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

Typical, axisymmetrical cup shaped cells have been carefully measured and the shapes analyzed mathematically. The results show that the strain energy of a cup shaped cell is always higher than that of a biconcave cell except when the two layers of the membrane involved in resistance to bending are free to slide over one another. This is true whether intrinsic curvature of the membrane is positive, negative or zero. If the two layers can slide over one another, the cup shape becomes the lower energy form. Shear resistance, if appreciable, must cause the cup cell to buckle. Photomicrographs of cup shaped cells show buckled configurations characteristic of those of a partly deflated thin-walled rubber ball, which is a similar object having a low ratio of bending/shear strength.In light of these findings, the cup shape of the red cell can no longer be considered as evidence of intrinsic membrane curvature of opposite sign to that of the crenated cell, but appears to indicate a phase change either in the hydrophobic interior of the bimolecular membrane or in some equivalent interface.     

Mechanical Properties of the Red Cell Membrane: I. Membrane Stiffness and Intracellular Pressure

The technique of Mitchison and Swann (1954) was modified for determining the resistance to deformation, or “stiffness,” of the red cell membrane and the pressure gradient across the cell wall. It requires a measure of the pressure needed to suck a portion of the cell into a micropipette. Stiffness of hypertonically crenated cells was less than that of biconcave discs or hypotonically swollen cells. Crenated cells showed zero pressure gradient and a stiffness, probably due to pure bending, equivalent to 0.007 ± 0.001 (SE) dynes/cm. Normal and swollen cells showed a pressure gradient of 2.3 ± 0.8 (SE) mm H₂O and a stiffness, due to bending and tension in the membrane, equivalent to 0.019 ± 0.002 (SE) dynes/cm. No difference in stiffness was found between the rim and the biconcavity of the cell or between biconcave discs and hypotonically swollen cells. Micromanipulation showed that the membrane can withstand large bending strains but limited tangential strains (stretching). These results have significant implications in any theory explaining the cell shape. For example, the data give no indication that the physical properties of the membrane are different at the rim from those of the biconcavities, and the existence of a positive pressure in the normal cell is established.     

The role of Mg++-ATPase (actomyosin-like protein) in maintaining the biconcave shape of erythrocytes.

Chemically different substances known to change the Mg++-ATPase activity in the red cell membrane, likewise alter the red cell shape. Normal human red cells retain their biconcave shape only when the activity of this enzyme remains unchanged. The present work deals with the possibility that Mg++-ATPase may cause certain tension in the membrane responsible for the biconcave shape of the erythrocyte.     

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.     

Native ultrastructure of the red cell cytoskeleton by cryo-electron tomography

Erythrocytes possess a spectrin-based cytoskeleton that provides elasticity and mechanical stability necessary to survive the shear forces within the microvasculature. The architecture of this membrane skeleton and the nature of its intermolecular contacts determine the mechanical properties of the skeleton and confer the characteristic biconcave shape of red cells. We have used cryo-electron tomography to evaluate the three-dimensional topology in intact, unexpanded membrane skeletons from mouse erythrocytes frozen in physiological buffer. The tomograms reveal a complex network of spectrin filaments converging at actin-based nodes and a gradual decrease in both the density and the thickness of the network from the center to the edge of the cell. The average contour length of spectrin filaments connecting junctional complexes is 46 ± 15 nm, indicating that the spectrin heterotetramer in the native membrane skeleton is a fraction of its fully extended length (∼190 nm). Higher-order oligomers of spectrin were prevalent, with hexamers and octamers seen between virtually every junctional complex in the network. Based on comparisons with expanded skeletons, we propose that the oligomeric state of spectrin is in a dynamic equilibrium that facilitates remodeling of the network as the cell changes shape in response to shear stress.     

Cyclic loading on the red cell membrane in a shear flow: a possible cause of hemolysis

The fluid force acting on single human red cells in a high shear flow was analyzed. A two-dimensional elliptical microcapsule as a model of the deformed red cells was adopted to numerically calculate the distributions of the shear forces on both sides of the cell membrane. It is theoretically shown that the cell membrane undergoes an unsteady cyclic loading under the rotational motion around the interior. The mechanism leading to blood cell trauma is examined by repeatedly loading the continuously moving cell membrane.     

Effects of membrane reference state on shape memory of a red blood cell

By using a three-dimensional continuum model, we simulate the shape memory of a red blood cell after the remove of external forces. The purpose of this study is to illustrate the effect of membrane reference state on cell behavior during the recovery process. The reference state of an elastic element is the geometry with zero stress. Since the cell membrane is composed of cytoskeleton and lipid bilayer, both the reference states of cytoskeleton (RSC) and lipid bilayer (RSL) are considered. Results show that a non-spherical RSC can result in shape memory. The energy barrier due to non-spherical RSC is determined by the ratio of the equator length to the meridian length of the RSC. Thus different RSCs can have similar energy barrier and leading to identical recovery response. A series of simulations of more intermediate RSCs show that the recovery time scale is inversely proportional to the energy barrier. Comparing to spherical RSL, a spheroid RSL contributes to the energy barrier and recovery time. Furthermore, we observe a folding recovery due to the biconcave RSL which is different from the tank treading recovery. These results may motivate novel numerical and experimental studies to determine the exact RSC and RSL.     

Shape and elasticity effects on erythrocyte electrostatic repulsion

It is shown that the double-layer interaction between two symmetrically parallel red blood cells depends largely on their shape and elasticity. Various shapes are examined, including the realistic shape of a biconcave discocyte with an elastic membrane. It is found that changes in the shape and the elasticity of the erythrocyte may result in orders of magnitude differences in the forces and energies of pair interaction.     

Remodeling the shape of the skeleton in the intact red cell.

The role of the membrane skeleton in determining the shape of the human red cell was probed by weakening it in situ with urea, a membrane-permeable perturbant of spectrin. Urea by itself did not alter the biconcave disk shape of the red cell; however, above threshold conditions (1.5 M, 37 degrees C, 10 min), it caused an 18% reduction in the membrane elastic shear modulus. It also potentiated the spiculation of cells by lysophosphatidylcholine. These findings suggest that the contour of the resting cell is not normally dependent on the elasticity of or tension in the membrane skeleton. Rather, the elasticity of the skeleton stabilizes membranes against deformation. Urea treatment also caused the projections induced both by micropipette aspiration and by lysophosphatidylcholine to become irreversible. Furthermore, urea converted the axisymmetric conical spicules induced by lysophosphatidylcholine into irregular, curved and knobby spicules; i.e., echinocytosis became acanthocytosis. Unlike controls, the ghosts and membrane skeletons obtained from urea-generated acanthocytes were imprinted with spicules. These data suggest that perturbing interprotein associations with urea in situ allowed the skeleton to evolve plastically to accommodate the contours imposed upon it by the overlying membrane.     

Evidence from studies of birefringence of structure across the dimple region of red cells

Analysis of the equilibrium of the normal biconcave human red cell, in terms of its tension and a pressure inside, suggests a force of attraction between the opposite membranes at the dimple regions. The analogous attraction that causes rouleaux formation is mediated by long-chain molecules. Single cells hanging on edge between polarizer and analyser, almost “crossed,” were photographed at different angles to the axis of the polarizer. Enlarged prints were scanned by a photometer. For single cells the records showed non-significant fluctuations of intensity, but mean values for 32 cells showed a very significant sinusoidal variation with angle, as predicted by theory for birefringence in the cell at the dimple region. For the rim region, the averaged data showed no variation with angle. In cells moderately osmotically swollen, birefringence in the centre of the dimple region was absent, but persisted close to the inside of the membranes. The latter disappeared in cells further swollen to a biconvex shape. The data is interpreted as indicating oriented chains of molecules across the interior of the cell at the dimple region. The behaviour on swelling was what had been seen in a model with nylon fibres oriented between the charged plates of a condenser, in which the variation of attractive force with distance was adequate to explain the equilibrium of the red cell membrane.     

Transformation pathways of liposomes

Liposomes undergoing transformation were observed by dark-field light microscopy in order to study the role of lipid in morphogenesis of biological vesicular structures. Liposomes were found to transform sequentially in a well-defined manner through one of several transformation pathways. A circular biconcave form was an initial shape in all the pathways and it transformed into a stable thin flexible filament or small spheres via a variety of regularly shaped vesicles which possessed geometrical symmetry. The transformation was reversible up to a certain point in each pathway. Osmotic pressure was found to be the driving force for the transformations. Biological membrane vesicles such as trypsinized red cell ghosts also transformed by similar pathways.     

Spin label study of erythrocyte deformability I. Electron spin resonance spectral change under shear flow

A spin labeling method in electron spin resonance spectroscopy (ESR) is applied for the first time to study the deformability of human red blood cells (RBC). ESR measurements of a RBC suspension incubated with a fatty acid spin label were performed, using a narrow-gap flat ESR sample cell under various flow shear stresses (tau). Remarkable changes were observed in ESR spectra with tau, indicating that RBC are oriented in such a way that the greater part of the membrane surface is aligned parallel to the ESR cell walls. The diamide-treated, hardened RBC, in which the biconcave discoid shape remains intact under no shear stress, exhibit a smaller ESR spectral change with tau than the intact, demonstrating that the present method can be used to assess the deformation of RBC occurring with flow orientation. In particular, the relative amplitude of an ESR difference spectrum may be used as a measure of the elongation of RBC. The conclusion is further supported by experiments using glutaraldehyde-treated or heat-denatured RBC. All these ESR results are in good agreement with the corresponding results obtained by several different methods. The present spin labeling technique is thus proven to be applicable for evaluating RBC deformability.     

Interaction forces between red cells agglutinated by antibody. I. Theoretical.

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutrally buoyant spheres suspended in viscous liquids undergoing a uniform shear flow has been given by Arp and Mason (1977). The method is based on the matrix formulation of hydrodynamic resistances in creeping flow by Brenner and O'Neill (1972). We describe the solution of the Brenner-O'Neill force-torque vector equation in terms of the particle and external flow field coordinates and derive expressions for the normal force acting along, and the shear force acting perpendicular to, the axis of the doublet of spheres, the latter explicitly given for the first time. The equations consist of a term comprising force and torque coefficients obtained from the matrices of the hydrodynamic resistances (functions of the distance h between sphere surfaces which have been computed), and terms comprising the orientation of the doublet axis relative to the coordinates of the external flow field and the shear stress (which can be experimentally determined). We have applied the theory to a system of doublets of sphered, hardened human red cells of group A or B antigenic type cross-linked by the corresponding antibody at a fixed interparticle distance. Working from studies of the breakup of doublets of red cells in an accelerating Poiseuille flow, given in the succeeding paper, we are able to compute the hydrodynamic force required to separate the two spheres. Previous work has shown that the theory can be applied to doublets in a variable shear, Poiseuille flow, provided the ratio of particle to tube diameter is small. In calculating the force-torque coefficients it was assumed that the cells are crosslinked by antibody with h = 20 nm.     

Axisymmetric optical-trap measurement of red blood cell membrane elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10(-6)?N/m over a 1-2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10(-19)?Nm and an upper limit to the in-plane shear modulus H<2×10(-6)?N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell's resistance to deformation on the scale and geometry of the deformation.     

Depletion of membrane skeleton in red blood cell vesicles.

A possible physical interpretation of the partial detachment of the membrane skeleton in the budding region of the cell membrane and consequent depletion of the membrane skeleton in red blood cell vesicles is given. The red blood cell membrane is considered to consist of the bilayer part and the membrane skeleton. The skeleton is, under normal conditions, bound to the bilayer over its whole area. It is shown that, when in such conditions it is in the expanded state, some cell shape changes can induce its partial detachment. The partial detachment of the skeleton from the bilayer is energetically favorable if the consequent decrease of the skeleton expansion energy is larger than the corresponding increase of the bilayer-skeleton binding energy. The effect of shape on the skeleton detachment is analyzed theoretically for a series of the pear class shapes, having decreasing neck diameter and ending with a parent-daughter pair of spheres. The partial detachment of the skeleton is promoted by narrowing of the cell neck, by increasing the lateral tension in the skeleton and its area expansivity modulus, and by diminishing the attraction forces between the skeleton and the bilayer. If the radius of the daughter vesicle is sufficiently small relative to the radius of the parent cell, the daughter vesicle can exist either completely underlaid with the skeleton or completely depleted of the skeleton.     

Reduction of the surface-volume ratio: a physical mechanism contributing to the loss of red cell deformability in malaria

Plasmodia and other intraerythrocytic parasites reduce the deformability of the red cells they infect. One mechanism potentially responsible for this reduction in deformability is the decrease in the surface:volume (S/V) ratio of the red cell which occurs with parasite growth. To examine this hypothesis, normal red cells were allowed to phagocytize polylysine-coated latex spheres 1.0 to 2.9 microns in diameter. Deformability decreased progressively with spheres of increasing size, consistent with the decreasing S/V ratios of those cells (from an initial length:width [L/W] ratio of 2.398 +/- 0.549 for normal red cells to 1.559 +/- 0.249 for red cells containing 2.92 microns latex spheres at 40 dynes per cm2, p less than 0.001). Nevertheless, red cells containing latex spheres 2.0-2.9 microns in diameter remained deformable and continued to tank tread, in contrast to red cells containing Plasmodium falciparum parasites of that size, which are not deformable and do not tank tread. The progressive decrease in S/V produced by the latex spheres is consistent with their effect on the L/W ratio. However, the total loss of deformability observed with red cells containing parasites of similar or smaller size cannot be explained on these grounds alone. It suggests an additional mechanism, such as calcium-induced crosslinking of the red cell cytoskeleton.     

Kinetics and locus of failure of receptor-ligand-mediated adhesion between latex spheres. I. Protein-carbohydrate bond.

We previously described the use of a counter-rotating cone and plate rheoscope to measure the time and force dependence of break-up of doublets of sphered, swollen, and fixed red cells (SSRC) cross-linked by monoclonal IgM antibody. It has been shown that doublet break-up can occur by extraction of receptors from the membrane, rather than by antibody-antigen bond break-up, and is a stochastic process. We therefore prepared 4.62-microns carboxyl modified latex spheres with a covalently coupled synthetic blood group B antigen trisaccharide. Using a two-step carbodiimide process, ethylene diamine was covalently linked to the carboxyl modified latex spheres, and the trisaccharide, having an eight carbon spacer modified to bear a terminal carboxyl group, was linked to the ethylene diamine. Using these antigen spheres we carried out studies in Couette flow, in a transparent cone and plate rheoscope, of the shear-induced break-up of doublets cross-linked by monoclonal IgM anti-B antibody in 19% and 15% Dextran 40. As previously found with SSRC, over a range of normal force from 55 to 175 pN, there was a distribution in times to break-up. However, the fraction of antigen sphere doublets broken up, which increased from 0.08 to 0.43 at 75 pM IgM, and from 0.06 to 0.20 at 150 pM IgM, was significantly lower than that for the SSRC, where the fraction broken up at 150 pM IgM increased from 0.10 to 0.47. Thus, significantly higher forces were required to achieve the same degree of break-up for doublets of antigen-linked spheres than for SSRC. Computer simulation using a stochastic model of break-up showed that the differences between antigen sphere and SSRC doublet break-up were due to a change in bond character (the range and depth of the bond energy minimum) rather than to an increase in the number of bonds linking antigen-sphere doublets. This supports the notion that antibody-antigen bonds are ruptured in the case of antigen spheres, whereas antigen is able to be extracted from the membrane of SSRC, although changes of receptor substrate from cell to latex and the possibility of latex strand extraction from the microspheres are potential complicating factors.     

Effective bilayer expansion and erythrocyte shape change induced by monopalmitoyl phosphatidylcholine. Quantitative light microscopy and nuclear magnetic resonance spectroscopy measurements.

When human erythrocytes are treated with exogenous monopalmitoyl phosphatidylcholine (MPPC), the normal biconcave disk shape red blood cells (RBC) become spiculate echinocytes. The present study examines the quantitative aspect of the relationship between effective bilayer expansion and erythrocyte shape change by a newly developed method. This method is based on the combination of direct surface area measurement of micropipette and relative bilayer expansion measurement of 13C crosspolarization/magic angle spinning nuclear magnetic resonance (NMR). Assuming that 13C NMR chemical shift of fatty acyl chain can be used as an indicator of lateral packing of membrane bilayers, it is possible for us to estimate the surface area expansion of red cell membrane induced by MPPC from that induced by ethanol. Partitions of lipid molecules into cell membrane were determined by studies of shape change potency as a function of MPPC and red cell concentration. It is found that 8(+/- 0.5) x 10(6) molecules of MPPC per cell will effectively induce stage three echinocytes and yield 3.2(+/- 0.2)% expansion of outer monolayer surface area. Surface area of normal cells determined by direct measurements from fixed geometry of red cells aspirated by micropipette was 118.7 +/- 8.5 microns2. The effective cross-sectional area of MPPC molecules in the cell membrane therefore was determined to be 48(+/- 4) A2, which is in agreement with those determined by x-ray from model membranes and crystals of lysophospholipids. We concluded that surface area expansion of RBC can be explained by a simple consideration of cross-sectional area of added molecules and that erythrocyte shape changes correspond quantitatively to the incorporated lipid molecules.     

Mechanism of red blood cell acanthocytosis and echinocytosis in vivo

Patients with abetalipoproteinemia have an inborn absence of the major apoprotein of low density plasma lipoproteins, an abnormal serum and red cell lipid profile, and spiny erythrocytes, called acanthocytes. We now show that these deformed cells are reversibly converted to a normal shape, that of a biconcave disk, by incubation with 3 to 10 X 10(-5) M chlorpromazine. We suppose that chlorpromazine acts by expanding the cytoplasmic leaflet of the bilayer, thus promoting inward curvature. Ghosts isolated from the acanthocytes are themselves spiny but are also converted to normal, concave disks by chlorpromazine or merely by a brief incubation at 37 degrees C in low ionic strength buffer. We attribute the latter to a redistribution of lipids between the two leaflets of the membrane bilayer. Similar observations were made with red cells and ghosts from a patient with benign echinocytosis. These observations suggest that the morphological abnormality in acanthocytes and echinocytes can be ascribed to the same mechanism as crenation in vitro; that is, a bilayer couple effect in which an excess of surface area in the outer leaflet over the inner leaflet of the membrane bilayer drives outward curvature. It is striking that cells which were chronically abnormal in shape in vivo contain the information to become biconcave disks immediately upon simple chemical treatment in vitro.