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.     

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.     

The human erythrocyte membrane skeleton may be an ionic gel

Biochemical and biophysical observations indicate that the erythrocyte membrane skeleton is composed of a swollen network of long, flexible and ionizable macromolecules located at the cytoplasmic surface of the fluid membrane lipid bilayer. We have analyzed the mechanochemical properties of the erythrocyte membrane assuming that the membrane skeleton constitutes an ionic gel (swollen ionic elastomer). Using recently established statistical thermodynamic theory for such gels, our analysis yields mathematical expressions for the mechanochemical properties of erythrocyte membranes that incorporate membrane molecular parameters to an extent not achieved previously. The erythrocyte membrane elastic shear modulus and maximum elastic extension ratio predicted by our membrane model are in quantitative agreement with reported values for these parameters. The gel theory predicts further that the membrane skeleton modulus of area compression, K _G, may be small as well as large relative to the membrane elastic shear modulus, G, depending on the environmental conditions. Our analysis shows that the ratio between these two parameters affects both the geometry and the stability of the favoured cell shapes.     

Red cell membrane elasticity as determined by flow channel technique.

The elasticity of red cell membrane was determined in a rectangular flow channel under controlled shear flow. The relation between shear stress and cell extension ratio (lambda) has been analyzed with the use of Evans' two-dimensional model. The deformed cell shapes observed experimentally agreed well with the model with lambda up to 1.4. The best correlation was found at lambda = 1.2. The analysis suggests a nonlinear extensional membrane modulus in the low stress range encountered in the flow channel. In terms of an appropriate strain parameter, the elastic modulus is shown to rise toward the level encountered in micropipette aspiration experiments. The implications of the present findings in modeling of cell mechanics and in cell hemolysis are discussed.     

Thermoelasticity of red blood cell membrane.

The elastic properties of the human red blood cell membrane have been measured as functions of temperature. The area compressibility modulus and the elastic shear modulus, which together characterize the surface elastic behavior of the membrane, have been measured over the temperature range of 2-50 degrees C with micropipette aspiration of flaccid and osmotically swollen red cells. In addition, the fractional increase in membrane surface area from 2-50 degrees C has been measured to give a value for the thermal area expansivity. The value of the elastic shear modulus at 25 degrees C was measured to be 6.6 X 10(-3) dyne/cm. The change in the elastic shear modulus with temperature was -6 X 10(-5) dyne/cm degrees C. Fractional forces were shown to be only on the order of 10-15%. The area compressibility modulus at 25 degrees C was measured to be 450 dyne/cm. The change in the area compressibility modulus with temperature was -6 dyne/cm degrees C. The thermal area expansivity for red cell membrane was measured to be 1.2 X 10(-3)/degrees C. With this data and thermoelastic relations the heat of expansion is determined to be 110-200 ergs/cm2; the heat of extension is 2 X 10(-2) ergs/cm2 for unit extension of the red cell membrane. The heat of expansion is of the order anticipated for a lipid bilayer idealized as twice the behavior of a monolayer at an oil-water interface. The observation that the heat of extension is positive demonstrates that the entropy of the material increases with extension, and that the dominant mechanism of elastic energy storage is energetic. Assuming that the red cell membrane shear rigidity is associated with "spectrin," unit extension of the membrane increases the configurational entropy of spectrin by 500 cal/mol.     

Cell poking. Determination of the elastic area compressibility modulus of the erythrocyte membrane

Cell poking, a new method for measuring mechanical properties of single cells was used to determine the elastic area compressibility modulus of osmotically swollen human erythrocytes. With this method we determined the force required to indent cells attached to a glass coverslip (Petersen, N.O., W. B. McConnaughey , and E. L. Elson , 1982, Proc. Natl. Acad. Sci. USA, 79:5327. Forces on the order of one millidyne and indentations on the order of one micron were detected. An analysis of these data in terms of a simplified mechanical model yielded the elastic area compressibility modulus. This analysis used a variational approach to minimize the isothermal elastic potential energy density function given by E. A. Evans and R. Skalak (Mechanics and Thermodynamics of Biomembranes, 1980, CRC Press, Boca Raton , FL). Measurements on swollen erythrocytes gave a range of values, depending in part on the osmotic conditions, of 17.9 +/- 8.2 to 34.8 +/- 12.0 mdyn /micron for the elastic area compressibility modulus at 25 degrees C. Fractional area expansion greater than 2.6 +/- 0.8% produced rapid cell lysis. These values were not corrected for the reversible movement of water across the cell membrane in response to hydrostatic pressure gradients. Our results agree reasonably with those obtained by Evans et al. (Evans, E.A., R. Waugh , and L. Melnick , 1976, Biophys. J., 16:585-595.) using micropipette aspiration under similar conditions.     

Time-dependent elastic extensional RBC deformation by micropipette aspiration: redistribution of the spectrin network?

The time dependence of small elastic extensional RBC deformation by micropipette aspiration has been analyzed. This process shows two-phases which are characterized by time constants of the order of some tenths of seconds and about ten seconds, respectively. The equilibrium tongue length is reached after about 30 s. For the first, fast step we assume that the membrane model of immobilized boundaries holds, i.e., the skeleton is tightly associated with the lipid bilayer and no redistribution of the skeleton with respect to the lipid bilayer is allowed. This lipid-spectrin interaction or anchorage is characterized by some association force density. It has been shown that at a given tongue length the force generated owing to the membrane deformation and acting to redistribute the spectrin, overcomes (in some membrane area) the association force density and results in an additional increase of the sucked membrane length. Equations have been derived to describe this process. From the experimental conditions of an RBC aspiration and the determined tongue length corresponding to the second slow aspiration step, the association force density between the lipid bilayer and the spectrin network may be determined. From literature data and our own results a force density of between 40 and 50 Pa has been estimated. Offprint requests to: D. Lerche     

An extended modeling of the micropipette aspiration experiment for the characterization of the Young's modulus and Poisson's ratio of adherent thin biological samples: numerical and experimental studies

The micropipette aspiration (MA) experiment remains a quite widely used micromanipulation technique for quantifying the elastic modulus of cells and, less frequently, of other biological samples. However, moduli estimations derived from MA experiments are only valid if the probed sample is non-adherent to the rigid substrate. This study extends this standard formulation by taking into account the influence of the sample adhesion. Using a finite element analysis of the sample aspiration into the micropipette, we derived a new expression of the aspirated length for linear elastic materials. Our results establish that (i) below a critical value, the thickness h of the probed sample must be considered to get an accurate value of its Young's modulus (ii) this critical value depends both on the Poisson's ratio and on the sample adhesivity. Additionally, we propose a novel method which allows the computation of the intrinsic Young's modulus of the adherent probed sample from its measured apparent elasticity modulus. Thanks to the set of computational graphs we derived from our theoretical analysis, we successfully validate this method by experiments performed on polyacrylamide gels. Interestingly, the original procedure we proposed allows a simultaneous quantification of the Young's modulus and of the Poisson's ratio of the adherent gel. Thus, our revisited analysis of MA experiments extends the application domain of this technique, while contributing to decrease the dispersion of elastic modulus values obtained by this method.     

Erythrocyte membrane elasticity during in vivo ageing

Changes in the ability of senescent erythrocytes to pass through the micro-circulation may cause them to be trapped in the spleen and removed from the blood. To help understand this process we have measured erythrocyte membrane elasticity, to see whether it changes during in vivo ageing. Human and rabbit red cells were fractionated by isopycnic sedimentation to obtain samples of aged and young cells. These were subjected to micropipette analysis in order to determine their membrane shear elastic modulus. We found that the membrane rigidity did not significantly alter as red cells aged. Previously we have also demonstrated that the changed size and shape of aged cells is unlikely to explain their removal from the circulation (Nash, G.B. and Wyard, S.J. (1981) Biorheology, in the press). Thus we conclude that the lifespan of erythrocytes is not determined by factors related to membrane flexibility or cell shape but may depend on changes in their viscous properties (as suggested by Williams, A.R. and Morris, D.R. (1980), Scand. J. Haematol. 24, 57–62).     

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.     

Adhesivity and rigidity of erythrocyte membrane in relation to wheat germ agglutinin binding

Binding of the plant lectin wheat germ agglutinin (WGA) to erythrocyte membranes causes membrane rigidification. One of our objectives has been to directly measure the effects of WGA binding on membrane rigidity and to relate rigidification to the kinetics and levels of WGA binding. Our other objective has been to measure the strength of adhesion and mechanics of cell separation for erythrocytes bound together by WGA. The erythrocyte membrane rigidity was measured on single cells by micropipette aspiration. The slope of the suction pressure-length data for entry into the pipette provided the measure of the membrane extensional modulus. Data were collected for cells equilibrated with WGA solutions in the range of concentrations of 0.01- 10 micrograms/ml. Erythrocyte-erythrocyte adherence properties were studied by micropipette separation of two-cell aggregates. First, a "test" cell was selected from a WGA solution by aspiration into a small micropipette, then transferred to a separate chamber that contained erythrocytes in WGA-free buffer. Here, a second cell was aspirated with another pipette and maneuvered into close proximity of the test cell surface, and adhesive contact was produced. The flaccid cell was separated from the test cell surface in steps at which the force of attachment was derived from the pipette suction pressure and cell geometry. In addition, we measured the time-dependent binding and release of fluorescently labeled WGA to single erythrocytes with a laser microfluorometry system. The results showed that the stiffening of the erythrocyte membrane and binding of fluorescently labeled WGA to the membrane surface followed the same concentration and time dependencies. The threshold concentration for membrane stiffening was at approximately 0.1 microgram/ml where the time course to reach equilibrium was close to 1 h. The maximal stiffening (almost 30-fold over the normal membrane elastic modulus) occurred in concentrations greater than 2 micrograms/ml where the time to reach equilibrium took less than 1 min. The WGA binding also altered the normal elastic membrane behavior into an inelastic, plastic-like response which indicated that mechanical extension of the membrane caused an increase in cross-linking within the surface plane. Similar to the stiffening effect, we observed that the membrane adhesivity of cells equilibrated with WGA solutions greatly increased with concentration greater than 0.1 microgram/ml.(ABSTRACT TRUNCATED AT 400 WORDS)     

Spectrin properties and the elasticity of the red blood cell membrane skeleton

Two models of spectrin elasticity are developed and compared to experimental measurements of the red blood cell (RBC) membrane shear modulus through the use of an elastic finite element model of the RBC membrane skeleton. The two molecular models of spectrin are: (i) An entropic spring model of spectrin as a flexible chain. This is a model proposed by several previous authors. (ii) An elastic model of a helical coiled-coil which expands by increasing helical pitch. In previous papers, we have computed the relationship between the stiffness of a single spectrin molecule (K) and the shear modulus of a network (μ), and have shown that this behavior is strongly dependent upon network topology. For realistic network models of the RBC membrane skeleton, we equate μ to micropipette measurements of RBCs and predict K for spectrin that is consistent with the coiled-coil molecular model. The value of spectrin stiffness derived from the entropic molecular model would need to be at least 30 times greater to match the experimental results. Thus, the conclusion of this study is that a helical coiled-coil model for spectrin is more realistic than a purely entropic model.     

Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study

We have used optical tweezers to study the elasticity of red cell membranes; force was applied to a bead attached to a permeabilized spherical ghost and the force-extension relation was obtained from the response of a second bead bound at a diametrically opposite position. Interruption of the skeletal network by dissociation of spectrin tetramers or extraction of the actin junctions engendered a fourfold reduction in stiffness at low applied force, but only a twofold change at larger extensions. Proteolytic scission of the ankyrin, which links the membrane skeleton to the integral membrane protein, band 3, induced a similar effect. The modified, unlike the native membranes, showed plastic relaxation under a prolonged stretch. Flaccid giant liposomes showed no measurable elasticity. Our observations indicate that the elastic character is at least as much a consequence of the attachment of spectrin as of a continuous membrane-bound network, and they offer a rationale for formation of elliptocytes in genetic conditions associated with membrane-skeletal perturbations. The theory of Parker and Winlove for elastic deformation of axisymmetric shells (accompanying paper) allows us to determine the function BH(2) for the spherical saponin-permeabilized ghost membranes (where B is the bending modulus and H the shear modulus); taking the literature value of 2 x 10(-19) Nm for B, H then emerges as 2 x 10(-6) Nm(-1). This is an order of magnitude higher than the value reported for intact cells from micropipette aspiration. Reasons for the difference are discussed.     

Theoretical model and experimental study of red blood cell (RBC) deformation in microchannels

The motion and deformation of red blood cells (RBCs) flowing in a microchannel were studied using a theoretical model and a novel automated rheoscope. The theoretical model was developed to predict the cells deformation under shear as a function of the cells geometry and mechanical properties. Fluid dynamics and membrane mechanics are incorporated, calculating the traction and deformation in an iterative manner. The model was utilized to evaluate the effect of different biophysical parameters, such as: inner cell viscosity, membrane shear modulus and surface to volume ratio on deformation measurements. The experimental system enables the measurement of individual RBCs velocity and their deformation at defined planes within the microchannel. Good agreement was observed between the simulation results, the rheoscope measurements and published ektacytometry results. The theoretical model results imply that such deformability measuring techniques are weakly influenced by changes in the inner viscosity of the cell or the ambient fluid viscosity. However, these measurements are highly sensitive to RBC shear modulus. The shear modulus, estimated by the model and the rheoscope measurements, falls between the values obtained by micropipette aspiration and laser trapping. The study demonstrates the integration of a theoretical model with a microfabricated device in order to achieve a better understanding of RBC mechanics and their measurement using microfluidic shear assays. The system and the model have the potential of serving as quantitative clinical tools for diagnosing deformability disorders in RBCs.     

Extensional flow of erythrocyte membrane from cell body to elastic tether. I. Analysis.

This is the first of two papers on an analytical and experimental study of the flow of the erythrocyte membrane. In the experiment to be discussed in detail in the second paper, preswollen human erythrocytes are sphered by aspirating a portion of the cell membrane into a small micropipette; and long, thin, membrane filaments or "tethers" are steadily withdrawn from the cell at a point diametrically opposite to the point of aspiration. The aspirated portion of the membrane furnished a "reservoir" of material that replaces the membrane as it flows as a liquid from the nearly spherical cell body to the cylindrical tether. In this paper we show that an application of the principle of conservation of mass permits the tether radius (approximately 200 A or less) to be measured with the light microscope as the tether is formed and extended at a constant rate. A static analysis of the axisymmetric cell deformation and tether formation process reveals that the tether radius is uniquely determined by the isotropic tension in the membrane and the elastic constitutive (material) behavior of the tether itself. A dynamic analysis of the extensional flow process reveals that the tether radius must decrease as the velocity of the tether is increased and that the decrease depends on both the viscosity of the membrane and the elasticity of the tether. The analysis also shows that these two factors (membrane viscosity and tether elasticity) are readily decomposed and determined separately when flow experiments are performed at different isotropic tensions.     

Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration

Chondrocytes in articular cartilage are regularly subjected to compression and recovery due to dynamic loading of the joint. Previous studies have investigated the elastic and viscoelastic properties of chondrocytes using micropipette aspiration techniques, but in order to calculate cell properties, these studies have generally assumed that cells are incompressible with a Poisson's ratio of 0.5. The goal of this study was to measure the Poisson's ratio and recovery properties of the chondrocyte by combining theoretical modeling with experimental measures of complete cellular aspiration and release from a micropipette. Chondrocytes isolated from non-osteoarthritic and osteoarthritic cartilage were fully aspirated into a micropipette and allowed to reach mechanical equilibrium. Cells were then extruded from the micropipette and cell volume and morphology were measured throughout the experiment. This experimental procedure was simulated with finite element analysis, modeling the chondrocyte as either a compressible two-mode viscoelastic solid, or as a biphasic viscoelastic material. By fitting the experimental data to the theoretically predicted cell response, the Poisson's ratio and the viscoelastic recovery properties of the cell were determined. The Poisson's ratio of chondrocytes was found to be 0.38 for non-osteoarthritic cartilage and 0.36 for osteoarthritic chondrocytes (no significant difference). Osteoarthritic chondrocytes showed an increased recovery time following full aspiration. In contrast to previous assumptions, these findings suggest that chondrocytes are compressible, consistent with previous studies showing cell volume changes with compression of the extracellular matrix.     

Micropipette aspiration on the outer hair cell lateral wall

The mechanical properties of the lateral wall of the guinea pig cochlear outer hair cell were studied using the micropipette aspiration technique. A fire-polished micropipette with an inner diameter of approximately 4 microm was brought into contact with the lateral wall and negative pressure was applied. The resulting deformation of the lateral wall was recorded on videotape and subjected to morphometric analysis. The relation between the length of the aspirated portion of the cell and aspiration pressure is characterized by the stiffness parameter, K(s) = 1.07 +/- 0.24 (SD) dyn/cm (n = 14). Values of K(s) do not correlate with the original cell length, which ranges from 29 to 74 microm. Theoretical analysis based on elastic shell theory applied to the experimental data yields an estimate of the effective elastic shear modulus, mu = 15.4 +/- 3.3 dyn/cm. These data were obtained at subcritical aspiration pressures, typically less than 10 cm H2O. After reaching a critical (vesiculation) pressure, the cytoplasmic membrane appeared to separate from the underlying structures, a vesicle with a length of 10-20 microm was formed, and the cytoplasmic membrane resealed. This vesiculation process was repeated until a cell-specific limit was reached and no more vesicles were formed. Over 20 vesicles were formed from the longest cells in the experiment.     

Effects of pressure on red blood cell geometry during micropipette aspiration

Micropipette aspiration is a potentially useful and accurate technique to measure red blood cell (RBC) geometry. Individual RBCs are partially aspirated and from the resulting sphere diameter, total cell length, and pipette diameter, membrane area and cell volume can be calculated. In this study we have focused on possible shape artifacts associated with the aspirated portion of RBC. We observed that the apparent RBC geometry (calculated area and volume) changed markedly (P < 0.001) with the applied aspiration pressure; for normal human RBC the area increased by 5.6 +/- 0.6% and volume decreased by 4.7 +/- 0.6% when the aspiration pressure was increased from 20 to 100 mm water. The calculated membrane area dilation modulus was 7.4 dyn/ cm, which is far below the expected value, and microscopic observations revealed a membrane folding artifact as a possible artifact. These assumptions were strengthened by using a short-duration (3 s) pressure peak of 20-100-20 mm water. The folding then disappeared permanently, but a small (0.31 +/- 0.09%; P < 0.001) area decrease was detected which yields a realistic dilation modulus of 215 dyn/cm. We conclude that membrane folding can critically affect RBC micropipette measurements and that a transient pressure peak can unfold the RBC membrane, thus allowing accurate measurements of RBC geometry.     

Constitutive equations of erythrocyte membrane incorporating evolving preferred configuration

The erythrocyte membrane is modeled as a two-dimensional viscoelastic continuum that evolves under the application of stress. The present analysis of the erythrocyte membrane is motivated by the recent development of knowledge about its molecular structure. The constitutive equations proposed in the present analysis explain in a consistent manner the data on both the deformation and recovery phases of the micropipette experiment. The rheological equations of the present study are applied in a later section to the analysis of a plane membrane deformation that is quantitatively similar to the tank-treading motion of the erythrocytes in a shear field. The computations yield useful information on how the membrane viscosity becomes a more dominant feature in tank-treading motion. The material constants appearing in the proposed constitutive equations may be useful indications of the biochemical state of the membrane in health and disease.     

Stress-free state of the red blood cell membrane and the deformation of its skeleton

The response of a red blood cell (RBC) to deformation depends on its membrane, a composite of a lipid bilayer and a skeleton, which is a closed, twodimensional network of spectrin tetramers as its bonds. The deformation of the skeleton and its lateral redistribution are studied in terms of the RBC resting state for a fixed geometry of the RBC, partially aspirated into a micropipette. The geometry of the RBC skeleton in its initial state is taken to be either two concentric circles, a references biconcave shape or a sphere. It is assumed that in its initial state the skeleton is distributed laterally in a homogeneous manner with its bonds either unstressed, presenting its stress-free state, or prestressed. The lateral distribution was calculated using a variational calculation. It was assumed that the spectrin tetramer bonds exhibit a linear elasticity. The results showed a significant effect of the initial skeleton geometry on its lateral distribution in the deformed state. The proposed model is used to analyze the measurements of skeleton extension ratios by the method of applying two modes of RBC micropipette aspiration.