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.     

Simulations of flexible membranes using a coarse-grained particle-based model with spontaneous curvature variables

We propose a simple mathematical model for lipid bilayer membranes of flexible fluid or solid state. The model consists of interacting coarse-grained particles with two extra variables. One indicates the spontaneous curvature at the particle position, and the other indicates the vector representing the direction normal to the membrane. When the spontaneous curvature variable is allowed to fluctuate significantly, the fluctuation causes a softening of the membrane and growth of large undulations as the amplitude of the fluctuation is increased. By changing the amplitude of the fluctuation in simulations, the bending rigidity of the membrane can be easily controlled. Because the proposed model includes anisotropic interactions between the particles, multilayered vesicles can be obtained through a reversible transition by weakening the strength of the anisotropic interactions.     

Organization in lipid membranes containing cholesterol

A fundamental attribute of raft formation in cell membranes is lateral separation of lipids into coexisting liquid phases. Using fluorescence microscopy, we observe spontaneous lateral separation in free-floating giant unilamellar vesicles. We record coexisting liquid domains over a range of composition and temperature significantly wider than previously reported. Furthermore, we establish correlations between miscibility in bilayers and in monolayers. For example, the same lipid mixtures that produce liquid domains in bilayer membranes produce two upper miscibility critical points in the phase diagrams of monolayers.     

Electrostatic origin of the Frank elastic energy and anisotropic line tension of the domains in monolayers at the air-water interface

The free energy of condensed phase domains in monolayers at the air-water interface has been analyzed by taking into account the surface pressure, line tension, and electrostatic energy due to the normal and in-plane spontaneous polarization. We found that this free energy was reduced to the sum of the Frank elastic energy and anisotropic line tension, which were used to reproduce the shapes of domains in fatty acid monolayers, in the limit of small orientational deformation. Domains in monolayers are interesting systems as a meeting point between the physics of liquid crystals and electrostatics.     

Inclusions induced phase separation in mixed lipid film

The effect of rigid inclusions on the phase behavior of a film containing a mixture of lipid molecules is investigated. In the proposed model, the inclusion-induced deformation of the film, and the resulting energy cost are strongly dependent upon the spontaneous curvature of the mixed film. The spontaneous curvature is in turn strongly influenced by the composition of film. This coupling between the film composition and the energy per inclusion leads to a lateral modulation of the composition, which follows the local curvature of the membrane. In particular, it is shown that inclusions may induce a global phase separation in a film which would otherwise be homogeneously mixed. The mixed film is then composed of patches of different average composition, separated by the inclusions. This process may be of relevance to explain some aspects of lipid-protein association in biological membranes. Received 8 April 1999 and Received in final form 4 October 1999     

On the organization of self-assembled actin networks in giant vesicles

We studied the formation of actin scaffolds in giant vesicles of dimyristoylphosphatidylcholine (DMPC). Polymerization of actin was induced at low ionic strength through ionophore-mediated influx of Mg²⁺ (2 mM). The spatial organization of the filamentous actin was visualized by confocal and epifluorescence microscopy as a function of the filaments length and membrane composition, by including various amounts of cholesterol or lipids with neutral and positively charged polyethyleneglycol headgroups (PEG lipopolymers). In vesicles of pure DMPC, the newly polymerized actin adsorbs to the membrane and forms a thin shell. In the presence of 2.5 mol% lipopolymers or of cholesterol at a molar fraction x = 0.37, formation of a thin adsorbed film is impeded. A fuzzy cortex is predominantly formed in vesicles of diameter d smaller than the filament persistence length ( d ⩽ 15μm) while for larger vesicles a homogeneous network formation is favoured in the bulk of the vesicle. The fuzzy-cortex formation is interpreted as a consequence of the reduction of the bending energy if the actin filaments accumulate close to the vesicle wall. Received: 17 January 2002 / Accepted: 21 March 2003 / Published online: 24 April 2003 RID="a" ID="a"e-mail: Laurent_Limozin@ph.tum.de     

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.     

Dynamic model and stationary shapes of fluid vesicles

A phase-field model that takes into account the bending energy of fluid vesicles is presented. The Canham-Helfrich model is derived in the sharp-interface limit. A dynamic equation for the phase-field has been solved numerically to find stationary shapes of vesicles with different topologies and the dynamic evolution towards them. The results are in agreement with those found by minimization of the Canham-Helfrich free energy. This fact shows that our phase-field model could be applied to more complex problems of instabilities.     

Phase transitions in a fluid surface model with a deficit angle term

Nambu-Goto model is investigated by using the canonical Monte Carlo simulation technique on dynamically triangulated surfaces of spherical topology. We find that the model has four distinct phases; crumpled, branched-polymer, linear, and tubular. The linear phase and the tubular phase appear to be separated by a first-order transition. It is also found that there is no long-range two-dimensional order in the model. In fact, no smooth surface can be seen in the whole region of the curvature modulus α, which is the coefficient of the deficit angle term in the Hamiltonian. The bending energy, which is not included in the Hamiltonian, remains large even at sufficiently large α in the tubular phase. On the other hand, the surface is spontaneously compactified into a one-dimensional smooth curve in the linear phase; one of the two degrees of freedom shrinks, and the other degree of freedom remains along the curve. Moreover, we find that the rotational symmetry of the model is spontaneously broken in the tubular phase just as in the same model on the fixed connectivity surfaces.     

A molecular model for the line tension of lipid membranes

The line tension of a symmetric, lipid bilayer in its liquid-crystalline state is calculated on the basis of a molecular lipid model. The lipid model extends the opposing forces model by an expression for the conformational free energy of the hydrocarbon chains. We consider a membrane edge that consists of a perturbed bilayer covered by a section of a cylinder-like micelle. The structural rearrangement of the lipids implies an excess free energy which we minimize with respect to the cross-sectional shape of the membrane edge, including both the micellar and the bilayer region. The line tension is derived as a function of molecular lipid properties, like the lipid chain length or the head group interaction strength. We also relate it to the spontaneous curvature of the lipid layer. We find the line tension to become smaller for lipid layers that tend to curve more towards the hydrophobic core. Our predictions for the line tension and their relation to experimentally derived values are discussed. Received 2 January 2000     

Physical model for the width distribution of axons

The distribution of widths of axons was recently investigated, and was found to have a distinct peak at an optimized value. The optimized axon width at the peak may arise from the conflicting demands of minimizing energy consumption and assuring signal transmission reliability. The distribution around this optimized value is found to have a distinct non-Gaussian shape, with an exponential “tail”. We propose here a mechanical model whereby this distribution arises from the interplay between the elastic energy of the membrane surrounding the axon core, the osmotic pressure induced by the neurofilaments inside the axon bulk, and active processes that remodel the microtubules and neurofilaments inside the axon. The axon’s radius of curvature can be determined by the cell’s control of the osmotic pressure difference across the membrane, the membrane tension or by changing the composition of the different components of the membrane. We find that the osmotic pressure, determined by the neurofilaments, seems to be the dominant control parameter.     

Elastic energy of tilt and bending of fluid membranes

Tilt of hydrocarbon chains of lipid molecules with respect to membrane plane is commonly considered to characterize the internal structure of a membrane in the crystalline state. However, membranes in the liquid state can also exhibit tilt resulting from packing constraints imposed on the lipid molecules in diverse biologically relevant structures such as intermediates of membrane fusion, pores in lipid bilayers and others. We analyze the energetics of tilt in liquid membranes and its coupling with membrane bending. We consider three contributions to the elastic energy: constant tilt, variation of tilt along the membrane surface and membrane bending. The major assumption of the model is that the core of a liquid membrane has the common properties of an elastic continuum. We show that the variation of tilt and membrane bending are additive and that their energy contributions are determined by the same elastic coefficient: the Helfrich bending modulus, the modulus of Gaussian curvature and the spontaneous curvature known from previous studies of pure bending. The energy of a combined deformation of bending and varying tilt is determined by an effective tensor accounting for the two factors. In contrast, the deformation of constant tilt does not couple with bending and its contribution to the elastic energy is determined by an independent elastic constant. While accurate determination of this constant requires additional measurements, we estimate its value using a simplified approach. We discuss the relationships between the obtained elastic Hamiltonian of a membrane and the previous models of membrane elasticity. Received 10 February 2000 and Received in final form 19 June 2000     

Nano-meter-sized domain formation in lipid membranes observed by small angle neutron scattering

Using a contrast matching technique of small angle neutron scattering (SANS), we have investigated a phase separation to liquid-disordered and liquid-ordered phases on ternary small unilamellar vesicles (SUVs) composed of deuterated-saturated, hydrogenated-unsaturated phosphatidylcholine lipids and cholesterol, where the equilibrium size of these domains is constrained to less than 10nm by the system size. Below a miscibility temperature, we observed characteristic scattering profiles with a maximum, indicating the formation of nano-meter-sized domains on the SUVs. The observed profiles can be described by a multi-domain model rather than a mono-domain model. The nano-meter-sized domain is agitated by thermal fluctuations and eventually ruptured, which may result in the multi-domain state. The kinetically trapped nano-meter-sized domains grow to a mono-domain state by decreasing temperature. Furthermore, between the miscibility and disorder-order transition temperature of saturated lipid, the integrated SANS intensity increased slightly, indicating the formation of nano-meter-sized heterogeneity prior to the domain nucleation.     

Phase structure of intrinsic curvature models on dynamically triangulated disk with fixed boundary length

A first-order phase transition is found in two types of intrinsic curvature models defined on dynamically triangulated surfaces of disk topology. The intrinsic curvature energy is included in the Hamiltonian. The smooth phase is separated from a non-smooth phase by the transition. The crumpled phase, which is different from the non-smooth phase, also appears at sufficiently small curvature coefficient α. The phase structure of the model on the disk is identical to that of the spherical surface model, which was investigated by us and reported previously. Thus, we found that the phase structure of the fluid surface model with intrinsic curvature is independent of whether the surface is closed or open.     

Scaling theory of mixed amphiphilic monolayers

We predict the elastic properties of mixed amphiphilic monolayers in the swollen state within the blob model using scaling arguments. First the elastic moduli and the spontaneous curvature of a bimodal brush are determined as a function of the composition and the relative chain length. We obtain simple and useful scaling functions which interpolate between the elastic moduli of a pure short-chain brush and a pure long-chain brush. By using the analogy between block copolymer interfaces and polymeric brushes, the effect of mixing on self-assembled diblock copolymer monolayers is investigated in the swollen state. We calculate various interfacial properties, such as the equilibrium surface coverage, interface curvature, and the mixing free energy as a function of the composition. In general, we find a nonlinear dependence on the composition, which deviates from the simple linear averaging of the properties of pure components. Our results are used to discuss a recent experiment on the effect of amphiphilic block copolymers on the efficiency of microemulsions. Received 29 December 2000 and Received in final form 19 March 2001     

Cooperativity of multiple kinesin-1 motors mechanically coupled through a shared load

Recent experiments using single-molecule techniques have characterized the mechanical properties of single kinesin molecules in vitro at a range of loads and ATP concentrations. These experiments have shown that kinesin moves processively along microtubules by alternately advancing each of its motor domains in a hand-over-hand fashion, using Brownian motion and the energy from ATP hydrolysis. We have extended the theoretical analysis of kinesin through a mechanistic model that is capable of describing transient and steady-state behavior. Transient dynamics are needed to describe the effect of external perturbations (e.g. interactions with other kinesin molecules). Quantitative metrics are tailored to characterize the synchronization of nonlinear, nonsmooth systems such as kinesin. These metrics are employed to analyze the simulation results and to quantify the effect of the cargo linker stiffness, the load, and the difference in intrinsic velocity on the synchronization of two coupled motor proteins. Herein, the mechanistic model and the new analysis techniques are demonstrated for the case of two coupled kinesin motors.     

Elastica hypoarealis

We examine the equilibria of a rigid loop in the plane, characterized by an energy functional quadratic in the curvature, subject to the constraints of fixed length and fixed enclosed area. Whereas the only non self-intersecting equilibrium corresponding to the fixed length constraint is the circle, the area constraint gives rise to distinct equilibria labeled by an integer. These configurations exhibit self-intersections and bifurcations as the area is reduced. In addition, not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but the embedding itself can be expressed as a local function of the curvature. Perturbations connecting equilibria are shown to satisfy a first order ODE which is readily solved. Analytical expressions for the energy as a function of the area are obtained in the limiting regimes. Received 18 October 2001 / Received in final form 31 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: capo@fis.cinvestav.mx RID="b" ID="b"e-mail: chryss@nuclecu.unam.mx RID="c" ID="c"e-mail: jemal@nuclecu.unam.mx     

The hemifused state on the pathway to membrane fusion

Fusion of compartments enclosed by membrane bilayers enables secretion and other vital cellular processes and is widely studied in model synthetic membrane systems. Experiments suggest the fusion pathway passes through a hemifused intermediate where only outer monolayers are fused. Here we show membrane tension and divalent cations drive vesicles to hemifused equilibrium with expanded hemifusion diaphragms (HDs) where inner monolayers engage. Predicted HD sizes agree with recent measurements of Nikolaus et al. [Biophys. J. 98, 1192 (2010).]. The fusion pathway is completed by HD lysis provided HD tension is sufficiently high.     

Shapes of lipid monolayer domains: Solutions using elliptic functions

Solid lipid monolayer domains surrounded by a fluid phase at an air-water interface exhibit complex shapes. These intriguing shapes can be understood in terms of a competition between line tension and long-range dipole-dipole interaction. The dipolar energy has recently been relevant to a negative line tension and a positive curvature energy at the boundary, and a corresponding shape equation was derived by the variation of the approximated domain energy (Phys. Rev. Lett. 93, 206101 (2004)). Here we further incorporate surface pressure into the shape equation and show that the equation can be analytically solved: the curvature of the domain boundary is exactly obtained as an elliptic function of arc-length. We find that a circular domain can grow into bean-and peach-like domains with pressure, i.e., dipping and cuspidal transitions of circle by compression. The comparison with the experimental observation shows nice agreement.