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

Dynamic mechanisms for membrane skeleton transitions

The plasma membrane and the underlying skeleton form a protective barrier for eukaryotic cells. The molecules forming this complex composite material constantly rearrange under mechanical stress to confer this protective capacity. One of those molecules, spectrin, is ubiquitous in the membrane skeleton and primarily located proximal to the inner leaflet of the plasma membrane and engages in protein-lipid interactions via a set of membrane-anchoring domains. Spectrin is linked by short actin filaments and its conformation varies in different types of cells. In this work, we developed a generalized network model for the membrane skeleton integrated with myosin contractility and membrane mechanics to investigate the response of the spectrin meshwork to mechanical loading. We observed that the force generated by membrane bending is important to maintain a smooth skeletal structure. This suggests that the membrane is not just supported by the skeleton, but has an active contribution to the stability of the cell structure. We found that spectrin and myosin turnover are necessary for the transition between stress and rest states in the skeleton. Our model reveals that the actin-spectrin meshwork dynamics are balanced by the membrane forces with area constraint and volume restriction promoting the stability of the membrane skeleton. Furthermore, we showed that cell attachment to the substrate promotes shape stabilization. Thus, our proposed model gives insight into the shared mechanisms of the membrane skeleton associated with myosin and membrane that can be tested in different types of cells.

A Simulation of the Mechanical Testing of the Cell Membrane and Cytoskeleton

Cell models play a crucial role in analyzing the mechanical response of cells and quantifying cellular damage incurred during micromanipulation. While traditional models can capture the overall mechanical behavior of cells, they often lack the ability to discern among distinct cellular components. Consequently, by employing dissipative particle dynamics, this study constructed a triangular network-like representation of the cell membrane along with cross-linked cytoskeletal chains. The mechanical properties of both the membrane and cytoskeleton were then analyzed through a series of simulated mechanical tests, validated against real-world experiments. The investigation utilized particle-tracking rheology to monitor changes in the mean square displacements of membrane particles over time, facilitating the analysis of the membrane's storage and loss moduli. Additionally, the cytoskeletal network's storage and loss moduli were examined via a double-plate oscillatory shear experiment. The simulation results revealed that both the membrane and cytoskeleton exhibit viscoelastic behavior, as evidenced by the power-law dependency of their storage and loss moduli on frequency. Furthermore, indentation and microinjection simulations were conducted to examine the overall mechanical properties of cells. In the indentation experiments, an increase in the shear modulus of the membrane's WLCs correlated with a higher Young's modulus for the entire cell. Regarding the microinjection experiment, augmenting the microinjection speed resulted in reduced deformation of the cell at the point of membrane rupture and a lower percentage of high strain.

Development of Mechanical Stability in Late-Stage Embryonic Erythroid Cells: Insights From Fluorescence Imaged Micro-Deformation Studies

The combined use of fluorescence labeling and micro-manipulation of red blood cells has proven to be a powerful tool for understanding and characterizing fundamental mechanisms underlying the mechanical behavior of cells. Here we used this approach to study the development of the membrane-associated cytoskeleton (MAS) in primary embryonic erythroid cells. Erythropoiesis comes in two forms in the mammalian embryo, primitive and definitive, characterized by intra- and extra-vascular maturation, respectively. Primitive erythroid precursors in the murine embryo first begin to circulate at embryonic day (E) 8.25 and mature as a semi-synchronous cohort before enucleating between E12.5 and E16.5. Previously, we determined that the major components of the MAS become localized to the membrane between E10.5 and E12.5, and that this localization is associated with an increase in membrane mechanical stability over this same period. The change in mechanical stability was reflected in the creation of MAS-free regions of the membrane at the tips of the projections formed when cells were aspirated into micropipettes. The tendency to form MAS-free regions decreases as primitive erythroid cells continue to mature through E14.5, at least 2 days after all detectable cytoskeletal components are localized to the membrane, indicating continued strengthening of membrane cohesion after membrane localization of cytoskeletal components. Here we demonstrate that the formation of MAS-free regions is the result of a mechanical failure within the MAS, and not the detachment of membrane bilayer from the MAS. Once a "hole" is formed in the MAS, the skeletal network contracts laterally along the aspirated projection to form the MAS-free region. In protein 4.1-null primitive erythroid cells, the tendency to form MAS-free regions is markedly enhanced. Of note, similar MAS-free regions were observed in maturing erythroid cells from human marrow, indicating that similar processes occur in definitive erythroid cells. We conclude that localization of cytoskeletal components to the cell membrane of mammalian erythroid cells during maturation is insufficient by itself to produce a mature MAS, but that subsequent processes are additionally required to strengthen intraskeletal interactions.

Red blood cell shape transitions and dynamics in time-dependent capillary flows

The dynamics of single red blood cells (RBCs) determine microvascular blood flow by adapting their shape to the flow conditions in the narrow vessels. In this study, we explore the dynamics and shape transitions of RBCs on the cellular scale under confined and unsteady flow conditions using a combination of microfluidic experiments and numerical simulations. Tracking RBCs in a comoving frame in time-dependent flows reveals that the mean transition time from the symmetric croissant to the off-centered, nonsymmetric slipper shape is significantly faster than the opposite shape transition, which exhibits pronounced cell rotations. Complementary simulations indicate that these dynamics depend on the orientation of the RBC membrane in the channel during the time-dependent flow. Moreover, we show how the tank-treading movement of slipper-shaped RBCs in combination with the narrow channel leads to oscillations of the cell's center of mass. The frequency of these oscillations depends on the cell velocity, the viscosity of the surrounding fluid, and the cytosol viscosity. These results provide a potential framework to identify and study pathological changes in RBC properties.

Organic and inorganic equivalent models for analysis of red blood cell mechanical behaviour

Experimental investigation into the mechanical response of red blood cells is presently impeded with the main impediments being the micro dimensions involved and ethical issues associated with in vivo testing. The widely employed alternative approach of computational modelling suffers from its own inherent limitations being reliant on precise constitutive and boundary information. Moreover, and somewhat critically, numerical computational models themselves are required to be validated by means of experimentation and hence suffer similar impediments. An alternative experimental approach is examined in this paper involving large-scale equivalent models manufactured principally from inorganic, and to lesser extent organic, materials. Although there presently exists no known method providing the means to investigate the mechanical response of red blood cells using scaled models simultaneously having different dimensions and materials, the present paper aims to develop a scaled framework based on the new finite-similitude theory that has appeared in the recent open literature. Computational models are employed to test the effectiveness of the proposed method, which in principle can provide experimental solution methods to a wide range of practical applications including the design of red-blood cell nanorobots and drug delivery systems. By means of experimentally validated numerical experiments under impact loading it is revealed that although exact prediction is not achieved good accuracy can nevertheless be obtained. Furthermore, it is demonstrated how the proposed approach for first time provides a means to relate models at different scales founded on different constitutive equations.

3D phonon microscopy with sub-micron axial-resolution

Brillouin light scattering (BLS) is an emerging method for cell imaging and characterisation. It allows elasticity-related contrast, optical resolution and label-free operation. Phonon microscopy detects BLS from laser generated coherent phonon fields to offer an attractive route for imaging since, at GHz frequencies, the phonon wavelength is sub-optical. Using phonon fields to image single cells is challenging as the signal to noise ratio and acquisition time are often poor. However, recent advances in the instrumentation have enabled imaging of fixed and living cells. This work presents the first experimental characterisation of phonon-based axial resolution provided by the response to a sharp edge. The obtained axial resolution is up to 10 times higher than that of the optical system used to take the measurements. Validation of the results are obtained with various polymer objects, which are in good agreement with those obtained using atomic force microscopy. Edge localisation, and hence profilometry, of a phantom boundary is measured with accuracy and precision of approximately 60 nm and 100 nm respectively. Finally, 3D imaging of fixed cells in culture medium is demonstrated.

Constitutive Model of Erythrocyte Membranes with Distributions of Spectrin Orientations and Lengths

We present an analytical hyperelastic constitutive model of the red blood cell (erythrocyte) membrane based on recently improved characterizations of density and microscopic structure of its spectrin network from proteomics and cryo-electron tomography. The model includes distributions of both orientations and natural lengths of spectrin and updated copy numbers of proteins. By applying finite deformation to the spectrin network, we obtain the total free energy and stresses in terms of invariants of shear and area deformation. We generalize an expression of the initial shear modulus, which is independent of the number of molecular orientations within the network and also derive a simplified version of the model. We apply the model and its simplified version to analyze micropipette aspiration computationally and analytically and explore the effect of local cytoskeletal density change. We also explore the discrepancies among shear modulus values measured using different experimental techniques reported in the literature. We find that the model exhibits hardening behavior and can explain many of these discrepancies. Moreover, we find that the distribution of natural lengths plays a crucial role in the hardening behavior when the correct copy numbers of proteins are used. The initial shear modulus values we obtain using our current model (5.9-15.6 pN/μm) are close to the early estimates (6-9 pN/μm). This new, to our knowledge, constitutive model establishes a direct connection between the molecular structure of spectrin networks and constitutive laws and also defines a new picture of a much denser spectrin network than assumed in prior studies.

Apparent attenuation by opto-acoustic defocus in phonon microscopy

Understanding the mechanical properties of biological cells is a challenging problem for the life sciences partly because there are limited methods for mapping elasticity with high resolution. Phonon microscopy is a form of Brillouin light scattering which uses coherent phonons for imaging with elasticity-related contrast, phonon resolution and without labels. It can measure material properties such as sound velocity, acoustic impedance and attenuation. To use it as a contrast mechanism in microscopy, high numerical aperture (NA) lenses are key to high resolution. However, increasing NA induces apparent attenuation, a premature decay of the detected signal. To reduce signal decay and quantify the sound attenuation coefficient in cells, it is necessary to understand the mechanisms that affect signal decay. Here we define opto-acoustic defocus as a signal decay mechanism and propose methods to achieve quantitative sound attenuation measurements, and to optimise in-depth imaging at high resolution which is crucial for cell imaging.

Theoretical Bases for the Role of Red Blood Cell Shape in the Regulation of Its Volume

The red blood cell (RBC) membrane contains a mechanosensitive cation channel Piezo1 that is involved in RBC volume homeostasis. In a recent model of the mechanism of its action it was proposed that Piezo1 cation permeability responds to changes of the RBC shape. The aim here is to review in a descriptive manner different previous studies of RBC behavior that formed the basis for this proposal. These studies include the interpretation of RBC and vesicle shapes based on the minimization of membrane bending energy, the analyses of various consequences of compositional and structural features of RBC membrane, in particular of its membrane skeleton and its integral membrane proteins, and the modeling of the establishment of RBC volume. The proposed model of Piezo1 action is critically evaluated, and a perspective presented for solving some remaining experimental and theoretical problems. Part of the discussion is devoted to the usefulness of theoretical modeling in studies of the behavior of cell systems in general.

Non-uniform distribution of myosin-mediated forces governs red blood cell membrane curvature through tension modulation

The biconcave disk shape of the mammalian red blood cell (RBC) is unique to the RBC and is vital for its circulatory function. Due to the absence of a transcellular cytoskeleton, RBC shape is determined by the membrane skeleton, a network of actin filaments cross-linked by spectrin and attached to membrane proteins. While the physical properties of a uniformly distributed actin network interacting with the lipid bilayer membrane have been assumed to control RBC shape, recent experiments reveal that RBC biconcave shape also depends on the contractile activity of nonmuscle myosin IIA (NMIIA) motor proteins. Here, we use the classical Helfrich-Canham model for the RBC membrane to test the role of heterogeneous force distributions along the membrane and mimic the contractile activity of sparsely distributed NMIIA filaments. By incorporating this additional contribution to the Helfrich-Canham energy, we find that the RBC biconcave shape depends on the ratio of forces per unit volume in the dimple and rim regions of the RBC. Experimental measurements of NMIIA densities at the dimple and rim validate our prediction that (a) membrane forces must be non-uniform along the RBC membrane and (b) the force density must be larger in the dimple than the rim to produce the observed membrane curvatures. Furthermore, we predict that RBC membrane tension and the orientation of the applied forces play important roles in regulating this force-shape landscape. Our findings of heterogeneous force distributions on the plasma membrane for RBC shape maintenance may also have implications for shape maintenance in different cell types.

The spectrin-actin network of the membrane skeleton plays an important role in controlling specialized cell membrane morphology. In the paradigmatic red blood cell (RBC), where actin filaments are present exclusively in the membrane skeleton, recent experiments reveal that nonmuscle myosin IIA (NMIIA) motor contractility maintains the RBC biconcave disk shape. In this study, we have identified criteria for micron-scale distributions of NMIIA forces at the membrane required to maintain the biconcave disk shape of an RBC in the resting condition. Supported by experimental measurements of RBC NMIIA distribution, we showed that a heterogeneous force distribution with a larger force density at the dimple is able to capture the experimentally observed biconcave morphology of an RBC with better accuracy compared to previous models that did not consider the heterogeneity in the force distribution. Furthermore, we showed that the biconcave geometry of the RBC is closely regulated by the effective membrane tension and the direction of applied forces on the membrane. These findings can be generalized to any force-mediated membrane shape, providing insight into the role of actomyosin forces in prescribing and maintaining the morphology of different cell types.

Comparison of cell mechanical measurements provided by Atomic Force Microscopy (AFM) and Micropipette Aspiration (MPA)

A comparative analysis of T-lymphocyte mechanical data obtained from Micropipette Aspiration (MPA) and Atomic Force Microscopy (AFM) is presented. Results obtained by fitting the experimental data to simple Hertz and Theret models led to non-Gaussian distributions and significantly different values of the elastic moduli obtained by both techniques. The use of more refined models, taking into account the finite size of cells (simplified double contact and Zhou models) reduces the differences in the values calculated for the elastic moduli. Several possible sources for the discrepancy between the techniques are considered. The analysis suggests that the local nature of AFM measurements compared with the more general character of MPA measurements probably contributed to the differences observed.

On Eulerian versus Lagrangian models of mechanical blood damage and the linearized damage function

Two limitations have been discovered in the derivation of the Eulerian method of hemolysis prediction using a linearized blood damage function. First is that in the transformation from the Lagrangian material volume of the original power-law model to a fixed Eulerian control volume, the spatial dependence of duration of exposure to fluid stress was neglected. This omission has the implication that the Eulerian method as reported is valid only for steady, uniaxial flow in which velocity is constant along streamlines. The second issue is related to linearization, which involves distributing an exponent across an integral. This operation is valid only for limited conditions that include the exponent being unity (which is not the case for any power-law hemolysis models) or the blood damage function being constant throughout the flow regime. These constraints severely restrict the applicability of the Eulerian method. An example problem is presented that demonstrates that the source term of the Eulerian method as reported does not account for differences in velocity between 2 similar flows. Correcting the source term to match the hemolysis prediction to that of the original, unlinearized method requires an analytical description of the flow field that may not be easily obtained for the complex flows in some cardiovascular devices.

A Model of Piezo1-Based Regulation of Red Blood Cell Volume

A red blood cell (RBC) performs its function of adequately carrying respiratory gases in blood by its volume being ∼60% of that of a sphere with the same membrane area. For this purpose, human and most other vertebrate RBCs regulate their content of potassium (K⁺) and sodium (Na⁺) ions. The focus considered here is on K⁺ efflux through calcium-ion (Ca²⁺)-activated Gárdos channels. These channels open under conditions that allow Ca²⁺ to enter RBCs through Piezo1 mechanosensitive cation-permeable channels. It is postulated that the fraction of open Piezo1 channels depends on the RBC shape as a result of the curvature-dependent Piezo1-bilayer membrane interaction. The consequences of this postulate are studied by introducing a simple model of RBC osmotic behavior supplemented by the dependence of RBC membrane K⁺ permeability on the reduced volume (i.e., the ratio of cell volume to its maximal possible volume) of RBC discoid shapes. It is assumed that because of its intrinsic curvature and strong interaction with the surrounding membrane, Piezo1 tends to concentrate in the dimple regions of these shapes, and the fraction of open Piezo1 channels depends on the membrane curvature in that region. It is shown that the properties of the described model can provide the basis for the formation of the negative feedback loop that interrelates cell volume and its content of potassium ions. The model predicts the relation, valid for each cell in an RBC population, between RBC volume and membrane area, thus explaining the large value of the measured membrane area versus the volume correlation coefficient. The mechanism proposed here for RBC volume regulation is in accord with the loss of this correlation in RBCs of Piezo1 knockout mice.

A New Concept of Biotensegrity Incorporating Liquid Tissues: Blood and Lymph

The definition of fascia includes tissues of mesodermal derivation, considered as specialized connective tissue: blood and lymph. As water shapes rocks, bodily fluids modify shapes and functions of bodily structures. Bodily fluids are silent witnesses of the mechanotransductive information, allowing adaptation and life, transporting biochemical and hormonal signals. While the solid fascial tissue divides, supports, and connects the different parts of the body system, the liquid fascial tissue feeds and transports messages for the solid fascia. The focus of this article is to reconsider the model of biotensegrity because it does not take into account the liquid fascia, and to try to integrate the fascial continuum with the lymph and the blood in a new model. The name given to this new model is RAIN—Rapid Adaptability of Internal Network.

Meaning of the Solid and Liquid Fascia to Reconsider the Model of Biotensegrity

The definition of fascia includes tissues of mesodermal derivation considered as specialized connective tissues: the blood and lymph. As water shapes rocks, bodily fluids modify the shape and functioning of bodily structures. Bodily fluids are silent witnesses to mechanotransductive information, allowing adaptation and life, transporting biochemical and hormonal signals. While the solid fascial tissue divides, supports, and connects the different parts of the body system, the liquid fascial tissue feeds and transports messages for the solid fascia. This article reconsiders the model of biotensegrity, by revising the definition of solid and liquid fascia, and tries to integrate the fascial continuum with the lymph and blood in a new model, because in the previous model, these two liquid elements were not taken into consideration. The name given to this new model is Rapid Adaptability of Internal Network (RAIN).

Study of in vitro RBCs membrane elasticity with AOD scanning optical tweezers

The elasticity of red cell membrane is a critical physiological index for the activity of RBC. Study of the inherent mechanism for RBCs membrane elasticity transformation is attention-getting all along. This paper proposes an optimized measurement method of erythrocytes membrane shear modulus incorporating acousto-optic deflector (AOD) scanning optical tweezers system. By use of this method, both membrane shear moduli and sizes of RBCs with different in vitro times were determined. The experimental results reveal that the RBCs membrane elasticity and size decline with in vitro time extension. In addition, semi quantitative measurements of S-nitrosothiol content in blood using fluorescent spectrometry during in vitro storage show that RBCs membrane elasticity change is positively associated with the S-nitrosylation level of blood. The analysis considered that the diminished activity of the nitric oxide synthase makes the S-nitrosylation of in vitro blood weaker gradually. The main reason for worse elasticity of the in vitro RBCs is that S-nitrosylation effect of spectrin fades. These results will provide a guideline for further study of in vitro cells activity and other clinical applications.