Continuum Models of Membrane Fusion: Evolution of the Theory.

Starting from fertilization, through tissue growth, hormone secretion, synaptic transmission, and sometimes morbid events of carcinogenesis and viral infections, membrane fusion regulates the whole life of high organisms. Despite that, a lot of fusion processes still lack well-established models and even a list of main actors. A merger of membranes requires their topological rearrangements controlled by elastic properties of a lipid bilayer. That is why continuum models based on theories of membrane elasticity are actively applied for the construction of physical models of membrane fusion. Started from the view on the membrane as a structureless film with postulated geometry of fusion intermediates, they developed along with experimental and computational techniques to a powerful tool for prediction of the whole process with molecular accuracy. In the present review, focusing on fusion processes occurring in eukaryotic cells, we scrutinize the history of these models, their evolution and complication, as well as open questions and remaining theoretical problems. We show that modern approaches in this field allow continuum models of membrane fusion to stand shoulder to shoulder with molecular dynamics simulations, and provide the deepest understanding of this process in multiple biological systems.

Undulations Drive Domain Registration from the Two Membrane Leaflets.

Phase separation in biological membranes plays an important role in protein targeting and transmembrane signaling. Its occurrence in both membrane leaflets commonly gives rise to matching liquid or liquid-ordered domains in the opposing monolayers. The underlying mechanism of such co-localization is not fully understood. The decrease of the line tension around the thicker ordered domain constitutes an important driving force. Yet, robust domain coupling requires an additional energy source, which we have now identified as thermal undulations. Our theoretical analysis of elastic deformations in a lipid bilayer shows that stiffer lipid domains tend to distribute into areas with lower fluctuations of monolayer curvature. These areas naturally align in the opposing monolayers. Thus, coupling requires both membrane leafs to display a heterogeneity in splay rigidities. The heterogeneity may either originate from intrinsic lipid properties or be acquired by adsorption of peripheral molecules. Undulations and line tension act synergistically: the gain in energy due a minimized line tension is proportional to domain radius and thus primarily fuels the registration of smaller domains; whereas the energetic contribution of undulations increases with membrane area and thus primarily acts to coalesce larger domains.

Elastic deformations of bolalipid membranes.

Archaeal membranes have unique mechanical properties that enable these organisms to survive under extremely aggressive environmental conditions. The so-called bolalipids contribute to this exceptional stability. They have two polar heads joined by two hydrocarbon chains. The two headgroups can face different sides of the membrane (O-shape conformation) or the same side (U-shape conformation). We have developed an elasticity theory for bolalipid membranes and show that the energetic contributions of (i) tilt deformations, (ii) area compression/stretching deformations, (iii) as well as those of Gaussian splay from the two membrane surfaces are additive, while splay deformations yield a cross-term. The presence of a small fraction of U-shaped molecules resulted in spontaneous membrane curvature. We estimated the tilt modulus to be approximately equal to that of membranes in eukaryotic cells. In contrast to conventional lipids, the bolalipid membrane possesses two splay moduli, one of which is estimated to be an order of magnitude larger than that of conventional lipids. The projected values of elastic moduli act to hamper pore formation and to decelerate membrane fusion and fission.

Human Immunodeficiency Virus Type 1 Gag Polyprotein Modulates Membrane Physical Properties like a Surfactant: Potential Implications for Virus Assembly.

Human immunodeficiency virus (HIV) assembly at an infected cell's plasma membrane requires membrane deformation to organize the near-spherical shape of an immature virus. While the cellular expression of HIV Gag is sufficient to initiate budding of virus-like particles, how Gag generates membrane curvature is not fully understood. Using highly curved lipid nanotubes, we have investigated the physicochemical basis of the membrane activity of recombinant nonmyristoylated Gag-Δp6. Gag protein, upon adsorption onto the membrane, resulted in the shape changes of both charged and uncharged nanotubes. This shape change was more pronounced in the presence of charged lipids, especially phosphatidylinositol bisphosphate (PI(4,5)P2). We found that Gag modified the interfacial tension of phospholipid bilayer membranes, as judged by comparison with the effects of amphipathic peptides and nonionic detergent. Bioinformatic analysis demonstrated that a region of the capsid and SP1 domains junction of Gag is structurally similar to the amphipathic peptide magainin-1. This region accounts for integral changes in the physical properties of the membrane upon Gag adsorption, as we showed with the synthetic CA-SP1 junction peptide. Phenomenologically, membrane-adsorbed Gag could diminish the energetic cost of increasing the membrane area in a way similar to foam formation. We propose that Gag acts as a surface-active substance at the HIV budding site that softens the membrane at the place of Gag adsorption, lowering the energy for membrane bending. Finally, our experimental data and theoretical considerations give a lipid-centric view and common mechanism by which proteins could bend membranes, despite not having intrinsic curvature in their molecular surfaces or assemblies.

Hydrophobic Mismatch Controls the Mode of Membrane-Mediated Interactions of Transmembrane Peptides.

Various cellular processes require the concerted cooperative action of proteins. The possibility for such synchronization implies the occurrence of specific long-range interactions between the involved protein participants. Bilayer lipid membranes can mediate protein-protein interactions via relatively long-range elastic deformations induced by the incorporated proteins. We considered the interactions between transmembrane peptides mediated by elastic deformations using the framework of the theory of elasticity of lipid membranes. An effective peptide shape was assumed to be cylindrical, hourglass-like, or barrel-like. The interaction potentials were obtained for membranes of different thicknesses and elastic rigidities. Cylindrically shaped peptides manifest almost neutral average interactions-they attract each other at short distances and repel at large ones, independently of membrane thickness or rigidity. The hourglass-like peptides repel each other in thin bilayers and strongly attract each other in thicker bilayers. On the contrary, the barrel-like peptides repel each other in thick bilayers and attract each other in thinner membranes. These results potentially provide possible mechanisms of control for the mode of protein-protein interactions in membrane domains with different bilayer thicknesses.

Molecular Shape Solution for Mesoscopic Remodeling of Cellular Membranes.

Cellular membranes self-assemble from and interact with various molecular species. Each molecule locally shapes the lipid bilayer, the soft elastic core of cellular membranes. The dynamic architecture of intracellular membrane systems is based on elastic transformations and lateral redistribution of these elementary shapes, driven by chemical and curvature stress gradients. The minimization of the total elastic stress by such redistribution composes the most basic, primordial mechanism of membrane curvature-composition coupling (CCC). Although CCC is generally considered in the context of dynamic compositional heterogeneity of cellular membrane systems, in this article we discuss a broader involvement of CCC in controlling membrane deformations. We focus specifically on the mesoscale membrane transformations in open, reservoir-governed systems, such as membrane budding, tubulation, and the emergence of highly curved sites of membrane fusion and fission. We reveal that the reshuffling of molecular shapes constitutes an independent deformation mode with complex rheological properties.This mode controls effective elasticity of local deformations as well as stationary elastic stress, thus emerging as a major regulator of intracellular membrane remodeling.

Additional contributions to elastic energy of lipid membranes: Tilt-curvature coupling and curvature gradient.

Lipid bilayer membranes under biologically relevant conditions are flexible thin laterally fluid films consisting of two unimolecular layers (monolayers) each about 2 nm thick. On spatial scales much larger than the bilayer thickness, the membrane elasticity is well determined by its shape. The classical Helfrich theory considers the membrane as an elastic two-dimensional (2D) film, which has no particular internal structure. However, various local membrane heterogeneities can result in a lipids tilt relative to the membrane surface normal. On the basis of the classical elasticity theory of 3D bodies, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)10.1007/s101890070003] derived the most general energy functional, taking into account the tilt and lipid monolayer curvature. Recently, Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)10.1063/1.4990404] showed that Hamm and Kozlov's derivation was incomplete because the tilt-curvature coupling term had been missed. However, the energy functional derived by Terzi and Deserno appeared to be unstable, thereby being invalid for applications that require minimizations of the overall energy of deformations. Here, we derive a stable elastic energy functional, showing that the squared gradient of the curvature was missed in both of these works. This change in the energy functional arises from a more accurate consideration of the transverse shear deformation terms and their influence on the membrane stability. We also consider the influence of the prestress terms on the stability of the energy functional, and we show that it should be considered small and the effective Gaussian curvature should be neglected because of the stability requirements. We further generalize the theory, including the stretching-compressing deformation modes, and we provide the geometrical interpretation of the terms that were previously missed by Hamm and Kozlov. The physical consequences of the new terms are analyzed in the case of a membrane-mediated interaction of two amphipathic peptides located in the same monolayer. We also provide the expression for director fluctuations, comparing it with that obtained by Terzi and Deserno.

Reconstitution and real-time quantification of membrane remodeling by single proteins and protein complexes.

Cellular membrane processes, from signal transduction to membrane fusion and fission, depend on acute membrane deformations produced by small and short-lived protein complexes working in conditions far from equilibrium. Real-time monitoring and quantitative assessment of such deformations are challenging; hence, mechanistic analyses of the protein action are commonly based on ensemble averaging, which masks important mechanistic details of the action. In this protocol, we describe how to reconstruct and quantify membrane remodeling by individual proteins and small protein complexes in vitro, using an ultra-short (80- to 400-nm) lipid nanotube (usNT) template. We use the luminal conductance of the usNT as the real-time reporter of the protein interaction(s) with the usNT. We explain how to make and calibrate the usNT template to achieve subnanometer precision in the geometrical assessment of the molecular footprints on the nanotube membrane. We next demonstrate how membrane deformations driven by purified proteins implicated in cellular membrane remodeling can be analyzed at a single-molecule level. The preparation of one usNT takes ~1 h, and the shortest procedure yielding the basic geometrical parameters of a small protein complex takes 10 h.

Dynamic constriction and fission of endoplasmic reticulum membranes by reticulon.

The endoplasmic reticulum (ER) is a continuous cell-wide membrane network. Network formation has been associated with proteins producing membrane curvature and fusion, such as reticulons and atlastin. Regulated network fragmentation, occurring in different physiological contexts, is less understood. Here we find that the ER has an embedded fragmentation mechanism based upon the ability of reticulon to produce fission of elongating network branches. In Drosophila, Rtnl1-facilitated fission is counterbalanced by atlastin-driven fusion, with the prevalence of Rtnl1 leading to ER fragmentation. Ectopic expression of Drosophila reticulon in COS-7 cells reveals individual fission events in dynamic ER tubules. Consistently, in vitro analyses show that reticulon produces velocity-dependent constriction of lipid nanotubes leading to stochastic fission via a hemifission mechanism. Fission occurs at elongation rates and pulling force ranges intrinsic to the ER, thus suggesting a principle whereby the dynamic balance between fusion and fission controlling organelle morphology depends on membrane motility.

Domain formation in membranes caused by lipid wetting of protein.

Formation of rafts and other domains in cell membranes is considered as wetting of proteins by lipids. The membrane is modeled as a continuous elastic medium. Thermodynamic functions of the lipid films that wet proteins are calculated using a mean-field theory of liquid crystals as adapted to biomembranes. This approach yields the conditions necessary for a macroscopic wetting film to form; its thickness could also be determined. It is shown that films of macroscopic thicknesses form around large (tens nanometers in diameter) lipid-protein aggregates; only thin adsorption films form around single proteins or small complexes. The means by which wetting films can facilitate the merger of these aggregates is considered. It is shown that a wetting film prevents a protein from leaving an aggregate. Using experimentally derived values of elastic moduli and spontaneous curvatures as well as height mismatch between aggregates and bulk membrane, we obtained numerical results, which can be compared with the experimental data.

Lateral tension increases the line tension between two domains in a lipid bilayer membrane.

The effect of an external applied lateral tension on the line tension between two domains of different thickness in a lipid bilayer membrane is calculated. The thick domain is treated as a liquid-ordered phase in order to model a raft in a biological membrane; the thin domain is considered a liquid-disordered phase to model the surrounding region. In our model, the monolayers elastically distort at the boundary to create a smooth rather than steplike boundary to avoid exposure of the hydrophobic interior of the thick raft to water. The energy of this distortion is described by the fundamental deformations of splay and tilt. This energy per unit length of boundary yields the line tension of the raft. Applying lateral tension alters the fundamental deformations such that line tension increases. This increase in line tension is larger when the spontaneous curvature of a raft is greater than that of the surround; if the spontaneous curvature of the raft is less than that of the surround, the increase of the line tension due to application of the lateral tension is more modest.

Pore formation in lipid membrane II: Energy landscape under external stress.

Lipid membranes are extremely stable envelopes allowing cells to survive in various environments and to maintain desired internal composition. Membrane permeation through formation of transversal pores requires substantial external stress. Practically, pores are usually formed by application of lateral tension or transmembrane voltage. Using the same approach as was used for obtaining continuous trajectory of pore formation in the stress-less membrane in the previous article, we now consider the process of pore formation under the external stress. The waiting time to pore formation proved a non-monotonous function of the lateral tension, dropping from infinity at zero tension to a minimum at the tension of several millinewtons per meter. Transmembrane voltage, on the contrary, caused the waiting time to decrease monotonously. Analysis of pore formation trajectories for several lipid species with different spontaneous curvatures and elastic moduli under various external conditions provided instrumental insights into the mechanisms underlying some experimentally observed phenomena.

Pore formation in lipid membrane I: Continuous reversible trajectory from intact bilayer through hydrophobic defect to transversal pore.

Lipid membranes serve as effective barriers allowing cells to maintain internal composition differing from that of extracellular medium. Membrane permeation, both natural and artificial, can take place via appearance of transversal pores. The rearrangements of lipids leading to pore formation in the intact membrane are not yet understood in details. We applied continuum elasticity theory to obtain continuous trajectory of pore formation and closure, and analyzed molecular dynamics trajectories of pre-formed pore reseal. We hypothesized that a transversal pore is preceded by a hydrophobic defect: intermediate structure spanning through the membrane, the side walls of which are partially aligned by lipid tails. This prediction was confirmed by our molecular dynamics simulations. Conversion of the hydrophobic defect into the hydrophilic pore required surmounting some energy barrier. A metastable state was found for the hydrophilic pore at the radius of a few nanometers. The dependence of the energy on radius was approximately quadratic for hydrophobic defect and small hydrophilic pore, while for large radii it depended on the radius linearly. The pore energy related to its perimeter, line tension, thus depends of the pore radius. Calculated values of the line tension for large pores were in quantitative agreement with available experimental data.

Line tension and interaction energies of membrane rafts calculated from lipid splay and tilt.

Membrane domains known as rafts are rich in cholesterol and sphingolipids, and are thought to be thicker than the surrounding membrane. If so, monolayers should elastically deform so as to avoid exposure of hydrophobic surfaces to water at the raft boundary. We calculated the energy of splay and tilt deformations necessary to avoid such hydrophobic exposure. The derived value of energy per unit length, the line tension gamma, depends on the elastic moduli of the raft and the surrounding membrane; it increases quadratically with the initial difference in thickness between the raft and surround; and it is reduced by differences, either positive or negative, in spontaneous curvature between the two. For zero spontaneous curvature, gamma is approximately 1 pN for a monolayer height mismatch of approximately 0.3 nm, in agreement with experimental measurement. Our model reveals conditions that could prevent rafts from forming, and a mechanism that can cause rafts to remain small. Prevention of raft formation is based on our finding that the calculated line tension is negative if the difference in spontaneous curvature for a raft and the surround is sufficiently large: rafts cannot form if gamma < 0 unless molecular interactions (ignored in the model) are strong enough to make the total line tension positive. Control of size is based on our finding that the height profile from raft to surround does not decrease monotonically, but rather exhibits a damped, oscillatory behavior. As an important consequence, the calculated energy of interaction between rafts also oscillates as it decreases with distance of separation, creating energy barriers between closely apposed rafts. The height of the primary barrier is a complex function of the spontaneous curvatures of the raft and the surround. This barrier can kinetically stabilize the rafts against merger. Our physical theory thus quantifies conditions that allow rafts to form, and further, defines the parameters that control raft merger.