Budding and domain shape transformations in mixed lipid films and bilayer membranes

Domain formation in bicomponent vesicles induced by composition-curvature coupling

Lipid vesicles composed of a mixture of two types of lipids are studied by intensive Monte Carlo numerical simulations. The coupling between the local composition and the membrane shape is induced by two different spontaneous curvatures of the components. We explore the various morphologies of these biphasic vesicles coupled to the observed patterns such as nano-domains or labyrinthine mesophases. The effect of the difference in curvatures, the surface tension, and the interaction parameter between components is thoroughly explored. Our numerical results quantitatively agree with the previous analytical results obtained by Gueguen et al. [Eur. Phys. J. E 37, 76 (2014)] in the disordered (high temperature) phase. Numerical simulations allow us to explore the full parameter space, especially close to and below the critical temperature, where analytical results are not accessible. Phase diagrams are constructed and domain morphologies are quantitatively studied by computing the structure factor and the domain size distribution. This mechanism likely explains the existence of nano-domains in cell membranes as observed by super-resolution fluorescence microscopy.

Nonaxisymmetric Shapes of Biological Membranes from Locally Induced Curvature

In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved morphologies. Previous models to study membrane morphologies resulting from locally induced curvature often only consider the possibility of axisymmetric shapes-an indeed unphysical constraint. Past studies predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction to axisymmetry to study all possible membrane morphologies. Only if the resting tension of the membrane is low, we reproduce axisymmetric membrane morphologies. When the resting tension is moderate to high, we show that 1) axisymmetric membrane pits are unstable and 2) nonaxisymmetric ridge-shaped structures are energetically favorable. Furthermore, we find the interplay between intramembrane viscous flow and the rate of induced curvature affects the membrane's ability to transition into nonaxisymmetric ridges and axisymmetric buds. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, whereas nonaxisymmetric ridges are favored when the curvature is slowly increased. Our results hold relevant implications for biological processes such as endocytosis and physical phenomena like phase separation in lipid bilayers.

Order–disorder skewness in alpha-synuclein: a key mechanism to recognize membrane curvature

Currently, membrane curvature is understood as an active mechanism to control cells spatial organization and activity.

A Rationale for Mesoscopic Domain Formation in Biomembranes

Cell plasma membranes display a dramatically rich structural complexity characterized by functional sub-wavelength domains with specific lipid and protein composition. Under favorable experimental conditions, patterned morphologies can also be observed in vitro on model systems such as supported membranes or lipid vesicles. Lipid mixtures separating in liquid-ordered and liquid-disordered phases below a demixing temperature play a pivotal role in this context. Protein-protein and protein-lipid interactions also contribute to membrane shaping by promoting small domains or clusters. Such phase separations displaying characteristic length-scales falling in-between the nanoscopic, molecular scale on the one hand and the macroscopic scale on the other hand, are named mesophases in soft condensed matter physics. In this review, we propose a classification of the diverse mechanisms leading to mesophase separation in biomembranes. We distinguish between mechanisms relying upon equilibrium thermodynamics and those involving out-of-equilibrium mechanisms, notably active membrane recycling. In equilibrium, we especially focus on the many mechanisms that dwell on an up-down symmetry breaking between the upper and lower bilayer leaflets. Symmetry breaking is an ubiquitous mechanism in condensed matter physics at the heart of several important phenomena. In the present case, it can be either spontaneous (domain buckling) or explicit, i.e., due to an external cause (global or local vesicle bending properties). Whenever possible, theoretical predictions and simulation results are confronted to experiments on model systems or living cells, which enables us to identify the most realistic mechanisms from a biological perspective.

Tuning Length Scales of Small Domains in Cell-Derived Membranes and Synthetic Model Membranes

Micron-scale, coexisting liquid-ordered (L_o) and liquid-disordered (L_d) phases are straightforward to observe in giant unilamellar vesicles (GUVs) composed of ternary lipid mixtures. Experimentally, uniform membranes undergo demixing when temperature is decreased: domains subsequently nucleate, diffuse, collide, and coalesce until only one domain of each phase remains. The sizes of these two domains are limited only by the size of the system. Under different conditions, vesicles exhibit smaller-scale domains of fixed sizes, leading to the question of what sets the length scale. In membranes with excess area, small domains are expected when coarsening is hindered or when a microemulsion or modulated phase arises. Here, we test predictions of how the size, morphology, and fluorescence levels of small domains vary with the membrane's temperature, tension, and composition. Using GUVs and cell-derived giant plasma membrane vesicles, we find that 1) the characteristic size of domains decreases when temperature is increased or membrane tension is decreased, 2) stripes are favored over circular domains for lipid compositions with low energy per unit interface, 3) fluorescence levels are consistent with domain registration across both monolayer leaflets of the bilayer, and 4) small domains form in GUVs composed of lipids both with and without ester-linked lipids. Our experimental results are consistent with several elements of current theories for microemulsions and modulated phases and inconsistent with others, suggesting a motivation to modify or enhance current theories.

Curvature-Induced Spatial Ordering of Composition in Lipid Membranes

Phase segregation of membranal components, such as proteins, lipids, and cholesterols, leads to the formation of aggregates or domains that are rich in specific constituents. This process is important in the interaction of the cell with its surroundings and in determining the cell's behavior and fate. Motivated by published experiments on curvature-modulated phase separation in lipid membranes, we formulate a mathematical model aiming at studying the spatial ordering of composition in a two-component biomembrane that is subjected to a prescribed (imposed) geometry. Based on this model, we identified key nondimensional quantities that govern the biomembrane response and performed numerical simulations to quantitatively explore their influence. We reproduce published experimental observations and extend them to surfaces with geometric features (imposed geometry) and lipid phases beyond those used in the experiments. In addition, we demonstrate the possibility for curvature-modulated phase separation above the critical temperature and propose a systematic procedure to determine which mechanism, the difference in bending stiffness or difference in spontaneous curvatures of the two phases, dominates the coupling between shape and composition.

General neck condition for the limit shape of budding vesicles

The shape equation and linking conditions for a vesicle with two phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. 70, 2964 (1993)PRLTAO0031-900710.1103/PhysRevLett.70.2964; Phys. Rev. E 53, 2670 (1996)1063-651X10.1103/PhysRevE.53.2670], and then we use the shape equation and linking conditions to prove that this conjecture holds not only for axisymmetric budding vesicles, but also for asymmetric ones. Our study reveals that the mean curvature at any point on the membrane segments adjacent to the neck satisfies the general neck condition for the limit shape of a budding vesicle when the length scale of the membrane segments is much larger than the characteristic size of the neck but still much smaller than the characteristic size of the vesicle.

Measuring the composition-curvature coupling in binary lipid membranes by computer simulations

The coupling between local composition fluctuations in binary lipid membranes and curvature affects the lateral membrane structure. We propose an efficient method to compute the composition-curvature coupling in molecular simulations and apply it to two coarse-grained membrane models-a minimal, implicit-solvent model and the MARTINI model. Both the weak-curvature behavior that is typical for thermal fluctuations of planar bilayer membranes as well as the strong-curvature regime corresponding to narrow cylindrical membrane tubes are studied by molecular dynamics simulation. The simulation results are analyzed by using a phenomenological model of the thermodynamics of curved, mixed bilayer membranes that accounts for the change of the monolayer area upon bending. Additionally the role of thermodynamic characteristics such as the incompatibility between the two lipid species and asymmetry of composition are investigated.

Budding of domains in mixed bilayer membranes

We propose a model that accounts for the budding behavior of domains in lipid bilayers, where each of the bilayer leaflets has a coupling between its local curvature and the local lipid composition. The compositional asymmetry between the two monolayers leads to an overall spontaneous curvature. The membrane free energy contains three contributions: the bending energy, the line tension, and a Landau free energy for a lateral phase separation. Within a mean-field treatment, we obtain various phase diagrams which contain fully budded, dimpled, and flat states. In particular, for some range of membrane parameters, the phase diagrams exhibit a tricritical behavior as well as a three-phase coexistence region. The global phase diagrams can be divided into three types and are analyzed in terms of the curvature-composition coupling parameter and domain size.

Raft formation in lipid bilayers coupled to curvature

We present computer simulations of a membrane in which the local composition is coupled to the local membrane curvature. At high temperatures (i.e., above the temperature of macroscopic phase separation), finite-sized transient domains are observed, reminiscent of lipid rafts. The domain size is in the range of hundred nanometers, and set by the membrane elastic properties. These findings are in line with the notion of the membrane as a curvature-induced microemulsion. At low temperature, the membrane phase separates. The transition to the phase-separated regime is continuous and belongs to the two-dimensional Ising universality class when the coupling to curvature is weak, but becomes first-order for strong curvature-composition coupling.

Mixed lipid bilayers with locally varying spontaneous curvature and bending

A model of lipid bilayers made of a mixture of two lipids with different average compositions on both leaflets, is developed. A Landau Hamiltonian describing the lipid-lipid interactions on each leaflet, with two lipidic fields ψ 1 and ψ 2, is coupled to a Helfrich one, accounting for the membrane elasticity, via both a local spontaneous curvature, which varies as C 0 + C 1(ψ 1 - ψ 2/2), and a bending modulus equal to κ 0 + κ 1(ψ 1 + ψ 2)/2. This model allows us to define curved patches as membrane domains where the asymmetry in composition, ψ 1 - ψ 2, is large, and thick and stiff patches where ψ 1 + ψ 2 is large. These thick patches are good candidates for being lipidic rafts, as observed in cell membranes, which are composed primarily of saturated lipids forming a liquid-ordered domain and are known to be thick and flat nano-domains. The lipid-lipid structure factors and correlation functions are computed for globally spherical membranes and planar ones and for a whole set of parameters including the surface tension and the coupling in the two leaflet compositions. Phase diagrams are established, within a Gaussian approximation, showing the occurrence of two types of Structure Disordered phases, with correlations between either curved or thick patches, and an Ordered phase, corresponding to the divergence of the structure factor at a finite wave vector. The varying bending modulus plays a central role for curved membranes, where the driving force κ 1 C 0 (2) is balanced by the line tension, to form raft domains of size ranging from 10 to 100 nm. For planar membranes, raft domains emerge via the cross-correlation with curved domains. A global picture emerges from curvature-induced mechanisms, described in the literature for planar membranes, to coupled curvature- and bending-induced mechanisms in curved membranes forming a closed vesicle.

Curvature-driven effective attraction in multicomponent membranes

We study closed liquid membranes that segregate into three phases due to differences in the chemical and physical properties of its components. The shape and in-plane membrane arrangement of the phases are coupled through phase-specific bending energies and line tensions. We use simulated annealing Monte Carlo simulations to find low-energy structures, allowing both phase arrangement and membrane shape to relax. The three-phase system is the simplest one in which there are multiple interface pairs, allowing us to analyze interfacial preferences and pairwise distinct line tensions. We observe the system's preference for interface pairs that maximize differences in spontaneous curvature. From a pattern selection perspective, this acts as an effective attraction between phases of most disparate spontaneous curvature. We show that this effective attraction is robust enough to persist even when the interface between these phases is the most penalized by line tension. This effect is driven by geometry and not by any explicit component-component interaction.

Polymer-hybridized liposomes anchored with alkyl grafted poly(asparagine)

Polymer-hybridized liposomes (PHLs) of saturated lecithin were formed by association of poly(asparagines) grafted with alkyl chains (PAsn-g-Cn). The thermal, physical, and surface properties of the polymer-hybridized liposomes were examined with varying polymer concentration, alkyl chain length (C(8), C(12), C(18), C(22)), and degree of substitution (DS) in the polymer. The inclusion of the polymer raised the membrane fluidity of liposomes. By the incorporation of small amount of polymer, the membrane rigidity of liposomes dropped sharply and then increased close to the original level as the polymer concentrations increased in the cases of PAsn-g-C(18) and PAsn-g-C(22). Also, the membrane rigidity and stability of PHLs increased with alkyl chain length at the same polymer concentration. The surface charge of PHL associated with PAsn-g-C(22) was changed by DS of alkyl chains. The polymer bearing long alkyl chains (C(12), C(18), C(22)) formed PHLs well at low polymer concentration and the number of disk-shaped polymer-lipid mixed micelles increased with polymer concentration. The anchored polymers induced shifts in gel-to-liquid crystal transition temperature (Tc) of the vesicles and Tc varied with polymer concentration, alkyl chain length, and DS of the polymer.

Defect dynamics in crystalline buckled membranes

We study the dynamics of defect annihilation in flexible crystalline membranes suffering a symmetry-breaking phase transition. The kinetic process leading the system toward equilibrium is described through a Brazovskii-Helfrich-Canham Hamiltonian. In membranes, a negative disclination has a larger energy than a positive disclination. Here we show that this energetic asymmetry does not only affect equilibrium properties, like the Kosterlitz-Thouless transition temperature, but also plays a fundamental role in the dynamic of defects. Both unbinding of dislocations and Carraro-Nelson "antiferromagnetic" interactions between disclinations slow down the dynamics below the Lifshitz-Safran regime observed in flat hexagonal systems.

Approximations, idealizations and 'experiments' at the physics-biology interface

This paper, which is based on recent empirical research at the University of Leeds, the University of Edinburgh, and the University of Bristol, presents two difficulties which arise when condensed matter physicists interact with molecular biologists: (1) the former use models which appear to be too coarse-grained, approximate and/or idealized to serve a useful scientific purpose to the latter; and (2) the latter have a rather narrower view of what counts as an experiment, particularly when it comes to computer simulations, than the former. It argues that these findings are related; that computer simulations are considered to be undeserving of experimental status, by molecular biologists, precisely because of the idealizations and approximations that they involve. The complexity of biological systems is a key factor. The paper concludes by critically examining whether the new research programme of 'systems biology' offers a genuine alternative to the modelling strategies used by physicists. It argues that it does not.

Lipid bilayer reorganization under extreme pH conditions

Supported lipid bilayers containing phosphatidylcholine headgroups are observed to undergo reorganization from a 2D fluid, lipid bilayer assembly into an array of complex 3D structures upon exposure to extreme pH environments. These conditions induce a combination of molecular packing and electrostatic interactions that can create dynamic morphologies of highly curved lipid membrane structures. This work demonstrates that fluid, single-component lipid bilayer assemblies can create complex morphologies, a phenomenon typically only associated with lipid bilayers of mixed composition.

The thermodynamics of simple biomembrane mimetic systems

Insight into the forces governing a system is essential for understanding its behavior and function. Thermodynamic investigations provide a wealth of information that is not, or is hardly, available from other methods. This article reviews thermodynamic approaches and assays to measure collective properties such as heat adsorption / emission and volume variations. These methods can be successfully applied to the study of lipid vesicles (liposomes) and biological membranes. With respect to instrumentation, differential scanning calorimetry, pressure perturbation calorimetry, isothermal titration calorimetry, dilatometry, and acoustic techniques aimed at measuring the isothermal and adiabatic processes, two- and three-dimensional compressibilities are considered. Applications of these techniques to lipid systems include the measurement of different thermodynamic parameters and a detailed characterization of thermotropic, barotropic, and lyotropic phase behavior. The membrane binding and / or partitioning of solutes (proteins, peptides, drugs, surfactants, ions, etc.) can also be quantified and modeled. Many thermodynamic assays are available for studying the effect of proteins and other additives on membranes, characterizing non-ideal mixing, domain formation, bilayer stability, curvature strain, permeability, solubilization, and fusion. Studies of membrane proteins in lipid environments elucidate lipid-protein interactions in membranes. Finally, a plethora of relaxation phenomena toward equilibrium thermodynamic structures can be also investigated. The systems are described in terms of enthalpic and entropic forces, equilibrium constants, heat capacities, partial volume changes, volume and area compressibility, and so on, also shedding light on the stability of the structures and the molecular origin and mechanism of the structural changes.

Phase diagrams for multi-component membrane vesicles: a coarse-grained modeling study

We develop a multicomponent membrane coarse-grained model in which the effects of spontaneous curvature and fluidity are included. This model is used to perform computer simulations to study the phase segregation, domain coarsening, budding and budding off of multicomponent membrane vesicles. Three types of phase diagram are presented with variation of composition, spontaneous curvature, and line tension (the unlike bond energy). Various phases have been observed, including, sphere, biconcave, starfish, capsule, budding, and budding off. Our simulations show that budding off occurs only when the size of a domain is larger than a critical value, and the critical condition is closely related to the spontaneous curvature and line tension. A continuum model is used to predict the critical condition for budding off. Quantitative comparisons are made between the present simulation results and the continuum model predictions, and good agreements have been achieved.

Dynamics of multicomponent vesicles in a viscous fluid

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibil-ity of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.

Bending stiffness depends on curvature of ternary lipid mixture tubular membranes

Lipid and protein sorting and trafficking in intracellular pathways maintain cellular function and contribute to organelle homeostasis. Biophysical aspects of membrane shape coupled to sorting have recently received increasing attention. Here we determine membrane tube bending stiffness through measurements of tube radii, and demonstrate that the stiffness of ternary lipid mixtures depends on membrane curvature for a large range of lipid compositions. This observation indicates amplification by curvature of cooperative lipid demixing. We show that curvature-induced demixing increases upon approaching the critical region of a ternary lipid mixture, with qualitative differences along two roughly orthogonal compositional trajectories. Adapting a thermodynamic theory earlier developed by M. Kozlov, we derive an expression that shows the renormalized bending stiffness of an amphiphile mixture membrane tube in contact with a flat reservoir to be a quadratic function of curvature. In this analytical model, the degree of sorting is determined by the ratio of two thermodynamic derivatives. These derivatives are individually interpreted as a driving force and a resistance to curvature sorting. We experimentally show this ratio to vary with composition, and compare the model to sorting by spontaneous curvature. Our results are likely to be relevant to the molecular sorting of membrane components in vivo.

Morphology and interaction between lipid domains

Cellular membranes are a heterogeneous mix of lipids, proteins and small molecules. Special groupings enriched in saturated lipids and cholesterol form liquid-ordered domains, known as "lipid rafts," thought to serve as platforms for signaling, trafficking and material transport throughout the secretory pathway. Questions remain as to how the cell maintains small fluid lipid domains, through time, on a length scale consistent with the fact that no large-scale phase separation is observed. Motivated by these examples, we have utilized a combination of mechanical modeling and in vitro experiments to show that membrane morphology plays a key role in maintaining small domain sizes and organizing domains in a model membrane. We demonstrate that lipid domains can adopt a flat or dimpled morphology, where the latter facilitates a repulsive interaction that slows coalescence and helps regulate domain size and tends to laterally organize domains in the membrane.

The multiple faces of self-assembled lipidic systems

Lipids, the building blocks of cells, common to every living organisms, have the propensity to self-assemble into well-defined structures over short and long-range spatial scales. The driving forces have their roots mainly in the hydrophobic effect and electrostatic interactions. Membranes in lamellar phase are ubiquitous in cellular compartments and can phase-separate upon mixing lipids in different liquid-crystalline states. Hexagonal phases and especially cubic phases can be synthesized and observed in vivo as well. Membrane often closes up into a vesicle whose shape is determined by the interplay of curvature, area difference elasticity and line tension energies, and can adopt the form of a sphere, a tube, a prolate, a starfish and many more. Complexes made of lipids and polyelectrolytes or inorganic materials exhibit a rich diversity of structural morphologies due to additional interactions which become increasingly hard to track without the aid of suitable computer models. From the plasma membrane of archaebacteria to gene delivery, self-assembled lipidic systems have left their mark in cell biology and nanobiotechnology; however, the underlying physics is yet to be fully unraveled.PACS Codes: 87.14.Cc, 82.70.Uv.

Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission

We develop a thermodynamically consistent phase-field model to simulate the dynamics of multicomponent vesicles. The model accounts for bending stiffness, spontaneous curvature, excess (surface) energy, and a line tension between the coexisting surface phases. Our approach is similar to that recently used by Wang and Du [J. Math. Biol. 56, 347 (2008)] with a key difference. Here, we concentrate on the dynamic evolution and solve the surface mass conservation equation explicitly; this equation was not considered by Wang and Du. The resulting fourth-order strongly coupled system of nonlinear nonlocal equations are solved numerically using an adaptive finite element numerical method. Although the system is valid for three dimensions, we limit our studies here to two dimensions where the vesicle is a curve. Differences between the spontaneous curvatures and the bending rigidities of the surface phases are found numerically to lead to the formation of buds, asymmetric vesicle shapes and vesicle fission even in two dimensions. In addition, simulations of configurations far from equilibrium indicate that phase separation via spinodal decomposition and coarsening not only affect the vesicle shape but also that the vesicle shape affects the phase separation dynamics, especially the coarsening and may lead to lower energy states than might be achieved by evolving initially phase-separated configurations.

Polymer-vesicle association

Mixed polymer-surfactant systems have been intensively investigated in the last two decades, with the main focus on surfactant micelles as the surfactant aggregate in interaction. The main types of phase behavior, driving forces and structural/rheological effects at stake are now fairly well understood. Polymer-vesicle systems, on the other hand, have received comparatively less attention from a physico-chemical perspective. In this review, our main goal has been to bridge this gap, taking a broad approach to cover a field that is in clear expansion, in view of its multiple implications for colloid and biological sciences and in applied areas. We start by a general background on amphiphile self-assembly and phase separation phenomena in mixed polymer-surfactant solutions. We then address vesicle formation, properties and stability not only in classic lipids, but also in various other surfactant systems, among which catanionic vesicles are highlighted. Traditionally, lipid and surfactant vesicles have been studied separately, with little cross-information and comparison, giving duplication of physico-chemical interpretations. This situation has changed in more recent times. We then proceed to cover more in-depth the work done on different aspects of the associative behavior between vesicles (of different composition and type of stability) and different types of polymers, including polysaccharides, proteins and DNA. Thus, phase behavior features, effects of vesicle structure and stability, and the forces/mechanisms of vesicle-macromolecule interaction are addressed. Such association may generate gels with interesting rheological properties and high potential for applications. Finally, special focus is also given to DNA, a high charge polymer, and its interactions with surfactants, and vesicles, in particular, in the context of gene transfection studies.

Modeling the phase separation in binary lipid membrane under externally imposed oscillatory shear flow

By adding external velocity terms, the two-dimensional time-dependent Ginzburg-Landau (TDGL) equations are modified. Based on this, the phase separation in binary lipid membrane under externally imposed oscillatory shear flow is numerically modeled employing the Cell Dynamical System (CDS) approach. Considering shear flows with different frequencies and amplitudes, several aspects of such a phase evolving process are studied. Firstly, visualized results are shown via snapshot figures of the membrane shape. And then, the simulated scattering patterns at typical moments are presented. Furthermore, in order to more quantitatively discuss this phase-separation process, the time growth laws of the characteristic domain sizes in both directions parallel and perpendicular to the flow are investigated for each case. Finally, the peculiar rheological properties of such binary lipid membrane system have been discussed, mainly the normal stress difference and the viscoelastic complex shear moduli.

Biophysical approaches to protein-induced membrane deformations in trafficking

Membrane traffic requires membrane deformation to generate vesicles and tubules. Strong evidence suggests that assembly of curvature-active proteins can drive such membrane shape changes. Well-documented pathways often involve protein scaffolds, in particular coats (clathrin or COP). However, membrane curvature should, in principle, be influenced by any protein binding asymmetrically on a membrane; large membrane morphological changes could result from their aggregation. In the case of Shiga toxin or viral matrix proteins, tubules and buds appear to result from the cargo-driven formation of protein-lipid nanodomains, showing that collective protein behaviour is crucial in the process. We argue here that a combination of in vitro experiments on giant unilamellar vesicles and theoretical modelling based on statistical physics is ideally suited to tackle these collective effects.

Harnessing janus nanoparticles to create controllable pores in membranes

We use a coarse-grained numerical simulation to design a synthetic membrane with stable pores that can be controllably opened and closed. Specifically, we use dissipative particle dynamics to probe the interactions between lipid bilayer membranes and nanoparticles. The particles are nanoscopic Janus beads that comprise both hydrophobic and hydrophilic portions. We demonstrate that when the membrane rips and forms a hole due to an external stress, these nanoparticles diffuse to the edge of the hole and form a stable pore, which persists after the stress is released. Once the particle-lined pore is formed, a small increase in membrane tension readily reopens the pore, allowing transport through the membrane. Besides the application of an external force, the membrane tension can be altered by varying, for example, temperature or pH. Thus, the findings provide guidelines for designing nanoparticle-bilayer assemblies for targeted delivery, where the pores open and the cargo is released only when the local environmental conditions reach a critical value.

The phase behavior of mixed lipid membranes in the presence of the rippled phase

We propose a model describing liquid-solid phase coexistence in mixed lipid membranes by including explicitly the occurrence of a rippled phase. For a single component membrane, we employ a previous model in which the membrane thickness is used as an order parameter. As function of temperature, this model properly accounts for the phase behavior of the three possible membrane phases: solid, liquid and the rippled phase. Our primary aim is to explore extensions of this model to binary lipid mixtures by considering the composition dependence of important model parameters. The obtained phase diagrams show various liquid, solid and rippled phase coexistence regions, and are in quantitative agreement with the experimental ones for some specific lipid mixtures.

Domain-induced budding in buckling membranes

We present a phase field model on buckling membranes to analyze phase separation and budding on soft membranes. By numerically integrating dynamic equations, it turns out that the formation of caps is greatly influenced by the presence of a little excess area due to the surface area constraint. When cap-shaped domains are created, domain coalescence is mainly observed not between domains with same budding directions, but between domains with opposite budding directions, because the bending energy between two domains is larger in the former case. Although we do not introduce spontaneous curvature like Helfrich model, we obtain some suggestions related to the slow dynamics of the phase separation on vesicles.

Effect of dipolar moments in domain sizes of lipid bilayers and monolayers

Lipid domains are found in systems such as multicomponent bilayer membranes and single component monolayers at the air-water interface. It was shown by Keller et al. [J. Phys. Chem. 91, 6417 (1987)] that in monolayers, the size of the domains results from balancing the line tension, which favors the formation of a large single circular domain, against the electrostatic cost of assembling the dipolar moments of the lipids. In this paper, we present an exact analytical expression for the electric potential, ion distribution, and electrostatic free energy for different problems consisting of three different slabs with different dielectric constants and Debye lengths, with a circular homogeneous dipolar density in the middle slab. From these solutions, we extend the calculation of domain sizes for monolayers to include the effects of finite ionic strength, dielectric discontinuities (or image charges), and the polarizability of the dipoles and further generalize the calculations to account for domains in lipid bilayers. In monolayers, the size of the domains is dependent on the different dielectric constants but independent of ionic strength. In asymmetric bilayers, where the inner and outer leaflets have different dipolar densities, domains show a strong size dependence with ionic strength, with molecular-sized domains that grow to macroscopic phase separation with increasing ionic strength. We discuss the implications of the results for experiments and briefly consider their relation to other two dimensional systems such as Wigner crystals or heteroepitaxial growth.

Coupled composition-deformation phase-field method for multicomponent lipid membranes

We present a method for modeling phase transitions and morphological evolution of binary lipid membranes with approximately planar geometries. The local composition and the shape of the membrane are coupled through composition-dependent spontaneous curvature in a Helfrich free energy. The evolution of the composition field is described by a Cahn-Hilliard-type equation, while shape changes are described by relaxation dynamics. Our method explicitly treats the full nonlinear form of the geometrical scalars, tensors, and differential operators associated with the curved shape of the membrane. The model is applied to examine morphological evolution and stability of lipid membranes initialized in a variety of compositional and geometric configurations. Specifically, we investigate the dynamics of systems which have a lamellar structure as their lowest energy state. We find that evolution is very sensitive to initial conditions; only membranes with sufficiently large lamellar-type compositional perturbations or ripple-type shape perturbations in their initial configuration can deterministically evolve into a lamellar equilibrium morphology. We also observe that rigid topographical surface patterns have a strong effect on the phase separation and compositional evolution in these systems.

Applying a potential across a biomembrane: electrostatic contribution to the bending rigidity and membrane instability

We investigate the effect on biomembrane mechanical properties due to the presence an external potential for a nonconductive incompressible membrane surrounded by different electrolytes. By solving the Debye-Hückel and Laplace equations for the electrostatic potential and using the relevant stress-tensor we find (1) in the small screening length limit, where the Debye screening length is smaller than the distance between the electrodes, the screening certifies that all electrostatic interactions are short range and the major effect of the applied potential is to decrease the membrane tension and increase the bending rigidity; explicit expressions for electrostatic contribution to the tension and bending rigidity are derived as a function of the applied potential, the Debye screening lengths, and the dielectric constants of the membrane and the solvents. For sufficiently large voltages the negative contribution to the tension is expected to cause a membrane stretching instability. (2) For the dielectric limit, i.e., no salt (and small wave vectors compared to the distance between the electrodes), when the dielectric constant on the two sides are different the applied potential induces an effective (unscreened) membrane charge density, whose long-range interaction is expected to lead to a membrane undulation instability.

Formation of three-dimensional structures in supported lipid bilayers

The creation of three-dimensional structures in supported lipid bilayers has been examined. In bilayers, shape transformations can be triggered by adjusting a variety of parameters. Here, it is shown that bilayers composed of phosphatidylcholine and phosphatidic acid can be induced to reversibly form cap structures when exposed to an asymmetry in ionic strength. The structures that form depend on the asymmetry in the ionic strength and the amount of anionic lipid. Other factors that may be of importance in the creation of the structures, expansion forces, osmotic forces, and the bilayer-support interaction are discussed. The cap structures have the potential to be of considerable utility in examining the effect that curvature has on membrane processes.

The interface width of separated two-component lipid membranes

We study two-component vesicles with coherent domains of the components, where the domains are separated by an interface. The components are characterized by different spontaneous curvatures. No line tension term or interactions between components are included in the model. The influence of the interface width and interface location on the bending energy and shape of the vesicles is studied. How the spontaneous curvature of one component influences the concentration profile is examined. The vesicles of oblate and prolate geometries are investigated.

Stability of biphasic vesicles with membrane embedded proteins

The basic physical properties of homogeneous membranes are relatively well known, while the effects of inhomogeneities with membranes are very much an active field of study. In this paper, a biphasic lipid vesicle with membrane embedded proteins is investigated. To take into account the influences of the proteins, a simple phenomenological coupling between the local fraction of proteins and the mean curvature square is suggested. By minimizing the energy of system, the E-L equations and boundary conditions are obtained and solved analytically for vesicle with a simple shape. Besides, stability phase diagrams and stability factor are put forward by linear perturbation analysis. Our results show two different situations which are strongly dependent on the nature of the proteins: a regime of easy instability when the proteins are strongly coupled to the membrane and a regime of difficult instability.

Mechanisms behind the Faceting of Catanionic Vesicles by Polycations: Chain Crystallization and Segregation

Vesicles composed of an anionic and a cationic surfactant, with a net negative charge, associate strongly with a hydrophobically modified polycation (LM200) and with an unmodified polycation with higher charge density (JR400), forming viscoelastic gel-like structures. Calorimetric results show that in these gels, LM200 induces a rise of the chain melting temperature (Tm) of the vesicles, whereas JR400 has the opposite effect. For both polymer-vesicle systems, the shear viscosity exhibits an inflection point at Tm, and for the LM200 system the measured relaxation times are significantly higher below Tm. The neat vesicles and the polycation-bound vesicles have a polygonal-like faceted shape when the surfactant chains in the bilayer are crystallized, as probed by cryo-transmission electron microscopy. Above Tm, the neat and the LM200-bound vesicles regain a spheroidal shape, whereas those in the JR400 system remain with a deformed faceted shape even above Tm. These shape changes are interpreted in terms of different mechanisms for the polymer-vesicle interaction, which seem to be highly dependent on polymer architecture, namely charge density and hydrophobic modification. A crystallization-segregation mechanism is proposed for the LM200-vesicle system, while, for the JR400-vesicle one, charge polarization-lateral segregation effects induced by the polycation in the catanionic bilayer are envisaged.

Inclusion-induced boundary layers in lipid vesicles

The equilibrium shapes of lipid vesicles are perturbed by rigid inclusions. In a two-dimensional vesicle, that may also model a cylindrically elongated tubule, the shape modifications can be determined analytically, and turn out to be significant even far from the inclusion. On the contrary, previous numerical work has given evidence that in the three-dimensional case the shape perturbations decay quite rapidly and are negligible a few inclusion radii away. In this paper, we use the tools of asymptotic analysis to derive analytically the shape of the boundary layer induced by the inclusion. As a result, we are able to determine the dominant part of the free-energy perturbation that, in turn, allows to identify the vesicle points where the inclusion prefers to sit.

Morphology and phase behavior of two-component lipid membranes

The stability and shapes of domains with different bending rigidities in lipid membranes are investigated. These domains can be formed from the inclusion of an impurity in a lipid membrane or from the phase separation within the membrane. We show that, for weak line tensions, surface tensions and finite spontaneous curvatures, an equilibrium phase of protruding circular domains or striped domains may be obtained. We also predict a possible phase transition between the investigated morphologies.

Tension-induced morphological transition in mixed lipid bilayers

Recently, Rozovsky et al. reported on the morphology and dynamics of superstructures in three-component lipid bilayers containing saturated lipid, unsaturated lipid, and cholesterol (Rozovsky, S.; Kaizuka, Y.; Groves, J. T. J. Am. Chem. Soc. 2005, 127, 36). We suggest that the observed sequence of the striped-to-hexagonal morphological transition in mixed bilayers can be attributed to an enhanced membrane surface tension that is induced by the vesicle adhesion onto the solid surface.

Coupling between vesicle shape and lateral distribution of mobile membrane inclusions

Membrane inclusions such as membrane-embedded peptides or proteins exhibit a curvature-dependent interaction with the surrounding lipid matrix due to the mismatch between their intrinsic curvature and the local membrane curvature. This interaction causes an inhomogeneous lateral distribution of the inclusions and a corresponding adjustment of the vesicle shape. We have studied theoretically the axisymmetric equilibrium shapes of lipid vesicles with mobile inclusions, taking into account that the membrane free energy includes the elastic energy of the lipid bilayer and a contribution due to an inclusion-membrane interaction. Equations describing the shape are derived by minimizing the total free energy at fixed membrane area, enclosed volume, and number of inclusions and are then solved numerically. It is shown that vesicle shape may assume a symmetry that differs from that of the vesicle with no inclusions. If the inclusion-membrane interaction exceeds a certain value, there is no axisymmetric solution of the equations with a continuous and derivable lateral density of inclusions over the whole area of the vesicle. When approaching the critical vesicle shape, the shapes obtained differ qualitatively from those described by the area difference elasticity model of the elastic properties of lipid membranes. In general, vesicle shapes adjust to the presence of inclusions by increasing regions with favorable curvature and decreasing regions of unfavorable curvature in a way such that the lateral distribution of inclusions becomes inhomogeneous.

Anomalously slow domain growth in fluid membranes with asymmetric transbilayer lipid distribution

The effect of asymmetry in the transbilayer lipid distribution on the dynamics of phase separation in fluid vesicles is investigated numerically. This asymmetry is shown to set a spontaneous curvature for the domains that alter the morphology and dynamics considerably. For moderate tension, the domains are capped and the spontaneous curvature leads to anomalously slow dynamics, as compared to the case of symmetric bilayers. In contrast, in the limiting cases of high and low tensions, the dynamics proceeds toward full phase separation.