Equilibrium shapes of erythrocytes in rouleau formation

Mechanical model of Golgi apparatus

The Golgi apparatus has intrigued researchers since its discovery and despite the advances, there are still many open questions in regards to its shape and function. We propose a mechanical model of Golgi apparatus stack and explain its most elementary geometrical properties: the equilibrium number of cisternae, the stack size, and its general equilibrium shape. Combining both analytical and numerical methods we successfully reconstruct the stack morphology within the theory of bending elasticity. We demonstrate that energy-wise the stack prefers an overall bent shape and show strong evidence that the adhesion strength determines the equilibrium number of cisternae per stack. We explore the morphological role of fenestrations and discuss their impact on the overall stack structure. We also comment on the effects of the asymmetry in the composition of membrane leaflets on the shape of the cisternae and thus offer a broad steady-state study of the stack morphology and present a method that can be used also for other membrane-bound organelles.

Characteristics of phospholipid vesicles enhanced by adhesion on an annular region

Phospholipid vesicle membranes are simple models used to study the mechanical properties of cell membranes. The shapes of flaccid vesicles can exhibit very diverse forms. When researching very flaccid vesicles, axisymmetrical vesicles with the membranes adhered to an annular region can also be observed. A phase diagram of such shapes was studied for different values of the vesicle parameters, i.e., the adhesion constant, the vesicle volume-to-membrane ratio, the volume ratio between the polar and the equatorial parts, and the equilibrium difference between the membrane monolayers. The energies of the annular shapes with respect to the vesicle parameters were closely examined and compared with the energies of the discocyte and stomatocyte shapes. The requirements for the existence of such annular shapes were also given for adhesion-free vesicle membranes. The results show that the adhesion between the lipid bilayers stabilizes the observed shapes, which belong to the locally stable branch of the annular vesicles. The value obtained for the adhesion constant of the SOPC membrane is 3×10^{-9}J/m^{2}.

On the Mechanism of Human Red Blood Cell Longevity: Roles of Calcium, the Sodium Pump, PIEZO1, and Gardos Channels

In a healthy adult, the transport of O₂ and CO₂ between lungs and tissues is performed by about 2 · 10¹³ red blood cells, of which around 1.7 · 10¹¹ are renewed every day, a turnover resulting from an average circulatory lifespan of about 120 days. Cellular lifespan is the result of an evolutionary balance between the energy costs of maintaining cells in a fit functional state versus cell renewal. In this Review we examine how the set of passive and active membrane transporters of the mature red blood cells interact to maximize their circulatory longevity thus minimizing costs on expensive cell turnover. Red blood cell deformability is critical for optimal rheology and gas exchange functionality during capillary flow, best fulfilled when the volume of each human red blood cell is kept at a fraction of about 0.55-0.60 of the maximal spherical volume allowed by its membrane area, the optimal-volume-ratio range. The extent to which red blood cell volumes can be preserved within or near these narrow optimal-volume-ratio margins determines the potential for circulatory longevity. We show that the low cation permeability of red blood cells allows volume stability to be achieved with extraordinary cost-efficiency, favouring cell longevity over cell turnover. We suggest a mechanism by which the interplay of a declining sodium pump and two passive membrane transporters, the mechanosensitive PIEZO1 channel, a candidate mediator of P_sickle in sickle cells, and the Ca²⁺-sensitive, K⁺-selective Gardos channel, can implement red blood cell volume stability around the optimal-volume-ratio range, as required for extended circulatory longevity.

Role of red blood cell elastic properties in capillary occlusions

The shape transformations of red blood cells stuck in capillary narrowings with radii close to the critical radius where the maximum deformations occur are analyzed. The membrane skeleton deformations are studied within the effective network model and the continuum elastic model, whereas the area-difference elasticity model is applied to describe the phospholipid bilayer. A minimization of the total free energy is performed to determine the cell shapes in a stopped flow, which are calculated by a triangulated representation of the membrane surface. The shapes are asymmetric, characterized by a single invagination, which decreases with decreasing radii of the narrowing and vanishes at its critical radius. The largest stretching deformations of the skeleton are at the ends of the elongated shape, and remarkable shear deformations appear around the invagination. The membrane's mechanical energy increases with the decreasing radius of the narrowing, predominantly due to the deformation of membrane skeleton. The increase in the shear energy is significantly larger than any other energy contribution within both models. The pressure differences needed for the penetration into the narrowing are strongly coupled with the membrane's mechanical energy. Their values were found to be of the order of 10 Pa. Both models correspond well.

Shapes of discoid intracellular compartments with small relative volumes

A prominent feature of many intracellular compartments is a large membrane surface area relative to their luminal volume, i.e., the small relative volume. In this study we present a theoretical analysis of discoid membrane compartments with a small relative volume and then compare the theoretical results to quantitative morphological assessment of fusiform vesicles in urinary bladder umbrella cells. Specifically, we employ three established extensions of the standard approach to lipid membrane shape calculation and determine the shapes that could be expected according to three scenarios of membrane shaping: membrane adhesion in the central discoid part, curvature driven lateral segregation of membrane constituents, and existence of stiffer membrane regions, e.g., support by protein scaffolds. The main characteristics of each scenario are analyzed. The results indicate that even though all three scenarios can lead to similar shapes, there are values of model parameters that yield qualitatively distinctive shapes. Consequently, a distinctive shape of an intracellular compartment may reveal its membrane shaping mechanism and the membrane structure. The observed shapes of fusiform vesicles fall into two qualitatively different classes, yet they are all consistent with the theoretical results and the current understanding of their structure and function.

Cellular uptake of elastic nanoparticles

A fundamental understanding of cell-nanomaterial interaction is of essential importance to nanomedicine and safe applications of nanotechnology. Here we investigate the adhesive wrapping of a soft elastic vesicle by a lipid membrane. We show that there exist a maximum of five distinct wrapping phases based on the stability of full wrapping, partial wrapping, and no wrapping states. The wrapping phases depend on the vesicle size, adhesion energy, surface tension of membrane, and bending rigidity ratio between vesicle and membrane. These results are of immediate interest to the study of vesicular transport and endocytosis or phagocytosis of elastic particles into cells.

Periodic three-dimensional assemblies of polyhedral lipid vesicles

We theoretically study the structure of periodic bulk assemblies of identical lipid vesicles. In our model, each vesicle is represented as a convex polyhedron with flat faces, rounded edges, and rounded vertices. Each vesicle carries an elastic and an adhesion energy and in the limit of strong adhesion, the minimal-energy shape of cells minimizes the weighted total edge length. We calculate exactly the shape of the rounded edge and show that it can be well described by a cylindrical surface. By comparing several candidate space-filling polyhedra, we find that the oblate shapes are preferred over prolate shapes for all volume-to-surface ratios. We also study periodic assemblies of vesicles whose adhesion strength on lateral faces is different from that on basal or apical faces. The anisotropy needed to stabilize prolate shapes is determined and it is shown that, at any volume-to-surface ratio, the transition between oblate and prolate shapes is very sharp. The geometry of the model vesicle assemblies reproduces the shapes of cells in certain simple animal tissues.

Degenerate polygonal tilings in simple animal tissues

The salient feature of one-cell-thick epithelia is their en face view, which reveals the polygonal cross section of the close-packed prismatic cells. The physical mechanisms that shape these tissues were hitherto explored using theories based on cell proliferation, which were either entirely topological or included certain morphogenetic forces. But mitosis itself may not be instrumental in molding the tissue. We show that the structure of simple epithelia can be explained by an equilibrium model where energy-degenerate polygons in an entropy-maximizing tiling are described by a single geometric parameter encoding their inflation. The two types of tilings found numerically--ordered and disordered--closely reproduce the patterns observed in Drosophila, Hydra, and Xenopus and they generalize earlier theoretical results. Free of a specific cell self-energy, cell-cell interaction, and cell division kinetics, our model provides an insight into the universality of living and inanimate two-dimensional cellular structures.

Amphipath-induced nanoscale changes in outer hair cell plasma membrane curvature

Outer hair cell (OHC) electromotility enables frequency selectivity and sensitivity in mammalian audition. Electromotility is generated by the transmembrane protein prestin and is sensitive to amphipathic compounds including salicylate, chlorpromazine (CPZ), and trinitrophenol (TNP). Although these compounds induce observable membrane curvature changes in erythrocytes, their effects on OHC membrane curvature are unknown. In this work, fluorescence polarization microscopy was applied to investigate the effects of salicylate, CPZ, and TNP on di-8-ANEPPS orientation in the OHC plasma membrane. Our results demonstrate the ability of fluorescence polarization microscopy to measure amphipath-induced changes in di-8-ANEPPS orientation, consistent with nanoscale changes in membrane curvature between regularly spaced proteins connecting the OHC plasma membrane and cytoskeleton. Simultaneous application of oppositely charged amphipaths generally results in no net membrane bending, consistent with predictions of the bilayer couple hypothesis; however, the application of salicylate (10 mM), which inhibits electromotility, is not reversed by the addition of CPZ. This result supports other findings that suggest salicylate primarily influences electromotiliy and OHC nonlinear capacitance via a direct interaction with prestin. In contrast, we find that CPZ and TNP influence the voltage sensitivity of prestin via membrane bending, demonstrating the mechanosensitivity of this unique membrane motor protein.

Morphology of small aggregates of red blood cells

Blood can be considered a two-phase liquid composed of plasma as well as cells and cell aggregates. The degree of cell aggregation is an important determinant of blood rheology: The size and shape of the aggregates affect blood viscosity. The microscopic mechanisms of red blood cell adhesion involve a complex interplay of electrostatic, van der Waals, and a range of specific biochemical inter-membrane interactions. Here we use an effective model of these interactions combined with the membrane elasticity theory to calculate the equilibrium shape of a red blood cell doublet and compare it with the experimentally observed red blood cell aggregates both in vitro and in vivo. Special attention is devoted to the shape of doublets formed by dissimilar cells. A possible effect of doublet shape on pathways of the formation of multicellular aggregates is discussed. Red blood cell rouleau formation is expected to take place at intermediate adhesion strengths where the outer doublet surfaces are either concave or flat, whereas in the strong-adhesion regime where the outer doublet surfaces are convex the cells should form rounded clump-like aggregates.

Aggregates of two-dimensional vesicles: rouleaux, sheets, and convergent extension

Using both numerical and variational minimization of the bending and adhesion energy of two-dimensional lipid vesicles, we study their aggregation, and we find that the stable aggregates include an infinite number of vesicles and that they arrange either in a columnar or in a sheetlike structure. We calculate the stability diagram and we show that the sheetlike aggregate can be transformed into the columnar aggregate via vesicle intercalation, which makes the transformation reminiscent of the process of convergent extension observed in some biological tissues.

Contact lines for fluid surface adhesion

When a fluid surface adheres to a substrate, the location of the contact line adjusts in order to minimize the overall energy. This implies boundary conditions which depend on the characteristic surface deformation energies. We develop a general geometrical framework within which these conditions can be derived in a completely systematic way. We treat both adhesion to a rigid substrate and adhesion between two fluid surfaces, and illustrate our general results for several important Hamiltonians involving both curvature and curvature gradients. Some of these have previously been studied using very different techniques. With the exception of capillary phenomena, the Hamiltonian will not only be sensitive to boundary translations, but may also respond to changes in slope and even in curvature. The functional form of the additional contributions will follow readily from our treatment. We will show that the boundary conditions describing adhesion between two fluid surfaces express the balance of stresses and torques, as one would expect. At a rigid substrate, however, this simple identification will generally fail. This is because local rotations of the surface normal will be entirely "enslaved" to translations on the substrate. As a consequence, stresses and torques enter a single balance condition and cannot be disentangled.

Flat and sigmoidally curved contact zones in vesicle-vesicle adhesion

Using the membrane-bending elasticity theory and a simple effective model of adhesion, we study the morphology of lipid vesicle doublets. In the weak adhesion regime, we find flat-contact axisymmetric doublets, whereas at large adhesion strengths, the vesicle aggregates are nonaxisymmetric and characterized by a sigmoidally curved, S-shaped contact zone with a single invagination and a complementary evagination on each vesicle. The sigmoid-contact doublets agree very well with the experimentally observed shapes of erythrocyte aggregates. Our results show that in identical vesicles with large to moderate surface-to-volume ratio, the sigmoid-contact shape is the only bound morphology. We also discuss the role of sigmoid contacts in the formation of multicellular aggregates such as erythrocyte rouleaux.

Computational fluid dynamic simulation of aggregation of deformable cells in a shear flow

We present computational fluid dynamic (CFD) simulation of aggregation of two deformable cells in a shearflow. This work is motivated by an attempt to develop computational models of aggregation of red blood cells (RBCs). Aggregation of RBCs is a major determinant of blood viscosity in microcirculation under physiological and pathological conditions. Deformability of the RBCs plays a major role in determining their aggregability. Deformability depends on the viscosity of the cytoplasmic fluid and on the rigidity of the cell membrane, in a macroscopic sense. This paper presents a computational study of RBC aggregation that takes into account the rheology of the cells as well as cell-cell adhesion kinetics. The simulation technique considered here is two dimensional and based on the front tracking/immersed boundary method for multiple fluids. Results presented here are on the dynamic events of aggregate formation between two cells, and its subsequent motion, rolling, deformation, and breakage. We show that the rheological properties of the cells have significant effects on the dynamics of the aggregate. A stable aggregate is formed at higher cytoplasmic viscosity and membrane rigidity. We also show that the bonds formed between the cells change in a cyclic manner as the aggregate rolls in a shearflow. The cyclic behavior is related to the rolling orientation of the aggregate. The frequency and amplitude of oscillation in the number of bonds also depend on the rheological properties.

Physical factors that affect the number and size of Golgi cisternae

Despite continual membrane reorganization in the Golgi complex, the number of cisternae in a Golgi stack is a stable parameter. The cisternal number is conserved within any given cell line and also after Golgi reassembly, e.g. following brefeldin-A-induced disruption. However, the factors that determine the cisternal number in a single Golgi stack remain to be fully determined. We propose a simple mechanical model of the Golgi stack and present a theoretical analysis of different physical factors that may affect the number of cisternae in a Golgi stack. The model takes into account the Golgi membrane bending elasticity, which is related to the membrane curvature, and the adhesion, which holds the cisternae together. The analysis shows that the equilibrium configuration of the Golgi stack can be regarded as a balance between these two effects - the adhesion, which tends to increase the number of cisternae, is opposed by the membrane resistance to bending, which does not favor highly curved cisternal rims. The adhesion strength that is needed to hold together a typical stack is calculated. In addition, the model is used to analyze changes in the cisternal numbers as a controlled traffic wave enters a Golgi stack and increases the amount of the membrane in that stack.

Erythrocyte rouleau formation under polarized electromagnetic fields

We study the influence of an external electromagnetic field of 1.8 GHz in the formation or disaggregation of long rouleau of identical erythrocyte cells. In particular we calculate the variation of the transmembrane potential of an individual erythrocyte illuminated by the external field due to the presence of the neighboring erythrocytes in the rouleau, and compare the total electric energy of isolated cells with the total electric energy of the rouleau. We show that the polarization of the external electromagnetic field plays a fundamental role in the total energy variation of the cell system, and consequently in the formation or disaggregation of rouleau.