Nonlocal membrane bending: a reflection, the facts and its relevance.

About forty years ago it was realized that phospholipid membranes, because they are composed of two layers, exhibit particular, and specific mechanical properties. This led to the concept of nonlocal membrane bending, often called area difference elasticity. We present a short history of the development of the concept, followed by arguments for a proper definition of the corresponding elastic constant. The effects of the nonlocal bending energy on vesicle shape are explained. It is demonstrated that lipid vesicles, cells and cellular aggregates exhibit phenomena that can only be described in a complete manner by considering nonlocal bending.

Screwlike order, macroscopic chirality, and elastic distortions in high-density DNA mesophases.

We investigate a new screwlike liquid-crystalline ordering in solutions of helical biopolymers and its influence on the state of individual molecules. In the resulting mesophase translational and rotational motions of molecules are coupled in screw fluctuations. We show that in contrast to the case of conventional chiral liquid crystals the elastic distortion does not twist the screw order but leads to overwinding of individual helical molecules. This explains the peculiarities of high-density DNA mesophases.

Phospholipid membrane bending as assessed by the shape sequence of giant oblate phospholipid vesicles.

Vesicle shape transformations caused by decreasing the difference between the equilibrium areas of membrane monolayers were studied on phospholipid vesicles with small volume to membrane area ratios. Slow transformations of the vesicle shape were induced by lowering of the concentration of lipid monomers in the solution outside the vesicle. The complete sequence of shapes consisted of a string of pearls, and wormlike, starfish, discocyte and stomatocyte shapes. The transformation from discocyte to stomatocyte vesicle shapes was analyzed theoretically to see whether these observations accord with the area difference elasticity (ADE) model. The membrane shape equation and boundary conditions were derived for axisymmetrical shapes for low volume vesicles, part of whose membranes are in contact. Calculated shapes were arranged into a phase diagram. The theory predicts that the transition between discocyte and stomatocyte shapes is discontinuous for relatively high volumes and continuous for low volumes. The calculated shape sequences matched well with the observed ones. By assuming a linear decrease of the equilibrium area difference with time, the ratio between the nonlocal and local bending constants is in agreement with reported values.

Elastic properties of the red blood cell membrane that determine echinocyte deformability.

The natural biconcave shape of red blood cells (RBC) may be altered by injury or environmental conditions into a spiculated form (echinocyte). An analysis is presented of the effect of such a transformation on the resistance of RBC to entry into capillary sized cylindrical tubes. The analysis accounts for the elasticity of the membrane skeleton in dilation and shear, and the local and nonlocal resistance of the bilayer to bending, the latter corresponding to different area strains in the two leaflets of the bilayer. The shape transformation is assumed to be driven by the equilibrium area difference (delta A(0), the difference between the equilibrium areas of the bilayer leaflets), which also affects the energy of deformation. The cell shape is approximated by a parametric model. Shape parameters, skeleton shear deformation, and the skeleton density of deformed membrane relative to the skeleton density of undeformed membrane are obtained by minimization of the corresponding thermodynamic potential. Experimentally, delta A(0) is modified and the corresponding discocyte-echinocyte shape transition obtained by high-pressure aspiration into a narrow pipette, and the deformability of the resulting echinocyte is examined by whole cell aspiration into a larger pipette. We conclude that the deformability of the echinocyte can be accounted for by the mechanical behavior of the normal RBC membrane, where the equilibrium area difference delta A(0) is modified.

Enhanced chirality by adding achiral molecules into the chiral system.

It is generally accepted that the doping of chiral materials with achiral molecules diminishes the chirality of the system. Here we report the opposite phenomenon. It was found that the structural chirality of smectic phases made of rodlike molecules, Sm-C(*) or Sm-C(*)(A) phases, measured as the reverse length of the helical pitch, is enhanced by adding small amount of achiral bent-shaped molecules. The effect is due to the chirality transfer between host and guest systems. Achiral bent-shaped molecules become structurally chiral due to the interactions with the chiral host that induces tilt and polar order of bent-shaped molecules. This induced chirality is then transferred back to the host.

Enantiomeric excess dependence of the phase diagram of antiferroelectric liquid crystals.

The phase diagram of the prototype antiferroelectric liquid crystal 4-(1-methylheptyloxycarbonyl)phenyl-4'-octyloxybiphengl-4-carboxylate (MHPOBC) in dependence of enantiomeric excess was measured. It was shown that the Sm-C*beta phase in very pure samples is the Sm-C*(FI2) phase with a four-layer structure, and only after small racemization it transforms into the ferroelectric Sm-C* phase. The phase diagram was theoretically explained by taking into account longer range bilinear and short range biquadratic interlayer interactions, that lead to the distorted clock structures and first-order transitions between them.

Mechanisms of equinatoxin II-induced transport through the membrane of a giant phospholipid vesicle.

Protein equinatoxin II from sea anemone Actinia equina L. was used to form pores in phospholipid membranes. We studied the effect of these pores on the net transmembrane transport of sucrose and glucose by observing single giant (cell-size) vesicles under the phase contrast microscope. Sugar composition in the vesicle was determined by measuring the width of the halo, which appears around the vesicle in the phase contrast image. The transport of sugars was induced when a vesicle, filled with the sucrose solution, was transferred into the isomolar environment of a glucose solution with added equinatoxin II. Typically, a vesicle grew to a critical size, then the membrane broke by bursting and the vesicle shrank, started to grow again, and the whole process was repeated. The consecutive membrane breaks occurred in the same spot. The observed behavior was interpreted by the diffusion flow of the glucose molecules through the equinatoxin II-induced pores and the consequent increase of the vesicle water content. The burst relaxed the critically strained membrane, which then apparently resealed. A mathematical model of the described behavior was developed and was used to obtain the equinatoxin II-induced membrane permeability for the glucose molecules. Its dependence on the equinatoxin II concentration is in agreement with the previous reports.

Fractional occurrence of defects in membranes and mechanically driven interleaflet phospholipid transport.

The picture of biological membranes as uniform, homogeneous bileaflet structures has been revised in recent times due to the growing recognition that these structures can undergo significant fluctuations both in local curvature and in thickness. In particular, evidence has been obtained that a temporary, localized disordering of the lipid bilayer structure (defects) may serve as a principal pathway for movement of lipid molecules from one leaflet of the membrane to the other. How frequently these defects occur and how long they remain open are important unresolved questions. In this report, we calculate the rate of molecular transport through a transient defect in the membrane and compare this result to measurements of the net transbilayer flux of lipid molecules measured in an experiment in which the lipid flux is driven by differences between the mechanical stress in the two leaflets of the membrane bilayer. Based on this comparison, we estimate the frequency of defect occurrence in the membrane. The occurrence of defects is rare: the probability of finding a defect in 1.0 microm2 of a lecithin membrane is estimated to be approximately 6.0x10(-6). Based on this fractional occurrence of defects, the free energy of defect formation is estimated to be approximately 1.0x10(-19) J. The calculations provide support for a model in which interleaflet transport in membranes is accelerated by mechanically driven lipid flow.

Positional, reorientational, and bond orientational order in DNA mesophases.

We investigate the orientational order of transverse polarization vectors of long, stiff polymer molecules and their coupling to bond orientational and positional order in high density mesophases. Homogeneous ordering of transverse polarization vector promotes distortions in the hexatic phase, whereas inhomogeneous ordering precipitates crystallization of the 2D sections with different orientations of the transverse polarization vector on each molecule in the unit cell. We propose possible scenarios for going from the hexatic phase, through the distorted hexatic phase, to the crystalline phase with an orthorhombic unit cell observed experimentally for the case of DNA.

Mechanical and functional aspects of membrane skeletons.

Membrane skeletons can be characterized as cytoskeletal structures lying parallel to the bilayer part of cellular and organelle membranes. Typical examples are spectrin network and actin-myosin cortex. We approach the problem of elucidating the function of membrane skeletons by theoretically analyzing mechanical models of the cellular behavior. Membranes of different physical and chemical properties are considered. In erythrocytes and some organelles membrane bilayers are smooth and simply underlaid or overlaid by membrane skeletons. It is argued that there the role of a membrane skeleton is, either, to keep the membrane composition laterally homogeneous as it is in the case of the erythrocyte, or, that it is involved in the processes of the lateral separation of integral membrane proteins as it is happening in the case of some intermediate steps of the vesicular membrane trafficking. In the second type of membranes the bilayer part is ruffled and folded, and there the membrane skeletons play a role in the determination of the cortical tension. Here we explore in more detail the mechanical behavior of a cell with such properties of its boundary. The shape transformations are described which occur under the influence (i) of different external forces, i.e., when an originally spherical cell is aspirated into the micropipette or when such a cell is adsorbed on a flat surface, and (ii) of different internal forces on the cell boundary exerted by the cytoskeletal elements.

Membrane active compounds that affect the shape of cells and cellular organelles.

Amphiphilic membrane active compounds are considered that affect the shapes of cells and cellular organelles by intercalation into the phospholipid part of their membranes. It is taken into consideration that amphiphile-membrane interaction modifies membrane mechanical properties. The relationship between membrane mechanical properties and vesicle shapes and the concept of the bilayer couple model are shortly reviewed. Then it is put forward that the strength of the amphiphile-membrane interaction may depend on the lateral packing of phospholipid molecules. It is shown that in such a case the amphiphile molecules bind to the membrane in a cooperative manner. Moreover, the amphiphile binding makes the ratio between the nonlocal and local membrane bending constants to be effectively larger and thus widens the range of possible stable vesicle and cellular shapes.

Flexoelectricity and piezoelectricity: the reason for the rich variety of phases in antiferroelectric smectic liquid crystals.

The free energy of antiferroelectric smectic liquid crystals which takes into account polar order explicitly is presented. Steric, van der Waals, piezoelectric, and flexoelectric interactions to the nearest layers, and dipolar electrostatic interactions to the nearest and to the next-nearest layers, induce indirect tilt interactions with chiral and achiral properties, which extend to the third- and to the fourth-nearest layers. Although the strength of microscopic interactions changes monotonically with decreasing temperature, the effective interlayer interactions change nonmonotonically and give rise to a nonmonotonic change of the modulation period through various phases. Increased chirality changes the phase sequence.

Reentrant ferroelectricity in liquid crystals.

The ferroelectric (Sm C*)-antiferroelectric (Sm C*A)-reentrant ferroelectric (re Sm C*) phase temperature sequence was observed for systems with competing synclinic-anticlinic interactions. The basic properties of this system are as follows: (i) the Sm C* phase is metastable in the temperature range of the Sm C*A; (ii) the helix handedness inverts at both Sm C*-Sm C*A and Sm C*A-re-Sm C* phase transitions; (iii) the threshold electric field that is necessary to induce synclinic ordering in the Sm C*A phase decreases near both Sm C*A-Sm C* and Sm C*A-re-Sm C* phase boundaries. All these properties are properly described by a simple Landau model that accounts for nearest neighboring layer steric interactions and quadrupolar ordering only.

Uniplanar smectic phases in free-standing films

Structures of various phases that appear in antiferroelectric liquid crystals can be described within a simple discrete phenomenological model of antiferroelectric liquid crystals, which for the smectic-C-alpha (SmC(alpha)) phase suggests a short-pitch helicoidal structure. In a paper presented here the same model is used to analyze theoretically free-standing films, formed by a finite number of smectic layers. In the bulk sample a transition from the smectic-A (SmA) phase is either to the ferroelectric SmC* phase, the SmC(*)(A) phase, or the antiferroelectric SmC(alpha) phase, as observed from experiments as well as comprehended in the model. However in free-standing films uniplanar structures, which in a bulk sample were not observed experimentally nor predicted by the model, are shown to appear immediately below the transition from the high-temperature SmA phase. In free-standing film uniplanar structure remains stable in a narrow temperature region between SmA and SmC(alpha) phase.

Effect of pH on red blood cell deformability.

The effect of pH on the red blood cell (RBC) deformability, which is a consequence of a change of cell membrane elastic properties is studied experimentally. With the intention to reduce the effects on deformability of cell geometry and cytoplasmic viscosity, we measured the deformability of the cells with the same volume at various pH of cell suspension from 6.2 to 8.0. Constant cell volume was achieved by varying osmolarity. Deformability was quantified by measuring the elongation of RBCs subjected to velocity gradient in a transparent cone-plate rheoscope. Observed significant decrease of deformability at lower pH leads to the conclusion that membrane elastic properties could be affected by pH changes in the range from 6.2 to 8.0.

The influence of the successive depolymerization and polymerization of cytoskeleton components on the fibroblast shapes.

Monitoring the influence of the cytoskeleton polymers on the shape of fibroblasts, performing the experiments of repeated degradation and polymerization of microtubules and microfilaments, we found out that the presence of microtubules is necessary in order to regenerate the proper functional structure of microfilaments, and vice versa.

Stability analysis of micropipette aspiration of neutrophils.

During micropipette aspiration, neutrophil leukocytes exhibit a liquid-drop behavior, i.e., if a neutrophil is aspirated by a pressure larger than a certain threshold pressure, it flows continuously into the pipette. The point of the largest aspiration pressure at which the neutrophil can still be held in a stable equilibrium is called the critical point of aspiration. Here, we present a theoretical analysis of the equilibrium behavior and stability of a neutrophil during micropipette aspiration with the aim to rigorously characterize the critical point. We take the energy minimization approach, in which the critical point is well defined as the point of the stability breakdown. We use the basic liquid-drop model of neutrophil rheology extended by considering also the neutrophil elastic area expansivity. Our analysis predicts that the behavior at large pipette radii or small elastic area expansivity is close to the one predicted by the basic liquid-drop model, where the critical point is attained slightly before the projection length reaches the pipette radius. The effect of elastic area expansivity is qualitatively different at smaller pipette radii, where our analysis predicts that the critical point is attained at the projection lengths that may significantly exceed the pipette radius.

The relation between the rod-like inner structures and the shapes of the cells.

The shapes of the cells with simple rod-like inner structures are studied theoretically. Since the cell with inner structure can be bent, the possibility of non-axisymmetric shapes is considered. The equilibrium shape of the cell, obtained by minimizing the sum of the membrane bending energy and the bending energy of the rod, depends on the ratio between the bending constant of the membrane and the bending rigidity of the polymer rod. The dependence of the cell shape on the length of the rod and on the difference between inner and outer membrane layer areas is presented.

The frequency dependence of phospholipid vesicle shapes in an external electric field.

Experiments show that phospholipid vesicles exposed to AC electric field undergo a shape transition from prolate to oblate ellipsoidal shape when the frequency of the field is increased. A theoretical model, based on the minimization of total free energy of the vesicle, was devised to explain this phenomenon. The model exhibits the same frequency-dependent prolate-to-oblate shape transition as observed in the experiment.

Rotation of giant phospholipid vesicles in an uniform shear flow.

Rotation of giant "point attached" phospholipid (POPC) vesicles in a shear flow was studied. The dependence of the angular velocity on the flow gradient was measured and the experimental results were compared to the predictions of a theoretical model. A good linear correlation between the angular velocity of the vesicle and the flow gradient, as predicted, was observed.