes induced by arc-shaped scaffolds such as proteins and proteincomplexes with coarse-grained modeling and simulations. We find that arc-shaped scaffolds induce membrane tubules atmembrane coverages larger than a threshold of about 40%, irrespective of their arc angle. The membrane morphologies atintermediate coverages below this tubulation threshold, in contrast, strongly depend on the arc angle. Scaffolds with arc anglesof about akin to N-BAR domains do not change the membrane shape at coverages below the tubulation threshold, whilescaffolds with arc angles larger than about induce double-membrane stacks at intermediate coverages. The scaffoldsstabilize the curved membrane edges that connect the membrane stacks, as suggested for complexes of reticulon proteins. Ourresults provide general insights on the determinants of membrane shaping by arc-shaped scaffolds .
INTRODUCTION
The shapes of biological membranes that surround cellsand cellular organelles are often highly curved (1–3). Themembrane curvature is induced and regulated by proteins suchas the arc-shaped BAR domains (4–8) or the reticulons (9, 10),which have been suggested to oligomerize into arc-shapedprotein complexes (11). Arc-shaped proteins and proteincomplexes can induce membrane tubules (12–14), but havealso been associated with other highly curved membranestructures. Reticulon proteins, for example, are involved inthe generation of the membrane tubules of the endoplasmicreticulum (ER) and have been suggested to stabilize thehighly curved edges (15, 16) that connect stacked membranesheets of the ER (3, 17). Electron microscopy indicates thatBAR domain proteins can form highly ordered helical coatsaround membrane tubules (12, 18, 19) that are apparentlyheld together by specific protein-protein interactions, as wellas rather loose, irregular arrangements (14). The variabilityof distances and angles between neighboring BAR domainsin these loose arrangements suggests that the arrangementsform without specific protein-protein interactions (14) and,thus, may be dominated by membrane-mediated interactionsbetween the proteins (20–22). These indirect, membrane-mediated interactions arise because the overall bending energyof the membrane depends on the distance and orientationof curvature-inducing proteins. In simulations with coarse-grained models, a variety of morphologies with tubular or disk-like membrane shapes have been observed (23–29). The disk-like shapes consist of a double-membrane stack connected bya curved edge and are counterparts of the connected, stackedmembrane sheets in the much larger membrane systemsinvestigated in experiments (3, 17). In this article, we systematically investigate the membranemorphologies induced by arc-shaped scaffold particles such asproteins or protein complexes with coarse-grained modelingand simulations. In our coarse-grained model of membraneshaping, the membrane is described as a triangulated elas-tic surface, and the particles as segmented arcs that inducemembrane curvature by binding to the membrane. The directparticle-particle interactions are purely repulsive and only pre-vent particle overlap. The particle arrangements in our modelare therefore governed by indirect, membrane-mediated inter-actions. These particle arrangements are essentially unaffectedby the membrane discretization because the particles are notembedded in the membrane, in contrast to previous elastic-membrane models. In previous models, curvature-inducingparticles have been described as nematic objects embeddedon the vertices of a triangulated membrane (25, 26), or ascurved chains of beads embedded in a two-dimensional sheetof beads that represents the membrane (27, 28).Our main aim here is to obtain general classification of themembrane morphologies induced by arc-shaped scaffold par-ticles that do not exhibit specific attractive interactions. Thisclassification is obtained from simulations in which the overallnumber of particles exceeds the number of membrane-boundparticles. The membrane coverage then is not constrainedby the number of available particles, which leads to rathersharp transitions between ‘pure’ spherical, tubular, or disk-likemorphologies in our simulations. Previous elastic membranemodels, in contrast, have been investigated for a fixed numberof membrane-embedded or bound particles, which typicallyleads to ‘mixed’ membrane morphologies, e.g. morphologieswith membrane tubules or disks protruding from a sphericalversicle. a r X i v : . [ q - b i o . S C ] F e b e find that the membrane shape is fully determined bythe arc angle and the membrane coverage of the particles. Forall considered arc angles of the particles between 60° and180°, membrane tubules are formed at particle coverages thatexceed about 40%. Arc angles of 60° roughly correspondto the angle enclosed by BAR domain proteins such as theArfaptin BAR domain and the endophilin and amphiphysin N-BAR domains (30, 31), while larger arc angles up to 180° havebeen postulated for reticulon scaffolds (15, 16). At smallermembrane coverages below 40%, particles with arc angles ofabout 60° do not change the membrane morphologies in ourmodel, while particles with arc angles larger than about 120°induce disk-like double-membrane stacks by stabilizing curvededges. Particles with arc angles around 90° lead to faceted,irregular membrane morphologies at smaller coverages. Thearrangements of particles with arc angles of 60° along tubulesin our simulations is similar to the rather loose arrangement ofN-BAR domains observed in electron microscopy experiments(14). This similarity supports the suggestion that these ratherloose arrangements of N-BAR domains are dominated bymembrane-mediated interactions. METHODSModel
We model the membrane as a discretized closed surface. Thebending energy of a closed continuous membrane withoutspontaneous curvature is the integral E be = κ ∮ M dS overthe membrane surface with local mean curvature M (32).We use the standard discretization of the bending energyfor triangulated membrane described in Ref. (33, 34) andchoose as typical bending rigidity the value κ = k B T (35). Our discretized membranes are composed of either n t = n t = [ a m , √ a m ] (33,34). The area of the membrane is constrained to ensurethe near incompressibility of lipid membranes (36). Thestrength of the harmonic constraining potential is chosensuch that the fluctuations of the membrane area are limitedto less than 0.5%. The enclosed volume is unconstrained toenable the full range of membrane morphologies. Coarse-grained molecular simulations indicate that the full spectrumof bending fluctuations can be described for a membranediscretization length a m of about 5 nm(37).Our arc-shaped particles are composed of 3 to 7 identicalplanar quadratic segments. Neighboring segments share aquadratic edge and enclose an angle of 30° in most of oursimulations (see Figure 1(a)). The arc angle of the particles,i.e. the angle between the first and last segment, then adoptsthe values 60°, 90°, 120°, 150°, and 180° for particles with3, 4, 5, 6, and 7 segments, respectively. In addition, weconsider particles composed of 5 segments with an angle of15° between neighboring segments. These particles have thesame arc angle of 60° as particles composed of 3 segments o o o o o o o o o o o o o o o induced angleinduced angle p r obab ili t y p r obab ili t y size 3 size 6size 4 size 5particle size 3binding cutoff θ c =10 o size 7binding cutoff θ c =10 o θ c =5 o θ c =3 o size 3 size 4 size 6 size 7size 5(a)(b)(c) Figure 1: (a) Arc-shaped particles composed of 3 to 7 planarsegments with an angle of 30° between neighboring segments.The angle between the two end segments of the particles is60° for particles of size 3, and 90°, 120°, 150°, and 180° forparticles of size 4, 5, 6, and 7, respectively. (b) Distributionsof angles between the membrane triangles that are boundto the end segments of the particles for the binding cutoff θ c =
10° of the particle-membrane adhesion potential (seeEquation (1)). The mean values of these angle distributions are52.5°, 82.6°, 112.4°, 143.2°, and 169.7° for particles of size3 to 7, respectively. (c) Distributions of angles between themembrane triangles bound to the end segments of particles ofsize 3 for different values of the binding cutoff θ c . The meanvalue of the distributions increases from 52.5° for θ c = θ c =
5° and 59.1° for θ c = V pm = − U f r ( r ) f θ ( θ ) , (1)where r is the distance between the center of the segment andthe center of the nearest triangle, θ is the angle between the nor-mals of the segments and this triangle, and U is the adhesionenergy per particle segment. The distance-dependent function f r is a square-well function that adopts the values f r ( r ) = r < r < r and f r ( r ) = f θ is square-well function with values f θ ( θ ) = | θ | < θ c and f θ ( θ ) = r = a m and r = a m in all our simulations, and the value θ c = θ c =
3° and 5°, besides θ c = a p . Thehard-core area of a particle segment thus is π a p /
4. We use thishard-core area in calculating the membrane coverage of boundparticles. We choose the value a p = a m for the linear sizeof the planar particle segments. The particle segments thenare slightly larger than the membrane triangles with minimumside length a m , which ensures that different particle segmentsbind to different triangles. Simulations
We have performed Metropolis Monte Carlo simulations in acubic box with periodic boundary conditions. The simulationsconsist of four different types of Monte Carlo steps: membranevertex translations, membrane edge flips, particle translations,and particle rotations. Vertex translations enable changesof the membrane shape, while edge flips ensure membranefluidity (38). In a vertex translation, a randomly selected vertexof the triangulated membrane is translated along a randomdirection in three-dimensional space by a distance that israndomly chosen from an interval between 0 and 0.1 a m . In aparticle translation, a randomly selected particle is translatedin random direction by a random distance between 0 and a m .In a particle rotation, a randomly selected particle is rotatedaround a rotation axis that passes trough the central pointalong the particle arc. For particles that consist of 3, 5, or7 segments, the rotation axis runs through the center of thecentral segments. For particles of 4 or 6 segments, the rotationaxis runs through the center of the edge that is shared bythe two central segments. The rotation axis is oriented ina random direction. The random rotations are implemented using quaternions (39, 40) with rotation angles between 0and a maximum angle of about 2.3°. Each of these types ofMonte Carlo steps occur with equal probabilities for singlemembrane vertices, edges, or particles.The membrane coverage x of the particles in our simula-tions depends on the overall number of particles and on theadhesion energy U per particle segment. The overall numberof particles in our simulations is either N =
200 or 400, andthe adhesion energy per particle segment is varied from U = k B T to obtain the full range of possible coverages. Foreach combination of particle number N and adhesion energy U , we have performed simulations starting from an initialspherical, disk-like, or tubular membrane shape (see FigureS1). The particles are initially randomly distributed in thesimulation box outside of the membrane. Simulations startingfrom initial disk-like and tubular shapes first include onlyparticle translations and rotations to stabilize these initialshapes by bound particles. All simulations then include allfour types of MC moves for total simulation lengths between1 · and 8 · MC steps per membrane vertex, dependingon convergence. To verify convergence, we divide the last10 MC steps per vertex of a simulation into ten intervals of10 steps and calculate the average coverage x and reducedvolume v of the membrane for each interval. For membranescomposed of n t = x and reduced volume v given in Figures2 and 3 are the mean values of these 10 averages. For ourlarger membranes composed of n t = x and v for the last 10 intervals of 10 MCsteps are both smaller than 0.01.
RESULTS
The arc-shaped particles of our model induce membranecurvature by binding to the membrane with their inner concavesides. We first consider particles composed of 3 to 7 planarsegments with an angle of 30° between adjacent segments(see Figure 1(a)). The arc angle of these particles dependson the particle size, i.e. on the number of planar segments.A particle segment is bound to the discretized, triangulatedmembrane of our model if its distance to the closest membranetriangle is within a given range, and if the particle segmentand membrane triangle are nearly parallel with an angle thatis smaller than a cutoff angle θ c (see Methods for details). Therelative area of the particle segments and membrane trianglesis chosen such that a particle segment can only be bound to asingle membrane triangle. Figures 1(b) and (c) illustrate thedistributions of angles between the two membrane trianglesthat are bound to the two end segments of the particles. Theseinduced membrane angles increase with increasing particlesize and with decreasing binding cutoff θ c .The membrane morphologies obtained in our simulations r edu c ed v o l u m e v r edu c ed v o l u m e v membrane coverage x membrane coverage x membrane coverage xsphere tubedisksphere facetedtubediskfaceted tubetube tubetube tubespheredisk diskparticle size 3 particle size 3 particle size 4particle size 5 particle size 6 particle size 7(a) (b) (c)(d) (e) (f) Figure 2: (a) Reduced volume versus membrane coverage for membrane morphologies induced by (a) particles of size 3 withbinding cutoff θ c = θ c =
3° (full circles) and θ c =
5° (open circles), and (c) to(f) particles of size 4 to 7 with binding cutoff θ c = A (cid:39) n t a m (cid:39) · a m where a m is the minimum edge length of the triangulated membrane. The overall numberof particles in our simulations is either N =
200 or 400, and the adhesion energy per particle segment adopts one of the values U =
3, 4, 5, . . . k B T (see Figure S2). For each combination of particle number N and adhesion energy U , we have performedsimulations starting from an initial spherical, disk-like, or tubular membrane shape (see Figure S1). The membrane and particlesare enclosed in a cubic box of volume V box (cid:39) · a m . This box volume is 27 times larger than the volume of a perfectsphere with the membrane area A given above.are determined by the size and membrane coverage of theparticles (see Figure 2). The overall number of particles inthe simulations is always larger than the number of boundparticles covering the membrane. The membrane coveragethen depends on the concentration and binding energy of theparticles, but is not limited or constrained by the number ofavailable particles. An overlap between particles is preventedby a hard-core repulsion potential. Without particles, theclosed membrane of our model adopts a spherical shapebecause the bending energy of such a membrane vesicle isminimal for the sphere. For the smallest particles of size 3and membranes composed of 2000 triangles, the membraneretains a spherical shape until coverages of about 50% forthe binding cutoff angle θ c =
10° (see Figure 2(a)), and until coverages of about 45% for the binding cutoffs θ c =
3° and5° (see Figure 2(b)). At larger coverages, the morphologyof the membrane changes from spherical to tubular. Thismorphology change leads to a drop in the reduced volume v = √ π V / A / ≤
1, which is a measure for the area-to-volume ratio of the membrane vesicle (41) and adopts itsmaximum value of 1 for an ideal sphere. The area A of themembrane is constrained in our simulations to ensure thenear incompressibility of lipid membranes (36), whereas thevolume V is unconstrained to allow for the full range ofmembrane morphologies.At intermediate coverages, the membrane morphologiesdepend on the particle size. For particles of size 4 to 7,spherical morphologies with bound particles do not occur, in o o o o o o o o m ean i ndu c ed ang l e o f pa r t i c l e s membrane coverage x of particlesspherefaceted tubedisk Figure 3: Morphology diagram with mean induced angleversus membrane coverage for the data points of Figure 2.The mean induced angles are the mean values of the angledistributions of Figure 1. The lines of full circles with thesame mean induced angle are from simulations with particlesof size 3 to 7 (bottom to top) and binding cutoff θ c = θ c = v ofthe membrane is a function of the membrane coverage. Foreach particle type, the membrane coverage determines thereduced volume v and, thus, the membrane morphology.All membrane morphologies of Figure 2 are summarizedin the diagram of Figure 3. In this diagram, the mean induced angle is displayed versus the membrane coverage for the datapoints of Figure 2. The mean induced angle varies from 52.5°for particles of size 3 with binding cutoff θ c =
10° to 169.7°for particles of size 7. Figure 3 illustrates that the thresholdvalues of the membrane coverage above which membranetubes are formed in our simulations is rather independent ofthe particle type. These threshold values range from about40% for particles of size 4 to 50% for particles of size 3and angle cutoff θ c = x , compared toparticles of size 6 and 7. This gap arises because particles ofsize 5 are arranged in three lines along the tubes (see Figures4 and S2), while elongated disks exhibit only two lines ofparticles at opposing sides (see Figures 5 and S3). Tubeswith particles of size 5 thus cannot be generated by simpleelongation of disks, in contrast to particles of size 6 and 7,which are arranged in two lines of particles along the tubes.Particles of size 4 induce tubular morphologies with four linesof particles along the tube. At intermediate coverages, theparticles lead to irregular, faceted morphologies with stronglycurved membrane ridges covered by lines of particles, andweakly curved, uncovered membrane segments in betweenthese ridges (see Figure 2, top). Particles of size 3 tend to alignside by side along the tubules, but do not form continuouslines along the whole tubule. The ordering of particles of size3 along the tubes is thus shorter ranged compared to largerparticles.In Figure 6, we compare simulation results for (a) particlescomposed of 3 segments with an angle of 30° between neigh-boring segments and (b) particles composed of 5 segmentswith an angle of 15° between neighboring segments. Bothtypes of particles enclose the same arc angle of 60° betweentheir terminal segments, but have different curvatures becauseof the different angles between their neighboring segments,and different sizes. The membrane in the simulations of Figure6 is composed of 5120 triangles and is, thus, significantlylarger than the membrane in the simulations of Figure 2. Formembranes composed of 2000 triangles as in Figure 2, themore weakly curved particles of Figure 6(b) do not induce aclear morphology transition from spherical to tubular becausethis smaller membrane size does not allow for sufficientlyelongated spherocylinders that are clearly distinguishablefrom spheres (data not shown). For the larger membrane sizeof 5120 triangles, however, both types of particles of Figure 6exhibit a rather sharp morphology transition from spherical totubular at membrane coverages of about 0.37. This identicalthreshold value for the sphere-to-tubule transition illustratesthat the overall membrane morphology is determined by thearc angle, which is identical for both types of particles, andnot by the size or curvature of the particles. As expected, themore weakly curved particles of Figure 6(b) induce thickertubules for membrane coverages beyond the threshold value ize 3 size 4size 6 size 5size 7 Figure 4: Exemplary tubular morphologies for particles composed of 3 to 7 segments with an angle of 30° between adjacentsegments. The membrane coverage is x = size 5 size 6 size 7top viewside view Figure 5: Exemplary disk-like morphologies for particles composed of 5 to 7 segments with an angle of 30° between adjacentsegments and membrane coverages of x = x close to the threshold value for tube formation (seeFigure S3). The membrane size of 2000 triangles therefore islikely too small for reliable estimates of the sphere-to-tubuletransition for these particles. The larger particles of Figure2, in contrast, induce tubules with significantly larger aspectratios, compared to particles of size 3 (see Figures 4 and S3).The threshold values for tube formation obtained for theseparticles from Figures 2(c) to (f) therefore should be onlyweakly affected by the membrane size. DISCUSSION AND CONCLUSIONS
In this article, we have investigated the transitions betweendifferent membrane morphologies induced by arc-shapedparticles with purely repulsive direct particle-particle inter-actions. Our aim was to classify the membrane morphologiesand to identify the particle properties that determine thesemorphologies. Arc-shaped particles can differ in their size, curvature, arc angle, adhesion energy, and overall number.Our central result is that the membrane morphologies in-duced by arc-shaped particles are determined by the arc angleand the membrane coverage of the particles. In our model,the particles are described as segmented arcs that adhere tothe triangulated membrane. The particle discretization thusis independent from the membrane discretization, and themembrane coverages obtained in our simulations are notaffected or limited by the membrane discretization. In oursimulations, the overall number of particles is larger thanthe number of adsorbed particles. Unbound particles thusconstitute a particle reservoir in our simulations, which leadsto rather sharp transitions and ‘pure’ morphologies, as typicalfor simulations in a grand-canonical ensemble. We find thatarc-shaped particles induce membrane tubules for membranecoverages larger than a threshold value of about 0.4, ratherindependent of their arc angle. At smaller coverages, parti-cles with arc angles larger than about 120° induce disk-likemembrane morphologies. These disk-like morphologies havecharacteristic membrane edges that connect the two opposing,nearly planar membrane segments of the disks. The disksare therefore ‘small-membrane equivalents’ of the stacked,connected membrane sheets observed in the endoplasmicreticulum (3, 17). On the coarse-graining level of our model,the curvature generation of the particles is captured by in-duced curvature angles of the particles. Proteins may induce sphere tubespheretube membrane coverage x of particles r edu c ed v o l u m e v r edu c ed v o l u m e v membrane coverage x of particles(a) (b) Figure 6: (a) Reduced volume v versus membrane coverage for membrane morphologies with 5120 membrane triangles and (a)particles composed of 3 segments with an angle of 30° between neighboring segments and (b) particles composed of 5 segmentswith an angle of 15° between neighboring segments. Spherical morphologies are indicated by blue points, tubular morphologiesby red points. The two grey points in (a) correspond to intermediate morphologies. The data result from simulations withoverall particle number N = x , we have run simulations with adhesion energy U =
3, 3.5, 4,4.5, 5, 5.5, 6, 6.5, 6.6, 6.8, 7, 7.2, 7.4, 7.5, 7.6, 7.8, 8, 8.2, 8.4, 8.5, 8.6, 9, 10, 11, 12, 13, and 14 k B T in (a) and U =
3, 3.5, 4, 4.1,4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 5, 6, 7, 8, 9, 10, 11, 12, and 13 k B T in (b). For each value of U , we have run 3 simulationsstarting from a spherical morphology and 2 to 3 simulations starting from a tubular morphology. Only points from simulationswith converged membrane coverage and reduced volume are included in the plots (see Methods). The mean induced angle ofthe particles in (b) is 56.1° and, thus, slightly larger than the mean induced angle of 52.5° of the particles in (a). The particlecoverage is x = x = ±
27% of endophilin N-BARdomains and 37 ±
9% of β √ A . In general, stacked membraneswith lateral extensions that are significantly larger than theirseparation repel each other sterically because of membraneshape fluctuations (50, 51). Therefore, stacked membranes oflarge area are presumably stabilized by additional attractiveinteractions between the membranes.The membrane morphologies in our model result from anintricate interplay of the bending free energy of the membraneand the overall adhesion free energy of the particles. Themembrane in our model is tensionless. In general, the bendingenergy dominates over a membrane tension σ on lengthscales smaller than the characteristic length (cid:112) κ / σ , whichadopts values between 100 and 400 nm for typical tensions σ of a few µ N / m (52–54) and typical bending rigidities κ between 10 and 40 k B T where k B T is the thermal energy(35, 55). Our results thus hold on length scales smaller thanthis characteristic length. In contrast, the overall membraneorphology on length scales larger than (cid:112) κ / σ depends onthe membrane tension (44, 56, 57). AUTHOR CONTRIBUTIONS
FB and TW designed the research. FB carried out all simula-tions. FB and TW analyzed the data and wrote the article.
ACKNOWLEDGMENTS
Financial support from the Deutsche Forschungsgemeinschaft(DFG) via the International Research Training Group 1524“Self-Assembled Soft Matter Nano-Structures at Interfaces"is gratefully acknowledged.
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Nat.Commun. UPPLEMENTARY FIGURES disksphere tube
Figure S1: Initial spherical, disk-like, and tubular membrane shapes of our MC simulations.
10 15 200.60.70.80.9 0 0.1 0.2 0.3 0.4 0.5 0.60.60.70.80.9 adhesion energy U per particle segment [ k B T ] r edu c ed v o l u m e v r edu c ed v o l u m e v membrane coverage x of particles N = 400N = 200N = 100N = 50N = 400N = 200N = 100N = 50 (a)(b)
Figure S2: Reduced volume v of the membrane vesicles versus (a) adhesion energy U per particle segment and (b) membranecoverage x for particles composed of 3 segments at different overall particle numbers N . The data for the overall particlenumbers N =
200 and 400 are identical to the data shown in Figure 2(a). For the particle numbers N =
50 and 100, themembrane coverage x is not sufficiently high to induce membrane tubules with v (cid:46) U . Asa function of x , all data collapse onto a single curve, which illustrates that the reduced volume v and, thus, the membranemorphology, is determined by the membrane coverage x of the particles. This coverage of membrane-bound particles dependsboth on the overall particle number N and on the adhesion energy U per particle segment. ize 3size 4size 6size 5size 7 x = 0.54 x = 0.59 x = 0.64x = 0.44 x = 0.53 x = 0.62x = 0.46 x = 0.54 x = 0.62x = 0.48 x = 0.52 x = 0.57x = 0.46 x = 0.52 x = 0.57 Figure S3: Exemplary tubular morphologies for particles of size 3 to 7 at different membrane coverages x . ize 6size 5size 7 top viewside viewtop viewside viewtop viewside view x = 0.19 x = 0.26 x = 0.33x = 0.29 x = 0.32 x = 0.35x = 0.34 x = 0.35 x = 0.36 Figure S4: Exemplary disk-like morphologies for particles of size 5 to 7 at different membrane coverages x . Figure S5: Time sequence of a morphology change from spherical to tubular induced by arc-shaped particles with 3 segments.The numbers indicate simulation times in units of 10 MC steps per membrane vertex. At time 0, the membrane has the initialspherical shape depicted in Figure S1, and all particles are unbound. In this simulation, the adhesion energy per particle segmentis U = k B T , the cutoff angle for binding is θ c =
10 20 301 2 74 80.04 0.05 0.09 0.2 0.4 0.6
Figure S6: Time sequence of a morphology change from spherical to tubular induced by arc-shaped particles with 4 segments.The numbers indicate simulation times in units of 10 MC steps per membrane vertex. At time 0, the membrane has the initialspherical shape depicted in Figure S1, and all particles are unbound. In this simulation, the adhesion energy per particle segmentis U = k B T , and the total number of particles is 400. Figure S7: Time sequence of a morphology change from spherical to tubular induced by arc-shaped particles with 5 segments.The numbers indicate simulation times in units of 10 MC steps per membrane vertex. At time 0, the membrane has the initialspherical shape depicted in Figure S1, and all particles are unbound. In this simulation, the adhesion energy per particle segmentis U = k B T , and the total number of particles is 400. tube diskparticle size t h i ck ne ss [ a m ] Figure S8: Thickness of disks and tubes in units of the minimum edge length a m of the triangulated membrane. To determinethe disk thickness, each membrane vertex is connected to an opposing vertex such that the line between the vertices is closer tothe center of mass of the membrane than lines connecting the vertix to other vertices. The disk thickness is defined as theminimum length of all lines between opposing pairs of vertices. To determine the tube thickness, we first project all vertices onthe axis of inertia parallel to the tube and select those vertices for which the distance to the center of mass along this axis issmaller than 2.3 a mm