DNA-lipid complexes: stability of honeycomb-like and spaghetti-like structures.

A molecular level theory is presented for the thermodynamic stability of two (similar) types of structural complexes formed by (either single strand or supercoiled) DNA and cationic liposomes, both involving a monolayer-coated DNA as the central structural unit. In the "spaghetti" complex the central unit is surrounded by another, oppositely curved, monolayer, thus forming a bilayer mantle. The "honeycomb" complex is a bundle of hexagonally packed DNA-monolayer units. The formation free energy of these complexes, starting from a planar cationic/neutral lipid bilayer and bare DNA, is expressed as a sum of electrostatic, bending, mixing, and (for the honeycomb) chain frustration contributions. The electrostatic free energy is calculated using the Poisson-Boltzmann equation. The bending energy of the mixed lipid layers is treated in the quadratic curvature approximation with composition-dependent bending rigidity and spontaneous curvature. Ideal lipid mixing is assumed within each lipid monolayer. We found that the most stable monolayer-coated DNA units are formed when the charged/neutral lipid composition corresponds (nearly) to charge neutralization; the optimal monolayer radius corresponds to close DNA-monolayer contact. These conclusions are also valid for the honeycomb complex, as the chain frustration energy is found to be negligible. Typically, the stabilization energies for these structures are on the order of 1 k(B)T/A of DNA length, reflecting mainly the balance between the electrostatic and bending energies. The spaghetti complexes are less stable due to the additional bending energy of the external monolayer. A thermodynamic analysis is presented for calculating the equilibrium lipid compositions when the complexes coexist with excess bilayer.     

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.     

Elastic, electrostatic and electrokinetic forces influencing membrane curvature

Many cellular and intracellular processes critically depend on membrane shape, but the shape generating mechanisms are still to be fully understood. In this study we evaluate how electrostatic/electrokinetic forces contribute to membrane curvature. Membrane bilayer had finite thickness and was either elastically anisotropic or anisotropic overall, but isotropic per sections (heads and tails). The physics of the situation was evaluated using a coupled system of elastic and electrostatic/electrokinetic (Poisson-Nernst-Planck) equations. The fixed charges present only on the upper membrane surface lead to the accumulation of counter-ions and depletion of co-ions that decay spatially very rapidly (Debye length<1nm), as does the potential and electric field. Spatially uneven electric field and the permittivity mismatch also induce charges at the membrane-solution interface, which are not fixed but influence the electrostatics nevertheless. Membrane bends due to - Coulomb force (caused by fixed membrane charges in the electric field) and the dielectric force (due to the non-uniform electric field and the permittivity mismatch between the membrane and the solution). Both act as membrane surface forces, and both depend supra-linearly on the fixed charge density. Regardless of sign of the fixed charges, the membrane bends toward the charged (upper) surface owing to the action of the Coulomb force, but this is opposed by the smaller dielectric force. The spontaneous membrane curvature becomes very pronounced at high fixed charge densities, leading to very small spontaneous radii (<50nm). In conclusion the electrostatic/electrokinetic forces contribute significantly to the membrane curvature.     

Modeling Membrane Deformations and Lipid Demixing upon Protein-Membrane Interaction: The BAR Dimer Adsorption

We use a self-consistent mean-field theory, designed to investigate membrane reshaping and lipid demixing upon interaction with proteins, to explore BAR domains interacting with large patches of lipid membranes of heterogeneous compositions. The computational model includes contributions to the system free energy from electrostatic interactions and elastic energies of the membrane, as well as salt and lipid mixing entropies. The results from our simulation of a single adsorbing Amphiphysin BAR dimer indicate that it is capable of stabilizing a significantly curved membrane. However, we predict that such deformations will occur only for membrane patches that have the inherent propensity for high curvature, reflected in the tendency to create local distortions that closely match the curvature of the BAR dimer itself. Such favorable preconditioning for BAR-membrane interaction may be the result of perturbations such as local lipid demixing induced by the interaction, or of a prior insertion of the BAR domain's amphiphatic N-helix. From our simulations it appears that local segregation of charged lipids under the influence of the BAR dimer cannot produce high enough asymmetry between bilayer leaflets to induce significant bending. In the absence of additional energy contributions that favor membrane asymmetry, the membrane will remain nearly flat upon single BAR dimer adsorption, relative to the undulation expected from thermal fluctuations. Thus, we conclude that the N-helix insertions have a critical mechanistic role in the local perturbation and curving of the membrane, which is then stabilized by the electrostatic interaction with the BAR dimer. We discuss how these results can be used to estimate the tendency of BARs to bend membranes in terms of a spatially nonisotropic spontaneous curvature.     

Lipid demixing and protein-protein interactions in the adsorption of charged proteins on mixed membranes

The adsorption free energy of charged proteins on mixed membranes, containing varying amounts of (oppositely) charged lipids, is calculated based on a mean-field free energy expression that accounts explicitly for the ability of the lipids to demix locally, and for lateral interactions between the adsorbed proteins. Minimization of this free energy functional yields the familiar nonlinear Poisson-Boltzmann equation and the boundary condition at the membrane surface that allows for lipid charge rearrangement. These two self-consistent equations are solved simultaneously. The proteins are modeled as uniformly charged spheres and the (bare) membrane as an ideal two-dimensional binary mixture of charged and neutral lipids. Substantial variations in the lipid charge density profiles are found when highly charged proteins adsorb on weakly charged membranes; the lipids, at a certain demixing entropy penalty, adjust their concentration in the vicinity of the adsorbed protein to achieve optimal charge matching. Lateral repulsive interactions between the adsorbed proteins affect the lipid modulation profile and, at high densities, result in substantial lowering of the binding energy. Adsorption isotherms demonstrating the importance of lipid mobility and protein-protein interactions are calculated using an adsorption equation with a coverage-dependent binding constant. Typically, at bulk-surface equilibrium (i.e., when the membrane surface is "saturated" by adsorbed proteins), the membrane charges are "overcompensated" by the protein charges, because only about half of the protein charges (those on the hemispheres facing the membrane) are involved in charge neutralization. Finally, it is argued that the formation of lipid-protein domains may be enhanced by electrostatic adsorption of proteins, but its origin (e.g., elastic deformations associated with lipid demixing) is not purely electrostatic.     

Lamellar cationic lipid‐DNA complexes from lipids with a strong preference for planar geometry: A Minimal Electrostatic Model

We formulate and analyze a minimal model, based on condensation theory, of the lamellar cationic lipid (CL)‐DNA complex of alternately charged lipid bilayers and DNA monolayers in a salt solution. Each lipid bilayer, composed by a random mixture of cationic and neutral lipids, is assumed to be a rigid uniformly charged plane. Each DNA monolayer, located between two lipid bilayers, is formed by the same number of parallel DNAs with a uniform separation distance. For the electrostatic calculation, the model lipoplex is collapsed to a single plane with charge density equal to the net lipid and DNA charge. The free energy difference between the lamellar lipoplex and a reference state of the same number of free lipid bilayers and free DNAs, is calculated as a function of the fraction of CLs, of the ratio of the number of CL charges to the number of negative charges of the DNA phosphates, and of the total number of planes. At the isoelectric point the free energy difference is minimal. The complex formation, already favoured by the decrease of the electrostatic charging free energy, is driven further by the free energy gain due to the release of counterions from the DNAs and from the lipid bilayers, if strongly charged. This minimal model compares well with experiment for lipids having a strong preference for planar geometry and with major features of more detailed models of the lipoplex. © 2014 Wiley Periodicals, Inc. Biopolymers 101: 1114–1128, 2014.     

Electroporation of curved lipid membranes in ionic strength gradients

A thermodynamic theory for the membrane electroporation of curved membranes such as those of lipid vesicles and cylindrical membrane tubes has been developed. The theory covers in particular the observation that electric pore formation and shape deformation of vesicles and cells are dependent on the salt concentration of the suspending solvent. It is shown that transmembrane salt gradients can appreciably modify the electrostatic part of Helfrich's spontaneous curvature, elastic bending rigidity and Gaussian curvature modulus of charged membranes. The Gibbs reaction energy of membrane electroporation can be explicitely expressed in terms of salt gradient-dependent contributions of bending, the ionic double layers and electric surface potentials and dielectric polarisation of aqueous pores. In order to cover the various physical contribution to the chemical process of electroporation-resealing, we have introduced a generalised chemophysical potential covering all generalised forces and generalised displacements in terms of a transformed Gibbs energy formalism. Comparison with, and analysis of, the data of electrooptical relaxation kinetic studies show that the Gibbs reaction energy terms can be directly determined from turbidity dichroism (Planck's conservative dichroism). The approach also quantifies the electroporative cross-membrane material exchange such as electrolyte release, electrohaemolysis of red blood cells or uptake of drugs and dyes and finally gene DNA by membrane electroporation.     

Evidence of cholesterol accumulated in high curvature regions: implication to the curvature elastic energy for lipid mixtures

Recent experiments suggested that cholesterol and other lipid components of high negative spontaneous curvature facilitate membrane fusion. This is taken as evidence supporting the stalk-pore model of membrane fusion in which the lipid bilayers go through intermediate structures of high curvature. How do the high-curvature lipid components lower the free energy of the curved structure? Do the high-curvature lipid components modify the average spontaneous curvature of the relevant monolayer, thereby facilitate its bending, or do the lipid components redistribute in the curved structure so as to lower the free energy? This question is fundamental to the curvature elastic energy for lipid mixtures. Here we investigate the lipid distribution in a monolayer of a binary lipid mixture before and after bending, or more precisely in the lamellar, hexagonal, and distorted hexagonal phases. The lipid mixture is composed of 2:1 ratio of brominated di18:0PC and cholesterol. Using a newly developed procedure for the multiwavelength anomalous diffraction method, we are able to isolate the bromine distribution and reconstruct the electron density distribution of the lipid mixture in the three phases. We found that the lipid distribution is homogenous and uniform in the lamellar and hexagonal phases. But in the distorted hexagonal phase, the lipid monolayer has nonuniform curvature, and cholesterol almost entirely concentrates in the high curvature region. This finding demonstrates that the association energies between lipid molecules vary with the curvature of membrane. Thus, lipid components in a mixture may redistribute under conditions of nonuniform curvature, such as in the stalk structure. In such cases, the spontaneous curvature depends on the local lipid composition and the free energy minimum is determined by lipid distribution as well as curvature.     

Electrostatics of lipid bilayer bending.

The electrostatic contribution to spontaneous membrane curvature is calculated within Poisson-Boltzmann theory under a variety of assumptions and emphasizing parameters in the physiological range. Asymmetrical surface charges can be fixed with respect to bilayer midplane area or with respect to the lipid-water area, but induce curvatures of opposite signs. Unequal screening layers on the two sides of a vesicle (e.g., multivalent cationic proteins on one side and monovalent salt on the other) also induce bending. For reasonable parameters, tubules formed by electrostatically induced bending can have radii in the 50-100-nm range, often seen in many intracellular organelles. Thus membrane associated proteins may induce curvature and subsequent budding, without themselves being intrinsically curved. Furthermore, we derive the previously unexplored effects of respecting the strict conservation of charge within the interior of a vesicle. The electrostatic component of the bending modulus is small under most of our conditions and is left as an experimental parameter. The large parameter space of conditions is surveyed in an array of graphs.     

Minimum Membrane Bending Energies of Fusion Pores

Membranes fuse by forming highly curved intermediates, culminating in structures described as fusion pores. These hourglass-like figures that join two fusing membranes have high bending energies, which can be estimated using continuum elasticity models. Fusion pore bending energies depend strongly on shape, and the present study developed a method for determining the shape that minimizes bending energy. This was first applied to a fusion pore modeled as a single surface and then extended to a more realistic model treating a bilayer as two monolayers. For the two-monolayer model, fusion pores were found to have metastable states with energy minima at particular values of the pore diameter and bilayer separation. Fusion pore energies were relatively insensitive to membrane thickness but highly sensitive to spontaneous curvature and membrane asymmetry. With symmetrical bilayers and monolayer spontaneous curvatures of ?0.1 nm^?1 (a typical value) separated by 6 nm (closest distance determined by repulsive hydration forces), fusion pore formation required 43–65 kT. The pore radius of ~2.25 nm fell within the range estimated from conductance measurements. With bilayer separation >6 nm, fusion pore formation required less energy, suggesting that protein scaffolds can promote fusion by bending membranes toward one another. With nonzero spontaneous monolayer curvature, the shape that minimized the energy change during fusion pore formation differed from the shape that minimized its energy after it formed. Thus, a nascent fusion pore will relax spontaneously to a new shape, consistent with the experimentally observed expansion of nascent fusion pores during viral fusion.     

Chained vesicles in vascular endothelial cells.

There is extensive ultrastructural evidence in endothelium for the presence of chained vesicles or clusters of attached vesicles, and they are considered to be involved in specific transport mechanisms, such as the formation of trans-endothelial channels. However, few details are known about their mechanical characteristics. In this study, the formation mechanism and mechanical aspects of vascular endothelial chained vesicles are investigated theoretically, based on membrane bending strain energy analysis. The shape of the axisymmetric vesicles was computed on the assumption that the cytoplasmic side of the vesicle has a molecular layer or cytoskeleton attached to the lipid bilayer, which induces a spontaneous curvature in the resting state. The bending strain energy is the only elasticity involved, while the shear elasticity is assumed to be negligible. The surface area of the membrane is assumed to be constant due to constant lipid bilayer thickness. Mechanically stable shapes of chained vesicles are revealed, in addition to a cylindrical tube shape. Unfolding of vesicles into a more flattened shape is associated with increase in bending energy without a significant increase in membrane tension. These results provide insights into the formation mechanism and mechanics of the chained vesicle.     

Electrostatic free energy and shift of the phase transition for charged lipid membranes

For a charged membrane in an electrolyte solution the electrostatic free energy is derived treating the system as a diffuse double layer. The dependence of the free energy on external parameters like surface charge density and temperature is obtained and the physical basis discussed. As an application the charges are shown to exert an electrostatic surface pressure on the lipid chain packing which leads to a shift in the phase transition of lipid membranes. The results confirm the interpretation of experimental data as given by Träuble et al. in the accompanying paper.     

Molecular theory of lipid-protein interaction and the Lalpha-HII transition.

We present a molecular-level theory for lipid-protein interaction and apply it to the study of lipid-mediated interactions between proteins and the protein-induced transition from the planar bilayer (Lalpha) to the inverse-hexagonal (HII) phase. The proteins are treated as rigid, membrane-spanning, hydrophobic inclusions of different size and shape, e.g., "cylinder-like," "barrel-like," or "vase-like." We assume strong hydrophobic coupling between the protein and its neighbor lipids. This means that, if necessary, the flexible lipid chains surrounding the protein will stretch, compress, and/or tilt to bridge the hydrophobic thickness mismatch between the protein and the unperturbed bilayer. The system free energy is expressed as an integral over local molecular contributions, the latter accounting for interheadgroup repulsion, hydrocarbon-water surface energy, and chain stretching-tilting effects. We show that the molecular interaction constants are intimately related to familiar elastic (continuum) characteristics of the membrane, such as the bending rigidity and spontaneous curvature, as well as to the less familiar tilt modulus. The equilibrium configuration of the membrane is determined by minimizing the free energy functional, subject to boundary conditions dictated by the size, shape, and spatial distribution of inclusions. A similar procedure is used to calculate the free energy and structure of peptide-free and peptide-rich hexagonal phases. Two degrees of freedom are involved in the variational minimization procedure: the local length and local tilt angle of the lipid chains. The inclusion of chain tilt is particularly important for studying noncylindrical (for instance, barrel-like) inclusions and analyzing the structure of the HII lipid phase; e.g., we find that chain tilt relaxation implies strong faceting of the lipid monolayers in the hexagonal phase. Consistent with experiment, we find that only short peptides (large negative mismatch) can induce the Lalpha --> HII transition. At the transition, a peptide-poor Lalpha phase coexists with a peptide-rich HII phase.     

Equilibrium surface charge distribution in phospholipid vesicles. I. Method of calculation

This paper presents a method of calculation of the surface charge equilibrium distribution between the two surfaces of a spherically closed phospholipid bilayer suspended in aqueous electrolyte solution. The net surface charge is supposed to be provided by the ionized polar groups of the phospholipid molecules. Its equilibrium distribution is found by minimization of the free electrostatic energy. The procedure of minimization utilizes the solution of the Poisson-Boltzmann equation which describes the double electric layers of the membrane and an expression for the membrane potential derived under the assumption of absence of charges in the membrane phase. An analytical solution of the problem in the range of validity of the linearized Poisson-Boltzman equation is obtained. It is shown that in this case an equilibrium transmembrane potential exists, and the surface charge density is greater at the outer surface of the vesicle.     

Amphiphile induced echinocyte-spheroechinocyte transformation of red blood cell shape

A possible physical explanation of the echinocyte-spheroechinocyte red blood cell (RBC) shape transformation induced by the intercalation of amphiphilic molecules into the outer layer of the RBC plasma membrane bilayer is given. The stable RBC shape is determined by the minimization of the membrane elastic energy, consisting of the bilayer bending energy, the bilayer relative stretching energy and the skeleton shear elastic energy. It is shown that for a given relative cell volume the calculated number of echinocyte spicula increases while their size decreases as the number of the intercalated amphiphilic molecules in the outer layer of the cell membrane bilayer is increased, which is in agreement with experimental observations. Further, it is shown that the equilibrium difference between the outer and the inner membrane leaflet areas of the stable RBC shapes increases if the amount of the intercalated amphiphiles is increased, thereby verifying theoretically the original bilayer couple hypothesis of Sheetz and Singer (1974) and Evans (1974). Received: 22 August 1997 / Revised version: 25 November 1997 / Accepted: 11 February 1998     

Conformational energetics of rhodopsin modulated by nonlamellar-forming lipids

Rhodopsin is an important example of a G protein-coupled receptor (GPCR) in which 11-cis-retinal is the ligand and acts as an inverse agonist. Photolysis of rhodopsin leads to formation of the activated meta II state from its precursor meta I. Various mechanisms have been proposed to explain how the membrane composition affects the meta I-meta II conformational equilibrium in the visual process. For rod disk membranes and recombinant membranes containing rhodopsin, the lipid properties have been discussed in terms of elastic deformation of the bilayer. Here we have investigated the relation of nonlamellar-forming lipids, such as dioleoylphosphatidylethanolamine (DOPE), together with dioleoylphosphatidylcholine (DOPC), to the photochemistry of membrane-bound rhodopsin. We conducted flash photolysis experiments for bovine rhodopsin recombined with DOPE/DOPC mixtures (0:100 to 75:25) as a function of pH to explore the dependence of the photochemical activity on the monolayer curvature free energy of the membrane. It is well-known that DOPC forms bilayers, whereas DOPE has a propensity to adopt the nonlamellar, reverse hexagonal (H(II)) phase. In the case of neutral DOPE/DOPC recombinants, calculations of the membrane surface pH confirmed that an increase in DOPE favored the meta II state. Moreover, doubling the PE headgroup content versus the native rod membranes substituted for the polyunsaturated, docosahexaenoic acyl chains (22:6 omega 3), suggesting rhodopsin function is associated with a balance of forces within the bilayer. The data are interpreted by applying a flexible surface model, in which the meta II state is stabilized by lipids tending to form the H(II) phase, with a negative spontaneous curvature. A simple theory, based on principles of surface chemistry, for coupling the energetics of membrane proteins to material properties of the bilayer lipids is described. For rhodopsin, the free energy balance of the receptor and the lipids is altered by photoisomerization of retinal and involves curvature stress/strain of the membrane (frustration). A new biophysical principle is introduced: matching of the spontaneous curvature of the lipid bilayer to the mean curvature of the lipid/water interface adjacent to the protein, which balances the lipid/protein solvation energy. In this manner, the thermodynamic driving force for the meta I-meta II conformational change of rhodopsin is tightly controlled by mixtures of nonlamellar-forming lipids having distinctive material properties.     

Molecular Modeling of Lipid Membrane Curvature Induction by a Peptide: More than Simply Shape

Molecular dynamics simulations of an amphipathic helix embedded in a lipid bilayer indicate that it will induce substantial positive curvature (e.g., a tube of diameter 20 nm at 16% surface coverage). The induction is twice that of a continuum model prediction that only considers the shape of the inclusion. The discrepancy is explained in terms of the additional presence of specific interactions described only by the molecular model. The conclusion that molecular shape alone is insufficient to quantitatively model curvature is supported by contrasting molecular and continuum models of lipids with large and small headgroups (choline and ethanolamine, respectively), and of the removal of a lipid tail (modeling a lyso-lipid). For the molecular model, curvature propensity is analyzed by computing the derivative of the free energy with respect to bending. The continuum model predicts that the inclusion will soften the bilayer near the headgroup region, an effect that may weaken curvature induction. The all-atom predictions are consistent with experimental observations of the degree of tubulation by amphipathic helices and variation of the free energy of binding to liposomes.     

Molecular Modeling of Lipid Membrane Curvature Induction by a Peptide: More than Simply Shape

Molecular dynamics simulations of an amphipathic helix embedded in a lipid bilayer indicate that it will induce substantial positive curvature (e.g., a tube of diameter 20 nm at 16% surface coverage). The induction is twice that of a continuum model prediction that only considers the shape of the inclusion. The discrepancy is explained in terms of the additional presence of specific interactions described only by the molecular model. The conclusion that molecular shape alone is insufficient to quantitatively model curvature is supported by contrasting molecular and continuum models of lipids with large and small headgroups (choline and ethanolamine, respectively), and of the removal of a lipid tail (modeling a lyso-lipid). For the molecular model, curvature propensity is analyzed by computing the derivative of the free energy with respect to bending. The continuum model predicts that the inclusion will soften the bilayer near the headgroup region, an effect that may weaken curvature induction. The all-atom predictions are consistent with experimental observations of the degree of tubulation by amphipathic helices and variation of the free energy of binding to liposomes.     

Syringomycin E channel: a lipidic pore stabilized by lipopeptide?

Highly reproducible ion channels of the lipopeptide antibiotic syringomycin E demonstrate unprecedented involvement of the host bilayer lipids. We find that in addition to a pronounced influence of lipid species on the open-channel ionic conductance, the membrane lipids play a crucial role in channel gating. The effective gating charge, which characterizes sensitivity of the conformational equilibrium of the syringomycin E channels to the transmembrane voltage, is modified by the lipid charge and lipid dipolar moment. We show that the type of host lipid determines not only the absolute value but also the sign of the gating charge. With negatively charged bilayers, the gating charge sign inverts with increased salt concentration or decreased pH. We also demonstrate that the replacement of lamellar lipid by nonlamellar with the negative spontaneous curvature inhibits channel formation. These observations suggest that the asymmetric channel directly incorporates lipids. The charges and dipoles resulting from the structural inclusion of lipids are important determinants of the overall energetics that underlies channel gating. We conclude that the syringomycin E channel may serve as a biophysical model to link studies of ion channels with those of lipidic pores in membrane fusion.     

Modeling morphological instabilities in lipid membranes with anchored amphiphilic polymers

Anchoring molecules, like amphiphilic polymers, are able to dynamically regulate membrane morphology. Such molecules insert their hydrophobic groups into the bilayer, generating a local membrane curvature. In order to minimize the elastic energy penalty, a dynamic shape instability may occur, as in the case of the curvature-driven pearling instability or the polymer-induced tubulation of lipid vesicles. We review recent works on modeling of such instabilities by means of a mesoscopic dynamic model of the phase-field kind, which take into account the bending energy of lipid bilayers.