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

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.

Dynamics of erythrocytes in oscillatory shear flows: effects of S/V ratio

By combining a multiscale structural model of erythrocyte with a fluid-cell interaction model based on the boundary-integral method, we numerically investigate the dynamic response of erythrocytes in oscillatory shear flows (OSFs). The goal is to develop a novel experimental method to test the structural robustness of erythrocytes in transient mechanical loads with small time scales, conditions closely imitating the mechanical environment in vivo. Following the discovery of multiple response modes (wheeling, mode 1 tank treading, and mode 2 tank treading) under these conditions (Zhu & Asaro, 2019), we concentrate on deformation and stress inside RBCs driven by OSF, especially shear deformation of the membrane and the skeleton-bilayer dissociation stress, parameters that are related to mechanically induced structural remodeling such as vesiculation. Effects related to changes in surface area-to-volume (S/V) ratio are considered. Our results show that with the variation of the S/V ratio there could be significant change in terms of the occurrence of response modes even if other parameters are kept unchanged. For example, by reducing the S/V ratio of the cell, an asymmetric mode featuring a mixture of the wheeling and mode 2 tank treading responses is discovered. This mode is found to be associated with large skeleton-bilayer dissociation stress so that its potential impact on OSF-driven vesiculation should not be overlooked. By systematically examining the dependencies of skeleton deformation and skeleton-bilayer dissociation stress upon S/V, this study is critical for the development of the OSF technique in applications such as diagnosis since cell conditions are often reflected in its geometric properties.

Erythrocyte Aging, Protection via Vesiculation: An Analysis Methodology via Oscillatory Flow

We demonstrate that erythrocyte deformations, specifically of a type as occur in splenic flow (Zhu et al., 2017), and of the type that promote vesiculation can be caused by simple, yet tailored, oscillatory shear flow. We show that such oscillatory shear flow provides an ideal environment to explore a wide variety of metabolic and biochemical effects that promote erythrocyte vesiculation. Deformation details, typical of splenic flow, such as in-folding and implications for membrane/skeleton interaction are demonstrated and quantitatively analyzed. We introduce a theoretical, essentially analytical, vesiculation model that directly couples to our more complex numerical, multilevel, model that clearly delineates various fundamental elements, i.e., sub-processes, that are involved and mediate the vesiculation process. This analytical model highlights particulary important vesiculation precursors such as areas of membrane/skeleton disruptions that trigger the vesiculation process. We demonstrate, using flow cytometry, that the deformations we experimentally induce on cells, and numerically simulate, do not induce lethal forms of cell damage but do induce vesiculation as theoretically forecasted. This, we demonstrate, provides a direct link to cell membrane/skeletal damage such as is associated with metabolic and aging damage. An additional noteworthy feature of this approach is the avoidance of artificial devices, e.g., micro-fluidic chambers, in which deformations and their time scales are often unrepresentative of physiological processes such as splenic flow.

Do Skeletal Dynamics Mediate Sugar Uptake and Transport in Human Erythrocytes?

We explore, herein, the hypothesis that transport of molecules or ions into erythrocytes may be affected and directly stimulated by the dynamics of the spectrin/actin skeleton. Skeleton/actin motions are driven by thermal fluctuations that may be influenced by ATP hydrolysis as well as by structural alterations of the junctional complexes that connect the skeleton to the cell's lipid membrane. Specifically, we focus on the uptake of glucose into erythrocytes via glucose transporter 1 and on the kinetics of glucose disassociation at the endofacial side of glucose transporter 1. We argue that glucose disassociation is affected by both hydrodynamic forces induced by the actin/spectrin skeleton and by probable contact of the swinging 37-nm-long F-actin protofilament with glucose, an effect we dub the "stickball effect." Our hypothesis and results are interpreted within the framework of the kinetic measurements and compartmental kinetic models of Carruthers and co-workers; these experimental results and models describe glucose disassociation as the "slow step" (i.e., rate-limiting step) in the uptake process. Our hypothesis is further supported by direct simulations of skeleton-enhanced transport using our molecular-based models for the actin/spectrin skeleton as well as by experimental measurements of glucose uptake into cells subject to shear deformations, which demonstrate the hydrodynamic effects of advection. Our simulations have, in fact, previously demonstrated enhanced skeletal dynamics in cells in shear deformations, as they occur naturally within the skeleton, which is an effect also supported by experimental observations.

Prospects for Human Erythrocyte Skeleton-Bilayer Dissociation during Splenic Flow

Prospects of vesiculation occurring during splenic flow of erythrocytes are addressed via model simulations of RBC flow through the venous slits of the human spleen. Our model is multiscale and contains a thermally activated rate-dependent description of the entropic elasticity of the RBC spectrin cytoskeleton, including domain unfolding/refolding. Our model also includes detail of the skeleton attachment to the fluidlike lipid bilayer membrane, including a specific accounting for the expansion/contraction of the skeleton that may occur via anchor protein diffusive motion, that is, band 3 and glycophorin, through the membrane. This ability allows us to follow the change in anchor density and thereby the strength of the skeleton/membrane attachment. We define a negative pressure between the skeleton/membrane connection that promotes separation; critical levels for this are estimated using published data on the work of adhesion of this connection. By following the maximum range of negative pressure, along with the observed slight decrease in skeletal density, we conclude that there must be biochemical influences that probably include binding of degraded hemoglobin, among other things, that significantly reduce effective attachment density. These findings are consistent with reported trends in vesiculation that are believed to occur in cases of various hereditary anemias and during blood storage. Our findings also suggest pathways for further study of erythrocyte vesiculation that point to the criticality of understanding the biochemical phenomena involved with cytoskeleton/membrane attachment.

Multiscale modeling methods in biomechanics

More and more frequently, computational biomechanics deals with problems where the portion of physical reality to be modeled spans over such a large range of spatial and temporal dimensions, that it is impossible to represent it as a single space-time continuum. We are forced to consider multiple space-time continua, each representing the phenomenon of interest at a characteristic space-time scale. Multiscale models describe a complex process across multiple scales, and account for how quantities transform as we move from one scale to another. This review offers a set of definitions for this emerging field, and provides a brief summary of the most recent developments on multiscale modeling in biomechanics. Of all possible perspectives, we chose that of the modeling intent, which vastly affect the nature and the structure of each research activity. To the purpose we organized all papers reviewed in three categories: 'causal confirmation,' where multiscale models are used as materializations of the causation theories; 'predictive accuracy,' where multiscale modeling is aimed to improve the predictive accuracy; and 'determination of effect,' where multiscale modeling is used to model how a change at one scale manifests in an effect at another radically different space-time scale. Consistent with how the volume of computational biomechanics research is distributed across application targets, we extensively reviewed papers targeting the musculoskeletal and the cardiovascular systems, and covered only a few exemplary papers targeting other organ systems. The review shows a research subdomain still in its infancy, where causal confirmation papers remain the most common. WIREs Syst Biol Med 2017, 9:e1375. doi: 10.1002/wsbm.1375 For further resources related to this article, please visit the WIREs website.

Effects of stiffness and volume on the transit time of an erythrocyte through a slit

By using a fully coupled fluid-cell interaction model, we numerically simulate the dynamic process of a red blood cell passing through a slit driven by an incoming flow. The model is achieved by combining a multiscale model of the composite cell membrane with a boundary element fluid dynamics model based on the Stokes flow assumption. Our concentration is on the correlation between the transit time (the time it takes to finish the whole translocation process) and different conditions (flow speed, cell orientation, cell stiffness, cell volume, etc.) that are involved. According to the numerical prediction (with some exceptions), the transit time rises as the cell is stiffened. It is also highly sensitive to volume increase inside the cell. In general, even slightly swollen cells (i.e., the internal volume is increased while the surface area of the cell kept unchanged) travel dramatically slower through the slit. For these cells, there is also an increased chance of blockage.

Deformation and internal stress in a red blood cell as it is driven through a slit by an incoming flow

To understand the deformation and internal stress of a red blood cell when it is pushed through a slit by an incoming flow, we conduct a numerical investigation by combining a fluid-cell interaction model based on boundary-integral equations with a multiscale structural model of the cell membrane that takes into account the detailed molecular architecture of this biological system. Our results confirm the existence of cell 'infolding', during which part of the membrane is inwardly bent to form a concave region. The time histories and distributions of area deformation, shear deformation, and contact pressure during and after the translocation are examined. Most interestingly, it is found that in the recovery phase after the translocation significant dissociation pressure may develop between the cytoskeleton and the lipid bilayer. The magnitude of this pressure is closely related to the locations of the dimple elements during the transit. Large dissociation pressure in certain cases suggests the possibility of mechanically induced structural remodeling and structural damage such as vesiculation. With quantitative knowledge about the stability of intra-protein, inter-protein and protein-to-lipid linkages under dynamic loads, it will be possible to achieve numerical prediction of these processes.

Continuum modeling of deformation and aggregation of red blood cells

In order to gain better understanding for rheology of an isolated red blood cell (RBC) and a group of multiple RBCs, new continuum models for describing mechanical properties of cellular structures of an RBC and inter-cellular interactions among multiple RBCs are developed. The viscous property of an RBC membrane, which characterizes dynamic behaviors of an RBC under stress loading and unloading processes, is determined using a generalized Maxwell model. The present model is capable of predicting stress relaxation and stress-strain hysteresis, of which prediction is not possible using the commonly used Kelvin-Voigt model. Nonlinear elasticity of an RBC is determined using the Yeoh hyperelastic material model in a framework of continuum mechanics using finite-element approximation. A novel method to model inter-cellular interactions among multiple adjacent RBCs is also developed. Unlike the previous modeling approaches for aggregation of RBCs, where interaction energy for aggregation is curve-fitted using a Morse-type potential function, the interaction energy is analytically determined. The present aggregation model, therefore, allows us to predict various effects of physical parameters such as the osmotic pressure, the thickness of a glycocalyx layer, the penetration depth, and the permittivity, on the depletion and electrostatic energy among RBCs. Simulations for elongation and recovery deformation of an RBC and for aggregation of multiple RBCs are conducted to evaluate the efficacy of the present continuum modeling methods.

Cell type-restricted expression of erythrocyte tropomodulin Isoform41 in exon 1 knockout/LacZ knock-in heterozygous mice

Full-length erythrocyte tropomodulin (E-Tmod or Tmod1) isoform of 41 kDa is an actin nucleation protein and caps the pointed end of tropomyosin-coated actin filaments. It participates in the length control of short actin protofilaments in the erythrocyte membrane skeletal network as well as the organization of microfilaments in non-erythroid cells. Recently we discovered and characterized a truncated isoform of 29 kDa, which lacks the N-terminal sequence encoded by exons 1 and 2 required for nucleation and capping. Thus, it is important to study the expression pattern of solely the E-Tmod41 isoform in tissues. We utilized our exon 1 knockout (KO) mouse model with a knock-in lacZ reporter gene which reports the expression of E-Tmod41, but not E-Tmod29. Because this homozygous isoform-specific KO is an embryonic lethal mutation, we used heterozygous mice. X-gal staining localized specific signals at the single cell level and revealed a timed expression during embryonic development and restricted expression in adult mice. Our results showed that E-Tmod41 expressing cells include developing and young erythroid cells, developing somites, young fiber cells in the lens, certain subtype(s) of tubular cells in the kidney, smooth muscle cells in various tissues, and horizontal cells in the retina. A comparison with previous studies revealed that most if not all tissues known to express E-Tmod contained lacZ-expressing cells. Interestingly, some tubular cells were lacZ-positive while others in the same renal tubule were not, indicating heterogeneity within the tubular cells. Combined with double immunocytochemistry, we further localized E-Tmod41 to dendritic spines of horizontal cells. These timed and cell-type restricted expressions of E-Tmod41 suggest a role of actin nucleation and/or short actin protofilaments in these cell types and sub-cellular structures.

Erythrocyte stiffness during morphological remodeling induced by carbon ion radiation

The adverse effect induced by carbon ion radiation (CIR) is still an unavoidable hazard to the treatment object. Thus, evaluation of its adverse effects on the body is a critical problem with respect to radiation therapy. We aimed to investigate the change between the configuration and mechanical properties of erythrocytes induced by radiation and found differences in both the configuration and the mechanical properties with involving in morphological remodeling process. Syrian hamsters were subjected to whole-body irradiation with carbon ion beams (1, 2, 4, and 6 Gy) or X-rays (2, 4, 6, and 12 Gy) for 3, 14 and 28 days. Erythrocytes in peripheral blood and bone marrow were collected for cytomorphological analysis. The mechanical properties of the erythrocytes were determined using atomic force microscopy, and the expression of the cytoskeletal protein spectrin-α1 was analyzed via western blotting. The results showed that dynamic changes were evident in erythrocytes exposed to different doses of carbon ion beams compared with X-rays and the control (0 Gy). The magnitude of impairment of the cell number and cellular morphology manifested the subtle variation according to the irradiation dose. In particular, the differences in the size, shape and mechanical properties of the erythrocytes were well exhibited. Furthermore, immunoblot data showed that the expression of the cytoskeletal protein spectrin-α1 was changed after irradiation, and there was a common pattern among its substantive characteristics in the irradiated group. Based on these findings, the present study concluded that CIR could induce a change in mechanical properties during morphological remodeling of erythrocytes. According to the unique characteristics of the biomechanical categories, we deduce that changes in cytomorphology and mechanical properties can be measured to evaluate the adverse effects generated by tumor radiotherapy. Additionally, for the first time, the current study provides a new strategy for enhancing the assessment of the curative effects and safety of clinical radiotherapy, as well as reducing adverse effects.

Continuum- and particle-based modeling of shapes and dynamics of red blood cells in health and disease

We review recent advances in multiscale modeling of the mechanics of healthy and diseased red blood cells (RBCs), and blood flow in the microcirculation. We cover the traditional continuum-based methods but also particle-based methods used to model both the RBCs and the blood plasma. We highlight examples of successful simulations of blood flow including malaria and sickle cell anemia.

The human erythrocyte plasma membrane: a Rosetta Stone for decoding membrane-cytoskeleton structure

The mammalian erythrocyte, or red blood cell (RBC), is a unique experiment of nature: a cell with no intracellular organelles, nucleus or transcellular cytoskeleton, and a plasma membrane with uniform structure across its entire surface. By virtue of these specialized properties, the RBC membrane has provided a template for discovery of the fundamental actin filament network machine of the membrane skeleton, now known to confer mechanical resilience, anchor membrane proteins, and organize membrane domains in all cells. This chapter provides a historical perspective and critical analysis of the biochemistry, structure, and physiological functions of this actin filament network in RBCs. The core units of this network are nodes of ~35-37 nm-long actin filaments, interconnected by long strands of (α1β1)₂-spectrin tetramers, forming a 2D isotropic lattice with quasi-hexagonal symmetry. Actin filament length and stability is critical for network formation, relying upon filament capping at both ends: tropomodulin-1 at pointed ends and αβ-adducin at barbed ends. Tropomodulin-1 capping is essential for precise filament lengths, and is enhanced by tropomyosin, which binds along the short actin filaments. αβ-adducin capping recruits spectrins to sites near barbed ends, promoting network formation. Accessory proteins, 4.1R and dematin, also promote spectrin binding to actin and, with αβ-adducin, link to membrane proteins, targeting actin nodes to the membrane. Dissection of the molecular organization within the RBC membrane skeleton is one of the paramount achievements of cell biological research in the past century. Future studies will reveal the structure and dynamics of actin filament capping, mechanisms of precise length regulation, and spectrin-actin lattice symmetry.

Red blood cell dynamics: from cell deformation to ATP release

The mechanisms of red blood cell (RBC) deformation under both static and dynamic, i.e., flow, conditions have been studied extensively since the mid 1960s. Deformation-induced biochemical reactions and possible signaling in RBCs, however, were proposed only fifteen years ago. Therefore, the fundamental relationship between RBC deformation and cellular signaling dynamics i.e., mechanotransduction, remains incompletely understood. Quantitative understanding of the mechanotransductive pathways in RBCs requires integrative studies of physical models of RBC deformation and cellular biochemical reactions. In this article we review the physical models of RBC deformation, spanning from continuum membrane mechanics to cellular skeleton dynamics under both static and flow conditions, and elaborate the mechanistic links involved in deformation-induced ATP release.

Continuum simulations of biomembrane dynamics and the importance of hydrodynamic effects

Traditional particle-based simulation strategies are impractical for the study of lipid bilayers and biological membranes over the longest length and time scales (microns, seconds and longer) relevant to cellular biology. Continuum-based models developed within the frameworks of elasticity theory, fluid dynamics and statistical mechanics provide a framework for studying membrane biophysics over a range of mesoscopic to macroscopic length and time regimes, but the application of such ideas to simulation studies has occurred only relatively recently. We review some of our efforts in this direction with emphasis on the dynamics in model membrane systems. Several examples are presented that highlight the prominent role of hydrodynamics in membrane dynamics and we argue that careful consideration of fluid dynamics is key to understanding membrane biophysics at the cellular scale.

Tank-treading and tumbling frequencies of capsules and red blood cells

This study is motivated in part by the discrepancy that exists in the literature with regard to the dependence of the tank-treading frequency of red blood cells on the shear rate and suspending medium viscosity. Here we consider three-dimensional numerical simulations of deformable capsules of initially spherical and oblate spheroidal shapes and biconcave discoid representing the red blood cell resting shape. By considering a much broader range of the viscosity ratio (ratio of capsule or cell interior to suspending fluid viscosity), shear rate, and aspect ratio (ratio of minor to major axes) than that considered in the previous experiments, we find several new characteristics of the tank-treading and tumbling frequencies that have not been reported earlier. These new characteristics are the result of the large shape deformation and the coupling between shape and angular oscillations of the capsules or cells. For the spherical and oblate spheroidal capsules, the tank-treading frequency shows a nonmonotonic trend that is characterized by an initial decrease leading to a minimum followed by an increase with increasing viscosity ratio. For red blood cells, we find two regimes of the viscosity dependence of the tank-treading frequency: an exponential regime in which the tank-treading frequency decreases at a slower rate with increasing viscosity ratio, and a logarithmic range in which it decreases at a much faster rate. While this trend agrees well with different theoretical models of shape-preserving capsules, it was not evident in previous experimental results. When the shear rate dependence is considered, the tank-treading frequency of red blood cells and capsules of highly elongated initial shapes exhibits a nonmonotonic trend that is characterized by an initial increase leading to a maximum followed by a sharp decrease with decreasing shear rate. This anomalous behavior of the tank-treading frequency is shown to be due to a breathing-like dynamics of the capsule or cell that is characterized by a repeated emergence and absence of deep, crater-like dimples, and a large swinging motion. We further observe that the tumbling frequency exhibits a decreasing trend with increasing viscosity ratio that is in contrast to the theoretical result for the shape-preserving capsules and is due to the periodic deformation and preferential alignment of the capsules in the extensional quadrant of the flow.

Intertwined αβ spectrin meeting helical actin protofilament in the erythrocyte membrane skeleton: wrap-around vs. point-attachment

Our 3-D model for a junctional complex (JC) in the erythrocyte membrane skeleton proposed that the helical actin protofilament functions as a mechanical axis for three pairs of αβ spectrin (Sp), and each pair wraps around the protofilament in a back-to-back fashion. The distal end of each Sp is further associated with the lipid bilayer by a suspension complex (SC). Here, we detail how splitting and rejoining of αβ Sp around a protofilament may form a loop that sustains and equilibrates tension. Sequential association of β and α Sp solves the challenge of constructing multiple loops along the protofilament, and topological connection facilitates their re-association. The wrap-around model minimizes the strain of the actin binding site on β Sp due to tension, redirection, or sliding of intertwined Sp. Pairing Sp balances the opposing forces and provides a mechanism for elastic recovery. The wrap-around junction thus provides mechanical advantages over a point-attachment junction in maintaining the integrity and functionality of the network. Severing α or β Sp may convert a wrapping-around junction to a point-attachment junction. In that case, a “bow up” motion of JC during deformation may disturb or flip the overlaid lipid bilayer, and mark stressed erythrocytes for phagocytosis.

Erythrocyte tropomodulin isoforms with and without the N-terminal actin-binding domain

Erythrocyte tropomodulin (E-Tmod or Tmod1) of 41 kDa is a tropomyosin (TM)-binding protein that caps the slow-growing end of the actin filaments. Its N-terminal half is flexible, whereas the C-terminal half has a single domain structure. E-Tmod/TM5 complex may function as a "molecular ruler" generating actin protofilaments of ∼37 nm. Here we report the discovery of a short isoform of 29 kDa that lacks the N-terminal actin-binding domain (N-ABD) but retains the C-terminal actin-binding domain (C-ABD). E-Tmod29 can be generated by alternative splicing from an upstream promoter or by multiple transcriptional start sites from a downstream promoter. Promoter switching leads to a surge of E-Tmod41 in reticulocytes, which degrades quickly in the cytosol. We expressed recombinant isoforms in Escherichia coli and tested their binding toward TM5, G-actin, and F-actin. Solid-phase binding assays show that, without the N-terminal 102 residues, E-Tmod29 binds to TM5 or G-actin more strongly than E-Tmod41 does, but barely binds to F-actin after TM5 binding. Differential bindings explain the distinct localizations of E-Tmod29 in the cytosol and E-Tmod41 on the membrane. Sequential bindings and immunofluorescent staining further suggest that 1) TM5 binding to E-Tmod41 may open up the flexible N-terminal half, exposing N-ABD and unblocking C-ABD; 2) N-ABD binds to F-actin and C-ABD binds to G-actin; and 3) F-actin binding to N-ABD may prevent G-actin from binding to C-ABD. E-Tmod29 may thus modulate the availability of TM5 and G-actin for E-Tmod41 to construct the protofilament-based membrane skeletal network for circulating erythrocytes.

Nanomechanics of multiple units in the erythrocyte membrane skeletal network

Erythrocytes undergo deformations when they transport O₂ and CO₂ across the membrane, yet the 3D nanomechanics of the skeletal network remains poorly understood. Expanding from our previous single isolated unit, we now simulate networks consisting of 1–10 concentric rings of repeating units in equibiaxial deformation. The networks are organized with (1) a 3D model for a single unit, (2) a wrap-around mode between Sp and actin protofilament in the intra-unit interaction, and (3) a random inter-unit connectivity. These assumptions permit efficient five-degrees-of-freedom (5DOF) simulations when up to 30 pN of radial forces are applied to the boundary spectrin (Sp) and the center and other units are analyzed. As 6 Sp balance their tensions, hexagonal units become irregular. While actin protofilaments remain tangent to the network, their yaw (Φ) angles change drastically with addition of neighboring units or an Sp unfolding. It is anticipated that during deformation, transmembrane complexes associated with the network move laterally through the lipid bilayer and increase the diffusion of molecules across the membrane. When protofilament/Sp sweeps under the lipid bilayer, they mix up the submembrane concentration gradient. Thus, the nanomechanics of actin protofilaments and Sp may enhance the transport of molecules during erythrocyte deformation.

Rheology of a dilute suspension of liquid-filled elastic capsules

Rheology of a dilute suspension of liquid-filled elastic capsules in linear shear flow is studied by three-dimensional numerical simulations using a front-tracking method. This study is motivated by a recent discovery that a suspension of viscous vesicles exhibits a shear viscosity minimum when the vesicles undergo an unsteady vacillating-breathing dynamics at the threshold of a transition between the tank-treading and tumbling motions. Here we consider capsules of spherical resting shape for which only a steady tank-treading motion is observed. A comprehensive analysis of the suspension rheology is presented over a broad range of viscosity ratio (ratio of internal-to-external fluid viscosity), shear rate (or, capillary number), and capsule surface-area dilatation. We find a result that the capsule suspension exhibits a shear viscosity minimum at moderate values of the viscosity ratio, and high capillary numbers, even when the capsules are in a steady tank-treading motion. It is further observed that the shear viscosity minimum exists for capsules with area-dilating membranes but not for those with nearly incompressible membranes. Nontrivial results are also observed for the normal stress differences which are shown to decrease with increasing capillary number at high viscosity ratios. Such nontrivial results neither can be predicted by the small-deformation theory nor can be explained by the capsule geometry alone. Physical mechanisms underlying these results are studied by decomposing the particle stress tensor into a contribution due to the elastic stresses in the capsule membrane and a contribution due to the viscosity differences between the internal and suspending fluids. It is shown that the elastic contribution is shear-thinning, but the viscous contribution is shear thickening. The coupling between the capsule geometry and the elastic and viscous contributions is analyzed to explain the observed trends in the bulk rheology.

Multiscale simulation of erythrocyte membranes

To quantitatively predict the mechanical response and mechanically induced remodeling of red blood cells, we developed a multiscale method to correlate distributions of internal stress with overall cell deformation. This method consists of three models at different length scales: in the complete cell level the membrane is modeled as two distinct layers of continuum shells using finite element method (Level III), in which the skeleton-bilayer interactions are depicted as a slide in the lateral (i.e., in-plane) direction (caused by the mobility of the skeleton-bilayer pinning points) and a normal contact force; the constitutive laws of the inner layer (the protein skeleton) are obtained from a molecular-based model (Level II); the mechanical properties of the spectrin (Sp, a key component of the skeleton), including its folding/unfolding reactions, are obtained with a stress-strain model (Level I). Model verification is achieved through comparisons with existing numerical and experimental studies in terms of the resting shape of the cell as well as cell deformations induced by micropipettes and optical tweezers. Detailed distributions of the interaction force between the lipid bilayer and the skeleton that may cause their dissociation and lead to phenomena such as vesiculation are predicted. Specifically, our model predicts correlation between the occurrence of Sp unfolding and increase in the mechanical load upon individual skeleton-bilayer pinning points. Finally a simulation of the necking process after skeleton-bilayer dissociation, a precursor of vesiculation, is conducted.

Dynamics of nonspherical capsules in shear flow

Three-dimensional numerical simulations using a front-tracking method are presented on the dynamics of oblate shape capsules in linear shear flow by considering a broad range of viscosity contrast (ratio of internal-to-external fluid viscosity), shear rate (or capillary number), and aspect ratio. We focus specifically on the coupling between the shape deformation and orientation dynamics of capsules, and show how this coupling influences the transition from the tank-treading to tumbling motion. At low capillary numbers, three distinct modes of motion are identified: a swinging or oscillatory (OS) mode at a low viscosity contrast in which the inclination angle theta(t) oscillates but always remains positive; a vacillating-breathing (VB) mode at a moderate viscosity contrast in which theta(t) periodically becomes positive and negative, but a full tumbling does not occur; and a pure tumbling mode (TU) at a higher viscosity contrast. At higher capillary numbers, three types of transient motions occur, in addition to the OS and TU modes, during which the capsule switches from one mode to the other as (i) VB to OS, (ii) TU to VB to OS, and (iii) TU to VB. Phase diagrams showing various regimes of capsule dynamics are presented. For all modes of motion (OS, VB, and TU), a large-amplitude oscillation in capsule shape and a strong coupling between the shape deformation and orientation dynamics are observed. It is shown that the coupling between the shape deformation and orientation is the strongest in the VB mode, and hence at a moderate viscosity contrast, for which the amplitude of shape deformation reaches its maximum. The numerical results are compared with the theories of Keller and Skalak, and Skotheim and Secomb. Significant departures from the two theories are discussed and related to the strong coupling between the shape deformation, inclination, and transition dynamics.

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.

Spectrin folding versus unfolding reactions and RBC membrane stiffness

Spectrin (Sp), a key component of the erythrocyte membrane, is routinely stretched to near its fully folded contour length during cell deformations. Such dynamic loading may induce domain unfolding as suggested by recent experiments. Herein we develop a model to describe the folding/unfolding of spectrin during equilibrium or nonequilibrium extensions. In both cases, our model indicates that there exists a critical extension beyond which unfolding occurs. We further deploy this model, together with a three-dimensional model of the junctional complex in the erythrocyte membrane, to explore the effect of Sp unfolding on the membrane's mechanical properties, and on the thermal fluctuation of membrane-attached beads. At large deformations our results show a distinctive strain-induced unstiffening behavior, manifested in the slow decrease of the shear modulus, and accompanied by an increase in bead fluctuation. Bead fluctuation is also found to be influenced by mode switching, a phenomenon predicted by our three-dimensional model. The amount of stiffness reduction, however, is modest compared with that reported in experiments. A possible explanation for the discrepancy is the occurrence of spectrin head-to-head disassociation which is also included within our modeling framework and used to analyze bead motion as observed via experiment.