^{1}

^{2}

^{*}

^{1}

^{2}

Edited by: Lars Kaestner, Saarland University, Germany

Reviewed by: Chaouqi Misbah, UMR 5588 Laboratoire Interdisciplinaire de Physique (LIPhy), France; Dmitry A. Fedosov, Helmholtz-Verband Deutscher Forschungszentren (HZ), Germany

This article was submitted to Red Blood Cell Physiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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.

The red blood cell (RBC) shape is, basically, assumed to depend on the cohesion and mechanical stability of its membrane (^{++} ions that enter the cell when it is open (

To understand a given cell process it is necessary to identify the structural elements responsible and to provide a description of the mode of their operation. The corresponding theoretical studies are aimed at obtaining their structure–function relationship in a quantitative manner. This task is, in general, difficult, since cells are complex. The only way to make progress is frequently by analyses of mathematical models. In modeling it is usually necessary first to identify the structural level that is proper for the description of different aspects and for the function of a treated physiological process, and then to reveal its essential features on the basis of the simplest possible system. Models are, as a rule, built on the basis of a set of assumptions that can then be tested experimentally. When these assumptions are found to be correct, and it is thus possible to obtain model predictions by exact either analytical or numerical calculations, a model becomes a theory. The modeling approach should be distinguished from the use of mathematics in the analysis of experimental results and from simulations where, on the basis of the already established theory, the system’s behavior can be described mathematically in an exact manner. When modeling the behavior of whole cells it is advantageous to study those that are simple. RBCs, although composed of several thousand different molecules and ions, are, in some aspects, extremely simple. Basically, they are constituted by a concentrated hemoglobin solution enclosed by an essentially smooth membrane. Moreover, they also have a well-defined main function of carrying respiratory gasses. Therefore, and because of its availability, the RBC served, and still serves, as an ideal system for developing the principles of modeling structure–function relationships in cell systems in general (

This review will be focused on the bases on which we recently developed a model of the role of Piezo1 in the regulation of the RBC volume (

The review is organized as follows. The treated model (

The function of RBC as the carrier of respiratory gasses led to its adoption throughout evolution of numerous specific mechanical and thermodynamic properties. In the absence of external forces, the normal RBCs of most vertebrates assume the shape of a disk that involves, at its poles, the presence of symmetrically indented dimples. RBCs are deformable, e.g., under microcirculatory flow conditions, at sufficiently high shear stress, deform into rolling stomatocytes and, finally, adopt polylobed shapes (

where _{s} = (^{1/2}. The mechanism for the establishment of RBC volume will be dealt with in section “RBC Volume and Related Aspects of the Variability of RBC Population.” Here our focus is on how, at a given value of

Red blood cell shape has been treated by assuming its membrane to be a single, thin, laterally homogeneous mechanical entity. RBC membrane can, at its reduced volume _{b}) that can be for symmetrical bilayer obtained by the integral of the square of the mean membrane curvature (_{1} + _{2})/2 where _{1} and _{2} are the principal membrane curvatures) over the whole membrane area expressed as

with _{c} membrane bending constant. In general membrane bending energy involves also a contribution due to Gaussian curvature (_{1}_{2}) (_{b} (Eq. 2) and found that, at _{0}). _{s}_{0}) beside the discocyte also several other shapes including cup shaped stomatocytes. At about the same time

The bilayer couple theory (_{s} with _{s}, as already defined, the radius of the sphere). The geometrical meaning of Δ_{s}

Demonstration of basic features of the bilayer couple theories of vesicle shapes [_{b} (defined relative to the bending energy of the sphere which is 8π_{c}) calculated at _{0} and _{r}/_{c} = 3 (reprinted with permission from

The described predictions of the bilayer couple model are strictly only valid if the two equally composed leaflets of a bilayer are incompressible. In reality they are compressible and therefore it has to be taken into account that, in general, in a given shape, they might be deformed differently, for example in that the area of one is extended and of the other compressed. In such cases the reduced area difference (Δ_{0}) which corresponds to the situation where leaflets are neither extended nor compressed. The bilayer thus exhibits, in addition to the already defined bending energy (Eq. 2), also the non-local bending energy (_{k}) (_{k} = _{k}/8π_{c}) as

where _{r} is the non-local bending constant. The derivation and consequences of non-local bending energy were comprehensively reviewed in

The solution of Eq. 4 for its unknown Δ_{0}, be obtained graphically as a point on the graph of _{b}/∂Δ_{0} depends on the slope of dashed curves that is proportional to the ratio _{r}/_{c}. There is only one solution if this slope is steeper than that of the largest derivative by Δ_{b}/∂Δ_{r}/_{c} that are smaller than above defined critical value of ∂_{b}/∂Δ_{0} (e.g., 0.82 in _{0}, a discontinuous shape transformation from the Δ_{r}/_{c} shape phase diagram. The example of the cross-section of this diagram is for _{b}/∂Δ_{r}/_{c} for RBC membrane is about 2 (_{0}. _{0} and Δ_{0}, in spite of having different physical background affect the shapes of vesicular objects with bilayer membranes in a similar manner. The stationary shapes obtained by solving the shape equation are the same as in the strict bilayer couple model if for the reduced equilibrium area difference is taken an effective one defined as

It has to be noted that in this case the region of stable shapes in the generalized shape phase diagram _{0,eff} – _{r}/_{c} depends on the relative contribution to Δ_{0,eff} of Δ_{0} and _{0}. It is because the energy term due to Δ_{0} (Eq. 3) involves Δ^{2} whereas the energy term due to _{0} is a linear function of Δ_{0}, the region of stable shapes is diminishing. The limit _{r}/_{c} = 0 represents the spontaneous curvature model of _{0} = Δ

Red blood cell membrane is, compared to phospholipid membranes, complex. Its bilayer part is crowded with integral membrane proteins such as band3 that is involved in RBC’s function of carrying carbon dioxide, different pumps and channels that take care of the establishment of RBC volume, and many other proteins serving in its protection (

Red blood cell membrane exhibits shear elasticity. Because its bilayer part can be considered as two-dimensional liquid, the shear elasticity can be ascribed solely to its membrane skeleton which is a two-dimensional pseudo-hexagonal network of spectrin tetramers as bonds and acting filaments as nodes. To understand the skeleton behavior it is crucial to realize that RBC membrane deformation may cause an alteration of local skeleton densities while the density of the lipids remains the same, as was observed by measuring skeleton lateral distribution in RBC partially aspirated into the micropipette (_{i}, the ratio between the final length and the initial length of the deformed skeleton material in the i-th direction) may reach a value of about 3, which corresponds to a fully extended spectrin tetramer of length ∼ 200 nm. Skeleton deformation may be described by shear and area compressibility deformational modes (

Deformation of RBC skeleton. _{p} ≈ 2 μm) (from _{p} (dashed) and along meridians _{m} (dots) and the density 1/_{p}_{m} (relative to its mean value; full line) calculated according to the described model (

Theoretical modeling of the RBC skeleton is developing in several different directions (e.g., _{0}(_{0} is the arc-length distance from the cell pole to a given contour point of the original skeleton state and _{0}(_{0} from the axis moves in the deformed state to the distance _{p} = _{0}. The corresponding compression (in case that _{0}) exerts a tendency to make _{m} larger than 1. The local magnitudes of the extensions along meridians are restricted by the requirement that the total skeleton area is constant. The effect of this requirement cannot be visualized so clearly; however, it can still be concluded that the skeleton prevents shape changes with significant changes of the distances of the membrane from the axis. In

Red blood cell shape behavior differs qualitatively from that of phospholipid vesicles in the region of the _{0}, the skeleton is not deformed so much because these two shapes are both oblate and the distances of skeleton elements from the axis in them do not differ appreciably. Therefore, the behavior of RBC shape in the “oblate” region of the

Another physiological role of the RBC membrane skeleton is that of the prevention of formation of risky budded shapes due to lateral segregation of its integral proteins. Membrane embedded proteins interact with the surrounding membrane when their intrinsic principal curvatures differ from those of the membrane at their location. For example, when the drastically curved protein Piezo1 (

Illustrations of effects of protein–membrane interaction.

where _{P,j} = (_{1,P,j} + _{2,P,j})/2 is the mean principal intrinsic curvature of the transmembrane part of the inclusion and Δ_{P,j} = (_{1,P,j} − _{2,P,j})/2 is a measure of the difference between its two principal curvatures. _{j} and _{j}^{∗} are independent interaction constants. The angle _{j} defines the mutual orientation of the coordinate systems of the intrinsic principal curvatures of the inclusion and the principal curvatures of the membrane. One consequence of such interaction term is curvature sensing, meaning that mobile membrane proteins, due to curvature dependent interaction energy term, accumulate in membrane regions where this mismatch is small and are depleted from regions where it is large. For example, it is reasonable to expect that it is more probable for the Piezo1, due to its curved structure, to reside in regions of RBC discocyte poles (dimples) than on its equator. The second possible consequence of the curvature dependent protein–membrane interaction is its effect on shape which, for a membrane with mobile proteins, corresponds to the minimum of the sum of their distributional free energy and the bending energy of the membrane (^{6} band3 proteins which could constitute a potential danger for the stability of the RBC if most of them were not linked to the skeleton.

Red blood cell membrane proteins that are not linked to the skeleton can, upon the deformation, redistribute over the membrane with a time constant that depends on their diffusion coefficient. The latter can be smaller than in a vesicle because of the corralling effect of the spectrin skeleton (

Red blood cell membrane is, as those of most mammalian cells, well permeable for water. Therefore the RBC’s water content, and thus also its volume, depend on its content of osmotically active substances and on the external tonicity (^{+} and Na^{+}, nevertheless kept at about 0.6. These cations are pumped by Na^{+}/K^{+}- ATPase which actively expels three sodium ions and takes in two potassium ions (^{+} and lower concentration of Na^{+}, both relative to their concentration in the environment, cause fluxes of these two cations in the direction of their concentration gradients. The pumping and leaking of K^{+} and Na^{+} eventually leads to the stationary volume level. There are several channels/transporters involved in the passive leakage of these two cations. The results of many studies of their action and of the data on cation pumping made it possible to formulate realistic mathematical modeling of RBC volume regulation (^{+} and Na^{+} attain their stationary value in a time scale which is several orders of magnitude larger than that of water and also of univalent anions Cl^{–} and HCO_{3}^{–} which can thus be treated at correspondingly short time scales as being in quasi-equilibrium between the inside and outside solutions. Because hemoglobin is charged, this equilibrium can be described by models that involve a version of the Donnan equilibrium in which cations do not exchange (

The issue here is the extension of already established models of RBC volume regulation that take into consideration the role of Piezo1 and Gárdos channels (_{A,V} ∼ 0.97 (^{2+} ionophore A23187, this correlation was lost (

Illustrations of consequences of the correlation between RBC area (_{v}) in dependence on the correlation coefficient _{A,V} obtained for the values of coefficients of variations of RBC volume and membrane area to be 0.12 and 0.13, respectively (reprinted with permission from _{v} at _{A,V} = 0.96 is about 0.06. ^{2+} ionophore A23187 (reprinted with permission from _{h}) obtained from steepness of osmotic fragility curves as indicated by red dashed lines.

The fact that RBC dehydration in hereditary xerocytosis can be caused by malfunctioning of a mechanosensitive protein Piezo1 indicates that RBC volume may also depend on mechanical properties of RBC membrane. Piezo1 system appeared to represent a relatively independent module of the otherwise complex regulation of RBC volume, and could thus be considered as an ideal candidate for application of the modeling approach. In the model under consideration (^{++} permeability on RBC discocyte shape. These ideas were supported by curved structure of the Piezo1 trimer, evidenced by its structural studies (

The essential ingredients of the proposed mechanism of the effect of Piezo1 on RBC volume are schematically represented by the cause-effect links shown in ^{+} and RBC volume is in a broad sense the consequence of the fact that RBC volume is established through osmotic equilibrium with the surrounding solution and that thus depends on the level of its cytoplasm cations. As discussed in section “RBC Volume and Related Aspects of the Variability of RBC Population” the regulation of cell cation content operates on the basis of active and passive membrane cation permeabilities (^{+}. The preceding two links are thus based on the experiments of ^{+} and Na^{+}. In the model it is assumed that, due to active Ca^{++} efflux, the Gárdos channels are on the average open only part of the time and that it is therefore reasonable to express membrane permeability coefficient for K^{+} (_{K}) as the sum two contributions

Schematic presentation of processes involved in the effect of RBC discocyte shape on RBC volume. The meanings of the links are described in the text. Because the volume affects the shape (dashed links) the described system as a whole represents a closed regulatory loop. It is indicated that Piezo1 Ca^{++} permeability can be affected either by membrane curvature or membrane lateral tension.

where _{K,G} is RBC K^{+} permeability of its Gárdos channels, _{G} the average fraction of them that are open, and _{K,0} the potassium permeability of its other K^{+} channels. Due to osmotic equilibrium between its interior and exterior (see section “RBC Volume and Related Aspects of the Variability of RBC Population”), RBC volume is at larger values of _{G} smaller. In the treated model we derived a relationship between _{G} and the reduced volume _{K,G}/_{K,0}, the relative amount of other RBC cytoplasm ingredients that cannot penetrate the membrane, and the reduced volume at _{G} = 0. The crucial task of the model was to reveal a plausible mechanism for the effect of RBC shape on the fraction of time that Piezo1 channels are open. In the model it was proposed that there is another relationship between _{G} and ^{++} permeability. The theory described in sub-section “Interpretation of RBC and Vesicle Shapes on the Basis of Membrane Bending” makes it possible to determine reduced mean membrane curvature (

where _{pole,r} is the reduced mean curvature at an arbitrarily chosen reference reduced volume _{r}. The value of the coefficient β_{pole} is 4.0. It was then taken into account that due to Piezo1 intrinsic curvature and its interaction with the membrane (Eq. 6), its molecules would tend to concentrate in the regions of RBC poles. On the basis of the assumption of that open Piezo1 conformation is less curved than its closed conformation it follows that, at the decrease of _{G}(_{G} and _{G}) and the RBC reduced volume (

The described model was meant primarily to serve as the proof of principle for Piezo1 based regulation of RBC volume. Therefore it involves many simplifications of the real system. For example, it was restricted to K^{+} homeostasis and did not take into consideration possible concomitant changes of RBC Na^{+} content; the fraction of open Piezo1 channels was calculated as if they would all be located at the RBC poles; it was assumed that there are only two relevant Piezo1 conformations, etc. However, some of the model predictions are general in that they do not depend on its specific features. The main outcome of the effect of the RBC discoid shape on its volume is that it implies the existence of a closed regulatory loop for RBC volume regulation. The Piezo1–Gárdos channel system can be considered as a complement to the mechanism for the regulation of cell volume that operates on the basis of active and passive membrane cation permeabilities (^{+} and Na^{+}, Piezo1 acts at the level of K^{+} efflux. The membrane permeability coefficient for K^{+} involves a contribution that depends on the RBC reduced volume ^{2+} ionophore A23187 which overrules its action (

The model presented here points to the involvement of the RBC discoid shape in the fine regulation of its volume in a rather consistent manner. However, there are still many unanswered questions that require further experimentation. One such concern is whether the response of Piezo1 to change of RBC shape is due to change of membrane curvature or to the change of membrane lateral tension (^{4} proteins per 1 μm^{2}) or because the Piezo1 molecule modifies the shape of the surrounding membrane to the size (

There are also many aspects of the proposed model that require further theoretical modeling. For example, there is the question as to what is causing, in the

The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author thanks Prof. Roger H. Pain for critical reading of the manuscript.