Formation of polyhedral vesicles and polygonal membrane tubes induced by banana-shaped proteins

Which Coverages of Arc-Shaped Proteins Are Required for Membrane Tubulation?

Arc-shaped BIN/Amphiphysin/Rvs (BAR) domain proteins generate curvature by binding to membranes and induce membrane tubulation at sufficiently large protein coverages. For the amphiphysin N-BAR domain, Le Roux et al., Nat. Commun. 2021, 12, 6550, measured a threshold coverage of 0.44 ± 0.097 for nanotubules emerging from the supported lipid bilayer. In this article, we systematically investigate membrane tubulation induced by arc-shaped protein-like particles with coarse-grained modeling and simulations and determine the threshold coverages at different particle-particle interaction strengths and membrane spontaneous curvatures. In our simulations, the binding of arc-shaped particles induces a membrane shape transition from spherical vesicles to tubules at a particle threshold coverage of about 0.5, which is rather robust to variations of the direct attractive particle interactions or spontaneous membrane curvature in the coarse-grained model. Our study suggests that threshold coverages of around or slightly below 0.5 are a general requirement for membrane tubulation by arc-shaped BAR domain proteins.

Membrane domain formation induced by binding/unbinding of curvature-inducing molecules on both membrane surfaces

The domain formation of curvature-inducing molecules, such as peripheral or transmembrane proteins and conical surfactants, is studied in thermal equilibrium and nonequilibrium steady states using meshless membrane simulations. These molecules can bind to both surfaces of a bilayer membrane and also move to the opposite leaflet by a flip-flop. Under symmetric conditions for the two leaflets, the membrane domains form checkerboard patterns in addition to striped and spot patterns. The unbound membrane stabilizes the vertices of the checkerboard. Under asymmetric conditions, the domains form kagome-lattice and thread-like patterns. In the nonequilibrium steady states, a flow of the binding molecules between the upper and lower solutions can occur via flip-flop.

Molecular Dynamics Simulations of Curved Lipid Membranes

Eukaryotic cells contain membranes with various curvatures, from the near-plane plasma membrane to the highly curved membranes of organelles, vesicles, and membrane protrusions. These curvatures are generated and sustained by curvature-inducing proteins, peptides, and lipids, and describing these mechanisms is an important scientific challenge. In addition to that, some molecules can sense membrane curvature and thereby be trafficked to specific locations. The description of curvature sensing is another fundamental challenge. Curved lipid membranes and their interplay with membrane-associated proteins can be investigated with molecular dynamics (MD) simulations. Various methods for simulating curved membranes with MD are discussed here, including tools for setting up simulation of vesicles and methods for sustaining membrane curvature. The latter are divided into methods that exploit scaffolding virtual beads, methods that use curvature-inducing molecules, and methods applying virtual forces. The variety of simulation tools allow researcher to closely match the conditions of experimental studies of membrane curvatures.

Membrane shape deformation induced by curvature-inducing proteins consisting of chiral crescent binding and intrinsically disordered domains

Curvature-inducing proteins containing a bin/amphiphysin/Rvs domain often have intrinsically disordered domains. Recent experiments have shown that these disordered chains enhance curvature sensing and generation. Here, we report on the modification of protein–membrane interactions by disordered chains using meshless membrane simulations. The protein and bound membrane are modeled together as a chiral crescent protein rod with two excluded-volume chains. As the chain length increases, the repulsion between them reduces the cluster size of the proteins. It induces spindle-shaped vesicles and a transition between arc-shaped and circular protein assemblies in a disk-shaped vesicle. For flat membranes, an intermediate chain length induces many tubules owing to the repulsion between the protein assemblies, whereas longer chains promote perpendicular elongation of tubules. Moreover, protein rods with zero rod curvature and sufficiently long chains stabilize the spherical buds. For proteins with a negative rod curvature, an intermediate chain length induces a rugged membrane with branched protein assemblies, whereas longer chains induce the formation of tubules with periodic concave-ring structures.

Binding of anisotropic curvature-inducing proteins onto membrane tubes

Bin/Amphiphysin/Rvs superfamily proteins and other curvature-inducing proteins have anisotropic shapes and anisotropically bend biomembranes. Here, we report how the anisotropic proteins bind the membrane tube and are orientationally ordered using mean-field theory including an orientation-dependent excluded volume. The proteins exhibit a second-order or first-order nematic transition with increasing protein density depending on the radius of the membrane tube. The tube curvatures for the maximum protein binding and orientational order are different and varied by the protein density and rigidity. As the external force along the tube axis increases, a first-order transition from a large tube radius with low protein density to a small radius with high density occurs once, and subsequently, the protein orientation tilts to the tube-axis direction. When an isotropic bending energy is used for the proteins with an elliptic shape, the force-dependence curves become symmetric and the first-order transition occurs twice. This theory quantitatively reproduces the results of meshless membrane simulation for short proteins, whereas deviations are seen for long proteins owing to the formation of protein clusters.

Binding of curvature-inducing proteins onto tethered vesicles

A tethered vesicle, which consists of a cylindrical membrane tube and a spherical vesicle, is produced by a mechanical force that is experimentally imposed by optical tweezers and a micropipette. This tethered vesicle is employed for examining the curvature sensing of curvature-inducing proteins. In this study, we clarify how the binding of proteins with a laterally isotropic spontaneous curvature senses and generates the membrane curvatures of the tethered vesicle using mean-field theory and meshless membrane simulation. The force-dependence curves of the protein density in the membrane tube and the tube curvature are reflection symmetric and point symmetric, respectively, from the force point, in which the tube has a sensing curvature. The bending rigidity and spontaneous curvature of the bound proteins can be estimated from these force-dependence curves. First-order transitions can occur between low and high protein densities in the tube at both low and high force amplitudes. The simulation results of the homogeneous phases agree very well with the theoretical predictions. In addition, beaded-necklace-like tubes with microphase separation are found in the simulation.

Vesicle budding induced by binding of curvature-inducing proteins

Vesicle budding induced by protein binding that generates an isotropic spontaneous curvature is studied using a mean-field theory. Many spherical buds are formed via protein binding. As the binding chemical potential increases, the proteins first bind to the buds and then to the remainder of the vesicle. For a high spontaneous curvature and/or high bending rigidity of the bound membrane, it is found that a first-order transition occurs between a small number of large buds and a large number of small buds. These two states coexist around the transition point. The proposed scheme is simple and easily applicable to many interaction types, so we investigate the effects of interprotein interactions, the protein-insertion-induced changes in area, the variation of the saddle-splay modulus, and the area-difference-elasticity energy. The differences in the preferred curvatures for curvature sensing and generation are also clarified.

Recent developments in membrane curvature sensing and induction by proteins

BACKGROUND

Membrane-bound intracellular organelles have characteristic shapes attributed to different local membrane curvatures, and these attributes are conserved across species. Over the past decade, it has been confirmed that specific proteins control the large curvatures of the membrane, whereas many others due to their specific structural features can sense the curvatures and bind to the specific geometrical cues. Elucidating the interplay between sensing and induction is indispensable to understand the mechanisms behind various biological processes such as vesicular trafficking and budding.

SCOPE OF REVIEW

We provide an overview of major classes of membrane proteins and the mechanisms of curvature sensing and induction. We then discuss the importance of membrane elastic characteristics to induce the membrane shapes similar to intracellular organelles. Finally, we survey recently available assays developed for studying the curvature sensing and induction by many proteins.

MAJOR CONCLUSIONS

Recent theoretical/computational modeling along with experimental studies have uncovered fascinating connections between lipid membrane and protein interactions. However, the phenomena of protein localization and synchronization to generate spatiotemporal dynamics in membrane morphology are yet to be fully understood.

GENERAL SIGNIFICANCE

The understanding of protein-membrane interactions is essential to shed light on various biological processes. This further enables the technological applications of many natural proteins/peptides in therapeutic treatments. The studies of membrane dynamic shapes help to understand the fundamental functions of membranes, while the medicinal roles of various macromolecules (such as proteins, peptides, etc.) are being increasingly investigated.

A theory of ordering of elongated and curved proteins on membranes driven by density and curvature

Cell membranes interact with a myriad of curvature-active proteins that control membrane morphology and are responsible for mechanosensation and mechanotransduction. Some of these proteins, such as those containing BAR domains, are curved and elongated, and hence may adopt different states of orientational order, from isotropic to maximize entropy to nematic as a result of crowding or to adapt to the curvature of the underlying membrane. Here, extending the classical work of Onsager for ordering in hard particle systems and that of [E. S. Nascimento et al., Phys. Rev. E, 2017, 96, 022704], we develop a mean-field density functional theory to predict the orientational order and evaluate the free energy of ensembles of elongated and curved objects on curved membranes. This theory depends on the microscopic properties of the particles and explains how a density-dependent isotropic-to-nematic transition is modified by anisotropic curvature. We also examine the coexistence of isotropic and nematic phases. This theory predicts how ordering depends on geometry but we assume here that the geometry is fixed. It also lays the ground to understand the interplay between membrane reshaping by BAR proteins and molecular order, examined by [Le Roux et al., submitted, 2020].

Binding, unbinding and aggregation of crescent-shaped nanoparticles on nanoscale tubular membranes

Using molecular dynamics simulations of a coarse-grained implicit solvent model, we investigate the binding of crescent-shaped nanoparticles (NPs) on tubular lipid membranes. The NPs adhere to the membrane through their concave side. We found that the binding/unbinding transition is first-order, with the threshold binding energy being higher than the unbinding threshold, and the energy barrier between the bound and unbound states at the transition that increases with increasing the NP's arclength L_np or curvature mismatch μ = R_c/R_np, where R_c and R_np are the radii of curvature of the tubular membrane and the NP, respectively. Furthermore, we found that the threshold binding energy increases with increasing either L_np or μ. NPs with curvature larger than that of the tubule (μ > 1) lie perpendicularly to the tubule's axis. However, for μ smaller than a specific arclength-dependent mismatch μ*, the NPs are tilted with respect to the tubule's axis, with the tilt angle that increases with decreasing μ. We also investigated the self-assembly of the NPs on the tubule at relatively weak adhesion strength and found that for μ > 1 and high values of L_np, the NPs self-assemble into linear chains, and lie side-by-side. For μ < μ* and high L_np, the NPs also self-assemble into chains, while being tilted with respect to the tubule's axis.

Membrane morphologies induced by mixtures of arc-shaped particles with opposite curvature

Biological membranes are shaped by various proteins that either generate inward or outward membrane curvature. In this article, we investigate 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 membrane curvature of opposite sign. We find that small fractions of convex-binding particles can stabilize three-way junctions of membrane tubules, as suggested for the protein lunapark in the endoplasmic reticulum of cells. For comparable fractions of concave-binding and convex-binding particles, we observe lines of particles of the same type, and diverse membrane morphologies with grooves and bulges induced by these particle lines. The alignment and segregation of the particles is driven by indirect, membrane-mediated interactions.

Radial aggregation of proteins prevails over axial aggregation on membrane tubes

Tubular membrane structures are abundant in living cells and participate in various cell activities with the help of specific proteins. This complicated protein-membrane interaction raises a largely unclear question of how cells create, maintain and eliminate membrane nanotubes with a variety of proteins involved. To simplify the question and to give a common mechanism, we simply model membrane proteins as various solid nanoparticles (NPs) of different shapes, instead of considering the detailed structure of different proteins. With this minimal model of membrane proteins, we use molecular simulations to study the common features for protein self-assembly on membrane tubes. Both molecular simulations and energy analysis reveal that on tubular membrane surfaces, the radial aggregation structure of spherical NPs prevails over axial aggregation. We demonstrate that anisotropic deformation of membrane tubes by NP adhesion leads to a direction-dependent (effective) inter-NP attraction, which controls the direction of NP assembly. Moreover, this radial aggregation morphology seems to be independent of the shape of NPs, except for NPs with a length comparable to the tube diameter. This observation indicates that proteins with strong adhesion to a membrane tube tend to form ring-like aggregates, and this finding offers an insight into how proteins play their roles in generating, maintaining and breaking tubular membrane structures.

Vesicle constriction by rings of Janus nanoparticles and aggregates of curved proteins

Membrane constriction and associated scission by proteins and nano structures are crucial to many processes in cellular and synthetic biology. We report mechanical constriction of vesicles by rings of adsorbed Janus nanoparticles that represent synthetic nano structures and mimic contractile proteins, and by aggregates of curved crescents that mimic scaffold proteins. Membrane energetics from Monte Carlo simulations and simulated annealing of the elastic membrane model confirms spontaneous vesicle constriction by aggregates of sufficiently-curved crescents of various lengths and by rings of Janus nanoparticles with a variety of ring lengths, particle sizes, and particle area fractions. We show that shorter rings of smaller particles with higher area fractions reinforce the constriction by increasing the energetic drive towards the constricted vesicle with smaller constriction radius. We demonstrate that vesicle constriction by crescent aggregates strongly depends on the crescent curvature. In contrast to aggregates of sufficiently-curved crescents that are capable of inducing full vesicle constriction, those of near flat crescents with negligible curvature leave the vesicle unconstricted. Our results offer promising perspectives for designing membrane-constricting nano structures such as nanoparticle aggregates and clusters of synthetic curved proteins such as DNA origami scaffolds with applications in synthetic biology. Our findings reveal the significant contribution of highly-curved F-BAR domains to cell division and explain how contractile protein rings such as dynamin GTPase, actomyosin rings, and endosomal sorting complexes required for transport constrict the membrane.

Shape transition from elliptical to cylindrical membrane tubes induced by chiral crescent-shaped protein rods

Proteins often form chiral assembly structures on a biomembrane. However, the role of the chirality in the interaction with an achiral membrane is poorly understood. Here, we report how chirality of crescent-shaped protein rods changes their assembly and tubulation using meshless membrane simulations. The achiral rods deformed the membrane tube into an elliptical shape by stabilizing the edges of the ellipse. In contrast, the chiral rods formed a helical assembly that generated a cylindrical membrane tube with a constant radius in addition to the elliptical tube. This helical assembly could be further stabilized by the direct side-to-side attraction between the protein rods. The chirality also promotes the tubulation from a flat membrane. These results agree with experimental findings of the constant radius of membrane tubules induced by the Bin/Amphiphysin/Rvs (BAR) superfamily proteins.

Characterization of Lipid-Protein Interactions and Lipid-Mediated Modulation of Membrane Protein Function through Molecular Simulation

The cellular membrane constitutes one of the most fundamental compartments of a living cell, where key processes such as selective transport of material and exchange of information between the cell and its environment are mediated by proteins that are closely associated with the membrane. The heterogeneity of lipid composition of biological membranes and the effect of lipid molecules on the structure, dynamics, and function of membrane proteins are now widely recognized. Characterization of these functionally important lipid-protein interactions with experimental techniques is however still prohibitively challenging. Molecular dynamics (MD) simulations offer a powerful complementary approach with sufficient temporal and spatial resolutions to gain atomic-level structural information and energetics on lipid-protein interactions. In this review, we aim to provide a broad survey of MD simulations focusing on exploring lipid-protein interactions and characterizing lipid-modulated protein structure and dynamics that have been successful in providing novel insight into the mechanism of membrane protein function.

Transition from curvature sensing to generation in a vesicle driven by protein binding strength and membrane tension

The ability of proteins to sense and/or generate membrane curvature is crucial for many biological processes inside the cell. We introduce a model for the binding and unbinding of curvature inducing proteins on vesicles using Dynamic Triangulation Monte Carlo (DTMC) simulations. In our study, the interaction between membrane curvature and protein binding is characterised by the binding affinity parameter μ, which indicates the interaction strength. We demonstrate that both sensing and generation of curvature can be observed in the same system as a function of the protein binding affinity on the membrane. Our results show that at low μ values, proteins only sense membrane curvature, whereas at high μ values, they induce curvature. The transition between sensing and generation regimes is marked by a sharp change in the μ-dependence of the protein bound fraction. We present ways to quantitatively characterise these two regimes. We also observe that imposing tension on the membrane (through internal excess pressure for liposomes) extends the region of curvature sensing in the parameter space.

Membrane Morphologies Induced by Arc-Shaped Scaffolds Are Determined by Arc Angle and Coverage

The intricate shapes of biological membranes such as tubules and membrane stacks are induced by proteins. In this article, we systematically investigate the membrane shapes induced by arc-shaped scaffolds such as proteins and protein complexes with coarse-grained modeling and simulations. We find that arc-shaped scaffolds induce membrane tubules at membrane coverages larger than a threshold of ∼40%, irrespective of their arc angle. The membrane morphologies at intermediate coverages below this tubulation threshold, in contrast, strongly depend on the arc angle. Scaffolds with arc angles of about 60°, akin to N-BAR domains, do not change the membrane shape at coverages below the tubulation threshold, whereas scaffolds with arc angles larger than about 120° induce double-membrane stacks at intermediate coverages. The scaffolds stabilize the curved membrane edges that connect the membrane stacks, as suggested for complexes of reticulon proteins. Our results provide general insights on the determinants of membrane shaping by arc-shaped scaffolds.

Acceleration and suppression of banana-shaped-protein-induced tubulation by addition of small membrane inclusions of isotropic spontaneous curvatures

Membrane tubulation induced by banana-shaped protein rods is investigated by using coarse-grained meshless membrane simulations. It is found that tubulation is promoted by laterally isotropic membrane inclusions that generate the same sign of spontaneous curvature as the adsorbed protein rods. The inclusions are concentrated in the tubules and reduce the bending energy of the tip of the tubules. On the other hand, inclusions with an opposite curvature suppress tubulation by percolated-network formation at a high protein-rod density while they induce the formation of a spherical membrane bud at a low rod density. When equal amounts of the two types of inclusions (with positive and negative curvatures) are added, their effects cancel each other for the first short period but later the tubulation is slowly accelerated. Positive surface tension suppresses tubulation. Our results suggest that the cooperation of scaffolding of BAR (Bin/Amphiphysin/Rvs) domains and isotropic membrane inclusions is important for tubulation.

Membrane structure formation induced by two types of banana-shaped proteins

The assembly of banana-shaped rodlike proteins on membranes and the associated membrane shape transformations are investigated by analytical theory and coarse-grained simulations. The membrane-mediated interactions between two banana-shaped inclusions are derived theoretically using a point-like formalism based on fixed anisotropic curvatures, both for zero surface tension and for finite surface tension. On a larger scale, the interactions between the assemblies of such rodlike inclusions are determined analytically. Meshless membrane simulations are performed in the presence of a large number of inclusions of two types, corresponding to the curved rods of opposite curvatures, both for flat membranes and vesicles. Rods of the same type aggregate into linear assemblies perpendicular to the rod axis, leading to membrane tubulation. However, rods of the other type, those of opposite curvature, are attracted to the lateral sides of these assemblies, and stabilize a straight bump structure that prevents tubulation. When the two types of rods have almost opposite curvatures, the bumps attract one another, forming a striped structure. Positive surface tension is found to stabilize stripe formation. The simulation results agree well with the theoretical predictions provided the point-like curvatures of the model are scaled-down to account for the effective flexibility of the simulated rods.

Golgi apparatus self-organizes into the characteristic shape via postmitotic reassembly dynamics

The Golgi apparatus is a membrane-bounded organelle with the characteristic shape of a series of stacked flat cisternae. During mitosis in mammalian cells, the Golgi apparatus is once fragmented into small vesicles and then reassembled to form the characteristic shape again in each daughter cell. The mechanism and details of the reassembly process remain elusive. Here, by the physical simulation of a coarse-grained membrane model, we reconstructed the three-dimensional morphological dynamics of the Golgi reassembly process. Considering the stability of the interphase Golgi shape, we introduce two hypothetical mechanisms-the Golgi rim stabilizer protein and curvature-dependent restriction on membrane fusion-into the general biomembrane model. We show that the characteristic Golgi shape is spontaneously organized from the assembly of vesicles by proper tuning of the two additional mechanisms, i.e., the Golgi reassembly process is modeled as self-organization. We also demonstrate that the fine Golgi shape forms via a balance of three reaction speeds: vesicle aggregation, membrane fusion, and shape relaxation. Moreover, the membrane fusion activity decreases thickness and the number of stacked cisternae of the emerging shapes.

From Cooperative Self-Assembly to Water-Soluble Supramolecular Polymers Using Coarse-Grained Simulations

Supramolecular polymers, formed via noncovalent self-assembly of elementary monomers, are extremely interesting for their dynamic bioinspired properties. In order to understand their behavior, it is necessary to access their dynamics while maintaining high resolution in the treatment of the monomer structure and monomer-monomer interactions, which is typically a difficult task, especially in aqueous solution. Focusing on 1,3,5-benzenetricarboxamide (BTA) water-soluble supramolecular polymers, we have developed a transferable coarse-grained model that allows studying BTA supramolecular polymerization in water, while preserving remarkable consistency with the atomistic models in the description of the key interactions between the monomers (hydrophobic, H-bonding, etc.), self-assembly cooperativity, and amplification of order into the growing fibers. This permitted us to monitor the amplification of the key interactions between the monomers (including H-bonding) in the BTA fibers during the dynamic polymerization process. Our molecular dynamics simulations provide a picture of a stepwise cooperative polymerization mechanism, where initial fast hydrophobic aggregation of the BTA monomers in water is followed by the slower reorganization of these disordered aggregates into ordered directional oligomers. Supramolecular polymer growth then proceeds on a slower time scale. We challenged our models via comparison with the experimental evidence, capturing the effect of temperature variations and subtle changes in the monomer structure on the polymerization and on the properties of the fibers seen in the real systems. This work provides a multiscale spatiotemporal characterization of BTA self-assembly in water and a useful platform to study a variety of BTA-based supramolecular polymers toward structure-property relationships.

Physical basis of some membrane shaping mechanisms

In vesicular transport pathways, membrane proteins and lipids are internalized, externalized or transported within cells, not by bulk diffusion of single molecules, but embedded in the membrane of small vesicles or thin tubules. The formation of these 'transport carriers' follows sequential events: membrane bending, fission from the donor compartment, transport and eventually fusion with the acceptor membrane. A similar sequence is involved during the internalization of drug or gene carriers inside cells. These membrane-shaping events are generally mediated by proteins binding to membranes. The mechanisms behind these biological processes are actively studied both in the context of cell biology and biophysics. Bin/amphiphysin/Rvs (BAR) domain proteins are ideally suited for illustrating how simple soft matter principles can account for membrane deformation by proteins. We review here some experimental methods and corresponding theoretical models to measure how these proteins affect the mechanics and the shape of membranes. In more detail, we show how an experimental method employing optical tweezers to pull a tube from a giant vesicle may give important quantitative insights into the mechanism by which proteins sense and generate membrane curvature and the mechanism of membrane scission.This article is part of the themed issue 'Soft interfacial materials: from fundamentals to formulation'.

Shape deformation of lipid membranes by banana-shaped protein rods: Comparison with isotropic inclusions and membrane rupture

The assembly of curved protein rods on fluid membranes is studied using implicit-solvent meshless membrane simulations. As the rod curvature increases, the rods on a membrane tube assemble along the azimuthal direction first and subsequently along the longitudinal direction. Here, we show that both transition curvatures decrease with increasing rod stiffness. For comparison, curvature-inducing isotropic inclusions are also simulated. When the isotropic inclusions have the same bending rigidity as the other membrane regions, the inclusions are uniformly distributed on the membrane tubes and vesicles even for large spontaneous curvature of the inclusions. However, the isotropic inclusions with much larger bending rigidity induce shape deformation and are concentrated on the region of a preferred curvature. For high rod density, high rod stiffness, and/or low line tension of the membrane edge, the rod assembly induces vesicle rupture, resulting in the formation of a high-genus vesicle. A gradual change in the curvature suppresses this rupture. Hence, large stress, compared to the edge tension, induced by the rod assembly is the key factor determining rupture. For rod curvature with the opposite sign to the vesicle curvature, membrane rupture induces inversion of the membrane, leading to division into multiple vesicles as well as formation of a high-genus vesicle.

Membrane tubule formation by banana-shaped proteins with or without transient network structure

In living cells, membrane morphology is regulated by various proteins. Many membrane reshaping proteins contain a Bin/Amphiphysin/Rvs (BAR) domain, which consists of a banana-shaped rod. The BAR domain bends the biomembrane along the rod axis and the features of this anisotropic bending have recently been studied. Here, we report on the role of the BAR protein rods in inducing membrane tubulation, using large-scale coarse-grained simulations. We reveal that a small spontaneous side curvature perpendicular to the rod can drastically alter the tubulation dynamics at high protein density, whereas no significant difference is obtained at low density. A percolated network is intermediately formed depending on the side curvature. This network suppresses tubule protrusion, leading to the slow formation of fewer tubules. Thus, the side curvature, which is generated by protein–protein and membrane–protein interactions, plays a significant role in tubulation dynamics. We also find that positive surface tensions and the vesicle membrane curvature can stabilize this network structure by suppressing the tubulation.