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.     

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.     

How malaria parasites reduce the deformability of infected red blood cells

The pathogenesis of malaria is largely due to stiffening of the infected red blood cells (RBCs). Contemporary understanding ascribes the loss of RBC deformability to a 10-fold increase in membrane stiffness caused by extra cross-linking in the spectrin network. Local measurements by micropipette aspiration, however, have reported only an increase of ～3-fold in the shear modulus. We believe the discrepancy stems from the rigid parasite particles inside infected cells, and have carried out numerical simulations to demonstrate this mechanism. The cell membrane is represented by a set of discrete particles connected by linearly elastic springs. The cytosol is modeled as a homogeneous Newtonian fluid, and discretized by particles as in standard smoothed particle hydrodynamics. The malaria parasite is modeled as an aggregate of particles constrained to rigid-body motion. We simulate RBC stretching tests by optical tweezers in three dimensions. The results demonstrate that the presence of a sizeable parasite greatly reduces the ability of RBCs to deform under stretching. With the solid inclusion, the observed loss of deformability can be predicted quantitatively using the local membrane elasticity measured by micropipettes.     

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.     

Elastic behavior of a red blood cell with the membrane’s nonuniform natural state: equilibrium shape,motion transition under shear flow,and elongation during tank-treading motion

Direct numerical simulations of the mechanics of a single red blood cell (RBC) were performed by considering the nonuniform natural state of the elastic membrane. A RBC was modeled as an incompressible viscous fluid encapsulated by an elastic membrane. The in-plane shear and area dilatation deformations of the membrane were modeled by Skalak constitutive equation, while out-of-plane bending deformation was formulated by the spring model. The natural state of the membrane with respect to in-plane shear deformation was modeled as a sphere ( \(\alpha =0\) ), biconcave disk shape ( \(\alpha =1\) ) and their intermediate shapes ( \(0<\alpha <1\) ) with the nonuniformity parameter \(\alpha \) , while the natural state with respect to out-of-plane bending deformation was modeled as a flat plane. According to the numerical simulations, at an experimentally measured in-plane shear modulus of \(2.5\times 10^{-6}\,\hbox {N}/\hbox {m}\) and an out-of-plane bending rigidity of \(2.0\times 10^{-19}\,\hbox {N}\cdot \hbox {m}\) of the cell membrane, the following results were obtained. (i) The RBC shape at equilibrium was biconcave discoid for \(\alpha >0.22\) and cupped otherwise; (ii) the experimentally measured fluid shear stress at the transition between tumbling and tank-treading motions under shear flow was reproduced for \(0.05<\alpha <0.34\) ; (iii) the elongation deformation of the RBC during tank-treading motion from the simulation was consistent with that from in vitro experiments, irrespective of the \(\alpha \) value. Based on our RBC modeling, the three phenomena (i), (ii), and (iii) were mechanically consistent for \(0.22<\alpha <0.34\) . The condition \(0.05<\alpha <0.22\) precludes a biconcave discoid shape at equilibrium (i); however, it gives appropriate fluid shear stress at the motion transition under shear flow (ii), suggesting that a combined effect of \(\alpha \) and the natural state with respect to out-of-plane bending deformation is necessary for understanding details of the RBC mechanics at equilibrium. Our numerical results demonstrate that moderate nonuniformity in a membrane’s natural state with respect to in-plane shear deformation plays a key role in RBC mechanics.     

Red Blood Cell Shape and Fluctuations: Cytoskeleton Confinement and ATP Activity

We review recent theoretical work that analyzes experimental measurements of the shape and fluctuations of red blood cells. Particular emphasis is placed on the role of the cytoskeleton and cell elasticity and we contrast the situation of elastic cells with that of fluid-filled vesicles. In red blood cells (RBCs), the cytoskeleton consists of a two-dimensional network of spectrin proteins. Our analysis of the wave vector and frequency dependence of the fluctuation spectrum of RBCs indicates that the spectrin network acts as a confining potential that reduces the fluctuations of the lipid bilayer membrane. However, since the cytoskeleton is only sparsely connected to the bilayer, one cannot regard the composite cytoskeleton membrane as a polymerized object with a shear modulus. The sensitivity of RBC fluctuations and shapes to ATP concentration may reflect the transient defects induced in the cytoskeleton network by ATP.     

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.     

An elastic network model based on the structure of the red blood cell membrane skeleton.

A finite element network model has been developed to predict the macroscopic elastic shear modulus and the area expansion modulus of the red blood cell (RBC) membrane skeleton on the basis of its microstructure. The topological organization of connections between spectrin molecules is represented by the edges of a random Delaunay triangulation, and the elasticity of an individual spectrin molecule is represented by the spring constant, K, for a linear spring element. The model network is subjected to deformations by prescribing nodal displacements on the boundary. The positions of internal nodes are computed by the finite element program. The average response of the network is used to compute the shear modulus (mu) and area expansion modulus (kappa) for the corresponding effective continuum. For networks with a moderate degree of randomness, this model predicts mu/K = 0.45 and kappa/K = 0.90 in small deformations. These results are consistent with previous computational models and experimental estimates of the ratio mu/kappa. This model also predicts that the elastic moduli vary by 20% or more in networks with varying degrees of randomness. In large deformations, mu increases as a cubic function of the extension ratio lambda 1, with mu/K = 0.62 when lambda 1 = 1.5.     

Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton.

The area expansion and the shear moduli of the free spectrin skeleton, freshly extracted from the membrane of a human red blood cell (RBC), are measured by using optical tweezers micromanipulation. An RBC is trapped by three silica beads bound to its membrane. After extraction, the skeleton is deformed by applying calibrated forces to the beads. The area expansion modulus K(C) and shear modulus mu(C) of the two-dimensional spectrin network are inferred from the deformations measured as functions of the applied stress. In low hypotonic buffer (25 mOsm/kg), one finds K(C) = 4.8 +/- 2.7 microN/m, mu(C) = 2.4 +/- 0.7 microN/m, and K(C)/mu(C) = 1.9 +/- 1.0. In isotonic buffer, one measures higher values for K(C), mu(C), and K(C)/mu(C), partly because the skeleton collapses in a high-ionic-strength environment. Some data concerning the time evolution of the mechanical properties of the skeleton after extraction and the influence of ATP are also reported. In the Discussion, it is shown that the measured values are consistent with estimates deduced from experiments carried out on the intact membrane and agree with theoretical and numerical predictions concerning two-dimensional networks of entropic springs.     

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.     

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.     

A tethered adhesive particle model of two-dimensional elasticity and its application to the erythrocyte membrane.

A new model of two-dimensional elasticity with application to the erythrocyte membrane is proposed. The system consists of a planar array of self-adhesive particles attached to nearest neighbors with flexible tethers. Stretching from the equilibrium dimension is resisted because force is required to dissociate the particle clusters and to decrease the distribution entropy. Release of the external force is accompanied by a contraction as thermal diffusion randomizes the particles and allows interparticle attachments to form again. Analysis of membrane thermodynamics and mechanics under the two-state particle assumption results in a shear softening stress-strain relation. The shear modulus is found proportional to the square root of the surface density of particles, the interparticle adhesive energy, and is inversely proportional to the tether length. Applied to the erythrocyte membrane under the assumption that band 3 tetramer represents the particle and spectrin the tether, the shear modulus predicted corresponds to the measured value when the interparticle adhesive energy is approximately 4.0-5.9 kT, where kT is the Boltzmann constant multiplied by the temperature. This model suggests a mechanism wherein erythrocyte membrane deformability depends on integral protein homomultimeric interactions and can be modulated from the external surface.     

Elastic thickness compressibilty of the red cell membrane

We have used an ultrasensitive force probe and optical interferometry to examine the thickness compressibility of the red cell membrane in situ. Pushed into the centers of washed-white red cell ghosts lying on a coverglass, the height of the microsphere-probe tip relative to its closest approach on the adjacent glass surface revealed the apparent material thickness, which began at approximately 90 nm per membrane upon detection of contact (force approximately 1-2 pN). With further impingement, the apparent thickness per membrane diminished over a soft compliant regime that spanned approximately 40 nm and stiffened on approach to approximately 50 nm under forces of approximately 100 pN. The same force-thickness response was obtained on recompression after retraction of the probe, which demonstrated elastic recoverability. Scaled by circumferences of the microspheres, the forces yielded energies of compression per area which exhibited an inverse distance dependence resembling that expected for flexible polymers. Attributed to the spectrin component of the membrane cytoskeleton, the energy density only reached one thermal energy unit (k(B)T) per spectrin tetramer near maximum compression. Hence, we hypothesized that the soft compliant regime probed in the experiments represented the compressibility of the outer region of spectrin loops and that the stiff regime < 50 nm was the response of a compact mesh of spectrin backed by a hardcore structure. To evaluate this hypothesis, we used a random flight theory for the entropic elasticity of polymer loops to model the spectrin network. We also examined the possibility that additional steric repulsion and apparent thickening could arise from membrane thermal-bending excitations. Fixing the energy scale to k(B)T/spectrin tetramer, the combined elastic response of a network of ideal polymer loops plus the membrane steric interaction correlated well with the measured dependence of energy density on distance for a statistical segment length of approximately 5 nm for spectrin (i.e., free chain end-to-end length of approximately 29 nm) and a hardcore limit of approximately 30 nm for underlying structure.     

Shear-response of the spectrin dimer-tetramer equilibrium in the red blood cell membrane

The red cell membrane derives its elasticity and resistance to mechanical stresses from the membrane skeleton, a network composed of spectrin tetramers. These are formed by the head-to-head association of pairs of heterodimers attached at their ends to junctional complexes of several proteins. Here we examine the dynamics of the spectrin dimer-dimer association in the intact membrane. We show that univalent fragments of spectrin, containing the dimer self-association site, will bind to spectrin on the membrane and thereby disrupt the continuity of the protein network. This results in impairment of the mechanical stability of the membrane. When, moreover, the cells are subjected to a continuous low level of shear, even at room temperature, the incorporation of the fragments and the consequent destabilization of the membrane are greatly accentuated. It follows that a modest shearing force, well below that experienced by the red cell in the circulation, is sufficient to sever dimer-dimer links in the network. Our results imply 1) that the membrane accommodates the enormous distortions imposed on it during the passage of the cell through the microvasculature by means of local dissociation of spectrin tetramers to dimers, 2) that the network in situ is in a dynamic state and undergoes a "breathing" action of tetramer dissociation and re-formation.     

A flow EPR study of deformation and orientation characteristics of erythrocyte ghosts: A possible effect of an altered state of cytoskeletal network

Summary Using the flow EPR technique, we investigated the resealed ghost deformability in shear flow and the effects of the altered state of cytoskeletal network induced by hypotonic incubation of ghosts. Isotonically resealed ghosts in the presence of Mg-ATP, in which alteration of cytoskeletal network is not effected, have smooth biconcave discoid shapes, and show a flow orientation and deformation behavior similar to that of erythrocytes, except that higher viscosities are required to induce the same degrees of deformation and orientation as in erythrocytes. The flow behavior of resealed ghosts is Mg-ATP dependent, and the shape of the ghosts resealed without Mg-ATP is echinocytic. In contrast, the ghosts resealed by hypotonic incubation show a markedly reduced deformability even with Mg-ATP present. Nonreducing, nondenaturing polyacrylamide gel electrophoresis (PAGE) of the low ionic strength extracts from hypotonically resealed ghosts reveals a shift of the spectrin tetramer-dimer equilibrium toward the dimers. In the maleimide spin-labeled ghosts, the ratios of the weakly immobilized to the strongly immobilized EPR intensities are larger in hypotonically resealed ghosts than in the isotonically resealed ghosts, indicating an enhanced mobility in the spectrin structure in the former. Photomicrographs of hypotonically resealed ghosts show slightly stomatocytic transformations. These data suggest that the shape and the deformability loss in hypotonically resealed ghosts are related to an alteration of the spectrin tetramer-dimer equilibrium in the membrane. Thus, the shift of the equilibrium is likely to affect the regulation of the membrane deformability both in normal and pathological cells such as hereditary elliptocytes.     

Human erythrocyte spectrin dimer intrinsic viscosity: temperature dependence and implications for the molecular basis of the erythrocyte membrane free energy

We have determined experimentally the temperature dependence of human erythrocyte spectrin dimer intrinsic viscosity at shear rates 8-12 s-1 using a Cartesian diver viscometer. We find that the intrinsic viscosity decreases from 43 +/- 3 ml/g at 4 degrees C to 34 +/- 3 ml/g when the temperature is increased to 38 degrees C. Our results show that spectrin dimers are flexible worm-like macromolecules with persistence length about 20 nm and that the mean square end-to-end distance for this worm-like macromolecules decreases when the temperature is increased. This implies that the spectrin dimer internal energy decreases when the end-to-end distance is increased and that the free energy increase associated with making the end-to-end distance longer than the equilibrium value for the free molecules is of entropic origin. The temperature dependence of the erythrocyte membrane shear modulus reported previously in the literature therefore appears mainly to be due to temperature dependent alterations in the membrane skeleton topology.     

On the measurement of shear elastic moduli and viscosities of erythrocyte plasma membranes by transient deformation in high frequency electric fields.

We present a new method to measure the shear elastic moduli and viscosities of erythrocyte membranes which is based on the fixation and transient deformation of cells in a high-frequency electric field. A frequency domain of constant force (arising by Maxwell Wagner polarization) is selected to minimize dissipative effects. The electric force is thus calculated by electrostatic principles by considering the cell as a conducting body in a dielectric fluid and neglecting membrane polarization effects. The elongation A of the cells perpendicular to their rotational axis exhibits a linear regime (A proportional to Maxwell tension or to square of the electric field E2) at small, and a nonlinear regime (A proportional to square root of Maxwell tension or to the electric field E) at large extensions with a cross-over at A approximately 0.5 micron. The nonlinearity leads to amplitude-dependent response times and to differences of the viscoelastic response and relaxation functions. The cells exhibit pronounced yet completely reversible tip formations at large extensions. Absolute values of the shear elastic modulus, mu, and membrane viscosity, eta, are determined by assuming that field-induced stretching of the biconcave cell may be approximately described in terms of a sphere to ellipsoid deformation. The (nonlinear) elongation-vs.-force relationship calculated by the elastic theory of shells agress well with the experimentally observed curves and the values of mu = 6.1 x 10(-6) N/m and eta = 3.4 x 10(-7) Ns/m are in good agreement with the micropipette results of Evans and co-workers. The effect of physical, biochemical, and disease-induced structural changes on the viscoelastic parameters is studied. The variability of mu and eta of a cell population of a healthy donor is +/- 45%, which is mainly due to differences in the cell age. The average mu value of cells of different healthy donors scatters by +/- 18%. Osmotic deflation of the cells leads to a fivefold increase of mu and 10-fold increase of eta at 500 mosm. The shear modulus mu increases with temperature showing that the cytoskeleton does not behave as a network of entropy elastic springs. Elliptic cells of patients suffering from elliptocytosis of the Leach phenotype exhibit a threefold larger value of mu than normal discocytes of control donors. Cross-linking of the spectrin by the divalent S-H agents diamide (1 mM, 15 min incubation) leads to an eightfold increase of mu whereas eta is essentially constant. The effect of diamide is reversed after treatment with S-S bond splitting agents.     

Euler buckling-induced folding and rotation of red blood cells in an optical trap

We investigate the physics of an optically driven micromotor of biological origin. When a single, live red blood cell (RBC) is placed in an optical trap, the normal biconcave disc shape of the cell is observed to fold into a rod-like shape. If the trapping laser beam is circularly polarized, the folded RBC rotates. A model based on geometric considerations, using the concept of buckling instabilities, captures the folding phenomenon; the rotation of the cell is rationalized using the Poincaré sphere. Our model predicts that (i) at a critical power of the trapping laser beam the RBC shape undergoes large fluctuations, and (ii) the torque that is generated is proportional to the power of the laser beam. These predictions are verified experimentally. We suggest a possible mechanism for the emergence of birefringent properties in the RBC in the folded state.     

Amphiphile induced echinocyte-spheroechinocyte transformation of red blood cell shape

A possible physical explanation of the echinocyte-spheroechinocyte red blood cell (RBC) shape transformation induced by the intercalation of amphiphilic molecules into the outer layer of the RBC plasma membrane bilayer is given. The stable RBC shape is determined by the minimization of the membrane elastic energy, consisting of the bilayer bending energy, the bilayer relative stretching energy and the skeleton shear elastic energy. It is shown that for a given relative cell volume the calculated number of echinocyte spicula increases while their size decreases as the number of the intercalated amphiphilic molecules in the outer layer of the cell membrane bilayer is increased, which is in agreement with experimental observations. Further, it is shown that the equilibrium difference between the outer and the inner membrane leaflet areas of the stable RBC shapes increases if the amount of the intercalated amphiphiles is increased, thereby verifying theoretically the original bilayer couple hypothesis of Sheetz and Singer (1974) and Evans (1974). Received: 22 August 1997 / Revised version: 25 November 1997 / Accepted: 11 February 1998