Internal forces, tension and energy density in tethered cellular membranes

We analyze tethered cellular membranes by considering the membrane resultants, tension and densities of two modes of energy, bending and adhesion. These characteristics are determined based on a computational (finite-difference) analysis of membrane shape. We analyze the relative contribution and distribution of the membrane characteristics in four typical zones of the membrane surface. Using an axisymmetric model, we found that the meridional and circumferential components of the resultant are different near the tether body and they converge to the value of membrane tension farther from the tether. At the beginning of the area of membrane detachment from the cytoskeleton, the density of bending energy is on the same order of magnitude as membrane tension (resultant). Away from the tether, the bending energy density quickly decreases and becomes of the same order as that of the adhesion energy in the membrane-cytoskeleton attachment area. In that area, both modes of energy are significantly smaller than the membrane tension. We also consider the effect of the membrane bending modulus on the distribution of the membrane characteristics. An increase in the bending modulus results in changing the length and position on the membrane surface of zone 1 characterized by significant evolution of the resultant components. It also results in shortening zone 2 that covers the rest of the area of membrane detachment. The obtained results can help in a better interpretation of the measurements of membrane mechanical properties as well as in analyses of proteins and channels in curved membranes.     

Role of the membrane cortex in neutrophil deformation in small pipets.

The simplest model for a neutrophil in its "passive" state views the cell as consisting of a liquid-like cytoplasmic region surrounded by a membrane. The cell surface is in a state of isotropic contraction, which causes the cell to assume a spherical shape. This contraction is characterized by the cortical tension. The cortical tension shows a weak area dilation dependence, and it determines the elastic properties of the cell for small curvature deformations. At high curvature deformations in small pipets (with internal radii less than 1 micron), the measured critical suction pressure for cell flow into the pipet is larger than its estimate from the law of Laplace. A model is proposed where the region consisting of the cytoplasm membrane and the underlying cortex (having a finite thickness) is introduced at the cell surface. The mechanical properties of this region are characterized by the apparent cortical tension (defined as a free contraction energy per unit area) and the apparent bending modulus (introduced as a bending free energy per unit area) of its middle plane. The model predicts that for small curvature deformations (in pipets having radii larger than 1.2 microns) the role of the cortical thickness and the resistance for bending of the membrane-cortex complex is negligible. For high curvature deformations, they lead to elevated suction pressures above the values predicted from the law of Laplace. The existence of elevated suction pressures for pipets with radii from 1 micron down to 0.24 micron is found experimentally. The measured excess suction pressures cannot be explained only by the modified law of Laplace (for a cortex with finite thickness and negligible bending resistance), because it predicts unacceptable high cortical thicknesses (from 0.3 to 0.7 micron). It is concluded that the membrane-cortex complex has an apparent bending modulus from 1 x 10(-18) to 2 x 10(-18) J for a cortex with a thickness from 0.1 micron down to values much smaller than the radius of the smallest pipet (0.24 micron) used in this study.     

Flickering Analysis of Erythrocyte Mechanical Properties: Dependence on Oxygenation Level, Cell Shape, and Hydration Level

Erythrocytes (red blood cells) play an essential role in the respiratory functions of vertebrates, carrying oxygen from lungs to tissues and CO₂ from tissues to lungs. They are mechanically very soft, enabling circulation through small capillaries. The small thermally induced displacements of the membrane provide an important tool in the investigation of the mechanics of the cell membrane. However, despite numerous studies, uncertainties in the interpretation of the data, and in the values derived for the main parameters of cell mechanics, have rendered past conclusions from the fluctuation approach somewhat controversial. Here we revisit the experimental method and theoretical analysis of fluctuations, to adapt them to the case of cell contour fluctuations, which are readily observable experimentally. This enables direct measurements of membrane tension, of bending modulus, and of the viscosity of the cell cytoplasm. Of the various factors that influence the mechanical properties of the cell, we focus here on: 1), the level of oxygenation, as monitored by Raman spectrometry; 2), cell shape; and 3), the concentration of hemoglobin. The results show that, contrary to previous reports, there is no significant difference in cell tension and bending modulus between oxygenated and deoxygenated states, in line with the softness requirement for optimal circulatory flow in both states. On the other hand, tension and bending moduli of discocyte- and spherocyte-shaped cells differ markedly, in both the oxygenated and deoxygenated states. The tension in spherocytes is much higher, consistent with recent theoretical models that describe the transitions between red blood cell shapes as a function of membrane tension. Cell cytoplasmic viscosity is strongly influenced by the hydration state. The implications of these results to circulatory flow dynamics in physiological and pathological conditions are discussed.     

Membrane Elastic Properties and Cell Function

Recent studies indicate that the cell membrane, interacting with its attached cytoskeleton, is an important regulator of cell function, exerting and responding to forces. We investigate this relationship by looking for connections between cell membrane elastic properties, especially surface tension and bending modulus, and cell function. Those properties are measured by pulling tethers from the cell membrane with optical tweezers. Their values are determined for all major cell types of the central nervous system, as well as for macrophage. Astrocytes and glioblastoma cells, which are considerably more dynamic than neurons, have substantially larger surface tensions. Resting microglia, which continually scan their environment through motility and protrusions, have the highest elastic constants, with values similar to those for resting macrophage. For both microglia and macrophage, we find a sharp softening of bending modulus between their resting and activated forms, which is very advantageous for their acquisition of phagocytic functions upon activation. We also determine the elastic constants of pure cell membrane, with no attached cytoskeleton. For all cell types, the presence of F-actin within tethers, contrary to conventional wisdom, is confirmed. Our findings suggest the existence of a close connection between membrane elastic constants and cell function.     

Effect of hydroperoxides on red blood cell membrane mechanical properties

We investigate the effect of oxidative stress on red blood cell membrane mechanical properties in vitro using detailed analysis of the membrane thermal fluctuation spectrum. Two different oxidants, the cytosol-soluble hydrogen peroxide and the membrane-soluble cumene hydroperoxide, are used, and their effects on the membrane bending elastic modulus, surface tension, strength of confinement due to the membrane skeleton, and 2D shear elastic modulus are measured. We find that both oxidants alter significantly the membrane elastic properties, but their effects differ qualitatively and quantitatively. While hydrogen peroxide mainly affects the elasticity of the membrane protein skeleton (increasing the membrane shear modulus), cumene hydroperoxide has an impact on both membrane skeleton and lipid bilayer mechanical properties, as can be seen from the increased values of the shear and bending elastic moduli. The biologically important implication of these results is that the effects of oxidative stress on the biophysical properties, and hence the physiological functions, of the cell membrane depend on the nature of the oxidative agent. Thermal fluctuation spectroscopy provides a means of characterizing these different effects, potentially in a clinical milieu.     

Minimum energy analysis of membrane deformation applied to pipet aspiration and surface adhesion of red blood cells.

An experimental procedure is demonstrated which can be used to determine the interfacial free energy density for red cell membrane adhesion and membrane elastic properties. The experiment involves micropipet aspiration of a flaccid red blood cell and manipulation of the cell proximal to a surface where adhesion occurs. A minimum free energy method is developed to model the equilibrium contour of unsupported membrane regions and to evaluate the partial derivatives of the total free energy, which correspond to the micropipet suction force and the interfacial free energy density of adhesion. It is shown that the bending elasticity of the red cell membrane does not contribute significantly to the pressure required to aspirate a flaccid red cell. Based on experimental evidence, the upper bound for the bending or curvature elastic modulus of the red cell membranes is 10-12 ergs (dyn-cm). Analysis of the adhesion experiment shows that interfacial free energy densities for red cell adhesion can be measured from a lower limit of 10-4 ergs/cm2 to an upper limit established by the membrane tension for lysis of 5-10 ergs/cm2.     

Spectrin, human erythrocyte shapes, and mechanochemical properties.

Physical studies of human erythrocyte spectrin indicate that isolated spectrin dimers and tetramers in solution are worm-like coils with a persistence length of approximately 20 nm. This finding, the known polyelectrolytic nature of spectrin, and other structural information about spectrin and the membrane skeleton molecular organization have lead us to the hypothesis that the human erythrocyte membrane skeleton constitutes a two-dimensional ionic gel (swollen ionic elastomer). This concept is incorporated in what we refer to as the protein gel-lipid bilayer membrane model. The model accounts quantitatively for red elastic shear modulus and the maximum elastic extension ratio reported for the human erythrocytes membrane. Gel theory further predicts that depending on the environmental conditions, the membrane skeleton modulus of area compression may be small or large relative to the membrane elastic shear modulus. Our analyses show that the ratio between these two parameters affects both the geometry and the stability of the favored cell shapes and that the higher the membrane skeleton compressibility the smaller the values of the gel tension needed to induce cell shape transformations. The main virtue of the protein gel-lipid bilayer membrane model is that it offers a novel theoretical and molecular basis for the various mechanical properties of the membrane skeleton such as the membrane skeleton modulus of area compression and osmotic tension, and the effects of these properties on local membrane skeleton density, cell shape, and shape transformations.     

The influence of short-chain alcohols on interfacial tension, mechanical properties, area/molecule, and permeability of fluid lipid bilayers

We used micropipette aspiration to directly measure the area compressibility modulus, bending modulus, lysis tension, lysis strain, and area expansion of fluid phase 1-stearoyl, 2-oleoyl phosphatidylcholine (SOPC) lipid bilayers exposed to aqueous solutions of short-chain alcohols at alcohol concentrations ranging from 0.1 to 9.8 M. The order of effectiveness in decreasing mechanical properties and increasing area per molecule was butanol>propanol>ethanol>methanol, although the lysis strain was invariant to alcohol chain-length. Quantitatively, the trend in area compressibility modulus follows Traube's rule of interfacial tension reduction, i.e., for each additional alcohol CH(2) group, the concentration required to reach the same area compressibility modulus was reduced roughly by a factor of 3. We convert our area compressibility data into interfacial tension values to: confirm that Traube's rule is followed for bilayers; show that alcohols decrease the interfacial tension of bilayer-water interfaces less effectively than oil-water interfaces; determine the partition coefficients and standard Gibbs adsorption energy per CH(2) group for adsorption of alcohol into the lipid headgroup region; and predict the increase in area per headgroup as well as the critical radius and line tension of a membrane pore for each concentration and chain-length of alcohol. The area expansion predictions were confirmed by direct measurements of the area expansion of vesicles exposed to flowing alcohol solutions. These measurements were fitted to a membrane kinetic model to find membrane permeability coefficients of short-chain alcohols. Taken together, the evidence presented here supports a view that alcohol partitioning into the bilayer headgroup region, with enhanced partitioning as the chain-length of the alcohol increases, results in chain-length-dependent interfacial tension reduction with concomitant chain-length-dependent reduction in mechanical moduli and membrane thickness.     

Measuring optical and mechanical properties of a living cell with defocusing microscopy

Defocusing microscopy (DM) is a recently developed technique that allows quantitative analysis of membrane surface dynamics of living cells using a simple bright-field optical microscope. According to DM, the contrast of defocused images is proportional to cell surface curvature. Although, until now, this technique was used mainly to determine size and amount of membrane shape fluctuations, such as ruffles and small random membrane fluctuations, in macrophages, its applications on cell biology extend beyond that. We show how DM can be used to measure optical and mechanical properties of a living macrophage, such as cell refractive index n, membrane bending modulus K(c), and effective cell viscosity eta for membrane-actin meshwork relaxation. Experimental data collected from defocused images of bone marrow-derived macrophages were used to evaluate these parameters. The obtained values, averaged over several different macrophages, are n = (1.384 +/- 0.015), K(c) approximately 3.2 x 10(-19) J, and eta approximately 459 Pa.s. We also estimate the amplitude of the small fluctuations to be of the order of 3 nm, which is around the step size of a polymerizing actin filament.     

Mechanics of curved plasma membrane vesicles: resting shapes, membrane curvature, and in-plane shear elasticity

Highly curved cell membrane structures, such as plasmalemmal vesicles (caveolae) and clathrin-coated pits, facilitate many cell functions, including the clustering of membrane receptors and transport of specific extracellular macromolecules by endothelial cells. These structures are subject to large mechanical deformations when the plasma membrane is stretched and subject to a change of its curvature. To enhance our understanding of plasmalemmal vesicles we need to improve the understanding of the mechanics in regions of high membrane curvatures. We examine here, theoretically, the shapes of plasmalemmal vesicles assuming that they consist of three membrane domains: an inner domain with high curvature, an outer domain with moderate curvature, and an outermost flat domain, all in the unstressed state. We assume the membrane properties are the same in these domains with membrane bending elasticity as well as in-plane shear elasticity. Special emphasis is placed on the effects of membrane curvature and in-plane shear elasticity on the mechanics of vesicle during unfolding by application of membrane tension. The vesicle shapes were computed by minimization of bending and in-plane shear strain energy. Mechanically stable vesicles were identified with characteristic membrane necks. Upon stretch of the membrane, the vesicle necks disappeared relatively abruptly leading to membrane shapes that consist of curved indentations. While the resting shape of vesicles is predominantly affected by the membrane spontaneous curvatures, the membrane shear elasticity (for a range of values recorded in the red cell membrane) makes a significant contribution as the vesicle is subject to stretch and unfolding. The membrane tension required to unfold the vesicle is sensitive with respect to its shape, especially as the vesicle becomes fully unfolded and approaches a relative flat shape.     

Membrane elasticity in giant vesicles with fluid phase coexistence

Biological membranes are known to contain compositional heterogeneities, often termed rafts, with distinguishable composition and function, and these heterogeneities participate in vigorous transport processes. Membrane lipid phase coexistence is expected to modulate these processes through the differing mechanical properties of the bulk domains and line tension at phase boundaries. In this contribution, we compare the predictions from a shape theory derived for vesicles with fluid phase coexistence to the geometry of giant unilamellar vesicles with coexisting liquid-disordered (L(d)) and liquid-ordered (L(o)) phases. We find a bending modulus for the L(o) phase higher than that of the L(d) phase and a saddle-splay (Gauss) modulus difference with the Gauss modulus of the L(o) phase being more negative than the L(d) phase. The Gauss modulus critically influences membrane processes that change topology, such as vesicle fission or fusion, and could therefore be of significant biological relevance in heterogeneous membranes. Our observations of experimental vesicle geometries being modulated by Gaussian curvature moduli differences confirm the prediction by the theory of Juelicher and Lipowsky.     

Movement of spermatozoa of Megaselia Scalaris (Diptera: Brachycera: Cyclorrhapha: Phoridae) in artificial and natural fluids

In artificial fluid, the spermatozoa move as linear cells or round up and rotate, propelled by spontaneous bending of their tails. Both linear and rounded cells can move forward and backward, but usually they move forward. The tails of all cells display, simultaneously, small primary bends and fewer, much larger secondary bends. Rounded cells form single secondary bends that remain unchanged as the cells rotate. They also form “node-like” primary bends that travel posteriorly or anteriorly as the cells rotate forward or backward, respectively. Linear cells move their anterior regions into and out of focus in a cyclic fashion. They form rather prominent primary bends, as well as two to four secondary bends that travel posteriorly as the cells move forward. Secondary bends change in shape continuously and are not sinusoidal. The cells follow approximately linear trajectories, but the distances traveled per cycle, speeds, and secondary bending patterns are variable. When methyl cellulose is added to artificial fluid, linear movement is improved, and forward speeds are approximately tripled. The movement of spermatozoa in natural fluid of the female reproductive tract is remarkably less stereotyped than that of cells in artificial fluid. The cells, usually resembling straight lines or arcs, are very flexible and active. They lack obvious cyclic activity and double bending patterns. They are capable of moving both forward and backward and of adjusting their bending activity and speed within rather wide limits. Their average forward speed is about nine times faster than that of cells in artificial fluid.     

Flexible wings and fins: bending by inertial or fluid-dynamic forces?

Flapping flight and swimming in many organisms is accompaniedby significant bending of flexible wings and fins. The instantaneousshape of wings and fins has, in turn, a profound effect on thefluid dynamic forces they can generate, with non-monotonic relationshipsbetween the pattern of deformation waves passing along the wingand the thrust developed. Many of these deformations arise,in part, from the passive mechanics of oscillating a flexibleair- or hydrofoil. At the same time, however, their instantaneousshape may well emerge from details of the fluid loading. Thisissue—the extent to which there is feedback between theinstantaneous wing shape and the fluid dynamic loading—iscore to understanding flight control. We ask to what extentsurface shape of wings and fins is controlled by structuralmechanics versus fluid dynamic loading. To address this issue,we use a combination of computational and analytic methods toexplore how bending stresses arising from inertial-elastic mechanismscompare to those stresses that arise from fluid pressure forces.Our analyses suggest that for certain combinations of wing stiffness,wing motions, and fluid density, fluid pressure stresses playa relatively minor role in determining wing shape. Nearly allof these combinations correspond to wings moving in air. Theexciting feature provided by this analysis is that, for highReynolds number motions where linear potential flow equationsprovide reasonable estimates of lift and thrust, we can finallyexamine how wing structure affects flight performance. Armedwith this approach, we then show how modest levels of passiveelasticity can affect thrust for a given level of energy inputin the form of an inertial oscillation of a compliant foil.     

Cochlear outer hair cell bending in an external electric field.

We have used a high-resolution motion analysis system to reinvestigate shape changes in isolated guinea pig cochlear outer hair cells (OHCs) evoked by low-frequency (2-3 Hz) external electric stimulation. This phenomenon of electromotility is presumed to result from voltage-dependent structural changes in the lateral plasma membrane of the OHC. In addition to well-known longitudinal movements, OHCs were found to display bending movements when the alternating external electric field gradients were oriented perpendicular to the cylindrical cell body. The peak-to-peak amplitude of the bending movement was found to be as large as 0.7 microm. The specific sulfhydryl reagents, p-chloromercuriphenylsulfonic acid and p-hydroxymercuriphenylsulfonic acid, that suppress electrically evoked longitudinal OHCs movements, also inhibit the bending movements, indicating that these two movements share the same underlying mechanism. The OHC bending is likely to result from an electrical charge separation that produces depolarization of the lateral plasma membrane on one side of the cell and hyperpolarization on the other side. In the cochlea, OHC bending could produce radial distortions in the sensory epithelium and influence the micromechanics of the organ of Corti.     

Red blood cell shape transitions and dynamics in time-dependent capillary flows

The dynamics of single red blood cells (RBCs) determine microvascular blood flow by adapting their shape to the flow conditions in the narrow vessels. In this study, we explore the dynamics and shape transitions of RBCs on the cellular scale under confined and unsteady flow conditions using a combination of microfluidic experiments and numerical simulations. Tracking RBCs in a comoving frame in time-dependent flows reveals that the mean transition time from the symmetric croissant to the off-centered, nonsymmetric slipper shape is significantly faster than the opposite shape transition, which exhibits pronounced cell rotations. Complementary simulations indicate that these dynamics depend on the orientation of the RBC membrane in the channel during the time-dependent flow. Moreover, we show how the tank-treading movement of slipper-shaped RBCs in combination with the narrow channel leads to oscillations of the cell's center of mass. The frequency of these oscillations depends on the cell velocity, the viscosity of the surrounding fluid, and the cytosol viscosity. These results provide a potential framework to identify and study pathological changes in RBC properties.     

Computational Estimates of Membrane Flow and Tension Gradient in Motile Cells

All parts of motile cells, including the plasma membrane, have to translocate in the direction of locomotion. Both directed intracellular membrane transport coupled with polarized endo- and exocytosis and fluid flow in the plane of the plasma membrane can contribute to this overall plasma membrane translocation. It remains unclear how strong a force is required to generate this flow. We numerically solve Stokes equations for the viscous membrane flow across a flat plasma membrane surface in the presence of transmembrane proteins attached to the cytoskeleton and find the membrane tension gradient associated with this flow. This gradient is sensitive to the size and density of the transmembrane proteins attached to the cytoskeleton and can become significant enough to slow down cell movement. We estimate the influence of intracellular membrane transport and actin growth and contraction on the tension gradient, and discuss possible ‘tank tread’ flow at ventral and dorsal surfaces.     

Pay attention to membrane tension: Mechanobiology of the cell surface

The cell surface is a mechanobiological unit that encompasses the plasma membrane, its interacting proteins, and the complex underlying cytoskeleton. Recently, attention has been directed to the mechanics of the plasma membrane, and in particular membrane tension, which has been linked to diverse cellular processes such as cell migration and membrane trafficking. However, how tension across the plasma membrane is regulated and propagated is still not completely understood. Here, we review recent efforts to study the interplay between membrane tension and the cytoskeletal machinery and how they control cell form and function. We focus on factors that have been proposed to affect the propagation of membrane tension and as such could determine whether it can act as a global or local regulator of cell behavior. Finally, we discuss the limitations of the available tool kit as new approaches that reveal its dynamics in cells are needed to decipher how membrane tension regulates diverse cellular processes.     

Renormalization of the tension and area expansion modulus in fluid membranes.

Renormalization of the membrane tension and elastic area expansion modulus by thermally induced bending fluctuations is treated in terms of the formalism of Brochard, De Gennes, and Pfeuty (J. de Phys. (France). 37:1099-1104, 1976). The dependence of the renormalized tension on the bare membrane tension parallels the dependence on the fractional area extension of giant vesicles found experimentally by Evans and Rawicz (Physiol. Rev. Lett. 64:2094-2097, 1990), and suggests conditions for molecular dynamics simulations with membrane patches of limited size that might best represent the properties of macroscopic vesicles.     

Cytoplasmic streaming enables the distribution of molecules and vesicles in large plant cells

Recent studies of aquatic and land plants show that similar phenomena determine intracellular transport of organelles and vesicles. This suggests that aspects of cell signaling involved in development and response to external stimuli are conserved across species. The movement of molecular motors along cytoskeletal filaments directly or indirectly entrains the fluid cytosol, driving cyclosis (i.e., cytoplasmic streaming) and affecting gradients of molecular species within the cell, with potentially important metabolic implications as a driving force for cell expansion. Research has shown that myosin XI functions in organelle movement driving cytoplasmic streaming in aquatic and land plants. Despite the conserved cytoskeletal machinery propelling organelle movement among aquatic and land plants, the velocities of cyclosis in plant cells varies according to cell types, developmental stage of the cell, and plant species. Here, we synthesize recent insights into cytoplasmic streaming, molecular gradients, cytoskeletal and membrane dynamics, and expand current cellular models to identify important gaps in current research.     

Stereocilia membrane deformation: implications for the gating spring and mechanotransduction channel

In hair cells, although mechanotransduction channels have been localized to tips of shorter stereocilia of the mechanically sensitive hair bundle, little is known about how force is transmitted to the channel. Here, we use a biophysical model of the membrane-channel complex to analyze the nature of the gating spring compliance and channel arrangement. We use a triangulated surface model and Monte Carlo simulation to compute the deformation of the membrane under the action of tip link force. We show that depending on the gating spring stiffness, the compliant component of the gating spring arises from either the membrane alone or a combination of the membrane and a tether that connects the channel to the actin cytoskeleton. If a bundle is characterized by relatively soft gating springs, such as those of the bullfrog sacculus, the need for membrane reinforcement by channel tethering then depends on membrane parameters. With stiffer gating springs, such as those from rat outer hair cells, the channel must be tethered for all biophysically realistic parameters of the membrane. We compute the membrane forces (resultants), which depend on membrane tension, bending modulus, and curvature, and show that they can determine the fate of the channel.