Membrane-mediated aggregation of curvature-inducing nematogens and membrane tubulation

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'.

Multiscale simulations of protein-facilitated membrane remodeling

Protein-facilitated shape and topology changes of cell membranes are crucial for many biological processes, such as cell division, protein trafficking, and cell signaling. However, the inherently multiscale nature of membrane remodeling presents a considerable challenge for understanding the mechanisms and physics that drive this process. To address this problem, a multiscale approach that makes use of a diverse set of computational and experimental techniques is required. The atomistic simulations provide high-resolution information on protein-membrane interactions. Experimental techniques, like electron microscopy, on the other hand, resolve high-order organization of proteins on the membrane. Coarse-grained (CG) and mesoscale computational techniques provide the intermediate link between the two scales and can give new insights into the underlying mechanisms. In this Review, we present the recent advances in multiscale computational approaches established in our group. We discuss various CG and mesoscale approaches in studying the protein-mediated large-scale membrane remodeling.

A Model for Shaping Membrane Sheets by Protein Scaffolds

Membranes of peripheral endoplasmic reticulum form intricate morphologies consisting of tubules and sheets as basic elements. The physical mechanism of endoplasmic-reticulum shaping has been suggested to originate from the elastic behavior of the sheet edges formed by linear arrays of oligomeric protein scaffolds. The heart of this mechanism, lying in the relationships between the structure of the protein scaffolds and the effective intrinsic shapes and elastic properties of the sheets' edges, has remained hypothetical. Here we provide a detailed computational analysis of these issues. By minimizing the elastic energy of membrane bending, we determine the effects of a rowlike array of semicircular arclike membrane scaffolds on generation of a membrane fold, which shapes the entire membrane surface into a flat double-membrane sheet. We show, quantitatively, that the sheet's edge line tends to adopt a positive or negative curvature depending on the scaffold's geometrical parameters. We compute the effective elastic properties of the sheet edge and analyze the dependence of the equilibrium distance between the scaffolds along the edge line on the scaffold geometry.

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.

Inter-tube adhesion mediates a new pearling mechanism

A common mechanism for intracellular transport is the controlled shape transformation, also known as pearling, of membrane tubes. Exploring how tube pearling takes place is thus of quite importance to not only understand the bio-functions of tubes, but also promote their potential biomedical applications. While the pearling mechanism of one single tube is well understood, both the pathway and the mechanism of pearling of multiple tubes still remain unclear. Herein, by means of computer simulations we show that the tube pearling can be mediated by the inter-tube adhesion. By increasing the inter-tube adhesion strength, each tube undergoes a discontinuous transition from no pearling to thorough pearling. The discontinuous pearling transition is ascribed to the competitive variation between tube surface tension and the extent of inter-tube adhesion. Besides, the final pearling instability is also affected by tube diameter and inter-tube orientation. Thinner tubes undergo inter-tube lipid diffusion before completion of pearling. The early lipid diffusion reduces the extent of inter-tube adhesion and thus restrains the subsequent pearling. Therefore, only partial or no pearling can take place for two thinner tubes. For two perpendicular tubes, the pearling is also observed, but with different pathways and higher efficiency. The finite size effect is discussed by comparing the pearling of tubes with different lengths. It is expected that this work will not only provide new insights into the mechanism of membrane tube pearling, but also shed light on the potential applications in biomaterials science and nanomedicine.

When Physics Takes Over: BAR Proteins and Membrane Curvature

Cell membranes become highly curved during membrane trafficking, cytokinesis, infection, immune response, or cell motion. Bin/amphiphysin/Rvs (BAR) domain proteins with their intrinsically curved and anisotropic shape are involved in many of these processes, but with a large spectrum of modes of action. In vitro experiments and multiscale computer simulations have contributed in identifying a minimal set of physical parameters, namely protein density on the membrane, membrane tension, and membrane shape, that control how bound BAR domain proteins behave on the membrane. In this review, we summarize the multifaceted coupling of BAR proteins to membrane mechanics and propose a simple phase diagram that recapitulates the effects of these parameters.

Protein kinase A activity is necessary for fission and fusion of Golgi to endoplasmic reticulum retrograde tubules

It is becoming increasingly accepted that together with vesicles, tubules play a major role in the transfer of cargo between different cellular compartments. In contrast to our understanding of the molecular mechanisms of vesicular transport, little is known about tubular transport. How signal transduction molecules regulate these two modes of membrane transport processes is also poorly understood. In this study we investigated whether protein kinase A (PKA) activity regulates the retrograde, tubular transport of Golgi matrix proteins from the Golgi to the endoplasmic reticulum (ER). We found that Golgi-to-ER retrograde transport of the Golgi matrix proteins giantin, GM130, GRASP55, GRASP65, and p115 was impaired in the presence of PKA inhibitors. In addition, we unexpectedly found accumulation of tubules containing both Golgi matrix proteins and resident Golgi transmembrane proteins. These tubules were still attached to the Golgi and were highly dynamic. Our data suggest that both fission and fusion of retrograde tubules are mechanisms regulated by PKA activity.

Monte Carlo simulations of fluid vesicles

Lipid vesicles are closed two dimensional fluid surfaces that are studied extensively as model systems for understanding the physical properties of biological membranes. Here we review the recent developments in the Monte Carlo techniques for simulating fluid vesicles and discuss some of their applications. The technique, which treats the membrane as an elastic sheet, is most suitable for the study of large scale conformations of membranes. The model can be used to study vesicles with fixed and varying topologies. Here we focus on the case of multi-component membranes with the local lipid and protein composition coupled to the membrane curvature leading to a variety of shapes. The phase diagram is more intriguing in the case of fluid vesicles having an in-plane orientational order that induce anisotropic directional curvatures. Methods to explore the steady state morphological structures due to active flux of materials have also been described in the context of Monte Carlo simulations.

Amphipathic DNA origami nanoparticles to scaffold and deform lipid membrane vesicles

We report a synthetic biology-inspired approach for the engineering of amphipathic DNA origami structures as membrane-scaffolding tools. The structures have a flat membrane-binding interface decorated with cholesterol-derived anchors. Sticky oligonucleotide overhangs on their side facets enable lateral interactions leading to the formation of ordered arrays on the membrane. Such a tight and regular arrangement makes our DNA origami capable of deforming free-standing lipid membranes, mimicking the biological activity of coat-forming proteins, for example, from the I-/F-BAR family.

Membrane-mediated interaction between strongly anisotropic protein scaffolds

Specialized proteins serve as scaffolds sculpting strongly curved membranes of intracellular organelles. Effective membrane shaping requires segregation of these proteins into domains and is, therefore, critically dependent on the protein-protein interaction. Interactions mediated by membrane elastic deformations have been extensively analyzed within approximations of large inter-protein distances, small extents of the protein-mediated membrane bending and small deviations of the protein shapes from isotropic spherical segments. At the same time, important classes of the realistic membrane-shaping proteins have strongly elongated shapes with large and highly anisotropic curvature. Here we investigated, computationally, the membrane mediated interaction between proteins or protein oligomers representing membrane scaffolds with strongly anisotropic curvature, and addressed, quantitatively, a specific case of the scaffold geometrical parameters characterizing BAR domains, which are crucial for membrane shaping in endocytosis. In addition to the previously analyzed contributions to the interaction, we considered a repulsive force stemming from the entropy of the scaffold orientation. We computed this interaction to be of the same order of magnitude as the well-known attractive force related to the entropy of membrane undulations. We demonstrated the scaffold shape anisotropy to cause a mutual aligning of the scaffolds and to generate a strong attractive interaction bringing the scaffolds close to each other to equilibrium distances much smaller than the scaffold size. We computed the energy of interaction between scaffolds of a realistic geometry to constitute tens of kBT, which guarantees a robust segregation of the scaffolds into domains.

Reshaping biological membranes in endocytosis: crossing the configurational space of membrane-protein interactions

Lipid membranes are highly dynamic. Over several decades, physicists and biologists have uncovered a number of ways they can change the shape of membranes or alter their phase behavior. In cells, the intricate action of membrane proteins drives these processes. Considering the highly complex ways proteins interact with biological membranes, molecular mechanisms of membrane remodeling still remain unclear. When studying membrane remodeling phenomena, researchers often observe different results, leading them to disparate conclusions on the physiological course of such processes. Here we discuss how combining research methodologies and various experimental conditions contributes to the understanding of the entire phase space of membrane-protein interactions. Using the example of clathrin-mediated endocytosis we try to distinguish the question 'how can proteins remodel the membrane?' from 'how do proteins remodel the membrane in the cell?' In particular, we consider how altering physical parameters may affect the way membrane is remodeled. Uncovering the full range of physical conditions under which membrane phenomena take place is key in understanding the way cells take advantage of membrane properties in carrying out their vital tasks.

DNA origami nanoneedles on freestanding lipid membranes as a tool to observe isotropic-nematic transition in two dimensions

We introduce a simple experimental system to study dynamics of needle-like nanoobjects in two dimensions (2D) as a function of their surface density close to the isotropic-nematic transition. Using fluorescence correlation spectroscopy, we find that translational and rotational diffusion of rigid DNA origami nanoneedles bound to freestanding lipid membranes is strongly suppressed upon an increase in the surface particle density. Our experimental observations show a good agreement with results of Monte Carlo simulations of Brownian hard needles in 2D.

Phenomenology based multiscale models as tools to understand cell membrane and organelle morphologies

An intriguing question in cell biology is "how do cells regulate their shape?" It is commonly believed that the observed cellular morphologies are a result of the complex interaction among the lipid molecules (constituting the cell membrane), and with a number of other macromolecules, such as proteins. It is also believed that the common biophysical processes essential for the functioning of a cell also play an important role in cellular morphogenesis. At the cellular scale-where typical dimensions are in the order of micrometers-the effects arising from the molecular scale can either be modeled as equilibrium or non-equilibrium processes. In this chapter, we discuss the dynamically triangulated Monte Carlo technique to model and simulate membrane morphologies at the cellular scale, which in turn can be used to investigate several questions related to shape regulation in cells. In particular, we focus on two specific problems within the framework of isotropic and anisotropic elasticity theories: namely, (i) the origin of complex, physiologically relevant, membrane shapes due to the interaction of the membrane with curvature remodeling proteins, and (ii) the genesis of steady state cellular shapes due to the action of non-equilibrium forces that are generated by the fission and fusion of transport vesicles and by the binding and unbinding of proteins from the parent membrane.

Terasaki spiral ramps in the rough endoplasmic reticulum

We present a model describing the morphology as well as the assembly of "Terasaki ramps," the recently discovered helicoidal connections linking adjacent sheets of the rough endoplasmic reticulum (ER). The fundamental unit is a localized symmetric double-ramped "parking garage" formed by two separated gently pitched, approximately helicoidal, ramps of opposite chiralities. This geometry is stabilized by a short-range repulsive interaction between ramps associated with bending energy which opposes the long-range attraction associated with tension. The ramp inner boundaries are themselves stabilized by the condensation of membrane-shaping proteins along their length. A mechanism for parking garage self-assembly is proposed involving the nucleation of dipoles at the center of tubular three-way junctions within the smooth ER. Our predictions are compared with the experimental data.

Multiscale computational models in physical systems biology of intracellular trafficking

In intracellular trafficking, a definitive understanding of the interplay between protein binding and membrane morphology remains incomplete. The authors describe a computational approach by integrating coarse-grained molecular dynamics (CGMD) simulations with continuum Monte Carlo (CM) simulations of the membrane to study protein-membrane interactions and the ensuing membrane curvature. They relate the curvature field strength discerned from the molecular level to its effect at the cellular length-scale. They perform thermodynamic integration on the CM model to describe the free energy landscape of vesiculation in clathrin-mediated endocytosis. The method presented here delineates membrane morphologies and maps out the free energy changes associated with membrane remodeling due to varying coat sizes, coat curvature strengths, membrane bending rigidities, and tensions; furthermore several constraints on mechanisms underlying clathrin-mediated endocytosis have also been identified, Their CGMD simulations have revealed the importance of PIP2 for stable binding of proteins essential for curvature induction in the bilayer and have provided a molecular basis for the positive curvature induction by the epsin N-terminal homology (EIMTH) domain. Calculation of the free energy landscape for vesicle budding has identified the critical size and curvature strength of a clathrin coat required for nucleation and stabilisation of a mature vesicle.

Mesoscale computational studies of membrane bilayer remodeling by curvature-inducing proteins

Biological membranes constitute boundaries of cells and cell organelles. These membranes are soft fluid interfaces whose thermodynamic states are dictated by bending moduli, induced curvature fields, and thermal fluctuations. Recently, there has been a flood of experimental evidence highlighting active roles for these structures in many cellular processes ranging from trafficking of cargo to cell motility. It is believed that the local membrane curvature, which is continuously altered due to its interactions with myriad proteins and other macromolecules attached to its surface, holds the key to the emergent functionality in these cellular processes. Mechanisms at the atomic scale are dictated by protein-lipid interaction strength, lipid composition, lipid distribution in the vicinity of the protein, shape and amino acid composition of the protein, and its amino acid contents. The specificity of molecular interactions together with the cooperativity of multiple proteins induce and stabilize complex membrane shapes at the mesoscale. These shapes span a wide spectrum ranging from the spherical plasma membrane to the complex cisternae of the Golgi apparatus. Mapping the relation between the protein-induced deformations at the molecular scale and the resulting mesoscale morphologies is key to bridging cellular experiments across the various length scales. In this review, we focus on the theoretical and computational methods used to understand the phenomenology underlying protein-driven membrane remodeling. Interactions at the molecular scale can be computationally probed by all atom and coarse grained molecular dynamics (MD, CGMD), as well as dissipative particle dynamics (DPD) simulations, which we only describe in passing. We choose to focus on several continuum approaches extending the Canham - Helfrich elastic energy model for membranes to include the effect of curvature-inducing proteins and explore the conformational phase space of such systems. In this description, the protein is expressed in the form of a spontaneous curvature field. The approaches include field theoretical methods limited to the small deformation regime, triangulated surfaces and particle-based computational models to investigate the large-deformation regimes observed in the natural state of many biological membranes. Applications of these methods to understand the properties of biological membranes in homogeneous and inhomogeneous environments of proteins, whose underlying curvature fields are either isotropic or anisotropic, are discussed. The diversity in the curvature fields elicits a rich variety of morphological states, including tubes, discs, branched tubes, and caveola. Mapping the thermodynamic stability of these states as a function of tuning parameters such as concentration and strength of curvature induction of the proteins is discussed. The relative stabilities of these self-organized shapes are examined through free-energy calculations. The suite of methods discussed here can be tailored to applications in specific cellular settings such as endocytosis during cargo trafficking and tubulation of filopodial structures in migrating cells, which makes these methods a powerful complement to experimental studies.

Defining the free-energy landscape of curvature-inducing proteins on membrane bilayers

Curvature-sensing and curvature-remodeling proteins, such as Amphiphysin, Epsin, and Exo70, are known to reshape cell membranes, and this remodeling event is essential for key biophysical processes such as tubulation, exocytosis, and endocytosis. Curvature-inducing proteins can act as curvature sensors; they aggregate to membrane regions matching their intrinsic curvature; as well as induce curvature in cell membranes to stabilize emergent high curvature, nonspherical, structures such as tubules, discs, and caveolae. A definitive understanding of the interplay between protein recruitment and migration, the evolution of membrane curvature, and membrane morphological transitions is emerging but remains incomplete. Here, within a continuum framework and using the machinery of Monte Carlo simulations, we introduce and compare three free-energy methods to delineate the free-energy landscape of curvature-inducing proteins on bilayer membranes. We demonstrate the utility of the Widom test particle (or field) insertion methodology in computing the excess chemical potentials associated with curvature-inducing proteins on the membrane-in particular, we use this method to track the onset of morphological transitions in the membrane at elevated protein densities. We validate this approach by comparing the results from the Widom method with those of thermodynamic integration and Bennett acceptance ratio methods. Furthermore, the predictions from the Widom method have been tested against analytical calculations of the excess chemical potential at infinite dilution. Our results are useful in precisely quantifying the free-energy landscape, and also in determining the phase boundaries associated with curvature-induction, curvature-sensing, and morphological transitions. This approach can be extended to studies exploring the role of thermal fluctuations and other external (control) variables, such as membrane excess area, in shaping curvature-mediated interactions on bilayer membranes.

Anisotropic spontaneous curvatures in lipid membranes

Symmetry restrictions due to fluidity require the strain energy in the Helfrich theory of lipid membranes to be locally isotropic in nature. Although this framework is suitable for modeling the interaction of membranes with proteins that generate spherical curvature such as clathrin, there are other important membrane-bending proteins such as BIN-amphiphysin-Rvs proteins that form a cylindrical coat with different curvatures in the longitudinal and the circumferential directions. In this work, we present a detailed mathematical treatment of the theory of lipid membranes incorporating anisotropic spontaneous curvatures. We derive the associated Euler-Lagrange equations and the edge conditions in a generalized setting that allows spatial heterogeneities in the properties of the membrane-protein system. We employ this theory to model the constriction of a membrane tubule by a cylindrical scaffold. In particular, we highlight the role of the equilibrium equation in the tangential plane in regulating the spatial variation of the surface tension field.

Dynamics and instabilities of lipid bilayer membrane shapes

Biological membranes undergo constant shape remodeling involving the formation of highly curved structures. The lipid bilayer represents the fundamental architecture of the cellular membrane with its shapes determined by the Helfrich curvature bending energy. However, the dynamics of bilayer shape transitions, especially their modulation by membrane proteins, and the resulting shape instabilities, are still not well understood. Here, we review in a unifying manner several theories that describe the fluctuations (i.e. undulations) of bilayer shapes as well as their local coupling with lipid or protein density variation. The coupling between local membrane curvature and lipid density gives rise to a 'slipping mode' in addition to the conventional 'bending mode' for damping the membrane fluctuation. This leads to a number of interesting experimental phenomena regarding bilayer shape dynamics. More importantly, curvature-inducing proteins can couple with membrane shape and eventually render the membrane unstable. A criterion for membrane shape instability is derived from a linear stability analysis. The instability criterion reemphasizes the importance of membrane tension in regulating the stability and dynamics of membrane geometry. Recent progresses in understanding the role of membrane tension in regulating dynamical cellular processes are also reviewed. Protein density is emphasized as a key factor in regulating membrane shape transitions: a threshold density of curvature coupling proteins is required for inducing membrane morphology transitions.

Protein-mediated transformation of lipid vesicles into tubular networks

Key cellular processes are frequently accompanied by protein-facilitated shape changes in the plasma membrane. N-BAR-domain protein modules generate curvature by means of complex interactions with the membrane surface. The way they assemble and the mechanism by which they operate are largely dependent on their binding density. Although the mechanism at lower densities has recently begun to emerge, how membrane scaffolds form at high densities remains unclear. By combining electron microscopy and multiscale simulations, we show that N-BAR proteins at high densities can transform a lipid vesicle into a 3D tubular network. We show that this process is a consequence of excess adhesive energy combined with the local stiffening of the membrane, which occurs in a narrow range of mechanical properties of both the membrane and the protein. We show that lipid diffusion is significantly reduced by protein binding at this density regime and even more in areas of high Gaussian curvature, indicating a potential effect on molecular transport in cells. Finally, we reveal that the breaking of the bilayer topology is accompanied by the nematic arrangement of the protein on the surface, a structural motif that likely drives the formation of reticular structures in living cells.