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.     

Vesicle self-reproduction: The involvement of membrane hydraulic and solute permeabilities

Conditions for self-reproduction are sought for a growing vesicle with its growth defined by an exponential increase of vesicle membrane area and by adequate flow of the solution across the membrane. In the first step of the presumed vesicle self-reproduction process, the initially spherical vesicle must double its volume in the doubling time of the membrane area and, through the appropriate shape transformations, attain the shape of two equal spheres connected by an infinitesimally thin neck. The second step involves separation of the two spheres and relies on conditions that cause the neck to be broken. In this paper we consider the first step of this self-reproduction process for a vesicle suspended in a solution whose solute can permeate the vesicle membrane. It is shown that vesicle self-reproduction occurs only for certain combinations of the values of membrane hydraulic and solute permeabilities and the external solute concentration, these quantities being related to the mechanical properties of the membrane and the membrane area doubling time. The analysis includes also the relaxation of a perturbed system towards stationary self-reproduction behavior and the case where the final shape consists of two connected spheres of different radii.     

Polymers grafted to a fluid and flexible membrane: Extreme sensitivity to the grafting density

Equilibrium phase coexistence between two chemical species implies the equality of the chemical potentials and of the osmotic pressures. We study this problem on a deformable membrane when one type of the molecules serves as anchor for polymeric chains immersed in the surrounding medium (considered as a good solvent). We derive the general conditions for phase coexistence when both the curvature of the membrane and the density field of the anchor molecule are free to adjust themselves. We show that curvature favors phase segregation. Our model predicts that membranes decorated with polymeric chains exhibit new shape bifurcations without equivalent in fixed density systems. Received: 26 November 2002 / Accepted: 2 April 2003 / Published online: 12 May 2003 RID="a" ID="a"e-mail: nicolas@drfmc.ceng.cea.fr RID="b" ID="b"e-mail: bfourcade@cea.fr     

Dynamics of viscous vesicles in shear flow

The dynamics of giant lipid vesicles under shear flow is experimentally investigated. Consistent with previous theoretical and numerical studies, two flow regimes are identified depending on the viscosity ratio between the interior and the exterior of the vesicle, and its reduced volume or excess surface. At low viscosity ratios, a tank-treading motion of the membrane takes place, the vesicle assuming a constant orientation with respect to the flow direction. At higher viscosity ratios, a tumbling motion is observed in which the whole vesicle rotates with a periodically modulated velocity. When the shear rate increases, this tumbling motion becomes increasingly sensitive to vesicle deformation due to the elongational component of the flow and significant deviations from simpler models are observed. A good characterization of these various flow regimes is essential for the validation of analytical and numerical models, and to relate microscopic dynamics to macroscopic rheology of suspensions of deformable particles, such as blood.     

New solutions of the shape equation

The general shape equation describing the forms of vesicles is a highly nonlinear partial differential equation for which only a few explicit solutions are known. These solvable cases are briefly reviewed and a new analytical solution which represents the class of the constant mean curvature surfaces is described. Pearling states of the tubular fluid membranes can be explained as a continuous deformation preserving membrane mean curvature. Received 2 February 2002 / Received in final form 4 February 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: mladenov@obzor.bio21.bas.bg     

Elasticity and shape equation of a liquid membrane

The density of the elastic energy of a deformed membrane in a liquid state is calculated. The thermodynamic equilibrium of its different parts is taken into account. The shape equation of a closed membrane is deduced. The quantity which keeps its value, when the variations of the energy of the system are calculated, is not the area of the deformed membrane, but its area in the flat tension free state. Because of this, additional terms appear in the second variation around the stable state. The case of a lipid bilayer and its fluctuations is examined for both free and blocked exchange of molecules between the monolayers, comprising the bilayer. Received 4 February 2002 / Received in final form 15 April 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: bivas@issp.bas.bg     

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     

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.     

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.     

Simulation study of lateral diffusion in lipid-sterol bilayer mixtures

We employ off-lattice Monte Carlo simulations to study lateral diffusion in lipid-sterol bilayers using a two-dimensional model system which has been designed to simulate the experimental phase diagrams of both lipid-cholesterol and lipid-lanosterol systems. We focus on the effects of varying sterol concentration and temperature on the tracer diffusion coefficient, D, which characterizes the lateral motion of single tagged lipids in a bilayer. Generally, we find that increasing the cholesterol concentration suppresses D due to an increased conformational ordering of lipid chains. We argue that this effect competes with an increase in the average free area per lipid, which favours an increase in D. At temperatures close to the main transition temperature, the competition between the two effects leads to intriguing behavior of D. Overall, the model results are in excellent qualitative agreement with available experimental results for lipid-cholesterol mixtures. Additional studies of a model lipid-lanosterol system, for which experimental diffusion results are not available, predict that the presence of lanosterol has a smaller effect than cholesterol on reducing D relative to the pure lipid system. We conclude that the molecular model employed contains the essential features required to describe many of the qualitative features of the lateral diffusion behavior in lipid-sterol systems. Received 24 November 2000 and Received in final form 30 April 2001     

Budding and fission of a multiphase vesicle

We present a model of bi-phasic vesicles in the limit of large surface tension. In this regime, the vesicle is completely stretched and well described by two spherical caps with a fold, which concentrates the membrane stress. The conservation laws and geometric constraints restrict the space of possible shapes to a pair of solutions labeled by a parameter given by line tension/pressure. For a given value of , the two solutions differ by the length of the interface between domains. For a critical value, , the two vesicle shapes become identical and no connected solution exists above this critical value. This model sheds new light on two proposed mechanisms (osmotic shocks and molecule absorption) to explain the budding and the fission in recent experiments.     

Size and stability of liposomes: A possible role of hydration and osmotic forces

Dynamic light scattering and electrophoretic mobility measurements have been used to characterize the size, size distribution and zeta potentials (ζ-potentials) of egg yolk phosphatidylcholine (EYPC) liposomes in the presence of monovalent ions ( Na⁺ and K⁺). To study the stability of liposomes the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory has been extended by introducing the hydrated radius of the adsorbed ions onto the liposome surfaces. The decrease of liposome size is explained on the basis of the membrane impermeability to some ions which generate osmotic forces, which leads to evacuate water from liposome inside.     

Structure and interaction potentials in solid-supported lipid membranes studied by X-ray reflectivity at varied osmotic pressure

Highly oriented solid-supported lipid membranes in stacks of controlled number N ≃ 16 (oligo-membranes) have been prepared by spin-coating using the uncharged lipid model system 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC). The samples have been immersed in aqueous polymer solutions for control of osmotic pressure and have been studied by X-ray reflectivity. The bilayer structure and fluctuations have been determined by modelling the data over the full q-range. Thermal fluctuations are described using the continuous smectic Hamiltonian with the appropriate boundary conditions at the substrate and at the free surface of the stack. The resulting fluctuation amplitudes and the pressure-distance relation are discussed in view of the inter-bilayer potential.     

Influence of cholesterol on the bilayer properties of monounsaturated phosphatidylcholine unilamellar vesicles

The influence of cholesterol on the structure of unilamellar-vesicle (ULV) phospholipid bilayers is studied using small-angle neutron scattering. ULVs made up of short-, mid- and long-chain monounsaturated phospholipids (diCn :1PC, n = 14 , 18, 22, respectively) are examined over a range (0-45mol %) of cholesterol concentrations. Cholesterol's effect on bilayer structure is characterized through changes to the lipid's transmembrane thickness, lateral area and headgroup hydration. For all three lipids, analysis of the experimental data shows that the addition of cholesterol results in a monotonic increase of these parameters. In the case of the short- and mid-chain lipids, this is an expected result, however, such a finding was unexpected for the long-chain lipid. This implies that cholesterol has a pronounced effect on the lipid's hydrocarbon chain organization.     

Fluctuation spectra of free and supported membrane pairs

Fluctuation spectra of fluid compound membrane systems are calculated. The systems addressed contain two (or more) almost parallel membranes that are connected by harmonic tethers or by a continuous, harmonic confining potential. Additionally, such a compound system can be attached to a supporting substrate. We compare quasi-analytical results for tethers with analytical results for corresponding continuous models and investigate under what circumstances the discrete nature of the tethers actually influences the fluctuations. A tethered, supported membrane pair with similar bending rigidities and stiff tethers can possess a nonmonotonic fluctuation spectrum with a maximum. A nonmonotonic spectrum with a maximum and a minimum can occur for an either free or supported membrane pair of rather different bending rigidities and for stiff tethers. Typical membrane displacements are calculated for supported membrane pairs with discrete or continuous interacting potentials. Thereby an estimate of how close the constituent two membranes and the substrate typically approach each other is given. For a supported membrane pair with discrete or continuous interactions, the typical displacements of each membrane are altered with respect to a single supported membrane, where those of the membrane near the substrate are diminished and those of the membrane further away are enhanced.     

On coupling between the orientation and the shape of a vesicle under a shear flow

A simple 2D model of deformable vesicles tumbling in a shear under flow is introduced in order to account for the main qualitative features observed experimentally as shear rates are increased. The simplicity of the model allows for a full analytical tractability while retaining the essential physical ingredients. The model reveals that the main axes of the vesicle undergo oscillations which are coupled to the vesicle orientation in the flow. The model reproduces and sheds light on the main novel features reported in recent experiments [M. Mader et al., Eur. Phys. J. E. 19, 389 (2006)], namely that both coefficients A and B that enter the Keller-Skalak equation, dψ/dt = A+Bcos(2 ψ) (ψ is the vesicle orientation angle in the shear flow), undergo a collapse upon increasing shear rate.     

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