Curvature-driven lipid sorting in a membrane tubule

Motivated by recent experiments that implicate the mechanical properties of membranes in lipid sorting, we examine the interplay of lipid composition and curvature in membrane tubules. We study how the coupling between membrane composition and membrane bending stiffness and tension affects tubule formation. Drawing a tubule from a vesicle leads to a rearrangement of composition in which the phase of higher flexibility segregates into the highly curved tubule. For point forcing, the force vs extension curve can have a sharp drop just as the tubule begins to form.     

Adhesion of fluid vesicles at chemically structured substrates

The adhesion of fluid vesicles at chemically structured substrates is studied theoretically via Monte Carlo simulations. The substrate surface is planar and repels the vesicle membrane apart from a single surface domain γ , which strongly attracts this membrane. If the vesicle is larger than the attractive γ domain, the spreading of the vesicle onto the substrate is restricted by the size of this surface domain. Once the contact line of the adhering vesicle has reached the boundaries of the γ domain, further deflation of the vesicle leads to a regime of low membrane tension with pronounced shape fluctuations, which are now governed by the bending rigidity. For a circular γ domain and a small bending rigidity, the membrane oscillates strongly around an average spherical cap shape. If such a vesicle is deflated, the contact area increases or decreases with increasing osmotic pressure, depending on the relative size of the vesicle and the circular γ domain. The lateral localization of the vesicle's center of mass by such a domain is optimal for a certain domain radius, which is found to be rather independent of adhesion strength and bending rigidity. For vesicles adhering to stripe-shaped surface domains, the width of the contact area perpendicular to the stripe varies nonmonotonically with the adhesion strength.     

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 incompressibility 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.     

Morphological phase diagram for lipid membrane domains with entropic tension

Circular domains in phase-separated lipid vesicles with symmetric leaflet composition commonly exhibit three stable morphologies: flat, dimpled, and budded. However, stable dimples (i.e., partially budded domains) present a puzzle since simple elastic theories of domain shape predict that only flat and spherical budded domains are mechanically stable in the absence of spontaneous curvature. We argue that this inconsistency arises from the failure of the constant surface tension ensemble to properly account for the effect of entropic bending fluctuations. Formulating membrane elasticity within an entropic tension ensemble, wherein tension represents the free energy cost of extracting membrane area from thermal bending of the membrane, we calculate a morphological phase diagram that contains regions of mechanical stability for each of the flat, dimpled, and budded domain morphologies.     

Cellular uptake of elastic nanoparticles

A fundamental understanding of cell-nanomaterial interaction is of essential importance to nanomedicine and safe applications of nanotechnology. Here we investigate the adhesive wrapping of a soft elastic vesicle by a lipid membrane. We show that there exist a maximum of five distinct wrapping phases based on the stability of full wrapping, partial wrapping, and no wrapping states. The wrapping phases depend on the vesicle size, adhesion energy, surface tension of membrane, and bending rigidity ratio between vesicle and membrane. These results are of immediate interest to the study of vesicular transport and endocytosis or phagocytosis of elastic particles into cells.     

Vesicle migration and spatial organization driven by flow line curvature

Cross-streamline migration of deformable entities is essential in many problems such as industrial particulate flows, DNA sorting, and blood rheology. Using two-dimensional numerical experiments, we have discovered that vesicles suspended in a flow with curved flow lines migrate towards regions of high flowline curvature, which are regions of high shear rates. The migration velocity of a vesicle is found to be a universal function of the normal stress difference and the flow curvature. This finding quantitatively demonstrates a direct coupling between a microscopic quantity (migration) and a macroscopic one (normal stress difference). Furthermore, simulations with multiple vesicles revealed a self-organization, which corresponds to segregation, in a rim closer to the inner cylinder, resulting from a subtle interaction among vesicles. Such segregation effects could have a significant impact on the rheology of vesicle flows.     

A phase field model for vesicle–substrate adhesion

In this paper, a phase field model is developed for vesicle adhesion involving complex substrate and vesicle geometries. The model takes into account an adhesion potential that depends on the distance of vesicle to the substrate. A variational problem is solved in a 3D computational domain by minimizing the contribution of bending elastic energy and the adhesion energy under the constraints of total surface area and volume, described via a phase function. An adaptive finite element method is used to efficiently compute the numerical solutions of the model. The computational results are validated through comparison of several axisymmetric shapes with the sharp-interface ODE solution. Moreover, we compute shapes for non-axisymmetric situations to support the observation that concave substrates favor adhesion.     

Bound states of 3He at the edge of a 4He drop on a cesium surface

We show that small amounts of 3He atoms, added to a 4He drop deposited on a flat cesium surface at zero temperature, populate bound states localized at the contact line. These edge states show up for drops large enough to develop well defined surface and bulk regions together with a contact line, and they are structurally different from the well-known Andreev states that appear at the free surface and at the liquid-solid interface of films. We illustrate the one-body density of 3He in a drop with 1000 4He atoms, and show that for a sufficiently large number of impurities the density profiles spread beyond the edge, coating both the curved drop surface and its flat base and eventually isolating it from the substrate.     

Accurate determination of elastic parameters for multicomponent membranes

Heterogeneities in the cell membrane due to coexisting lipid phases have been conjectured to play a major functional role in cell signaling and membrane trafficking. Thereby the material properties of multiphase systems, such as the line tension and the bending moduli, are crucially involved in the kinetics and the asymptotic behavior of phase separation. In this Letter we present a combined analytical and experimental approach to determine the properties of phase-separated vesicle systems. First we develop an analytical model for the vesicle shape of weakly budded biphasic vesicles. Subsequently experimental data on vesicle shape and membrane fluctuations are taken and compared to the model. The parameters obtained set limits for the size and stability of nanodomains in the plasma membrane of living cells.     

Universality classes of self-avoiding fixed-connectivity membranes

We present an analysis of extensive large-scale Monte Carlo simulations of self-avoiding fixed-connectivity membranes for sizes (number of faces) ranging from 512 to 17672 (triangular) plaquettes. Self-avoidance is implemented via impenetrable plaquettes. We simulate the impenetrable plaquette model in both three and four bulk dimensions. In both cases we find the membrane to be flat for all temperatures: the size exponent in three dimensions is ν = 0.95(5) (Hausdorff dimension d _H = 2.1(1)). The single flat phase appears, furthermore, to be equivalent to the large bending rigidity phase of non-self-avoiding fixed-connectivity membranes --the roughness exponent in three dimensions is ξ = 0.63(4). This suggests that there is a unique universality class for flat fixed-connectivity membranes without attractive interactions. Finally, we address some theoretical and experimental implications of our work. Received 23 June 2000 and Received in final form 25 October 2000     

Stable patterns of membrane domains at corrugated substrates

Multicomponent membranes such as ternary mixtures of lipids and cholesterol can exhibit coexistence regions between two liquid phases. When such membranes adhere to a corrugated substrate, the phase separation process strongly depends on the interplay between substrate topography, bending rigidities, and line tension of the membrane domains as we show theoretically via energy minimization and Monte Carlo simulations. For sufficiently large bending rigidity contrast between the two membrane phases, the corrugated substrate truncates the phase separation process and leads to a stable pattern of membrane domains. Our theory is consistent with recent experimental observations and provides a possible control mechanism for domain patterns in biological membranes.     

Continuous melting of compact polymers

The competition between chain entropy and bending rigidity in compact polymers can be addressed within a lattice model introduced by Flory in 1956 [Proc. R. Soc. London A 234, 60 (1956)]]. It exhibits a transition between an entropy dominated disordered phase and an energetically favored crystalline phase. The nature of this order-disorder transition has been debated ever since the introduction of the model. Here we present exact results for the Flory model in two dimensions relevant for polymers on surfaces, such as DNA adsorbed on a lipid bilayer. We predict a continuous melting transition and compute exact values of critical exponents at the transition point.     

Depletion versus deflection: how membrane bending can influence adhesion

During depletion-driven vesicle adhesion, a stiff membrane's resistance to bending at fixed tension prevents contact angle equilibrium and vesicle spreading over an opposing vesicle, while more flexible vesicles partially engulf opposing vesicles. Estimates of the bending cost associated with the spreading contact line, relative to the adhesion energy, were consistent with the observed spreading or lack of spreading for the flexible and stiff membranes, respectively, and predicted a lag time sometimes preceding spreading.     

Shape fluctuation phase transition of elastic membranes with fluidity

It is confirmed by Monte Carlo (MC) simulation that the elastic membrane corresponding to discrete Polyakov–Kleinert string undergoes a second-order phase transition at finite bending rigidity. This phase transition is considered to be associated with a non-trivial fixed point of the β-function for the inverse bending rigidity.     

1H-13C polarization transfer in membranes: a tool for probing lipid dynamics and the effect of cholesterol

Phospholipid bilayers with over 20% cholesterol can form a liquid-ordered (l(o)) phase, which can be found in lateral domains, called rafts, in biomembranes. We show here that high-resolution (13)C and (1)H solid-state NMR are well suited to explore this phase, intermediate between gel and fluid. This approach can be applied to artificial or natural membranes, with no isotopic enrichment and with the help of magic-angle spinning (MAS), taking advantage of the high resolution and sensitivity of these nuclei. The sensitivity of magnetization transfer schemes to different lipid states has allowed us here to discriminate between various phases. We show that the phase composed of unsaturated phospholipids and cholesterol differs, in terms of lipid dynamics, both from the previously described l(o) phase and from the liquid-disordered phase.     

Complete wetting characteristics of micro/nano dual-scale surface by plasma etching to give nanohoneycomb structure

A variety of flat superhydrophilic surfaces have been fabricated for biological and industrial applications. We report here the preparation of a simple and inexpensive non-polar curved superhydrophilic surface. This surface has dual-scale surface roughness, on both micro- and nanoscales. Curved surfaces with a near-zero water contact angle and ‘complete wetting’ are demonstrated. By using a conventional plasma etching process, which creates microscale irregularity on an aluminum surface, followed by an anodization process which further modifies the plasma etched surface by creating nanoscale structures, we generate a surface having irregularities on two-scales. This surface displays a semi-permanent superhydrophilic property (if the surface has no damage by the exterior failure), having a near-zero contact angle with water drops. We further report a simple and inexpensive curved (i.e., non-planar) superhydrophilic structure with a near-zero contact angle. The dual-scale character of the surface increases the capillary force effect and reduces the energy barriers of the nanostructures.     

Mesoscale simulations of curvature-inducing protein partitioning on lipid bilayer membranes in the presence of mean curvature fields

The membrane-surface migration of curvature-inducing proteins in response to membrane curvature gradients has been investigated using Monte Carlo simulations of a curvilinear membrane model based on the Helfrich Hamiltonian. Consistent with theoretical and experimental data, we find the proteins that generate curvature can also sense the background membrane curvature, wherein they preferentially partition to the high curvature regions. The partitioning strength depends linearly on local membrane curvature and the slope (or the coupling constant) of the partitioning probability versus mean curvature depends on the membrane bending rigidity and instantaneous curvature field caused by different proteins. Our simulation study allows us to quantitatively characterize and identify the important factors affecting the coupling constant (slope), which may be difficult to determine in experiments. Furthermore, the membrane model is used to study budding of vesicles where it is found that in order to stabilize a mature vesicle with a stable ‘neck-region’ (or stable membrane overhangs), the area (extent) of the intrinsic curvature region needs to exceed a threshold-critical value. The migration and partitioning of curvature-inducing proteins in a budding vesicle with a stable neck (with a characteristic negative value of the Gaussian curvature) is investigated.     

Phase separation between phospholipids and grafted polymer chains onto a fluctuating membrane

We re-examine here the theoretical study of the phase separation between phospholipids and grafted long polymer chains onto a fluid membrane. The polymer chains are assumed to be anchored to the membrane by one extremity (anchor). The anchors are big amphiphile lipid molecules. The anchors and phospholipids forming the bilayer phase separate under the variation of a suitable parameter (temperature, pressure, membrane environment, ...). To investigate the demixtion transition, we elaborate a new approach that takes into account the membrane undulations. We show that these undulations have the tendency to induce additional attractive forces between anchors, and consequently, the separation transition is accentuated and occurs at high temperature. Quantitatively, we show that the membrane undulations contribute with an extra positive segregation parameter χ_m > 0 , which scales as χ_m κ^-2 , where κ is the bending rigidity constant. Therefore, the attraction phenomenon between species of the same kind is significant only for those membranes of small bending rigidity constant. Finally, the study is extended to the case where the lengths of the anchored polymer chains are randomly distributed. To achieve calculations, we choose a length distribution of fractal form. The essential conclusion is that the polydispersity increases the size of domains alternatively rich in phospholipids and anchors.     

Line tension at fluid membrane domain boundaries measured by micropipette aspiration

Line tension is a determinant of fluid phase domain formation kinetics and morphology in lipid bilayer membranes, which are models for biological membrane heterogeneity. We describe the first direct measurement of this line tension by micropipette aspiration. Our data are analyzed with a model that does not rely on independently measured (and composition dependent) secondary parameters, such as bending stiffness or membrane viscosities. Line tension is strongly composition dependent and decreases towards a critical consolute point in a quasiternary room temperature phase diagram.