Computer simulation of a model network for the erythrocyte cytoskeleton

A cell-and-plasma numerical model reveals hemodynamic stress and flow adaptation in zebrafish microvessels after morphological alteration

The development of a functional cardiovascular system ensures a sustainable oxygen, nutrient and hormone delivery system for successful embryonic development and homeostasis in adulthood. While early vessels are formed by biochemical signaling and genetic programming, the onset of blood flow provides mechanical cues that participate in vascular remodeling of the embryonic vascular system. The zebrafish is a prolific animal model for studying the quantitative relationship between blood flow and vascular morphogenesis due to a combination of favorable factors including blood flow visualization in optically transparent larvae. In this study, we have developed a cell-and-plasma blood transport model using computational fluid dynamics (CFD) to understand how red blood cell (RBC) partitioning affect lumen wall shear stress (WSS) and blood pressure in zebrafish trunk blood vascular networks with altered rheology and morphology. By performing live imaging of embryos with reduced hematocrit, we discovered that cardiac output and caudal artery flow rates were maintained. These adaptation trends were recapitulated in our CFD models, which showed reduction in network WSS via viscosity reduction in the caudal artery/vein and via pressure gradient weakening in the intersegmental vessels (ISVs). Embryos with experimentally reduced lumen diameter showed reduced cardiac output and caudal artery flow rate. Factoring in this trend into our CFD models, simulations highlighted that lumen diameter reduction increased vessel WSS but this increase was mitigated by flow reduction due to the adaptive network pressure gradient weakening. Additionally, hypothetical network CFD models with different vessel lumen diameter distribution characteristics indicated the significance of axial variation in lumen diameter and cross-sectional shape for establishing physiological WSS gradients along ISVs. In summary, our work demonstrates how both experiment-driven and hypothetical CFD modeling can be employed for the study of blood flow physiology during vascular remodeling.

Statistical mechanics of phase transitions in elastic media with vanishing thermal expansion

We consider a minimal spin model for Ising transitions in an isotropic elastic medium in the zero thermal expansion (ZTE) limit. We set up the elastic theory for this system. We use this theory to identify and study the nature of the fluctuations in the system near the second order phase transitions at T_{c} in the ZTE limit given by dT_{c}/dV=0, where V is the system volume, and explore anomalous elasticity. Allowing for the local strain to couple asymmetrically or selectively with the states of the order parameter, we uncover the dramatic effects of these couplings on the fluctuations of the local displacements near T_{c}, and also on the nature of the phase transition itself. Near second-order phase transitions and with weak asymmetry in the order parameter-strain couplings, the variance of the displacement fluctuations in two dimensions scale with the system size L in a universal fashion as [ln(L/a_{0})]^{2/3}; a_{0} is a small-scale cutoff. Likewise, the correlation functions of the difference of the local displacements at two different points separated by r scale as [ln(r/a_{0})]^{2/3} for large r. For stronger selectivity above a finite threshold, this variance diverge as L exceeds beyond a (nonuniversal) size, determined by the model parameters, signaling a transition to a phase with only short-range order or the loss of the positional order of the elastic medium. At dimensions higher than two, for sufficiently weak selectivity, the variance of the displacement fluctuations is L-independent corresponding to long-range order. However, if the selectivity parameters rise beyond a dimension-dependent threshold value, then again the positional order is lost with a concomitant transition to a phase with short-range order. Large values of the order parameter-strain couplings can turn the phase transition into a first order as well. Our theory establishes a one-to-one correspondence between the order of phase transitions and anomalous elasticity near the transitions. Our theory should be a useful guide to possible synthesis of appropriate ZTE materials.

A particle-based computational model to analyse remodelling of the red blood cell cytoskeleton during malaria infections

Red blood cells can withstand the harsh mechanical conditions in the vasculature only because the bending rigidity of their plasma membrane is complemented by the shear elasticity of the underlying spectrin-actin network. During an infection by the malaria parasite Plasmodium falciparum, the parasite mines host actin from the junctional complexes and establishes a system of adhesive knobs, whose main structural component is the knob-associated histidine rich protein (KAHRP) secreted by the parasite. Here we aim at a mechanistic understanding of this dramatic transformation process. We have developed a particle-based computational model for the cytoskeleton of red blood cells and simulated it with Brownian dynamics to predict the mechanical changes resulting from actin mining and KAHRP-clustering. Our simulations include the three-dimensional conformations of the semi-flexible spectrin chains, the capping of the actin protofilaments and several established binding sites for KAHRP. For the healthy red blood cell, we find that incorporation of actin protofilaments leads to two regimes in the shear response. Actin mining decreases the shear modulus, but knob formation increases it. We show that dynamical changes in KAHRP binding affinities can explain the experimentally observed relocalization of KAHRP from ankyrin to actin complexes and demonstrate good qualitative agreement with experiments by measuring pair cross-correlations both in the computer simulations and in super-resolution imaging experiments.

Malaria is one of the deadliest infectious diseases and its symptoms are related to the blood stage, when the parasite multiplies within red blood cells. In order to avoid clearance by the spleen, the parasite produces specific factors like the adhesion receptor PfEMP1 and the multifunctional protein KAHRP that lead to the formation of adhesive knobs on the surface of the red blood cells and thus increase residence time in the vasculature. We have developed a computational model for the parasite-induced remodelling of the actin-spectrin network to quantitatively predict the dynamical changes in the mechanical properties of the infected red blood cells and the spatial distribution of the different protein components of the membrane skeleton. Our simulations show that KAHRP can relocate to actin junctions due to dynamical changes in binding affinities, in good qualitative agreement with super-resolution imaging experiments. In the future, our simulation framework can be used to gain further mechanistic insight into the way malaria parasites attack the red blood cell cytoskeleton.

Structural modelling of the cardiovascular system

Computational modelling of the cardiovascular system offers much promise, but represents a truly interdisciplinary challenge, requiring knowledge of physiology, mechanics of materials, fluid dynamics and biochemistry. This paper aims to provide a summary of the recent advances in cardiovascular structural modelling, including the numerical methods, main constitutive models and modelling procedures developed to represent cardiovascular structures and pathologies across a broad range of length and timescales; serving as an accessible point of reference to newcomers to the field. The class of so-called hyperelastic materials provides the theoretical foundation for the modelling of how these materials deform under load, and so an overview of these models is provided; comparing classical to application-specific phenomenological models. The physiology is split into components and pathologies of the cardiovascular system and linked back to constitutive modelling developments, identifying current state of the art in modelling procedures from both clinical and engineering sources. Models which have originally been derived for one application and scale are shown to be used for an increasing range and for similar applications. The trend for such approaches is discussed in the context of increasing availability of high performance computing resources, where in some cases computer hardware can impact the choice of modelling approach used.

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.

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.

Computation of the effective mechanical response of biological networks accounting for large configuration changes

The asymptotic homogenization technique is involved to derive the effective elastic response of biological membranes viewed as repetitive beam networks. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes in the nonlinear regime, reflecting the influence of the geometrical and mechanical micro-parameters of the network structure on the overall response of the equivalent continuum. Biomembranes networks are classified based on nodal connectivity, so that we analyze in this work 3, 4 and 6-connectivity networks, which are representative of most biological networks. The individual filaments of the network are described as undulated beams prone to entropic elasticity, with tensile moduli determined from their persistence length. The effective micropolar continuum evaluated as a continuum substitute of the biological network has a kinematics reflecting the discrete network deformation modes, involving a nodal displacement and a microrotation. The statics involves the classical Cauchy stress and internal moments encapsulated into couple stresses, which develop internal work in duality to microcurvatures reflecting local network undulations. The relative ratio of the characteristic bending length of the effective 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 biomembranes. The peptidoglycan network may exhibit a re-entrant hexagonal configuration due to thermal or pressure fluctuations, for which micropolar effects become important. The homogenized responses are in good agreement with FE simulations performed over the whole network. The predictive nature of the employed homogenization technique allows the identification of a strain energy density of a hyperelastic model, for the purpose of performing structural calculations of the shape evolutions of biomembranes.

Erythrocyte membrane model with explicit description of the lipid bilayer and the spectrin network

The membrane of the red blood cell (RBC) consists of spectrin tetramers connected at actin junctional complexes, forming a two-dimensional (2D) sixfold triangular network anchored to the lipid bilayer. Better understanding of the erythrocyte mechanics in hereditary blood disorders such as spherocytosis, elliptocytosis, and especially, sickle cell disease requires the development of a detailed membrane model. In this study, we introduce a mesoscale implicit-solvent coarse-grained molecular dynamics (CGMD) model of the erythrocyte membrane that explicitly describes the phospholipid bilayer and the cytoskeleton, by extending a previously developed two-component RBC membrane model. We show that the proposed model represents RBC membrane with the appropriate bending stiffness and shear modulus. The timescale and self-consistency of the model are established by comparing our results with experimentally measured viscosity and thermal fluctuations of the RBC membrane. Furthermore, we measure the pressure exerted by the cytoskeleton on the lipid bilayer. We find that defects at the anchoring points of the cytoskeleton to the lipid bilayer (as in spherocytes) cause a reduction in the pressure compared with an intact membrane, whereas defects in the dimer-dimer association of a spectrin filament (as in elliptocytes) cause an even larger decrease in the pressure. We conjecture that this finding may explain why the experimentally measured diffusion coefficients of band-3 proteins are higher in elliptocytes than in spherocytes, and higher than in normal RBCs. Finally, we study the effects that possible attractive forces between the spectrin filaments and the lipid bilayer have on the pressure applied on the lipid bilayer by the filaments. We discover that the attractive forces cause an increase in the pressure as they diminish the effect of membrane protein defects. As this finding contradicts with experimental results, we conclude that the attractive forces are moderate and do not impose a complete attachment of the filaments to the lipid bilayer.

Multiple stiffening effects of nanoscale knobs on human red blood cells infected with Plasmodium falciparum malaria parasite

Our coarse-grained molecular dynamics (CGMD) simulations show that the deposition of nanoscale knobs, rather than spectrin network remodeling, is the primary cause of the dramatically increased stiffness of the Plasmodium falciparum (Pf)-infected red blood cell (RBC) membranes. Our analyses further reveal that the knobs stiffen the RBC membrane in a unique manner by simultaneously harnessing composite strengthening, strain hardening, and knob density-dependent vertical coupling effects. In addition to providing a fundamental understanding of the stiffening mechanism of Pf-infected RBCs, our simulation results suggest potential targets for antimalarial therapies.

During its asexual development within the red blood cell (RBC), Plasmodium falciparum (Pf), the most virulent human malaria parasite, exports proteins that modify the host RBC membrane. The attendant increase in cell stiffness and cytoadherence leads to sequestration of infected RBCs in microvasculature, which enables the parasite to evade the spleen, and leads to organ dysfunction in severe cases of malaria. Despite progress in understanding malaria pathogenesis, the molecular mechanisms responsible for the dramatic loss of deformability of Pf-infected RBCs have remained elusive. By recourse to a coarse-grained (CG) model that captures the molecular structures of Pf-infected RBC membrane, here we show that nanoscale surface protrusions, known as “knobs,” introduce multiple stiffening mechanisms through composite strengthening, strain hardening, and knob density-dependent vertical coupling. On one hand, the knobs act as structural strengtheners for the spectrin network; on the other, the presence of knobs results in strain inhomogeneity in the spectrin network with elevated shear strain in the knob-free regions, which, given its strain-hardening property, effectively stiffens the network. From the trophozoite to the schizont stage that ensues within 24–48 h of parasite invasion into the RBC, the rise in the knob density results in the increased number of vertical constraints between the spectrin network and the lipid bilayer, which further stiffens the membrane. The shear moduli of Pf-infected RBCs predicted by the CG model at different stages of parasite maturation are in agreement with experimental results. In addition to providing a fundamental understanding of the stiffening mechanisms of Pf-infected RBCs, our simulation results suggest potential targets for antimalarial therapies.

Atomic force microscopy demonstration of cytoskeleton instability in mouse erythrocytes with dematin-headpiece and β-adducin deficiency

The pattern of disassembly of the cytoskeletal network of murine erythrocytes with deficiency of either dematin-headpiece or β-adducin or both proteins were investigated using atomic force microscopy. A heterogeneous complex structure with fine filament features and coarse features was observed in the cytoskeleton of wild type mouse erythrocytes, whereas a significant amount of rearrangement and aggregation occurred in the mutants, particularly in the cells carrying double gene mutations. These results are consistent with the cellular and biochemical phenotype of the mutant cell membranes as being more fragile due to weakened vertical connections with the plasma membrane.

Cytoskeleton mediated effective elastic properties of model red blood cell membranes

The plasma membrane of human red blood cells consists of a lipid bilayer attached to a regular network of underlying cytoskeletal polymers. We model this system at a dynamic coarse-grained level, treating the bilayer as an elastic sheet and the cytoskeletal network as a series of phantom entropic springs. In contrast to prior simulation efforts, we explicitly account for dynamics of the cytoskeletal network, both via motion of the protein anchors that attach the cytoskeleton to the bilayer and through breaking and reconnection of individual cytoskeletal filaments. Simulation results are explained in the context of a simple mean field percolation model and comparison is made to experimental measurements of red blood cell fluctuation amplitudes.

Dynamics of pinned membranes with application to protein diffusion on the surface of red blood cells

We present a theoretical treatment and simulation algorithm for the dynamics of Helfrich elastic membrane surfaces in the presence of general harmonic perturbations and hydrodynamic coupling to the surrounding solvent. In the limit of localized and strong interactions, this harmonic model can be used to pin the membrane to intracellular/intercellular structures. We consider the case of pinning to the cytoskeleton and use such a model to estimate the macroscopic diffusion constant for band 3 protein on the surface of human erythrocytes. Comparison to experimental results suggests that thermal undulations of the membrane surface should play a significant role in protein mobility on the red blood cell.

One-dimensional transport with dynamic disorder

We study the mean quenching time distribution and its moments in a one-dimensional N-site donor-bridge-acceptor system where all sites are coupled to a two-state jump bath for arbitrary disorder and an arbitrary ratio kappa identical with <k>/R of the bath jump rate R and the average hopping rate <k>. When kappaN approximately 1, the quenching time distribution has long power-law tails even when the waiting times are exponentially distributed. These disappear for kappaN<<1 where the hopping rate self-averages on the bath relaxation time scale. In the absence of disorder or for small kappa, the mean quenching time scales linearly with N. Otherwise, we observe a power law, approximately N1+gamma, with a crossover to linear scaling (gamma=0) for large N. Distributions of particle position, its second moment, velocity and diffusion coefficient are computed in the infinite N limit. For times longer than R-1, the dynamic disorder self-averages and the average position, velocity, and diffusion coefficient scale linearly in time.

Regulation of protein mobility via thermal membrane undulations

The in-plane diffusivelike motion of membrane bound proteins on the surface of cells is considered. We suggest, on the basis of theoretical arguments and simulation, that thermally excited undulations of the lipid bilayer may serve as a mechanism for proteins to hop between adjacent regions on the cell surface separated by barriers composed of internal cellular structure (e.g., the cytoskeleton). We specifically investigate the mobility of band 3 dimer on the surface of red blood cells where the spectrin cytoskeletal meshwork defines a series of "corrals" on the cell surface known to hinder protein motion. Previous models of this system have postulated that the cytoskeleton must deform to allow passage of membrane bound proteins out of these corral regions and have ignored fluctuations of the bilayer. Our model provides a complementary mechanism and we posit that the mobility of real proteins in real cells is likely the result of several mechanisms acting in parallel.

Echinocyte shapes: bending, stretching, and shear determine spicule shape and spacing

We study the shapes of human red blood cells using continuum mechanics. In particular, we model the crenated, echinocytic shapes and show how they may arise from a competition between the bending energy of the plasma membrane and the stretching/shear elastic energies of the membrane skeleton. In contrast to earlier work, we calculate spicule shapes exactly by solving the equations of continuum mechanics subject to appropriate boundary conditions. A simple scaling analysis of this competition reveals an elastic length Lambda(el), which sets the length scale for the spicules and is, thus, related to the number of spicules experimentally observed on the fully developed echinocyte.

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.

Coupled multi-component systems: A simple membrane model

We present initial results regarding the existence, stability and interactionof linear and nonlinear vibrational modes in a system of two coupled, onedimensional lattices with unequal numbers of masses. The effects on thesenonlinear modes of coupling a near continuum system to a discrete systemusing a nonlinear coupling are examined. This numerical model is a firststep towards investigating the dynamical behavior of a flexible sheetcoupled nonlinearly to a semi-rigid support, a system which couldconceivably represent a biological cell membrane with a supporting proteinnetwork. General implications for the dynamical behavior of continuumsystems coupled nonlinearly to discrete systems are introduced.

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.

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.

Lateral diffusion of membrane proteins in the presence of static and dynamic corrals: suggestions for appropriate observables

We consider the possibility of inferring the nature of cytoskeletal interaction with transmembrane proteins via optical experiments such as single-particle tracking (SPT) and near-field scanning optical microscopy (NSOM). In particular, we demonstrate that it may be possible to differentiate between static and dynamic barriers to diffusion by examining the time-dependent variance and higher moments of protein population inside cytoskeletal "corrals." Simulations modeling Band 3 diffusion on the surface of erythrocytes provide a concrete demonstration that these statistical tools might prove useful in the study of biological systems.

New insights into erythrocyte membrane organization and microelasticity

The erythrocyte membrane's ability to withstand the stresses of circulation has its origins in various levels of structural organization. Central to this membrane's structure-function relationships is a quasi-two-dimensional meshwork of spectrin-actin-protein 4.1 that imparts a resilence to the overlying plasma membrane. New insights into the nonlinear microelasticity of this substructure are being provided by experiments that range from elegant atomic force microscopy tests of single spectrin chains to patterned photobleaching of the micropipette-deformed network. Breakthroughs in atomic level structure determinations are further complemented by emerging biophysical studies of transgenically engineered mice lacking specific erythrocyte membrane proteins. Recent theoretical efforts (computational approaches most notably) also have begun to correlate molecular scale aspects of structure with mechanical measures. All of this recent activity in the biophysics of erythrocyte structure-function is certain to challenge and refine some of the most basic tenets in cell membrane structure-function.

Regulation of protein mobility in cell membranes: a dynamic corral model

We analyze a two-state stochastic corral model for regulation of protein diffusion in a cell membrane. This model could mimic control of protein transport in the membrane by the cytoskeleton. The dynamic corral acts as a gate which when open permits an otherwise trapped protein to escape to a neighboring corral in the cytoskeletal network. We solve for the escape rate over a wide range of parameters of the model, and compare these results with Monte Carlo simulations. Upon introducing measured values of the model parameters for Band 3 in erythrocyte membranes, we are able to estimate the value for one unknown parameter, the average rate at which the corral closes. The ratio of calculated closing rate to measured opening rate is roughly 100:1, consistent with a gating mechanism whereby protein mobility is regulated by dissociation and reassociation of segments of the cytoskeletal network.

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.

Dynamics of oscillating erythrocyte doublets after electrofusion

Erythrocytes were electrofused with multiple rectangular voltage pulses to show an oscillatory movement, divided into swell phases and pump events. During each swell phase, which lasted from 0.5 s to more than 180 s, the fused cells' (doublets') volume increased by colloid osmotic swelling, and the membrane area was expanded until rupture. Membrane rupture initiated the pump event, where the doublets' volume and membrane area decreased with an almost exponential time course and time constants between 2 ms and 8 ms. Simultaneously, a portion of cytosolic hemoglobin solution was ejected into extracellular space ("jet"). Pump event time constants and swell phase durations decreased with rising chamber temperature, indicating that both parts of the oscillatory movements were determined by physical properties of membrane and liquids. Relative volume change developments express a gradual loss of membrane elasticity during the oscillation, decreasing the elastic forces stored in the membrane. Evidence is given that the first rupture causes a weakening of the membrane at the rupture site. Heat treatment up to 45 degrees C had a negligible effect on swell times, pump time constants, and relative volume changes. A heat treatment of 50 degrees C prevented oscillatory movements. The rupture location accorded with theories of potential induced membrane electropermeabilization.

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.

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. 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.

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.

Structure of the erythrocyte membrane skeleton as observed by atomic force microscopy

The structure of the membrane skeleton on the cytoplasmic surface of the erythrocyte plasma membrane was observed in dried human erythrocyte ghosts by atomic force microscopy (AFM), taking advantage of its high sensitivity to small height variations in surfaces. The majority of the membrane skeleton can be imaged, even on the extracellular surface of the membrane. Various fixation and drying methods were examined for preparation of ghost membrane samples for AFM observation, and it was found that freeze-drying (freezing by rapid immersion in a cryogen) of unfixed specimens was a fast and simple way to obtain consistently good results for observation without removing the membrane or extending the membrane skeleton. Observation of the membrane skeleton at the external surface of the cell was possible mainly because the bilayer portion of the membrane sank into the cell during the drying process. The average mesh size of the spectrin network observed at the extracellular and cytoplasmic surfaces of the plasma membrane was 4800 and 3000 nm2, respectively, which indicates that spectrin forms a three-dimensionally folded meshwork, and that 80% of spectrin can be observed at the extracellular surface of the plasma membrane.

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

A finite-element network model is used to investigate the influence of the topology of the red blood cell membrane skeleton on its macroscopic mechanical properties. Network topology is characterized by the number of spectrin oligomers per actin junction (phi a) and the number of spectrin dimers per self-association junction (phi s). If it is assumed that all associated spectrin is in tetrameric form, with six tetramers per actin junction (i.e., phi a = 6.0 and phi s = 2.0), then the topology of the skeleton may be modeled by a random Delaunay triangular network. Recent images of the RBC membrane skeleton suggest that the values for these topological parameters are in the range of 4.2 < phi a < 5.5 and 2.1 < phi s < 2.3. Model networks that simulate these realistic topologies exhibit values of the shear modulus that vary by more than an order of magnitude relative to triangular networks. This indicates that networks with relatively sparse nontriangular topologies may be needed to model the RBC membrane skeleton accurately. The model is also used to simulate skeletal alterations associated with hereditary spherocytosis and Southeast Asian ovalocytosis.

Single-particle tracking: applications to membrane dynamics

Measurements of trajectories of individual proteins or lipids in the plasma membrane of cells show a variety of types of motion. Brownian motion is observed, but many of the particles undergo non-Brownian motion, including directed motion, confined motion, and anomalous diffusion. The variety of motion leads to significant effects on the kinetics of reactions among membrane-bound species and requires a revision of existing views of membrane structure and dynamics.

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.

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.

Barrier-free paths of directed protein motion in the erythrocyte plasma membrane

A model is presented for the steric interaction between a plasma membrane protein and the membrane cytoskeleton in the human erythrocyte. The cytoskeleton is treated as a network of polymer chains attached to a flat bilayer, and the membrane protein is a hemisphere of effective radius R(e) with center on the bilayer edge. The simulation is used to investigate the barrier-free path L for linear guided motion of a protein in the bilayer plane. It is shown that the barrier-free paths of small proteins can be used to extract the effective in-plane diameter of cytoskeletal components. For example, the in-plane diameter of an ankyrin attachment site is found to be approximately 12 nm in the simulation, or twice the computational spectrin diameter. The barrier-free paths of large proteins (R(e) > 23 nm) vanish when the proteins are corralled by the cytoskeleton. For intermediate size proteins, L decreases approximately as L is directly proportional to S-1.4 where S is proportional to the sum of the protein and cytoskeleton chain radii.

Measurement of erythrocyte membrane elasticity by flicker eigenmode decomposition

We have studied the flickering of erythrocytes at wavelengths comparable to the cell dimension. To do this we have analyzed the edge fluctuations of the cell to a resolution of 5 nm by combining phase contrast microscopy with fast image processing. By measuring the edge excitations simultaneously at four orthogonal positions around the cell, the eigenmodes of equal azimuthal mode numbers m = 0,1,2 could be separated. From a continuous time sequence of 100 s of video frames taken at 40 ms time intervals, we determined the time-auto correlation function for the modes m = 0,1,2 and calculated their mean square amplitudes <delta n2m> as well as their decay times tau m. To explain the results we also present the theoretically calculated energy eigenmodes of an erythrocyte, accounting for the constraint that the cell is in contact with the substrate along an annular ring, which agreed well with the experimental findings. We found that the softest mode is a "hindered translational" mode with m = 1 of the adhered cell, which is almost insensitive to the shear elastic modulus. Comparison of the calculated and measured amplitudes yielded an average value for the bending stiffness of kc = 4 x 10(-19) J, which is much larger than the value obtained by flicker analysis at short wavelengths (kc = 2.3 x 10(-20) J). It would, however, agree well with the value expected from the red cell membrane area compressibility modulus of K = 4.5 x 10(-1)N/m, which corresponds to a lipid bilayer containing approximately 50 mol % of cholesterol. In contradiction to our theoretical expectations we found that the flicker eigenmodes seemed not to be influenced by the membrane shear elasticity, which will be discussed in terms of an unusual coupling between the lipid bilayer and the cytoskeleton.

Deformation and flow of red blood cells in a synthetic lattice: evidence for an active cytoskeleton

We introduce the use of microfabrication techniques to construct on a silicon wafer a synthetic capillary bed with 2.5- to 4-micron (mu)-wide channels. Establishment of a fluid pressure gradient allowed us to observe simultaneously using optical microscopy hundreds of cells flowing through the bed at physiological speeds. We find a large distribution of mobilities among red cells flowing through the structure; smaller channels provide a greater impedance to flow than larger ones, indicating that kinetic drag variations provide the origin of the distribution. The mobility of a particular cell is not correlated with the cell diameter but appears to be inversely correlated with intracellular calcium concentration of the cell, as determined by fluorescence of the calcium-binding dye fluo-3 AM. Also, we are able to use the parallel processing nature of our arrays to observe isolated events where the rigidity of the red cell seems to change suddenly over several orders of magnitude as it blocks a channel in the array.

Molecular maps of red cell deformation: hidden elasticity and in situ connectivity

Fluorescence-imaged micropipette aspiration was used to map redistribution of the proteins and lipids in highly extended human red blood cell membranes. Whereas the fluid bilayer distributed uniformly (+/- 10 percent), the underlying, solidlike cytoskeleton of spectrin, actin, and protein 4.1 exhibited a steep gradient in density along the aspirated projection, which was reversible on release from deformation. Quantitation of the cytoskeletal protein density gradients showed that skeletal elasticity is well represented by a grafted polymer network with a ratio of surface dilation modulus to shear modulus of approximately 2:1. Fractionally mobile integral proteins, such as band 3, and highly mobile receptors, such as CD59 as well as glycophorin C in protein 4.1-deficient cells, appeared to be squeezed out of areas dense in the underlying network and enriched in areas of network dilation. This complementary segregation demonstrates patterning of cell surface components by cytoskeletal dilation.