Membrane Localization of Piezo1 in the Context of Its Role in the Regulation of Red Blood Cell Volume.

Piezo1 is a membrane nonspecific cation channel involved in red blood cells (RBCs) in the regulation of their volume. Recently, it was shown that it is distributed on the RBC membrane in a nonuniform manner. Here it is shown that it is possible to interpret the lateral distribution of Piezo1 molecules on RBC membrane by the curvature dependent Piezo1-bilayer interaction which is the consequence of the mismatch between the intrinsic principal curvatures of the Piezo1 trimer and the principal curvatures of the membrane at Piezo1's location but without its presence. This result supports the previously proposed model for the role of Piezo1 in the regulation of RBC volume.

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.

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.

Investigating cell functioning by theoretical analysis of cell-to-cell variability.

Here we discuss cell-to-cell variability in isogenic cell populations on the basis of an analogy between the processes of vesicle self-reproduction and cell self-replication. A short review of the theoretical analysis of vesicle self-reproduction is presented to indicate that this process only occurs under the fulfillment of specific criteria: causal relations between the values of vesicle variables involved in its growth and division, and the parameters of the environment. It is shown that when division is asymmetric, both vesicle birth size and interdivision times are variable. We argue that during cell self-replication, the balance between processes of cell growth and division also relies on causal relations between the corresponding cellular variables. A possible method is suggested to unravel previously unidentified causal relations between cell variables from the relationships between their variability parameters such as the widths of their probability distributions and their correlation coefficients. The method is outlined by reviewing the results of the corresponding analysis applied to a population of red blood cells. Some novel research directions are suggested that could lead from the analysis of cell-to-cell variability to a better understanding of the organizational structure of cells and possibly also their evolutionary origin.

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

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

Curvature-dependent protein-lipid bilayer interaction and cell mechanosensitivity.

Cells respond to applied external forces through different mechanosensitive processes, with many of them based on the interaction between membrane-embedded proteins and their lipid environments. This interaction can depend on membrane curvature at the location of such proteins. Here we elucidate the general characteristics of a macroscopically defined protein-lipid bilayer interaction based on a mismatch between the shape of the protein surface and the surrounding membrane curvature. It is shown how the parameters of this interaction are related to the experimentally determined distribution of membrane-embedded proteins between highly curved tubular and flat membrane regions of a giant phospholipid vesicle. The results obtained for such distribution of potassium channel KvAP are presented as an example. Possible participation of the curvature-dependent protein-lipid bilayer interaction in mechanosensitive processes is indicated.

Mechanical and molecular basis for the symmetrical division of the fission yeast nuclear envelope.

In fission yeast Schizosaccharomyces pombe, the nuclear envelope remains intact throughout mitosis and undergoes a series of symmetrical morphological changes when the spindle pole bodies (SPBs), embedded in the nuclear envelope, are pushed apart by elongating spindle microtubules. These symmetrical membrane shape transformations do not correspond to the shape behavior of an analogous system based on lipid vesicles. Here we report that the symmetry of the dividing fission yeast nucleus is ensured by SPB-chromosome attachments, as loss of kinetochore clustering in the vicinity of SPBs results in the formation of abnormal asymmetric shapes with long membrane tethers. We integrated these findings in a biophysical model, which explains the symmetry of the nuclear shapes on the basis of forces exerted by chromosomes clustered at SPBs on the extending nuclear envelope. Based on this analysis we conclude that the fission yeast nuclear envelope exhibits the same mechanical properties as simple lipid vesicles, but interactions with other cellular components, such as chromosomes, influence the nuclear shape during mitosis, allowing the formation of otherwise energetically unfavorable symmetrical dumbbell structures upon spindle elongation. The model allows us to explain the appearance of abnormal asymmetric shapes in fission yeast mutants with mis-segregated chromosomes as well as with altered nuclear membrane composition.

Sorting of integral membrane proteins mediated by curvature-dependent protein-lipid bilayer interaction.

Cell membrane proteins, both bound and integral, are known to preferentially accumulate at membrane locations with curvatures favorable to their shape. This is mainly due to the curvature dependent interaction between membrane proteins and their lipid environment. Here, we analyze the effects of the protein-lipid bilayer interaction energy due to mismatch between the protein shape and the principal curvatures of the surrounding bilayer. The role of different macroscopic parameters that define the interaction energy term is elucidated in relation to recent experiment in which the lateral distribution of a membrane embedded protein potassium channel KvAP is measured on a giant unilamellar lipid vesicle (reservoir) and a narrow tubular extension - a tether - kept at constant length. The dependence of the sorting ratio, defined as the ratio between the areal density of the protein on the tether and on the vesicle, on the inverse tether radius is influenced by the strength of the interaction, the intrinsic shape of the membrane embedded protein, and its abundance in the reservoir. It is described how the values of these constants can be extracted from experiments. The intrinsic principal curvatures of a protein are related to the tether radius at which the sorting ratio attains its maximum value. The estimate of the principal intrinsic curvature of the protein KvAP, obtained by comparing the experimental and theoretical sorting behavior, is consistent with the available information on its structure.

Direct and remote constriction of membrane necks.

The physical properties of membrane necks are relevant in vesiculation, a process that plays an essential role in cellular physiology. Because the neck's radius is, in general, finite, membrane scission and the consequent pinching off of the vesicle can only occur if it is narrowed to permit the necessary membrane topological reformation. Here we examine, in a simple single phase lipid vesicle, how external forces can promote neck constriction not only by direct compression at the neck but also, counterintuitively, by dilation at remote locations. These results provide a new perspective on the role played by actin polymerization in the process of endocytosis.

Partitioning of oleic acid into phosphatidylcholine membranes is amplified by strain.

Partitioning of fatty acids into phospholipid membranes is studied on giant unilamellar vesicles (GUVs) utilizing phase-contrast microscopy. With use of a micropipet, an individual GUV is transferred from a vesicle suspension in a mixed glucose/sucrose solution into an isomolar glycerol solution with a small amount of oleic acid added. Oleic acid molecules intercalate into the phospholipid membrane and thus increase the membrane area, while glycerol permeates into the vesicle interior and thus via osmotic inflation causes an increase of the vesicle volume. The conditions are chosen at which a vesicle swells as a sphere. At sufficiently low oleic acid concentrations, when the critical membrane strain is reached, the membrane bursts and part of vesicle content is ejected, upon which the membrane reseals and the swelling commences again. The radius of the vesicle before and after the burst is determined at different concentrations of oleic acid in suspension. The results of our experiments show that the oleic acid partitioning increases when the membrane strain is increased. The observed behavior is interpreted on the basis of a tension-dependent intercalation of oleic acid into the membrane.

Positioning of integrin ß1, caveolin-1 and focal adhesion kinase on the adhered membrane of spreading cells.

We have investigated the relationship between the spreading of anchorage-dependent cells and the surface-density distribution of plasma membrane adhesion proteins. The surface positioning and density of integrin ß1, caveolin-1 (cav-1), the phosphorylated caveolin-1 (p-cav-1) and the focal adhesion kinase (FAK) located on the adhering cell membrane (ACM) of HUVEC cells was studied. Imaging with TIRF microscopy was used, which enabled us to observe a few-nanometers-thin section of the cell above the plasma membrane in combination with image-based analyses. Integrin ß1 and cav-1 have spatial interdependence on the ACM. Cells treated with substances that act on cell spreading caused changes in the size of the ACM area, as well as a redistribution of several proteins under investigation. Changes to the ACM area correlated positively with those to the surface density of the cav-1. The high integrin ß1 and the low cav-1 surface density, and vice versa, following the treatments show that the presence of one of them not only spatially excludes, but also reduces, the occurrence of the other protein on the ACM, which indicates a regulative mechanism between integrin ß1 and cav-1.

Cellular life could have emerged from properties of vesicles.

It is indicated why it is plausible to assume that the initiation of cellular life was based on properties of vesicles. Vesicle properties relevant for the process of vesicle self-reproduction are revealed. Some open questions related to the idea that vesicle self-reproduction is an evolutionary process that includes the elements of the Darwinian selection are put forward, and some suggestions are made for possible directions of further research.

The dynamics of melittin-induced membrane permeability.

The transport of co-encapsulated solutes through the melittin-induced pores in the membrane of giant phospholipid vesicles was studied, and the characteristics of the pore formation process were modeled. Molecules of two different sizes (dextran and the smaller, fluorescent marker Alexa Fluor) were encapsulated inside the vesicles. The chosen individual vesicles were then transferred by micromanipulation from the stock suspension to the environment with the melittin (MLT). The vesicles were observed optically with a phase-contrast microscope and by monitoring the fluorescence signal. Such an experimental setup enabled an analysis of a single vesicle's response to the MLT on the basis of simultaneous, separate measurements of the outflow of both types of encapsulated molecules through the MLT-induced pores in the membrane. The mechanisms of the MLT's action were suggested in a model for MLT pore formation, with oligomeric pores continuously assembling and dissociating in the membrane. Based on the model, the results of the experiments were explained as a consequence of the membrane's permeability dynamics, with a continuously changing distribution of pores in the membrane with regard to their size and number. The relatively stable "average MLT pore" characteristics can be deduced from the proposed model.

Stress-free state of the red blood cell membrane and the deformation of its skeleton.

The response of a red blood cell (RBC) to deformation depends on its membrane, a composite of a lipid bilayer and a skeleton, which is a closed, two-dimensional network of spectrin tetramers as its bonds. The deformation of the skeleton and its lateral redistribution are studied in terms of the RBC resting state for a fixed geometry of the RBC, partially aspirated into a micropipette. The geometry of the RBC skeleton in its initial state is taken to be either two concentric circles, a references biconcave shape or a sphere. It is assumed that in its initial state the skeleton is distributed laterally in a homogeneous manner with its bonds either unstressed, presenting its stress-free state, or prestressed. The lateral distribution was calculated using a variational calculation. It was assumed that the spectrin tetramer bonds exhibit a linear elasticity. The results showed a significant effect of the initial skeleton geometry on its lateral distribution in the deformed state. The proposed model is used to analyze the measurements of skeleton extension ratios by the method of applying two modes of RBC micropipette aspiration.

Red blood cell shape and deformability in the context of the functional evolution of its membrane structure.

It is proposed that it is possible to identify some of the problems that had to be solved in the course of evolution for the red blood cell (RBC) to achieve its present day effectiveness, by studying the behavior of systems featuring different, partial characteristics of its membrane. The appropriateness of the RBC volume to membrane area ratio for its circulation in the blood is interpreted on the basis of an analysis of the shape behavior of phospholipid vesicles. The role of the membrane skeleton is associated with preventing an RBC from transforming into a budded shape, which could form in its absence due to curvature-dependent transmembrane protein-membrane interaction. It is shown that, by causing the formation of echinocytes, the skeleton also acts protectively when, in vesicles with a bilayer membrane, the budded shapes would form due to increasing difference between the areas of their outer and inner layers.

Effects of the pore-forming agent nystatin on giant phospholipid vesicles.

The effects of the polyene pore-forming agent nystatin were investigated on individual giant unilamellar phospholipid vesicles (GUVs), made of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), in different methanol-water solutions using phase-contrast optical microscopy. Three characteristic effects were detected in three different nystatin concentration ranges: vesicle shape changes (between 150 and 250µM); transient, nonspecific, tension pores (between 250 and 400µM); and vesicle ruptures (above 400µM). Both the appearance of the transient tension pores and the vesicle ruptures were explained as being a consequence of the formation of size-selective nystatin channels, whose membrane area density increases with the increasing nystatin concentrations. Our results also show that nystatin is able to form pores in the absence of sterols. In addition, study of the cross-interactions between nystatin and methanol revealed mutually antagonizing effects on the vesicle behavior for methanol volume fractions higher than 10%.

Budding of giant unilamellar vesicles induced by an amphitropic protein ß2-glycoprotein I.

ß(2)-glycoprotein I (ß(2)GPI) is a plasma protein capable of binding reversibly to membranes, and is classified among the amphitropic proteins. Part of the protein intercalates into the outer membrane leaflet, altering the difference between the preferred areas of the membrane leaflets, which results in membrane shape transformations. Budding, as a specific example of such shape transformations, was studied using giant unilamellar vesicles. Our aim was to identify the vesicle parameters that influence the degree of membrane budding by studying this process qualitatively and quantitatively. A simple theoretical model has been developed and assessed against the experimental observations. The results show that ß(2)GPI binds in a concentration dependent manner, causing transitions between vesicle shapes with increasing numbers of buds. Higher numbers of buds are characteristic of larger and/or more flaccid vesicles. When the vesicle membrane is strained, a higher ß(2)GPI concentration is needed to produce the same effects as on the unstrained vesicle. Vesicles were found to be highly individual in their behaviour, so each was treated individually. Specific vesicle behaviour was found to be the consequence of the neck between the main vesicle body and the buds, which could be either open, closed for the exchange of solution, or closed for the exchange of both solution and membrane.

Vesicle budding and the origin of cellular life.

This Minireview provides an appropriate opportunity to demonstrate the connection between the results of some early experimental and theoretical investigations of vesicle budding and the more recent application of the concepts developed there to the process of vesicle self-reproduction. Herein, we also explain why vesicle budding could have preceded the establishment of cellular life.

Comment on "Thermodynamics of vesicle growth and instability".

Fanelli and McKane [Phys. Rev. E 78, 051406 (2008)] recently described the growth of vesicles due to the accretion of lipid molecules onto their surface in terms of linear irreversible thermodynamics. They calculated the critical radius at which the shape of a spherical vesicle becomes unstable. Their treatment is different from those previously put forward and, in the following, we explain why we regard their thermodynamic description to be deficient.

Equilibrium mechanics of monolayered epithelium.

In order to fully understand the epithelial mechanics it is essential to integrate different levels of epithelial organization. In this work, we propose a theoretical approach for connecting the macroscopic mechanical properties of a monolayered epithelium to the mechanical properties at the cellular level. The analysis is based on the established mechanical models-at the macroscopic scale the epithelium is described within the mechanics of thin layers, while the cellular level is modeled in terms of the cellular surface (cortical) tension and the intercellular adhesion. The macroscopic elastic energy of the epithelium is linked to the energy of an average epithelial cell. The epithelial equilibrium state is determined by energy minimization and the macroscopic elastic moduli are calculated from deformations around the equilibrium. The results indicate that the epithelial equilibrium state is defined by the ratio between the adhesion strength and the cellular surface tension. The lower and the upper bounds for this ratio are estimated. If the ratio is small, the epithelium is cuboidal, if it is large, the epithelium becomes columnar. Importantly, it is found that the cellular cortical tension and the intercellular adhesion alone cannot produce the flattened squamous epithelium. Any difference in the surface tension between the apical and basal cellular sides bends the epithelium towards the side with the larger surface tension. Interestingly, the analysis shows that the epithelial area expansivity modulus and the shear modulus depend only on the cellular surface tension and not on the intercellular adhesion. The results are presented in a general analytical form, and are thus applicable to a variety of monolayered epithelia, without relying on the specifics of numerical finite-element methods. In addition, by using the standard theoretical tools for multi-laminar systems, the results can be applied to epithelia consisting of layers with different mechanical properties.