Membrane morphologies induced by mixtures of arc-shaped particles with opposite curvature
MMembrane morphologies induced by mixtures of arc-shaped particles withopposite curvature
Francesco Bonazzi, a Carol K. Hall, b and Thomas R. Weikl a Biological membranes are shaped by various proteins that either generate inward or outward membrane curvature. In this article, weinvestigate the membrane morphologies induced by mixtures of arc-shaped particles with coarse-grained modeling and simulations.The particles bind to the membranes either with their inward, concave side or their outward, convex side and, thus, generate membranecurvature of opposite sign. We find that small fractions of convex-binding particles can stabilize three-way junctions of membranetubules, as suggested for the protein lunapark in the endoplasmic reticulum of cells. For comparable fractions of concave-binding andconvex-binding particles, we observe lines of particles of the same type, and diverse membrane morphologies with grooves and bulgesinduced by these particle lines. The alignment and segregation of the particles is driven by indirect, membrane-mediated interactions.
The intricately curved shapes of biological membranes are inducedand maintained by a variety of proteins . The arc-shaped BARdomain proteins, for example, induce membrane curvature bybinding to membranes . Different BAR domain proteins bind tomembranes either with their inward curved, concave side or withtheir outward bulged, convex side and, thus, impose membranecurvature of opposite sign . Spherical and tubular membraneshapes only exhibit curvature of one sign and can be induced bya single type of proteins . Three-way junctions of tubules, incontrast, contain membrane segments with curvatures of differentsign and are induced and stabilized by several proteins .The ubiquitous three-way junctions of tubules in the endoplasmicreticulum (ER) are stabilized by the protein lunapark , whilethe tubules of the ER are generated by reticulon and REEP pro-teins . The protein lunapark presumably induces a mem-brane curvature that is opposite to the tubular curvature generatedby reticulon and REEP proteins .In this article, we investigate the membrane morphologies in-duced by mixtures of arc-shaped particles that can either bindwith their inward curved, concave side (“concave particles") orwith their outward bulged, convex side (“convex particles"). Inour coarse-grained model of membrane shaping, the membraneis described as a triangulated elastic surface, and the particles assegmented arcs. In previous Monte Carlo (MC) simulations , wefound that the membrane morphologies induced by concave par-ticles are determined by the arc angle and membrane coverage ofthe particles. At membrane coverages that exceed about 40%, con-cave particles induce membrane tubules, irrespective of their arcangle. In MC simulations with mixtures of concave and convexparticles, in contrast, we observe a large variety of morphologiesthat depends on the relative coverage of the different types of par-ticles. If the membrane coverage of concave particles greatly ex-ceeds the coverage of convex particles, we either find single mem-brane tubules or three tubules connected by a three-way junction. a Max Planck Institute of Colloids and Interfaces, Department of Theory and Bio-Systems,Am Mühlenberg 1, 14476 Potsdam, Germany b North Carolina State University, Department of Chemical and Biomolecular Engineer-ing, Engineering Building I, 911 Partners Way, Raleigh, NC 27695-7905, USA.
The few convex particles cluster at the three-way junctions and ap-pear to stabilize the junction, or distort the single tubules locally.For larger fractions of convex particles, we observe lines of convexparticles segregated from lines of concave particles, and membranemorphologies with grooves and bulges induced by these lines. Thealignment and segregation of the convex and concave particles isdriven by indirect, membrane-mediated interactions becausethe direct particle-particle interactions are purely repulsive in ourmodel. A similar alignment and segregation has been previouslyobserved in simulations with mixtures of arc-shaped inclusions ofopposite curvature . We model the membrane as a discretized closed surface. Thebending energy of a closed continuous membrane without spon-taneous curvature is the integral E be = κ (cid:72) M dS over the mem-brane surface with local mean curvature M . We use the standarddiscretization of the bending energy for triangulated membranedescribed in refs. 30, 31 and choose as typical bending rigiditythe value κ = k B T . Our discretized membranes are composedof n t = triangles. The edge lengths of the triangles are keptwithin an interval [ a m , √ a m ] , and the area of the membrane is con-strained to A (cid:39) . n t a m to ensure the near incompressibility oflipid membranes. The strength of the harmonic constraining po-tential is chosen such that the fluctuations of the membrane areaare limited to less than . The enclosed volume is unconstrainedto enable a wide range of membrane morphologies with differentvolume-to-area ratios.The discretized particles in our model are linear chains of 3 to 5identical planar quadratic segments, with an angle of ◦ betweenneighboring segments that share a quadratic edge. The arc angleof the particles, i.e. the angle between the first and last segment,then adopts the values ◦ , ◦ , ◦ for particles composed of3, 4, and 5 segments respectively. Each planar segment of a par-ticle interacts with the nearest triangle of the membrane via theparticle-membrane adhesion potential V pm = ± U f r ( r ) f θ ( θ ) (1)1 a r X i v : . [ q - b i o . S C ] A p r .2 Simulations 2 METHODS top view bottom view time = 0.1 1 10 40 Fig. 1
Time sequence of morphologies for a mixture of concave and convex particles with arc angle ◦ . The numbers indicate simulation times inunits of MC steps per membrane vertex. At time t = , the membrane has a spherical shape, and all particles are unbound. In this simulation, theadhesion energy per particle segment is U = k B T , the total number of concave, orange particles is 320, and total number of convex, blue particlesis 80. Only membrane-bound particles are shown in the MC snapshots. In the final morphology, 243 out of the 320 concave particles and 79 out ofthe 80 convex particles are bound, which leads to membrane coverages of x orange = . and x blue = . of the particles. The reduced volume of themembrane in the final morphology is v = . . Here, r is the distance between the center of the segment and thecenter of the nearest triangle, θ is the angle between the nor-mals of the particle segment and this membrane triangle, and U > is the adhesion energy per particle segment. The distance-dependent function f r is a square-well function that adopts thevalues f r ( r ) = for . a m < r < . a m and f r ( r ) = otherwise.The angle-dependent function f θ is a square-well function withvalues f θ ( θ ) = for | θ | < ◦ and f θ ( θ ) = otherwise. By conven-tion, the normals of the membrane triangles are oriented outwardfrom the enclosed volume of the membrane, and the normals ofthe particle segments are oriented away from the center of theparticle arc. For a negative sign in Eq. (1), the particles bind withtheir inward curved, concave surface to the membrane (“concaveparticles"). For a positive sign in Eq. (1), the particles bind withtheir outward bulged, convex surface to the membrane (“convexparticles"). The overlapping of particles is prevented by a purelyrepulsive hard-core interaction that only allows distances betweenthe centers of the planar segments of different particles that arelarger than a p . The hard-core area of a particle segment thus is π a p / . We choose the value a p = . a m for the linear size of theplanar particle segments. The particle segments then are slightlylarger than the membrane triangles with minimum side length a m ,which ensures that different particle segments bind to different tri-angles. We have performed Metropolis Monte Carlo (MC) simulations ina cubic box with periodic boundary conditions. The simulationsconsist of four different types of MC steps: membrane vertex trans-lations, membrane edge flips, particle translations, and particlerotations. Vertex translations enable changes of the membrane shape, while edge flips ensure membrane fluidity. In a vertextranslation, a randomly selected vertex of the triangulated mem-brane is translated along a random direction in three-dimensionalspace by a distance that is randomly chosen from an interval be-tween and . a m . In a particle translation, a randomly selectedparticle is translated in random direction by a random distancebetween and a m . In a particle rotation, a randomly selectedparticle is rotated around a rotation axis that passes trough thecentral point along the particle arc. For particles that consist of3 or 5 segments, the rotation axis runs through the center of thecentral segments. For particles composed of 4 segments, the rota-tion axis runs through the center of the edge that is shared by thetwo central segments. The rotation axis is oriented in a randomdirection. The random rotations are implemented using quater-nions with rotation angles between 0 and a maximum angleof about . ◦ . Each of these types of MC steps occur with equalprobabilities for single membrane vertices, edges, or particles. We have run simulations with identical arc angles of either ◦ , ◦ , or ◦ of the concave and convex particles. The overall num-ber of concave and convex particles in our simulations is , andthe initial shape of the membrane is spherical, with all particlesunbound. For particles with arc angles of ◦ and ◦ , we haverun simulations with , , , , , , and convex particlesout of particles in total. The adhesion energy per segment isidentical for the concave and convex particles in these simulationsand has the value U = , , , , or k B T . In the case of 8 or20 convex particles, we have also run simulations with U = or k B T . The membrane and particles are enclosed in a cubic sim-ulation box of volume V box (cid:39) . · a m . To verify convergence,we divide the last MC steps per vertex of a simulation into tenintervals of steps and calculate the reduced volume v of the RESULTS
Fig. 2
Representative converged morphologies for mixtures of concave and convex particles with arc angle ◦ . The morphologies are ar-ranged in ascending order of the membrane coverage of convex, blue particles, which increases from top left to bottom right as x blue = . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , and . . The membrane coverage of concave, orange particles is x orange = . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , and . from top left to bottom right, and the reduced vol-ume of the membrane is v = . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , and . . The morphologies result fromsimulations with an initially spherical membrane and the adhesion energy per particle segment U = , , , , , , , , , , , , , , , ,and k B T . The overall number of bound and unbound concave particles is , , , , , , , , , , , , , , , , and in these simulations. The total number of concave and convex particles is in all simulations. membrane for each interval. We take a simulation to be convergedif the standard deviation of the 10 averages of v for the last 10intervals of MC steps is smaller than 0.03. The morphologiesobtained from converged simulations correspond to metastable orstable states. For the adhesion energies U ≥ k B T per particle seg-ment considered here, the total membrane coverage by concaveand convex particles after convergence is on average larger than % for the chosen box size V box and total particle number ofour simulations. For total coverages larger than 40%, the mem-branes are fully covered by particles. For smaller adhesion ener-gies of U = , , or k B T , the membranes are only partially coveredby the particles after convergence, with average total membranecoverages of . %, %, and %, respectively. For all adhesionenergies, intermediate morphologies with partial particle coverageoccur in our simulations at early time points prior to convergence,because the particles are initially unbound (see e.g. Fig. 1). ◦ Fig. 1 illustrates the segregation and alignment of particles with arcangle ◦ in a simulation with 320 concave, orange and 80 convex,blue particles. All particles are initially unbound in this simulation.After a simulation time of t = . · MC steps per membrane ver-tex, relatively few particles are bound. Some of the bound convex,blue particles are aligned side-to-side in groups of two or three par-ticles, and some of the bound concave, orange particles are alignedin pairs. The alignment of particles of the same type is driven byindirect, membrane-mediated interactions because the directparticle-particle interactions are purely repulsive in our model. Af-ter a simulation time of MC steps per vertex, bound convex,blue particles form continuous lines or grooves along the mem-brane, and the membrane bulges between these grooves are moresparsely covered by concave, orange particles. The overall cover-age of the membrane by particles then increases with time, and .2 Particles with arc angles of ◦ and more 3 RESULTS time = 1 4 10 17 22 Fig. 3
Time sequence of morphologies for a mixture of many concave and few convex particles with arc angle ◦ . The numbers indicate simulationtimes in units of MC steps per membrane vertex. At time t = , the membrane has spherical shape, and all particles are unbound. In this simulation,the adhesion energy per particle segment is U = k B T , the total number of concave, orange particles is 392, and total number of convex, blue particlesis 8. Only membrane-bound particles are shown in the MC snapshots. In the final morphology, 206 out of the 392 concave particles and all 8 convexparticles are bound, which leads to membrane coverages of x orange = . and x blue = . of the particles. The reduced volume of the membrane inthe final morphology is v = . . the membrane bulges between the grooves of single lines of con-vex particles are eventually covered by two or three partly irregu-lar lines of concave particles. During the simulation, the reducedvolume v = √ π V / A / ≤ of the closed membrane with area A and volume V decreases from values close to 1 to a final value of v = . . The reduced volume is a measure for the volume-to-arearatio of the closed membrane and adopts its maximum value of1 for an ideal sphere.The final, converged membrane morphologies depend on therelative coverages of concave and convex particles (see Fig. 2).Membranes that are predominantly covered with concave, orangeparticles as in the first two morphologies of Fig. 2 adopt a tubularshape. Concave particles with an arc angle of ◦ induce a transi-tion from a spherical to a tubular membrane shape at a coverageof about 0.4 in the absence of convex particles. In the first twomorphologies, the coverage of concave particles is x orange = . and . , respectively, while the coverage of convex particles is x blue = . . At these small coverages, the convex particles arebound as single particles or pairs in between the concave particlesand do not distort the overall tubular shape of the membrane. Atthe coverage x blue = . of the third morphology of Fig. 2, thetubular shape of the membrane is distorted by a larger cluster ofconvex, blue particles. At the larger coverages x blue of the remain-ing morphologies of Fig. 2, the convex particles form lines alongthe membrane. If the coverage x orange of the concave particles ex-ceeds the coverage x blue of the convex particles, the membranemorphologies exhibit grooves of single lines of convex particles,and bulges covered by several lines of concave particles in betweenthese grooves. For a coverage x orange of concave particles that issmaller than the coverage x blue of convex particle, grooves are alsoformed by two parallel lines of convex particles, while bulges in thebetween the groves can be covered by single lines of concave par-ticles. The particle lines need to branch or end because the closedmembrane vesicle cannot be covered by regular, parallel lines ofparticles. ◦ and more In the absence of convex particles, concave particles with arc an-gles of ◦ induce tubules covered by four lines of particles at mem-brane coverages larger than about . . For mixtures of manyconcave and few convex particles, we observe branched tubulestructures as in Fig. 3, with small clusters of convex particles ata three-way junction as branching point. In the simulation of Fig.3, the number of convex particles is 8, and the total number ofbound and unbound concave particles is 392. The bound concaveand convex particle have a rather strong tendency to align side-to-side with particles of the same type due to indirect, membrane-mediated interactions. At the simulation time t = · MC stepsper membrane vertex, bound concave particles form short lines,while convex particles are bound as single particles or in pairs. Attime t = · MC steps per vertex, a line of 5 convex particle isformed. This linear cluster of 5 convex particle remains until time t = · MC steps per vertex and eventually gains a sixth con-vex particle at time t = · MC steps. From time t = · to t = · MC steps per vertex, more and more concave particlesbind to the membrane by elongating lines of particles, and theseparticle lines eventually lead to three tubules protruding from athree-way junction at which the small cluster of convex particles islocated.The first three of the final, converged morphologies shown inFig. 4 result from simulations with same total numbers of 392 con-cave and 8 convex particles with arc angle ◦ as in the simula-tion of Fig. 3. In all three morphologies, the 8 convex particlesare bound, which leads to the membrane coverage x blue = . ofthese particles. In the first morphology, the 8 convex particles arebound in a cluster of 4 particles, a cluster of 3 particles, and as asingle particle, and induce a distortion or twist in the overall tubu-lar structure induced by the many bound concave particles. In thesecond and third morphology, the 8 convex particles are bound intwo clusters of 4 particles and a single cluster of 8 particles, respec-tively, which are located at a three-way junction as in Fig. 3. At thelarger membrane coverages x blue = . of convex particles in themorphologies 4 to 7 of Fig. 4, a tubular protrusion is formed at oneend of the closed membrane by bound concave particles, while theremaining membrane is covered by lines of convex and concave RESULTS 3.2 Particles with arc angles of ◦ and more Fig. 4
Representative converged morphologies for mixtures of concave and convex particles with arc angle ◦ . The morphologiesare arranged in ascending order of the membrane coverage of convex, blue particles, which increases from top left to bottom right as x blue = . , . , . , . , . , . , . , . , . , . , . , . , . , . , and . . The membrane coverage of concave, orange parti-cles is x orange = . , . , . , . , . , . , . , . , . , . , . , . , . , . , and . , and the reduced volume of the membrane is v = . , . , . , . , . , . , . , . , . , . , . , . , . , . , and . . The morphologies result from simulations with an initially sphericalmembrane and the adhesion energy per particle segment U = , , , , , , , , , , , , , , and k B T . The overall number of boundand unbound concave particles is , , , , , , , , , , , , , , and in these simulations. The total number of con-cave and convex particles is in all simulations. particles that induce grooves and bulges. In the remaining mor-phologies of Fig. 4, the membrane is covered by alternating andlocally parallel lines of convex and concave particles. Grooves aretypically formed by single lines of convex particles, while bulgesare covered by either one line or by two parallel lines of concaveparticles, depending on the relative coverages of the two particletypes.For mixtures of concave and convex particles with an arc angleof ◦ , we observe a temporal ordering in the binding of the twoparticle types to an initially spherical membrane (see Fig. 5). Atthe simulation time t = . · MC steps per vertex, only convexparticles are bound, and these particles are partially bound with typically one or two of the five segments of which the particles arecomposed. The partially bound convex particles are not yet alignedand deform the initially spherical membrane only rather slightly.At the simulation time t = · MC steps per vertex, the majorityof bound convex particles is fully bound and tightly aligned, whichleads to rather deep grooves on the vesicle membrane, and the firstconcave particles bind to the bulges emerging adjacent to thesegroves. At time t = · MC steps per vertex, small linear clustersof concave particles form on the bulges, which eventually grow andcoalesce into a single spiral of concave particles that is intertwinedwith a spiral of convex particles. DISCUSSION AND CONCLUSIONS time = 0.1 1 10 16 41
Fig. 5
Time sequence of morphologies for a mixture of concave and convex particles with arc angle ◦ . The numbers indicate simulation times inunits of MC steps per membrane vertex. In this simulation, the adhesion energy per particle segment is U = k B T , the total number of concave,orange particles is 359, and total number of convex, blue particles is 41. In the final morphology, 88 out of the 359 concave particles and 40 out of 41convex particles are bound, which leads to membrane coverages of x orange = . and x blue = . of the particles. The reduced volume of the membranein the final morphology is v = . . The arc-shaped particles of our model generate membrane curva-ture by imposing their shape on the membrane upon binding .The arrangements of these particles on the membranes are essen-tially unaffected by the membrane discretization because the parti-cles are not embedded in the membrane. In other models of mem-brane shaping , curvature-inducing particles and proteinshave been described as nematic objects embedded on the verticesof a triangulated membrane , as curved chains of beads em-bedded in a two-dimensional sheet of beads that represents themembrane , as curved chains of spheres adhered to a tri-angulated membrane , or as coarse-grained proteins or particlesin molecular dynamics simulations. Proteins can generatemembrane curvature via different mechanisms.
Arc-shapedscaffolding proteins impose curvature on the membrane by bind-ing to the lipid bilayer, transmembrane proteins with a conicalor wedged shape induce a curvature on the lipid bilayer that sur-rounds the proteins, and hydrophobic protein motifs that arepartially inserted into the lipid bilayer can act as wedges to gener-ate membrane curvature.
A central parameter for membrane shaping is the induced cur-vature angle of the particles or proteins.
For our arc-shapedparticles, the induced angle of the curved membrane segments towhich the particles are bound is close to the arc angle of the par-ticles, which varies here from ◦ to ◦ . Arc angles of ◦ roughly correspond to the angle enclosed by concave-binding BARdomain proteins such as the Arfaptin BAR domain and the en-dophilin and amphiphysin N-BAR domains, while larger arcangles up to ◦ have been postulated for reticulon scaffolds. The structural details of the curvature generation by transmem-brane proteins such as reticulon and lunapark proteins are not fullyknown, in contrast to soluble scaffold proteins such as BAR do-mains. Besides reticulon and lunapark proteins, the generation ofthe tubular membrane network of the endoplasmic reticulum alsorequires atlastin proteins, which appear to generate tubular junc-tions by tethering and fusing tubules .The membrane morphologies induced by mixtures of concaveand convex particles depend on the relative coverage of these par-ticles, besides the particles’ arc angle. For mixtures of few convexand many concave particles with arc angles of ◦ , we either find single membrane tubules as in the first morphology of Fig. 4, orthree tubules connected by a three-way junction as in Fig. 3 andin the second and third morphology of Fig. 4. These morphologiesare formed in simulations with 8 convex and 392 concave particlesin total. We have run 7 simulations with these particle numbers forthe adhesion energies per segment U = , , , , , , and ,respectively. In 5 of these 7 simulations, three-way junctions areformed. The few convex particles are bound and clustered in mem-brane regions of the three-way junction in which the curvature isopposite to the curvature of the tubules that emerge from the junc-tion. The convex particles thus appear to stabilize three-way junc-tions as suggested for lunapark proteins, which presumably prefermembrane curvature opposite to the tubular curvature. For par-ticles with arc angles of ◦ , we do not observe the formation ofthree-way junctions. One reason may be that the tubes formed byconcave particles with an arc angle of ◦ are thicker than tubesinduced by concave particles with arc angle ◦ . For the samemembrane area, tubes formed by concave particles with arc angle ◦ therefore are shorter, and the finite membrane area in our sim-ulations may impede morphologies with three such thicker tubulesemerging from a three-way junction. Another reason is that a fewconvex particles with arc angle ◦ lead to rather small perturba-tions of the tubules induced by many concave particles, see thefirst two morphologies in Fig. 2. The convex particles with arc an-gle ◦ thus are less ‘disruptiveâ˘AŸ for the tubules, compared toconvex particles with arc angle ◦ .For comparable fractions of concave and convex particles, weobserve lines of particles of the same type. Lines of convex par-ticles induce membrane grooves, and adjacent, locally parallellines of concave particles induce bulges next to these grooves. Inthese lines, the particles are oriented side-to-side. The side-to-side alignment and segregation of the concave and convex par-ticles is driven by indirect, membrane-mediated interactions be-cause the direct particle-particle interactions are purely repulsivein our model. The segregation patterns of particle lines are remi-niscent of the stripe morphologies observed for modulated phasesand microphase separation , which arise from a competition ofshort-range attractive and long-range repulsive interactions. Here,the segregation into lines of convex and concave particles resultsfrom an interplay of particle composition and membrane curva- OTES AND REFERENCES NOTES AND REFERENCES ture. The segregation into alternating lines of concave and convexparticles appears to be favourable at sufficiently large adhesionenergies, because the membrane vesicle can be rather densely cov-ered by the particles of the alternating lines. In addition, there isno line tension between clusters of different particles as drivingforce for full segregation into two domains of concave and convexparticles because of the purely repulsive direct particle-particle in-teractions in our model. A caveat is that the converged morpholo-gies observed in our simulations correspond to metastable or stablestates and, thus, not necessarily to equilibrium states.In previous work, both side-to-side and tip-to-tip alignmentof arc-shaped proteins or particles at membranes has been re-ported. Attractive membrane-mediated side-to-side pair inter-actions of arc-shaped particles have been obtained from energyminimization . Side-to-side alignment has also been observedin simulations with arc-shaped inclusions in membranes .In molecular dynamics (MD) simulations with a coarse-grainedmolecular model of N-BAR domains proteins on DLPC lipid vesi-cles, in contrast, a tip-to-tip alignment of proteins has been ob-served , which may be affected by the direct, coarse-grainedprotein-protein interactions of the model. A tip-to-tip alignmenthas also been reported for MD simulations with a coarse-grainedmodel of I-BAR domains and for coarse-grained MD simulationsof arc-shaped nanoaparticles on lipid vesicles at large adhesion en-ergies of the nanoparticles . At these large adhesion energies, thenanoparticles are partially wrapped by the membrane, which leadsto saddle-like membrane curvature around nanoparticles that maycause side-to-side repulsion. At smaller adhesion energies, the arc-shaped nanoparticles induce membrane curvature only along theirarcs and align side-to-side, similar to our arc-shaped particles. Insimulations with mixtures of arc-shaped and conical inclusions inmembranes, the tubulation caused by the arc-shaped particles hasbeen found to be accelerated if the conical inclusions induce cur-vature of the same sign, and suppressed if the conical inclusionsinduce curvature of opposite sign . For mixtures of arc-shapedinclusions with opposite curvatures, adjacent lines of the differentparticles have also been observed at overall relatively low densitiesof the particles .The morphologies in our simulations result from an interplayof the bending energy of the membrane and the overall adhesionfree energy of the particles. In these simulations, the membranesare tensionless because the volume enclosed by the membrane isnot constrained, in order to allow for a wide range of morpholo-gies with different volume-to-area ratios. In general, the bend-ing energy dominates over the membrane tension σ on lengthscales smaller than the characteristic length (cid:112) κ / σ , which adoptsvalues between 100 and 400 nm for typical tensions σ of a few µ N / m and typical bending rigidities κ between and k B T . Our results thus hold on length scales smaller than thischaracteristic length. In contrast, the overall membrane morphol-ogy on length scales larger than (cid:112) κ / σ depends on the membranetension . Conflicts of interest
There are no conflicts to declare.
Acknowledgements
Financial support from the Deutsche Forschungsgemeinschaft(DFG) via the International Research Training Group 1524 “Self-Assembled Soft Matter Nano-Structures at Interfaces" is gratefullyacknowledged.
Notes and references
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