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.     

Phase separation in lipid membranes induced by the elastic properties of components

The role of the elastic properties of multicomponent lipid membranes during their phase separation has been studied. The phase diagram of a three-component membrane, which consists of cholesterol and unsaturated lipids, has been derived. The calculation has been performed using the theory of regular solutions, taking into account the contribution of the mechanical energy to the parameters of pair interactions between the components. It has been shown that the difference in the spontaneous curvatures of the components plays the major role in phase separation. The effective bending modulus has been calculated for the membrane whose components have different bending rigidities.     

Switching mechanics with chemistry: a model for the bending stiffness of amphiphilic bilayers with interacting headgroups in crystalline order

Bilayer structures in catanionic systems experimentally showed peculiar mechanical behavior. The observed increase in the bending stiffness is supposedly connected to additional hydrogen bonds forming between anionic headgroups. With a simple model, we can explain the extreme sensitivity of the bending stiffness of the membrane on the molar ratio of the charged molecules. This effect is further amplified by the sandwichlike structure of the membrane, where the apolar core separating the headgroups acts via a kind of lever-arm principle. As a consequence of these combined effects, the model membrane changes from a soft behavior with bending rigidities on the order of 10k(B)T to an extremely stiff membrane with a bending stiffness more than 2 orders of magnitude larger where most of this change occurs within a molar ratio interval smaller than 0.1.     

Stiffening of fluid membranes and entropy loss of membrane closure: Two effects of thermal undulations

The problem of membrane softening by thermal undulations is revisited. In contrast to general belief, fluid membranes are predicted to be stiffened, not softened, by their undulations. Equal values of the effective bending rigidity are calculated from the interplay of local mean curvature modes (hats) on the basically flat membrane and from the coupling of spherical harmonic modes with spherical curvature. In addition, a conjecture is made on the entropy of membrane closure. It relies on a similarity of membrane closure to periodic boundary conditions. Received: 10 June 1997 / Revised: 7 October 1997 / Accepted: 2 December 1997     

Sticking of hydrogen on supported and suspended graphene at low temperature

The physisorption of atomic hydrogen on graphene is investigated quantum mechanically using a semiempirical model for the lattice dynamics. A thermally averaged wave packet propagation describes the motion of the H atoms with respect to the membrane. Two graphene configurations, either supported on a silicone oxide substrate or suspended over a hole in the substrate, are considered. In both cases, the phonon spectrum is modified in such a way that graphene is stabilized with respect to thermal fluctuations. The sticking probabilities of hydrogen on these stabilized membranes at 10 K are high at low collision energies, and larger than on graphite.     

Elastic property of bilayer membrane in copolymer-homopolymer mixtures

We study the domain morphology in a phase separated state of diblock copolymer-homopolymer mixtures. In the situation that one of the blocks of copolymers is incompatible with homopolymers, formation of a bilayer membrane of block copolymers is shown to be possible in the matrix of homopolymers. Starting with the free energy functional in terms of the local volume fractions of each monomer, we derive the bending and curvature rigidities in the Helfrich free energy for a membrane. It is found that the curvature modulus is negative only in a limited region of the parameters, where a closed shape of a membrane like a vesicle is possibly formed. We establish a method to calculate the rigidities without a molecular picture in a consistent way with the field theoretic model free energy. Stability of a bilayer membrane compared with micelles is also investigated. Received: 22 July 1997 / Received in final form: 1 December 1997 / Accepted: 22 January 1998     

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.     

Formation and interaction of membrane tubes

We show that the formation of membrane tubes (or membrane tethers), which is a crucial step in many biological processes, is highly nontrivial and involves first-order shape transitions. The force exerted by an emerging tube is a nonmonotonic function of its length. We point out that tubes attract each other, which eventually leads to their coalescence. We also show that detached tubes behave like semiflexible filaments with a rather short persistence length. We suggest that these properties play an important role in the formation and structure of tubular organelles.     

Optimum distribution of additive damping for vibrating beams

Cost and weight effectiveness of concentrated and distributed additive damping is studied for linear systems (discrete and continuous) under prescribed harmonic loads and/or displacements. Stiffness and mass changes due to additive damping are included. From a numerical example studied it can be concluded that optimal damping distributions can reduce resonant responses by about 40% as compared to uniformly distributed damping of the same cost or weight. The optimization technique as well as an exact displacement method for analysis of harmonically vibrating beams and frames are presented.     

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.     

Fluid membranes can drive linear aggregation of adsorbed spherical nanoparticles

Using computer simulations, we show that lipid membranes can mediate linear aggregation of spherical nanoparticles binding to it for a wide range of biologically relevant bending rigidities. This result is in net contrast with the isotropic aggregation of nanoparticles on fluid interfaces or the expected clustering of isotropic insertions in biological membranes. We present a phase diagram indicating where linear aggregation is expected and compute explicitly the free-energy barriers associated with linear and isotropic aggregation. Finally, we provide simple scaling arguments to explain this phenomenology.     

Adhesion of membranes with competing specific and generic interactions

Biomimetic membranes in contact with a planar substrate or a second membrane are studied theoretically. The membranes contain specific adhesion molecules (stickers) which are attracted by the second surface. In the absence of stickers, the trans-interaction between the membrane and the second surface is assumed to be repulsive at short separations. It is shown that the interplay of specific attractive and generic repulsive interactions can lead to the formation of a potential barrier. This barrier induces a line tension between bound and unbound membrane segments which results in lateral phase separation during adhesion. The mechanism for adhesion-induced phase separation is rather general, as is demonstrated by considering two distinct cases involving: i) stickers with a linear attractive potential, and ii) stickers with a short-ranged square-well potential. In both cases, membrane fluctuations reduce the potential barrier and, therefore, decrease the tendency of phase separation. Received 24 January 2002 and Received in final form 24 April 2002     

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     

On Husimi Distribution for Systems with Continuous Spectrum

We propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. In the present article, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. There are other ways to obtain it, but we do not consider here. We specially discuss the problem of the continuous harmonic oscillator.     

Rosensweig instability of ferrogel thin films or membranes

We derive the dispersion relation of surface waves for magnetic gel membranes or thin films at the interface between two fluids in the presence of an external magnetic field normal to the free surface. Above a critical field strength surface waves become linearly unstable with respect to a stationary pattern of surface protuberances. This linear stability criterion generalizes that of the Rosensweig instability for ferrofluid and ferrogel free surfaces to take into account bending elasticity and intrinsic elastic and magnetic surface properties of the film or membrane, additionally. The latter is of interest for uniaxial ferrogel film or membranes, which show a locked-in permanent magnetization.     

Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Quantum Brownian motion and the second law of thermodynamics

We consider a single harmonic oscillator coupled to a bath at zero temperature. As is well-known, the oscillator then has a higher average energy than that given by its ground state. Here we show analytically that for a damping model with arbitrarily discrete distribution of bath modes and damping models with continuous distributions of bath modes with cut-off frequencies, this excess energy is less than the work needed to couple the system to the bath, therefore, the quantum second law is not violated. On the other hand, the second law may be violated for bath modes without cut-off frequencies, which are, however, physically unrealistic models. An erratum to this article is available at .     

Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein-Gordon lattice

We study the existence and stability of two-dimensional discrete breathers in a two-dimensionai discrete diatomic Klein-Gordon lattice consisting of alternating light and heavy atoms, with nearest-neighbor harmonic coupling. Localized solutions to the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum, i.e. two-dimensional gap breathers, are investigated numerically. The numerical results of the corresponding algebraic equations demonstrate the possibility of the existence of two-dimensional gap breathers with three types of symmetries, i.e., symmetric, twin-antisymmetric and single-antisymmetric. Their stability depends on the nonlinear on-site potential （soft or hard）, the interaction potential （attractive or repulsive） and the center of the two-dimensional gap breathers （on a light or a heavy atom）.     

Coherent control of harmonic generation in superlattices: single-mode response

We study the harmonic generation spectrum in a semiconductor superlattice (a quantum-dot array) at slow relaxation. The effect of a single-mode response in an alternating rectangular (meander) electric field is demonstrated: For certain values of field parameters, the extremely wide discrete output spectrum with slowly decaying tails (multiharmonic generation) shrinks to one single harmonic (single-harmonic generation). Similarly, the effect is manifested in the transient continuous spectrum by diminishing the divergencies (peaks) at all odd harmonics but one. Substantial control over the spectrum is demonstrated.