Simulations of the erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration.

Coarse-grained molecular models of the erythrocyte membrane's spectrin cytoskeleton are presented in Monte Carlo simulations of whole cells in micropipette aspiration. The nonlinear chain elasticity and sterics revealed in more microscopic cytoskeleton models (developed in a companion paper; Boey et al., 1998. Biophys. J. 75:1573-1583) are faithfully represented here by two- and three-body effective potentials. The number of degrees of freedom of the system are thereby reduced to a range that is computationally tractable. Three effective models for the triangulated cytoskeleton are developed: two models in which the cytoskeleton is stress-free and does or does not have internal attractive interactions, and a third model in which the cytoskeleton is prestressed in situ. These are employed in direct, finite-temperature simulations of erythrocyte deformation in a micropipette. All three models show reasonable agreement with aspiration measurements made on flaccid human erythrocytes, but the prestressed model alone yields optimal agreement with fluorescence imaging experiments. Ensemble-averaging of nonaxisymmetrical, deformed structures exhibiting anisotropic strain are thus shown to provide an answer to the basic question of how a triangulated mesh such as that of the red cell cytoskeleton deforms in experiment.     

Direct measures of large, anisotropic strains in deformation of the erythrocyte cytoskeleton.

The erythrocyte's spectrin-actin membrane skeleton is directly shown to be capable of sustaining large, anisotropic strains. Photobleaching of an approximately 1-micrometer stripe in rhodamine phalloidin-labeled actin appears stable up to at least 37 degrees C, and is used to demonstrate large in-surface stretching during elastic deformation of the skeleton. Principal extension or stretch ratios of at least approximately 200% and contractions down to approximately 40%, both referenced to an essentially undistorted cell, are visually demonstrated in micropipette-imposed deformation. Such anisotropic straining is seen to be consistent at a qualitative level with now classic analyses (Evans. 1973. Biophys. J. 13:941-954) and is generally nonhomogeneous though axisymmetric down to the submicron scale. Local, direct measurements of stretching prove quantitatively consistent (within approximately 10%) with integrated estimates that are based simply on a measured relative density distribution of actin. The measurements are also in close agreement with direct computation of mean spectrin chain extension in full statistical mechanical simulations of a coarse-grained network held in a micropipette. Finally, as a cell thermally fragments near approximately 48 degrees C, the patterned photobleaching demonstrates a destructuring of the surface network in a process that is more readily attributable to transitions in spectrin than in F-actin.     

Microscopic investigation of erythrocyte deformation dynamics

The understanding of erythrocyte deformation under conditions of high shear stress and short exposure time is central to the study of hemorheology and hemolysis within prosthetic blood contacting devices. A combined computational and experimental microscopic study was conducted to investigate the erythrocyte deformation and its relation to transient stress fields. A microfluidic channel system with small channels fabricated using polydimethylsiloxane on the order of 100 mum was designed to generate transient stress fields through which the erythrocytes were forced to flow. The shear stress fields were analyzed by three-dimensional computational fluid dynamics. Microscopic images of deforming erythrocytes were experimentally recorded to obtain the changes in cell morphology over a wide range of fluid dynamic stresses. The erythrocyte elongation index (EI) increased from 0 to 0.54 with increasing shear stress up to 123 Pa. In this shear stress range, erythrocytes behaved like fluid droplets, and deformed and flowed following the surrounding fluid. Cells exposed to shear stress beyond 123 Pa (up to 5170 Pa) did not exhibit additional elongation beyond EI=0.54. Two-stage deformation of erythrocytes in response to shear stress was observed: an initial linear elongation with increasing shear stress and a plateau beyond a critical shear stress.     

Calmodulin-binding protein of erythrocyte cytoskeleton

Previously we have shown that purified spectrin binds calmodulin in the presence of Ca²⁺ with a Kd value of 3 μM (Sobue, K. et al. (1980) Biochemistry International 1, 561–566). We now provide evidence that the calmodulin-binding activity found in the human erythrocyte cytoskeleton is indeed due to spectrin and no other binding proteins are involved, i.e. the binding activity was purified from the erythrocyte cytoskeleton quantitatively and the purified peak contained spectrin as the only protein constituent. Moreover, Kd value (2.8 μM) and the maximum binding capacity (160,000 – 200,000 calmodulin per cell) obtained from the kinetic analysis of the binding activity in the crude cytoskeleton agreed with the corresponding values reported for purified spectrin. Since the concentration of calmodulin in the erythrocyte cell, which was 2.5 μM or 1.6 × 10⁵ molecules per cell, is close to both the Kd value and the number of the binding sites in the cell, respectively, free calmodulin in the erythrocyte cell may be in a dynamic equilibrium with the spectrin-bound form in vivo depending upon the intracellular concentration of Ca²⁺.     

Computer simulation of a model network for the erythrocyte cytoskeleton.

The geometry and mechanical properties of the human erythrocyte membrane cytoskeleton are investigated by a computer simulation in which the cytoskeleton is represented by a network of polymer chains. Four elastic moduli as well as the area and thickness are predicted for the chain network as a function of temperature and the number of segments in each chain. Comparisons are made with mean field arguments to examine the importance of steric interactions in determining network properties. Applied to the red blood cell, the simulation predicts that in the bilayer plane the membrane cytoskeleton has a shear modulus of 10 +/- 2 x 10(-6) J/m2 and an areal compression modulus of 17 +/- 2 x 10(-6) J/m2. The volume compression modulus and the transverse Young's modulus of the cytoskeleton are predicted to be 1.2 +/- 0.1 x 10(3) J/m3 and 2.0 +/- 0.1 x 10(3) J/m3, respectively. Elements of the cytoskeleton are predicted to have a mean displacement from the bilayer plane of 15 nm. The simulation agrees with some, but not all, of the shear modulus measurements. The other predicted moduli have not been measured.     

Interfacial models of nerve fiber cytoskeleton.

A new approach, basing on a resemblance between cytoskeleton structures associated with plasma membranes and interfacial layers of coexisting phases, is proposed. In particular, a lattice model, similar to those of the theory of surface properties of pure liquids and nonelectrolyte solutions (Ono, S., and S. Kondo. 1960. Handbuch der Physik.), has been developed to describe nerve fiber cytoskeleton. The preliminary consideration of the model shows the existence of submembrane cytoskeleton having increased peripheral densities of microtubules (compared with the bulk density) which is in qualitative agreement with the data in literature. Some additional possibilities of the approach proposed are briefly discussed.     

Structural unit of the erythrocyte cytoskeleton. Isolation and electron microscopic examination

We isolated a protein complex containing major cytoskeletal components from the Triton shell of bovine erythrocytes. This protein complex, which we called the 26-S complex, consisted of three major components, spectrin, band-4.1 protein and actin, and one minor component, band-4.9 protein. The molar ratio of spectrin heterodimer:band 4.1:actin was determined by sodium dodecyl sulfate (SDS) gel electrophoresis to be about 1:2:2, approximately the same as that for the Triton shell. By electron microscopic examinations of rotary-shadowed specimens, it was revealed that the 26-S complex had a "spider-like" morphology with a central core and several spectrin heterodimers radiating from it. The number of spectrin arms in the complex was not constant but was in the range between 3 and 6. The complexes with five spectrin heterodimers were the most numerous. The results showed that the 26-S complex contained on the average five spectrin heterodimers, ten band-4.1 polypeptides and ten actin monomers. As judged from the formation of oligomeric 26-S complexes through spectrin arms, the central core of the complex presumably contains band 4.1 and actin. Supporting this conclusion, the central core acted as a nucleus for actin polymerization when the 26-S complex was mixed with G-actin under an actin-polymerizing condition. The 26-S complex could form large aggregates under a certain condition that spectrin was promoted to associate from dimer to tetramer. We conclude that the 26-S complex is the structural unit of the erythrocyte cytoskeleton.     

Force relaxation and permanent deformation of erythrocyte membrane.

Force relaxation and permanent deformation processes in erythrocyte membrane were investigated with two techniques: micropipette aspiration of a portion of a flaccid cell, and extension of a whole cell between two micropipettes. In both experiments, at surface extension ratios less than 3:1, the extent of residual membrane deformation is negligible when the time of extension is less than several minutes. However, extensions maintained longer result in significant force relaxation and permanent deformation. The magnitude of the permanent deformation is proportional to the total time period of extension and the level of the applied force. Based on these observations, a nonlinear constitutive relation for surface deformation is postulated that serially couples a hyperelastic membrane component to a linear viscous process. In contrast with the viscous dissipation of energy as heat that occurs in rapid extension of a viscoelastic solid, or in plastic flow of a material above yield, the viscous process in this case represents dissipation produced by permanent molecular reorganization through relaxation of structural membrane components. Data from these experiments determine a characteristic time constant for force relaxation, tau, which is the ratio of a surface viscosity, eta to the elastic shear modulus, mu. Because it was found that the concentration of albumin in the cell suspension strongly mediates the rate of force relaxation, values for tau of 10.1, 40.0, 62.8, and 120.7 min are measured at albumin concentrations of 0.0, 0.01, 0.1, and 1.% by weight in grams, respectively. The surface viscosity, eta, is calculated from the product of tau and mu. For albumin concentrations of 0.0, 0.01, 0.1, and 1% by weight in grams, eta is equal to 3.6, 14.8, 25.6, and 51.9 dyn s/cm, respectively.     

Spectrin oligomers: A structural feature of the erythrocyte cytoskeleton

Spectrin reversibly self-associates to high molecular weight oligomers through a concentration-driven process characterized by association constants of about 10⁵ mol^?1. This association is prominent under physiological conditions of pH, ionic strength, and temperature. It is disrupted by urea, but not Triton X-100. The process of spectrin association appears mathematically to resemble that for tropomyosin, although the mechanism is probably different. Spectrin association is weak compared to other prominent protein–protein associations in the red cell membrane skeleton. The linkage of these weak and strong associations suggests a process whereby the membrane skelton spontaneously assembles. Such affinity-modulated assembly involving weak associations is likely to be the focus of numerous membrane control mechanisms.     

A computational method for developing hierarchical large deformation viscoelastic models of the contracting heart

In this study, a new computational method for modelling the contracting heart is described. Using this method, the cardiac wall is constructed from basic, repeating contractile units that represent individual myocardial units and collagen, each with its own set of parameters, including orientation, passive and active behaviour and stimulation propagation. The method allows individual control of each structural unit (e.g. at the level of a single myocardial unit). Feasibility of modelling dynamic heart contraction using this method is demonstrated using 2D cross-sections and simplified 3D geometries. Effects of non-contractile scar and myopathic tissue were also tested in these geometrical configurations. Results from the 2D and 3D simulations were, overall, in agreement with well-established physiological data. The present method holds promise for modelling complex heart pathologies, abnormal mechanical properties (e.g. myocardial infarcts) and electrical conduction properties (branch blocks), and their spatial distributions across the myocardial tissues.     

Malarial proteins that interact with the erythrocyte membrane and cytoskeleton

Several distinct classes of Plasmodium proteins have been proposed to interact with the submembrane skeleton of the erythrocyte based upon differential solubility and subcellular localization studies. That the parasite affects the erythrocyte membrane by interacting with the submembrane skeleton is an attractive hypothesis since the membrane skeleton likely regulates many aspects of membrane topography and function. The precise interactions between host and parasite proteins at the molecular level and how the parasite proteins are transported to the erythrocyte membrane are not completely understood. Experiments addressing these questions are under way, and such studies will provide valuable information about the host-parasite interface. In addition, the characterization of the interaction of Plasmodium proteins with the host erythrocyte membrane may also provide new insight into the structure and function of the erythrocyte membrane or membranes in general.     

Simulations of the dynamics at an RNA-protein interface.

Molecular dynamics simulations of the RNA-binding domain of the U1A spliceosomal protein in complex with its cognate RNA hairpin, performed at low and high ionic strength in aqueous solution, suggest a pathway for complex dissociation. First, cations condense around the RNA and compete with the protein for binding sites. Then solvated ions specifically destabilize residues at the RNA-protein interface. For a discrete cluster of residues at the complex interface, the simulations reveal an increased deviation from the crystal structure at high salt concentrations while the remaining protein scaffold is stabilized under these conditions. The microscopic picture of salt influence on the complex suggests guidelines for rational design of interface inhibitors targeted at RNA-protein complexes.     

Phospholipid asymmetry during erythrocyte deformation: maintenance of the unit membrane.

To assess the red blood cell (RBC) membrane's ability to maintain normal phospholipid orientation in the face of deforming stress, we examined RBC subjected to elliptical, tank-treading deformation. As determined by accessibility to phospholipase digestion and by labelling with fluorescamine, normal RBC are able to fully preserve their phospholipid asymmetry despite attaining over 96% of their maximal possible deformation. Phospholipid orientation is unchanged during deformation even for RBC that are ATP-depleted or vanadate-treated and for RBC that already have destabilized phospholipids due to treatment with t-butyl hydroperoxide. These data indicate that maintenance of phospholipid organization during marked deforming stress and tank-treading motion of the membrane is ascribable predominantly to the passive stabilizing effect of membrane proteins. This provides additional evidence for the concept of a unit membrane characterized by intimate associations between lipid and protein.     

Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte

We present a three-dimensional computational study of whole-cell equilibrium shape and deformation of human red blood cell (RBC) using spectrin-level energetics. Random network models consisting of degree-2, 3, ..., 9 junction complexes and spectrin links are used to populate spherical and biconcave surfaces and intermediate shapes, and coarse-grained molecular dynamics simulations are then performed with spectrin connectivities fixed. A sphere is first filled with cytosol and gradually deflated while preserving its total surface area, until cytosol volume consistent with the real RBC is reached. The equilibrium shape is determined through energy minimization by assuming that the spectrin tetramer links satisfy the worm-like chain free-energy model. Subsequently, direct stretching by optical tweezers of the initial equilibrium shape is simulated to extract the variation of axial and transverse diameters with the stretch force. At persistence length p = 7.5 nm for the spectrin tetramer molecule and corresponding in-plane shear modulus mu(0) approximately 8.3 microN/m, our models show reasonable agreement with recent experimental measurements on the large deformation of RBC with optical tweezers. We find that the choice of the reference state used for the in-plane elastic energy is critical for determining the equilibrium shape. If a position-independent material reference state such as a full sphere is used in defining the in-plane energy, then the bending modulus kappa needs to be at least a decade larger than the widely accepted value of 2 x 10(-19) J to stabilize the biconcave shape against the cup shape. We demonstrate through detailed computations that this paradox can be avoided by invoking the physical hypothesis that the spectrin network undergoes constant remodeling to always relax the in-plane shear elastic energy to zero at any macroscopic shape, at some slow characteristic timescale. We have devised and implemented a liquefied network structure evolution algorithm that relaxes shear stress everywhere in the network and generates cytoskeleton structures that mimic experimental observations.     

Ultrastructure of the human erythrocyte cytoskeleton and its attachment to the membrane

We attached paraformaldehyde-fixed human erythrocyte ghosts to coated coverslips and sheared them to expose the cytoskeleton. Quick-freeze, deep-etch, rotary-replication, or tannic acid/osmium fixation and plastic embedding revealed the cytoskeleton as a dense network of intersecting straight filaments. Previous negative stain studies on spread skeletons found 5-6 spectrin tetramers intersecting at each actin oligomer, with an estimated 250 such intersections/microns 2 of membrane. In contrast, we found 3-4 filaments at each intersection and approximately 400 intersections/microns 2 of membrane. Immunogold labeling verified that the filaments were spectrin, but their lengths (29-37 nm) were approximately one-third that of extended spectrin dimers. The length and diameter of the filaments were sufficient to accommodate spectrin dimers, but not spectrin tetramers. Our results suggest that, in situ, spectrin dimers may associate as hexamers and octamers, rather than tetramers. We present several explanations that can reconcile our observations on intact cytoskeletons with previous reports on spread material. Extracting sheared ghosts with solutions of low ionic strength removed the cytoskeleton to reveal projections from the cytoplasmic surface of the membrane. These projections contained band 3, as shown by immunogold labeling, and they aggregated to a similar extent as intramembrane particles (IMP) when the cytoskeleton was removed, suggesting a direct relationship between these structures. Quantification indicated a stoichiometry of 2 IMP for each cytoplasmic projection. Cytoplasmic projections presumably contain other proteins besides band 3 since further treatment with high ionic strength solutions extracts peripheral proteins and reduces the diameter of projections by approximately 3 nm.     

Simulations of the solvent structure for macromolecules. I. Solvation of B-DNA double helix at T = 300 K

Monte Carlo simulations are reported for a system of 447 water molecules enclosing a B-DNA double-helix fragment with 12 base pairs and the corresponding sugar and phosphate units. From a detailed analysis on the interaction energies and probability distributions (at a simulated temperature of 300 K), the water molecules can be partitioned into clusters strongly interacting with (1) the phosphates, (2) the sugars, (3) the sugars and the bases, and (4) the base pairs. In addition, transgroove and interphosphate filament of hydrogen-bonded water molecules have been detected. From simulations performed with variable numbers of water molecules, a theoretical isotherm has been obtained, with the characteristic sigmoidal shape, known from absorption–desorption experiments on related systems. The expected main features for the structure of water molecules solvating B-DNA with Na⁺ counterions are briefly discussed at the end of the paper.     

Sample preparation and imaging of erythrocyte cytoskeleton with the atomic force microscopy

A novel method for the covalent attachment of erythrocytes to glass microscope coverslips that can be used to image intact cells and the cytoplasmic side of the cell membrane with either solid or liquid mode atomic force microscopy (AFM) is described. The strong binding of cells to the glass surface is achieved by the interaction of cell membrane carbohydrates to lectin, which is bound to N-5-azido-2-nitrobenzoyloxysuccinimide (ANBNOS)-coated coverslips (1). The effectiveness of this method is compared with the other commonly used methods of immobilizing intact erythrocytes on glass coverslips for AFM observations. Experimental conditions of AFM imaging of biologic tissue are discussed, and typical topographies of the extracellular and the cytoplasmic surfaces of the plasma membrane in the dry state and in the liquid state are presented. Comparison of the spectrin network of cell age-separated erythrocytes has demonstrated significant loss in the network order in older erythrocytes. The changes are quantitatively described using the pixel height histogram and window size grain analysis.     

Recombination of human erythrocyte apoprotein and lipid. I. Interaction of apoprotein and lipid at the air-water interface

The formation and stabilization of a complex between total erythrocyte apoprotein and monolayers of total erythrocyte lipid as measured by changes of surface pressure (Δπ) and rate of change of surface pressure (dπ/dt) was studied as a function of pH, ionic strength, and lipid surface pressure. Penetration of apoprotein into lipid monolayers was favored by conditions in which lipid and apoprotein were oppositely charged. Once the interaction was completed, the resultant surface complex was resistant to large changes in subphase pH and ionic strength as shown by the insensitivity of Δπ to these parameters. The dπ/dt, however, showed strong dependence on pH and ionic strength, but not on lipid surface pressure. A sharp decrease in dπ/dt around pH 3.5–4.5 is associated with the change in apoprotein charge from (+) to (?). Comparison of complex formation between apoprotein and bovine serum albumin, cytochrome c, and human hemoglobin suggests that erythrocyte apoprotein was specialized in its interaction with erythrocyte lipids. The data show that formation of an apoprotein-lipid complex at the air-water interface has both electrostatic and hydrophobic components. This contradicts results from other laboratories studying erythrocyte membrane recombination by bulk methods.     

Actin protofilament orientation at the erythrocyte membrane.

The short actin filaments in the erythrocyte's membrane skeleton are shown to be largely oriented tangent to the lipid bilayer. Actin "proto"-filaments have previously been described as junctional centers intertriangulated by spectrin; however, the protofilaments may simultaneously serve as pinning centers between the network and the overlying bilayer. The latter function now seems of particular importance because near-normal network assembly has been reported with transgenic mouse sphero-erythrocytes that lack the primary linkage protein Band 3. To assess possible physical constraints on actin protofilaments in intact membranes, fluorescence polarization microscopy (FPM) has been used to study rhodamine phalloidin-labeled red cell ghosts. A basis for interpreting FPM images of cells is provided by FPM applied to isolated actin filaments. These are labeled with the same rhodamine probes and imaged at various orientations with respect to the polarizers, including filament orientations perpendicular to the image plane. High aperture and fluorophore conjugation effects are found to be minimal, enabling development of a simple, semi-empirical model which indicates that protofilaments are generally within approximately 20 degrees of the membrane tangent plane.