Influence of network topology on the elasticity of the red blood cell membrane skeleton

αI-spectrin represents evolutionary optimization of spectrin for red blood cell deformability

Spectrin tetramers of the membranes of enucleated mammalian erythrocytes play a critical role in red blood cell survival in circulation. One of the spectrins, αI, emerged in mammals with enucleated red cells after duplication of the ancestral α-spectrin gene common to all animals. The neofunctionalized αI-spectrin has moderate affinity for βI-spectrin, whereas αII-spectrin, expressed in nonerythroid cells, retains ancestral characteristics and has a 10-fold higher affinity for βI-spectrin. It has been hypothesized that this adaptation allows for rapid make and break of tetramers to accommodate membrane deformation. We have tested this hypothesis by generating mice with high-affinity spectrin tetramers formed by exchanging the site of tetramer formation in αI-spectrin (segments R0 and R1) for that of αII-spectrin. Erythrocytes with αIIβI presented normal hematologic parameters yet showed increased thermostability, and their membranes were significantly less deformable; under low shear forces, they displayed tumbling behavior rather than tank treading. The membrane skeleton is more stable with αIIβI and shows significantly less remodeling under deformation than red cell membranes of wild-type mice. These data demonstrate that spectrin tetramers undergo remodeling in intact erythrocytes and that this is required for the normal deformability of the erythrocyte membrane. We conclude that αI-spectrin represents evolutionary optimization of tetramer formation: neither higher-affinity tetramers (as shown here) nor lower affinity (as seen in hemolytic disease) can support the membrane properties required for effective tissue oxygenation in circulation.

Nanoerythrosomes tailoring: Lipid induced protein scaffolding in ghost membrane derived vesicles

A peculiar polygonal protein scaffolding that resembles to spectrin-based skeleton of red blood cells can be reconstructed on the outer surface of vesicle-like nanoerythrosomes. The approximately 130 nm sized nanoerythrosomes are produced from red blood cell ghosts by addition of phospholipids (dipalmitoylphosphatidylcholine, DPPC). The scaffolding, constructed from the structural proteins of the cell membrane skeleton, covers the whole object resulting an enhanced stiffness. The protein pattern of the scaffolding is thermosensitive, reversible transformable in the biologically relevant temperature range. When the lipid additive is changed from DPPC to lysophospholipid (LPC), the protein network/scaffolding ceases to exist. By the variation of lipid type and ratio, a tailoring of the nanoerythrosomes can be achieved. During the tailoring process nanoerythrosomes or micelles, in a wide size range from 200 to 30 nm, are produced.

Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones

Hemostasis is a complex physiological mechanism that functions to maintain vascular integrity under any conditions. Its primary components are blood platelets and a coagulation network that interact to form the hemostatic plug, a combination of cell aggregate and gelatinous fibrin clot that stops bleeding upon vascular injury. Disorders of hemostasis result in bleeding or thrombosis, and are the major immediate cause of mortality and morbidity in the world. Regulation of hemostasis and thrombosis is immensely complex, as it depends on blood cell adhesion and mechanics, hydrodynamics and mass transport of various species, huge signal transduction networks in platelets, as well as spatiotemporal regulation of the blood coagulation network. Mathematical and computational modeling has been increasingly used to gain insight into this complexity over the last 30 years, but the limitations of the existing models remain profound. Here we review state-of-the-art-methods for computational modeling of thrombosis with the specific focus on the analysis of unresolved challenges. They include: a) fundamental issues related to physics of platelet aggregates and fibrin gels; b) computational challenges and limitations for solution of the models that combine cell adhesion, hydrodynamics and chemistry; c) biological mysteries and unknown parameters of processes; d) biophysical complexities of the spatiotemporal networks' regulation. Both relatively classical approaches and innovative computational techniques for their solution are considered; the subjects discussed with relation to thrombosis modeling include coarse-graining, continuum versus particle-based modeling, multiscale models, hybrid models, parameter estimation and others. Fundamental understanding gained from theoretical models are highlighted and a description of future prospects in the field and the nearest possible aims are given.

Image-based model of the spectrin cytoskeleton for red blood cell simulation

We simulate deformable red blood cells in the microcirculation using the immersed boundary method with a cytoskeletal model that incorporates structural details revealed by tomographic images. The elasticity of red blood cells is known to be supplied by both their lipid bilayer membranes, which resist bending and local changes in area, and their cytoskeletons, which resist in-plane shear. The cytoskeleton consists of spectrin tetramers that are tethered to the lipid bilayer by ankyrin and by actin-based junctional complexes. We model the cytoskeleton as a random geometric graph, with nodes corresponding to junctional complexes and with edges corresponding to spectrin tetramers such that the edge lengths are given by the end-to-end distances between nodes. The statistical properties of this graph are based on distributions gathered from three-dimensional tomographic images of the cytoskeleton by a segmentation algorithm. We show that the elastic response of our model cytoskeleton, in which the spectrin polymers are treated as entropic springs, is in good agreement with the experimentally measured shear modulus. By simulating red blood cells in flow with the immersed boundary method, we compare this discrete cytoskeletal model to an existing continuum model and predict the extent to which dynamic spectrin network connectivity can protect against failure in the case of a red cell subjected to an applied strain. The methods presented here could form the basis of disease- and patient-specific computational studies of hereditary diseases affecting the red cell cytoskeleton.

Red blood cells are responsible for delivering oxygen to tissues throughout the body. These terminally differentiated cells have developed a fascinating flexibility and resiliency that is critical to navigating the circulatory system. Far from being rigid bodies, red blood cells adopt biconcave disk shapes at equilibrium, parachute-like shapes as they move between large vessels and small capillaries, and more extreme shapes as they traverse the endothelial slits of the spleen. Understanding the remarkable mechanical properties that allow red cells to experience such large deformations while maintaining structural integrity is a fundamental question in physiology that may help advance treatments of genetic disorders such as hereditary spherocytosis and elliptocytosis that affect red cell flexibility and can lead to severe anemia. In this work, we present a model of the red blood cell cytoskeleton based on cryoelectron tomography data. We develop an image processing technique to gather statistics from these data and use these statistics to generate a random entropic network to model the cytoskeleton. We then simulate the behavior of the resulting red blood cells in flow. As we demonstrate through simulations, this method makes it possible to examine the consequences of changes in microstructural properties such as the rate of cytoskeletal remodeling.

The effect of a high frequency electromagnetic field in the microwave range on red blood cells

The effect of red blood cells (RBC) exposed to an 18 GHz electromagnetic field (EMF) was studied. The results of this study demonstrated for the first time that exposure of RBCs to 18 GHz EMF has the capacity to induce nanospheres uptake in RBCs. The uptake of nanospheres (loading efficiency 96% and 46% for 23.5 and 46.3 nm nanospheres respectively), their presence and locality were confirmed using three independent techniques, namely scanning electron microscopy, confocal laser scanning microscopy and transmission electron microscopy. It appeared that 23.5 nm nanospheres were translocated through the membrane into the cytosol, while the 46.3 nm-nanospheres were mostly translocated through the phospholipid-cholesterol bilayer, with only some of these nanospheres passing the 2D cytoskeleton network. The nanospheres uptake increased by up to 12% with increasing temperature from 33 to 37 °C. The TEM analysis revealed that the nanospheres were engulfed by the cell membrane itself, and then translocated into the cytosol. It is believed that EMF-induced rotating water dipoles caused disturbance of the membrane, initiating its deformation and result in an enhanced degree of membrane trafficking via a quasi-exocytosis process.

Stiffening of Red Blood Cells Induced by Cytoskeleton Disorders: A Joint Theory-Experiment Study

The functions and elasticities of the cell are largely related to the structures of the cytoskeletons underlying the lipid bilayer. Among various cell types, the red blood cell (RBC) possesses a relatively simple cytoskeletal structure. Underneath the membrane, the RBC cytoskeleton takes the form of a two-dimensional triangular network, consisting of nodes of actins (and other proteins) and edges of spectrins. Recent experiments focusing on the malaria-infected RBCs (iRBCs) show that there is a correlation between the elongation of spectrins in the cytoskeletal network and the stiffening of the iRBCs. Here we rationalize the correlation between these two observations by combining the wormlike chain model for single spectrins and the effective medium theory for the network elasticity. We specifically focus on how the disorders in the cytoskeletal network affect its macroscopic elasticity. Analytical and numerical solutions from our model reveal that the stiffness of the membrane increases with increasing end-to-end distances of spectrins, but has a nonmonotonic dependence on the variance of the end-to-end distance distributions. These predictions are verified quantitatively by our atomic force microscopy and micropipette aspiration measurements of iRBCs. The model may, from a molecular level, provide guidelines for future identification of new treatment methods for RBC-related diseases, such as malaria infection.

A novel strain energy relationship for red blood cell membrane skeleton based on spectrin stiffness and its application to micropipette deformation

Red blood cell (RBC) membrane skeleton is a closed two-dimensional elastic network of spectrin tetramers with nodes formed by short actin filaments. Its three-dimensional shape conforms to the shape of the bilayer, to which it is connected through vertical linkages to integral membrane proteins. Numerous methods have been devised over the years to predict the response of the RBC membrane to applied forces and determine the corresponding increase in the skeleton elastic energy arising either directly from continuum descriptions of its deformation, or seeking to relate the macroscopic behavior of the membrane to its molecular constituents. In the current work, we present a novel continuum formulation rooted in the molecular structure of the membrane and apply it to analyze model deformations similar to those that occur during aspiration of RBCs into micropipettes. The microscopic elastic properties of the skeleton are derived by treating spectrin tetramers as simple linear springs. For a given local deformation of the skeleton, we determine the average bond energy and define the corresponding strain energy function and stress-strain relationships. The lateral redistribution of the skeleton is determined variationally to correspond to the minimum of its total energy. The predicted dependence of the length of the aspirated tongue on the aspiration pressure is shown to describe the experimentally observed system behavior in a quantitative manner by taking into account in addition to the skeleton energy an energy of attraction between RBC membrane and the micropipette surface.

Fluctuations of red blood cell membranes: The role of the cytoskeleton

We theoretically investigate the membrane fluctuations of red blood cells with focus laid on the role of the cytoskeleton, viewing the system as a membrane coupled to a sparse spring network. This model is exactly solvable and enables us to examine the coupling strength dependence of the membrane undulation. We find that the coupling modifies the fluctuation spectrum at wavelengths longer than the mesh size of the network, while leaving the fluid-like behavior of the membrane intact at shorter wavelengths. The fluctuation spectra can be markedly different, depending on not only the relative amplitude of the bilayer bending energy with respect to the cytoskeleton deformation energy but also the bilayer-cytoskelton coupling strength.

Tensegrity, cellular biophysics, and the mechanics of living systems

The recent convergence between physics and biology has led many physicists to enter the fields of cell and developmental biology. One of the most exciting areas of interest has been the emerging field of mechanobiology that centers on how cells control their mechanical properties, and how physical forces regulate cellular biochemical responses, a process that is known as mechanotransduction. In this article, we review the central role that tensegrity (tensional integrity) architecture, which depends on tensile prestress for its mechanical stability, plays in biology. We describe how tensional prestress is a critical governor of cell mechanics and function, and how use of tensegrity by cells contributes to mechanotransduction. Theoretical tensegrity models are also described that predict both quantitative and qualitative behaviors of living cells, and these theoretical descriptions are placed in context of other physical models of the cell. In addition, we describe how tensegrity is used at multiple size scales in the hierarchy of life—from individual molecules to whole living organisms—to both stabilize three-dimensional form and to channel forces from the macroscale to the nanoscale, thereby facilitating mechanochemical conversion at the molecular level.

Nanotopography as a trigger for the microscale, autogenous and passive lysis of erythrocytes

A mechano-responsive topology provides a highly active yet autogenous surface for erythrocyte lysis towards microfluidic haematology platforms.

Contractile network models for adherent cells

Cells sense the geometry and stiffness of their adhesive environment by active contractility. For strong adhesion to flat substrates, two-dimensional contractile network models can be used to understand how force is distributed throughout the cell. Here we compare the shape and force distribution for different variants of such network models. In contrast to Hookean networks, cable networks reflect the asymmetric response of biopolymers to tension versus compression. For passive networks, contractility is modeled by a reduced resting length of the mechanical links. In actively contracting networks, a constant force couple is introduced into each link in order to model contraction by molecular motors. If combined with fixed adhesion sites, all network models lead to invaginated cell shapes, but only actively contracting cable networks lead to the circular arc morphology typical for strongly adhering cells. In this case, shape and force distribution are determined by local rather than global determinants and thus are suited to endow the cell with a robust sense of its environment. We also discuss nonlinear and adaptive linker mechanics as well as the relation to tissue shape.

An Eulerian/XFEM formulation for the large deformation of cortical cell membrane

Most animal cells are surrounded by a thin layer of actin meshwork below their membrane, commonly known as the actin cortex (or cortical membrane). An increasing number of studies have highlighted the role of this structure in many cell functions including contraction and locomotion, but modelling has been limited by the fact that the membrane thickness (about 1 μm) is usually much smaller than the typical size of a cell (10-100 μm). To overcome theoretical and numerical issues resulting from this observation, we introduce in this paper a continuum formulation, based on surface elasticity, that views the cortex as an infinitely thin membrane that can resists tangential deformation. To accurately model the large deformations of cells, we introduced equilibrium equations and constitutive relations within the Eulerian viewpoint such that all quantities (stress, rate of deformation) lie in the current configuration. A solution procedure is then introduced based on a coupled extended finite element approach that enables a continuum solution to the boundary value problem in which discontinuities in both strain and displacement (due to cortical elasticity) are easily handled. We validate the approach by studying the effect of cortical elasticity on the deformation of a cell adhering on a stiff substrate and undergoing internal contraction. Results show very good prediction of the proposed method when compared with experimental observations and analytical solutions for simple cases. In particular, the model can be used to study how cell properties such as stiffness and contraction of both cytoskeleton and cortical membrane lead to variations in cell's surface curvature. These numerical results show that the proposed method can be used to gain critical insights into how the cortical membrane affects cell deformation and how it may be used as a means to determine a cell's mechanical properties by measuring curvatures of its membrane.

Equivalent mechanical properties of biological membranes from lattice homogenization

The goal of this manuscript is to set up a novel methodology for the calculation of the effective mechanical properties of biological membranes viewed as repetitive networks of elastic filaments, based on the discrete asymptotic homogenization method. We will show that for some lattice configurations, flexional effects due to internal structure mechanisms at the unit cell scale lead to additional flexional effects at the continuum scale, accounted for by an internal length associated to a micropolar behavior. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes for a given network topology, as closed form expressions of the geometrical and mechanical micro-parameters. The peptidoglycan and the erythrocyte have been analyzed using this methodology, and their effective moduli are calculated and recorded versus the geometrical and mechanical lattice parameters. A classification of lattices with respect to the choice of the equivalent continuum model is proposed: The Cauchy continuum and a micropolar continuum are adopted as two possible effective medium, for a given beam model. The relative ratio of the characteristic length of the micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. In most cases, the Cauchy continuum is sufficient to model membranes in most of their configurations. The peptidoglycan network may exhibit a re-entrant hexagonal lattice, for which micropolar effects become important. This is attested by the characteristic length becoming larger than the beam length for such configurations. The homogenized moduli give accurate results for both membranes, as revealed by comparison with experimental measurements or simulation results from the literature at the network scale. A first insight into the nonlinear mechanical behavior of the hexagonal and triangular networks is lastly investigated using a perturbative method.

Accurate coarse-grained modeling of red blood cells

We develop a systematic coarse-graining procedure for modeling red blood cells (RBCs) using arguments based on mean-field theory. The three-dimensional RBC membrane model takes into account the bending energy, in-plane shear energy, and constraints of fixed surface area and fixed enclosed volume. The coarse-graining procedure is general, it can be used for arbitrary level of coarse-graining and does not employ any fitting parameters. The sensitivity of the coarse-grained model is investigated and its behavior is validated against available experimental data and in dissipative particle dynamics (DPD) simulations of RBCs in capillary and shear flows.

Sticky chain model for shear response of red blood cells

A sticky chain model has been proposed to describe the unfolding of spectrin network under applied mechanical loads. With the model, the response of a red blood cell (RBC) under static and cyclic shear loading has been predicted, which agrees qualitatively with relevant experimental results.

Propagation of mechanical stress through the actin cytoskeleton toward focal adhesions: model and experiment

We investigate both theoretically and experimentally how stress is propagated through the actin cytoskeleton of adherent cells and consequentially distributed at sites of focal adhesions (FAs). The actin cytoskeleton is modeled as a two-dimensional cable network with different lattice geometries. Both prestrain, resulting from actomyosin contractility, and central application of external force, lead to finite forces at the FAs that are largely independent of the lattice geometry, but strongly depend on the exact spatial distribution of the FAs. The simulation results compare favorably with experiments with adherent fibroblasts onto which lateral force is exerted using a microfabricated pillar. For elliptical cells, central application of external force along the long axis leads to two large stress regions located obliquely opposite to the pulling direction. For elliptical cells pulled along the short axis as well as for circular cells, there is only one region of large stress opposite to the direction of pull. If in the computer simulations FAs are allowed to rupture under force for elliptically elongated and circular cell shapes, then morphologies arise which are typical for migrating fibroblasts and keratocytes, respectively. The same effect can be obtained also by internally generated force, suggesting a mechanism by which cells can control their migration morphologies.

Fluctuations of coupled fluid and solid membranes with application to red blood cells

The fluctuation spectra and the intermembrane interaction of two membranes at a fixed average distance are investigated. Each membrane can either be a fluid or a solid membrane, and in isolation, its fluctuations are described by a bare or a wave-vector-dependent bending modulus, respectively. The membranes interact via their excluded-volume interaction; the average distance is maintained by an external, homogeneous pressure. For strong coupling, the fluctuations can be described by a single, effective membrane that combines the elastic properties. For weak coupling, the fluctuations of the individual, noninteracting membranes are recovered. The case of a composite membrane consisting of one fluid and one solid membrane can serve as a microscopic model for the plasma membrane and cytoskeleton of the red blood cell. We find that, despite the complex microstructure of bilayers and cytoskeletons in a real cell, the fluctuations with wavelengths lambda greater, similar 400 nm are well described by the fluctuations of a single, polymerized membrane (provided that there are no inhomogeneities of the microstructure). The model is applied to the fluctuation data of discocytes ("normal" red blood cells), a stomatocyte, and an echinocyte. The elastic parameters of the membrane and an effective temperature that quantifies active, metabolically driven fluctuations are extracted from the experiments.

A hybrid model for erythrocyte membrane: a single unit of protein network coupled with lipid bilayer

To investigate the nanomechanics of the erythrocyte membrane we developed a hybrid model that couples the actin-spectrin network to the lipid bilayer. This model features a Fourier space Brownian dynamics model of the bilayer, a Brownian dynamics model of the actin protofilament, and a modified wormlike-chain model of the spectrin (including a cable-dynamics model to predict the oscillation in tension). This model enables us to predict the nanomechanics of single or multiple units of the protein network, the lipid bilayer, and the effect of their interactions. The present work is focused on the attitude of the actin protofilament at the equilibrium states coupled with the elevations of the lipid bilayer through their primary linkage at the suspension complex in deformations. Two different actin-spectrin junctions are considered at the junctional complex. With a point-attachment junction, large pitch angles and bifurcation of yaw angles are predicted. Thermal fluctuations at bifurcation may lead to mode-switching, which may affect the network and the physiological performance of the membrane. In contrast, with a wrap-around junction, pitch angles remain small, and the occurrence of bifurcation is greatly reduced. These simulations suggest the importance of three-dimensional molecular junctions and the lipid bilayer/protein network coupling on cell membrane mechanics.

Active elastic network: cytoskeleton of the red blood cell

In red blood cells there is a cortical cytoskeleton; a two-dimensional elastic network of membrane-attached proteins. We describe, using a simple model, how the metabolic activity of the cell, through the consumption of ATP, controls the stiffness of this elastic network. The unusual mechanical property of active strain softening is described and compared to experimental data. As a by-product of this activity there is also an active contribution to the amplitude of membrane fluctuations. We model this membrane as a field of independent "curvature motors," and calculate the spectrum of active fluctuations. We find that the active cytoskeleton contributes to the amplitude of the membrane height fluctuations at intermediate wavelengths, as observed experimentally.

Mechanical properties of the cell nucleus and the effect of emerin deficiency

Nuclear structure and mechanics are gaining recognition as important factors that affect gene expression, development, and differentiation in normal function and disease, yet the physical mechanisms that govern nuclear mechanical stability remain unclear. Here we examined the physical properties of the cell nucleus by imaging fluorescently labeled components of the inner nucleus (chromatin and nucleoli) and the nuclear envelope (lamins and membranes) in nuclei deformed by micropipette aspiration (confocal imaged microdeformation). We investigated nuclei, both isolated and in intact, living cells, and found that nuclear volume significantly decreased by 60-70% during aspiration. While nuclear membranes exhibited blebbing and fluid characteristics during aspiration, the nuclear lamina exhibited behavior of a solid-elastic shell. Under large deformations of GFP-lamin A-labeled nuclei, we observed a decay of fluorescence intensity into the tip of the deformed tongue that we interpreted in terms of nonlinear, two-dimensional elasticity theory. Here we applied this method to study nuclear envelope stability in disease and found that mouse embryo fibroblasts lacking the inner nuclear membrane protein, emerin, had a significantly decreased ratio of the area expansion to shear moduli (K/mu) compared to wild-type cells (2.1 +/- 0.2 versus 5.1 +/- 1.3). These data suggest that altered nuclear envelope elasticity caused by loss of emerin could contribute to increased nuclear fragility in Emery-Dreifuss muscular dystrophy patients with mutations in the emerin gene. Based on our experimental results and theoretical considerations, we present a model describing how the nucleus is stabilized in the pipette. Such a model is essential for interpreting the results of any micropipette study of the nucleus and porous materials in general.

3-D nanomechanics of an erythrocyte junctional complex in equibiaxial and anisotropic deformations

The erythrocyte membrane skeleton deforms constantly in circulation, but the mechanics of a junctional complex (JC) in the network is poorly understood. We previously proposed a 3-D mechanical model for a JC (Sung, L. A., and C. Vera. Protofilament and hexagon: A three-dimensional mechanical model for the junctional complex in the erythrocyte membrane skeleton. Ann Biomed Eng 31:1314-1326, 2003) and now developed a mathematical model to compute its equilibrium by dynamic relaxation. We simulated deformations of a single unit in the network to predict the tension of 6 alphabeta spectrin (Sp) (top, middle, and bottom pairs), and the attitude of the actin protofilament [pitch (theta), yaw (phi) and roll (psi) angles]. In equibiaxial deformation, 6 Sp would not begin their first round of "single domain unfolding in cluster" until the extension ratio (lambda) reach approximately 3.6, beyond the maximal sustainable lambda of approximately 2.67. Before Sp unfolds, the protofilament would gradually raise its pointed end away from the membrane, while phi and psi remain almost unchanged. In anisotropic deformation, protofilaments would remain tangent but swing and roll drastically at least once between lambda(i) = 1.0 and approximately 2.8, in a deformation angle- and lambda(i)-dependent fashion. This newly predicted nanomechanics in response to deformations may reveal functional roles previous unseen for a JC, and molecules associated with it, during erythrocyte circulation.

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.

Finite element modelling and simulations in cardiovascular mechanics and cardiology: A bibliography 1993–2004

The paper gives a bibliographical review of the finite element modelling and simulations in cardiovascular mechanics and cardiology from the theoretical as well as practical points of views. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1993 and 2004. At the end of this paper, more than 890 references are given dealing with subjects as: Cardiovascular soft tissue modelling; material properties; mechanisms of cardiovascular components; blood flow; artificial components; cardiac diseases examination; surgery; and other topics.

Characterization of the Nuclear Deformation Caused by Changes in Endothelial Cell Shape

We investigated the mechanotransduction pathway in endothelial cells between their nucleus and adhesions to the extracellular matrix. First, we measured nuclear deformations in response to alterations of cell shape as cells detach from a flat surface. We found that the nuclear deformation appeared to be in direct and immediate response to alterations of the cell adhesion area. The nucleus was then treated as a neo-Hookean compressible material, and we estimated the stress associated with the cytoskeleton and acting on the nucleus during cell rounding. With the obtained stress field, we estimated the magnitude of the forces deforming the nucleus. Considering the initial and final components of this adhesion-cytoskeleton-nucleus force transmission pathway, we found our estimate for the internal forces acting on the nucleus to be on the same order of magnitude as previously measured traction forces, suggesting a direct mechanical link between adhesions and the nucleus.

A prestressed cable network model of the adherent cell cytoskeleton

A prestressed cable network is used to model the deformability of the adherent cell actin cytoskeleton. The overall and microstructural model geometries and cable mechanical properties were assigned values based on observations from living cells and mechanical measurements on isolated actin filaments, respectively. The models were deformed to mimic cell poking (CP), magnetic twisting cytometry (MTC) and magnetic bead microrheometry (MBM) measurements on living adherent cells. The models qualitatively and quantitatively captured the fibroblast cell response to the deformation imposed by CP while exhibiting only some qualitative features of the cell response to MTC and MBM. The model for CP revealed that the tensed peripheral actin filaments provide the key resistance to indentation. The actin filament tension that provides mechanical integrity to the network was estimated at approximately 158 pN, and the nonlinear mechanical response during CP originates from filament kinematics. The MTC and MBM simulations revealed that the model is incomplete, however, these simulations show cable tension as a key determinant of the model response.

Nanostructure, effective properties, and deformation pattern of the cochlear outer hair cell cytoskeleton

We consider the mechanical properties of the outer hair cell cytoskeleton. The cytoskeleton is represented as a set of microdomains of different sizes and orientations composed of actin filaments and spectrin crosslinks. An intermediate material between domains is also introduced. The domain characteristics are randomly generated and the histograms of the cytoskeleton stiffness moduli are obtained. We solve an inverse problem and estimate the stiffness of the crosslink and connective molecule in the intermediate material. We discovered a pattern of highly inhomogeneous deformation of the cytoskeleton where the circumferential strain is primarily determined by the deformation of the intermediate material.

Theoretical predictions of electromechanical deformation of cells subjected to high voltages for membrane electroporation

An electromechanical analysis based on thin-shell theory is presented to analyze cell shape changes in response to external electric fields. This approach can be extended to include osmotic-pressure changes. Our calculations demonstrate that at large fields, the spherical cell geometry can be significantly modified, and even ellipsoidal forms would be inappropriate to account for the deformation. Values of the surface forces obtained from our calculations are in very good agreement with the 1--10 mN/m range for membrane rupture reported in the literature. The results, in keeping with reports in the literature, demonstrate that the final shape depends on membrane thickness. This has direct implications for tissues in which significant molecular restructuring can occur. It is also shown that, at least for the smaller electric fields, both the cellular surface area and volume change roughly in a quadratic manner with the electric field. Finally, it is shown that the bending moments are generally quite small and can be neglected for a simpler analysis.

Deformation-enhanced fluctuations in the red cell skeleton with theoretical relations to elasticity, connectivity, and spectrin unfolding

To assess local elasticity in the red cell's spectrin-actin network, nano-particles were tethered to actin nodes and their constrained thermal motions were tracked. Cells were both immobilized and controllably deformed by aspiration into a micropipette. Since the network is well-appreciated as soft, thermal fluctuations even in an unstressed portion of network were expected to be many tens of nanometers based on simple equipartition ideas. Real-time particle tracking indeed reveals such root-mean-squared motions for 40-nm fluorescent beads either tethered to actin directly within a cell ghost or connected to actin from outside a cell via glycophorin. Moreover, the elastically constrained displacements are significant on the scale of the network's internodal distance of approximately 60-80 nm. Surprisingly, along the aspirated projection-where the network is axially extended by as much as twofold or more-fluctuations in the axial direction are increased by almost twofold relative to motions in the unstressed network. The molecular basis for such strain softening is discussed broadly in terms of force-driven transitions. Specific considerations are given to 1) protein dissociations that reduce network connectivity, and 2) unfolding kinetics of a localized few of the red cell's approximately 10(7) spectrin repeats.

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.

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.

The deformation of spherical vesicles with permeable, constant-area membranes: application to the red blood cell

The deformation of an initially spherical vesicle of radius a with a permeable membrane under extensive forces applied at its poles is calculated as a function of the in-plane shear modulus, H, and the out-of-plane bending modulus, B, using an axisymmetric theory that is valid for large deformations. Suitably nondimensionalized, the results depend upon a single nondimensional parameter, C identical with a(2)H/B. For small deformations, the calculated force-polar strain curves are linear and, under these conditions, the slope of the curve determines only C, not the values of H and B separately. Independent determination of H and B from experimental measurements require deformations that are large enough to produce nonlinear behavior. Simple approximations for large and small C are given, which are applied to experimental measurements on red blood cell ghosts that have been made permeable by treatment with saponin.

Calculation of a Gap restoration in the membrane skeleton of the red blood cell: possible role for myosin II in local repair

Human red blood cells contain all of the elements involved in the formation of nonmuscle actomyosin II complexes (V. M. Fowler. 1986. J. Cell. Biochem. 31:1-9; 1996. Curr. Opin. Cell Biol. 8:86-96). No clear function has yet been attributed to these complexes. Using a mathematical model for the structure of the red blood cell spectrin skeleton (M. J. Saxton. 1992. J. Theor. Biol. 155:517-536), we have explored a possible role for myosin II bipolar minifilaments in the restoration of the membrane skeleton, which may be locally damaged by major mechanical or chemical stress. We propose that the establishment of stable links between distant antiparallel actin protofilaments after a local myosin II activation may initiate the repair of the disrupted area. We show that it is possible to define conditions in which the calculated number of myosin II minifilaments bound to actin protofilaments is consistent with the estimated number of myosin II minifilaments present in the red blood cells. A clear restoration effect can be observed when more than 50% of the spectrin polymers of a defined area are disrupted. It corresponds to a significant increase in the spectrin density in the protein free region of the membrane. This may be involved in a more complex repair process of the red blood cell membrane, which includes the vesiculation of the bilayer and the compaction of the disassembled spectrin network.

Simulations of the erythrocyte cytoskeleton at large deformation. I. Microscopic models

Three variations of a polymer chain model for the human erythrocyte cytoskeleton are used in large deformation simulations of microscopic membrane patches. Each model satisfies an experimental observation that the contour length of the spectrin tetramers making up the erythrocyte cytoskeleton is roughly square root of 7 times the end-to-end distance of the tetramer in vivo. Up to modest stress, each brushy cytoskeletal network behaves, consistently, like a low-temperature, planar network of Hookean springs, with a model-dependent effective spring constant, keff, in the range of 20-40 kBT/s(o)2, where T is the temperature and s(o) is the force-free spring length. However, several features observed at large deformation distinguish these models from spring networks: 1) Network dimensions do not expand without bound in approaching a critical isotropic tension (square root of 3 keff) that is a characteristic limit of Hookean spring nets. 2) In surface compression, steric interactions among the chain elements prevent a network collapse that is otherwise observed in compression of planar triangulated networks of springs. 3) Under uniaxial surface tension, isotropy of the network disappears only as the network is stretched by more than 50% of its equilibrium dimensions. Also found are definitively non-Hookean regimes in the stress dependence of the elastic moduli. Lastly, determinations of elastic moduli from both fluctuations and stress/strain relations prove to be consistent, implying that consistency should be expected among experimental determinations of these quantities.