We evaluate the effective Hamiltonian governing, at the optically resolved scale, the elastic properties of micro-manipulated membranes. We identify floppy, entropic-tense and stretched-tense regimes, representing different behaviors of the effective area-elasticity of the membrane. The corresponding effective tension depends on the microscopic parameters (total area, bending rigidity) and on the optically visible area, which is controlled by the imposed external constraints. We successfully compare our predictions with recent data on micropipette experiments.
We amplify previous arguments why mean curvature should be used as measure of integration in calculating the effective bending rigidity of fluid membranes subjected to a weak background curvature. The stiffening of the membrane by its fluctuations, recently derived for spherical shapes, is recovered for cylindrical curvature. Employing curvilinear coordinates, we then discuss stiffening for arbitrary shapes, confirm that the elastic modulus of Gaussian curvature is not renormalized in the presence of fluctuations, and show for the first time that any spontaneous curvature also remains unchange…
Fluid membranes endowed with tangent-plane order (TPO) such as tilt- and hexatic order afford unique soft matter systems for investigating the interplay between elasticity, shape, topology, and thermal fluctuations. Using the spin-connection formulation of membrane energy we obtain equa- tions of equilibrium together with free boundary conditions for ground states of such membranes. We extend the spin-connection formulation to smectic liquid crystals with TPO and show that for chiral smectics-C* this generalization leads to experimentally verifiable consequences for dispirations having topolog…
One of the most widely used methods for determination of the bending elasticity modulus of model lipid membranes is the analysis of the shape fluctuations of nearly spherical lipid vesicles. The theoretical basis of this analysis is given by Milner and Safran. In their theory the stretching effects are not considered. In the present study we generalized their approach including the stretching effects deduced after an application of statistical mechanics of vesicles.
A phenomenological Landau elasticity for the shape, dilation, and lipid-tilt of bilayer membranes is developed. The shape mode couples with the sum of the monolayers' tilt, while the dilation mode couples with the difference of the monolayers' tilts. Interactions among membrane inclusions within regular arrays are discussed. Inclusions modifying the membrane thickness and/or inducing a tilt-difference due to their convex or concave shape yield a dilation-induced attraction and a tilt-difference-induced repulsion. The resulting interaction can stabilize 2D crystal phases, with the possible coex…
Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solutions to these equations and their corresponding configurations are demonstrated.
We study the shape, elasticity and fluctuations of the recently predicted (cond-mat/9510172) and subsequently observed (in numerical simulations) (cond-mat/9705059) tubule phase of anisotropic membranes, as well as the phase transitions into and out of it. This novel phase lies between the previously predicted flat and crumpled phases, both in temperature and in its physical properties: it is crumpled in one direction, and extended in the other. Its shape and elastic properties are characterized by a radius of gyration exponent $ν$ and an anisotropy exponent $z$. We derive scaling laws for the…
Using computational and theoretical approaches, we investigate the snap-through transition of buckled graphene membranes. Our main interest is related to the possibility of using the buckled membrane as a plate of capacitor with memory (memcapacitor). For this purpose, we performed molecular-dynamics (MD) simulations and elasticity theory calculations of the up-to-down and down-to-up snap-through transitions for membranes of several sizes. We have obtained expressions for the threshold switching forces for both up-to-down and down-to-up transitions. Moreover, the up-to-down threshold switching…
The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed lipid bilayers, the shape equation and boundary conditions of open lipid bilayers and two-component membranes, the shape equation and in-plane strain equations of cell membranes with cross-linking structures, and the stabilities of closed lipid bilayers and cell membranes. The k…
Cell-cell adhesion is established by specific binding of receptor and ligand proteins. The adhesion bonds attract each other and often aggregate into large clusters that are central to many biological processes. One possible origin of attractive interactions between adhesion bonds is the elastic response of the membranes to their deformation by the bonds. Here, we analyze these elasticity-mediated interactions using a novel mean-field approach. Analysis of systems at different densities of bonds, $ϕ$, reveals that the phase diagram exhibits a nearly-universal behavior when the temperature $T$ …
Energies and equilibrium equations for thin elastic plates are discussed, with emphasis on several issues pertinent to recent approaches in soft condensed matter. Consequences of choice of basis, choice of invariant strain measures, and of approximating material energies as purely geometric in nature, are detailed. Ambiguities in the definitions of energies based on small-strain expansions, and in a typical informal process of dimensional reduction, are noted. A simple example serves to demonstrate that a commonly used bending energy has undesirable features, and it is suggested that a new the…
We have studied the electrostatic screening effect of NaCl solutions on the interactions between anionic lipid bilayers in the fluid lamellar phase using a Poisson-Boltzmann based mean-field approach with constant charge and constant potential limiting charge regulation boundary conditions. The full DLVO potential, including the electrostatic, hydration and van der Waals interactions, was coupled to thermal bending fluctuations of the membranes via a variational Gaussian Ansatz. This allowed us to analyze the coupling between the osmotic pressure and the fluctuation amplitudes and compare them…
The entropy of a polymer confined in a curved surface and the elastic free energy of a membrane consisting of polymers are obtained by scaling analysis. It is found that the elastic free energy of the membrane has the form of the in-plane strain energy plus Helfrich's curvature energy [Z. Naturforsch. C \textbf{28}, 693 (1973)]. The elastic constants in the free energy are obtained by discussing two simplified models: one is the polymer membrane without in-plane strains and asymmetry between its two sides, which is the counterpart of quantum mechanics in curved surface [Jensen and Koppe, Ann. …
Vesicles prepared in water from a series of diblock copolymers and termed "polymersomes" are physically characterized. With increasing molecular weight $\bar{M}_n$, the hydrophobic core thickness $d$ for the self-assembled bilayers of polyethyleneoxide - polybutadiene (PEO-PBD) increases up to 20 $nm$ - considerably greater than any previously studied lipid system. The mechanical responses of these membranes, specifically, the area elastic modulus $K_a$ and maximal areal strain $α_c$ are measured by micromanipulation. As expected for interface-dominated elasticity, $K_a$ ($\simeq$ 100 $pN/nm$)…
We have studied two types of meshwork models by using the canonical Monte Carlo simulation technique. The first meshwork model has elastic junctions, which are composed of vertices, bonds, and triangles, while the second model has rigid junctions, which are hexagonal (or pentagonal) rigid plates. Two-dimensional elasticity is assumed only at the elastic junctions in the first model, and no two-dimensional bending elasticity is assumed in the second model. Both of the meshworks are of spherical topology. We find that both models undergo a first-order collapsing transition between the smooth sph…
Cellular membranes are elastic lipid bilayers that contain a variety of proteins, including ion channels, receptors, and scaffolding proteins. These proteins are known to diffuse in the plane of the membrane and to influence the bending of the membrane. Experiments have shown that lipid flow in the plane of the membrane is closely coupled with the diffusion of proteins. Thus, there is a need for a comprehensive framework that accounts for the interplay between these processes. Here, we present a theory for the coupled in-plane viscous flow of lipids, diffusion of transmembrane proteins, and el…
We use coarse-grained Monte Carlo simulations to study the elastic properties of charged membranes in solutions of monovalent and pentavalent counterions. The simulation results of the two cases reveal trends opposite to each other. The bending rigidity and projected area increase with the membrane charge density for monovalent counterions, while they decrease for the pentavalent ions. These observations can be related to the counterion screening of the lipid charges. While the monovalent counterions only weakly screen the Coulomb interactions, which implies a repulsive Coulomb system, the mul…
FoF1-ATP synthase is the essential membrane enzyme maintaining the cellular level of adenosine triphosphate (ATP) and comprises two rotary motors. We measure subunit rotation in FoF1-ATP synthase by intramolecular Foerster resonance energy transfer (FRET) between two fluorophores at the rotor and at the stator of the enzyme. Confocal FRET measurements of freely diffusing single enzymes in lipid vesicles are limited to hundreds of milliseconds by the transit times through the laser focus. We evaluate two different methods to trap the enzyme inside the confocal volume in order to extend the obse…
One problem of non-crystalline condensed matter (soft matter) is creating the right equilibrium between elasticity and viscosity, referred to as viscoelasticity. Manifestations of that can be found in everyday live, where the viscoelasticity in a tire needs to be balanced so it is still flexible and can dissipate shock-energy, yet hard enough for energy-saving operation. Similarly, the cartilage in joints needs to absorb shocks while operating at low- level friction with high elasticity. Two such examples with a biological applicability are stiff membranes, which allow for the sliding of joint…
We preset a computational study of bending models for the curvature elasticity of lipid bilayer membranes that are relevant for simulations of vesicles and red blood cells. We compute bending energy and forces on triangulated meshes and evaluate and extend four well established schemes for their approximation: Kantor and Nelson 1987, Phys. Rev. A 36, 4020, Jülicher 1996, J. Phys. II France 6, 1797, Gompper and Kroll 1996, J. Phys. I France 6, 1305, and Meyer et. al. 2003 in Visualization and Mathematics III, Springer, p35, termed A, B, C, D. We present a comparative study of these four schemes…
Motivated by a freely suspended graphene and polymerized membranes in soft and biological matter we present a detailed study of a tensionless elastic sheet in the presence of thermal fluctuations and quenched disorder. The manuscript is based on an extensive draft dating back to 1993, that was circulated privately. It presents the general theoretical framework and calculational details of numerous results, partial forms of which have been published in brief Letters (Le Doussal and Radzihovsky 1992). The experimental realization of atom-thin graphene sheets has driven a resurgence in this fasci…
Continuum elastic models that account for membrane thickness variations are especially useful in the description of nanoscale deformations due to the presence of membrane proteins with hydrophobic mismatch. We show that terms involving the gradient and the Laplacian of the area per lipid are significant and must be retained in the effective Hamiltonian of the membrane. We reanalyze recent numerical data, as well as experimental data on gramicidin channels, in light of our model. This analysis yields consistent results for the term stemming from the gradient of the area per molecule. The order …
We study anomalous elasticity in the tubule phases of nematic and smectic elastomer membranes, which are flat in one direction and crumpled in another. These phases share the same macroscopic symmetry properties including spontaneously-broken in-plane isotropy and hence belong to the same universality class. Below an upper critical value $D_c =3$ of the membranes' intrinsic dimension D, thermal fluctuations renormalize the elasticity with respect to elastic displacements along the tubule axis so that elastic moduli for compression along the tubule axis and for bending the tubule axis become le…
We study the force generation by a set of parallel actin filaments growing against an elastic membrane. The elastic membrane tries to stay flat and any deformation from this flat state, either caused by thermal fluctuations or due to protrusive polymerization force exerted by the filaments, costs energy. We study two lattice models to describe the membrane dynamics. In one case, the energy cost is assumed to be proportional to the absolute magnitude of the height gradient (gradient model) and in the other case it is proportional to the square of the height gradient (Gaussian model). For the gr…
Adhesion between a T cell and an antigen presenting cell is achieved by TCR-pMHC and LFA1-ICAM1 protein complexes. These segregate to form a special pattern, known as the immunological synapse (IS), consisting of a central quasi-circular domain of TCR-pMHC bonds surrounded by a peripheral domain of LFA1-ICAM1 complexes. Insights gained from imaging studies had led to the conclusion that the formation of the central adhesion domain in the IS is driven by active (ATP-driven) mechanisms. Recent studies, however, suggested that passive (thermodynamic) mechanisms may also play an important role in …
Actomyosin actively generates contractile forces that provide the plasma membrane with the deformation stresses essential to carry out biological processes. Although the contractile property of purified actomyosin has been extensively studied, to understand the physical contribution of the actiomyosin contractile force on a deformable membrane is still a challenging problem and of great interest in the field of biophysics. Here, we reconstituted a model system with a cell-sized deformable interface that exhibits anomalous curvature dependent wrinkling caused by actomyosin cortex underneath the…
We use computer simulations and a simple free energy model to study the response of a bilayer membrane to the application of a negative (compressive) mechanical tension. Such a tension destabilizes the long wavelength undulation modes of giant vesicles, but it can be sustained when small membranes and vesicles are considered. Our negative tension simulation results reveal two regimes - (i) a weak negative tension regime characterized by stretching-dominated elasticity, and (ii) a strong negative tension regime featuring bending-dominated elastic behavior. This resembles the findings of the cla…
A continuum Landau theory for the micro-elasticity of membranes is discussed, which incorporates a coupling between the bilayer thickness variation and the difference in the two monolayers' tilts. This coupling stabilizes a new phase with a rippled micro-structure. Interactions among membrane inclusions combine a dilation-induced attraction and a tilt-difference-induced repulsion that yield 2D crystal phases, with possible coexistence of different lattice spacings for large couplings. Inclusions favoring crystals are those with either a long-convex or a short-concave hydrophobic core.
The mechanical properties of biological membranes play an important role in the structure and the functioning of living organisms. One of the most widely used methods for determination of the bending elasticity modulus of the model lipid membranes (simplified models of the biomembranes with similar mechanical properties) is analysis of the shape fluctuations of the nearly spherical lipid vesicles. A theoretical basis of such an analysis is developed by Milner and Safran. In the present studies we analyze their results using an approach based on the Bogoljubov inequalities and the approximating…
We propose a mechanism for mechanical regulation at the membrane of living cells, based on the exchange of membrane area between the cell membrane and a membrane reservoir. The reservoir is composed of invaginated membrane microdomains which are liable to flatten upon increase of membrane strain, effectively controlling membrane tension. We show that the domain shape transition is first order, allowing for coexistence between flat and invaginated domains. During coexistence, the membrane tension is controlled by the domains elasticity and by the kinetics of the shape transition. We show that t…
The dynamics of microcapsules in steady shear flow was studied using a theoretical approach based on three variables: The Taylor deformation parameter $α_{\rm D}$, the inclination angle $θ$, and the phase angle $ϕ$ of the membrane rotation. It is found that the dynamic phase diagram shows a remarkable change with an increase in the ratio of the membrane shear and bending elasticities. A fluid vesicle (no shear elasticity) exhibits three dynamic modes: (i) Tank-treading (TT) at low viscosity $η_{\rm {in}}$ of internal fluid ($α_{\rm D}$ and $θ$ relaxes to constant values), (ii) Tumbling (TB) at…
We study a Gaussian model of the membrane of red blood cells: a "phantom" triangular network of springs attached at its vertices to a fluid bilayer with curvature elasticity and tension. We calculate its fluctuation spectrum and we discuss the different regimes and non-monotonic features, including the precise crossover at the mesh size between the already known limits with two different tensions and the renormalisation of the bending rigidity at low wavevectors. We also show that the non-diagonal correlations reveal, in "dark field", the cytoskeletal defects. As a first step toward a non-inva…
The dynamical phase-space of axisymmetric Canham-Helfrich (CH) cells is constructed from a Hamiltonian field recapitulating membrane curvature-elasticity and systemic restrictions. Guiding principles are reparametrization to convert a static geometric system into a dynamical system, and Galilean transformation, to build a transformed Lagrangian invariant with respect to the action described by the CH free-energy. Building on the fluidity postulate, this Lagrangian describes the cellular membrane as an inverted harmonic oscillator driven by bending elasticity and effective friction governed by …
In practice, physical membranes are exposed to a certain amount of external strain (tension or compression), due to the environment where they are placed. As a result, the behavior of the phonon modes of the membrane is modified. We show that anharmonic effects in stiff two-dimensional membranes are highly suppressed under the application of tension. For this, we consider the anharmonic coupling between bending and stretching modes in the self-consistent screening approximation (SCSA), and compare the obtained height-height correlation function in the SCSA to the corresponding harmonic propaga…
The elastic properties of a self-assembled bilayer membrane are studied using the self-consistent field theory, applied to a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents. Examining the free energy of bilayer membranes with different geometries allows us to calculate their bending modulus, Gaussian modulus, two fourth-order membrane moduli, and the line tension. The dependence of these parameters on the microscopic characteristics of the amphiphilic chain, characterized by the volume fraction of the hydrophilic component, is systematically stu…
The effect of AC electric fields on the elasticity of supported lipid bilayers has been investigated at the microscopic level using grazing incidence synchrotron x-ray scattering. A strong decrease in the membrane tension up to 1mN/m and a dramatic increase of its effective rigidity up to 300kBT are observed for local electric potentials seen by the membrane < 1V. The experimental results were analyzed using detailed electrokinetic modeling and non-linear Poisson-Boltzmann theory. Based on a modeling of the electromagnetic stress which provides an accurate description of bilayer separation vs …
We explore micropatterned director structures of aqueous lyotropic chromonic liquid crystal (LCLC) films created on square lattice cylindrical-micropost substrates. The structures are manipulated by modulating the LCLC mesophases and their elastic properties via concentration through drying. Nematic LCLC films exhibit preferred bistable alignment along the diagonals of the micropost lattice. Columnar LCLC films, dried from nematics, form two distinct director and defect configurations: a diagonally aligned director pattern with local squares of defects, and an off-diagonal configuration with z…
The morphologies of genus-2 to -8 fluid vesicles are studied by using dynamically triangulated membrane simulations with area-difference elasticity. It is revealed that the alignments of the membrane pores alter the vesicle shapes and the types of shape transitions for the genus $g \ge 3$. At a high reduced volume, a stomatocyte with a circular alignment of $g+1$ pores continuously transforms into a discocyte with a line of $g$ pores with increasing intrinsic area difference. In contrast, at a low volume, a stomatocyte transforms into a ($g+1$)-hedral shape and subsequently exhibits a discrete…
