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.     

Coupling field theory with mesoscopic dynamical simulations of multicomponent lipid bilayers

A method for simulating a two-component lipid bilayer membrane in the mesoscopic regime is presented. The membrane is modeled as an elastic network of bonded points; the spring constants of these bonds are parameterized by the microscopic bulk modulus estimated from earlier atomistic nonequilibrium molecular dynamics simulations for several bilayer mixtures of DMPC and cholesterol. The modulus depends on the composition of a point in the elastic membrane model. The dynamics of the composition field is governed by the Cahn-Hilliard equation where a free energy functional models the coupling between the composition and curvature fields. The strength of the bonds in the elastic network are then modulated noting local changes in the composition and using a fit to the nonequilibrium molecular dynamics simulation data. Estimates for the magnitude and sign of the coupling parameter in the free energy model are made treating the bending modulus as a function of composition. A procedure for assigning the remaining parameters in the free energy model is also outlined. It is found that the square of the mean curvature averaged over the entire simulation box is enhanced if the strength of the bonds in the elastic network are modulated in response to local changes in the composition field. We suggest that this simulation method could also be used to determine if phase coexistence affects the stress response of the membrane to uniform dilations in area. This response, measured in the mesoscopic regime, is already known to be conditioned or renormalized by thermal undulations.     

Membrane Mechanics of Endocytosis in Cells with Turgor

Endocytosis is an essential process by which cells internalize a piece of plasma membrane and material from the outside. In cells with turgor, pressure opposes membrane deformations, and increases the amount of force that has to be generated by the endocytic machinery. To determine this force, and calculate the shape of the membrane, we used physical theory to model an elastic surface under pressure. Accurate fits of experimental profiles are obtained assuming that the coated membrane is highly rigid and preferentially curved at the endocytic site. The forces required from the actin machinery peaks at the onset of deformation, indicating that once invagination has been initiated, endocytosis is unlikely to stall before completion. Coat proteins do not lower the initiation force but may affect the process by the curvature they induce. In the presence of isotropic curvature inducers, pulling the tip of the invagination can trigger the formation of a neck at the base of the invagination. Hence direct neck constriction by actin may not be required, while its pulling role is essential. Finally, the theory shows that anisotropic curvature effectors stabilize membrane invaginations, and the loss of crescent-shaped BAR domain proteins such as Rvs167 could therefore trigger membrane scission.     

A Multiscale Red Blood Cell Model with Accurate Mechanics, Rheology, and Dynamics

Red blood cells (RBCs) have highly deformable viscoelastic membranes exhibiting complex rheological response and rich hydrodynamic behavior governed by special elastic and bending properties and by the external/internal fluid and membrane viscosities. We present a multiscale RBC model that is able to predict RBC mechanics, rheology, and dynamics in agreement with experiments. Based on an analytic theory, the modeled membrane properties can be uniquely related to the experimentally established RBC macroscopic properties without any adjustment of parameters. The RBC linear and nonlinear elastic deformations match those obtained in optical-tweezers experiments. The rheological properties of the membrane are compared with those obtained in optical magnetic twisting cytometry, membrane thermal fluctuations, and creep followed by cell recovery. The dynamics of RBCs in shear and Poiseuille flows is tested against experiments and theoretical predictions, and the applicability of the latter is discussed. Our findings clearly indicate that a purely elastic model for the membrane cannot accurately represent the RBC's rheological properties and its dynamics, and therefore accurate modeling of a viscoelastic membrane is necessary.     

The influence of curvature on membrane domains

An interdependence between local curvature and domain formation has been observed in both cell and model membranes. An implication of this observation is that domain formation in model membranes may be modulated by membrane curvature. In this paper, small-angle neutron scattering (SANS) is used to examine the influence of membrane curvature (i.e., vesicle size) on the formation of membrane domains. It is found that, although vesicle size and polydispersity are not significantly altered by the formation of membrane domains, the area fraction occupied by domains depends on the overall vesicle size. In particular, increasing membrane curvature (i.e., decreasing vesicle size) results in increased area fractions of membrane domains.     

A consistent model for thermal fluctuations and protein-induced deformations in lipid bilayers

We present an elastic Hamiltonian for membrane energetics that captures bilayer undulation and peristaltic deformations over all wavelengths, including the short wavelength protrusion regime. The model implies continuous functional forms for thermal undulation and peristaltic amplitudes as a function of wavelength and predicts previously overlooked relationships between these curves. Undulation and peristaltic spectra display excellent agreement with data from both atomistic and coarse-grained models over all simulated length scales. Additionally, the model accurately predicts the bilayer's response to a cylindrical protein inclusion as observed in coarse-grained simulation. This elastic response provides an explanation for gramicidin ion channel lifetime versus membrane thickness data that requires no fit constants. The physical parameters inherent to this picture may be expressed in terms of familiar material properties associated with lipid monolayers. Inclusion of a finite monolayer spontaneous curvature is essential to obtain fully consistent agreement between theory and the full range of available simulation/experimental data.     

Confocal microscopy-based three-dimensional cell-specific modeling for large deformation analyses in cellular mechanics

This study introduces a new confocal microscopy-based three-dimensional cell-specific finite element (FE) modeling methodology for simulating cellular mechanics experiments involving large cell deformations. Three-dimensional FE models of undifferentiated skeletal muscle cells were developed by scanning C2C12 myoblasts using a confocal microscope, and then building FE model geometries from the z-stack images. Strain magnitudes and distributions in two cells were studied when the cells were subjected to compression and stretching, which are used in pressure ulcer and deep tissue injury research to induce large cell deformations. Localized plasma membrane and nuclear surface area (NSA) stretches were observed for both the cell compression and stretching simulation configurations. It was found that in order to induce large tensile strains (>5%) in the plasma membrane and NSA, one needs to apply more than ～15% of global cell deformation in cell compression tests, or more than ～3% of tensile strains in the elastic plate substrate in cell stretching experiments. Utilization of our modeling can substantially enrich experimental cellular mechanics studies in classic cell loading designs that typically involve large cell deformations, such as static and cyclic stretching, cell compression, micropipette aspiration, shear flow and hydrostatic pressure, by providing magnitudes and distributions of the localized cellular strains specific to each setup and cell type, which could then be associated with the applied stimuli.     

The mouth of a dense-core vesicle opens and closes in a concerted action regulated by calcium and amphiphysin

Secretion of hormones and peptides by neuroendocrine cells occurs through fast and slow modes of vesicle fusion but the mechanics of these processes are not understood. We used interference reflection microscopy to monitor deformations of the membrane surface and found that both modes of fusion involve the tightly coupled dilation and constriction of the vesicle. The rate of opening is calcium dependent and occurs rapidly at concentrations <5 μM. The fast mode of fusion is blocked selectively by a truncation mutant of amphiphysin. Vesicles do not collapse when fusion is triggered by strontium, rather they remain locked open and membrane scission is blocked. In contrast, constriction of the vesicle opening continues when endocytosis is blocked by inhibiting the function of dynamin. Thus, fast and slow modes of fusion involve similar membrane deformations and vesicle closure can be uncoupled from membrane scission. Regulation of these processes by calcium and amphiphysin may provide a mechanism for controlling the release of vesicle contents.     

The mouth of a dense-core vesicle opens and closes in a concerted action regulated by calcium and amphiphysin

Secretion of hormones and peptides by neuroendocrine cells occurs through fast and slow modes of vesicle fusion but the mechanics of these processes are not understood. We used interference reflection microscopy to monitor deformations of the membrane surface and found that both modes of fusion involve the tightly coupled dilation and constriction of the vesicle. The rate of opening is calcium dependent and occurs rapidly at concentrations <5 microM. The fast mode of fusion is blocked selectively by a truncation mutant of amphiphysin. Vesicles do not collapse when fusion is triggered by strontium, rather they remain locked open and membrane scission is blocked. In contrast, constriction of the vesicle opening continues when endocytosis is blocked by inhibiting the function of dynamin. Thus, fast and slow modes of fusion involve similar membrane deformations and vesicle closure can be uncoupled from membrane scission. Regulation of these processes by calcium and amphiphysin may provide a mechanism for controlling the release of vesicle contents.     

Nonlinear sorting, curvature generation, and crowding of endophilin N-BAR on tubular membranes

The curvature of biological membranes is controlled by membrane-bound proteins. For example, during endocytosis, the sorting of membrane components, vesicle budding, and fission from the plasma membrane are mediated by adaptor and accessory proteins. Endophilin is a peripherally binding membrane protein that functions as an endocytic accessory protein. Endophilin's membrane tubulation capacity is well known. However, to understand the thermodynamic and mechanical aspects of endophilin function, experimental measurements need to be compared to quantitative theoretical models. We present measurements of curvature sorting and curvature generation of the endophilin A1 N-BAR domain on tubular membranes pulled from giant unilamellar vesicles. At low concentration, endophilin functions primarily as a membrane curvature sensor; at high concentrations, it also generates curvature. We determine the spontaneous curvature induced by endophilin and observe sigmoidal curvature/composition coupling isotherms that saturate at high membrane tensions and protein solution concentrations. The observation of saturation is supported by a strong dependence of lateral diffusion coefficients on protein density on the tether membrane. We develop a nonlinear curvature/composition coupling model that captures our experimental observations. Our model predicts a curvature-induced phase transition among two states with varying protein density and membrane curvature. This transition could act as a switch during endocytosis.     

Contributions of Gaussian curvature and nonconstant lipid volume to protein deformation of lipid bilayers

An elastic model for membrane deformations induced by integral membrane proteins is presented. An earlier theory is extended to account for nonvanishing saddle splay modulus within lipid monolayers and perturbations to lipid volume proximal to the protein. Analytical results are derived for the deformation profile surrounding a single cylindrical protein inclusion, which compare favorably to coarse-grained simulations over a range of protein sizes. Numerical results for multi-protein systems indicate that membrane-mediated interactions between inclusions are strongly affected by Gaussian curvature and display nonpairwise additivity. Implications for the aggregation of proteins are discussed.     

Simulation of gravitational field variation on fluid-filled biological membranes

The knowledge of the behavior of biological organs in a gravitational field is important to understand the functioning of the human body in the aerospace environment. The disturbances in biological transport processes in microgravity have indicated adverse effects on humans engaged in space operations. The relationship between the deformations in the biological organs and the transport phenomena that take place in them has been long established and widely reported in biological sciences and engineering literature. A number of soft tissue organs such as brain, lungs, heart, kidney, bladder, stomach, and the circulatory system can be modeled as fluid-filled membranes. In this investigation, a mathematical model of a fluid-filled biological membrane is developed, and its deformation and spatial configuration in a variable gravitational field are calculated. The variation in the gravitational field in the range 1g to zero-g is simulated by partial submergence of the fluid-filled membrane which, by virtue of buoyancy, gains an effective density as if it is in a different gravitational field. The equations of motion are derived using the theory of large elastic deformations and numerically solved in conjunction with a constitutive equation suitably selected for the biological membrane.     

Slow Sedimentation and Deformability of Charged Lipid Vesicles

The study of vesicles in suspension is important to understand the complicated dynamics exhibited by cells in in vivo and in vitro. We developed a computer simulation based on the boundary-integral method to model the three dimensional gravity-driven sedimentation of charged vesicles towards a flat surface. The membrane mechanical behavior was modeled using the Helfrich Hamiltonian and near incompressibility of the membrane was enforced via a model which accounts for the thermal fluctuations of the membrane. The simulations were verified and compared to experimental data obtained using suspended vesicles labelled with a fluorescent probe, which allows visualization using fluorescence microscopy and confers the membrane with a negative surface charge. The electrostatic interaction between the vesicle and the surface was modeled using the linear Derjaguin approximation for a low ionic concentration solution. The sedimentation rate as a function of the distance of the vesicle to the surface was determined both experimentally and from the computer simulations. The gap between the vesicle and the surface, as well as the shape of the vesicle at equilibrium were also studied. It was determined that inclusion of the electrostatic interaction is fundamental to accurately predict the sedimentation rate as the vesicle approaches the surface and the size of the gap at equilibrium, we also observed that the presence of charge in the membrane increases its rigidity.     

Prefission constriction of Golgi tubular carriers driven by local lipid metabolism: a theoretical model

Membrane transport within mammalian cells is mediated by small vesicular as well as large pleiomorphic transport carriers (TCs). A major step in the formation of TCs is the creation and subsequent narrowing of a membrane neck connecting the emerging carrier with the initial membrane. In the case of small vesicular TCs, neck formation may be directly induced by the coat proteins that cover the emerging vesicle. However, the mechanism underlying the creation and narrowing of a membrane neck in the generation of large TCs remains unknown. We present a theoretical model for neck formation based on the elastic model of membranes. Our calculations suggest a lipid-driven mechanism with a central role for diacylglycerol (DAG). The model is applied to a well-characterized in vitro system that reconstitutes TC formation from the Golgi complex, namely the pearling and fission of Golgi tubules induced by CtBP/BARS, a protein that catalyzes the conversion of lysophosphatidic acid into phosphatidic acid. In view of the importance of a PA-DAG cycle in the formation of Golgi TCs, we assume that the newly formed phosphatidic acid undergoes rapid dephosphorylation into DAG. DAG possesses a unique molecular shape characterized by an extremely large negative spontaneous curvature, and it redistributes rapidly between the membrane monolayers and along the membrane surface. Coupling between local membrane curvature and local lipid composition results, by mutual enhancement, in constrictions of the tubule into membrane necks, and a related inhomogeneous lateral partitioning of DAG. Our theoretical model predicts the exact dimensions of the constrictions observed in the pearling Golgi tubules. Moreover, the model is able to explain membrane neck formation by physiologically relevant mole fractions of DAG.     

Increased vesicle endocytosis due to an increase in the plasma membrane phosphatidylserine concentration.

Endocytosis vesiculation consists of local membrane invaginations, continuously generated on the plasma membrane surface of living cells. This vesiculation process was found to be activated in vivo by the generation of a transmembrane surface area asymmetry in the plasma membrane bilayer, after enhancement of transbilayer phospholipid translocation. The observed enhancement was shown to be in good quantitative agreement with a theoretical model of elastic equilibrium describing stabilization of 100-nm vesicles in response to phospholipid redistribution. Very rapid dynamic vesiculation and direct re-fusion of the vesicles, both dependent on the phospholipid translocation activity, were found on a time scale of seconds. Both vesiculation and re-fusion were shown to result in a steady-state population of internal vesicles at long time points. The plasma membrane appears to be a dynamic structure, oscillating between two distinct curvature states, the 10 microns-1 "vesicle" and the 0.1 micron-1 "plasma membrane" curvature states. This dynamic behavior is discussed in terms of an elastic control of the membranes curvature state by the phospholipid translocation activity.     

Frequency-dependent electrodeformation of giant phospholipid vesicles in AC electric field

A model of vesicle electrodeformation is described which obtains a parametrized vesicle shape by minimizing the sum of the membrane bending energy and the energy due to the electric field. Both the vesicle membrane and the aqueous media inside and outside the vesicle are treated as leaky dielectrics, and the vesicle itself is modeled as a nearly spherical shape enclosed within a thin membrane. It is demonstrated (a) that the model achieves a good quantitative agreement with the experimentally determined prolate-to-oblate transition frequencies in the kilohertz range and (b) that the model can explain a phase diagram of shapes of giant phospholipid vesicles with respect to two parameters: the frequency of the applied alternating current electric field and the ratio of the electrical conductivities of the aqueous media inside and outside the vesicle, explored in a recent paper (S. Aranda et al., Biophys J 95:L19–L21, 2008). A possible use of the frequency-dependent shape transitions of phospholipid vesicles in conductometry of microliter samples is discussed.     

The Vesicle Trafficking Protein Sar1 Lowers Lipid Membrane Rigidity

The sculpting of membranes into dynamic, curved shapes is central to intracellular cargo trafficking. Though the generation of membrane curvature during trafficking necessarily involves both lipids and membrane-associated proteins, current mechanistic views focus primarily on the formation of rigid cages and curved scaffolds by protein assemblies. Here we report on a different mechanism for the control of membrane deformation, unrelated to the imposition of predefined curvature, involving modulation of membrane material properties: Sar1, a GTPase that regulates vesicle trafficking from the endoplasmic reticulum, lowers the rigidity of the lipid bilayer membrane to which it binds. In vitro assays in which optically trapped microspheres create controlled membrane deformations revealed a monotonic decline in bending modulus as a function of Sar1 concentration, down to nearly zero rigidity, indicating a dramatic lowering of the energetic cost of curvature generation. This is the first demonstration that a vesicle trafficking protein lowers the rigidity of its target membrane, leading to a new conceptual framework for vesicle biogenesis.     

Theory of cation-phospholipid-induced shape changes in a lipid bilayer couple

A quantitative model of ion binding and molecular interactions in the lipid bilayer membrane is proposed and found to be useful in examining the factors underlying such membrane characteristics as shape, sidedness, stability and vesicle size at various cation concentrations. The lipid membrane behaves as a bilayer couple whose preferential radius of curvature depends on the expansion or contraction of one monolayer relative to the other. It is proposed that molecular packing may be altered by electrostatic repulsion of adjacent like-charged phospholipid headgroups, or by bringing two headgroups closer together by divalent cation crossbridging. The surface concentrations of each type of cation-phospholipid complex can be described by simple binding equilibria and the Gouy-Chapman-Stern formulation for the surface potential in a diffuse double layer. The asymmetric distribution of acidic phospholipids in most biological membranes can account for the differential effects of identical ionic environments on either side of the bilayer. The fraction of vesicle material which tends to have a right-side-out orientation may be approximated by a normal distribution about the mean curvature. The theory generates vesicle sidedness distributions that, when fitted to experimental results from human erythrocyte membranes, provide an alternative method of estimating intrinsic cationphospholipid dissociation constants and other molecular parameters of the bilayer. The results also corroborate earlier suggestions that the Gouy-Chapman theory tends to overestimate free counter-ion concentrations at the surface under large surface potentials.     

Roles of Amphipathic Helices and the Bin/Amphiphysin/Rvs (BAR) Domain of Endophilin in Membrane Curvature Generation

Control of membrane curvature is required in many important cellular processes, including endocytosis and vesicular trafficking. Endophilin is a bin/amphiphysin/rvs (BAR) domain protein that induces vesicle formation by promotion of membrane curvature through membrane binding as a dimer. Using site-directed spin labeling and EPR spectroscopy, we show that the overall BAR domain structure of the rat endophilin A1 dimer determined crystallographically is maintained under predominantly vesiculating conditions. Spin-labeled side chains on the concave surface of the BAR domain do not penetrate into the acyl chain interior, indicating that the BAR domain interacts only peripherally with the surface of a curved bilayer. Using a combination of EPR data and computational refinement, we determined the structure of residues 63–86, a region that is disordered in the crystal structure of rat endophilin A1. Upon membrane binding, residues 63–75 in each subunit of the endophilin dimer form a slightly tilted, amphipathic α-helix that directly interacts with the membrane. In their predominant conformation, these helices are located orthogonal to the long axis of the BAR domain. In this conformation, the amphipathic helices are positioned to act as molecular wedges that induce membrane curvature along the concave surface of the BAR domain.     

Role of the membrane cortex in neutrophil deformation in small pipets.

The simplest model for a neutrophil in its "passive" state views the cell as consisting of a liquid-like cytoplasmic region surrounded by a membrane. The cell surface is in a state of isotropic contraction, which causes the cell to assume a spherical shape. This contraction is characterized by the cortical tension. The cortical tension shows a weak area dilation dependence, and it determines the elastic properties of the cell for small curvature deformations. At high curvature deformations in small pipets (with internal radii less than 1 micron), the measured critical suction pressure for cell flow into the pipet is larger than its estimate from the law of Laplace. A model is proposed where the region consisting of the cytoplasm membrane and the underlying cortex (having a finite thickness) is introduced at the cell surface. The mechanical properties of this region are characterized by the apparent cortical tension (defined as a free contraction energy per unit area) and the apparent bending modulus (introduced as a bending free energy per unit area) of its middle plane. The model predicts that for small curvature deformations (in pipets having radii larger than 1.2 microns) the role of the cortical thickness and the resistance for bending of the membrane-cortex complex is negligible. For high curvature deformations, they lead to elevated suction pressures above the values predicted from the law of Laplace. The existence of elevated suction pressures for pipets with radii from 1 micron down to 0.24 micron is found experimentally. The measured excess suction pressures cannot be explained only by the modified law of Laplace (for a cortex with finite thickness and negligible bending resistance), because it predicts unacceptable high cortical thicknesses (from 0.3 to 0.7 micron). It is concluded that the membrane-cortex complex has an apparent bending modulus from 1 x 10(-18) to 2 x 10(-18) J for a cortex with a thickness from 0.1 micron down to values much smaller than the radius of the smallest pipet (0.24 micron) used in this study.