Membrane bending energy and shape determination of phospholipid vesicles and red blood cells

A procedure is developed to calculate red blood cell and phospholipid vesicle shapes within the bilayer couple model of the membrane. The membrane is assumed to consist of two laterally incompressible leaflets which are in close contact but unconnected. Shapes are determined by minimizing the membrane bending energy at a given volume of a cell (V), given average membrane area (A) and given difference of the areas of two leaflets (A). Different classes of shapes exist in parts of the v/a phase diagram, where v and a are the volume and the leaflet area difference relative to the sphere with area A. The limiting shapes are composed of sections of spheres with only two values allowed for their radii. Two low energy axisymmetrical classes, which include discocyte and stomatocyte shapes are studied and their phase diagrams are analyzed. For v=0.6, the discocyte is the lowest energy shape, which transforms by decreasing a continuously into a stomatocyte. The spontaneous membrane curvature (C ₀) and compressibility of membrane leaflest can be incorporated into the model.A model, where A is free and C ₀ determines the shapes at given V and A, is also studied. In this case, by decreasing C ₀, a discocyte transforms discontinuously into an almost closed stomatocyte.     

Elastic energy of the discocyte–stomatocyte transformation

The aim of this study is to calculate the membrane elastic energy for the different shapes observed in the discocyte–stomatocyte sequence. This analysis can provide a better quantitative understanding of the hypothesis put forward over the last decades to explain how red blood cells produce and maintain their typical shape. For this purpose, we use geometrical models based on parametric equations. The energy model considered for the elastic properties of RBC membrane includes the local and nonlocal resistance effects of the bilayer to bending. In particular, the results confirm the discocyte as the lowest energy value configuration among the sets of different red blood cell deformations considered in the sequence.     

Shape and elasticity effects on erythrocyte electrostatic repulsion

It is shown that the double-layer interaction between two symmetrically parallel red blood cells depends largely on their shape and elasticity. Various shapes are examined, including the realistic shape of a biconcave discocyte with an elastic membrane. It is found that changes in the shape and the elasticity of the erythrocyte may result in orders of magnitude differences in the forces and energies of pair interaction.     

Electrostatic free energy and spontaneous curvature of spherical charged layered membrane

A biophysical model for the equilibrium curvature of a composite membrane element is derived taking into account the mechanical bilayer properties and the adjacent charged protein layers. The minimum of the total free energy density with respect to the curvature of such a membrane curved was estimated from the sum of the electrostatic free energy density of the charges of the membrane and the elastic surface energy density due to bending the lipid bilayer membrane. It was shown that the equilibrium curvature, i.e. the spontaneous curvature, of such a charged composite sandwich-like membrane depends inversely on the bending stiffness of the lipid membrane itself and directly on the charge amount inside and outside the membrane to the second power. Furthermore the geometric and electrostatic structure of the protein layers and the physico-chemical environment conditions are involved. Corresponding to the model developed a "standard RBC" membrane element has a negative spontaneous curvature, accounting for a discocyte RBC shape. The shape change from a discocyte to a more stomatocytic shape (increase in the negative spontaneous curvature) after reducing the charges in the glycocalyx is also explained within this model.     

Phospholipid membrane bending as assessed by the shape sequence of giant oblate phospholipid vesicles

Vesicle shape transformations caused by decreasing the difference between the equilibrium areas of membrane monolayers were studied on phospholipid vesicles with small volume to membrane area ratios. Slow transformations of the vesicle shape were induced by lowering of the concentration of lipid monomers in the solution outside the vesicle. The complete sequence of shapes consisted of a string of pearls, and wormlike, starfish, discocyte and stomatocyte shapes. The transformation from discocyte to stomatocyte vesicle shapes was analyzed theoretically to see whether these observations accord with the area difference elasticity (ADE) model. The membrane shape equation and boundary conditions were derived for axisymmetrical shapes for low volume vesicles, part of whose membranes are in contact. Calculated shapes were arranged into a phase diagram. The theory predicts that the transition between discocyte and stomatocyte shapes is discontinuous for relatively high volumes and continuous for low volumes. The calculated shape sequences matched well with the observed ones. By assuming a linear decrease of the equilibrium area difference with time, the ratio between the nonlocal and local bending constants is in agreement with reported values.     

Phospholipid membrane bending as assessed by the shape sequence of giant oblate phospholipid vesicles

Vesicle shape transformations caused by decreasing the difference between the equilibrium areas of membrane monolayers were studied on phospholipid vesicles with small volume to membrane area ratios. Slow transformations of the vesicle shape were induced by lowering of the concentration of lipid monomers in the solution outside the vesicle. The complete sequence of shapes consisted of a string of pearls, and wormlike, starfish, discocyte and stomatocyte shapes. The transformation from discocyte to stomatocyte vesicle shapes was analyzed theoretically to see whether these observations accord with the area difference elasticity (ADE) model. The membrane shape equation and boundary conditions were derived for axisymmetrical shapes for low volume vesicles, part of whose membranes are in contact. Calculated shapes were arranged into a phase diagram. The theory predicts that the transition between discocyte and stomatocyte shapes is discontinuous for relatively high volumes and continuous for low volumes. The calculated shape sequences matched well with the observed ones. By assuming a linear decrease of the equilibrium area difference with time, the ratio between the nonlocal and local bending constants is in agreement with reported values.     

The hydrophobic insertion mechanism of membrane curvature generation by proteins

A wide spectrum of intracellular processes is dependent on the ability of cells to dynamically regulate membrane shape. Membrane bending by proteins is necessary for the generation of intracellular transport carriers and for the maintenance of otherwise intrinsically unstable regions of high membrane curvature in cell organelles. Understanding the mechanisms by which proteins curve membranes is therefore of primary importance. Here we suggest, for the first time to our knowledge, a quantitative mechanism of lipid membrane bending by hydrophobic or amphipathic rodlike inclusions which simulate amphipathic α-helices—structures shown to sculpt membranes. Considering the lipid monolayer matrix as an anisotropic elastic material, we compute the intramembrane stresses and strains generated by the embedded inclusions, determine the resulting membrane shapes, and the accumulated elastic energy. We characterize the ability of an inclusion to bend membranes by an effective spontaneous curvature, and show that shallow rodlike inclusions are more effective in membrane shaping than are lipids having a high propensity for curvature. Our computations provide experimentally testable predictions on the protein amounts needed to generate intracellular membrane shapes for various insertion depths and membrane thicknesses. We also predict that the ability of N-BAR domains to produce membrane tubules in vivo can be ascribed solely to insertion of their amphipathic helices.     

Mechanics of curved plasma membrane vesicles: resting shapes, membrane curvature, and in-plane shear elasticity

Highly curved cell membrane structures, such as plasmalemmal vesicles (caveolae) and clathrin-coated pits, facilitate many cell functions, including the clustering of membrane receptors and transport of specific extracellular macromolecules by endothelial cells. These structures are subject to large mechanical deformations when the plasma membrane is stretched and subject to a change of its curvature. To enhance our understanding of plasmalemmal vesicles we need to improve the understanding of the mechanics in regions of high membrane curvatures. We examine here, theoretically, the shapes of plasmalemmal vesicles assuming that they consist of three membrane domains: an inner domain with high curvature, an outer domain with moderate curvature, and an outermost flat domain, all in the unstressed state. We assume the membrane properties are the same in these domains with membrane bending elasticity as well as in-plane shear elasticity. Special emphasis is placed on the effects of membrane curvature and in-plane shear elasticity on the mechanics of vesicle during unfolding by application of membrane tension. The vesicle shapes were computed by minimization of bending and in-plane shear strain energy. Mechanically stable vesicles were identified with characteristic membrane necks. Upon stretch of the membrane, the vesicle necks disappeared relatively abruptly leading to membrane shapes that consist of curved indentations. While the resting shape of vesicles is predominantly affected by the membrane spontaneous curvatures, the membrane shear elasticity (for a range of values recorded in the red cell membrane) makes a significant contribution as the vesicle is subject to stretch and unfolding. The membrane tension required to unfold the vesicle is sensitive with respect to its shape, especially as the vesicle becomes fully unfolded and approaches a relative flat shape.     

Synergetics of the Membrane Self-Assembly: A Micelle-to-Vesicle Transition

A model approach is developed to study intermediate steps and transientstructures in a course of the membrane self-assembly. The approach isbased on investigation of mixed lipid/protein-detergent systems capable ofthe temperature induced transformation from a solubilized micellar stateto closed membrane vesicles. We performed a theoretical analysis ofself-assembling molecular structures formed in binary mixtures ofdimyristoylphosphatidylcholine (DMPC) and sodium cholate (NaC). Thetheoretical model is based on the Helfrich theory of curvature elasticity,which relates geometrical shapes of the structures to their free energy inthe Ginzburg-Landau approximation. The driving force for the shapetransformation is spontaneous curvature of amphiphilic aggregates which isnonlinearly dependent on the lipid/detergent composition. An analysis ofthe free energy in the regular solution approximation shows that theformation of mixed structures of different shapes (discoidal micelles,rod-like micelles, multilayer membrane structures and vesicles) ispossible in a certain range of detergent/lipid ratios. A transition fromthe flat discoidal micelles to the rod-like cylindrical micelles isinduced by curvature instabilities resulting from acyl chain melting andinsertion of detergent molecules into the lipid phase. Nonideal mixing ofthe NaC and DMPC molecules results in formation of nonideal cylindricalaggregates with elliptical cross section. Further dissolution of NaCmolecules in DMPC may be accompanied with a change of their orientation inthe lipid phase and leads to temperature-induced curvature instabilitiesin the highly curved cylindrical geometry. As a result the rod-likemicelles fuse into less curved bilayer structures which transformeventually to the unilamellar and multilamellar membrane vesicles. Thetheoretical analysis performed shows that a sequence of shapetransformations in the DMPC/NaC mixed systems is determined by thesynergism of four major factors: detergent/lipid ratio, temperature (acylchain melting), DMPC and NaC mixing, and reorientation of NaC molecules inmixed aggregates.     

Diffusion on Membrane Domes,Tubes, and Pearling Structures

Diffusion is a fundamental mechanism for protein distribution in cell membranes. These membranes often exhibit complex shapes, which range from shallow domes to elongated tubular or pearl-like structures. Shape complexity of the membrane influences the diffusive spreading of proteins and molecules. Despite the importance membrane geometry plays in these diffusive processes, it is challenging to establish the dependence between diffusion and membrane morphology. We solve the diffusion equation numerically on various static curved shapes representative for experimentally observed membrane shapes. Our results show that membrane necks become diffusion barriers. We determine the diffusive half-time, i.e., the time that is required to reduce the amount of protein in the budded region by one half, and find a quadratic relation between the diffusive half-time and the averaged mean curvature of the membrane shape, which we rationalize by a scaling law. Our findings thus help estimate the characteristic diffusive timescale based on the simple measure of membrane mean curvature.     

Mechanical Deformability of Biological Membranes and the Sphering of the Erythrocyte

Equations of mechanical equilibrium are applied to the erythrocyte membrane in the normal, hypotonically swollen, and sphered configurations. The hydrostatic pressure drop across the normal cell membrane is shown to be zero for all biconcave shapes if the membrane thickness is uniform. This result leads to the conclusion that the membrane tension is uniform and is a function of membrane potential. A two-dimensional fluid film model for the membrane is introduced to describe the unusual deformability of the erythrocyte during sphering in hypotonic solutions. The model predicts a smooth transition from the biconcave shape to a perfect sphere.     

Exploring the binding dynamics of BAR proteins

We used a continuum model based on the Helfrich free energy to investigate the binding dynamics of a lipid bilayer to a BAR domain surface of a crescent-like shape of positive (e.g. I-BAR shape) or negative (e.g. F-BAR shape) intrinsic curvature. According to structural data, it has been suggested that negatively charged membrane lipids are bound to positively charged amino acids at the binding interface of BAR proteins, contributing a negative binding energy to the system free energy. In addition, the cone-like shape of negatively charged lipids on the inner side of a cell membrane might contribute a positive intrinsic curvature, facilitating the initial bending towards the crescent-like shape of the BAR domain. In the present study, we hypothesize that in the limit of a rigid BAR domain shape, the negative binding energy and the coupling between the intrinsic curvature of negatively charged lipids and the membrane curvature drive the bending of the membrane. To estimate the binding energy, the electric potential at the charged surface of a BAR domain was calculated using the Langevin-Bikerman equation. Results of numerical simulations reveal that the binding energy is important for the initial instability (i.e. bending of a membrane), while the coupling between the intrinsic shapes of lipids and membrane curvature could be crucial for the curvature-dependent aggregation of negatively charged lipids near the surface of the BAR domain. In the discussion, we suggest novel experiments using patch clamp techniques to analyze the binding dynamics of BAR proteins, as well as the possible role of BAR proteins in the fusion pore stability of exovesicles.     

Coarse-grained Brownian ratchet model of membrane protrusion on cellular scale

Membrane protrusion is a mechanochemical process of active membrane deformation driven by actin polymerization. Previously, Brownian ratchet (BR) was modeled on the basis of the underlying molecular mechanism. However, because the BR requires a priori load that cannot be determined without information of the cell shape, it cannot be effective in studies in which resultant shapes are to be solved. Other cellular-scale models describing the protrusion have also been suggested for modeling a whole cell; however, these models were not developed on the basis of coarse-grained physics representing the underlying molecular mechanism. Therefore, to express the membrane protrusion on the cellular scale, we propose a novel mathematical model, the coarse-grained BR (CBR), which is derived on the basis of nonequilibrium thermodynamics theory. The CBR can reproduce the BR within the limit of the quasistatic process of membrane protrusion and can estimate the protrusion velocity consistently with an effective elastic constant that represents the state of the energy of the membrane. Finally, to demonstrate the applicability of the CBR, we attempt to perform a cellular-scale simulation of migrating keratocyte in which the proposed CBR is used for the membrane protrusion model on the cellular scale. The results show that the experimentally observed shapes of the leading edge are well reproduced by the simulation. In addition, The trend of dependences of the protrusion velocity on the curvature of the leading edge, the temperature, and the substrate stiffness also agreed with the other experimental results. Thus, the CBR can be considered an appropriate cellular-scale model to express the membrane protrusion on the basis of its underlying molecular mechanism.     

Asymptotic solution of the model of the erythrocyte shape as an autowave process

An asymptotic solution was plotted for a model of erythrocyte forms assuming that the biomembrane is anisotropic and of "small" thickness. This leads to small non-linearity and low diffusion, therefore the solution is unrelaxational. The model was investigated qualitatively assuming that the liquid current directed inside the spheric membrane induces its "distension", while that directed outside-its "crumpling". In the spherical system of coordinates the lines of solution level at theta = const are circumferences, while at phi-const-trochoids (Pascal coil, for example). Trochoids rotation areas show stomacyte and discocyte forms. Several hypotheses based on the analysis performed are advanced.     

Spontaneous curvature of ganglioside GM1--effect of cross-linking

The membrane-curvature dependent lateral distribution of outer leaflet ganglioside GM1 (GM1) and the influence of GM1 cross-linking induced by fluorophore-tagged cholera toxin subunit B (CTB) plus anti-CTB was analysed in cell membranes by fluorescence microscopy. Data are presented indicating that cross-linked GM1-ligand patches accumulated at the tips of human erythrocyte echinocytic spiculae induced by Ca(2+)/ionophore A23187. However, when lipid fixative osmium tetroxide was added prior to the ligand no accumulation in spiculae occurred. GM1-staining remained here distributed over the spheroid cell body and in spiculae. Similarly, osmium tetroxide completely prohibited CTB plus anti-CTB-induced GM1 patching in representatives for flat membrane, i.e. discoid erythrocytes and K562 cells. Our results demonstrate that GM1 per se shows low membrane curvature dependent distribution and therefore holds flexible spontaneous curvature. In contrast, the cross-linked GM1-ligand complex has a strong preference for highly outward curved membrane and possesses overall positive spontaneous curvature. Osmium tetroxide efficiently immobilises GM1.     

Lateral organization of membranes and cell shapes.

The relations among membrane structure, mechanical properties, and cell shape have been investigated. The fluid mosaic membrane models used contains several components that move freely in the membrane plane. These components interact with each other and determine properties of the membrane such as curvature and elasticity. A free energy equation is postulated for such a multicomponent membrane and the condition of free energy minimum is used to obtain differential equations relating the distribution of membrane components and the local membrane curvature. The force that moves membrane components along the membrane in a variable curvature field is calculated. A change in the intramembrane interactions can bring about phase separation or particle clustering. This, in turn, may strongly affect the local curvature. The numerical solution of the set of equations for the two dimensional case allows determination of the cell shape and the component distribution along the membrane. The model has been applied to describe certain erythrocytes shape transformations.     

Physical Model of the Dynamic Instability in an Expanding Cell Culture

Collective cell migration is of great significance in many biological processes. The goal of this work is to give a physical model for the dynamics of cell migration during the wound healing response. Experiments demonstrate that an initially uniform cell-culture monolayer expands in a nonuniform manner, developing fingerlike shapes. These fingerlike shapes of the cell culture front are composed of columns of cells that move collectively. We propose a physical model to explain this phenomenon, based on the notion of dynamic instability. In this model, we treat the first layers of cells at the front of the moving cell culture as a continuous one-dimensional membrane (contour), with the usual elasticity of a membrane: curvature and surface-tension. This membrane is active, due to the forces of cellular motility of the cells, and we propose that this motility is related to the local curvature of the culture interface; larger convex curvature correlates with a stronger cellular motility force. This shape-force relation gives rise to a dynamic instability, which we then compare to the patterns observed in the wound healing experiments.     

The human erythrocyte membrane skeleton may be an ionic gel

In the first paper in this series (Stokke et al. Eur Biophys J 1986, 13:203-218) we developed the general theory of the mechanochemical properties and the elastic free energy of the protein gel--lipid bilayer membrane model. Here we report on an extensive numerical analysis of the human erythrocyte shapes and shape transformations predicted by this new cell membrane model. We have calculated the total elastic free energy of deformation of four different cell shape classes: disc-shaped cells, cup-shaped cells, crenated cells, and cells with membrane invaginations. We find that which of these shape classes is favoured depends strongly on the spectrin gel osmotic tension, IIGu, and the surface tensions, IIEu and IIPu, of the extracellular and protoplasmic halves of the membrane lipid bilayer, respectively. For constant ratio IIEu/IIPu greater than O large negative or positive values of IIGu favour respectively the crenated and invaginated cell shape classes. For small absolute values of IIGu, IIEu, and IIPu, biconcave or cup-shaped cells are the stable ones. Our numerical analysis shows that the higher the membrane skeleton compressibility is, the smaller are the values of IIGu needed to induce cell shape transformation. We find that the stable and metastable shapes of discocytes and stomatocytes generally depend both on the shape of the stressfree membrane skeleton and the membrane skeleton compressibility.     

球形拓扑下对budding和multi-budding生物膜泡形状的研究

流体相磷脂双分子层在水溶液中能自组织的形成球形拓扑的膜泡,膜泡的平衡形状是由其弯曲弹性能决定的.Budding是流体相磷脂双分子膜泡的一个显著的形状.从自发曲率(SC)模型的弯曲能量和双层(BC)模型的弯曲能量在参数替换情况下,我们看到自发曲率(SC)和双层(BC)模型具有相同的形状方程.本文从双层(BC)模型的相图出发,在双层(BC)模型下,通过建立一些与目标形状相近似的初始形状,在面积A,体积V和平均曲率的积分M的约束条件和使用约化量的情况下,经Surface Evolver软件的逐次细分并长时间演化,得到了一些budding和multi-budding型的生物膜泡形状.