Renormalization-group calculation of the dependence on gravity of the surface tension and bending rigidity of a fluid interface

The surface tension and the bending rigidity of a planar liquid-vapor interface in the presence of vanishing gravity are analyzed using the renormalization group. Based on the density functional theory of inhomogeneous fluids, we show that a term, quartic in the density fluctuations, can be added to the classical capillary-wave model so that a renormalization-group calculation can be performed. By comparing the outcome of such a calculation with rigorous results relating the direct correlation function with surface tension and bending rigidity, we find the scaling dependence of the latter on gravity. The results agree with the expected fact that the interface should become unstable as gravity vanishes.     

Dispersion relation of lipid membrane shape fluctuations by neutron spin-echo spectrometry

We have studied the mesoscopic shape fluctuations in aligned multilamellar stacks of 1,2-dimyristoyl-sn-glycero-3-phoshatidylcholine bilayers using the neutron spin-echo technique. The corresponding in plane dispersion relation tau-1(q//) at different temperatures in the gel (ripple, Pbeta') and the fluid (Lalpha) phase of this model system has been determined. Two relaxation processes, one at about 10 ns and a second, slower process at about 100 ns can be quantified. The dispersion relation in the fluid phase is fitted to a smectic hydrodynamic theory, with a correction for finite qz resolution. We extract values for the bilayer bending rigidity kappa, the compressional modulus of the stacks B, and the effective sliding viscosity eta3. The softening of a mode which can be associated with the formation of the ripple structure is observed close to the main phase transition.     

Effect of thermal undulations on the bending elasticity and spontaneous curvature of fluid membranes

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 unchanged. Received 19 April 1999 and in Received in final form 7 January 2000     

The concept of effective tension for fluctuating vesicles

Vesicles are closed surfaces of bilayer membranes. Their mean shapes and fluctuations are governed by the competition of curvature energy and geometrical constraints on the enclosed volume and total surface area. A scheme to calculate these fluctuations to lowest order in the ratio of temperature to bending rigidity is developed. It is shown that for fluctuations that break a symmetry of the mean shape the area constraint indeed acts like a tension whose value is given by the Lagrange multiplier used to enforce the area constraint in the first place. As a consequence, these fluctuations are also insensitive to the specific variants of the curvature model. For fluctuations that preserve the symmetry of the mean shape the role of the area constraint is more subtle. The low temperature expansion breaks down in the spherical limit where with the excess area another small parameter enters. By incorporating the area constraint in this limit exactly, the validity of the conventional approach using an effective tension for fluctuations of quasi-spherical vesicles can be assessed.Dedicated to Prof. Herbert Wagner on the occasion of his 60th birthday     

Exact treatment of interface fluctuations in a two-dimensional emulsion

Summary We present a model for a polydisperse ensemble of two-dimensional droplets wich accounts for the effects of arbitrarily large distortions of the droplet shape. Interactions within a droplet include bending rigidity and spontaneous curvature. Interactions between droplets are omitted. Even at high temperatures, the effects of the shape fluctuations on the droplet size distribution remain small, as they are dominated by the contributions from the mixing entropy. In contrast, shape fluctuations lead to a pronounced peak in thermodynamic quantities like specific heat. This peak occurs at temperatures where thermal excitations become of the order of the bending energies of the droplet surface. The fluctuation-dominated regime extends to temperatures far lower than expected from a mean-field calculation. Paper presented at the I International Conference on Scaling, Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

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.     

Fluctuation-induced forces between inclusions in a fluid membrane under tension

We develop an exact method to calculate thermal Casimir forces between inclusions of arbitrary shapes and separation, embedded in a fluid membrane whose fluctuations are governed by the combined action of surface tension, bending modulus, and Gaussian rigidity. Each object's shape and mechanical properties enter only through a characteristic matrix, a static analog of the scattering matrix. We calculate the Casimir interaction between two elastic disks embedded in a membrane. In particular, we find that at short separations the interaction is strong and independent of surface tension.     

Stretching instability of intrinsically curved semiflexible biopolymers: A lattice model approach

We apply a Monte Carlo simulation method to lattice systems to study the effect of an intrinsic curvature on the mechanical property of a semiflexible biopolymer.We find that when the intrinsic curvature is sufficiently large,the extension of a semiflexible biopolymer can undergo a first-order transition at finite temperature.The critical force increases with increasing intrinsic curvature.However,the relationship between the critical force and the bending rigidity is structuredependent.In a triangle lattice system,when the intrinsic curvature is smaller than a critical value,the critical force increases with the increasing bending rigidity first,and then decreases with the increasing bending rigidity.In a square lattice system,however,the critical force always decreases with the increasing bending rigidity.In contrast,when the intrinsic curvature is greater than the critical value,the larger bending rigidity always results in a larger critical force in both lattice systems.     

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.     

In-plane correlations in a polymer-supported lipid membrane measured by off-specular neutron scattering

Polymer-supported single lipid bilayers are models to study configurations of cell membranes. We used off-specular neutron scattering to quantify in-plane height-height correlations of interfacial fluctuations of such a lipid bilayer. As temperature decreased from 37 °C to 25 °C, the polymer swells and the polymer-supported lipid membrane deviates from its initially nearly planar structure. A correlation length characteristic of capillary waves changes from 30 μm at 37 °C to 11 μm at 25 °C, while the membrane bending rigidity remains roughly constant in this temperature range.     

Self-consistent interpretation of the 2D structure of the liquid Au82Si18 surface: bending rigidity and the Debye-Waller effect

The structural and mechanical properties of 2D crystalline surface phases that form at the surface of liquid eutectic Au82Si18 are studied using synchrotron x-ray scattering over a large temperature range. In the vicinity of the eutectic temperature the surface consists of a 2D atomic bilayer crystalline phase that transforms into a 2D monolayer crystalline phase during heating. The latter phase eventually melts into a liquidlike surface on further heating. We demonstrate that the short wavelength capillary wave fluctuations are suppressed due to the bending rigidity of 2D crystalline phases. The corresponding reduction in the Debye-Waller factor allows for measured reflectivity to be explained in terms of an electron density profile that is consistent with the 2D surface crystals.     

On the estimation of the curvatures and bending rigidity of membrane networks via a local maximum-entropy approach

We present a meshfree method for the curvature estimation of membrane networks based on the local maximum entropy approach recently presented in [1]. A continuum regularization of the network is carried out by balancing the maximization of the information entropy corresponding to the nodal data, with the minimization of the total width of the shape functions. The accuracy and convergence properties of the given curvature prediction procedure are assessed through numerical applications to benchmark problems, which include coarse grained molecular dynamics simulations of the fluctuations of red blood cell membranes [2], [3]. We also provide an energetic discrete-to-continuum approach to the prediction of the zero-temperature bending rigidity of membrane networks, which is based on the integration of the local curvature estimates. The local maximum entropy approach is easily applicable to the continuum regularization of fluctuating membranes, and the prediction of membrane and bending elasticities of molecular dynamics models.     

Limits of filopodium stability

Filopodia are long, fingerlike membrane tubes supported by cytoskeletal filaments. Their shape is determined by the stiffness of the actin filament bundles found inside them and by the interplay between the surface tension and bending rigidity of the membrane. Although one might expect the Euler buckling instability to limit the length of filopodia, we show through simple energetic considerations that this is in general not the case. By further analyzing the statics of filaments inside membrane tubes, and through computer simulations that capture membrane and filament fluctuations, we show under which conditions filopodia of arbitrary lengths are stable. We discuss several in vitro experiments where this kind of stability has already been observed. Furthermore, we predict that the filaments in long, stable filopodia adopt a helical shape.     

Supported bilayers: Combined specular and diffuse X-ray scattering

A method is proposed for the analysis of specular and off-specular reflectivity from supported lipid bilayers. Both thermal fluctuations and the “static” roughness induced by the substrate are carefully taken into account. Examples from supported bilayers and more complex systems comprising a bilayer adsorbed or grafted on the substrate and another “floating” bilayer are given. The combined analysis of specular and off-specular reflectivity allows the precise determination of the structure of adsorbed and floating bilayers, their tension, bending rigidity and interaction potentials. We show that this new method gives a unique opportunity to investigate phenomena like protrusion modes of adsorbed bilayers and opens the way to the investigation of more complex systems including different kinds of lipids, cholesterol or peptides.     

A continuum model of a multilayer nanosheet

A continuum model for describing the bending and free vibrations of a crystalline graphite sheet consisting of graphene layers is proposed. Graphene is modeled by a two-dimensional layer having a finite rigidity under extension and bending. The interval between graphene layers through which their Van-der-Waals interaction occurs is modeled by a fictitious layer with relatively low rigidity. In the solution, formulas describing the bending of a multilayer sheet with alternating rigid and soft layers are used.     

Nonequilibrium microtubule fluctuations in a model cytoskeleton

Biological activity gives rise to nonequilibrium fluctuations in the cytoplasm of cells; however, there are few methods to directly measure these fluctuations. Using a reconstituted actin cytoskeleton, we show that the bending dynamics of embedded microtubules can be used to probe local stress fluctuations. We add myosin motors that drive the network out of equilibrium, resulting in an increased amplitude and modified time dependence of microtubule bending fluctuations. We show that this behavior results from steplike forces on the order of 10 pN driven by collective motor dynamics.     

Gradient-driven fluctuations in microgravity

Equilibrium fluctuations of thermodynamic variables, such as density or concentration, are known to be small and typically occur at a molecular length scale. In contrast, theory predicts that non-equilibrium fluctuations grow very large both in amplitude and spatial size. On earth, the presence of gravity and buoyancy forces severely limits the development of the fluctuations. We will present the results of a 14-year long international collaboration on an experiment on non-equilibrium fluctuations in a single liquid and in a polymer solution under microgravity conditions. Non-equilibrium conditions are generated by applying a temperature gradient across millimetre-size liquid slabs. Phase modulations introduced by fluctuations are measured using a quantitative shadowgraph method, with the optical axis parallel to the temperature gradient. Thousands of images are analysed and their two-dimensional power spectra yield the fluctuation structure function S(q), once data are reduced accounting for the instrumental transfer function T(q). The mean-squared amplitude of the fluctuations exhibits an impressive power-law dependence at larger q and a crossover at low q showing that the fluctuation size is limited by the sample thickness. The shape of the structure function, its increase due to removing gravity, and its dependence on applied gradient are in reasonable agreement with available theoretical predictions.     

Crossover from intermittent to continuum dynamics for locally driven colloids

We simulate a colloid with charge q(d) driven through a disordered assembly of interacting colloids with charge q and show that, for q(d) approximately q, the velocity-force relation is nonlinear and the velocity fluctuations of the driven particle are highly intermittent with a 1/f characteristic. When g(d) >q , the average velocity drops, the velocity-force relation becomes linear, and the velocity fluctuations are Gaussian. We discuss the results in terms of a crossover from strongly intermittent heterogeneous dynamics to continuum dynamics. We also make several predictions for the transient response in the different regimes.     

Simulations of the bending rigidity of graphene

Molecular mechanics simulations for graphene bending rigidity are reported through calculations of the strain energy for graphene sheets subjected to a point loading. The rigidity is found to be dependent on the size and the shape of graphene sheets. Moreover, dependence of the rigidity on the deflection is found.     

Effect of internal friction on biofilament dynamics

We consider biofilaments-flexible multimolecule structures common in cell biology-and show how "internal" friction associated with either conformational fluctuations or with fluid flow through narrow pores inside the filaments can dominate the external hydrodynamic friction usually considered to be the main energy dissipation process. The signature of this is wave-number-independent relaxation time of bending fluctuations. Preliminary experimental data for bending fluctuations of single folded (mitotic) chromosomes display these dynamics.