Elastic energy of curvature-driven bump formation on red blood cell membrane.

Model calculations were performed to explore quantitative aspects of the discocyte-echinocyte shape transformation in red blood cells. The shape transformation was assumed to be driven by changes in the preferred curvature of the membrane bilayer and opposed by the elastic shear rigidity of the membrane skeleton. The energy required for echinocyte bump formation was calculated for a range of bump shapes for different preferred curvatures. Energy minima corresponding to nonzero bump heights were found when the stress-free area difference between the membrane leaflets or the spontaneous curvature of the membrane became sufficiently large, but the calculations predict that the membrane can tolerate significant differences in the resting areas of the inner and outer leaflets or significant spontaneous curvature without visible changes in shape. Thus, if the cell is near the threshold for bump formation, the calculations predict that small changes in membrane properties would produce large changes in cellular geometry. These results provide a rational framework for interpreting observations of shape transformations in red cells and for understanding the mechanism by which small changes in membrane elastic properties might lead to significant changes in geometry.     

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.     

Bending Resistance and Chemically Induced Moments in Membrane Bilayers

Pure bending of a membrane bilayer is developed including different properties for each membrane half. Both connected and unconnected bilayer surfaces are treated. The bilayer bending resistance is the resultant of parallel surface compression “resistances.” The neutral surface is a function of the upper and lower surface compressibility moduli and does not necessarily coincide with the mid-surface. Alterations in the interfacial chemical free energy density (surface tension) on either face can create induced bending moments and produce curvature; even small changes can have a pronounced curvature effect. Chemically induced moments are considered as a possible mechanism for crenation of red blood cells.     

Inner but Not Outer Membrane Leaflets Control the Transition from Glycosylphosphatidylinositol-anchored Influenza Hemagglutinin-induced Hemifusion to Full Fusion

Cells that express wild-type influenza hemagglutinin (HA) fully fuse to RBCs, while cells that express the HA-ectodomain anchored to membranes by glycosylphosphatidylinositol, rather than by a transmembrane domain, only hemifuse to RBCs. Amphipaths were inserted into inner and outer membrane leaflets to determine the contribution of each leaflet in the transition from hemifusion to fusion. When inserted into outer leaflets, amphipaths did not promote the transition, independent of whether the agent induces monolayers to bend outward (conferring positive spontaneous monolayer curvature) or inward (negative curvature). In contrast, when incorporated into inner leaflets, positive curvature agents led to full fusion. This suggests that fusion is completed when a lipidic fusion pore with net positive curvature is formed by the inner leaflets that compose a hemifusion diaphragm. Suboptimal fusion conditions were established for RBCs bound to cells expressing wild-type HA so that lipid but not aqueous dye spread was observed. While this is the same pattern of dye spread as in stable hemifusion, for this “stunted” fusion, lower concentrations of amphipaths in inner leaflets were required to promote transfer of aqueous dyes. Also, these amphipaths induced larger pores for stunted fusion than they generated within a stable hemifusion diaphragm. Therefore, spontaneous curvature of inner leaflets can affect formation and enlargement of fusion pores induced by HA. We propose that after the HA-ectodomain induces hemifusion, the transmembrane domain causes pore formation by conferring positive spontaneous curvature to leaflets of the hemifusion diaphragm.     

A computational procedure for prediction of structural effects of edge-to-edge repair on mitral valve

Edge-to-edge technique is a surgical procedure for the correction of mitral valve leaflets prolapse by suturing the edge of the prolapsed leaflet to the free edge of the opposing one. Suture presence modifies valve mechanical behavior and orifice flow area in the diastolic phase, when the valve opens and blood flows into the ventricle. In the present work, in order to support identification of potentially critical conditions, a computational procedure is described to evaluate the effects of changing suture length and position in combination with valve size and shape. The procedure is based on finite element method analyses applied to a range of different mitral valves, investigating for each configuration the influence of repair on functional parameters, such as mitral valve orifice area and transvalvular pressure gradient, and on structural parameters, such as stress in the leaflets and stitch tension. This kind of prediction would ideally require a coupled fluid-structural analysis, where the interactions between blood flows and mitral apparatus deformation are simultaneously considered. In the present study, however, an alternative approach is proposed, in which results obtained by purely structural finite element analyses are elaborated and interpreted taking into account the Bernoulli type equations available in literature to describe blood flow through mitral orifice. In this way, the effects of each parameter in terms of orifice flow area, suture loads, and leaflets stresses can be expressed as functions of atrioventricular pressure gradient and then correlated to blood flow rate. Results obtained by using this procedure for different configurations are finally discussed.     

Theoretical model of reticulocyte to erythrocyte shape transformation

A theoretical model describing the kinetics of reticulocyte shape transformation was developed. The model considers the evolution of a simple cellular shape under transmembrane pressure difference, and proposes a four-parameter axisymmetric approximation of the cell surface. The mathematical analysis considers plasma membrane tension in the plane of bilayer leaflets, membrane spontaneous curvature and transmembrane transport of water. Cytoskeleton dilatational and shear rigidity, and the energetic barrier preventing the decrease of cell volume below a certain minimum are also incorporated. The set of adequate physical assumptions allowed for formulation of the equation for free energy of the investigated system. Computer simulations of cell shape changes, down to the state of free energy minimum, together with estimation of the time needed for the resulting transport of water, revealed a complex, three-phase picture of temporal alterations in cellular geometry with a wide spectrum of final results, and led to propose a standard model of reticulocyte-erythrocyte transformation. According to the model, both cell volume and surface undergo changes, and the work of the pressure, initially accumulated in the cytoskeleton, is consumed for local bending of the cell membrane. Further simulations with modified initial shape or parameters of the standard model show the trajectories of system evolution and help in better understanding the conditions for the erythro-, sphero-, ovalo-, stomato-, and leptoidal metamorphosis of maturing red blood cells. The stability of the final biconcave shape was also verified. Spherogenic modifications were discussed in the context of spherocytosis. Future development of the model was proposed.     

The bulk rheology of close-packed red blood cells in shear flow

A theoretical analysis is made of the dynamical behavior and bulk rheology of close-packed red blood cell suspensions subjected to simple shear flow. The model for the polyhedral cell shapes and tank-treading membrane motion developed in the companion paper (1) is used. The flow in the thin lubricating plasma layers between cells is analyzed taking into account the mechanical properties of the membrane at the corner regions of sharp membrane curvature. This leads to predictions for the apparent viscosity as a function of hematocrit and shear rate. Good agreement with experimental results is obtained at moderate and high shear rates (above 20 s-1). At lower shear rates, a rapid rise in apparent viscosity has been found experimentally, and the mechanisms leading to this behavior are examined.     

Crenation and cupping of the red cell: A new theoretical approach. Part I. Crenation

Crenation can be thought of as a surface instability caused by intrinsic precurvature of the membrane. Mathematical modeling, on the presupposition that the red blood cell is a thin shell consisting of a connected (coupled) bilayer having uniformly distributed elastic properties shows that crenation can be initiated by negative precurvature, that is, intrinsic curvature having its concavity directed towards the outside of the cell. This is contrary to the currently accepted view which attributes the effect to positive precurvature of an unconnected bilayer. Crenation and the biconcave shape can coexist in the red cell. This suggests that the bilayer must be connected even when the cell is crenated because the biconcave shape could not otherwise be maintained. The progressive development of crenation to more advanced stages, such as the echinocyte type III and the spheroechinocyte can be accounted for if the outer layer of the membrane is stressed beyond the range where strain is proportional to stress. This is consistent with the extremely small radius of curvature at the tips of the crenations.Certain small variations in the uncrenated biconcave shape of the red cell can be interpreted mathematically as due either to negative intrinsic curvature or to shear resistance. Since, however, a small amount of negative precurvature has been shown to be capable of inducing crenation, it is unlikely to be the cause of the variations in the biconcave shape. These must therefore be due to shear resistance.In the light of this new approach, membrane molecular models based on the assumption that crenation is due to positive precurvature need reconsideration.     

Membrane stress increases cation permeability in red cells.

The human red cell is known to increase its cation permeability when deformed by mechanical forces. Light-scattering measurements were used to quantitate the cell deformation, as ellipticity under shear. Permeability to sodium and potassium was not proportional to the cell deformation. An ellipticity of 0.75 was required to increase the permeability of the membrane to cations, and flux thereafter increased rapidly as the limits of cell extension were reached. Induction of membrane curvature by chemical agents also did not increase cation permeability. These results indicate that membrane deformation per se does not increase permeability, and that membrane tension is the effector for increased cation permeability. This may be relevant to some cation permeabilities observed by patch clamping.     

Pulsatile flow in a coronary artery using multiphase kinetic theory

Pulsatile flow in a model of a right coronary artery (RCA) was previously modeled as a single-phase fluid and as a two-phase fluid using experimental rheological data for blood as a function of hematocrit and shear rate. Here we present a multiphase kinetic theory model which has been shown to compute correctly the viscosity of red blood cells (RBCs) and their migration away from vessel walls: the Fahraeus–Lindqvist effect. The computed RBC viscosity decreases with shear rate and vessel size, consistent with measurements. The pulsatile computations were performed using a typical cardiac waveform until a limit cycle was well established. The RBC volume fractions, shear stresses, shear stress gradients, granular temperatures, viscosities, and phase velocities varied with time and position during each cardiac cycle. Steady-state computations were also performed and were found to compare well with time-averaged transient results. The wall shear stress and wall shear stress gradients (both spatial and temporal) were found to be highest on the inside area of maximum curvature. Potential atherosclerosis sites are identified using these computational results.     

Membrane viscoplastic flow.

In this paper, a theory of viscoplasticity formulated by Prager and Hohenemser is developed for a two-dimensional membrane surface and applied to the analysis of the flow of "microtethers" pulled from red blood cells attached to glass substrates. The viscoplastic flow involves two intrinsic material constants: yield shear and surface viscosity. The intrinsic viscosity for plastic flow of membrane is calculated to be 1 X 10(-2) dyn-s/cm from microtether flow experiments, three orders of magnitude greater than surface viscosities of lipid membrane components. The fluid dissipation is dominated by the flow of a structural matrix which has exceeded its yield shear. The yield shear is the maximum shear resultant that the membrane can sustain before it begins to deform irreversibly. The yield shear is found to be in the range 2-8 X 10(-2) dyn/cm, two or three orders of magnitude smaller than the isotropic tension required to lyse red cells.     

Pericardial heterograft valves: An assessment of leaflet stresses and their implications for heart valve design

Bovine pericardium, stabilized with glutaraldehyde, is used widely in the construction of heart valve substitutes, but the design and construction of valve substitutes from this material are empirically based. Collagenous tissue can support tension, but experimental evidence indicates that flexure-induced compressive stresses can lead to fatigue failure. This study uses experimental results obtained from cyclic uniaxial load tests to predict the type and magnitude of operational stresses which occur in pericardial heterograft leaflets. Both Young's modulus and Poisson's ratio varied with uniaxial loading in pericardium, chemically modified free of tension. Leaflet stresses were analysed by using effective incremental representations of these parameters. In leaflets with unrestricted rotation at the point of attachment to the stent, the mid-plane tensions always exceeded the bending stresses, and no zones of leaflet compression were predicted. In contrast, with totally restricted leaflet rotation induced by clamping (possibly between a male and female frame) the bending stresses were greater than the mid-plane tensions at the hinge line and significant compressive stresses were predicted at this site. If elastic boundary conditions were introduced at the stent (possibly by wrapping the stent in pericardium) then the compressive stresses were reduced as the degree of elasticity was increased. Glutaraldehyde fixation of the pericardium under load produced a stiffer material; higher compressive stresses at the stent and significant increases in total stress were predicted for this tissue. The application of elevated pressure loading also increased the compressive and total stresses in the leaflet. Finally, it was shown that bicuspid leaflets were likely to experience higher stresses than tricuspid leaflets. This simple stress analysis should help valve designers of pericardial heterografts to identify those conditions which lead to tissue compression, high total stress, and ultimately material fatigue.     

Mechanical properties of hybridoma cells in batch culture

Summary Direct measurements throughout a batch culture of bursting force, bursting membrane tension, elastic area compressibility modulus, and the size of single hybridoma cells have been made by a novel micromanipulation technique. It has been found that the bursting membrane tension and compressibility modulus rise significantly in the rapid growth phase, and fall in the death phase. An approach is suggested for relating these mechanical properties to the shear sensitivity of the cells when they are exposed to shear stresses in flow fields. It is shown that reports of changes in hybridoma fragility during batch cultures, as measured using viscometers, might be explained using more fundamental micromanipulation measurements.     

Effect of the endothelial surface layer on transmission of fluid shear stress to endothelial cells

Responses of vascular endothelial cells to mechanical shear stresses resulting from blood flow are involved in regulation of blood flow, in structural adaptation of vessels, and in vascular disease. Interior surfaces of blood vessels are lined with a layer of bound or adsorbed macromolecules, known as the endothelial surface layer (ESL). In vivo investigations have shown that this layer has a width of order 1 microm, that it substantially impedes plasma flow, and that it excludes flowing red blood cells. Here, the effect of the ESL on transmission of shear stress to endothelial cells is examined using a theoretical model. The layer is assumed to consist of a matrix of molecular chains extending from the surface, held in tension by a slight increase in colloid osmotic pressure relative to that in free-flowing plasma. It is shown that, under physiological conditions, shear stress is transmitted to the endothelial surface almost entirely by the matrix, and fluid shear stresses on endothelial cell membranes are very small. Rapid fluctuations in shear stress are strongly attenuated by the layer. The ESL may therefore play an important role in sensing of shear stress by endothelial cells.     

Accelerated interleaflet transport of phosphatidylcholine molecules in membranes under deformation.

Biological membranes are lamellar structures composed of two leaflets capable of supporting different mechanical stresses. Stress differences between leaflets were generated during micromechanical experiments in which long thin tubes of lipid (tethers) were formed from the surfaces of giant phospholipid vesicles. A recent dynamic analysis of this experiment predicts the relaxation of local differences in leaflet stress by lateral slip between the leaflets. Differential stress may also relax by interleaflet transport of lipid molecules ("flip-flop"). In this report, we extend the former analysis to include interleaflet lipid transport. We show that transmembrane lipid flux will evidence itself as a linear increase in tether length with time after a step reduction in membrane tension. Multiple measurements were performed on 24 different vesicles composed of stearoyl-oleoyl-phosphatidylcholine plus 3% dinitrophenol-linked di-oleoyl-phosphatidylethanolamine. These tethers all exhibited a linear phase of growth with a mean value of the rate of interlayer permeation, cp = 0.009 s-1. This corresponds to a half-time of approximately 8 min for mechanically driven interleaflet transport. This value is found to be consistent with longer times obtained for chemically driven transport if the lipids cross the membrane via transient, localized defects in the bilayer.     

Characterization of erythrocyte membrane tension for hemolysis prediction in complex flows

Hemolysis is a persistent issue with blood-contacting devices. Many experimental and theoretical contributions over the last few decades have increased insight into the mechanisms of hemolysis in both laminar and turbulent flows, with the ultimate goal of developing a comprehensive, mechanistic hemolysis model. Many models assume that hemolysis scales with a resultant, scalar stress representing all components of the fluid stress tensor. This study critically evaluates this scalar stress hypothesis by calculating the response of the red blood cell membrane to different types of fluid stress (laminar shear and extension, and three turbulent shear and extension cases), each with the same scalar stress. It was found that even though the scalar stress is the same for all cases, membrane tension varied by up to three orders of magnitude. In addition, extensional flow causes constant tension, while tank-treading in shear flow causes periodic tension, with tank-treading frequency varying by three orders of magnitude among the cases. For turbulent flow, tension also depends on eddy size. It is concluded, therefore, that scalar stress alone is inadequate for scaling hemolysis. Fundamental investigations are needed to establish a new index of the fluid stress tensor that provides reliable hemolysis prediction across the wide range of complex flows that occur in cardiovascular devices.     

Transcellular cross bonding of the red blood cell membrane

When red blood cells are osmotically shrunk, opposing regions of the inner membrane surface touch each other in the dimple area. In normal red cells such a mechanical contact is undone by reswelling the cells. When the cells are treated with the SH reagents diamide or N-ethylmaleimide, or simply heated to temperatures between 42 and 48 degrees C such a mechanical contact can be made permanent by a process termed 'membrane cross bonding'. Cross bonding also occurred when the cells were treated before mechanical contact was established. The bridge between the two cross-bonded membrane regions may be assumed to be formed by membrane skeletal material. Membrane bridges become visible microscopically when the cells are swollen. These bridges are strong enough to resist the membrane tensions occurring at osmotic lysis. Bridged red cells can be a useful tool in rheology, since they are deformable but cannot adapt to shear flows by membrane tank treading.     

Is the surface area of the red cell membrane skeleton locally conserved?

The incompressibility of the lipid bilayer keeps the total surface area of the red cell membrane constant. Local conservation of membrane surface area requires that each surface element of the membrane skeleton keeps its area when its aspect ratio is changed. A change in area would require a flow of lipids past the intrinsic proteins to which the skeleton is anchored. in fast red cell deformations, there is no time for such a flow. Consequently, the bilayer provides for local area conservation. In quasistatic deformations, the extent of local change in surface area is the smaller the larger the isotropic modulus of the skeleton in relation to the shear modulus. Estimates indicate: (a) the velocity of relative flow between lipid and intrinsic proteins is proportional to the gradient in normal tension within the skeleton and inversely proportional to the viscosity of the bilayer; (b) lateral diffusion of lipids is much slower than this flow; (c) membrane tanktreading at frequencies prevailing in vivo as well as the release of a membrane tongue from a micropipette are fast deformations; and (d) the slow phase in micropipette aspiration may be dominated by a local change in skeleton surface.     

An analysis of the sluicing gate in pulmonary blood flow

For pulmonary blood flow in zone 2 condition, in which the blood pressure in the venule (pven) is lower than the alveolar gas pressure (pA), the blood exiting from the capillary sheet and entering a venule must go through a sluicing gate. The sluicing gate exists because the venule remains patent while the capillaries will collapse when the static pressure of blood falls below the alveolar gas pressure. In the original theory of sheet flow the effect of the tension in the interalveolar septa on the flow through the sluicing gate was ignored. Since the tension multiplied by the curvature of the membrane is equivalent to a lateral pressure tending to open the gate, and since the curvature of the capillary wall is high in the gate region, this effect may be important. The present analysis improves the original theory and demonstrates that the effect of membrane tension is to cause flow to increase when the venous pressure continues to decrease. The shape of the sluicing gate resembles that of a venturi tube, and can be determined by an iterative integration of the differential equations. The result forms an important link in the theory of pulmonary blood flow in zone 2 condition.