The human erythrocyte membrane skeleton may be an ionic gel. II. Numerical analyses of cell shapes and shape transformations

Systems biology and physical biology of clathrin-mediated endocytosis

In this review, we describe the application of experimental data and modeling of intracellular endocytic trafficking mechanisms with a focus on the process of clathrin-mediated endocytosis. A detailed parts-list for the protein-protein interactions in clathrin-mediated endocytosis has been available for some time. However, recent experimental, theoretical, and computational tools have proved to be critical in establishing a sequence of events, cooperative dynamics, and energetics of the intracellular process. On the experimental front, total internal reflection fluorescence microscopy, photo-activated localization microscopy, and spinning-disk confocal microscopy have focused on assembly and patterning of endocytic proteins at the membrane, while on the theory front, minimal theoretical models for clathrin nucleation, biophysical models for membrane curvature and bending elasticity, as well as methods from computational structural and systems biology, have proved insightful in describing membrane topologies, curvature mechanisms, and energetics.

Landscape of finite-temperature equilibrium behaviour of curvature-inducing proteins on a bilayer membrane explored using a linearized elastic free energy model

Using a recently developed multiscale simulation methodology, we describe the equilibrium behaviour of bilayer membranes under the influence of curvature-inducing proteins using a linearized elastic free energy model. In particular, we describe how the cooperativity associated with a multitude of protein-membrane interactions and protein diffusion on a membrane-mediated energy landscape elicits emergent behaviour in the membrane phase. Based on our model simulations, we predict that, depending on the density of membrane-bound proteins and the degree to which a single protein molecule can induce intrinsic mean curvature in the membrane, a range of membrane phase behaviour can be observed including two different modes of vesicle-bud nucleation and repressed membrane undulations. A state diagram as a function of experimentally tunable parameters to classify the underlying states is proposed.

Spectrin tethers and mesh in the biosynthetic pathway

The paradox of how the Golgi and other organelles can sort a continuous flux of protein and lipid but maintain temporal and morphological stability remains unresolved. Recent discoveries highlight a role for the cytoskeleton in guiding the structure and dynamics of organelles. Perhaps one of the more striking, albeit less expected, of these discoveries is the recognition that a spectrin skeleton associates with many organelles and contributes to the maintenance of Golgi structure and the efficiency of protein trafficking in the early secretory pathway. Spectrin interacts directly with phosphoinositides and with membrane proteins. The small GTPase ARF, a key player in Golgi dynamics, regulates the assembly of the Golgi spectrin skeleton through its ability to control phosphoinositide levels in Golgi membranes, whereas adapter molecules such as ankyrin link spectrin to other membrane proteins. Direct interactions of spectrin with actin and centractin (ARP1) provide a link to dynein, myosin and presumably other motors involved with intracellular transport. Building on the recognized ability of spectrin to organize macromolecular complexes of membrane and cytosolic proteins into a multifaceted scaffold linked to filamentous structural elements (termed linked mosaics), recent evidence supports a similar role for spectrin in organelle function and the secretory pathway. Two working models accommodate much of the available data: the Golgi mesh hypothesis and the spectrin ankyrin adapter protein tethering system (SAATS) hypothesis.

Stability of spiculated red blood cells induced by intercalation of amphiphiles in cell membrane

The stability of spiculated red blood cells, induced by intercalation of amphiphilic molecules into the cell membrane, is studied. It is assumed that the stable red blood cell shape corresponds to the minimum of its membrane elastic energy, which consists of the local and non-local bilayer bending energies and of the skeleton shear elastic energy. The cell, volume and the membrane area are kept constant. It is calculated that the number of spicules of the stable echinocytic shape is larger when the amphiphile concentration is higher, which is in agreement with experimental observations. Also, it is established that, in explaining the stability of the echinocytic shape of the red blood cell, it is necessary to include the membrane skeleton shear elasticity.

A possible mechanism determining the stability of spiculated red blood cells

In this work the stability of spiculated red blood cells, called echinocytes, is studied. It is assumed that the stable echinocyte shape corresponds to the minimum of its membrane elastic energy. It is shown that if the membrane skeleton shear elasticity is not taken into account the calculated stable echinocyte shapes always have only one spicule. However, by considering the skeleton shear elastic energy also, the calculated stable echinocyte shapes have many spicula in agreement with experimental observations.

Shapes of bilayer vesicles with membrane embedded molecules

The interdependence of the lateral distribution of molecules which are embedded in a membrane (such as integral membrane proteins) and the shape of a cell with no internal structure (such as phospholipid vesicles or mammalian erythrocytes) has been studied. The coupling of the lateral distribution of the molecules and the cell shape is introduced by considering that the energy of the membrane embedded molecule at a given site of the membrane depends on the curvature of the membrane at that site. Direct interactions between embedded molecules are not considered. A simple expression for the interaction of the membrane embedded molecule with the local membrane curvature is proposed. Starting from this interaction, the consistently related expressions for the free energy and for the distribution function of the embedded molecules are derived. The equilibrium cell shape and the corresponding lateral distribution of the membrane embedded molecules are determined by minimization of the membrane free energy which includes the free energy of the membrane embedded molecules and the membrane elastic energy. The resulting inhomogeneous distribution of the membrane embedded molecules affects the cell shape in a nontrivial manner. In particular, it is shown that the shape corresponding to the absolute energy minimum at given cell volume and membrane area may be elliptically non-axisymmetric, in contrast to the case of a laterally homogeneous membrane where it is axisymmetric.

Elastic energy of curvature-driven bump formation on red blood cell membrane

Model calculations were performed to explore quantitative aspects of the discocyte-echinocyte shape transformation in red blood cells. The shape transformation was assumed to be driven by changes in the preferred curvature of the membrane bilayer and opposed by the elastic shear rigidity of the membrane skeleton. The energy required for echinocyte bump formation was calculated for a range of bump shapes for different preferred curvatures. Energy minima corresponding to nonzero bump heights were found when the stress-free area difference between the membrane leaflets or the spontaneous curvature of the membrane became sufficiently large, but the calculations predict that the membrane can tolerate significant differences in the resting areas of the inner and outer leaflets or significant spontaneous curvature without visible changes in shape. Thus, if the cell is near the threshold for bump formation, the calculations predict that small changes in membrane properties would produce large changes in cellular geometry. These results provide a rational framework for interpreting observations of shape transformations in red cells and for understanding the mechanism by which small changes in membrane elastic properties might lead to significant changes in geometry.

Remodeling the shape of the skeleton in the intact red cell

The role of the membrane skeleton in determining the shape of the human red cell was probed by weakening it in situ with urea, a membrane-permeable perturbant of spectrin. Urea by itself did not alter the biconcave disk shape of the red cell; however, above threshold conditions (1.5 M, 37 degrees C, 10 min), it caused an 18% reduction in the membrane elastic shear modulus. It also potentiated the spiculation of cells by lysophosphatidylcholine. These findings suggest that the contour of the resting cell is not normally dependent on the elasticity of or tension in the membrane skeleton. Rather, the elasticity of the skeleton stabilizes membranes against deformation. Urea treatment also caused the projections induced both by micropipette aspiration and by lysophosphatidylcholine to become irreversible. Furthermore, urea converted the axisymmetric conical spicules induced by lysophosphatidylcholine into irregular, curved and knobby spicules; i.e., echinocytosis became acanthocytosis. Unlike controls, the ghosts and membrane skeletons obtained from urea-generated acanthocytes were imprinted with spicules. These data suggest that perturbing interprotein associations with urea in situ allowed the skeleton to evolve plastically to accommodate the contours imposed upon it by the overlying membrane.

Computer simulations of self-assembled membranes

Molecular dynamics simulations in three dimensions of particles that self-assemble to form two-dimensional, membrane-like objects are presented. Anisotropic, multibody forces, chosen so as to mimic real interactions between amphiphilic molecules, generate a finite rigidity and compressibility of the assembled membranes, as well as a finite line tension at their free edges. This model and its generalizations can be used to study a large class of phenomena taking place in fluctuating membranes. For instance, both fluid and solid-like phases, separated by a phase transition, are obtained and some of the large-scale properties of these membranes studied. In particular, thermal undulations of quasi-spherical fluid vesicles are analyzed, in a manner similar to recent experiments in lipid systems.

Effects of intramembrane particle aggregation on erythrocyte membrane fluidity: an electron spin resonance study in normal and in dystrophic subjects

Mobilization and aggregation of intramembrane particles (IMPs) are physiological events observed in various cells. In erythrocyte membranes, aggregation of IMPs can be induced by the exposure of partially desprectrinized erythrocyte membranes to acidic pH. We investigated the association between IMPs aggregation, protein mobility, and membrane fluidity in erythrocyte membranes of healthy controls and Duchenne muscular dystrophy (DMD) patients by using electron spin resonance and specific spin labels for membrane proteins and lipids. In erythrocyte membranes of control subjects, the partial spectrin removal induced a decreased segmental motion of protein spin label indicating an increase of protein-protein interactions. Stearic acid spin labels 5- and 16-(N-oxyl-4,4'-dimethyloxazolidine) showed that the treatment induces an increase of membrane fluidity. In DMD patients, both treated and untreated erythrocyte membranes showed changes of membrane fluidity when compared to those of the controls. Our results suggest that defects in the interactions between skeletal proteins and/or between membrane and skeleton components may contribute to the alterations of erythrocyte membranes in DMD.

Actin polymerization induces a shape change in actin-containing vesicles

We have encapsulated actin filaments in the presence and absence of various actin-binding proteins into lipid vesicles. These vesicles are approximately the same size as animal cells and can be characterized by the same optical microscopic and mechanical techniques used to study cells. We demonstrate that the initially spherical vesicles can be forced into asymmetric, irregular shapes by polymerization of the actin that they contain. Deformation of the vesicles requires that the actin filaments be on average at least approximately 0.5 micron long as shown by the effects of gelsolin, an actin filament-nucleating protein. Filamin, a filament-crosslinking protein, caused the surfaces of the vesicles to have a smoother appearance. Heterogeneous distribution of actin filaments within the vesicles is caused by interfilament interactions and modulated by gelsolin and filamin. The vesicles provide a model system to study control of cell shape and cytoskeletal organization, membrane-cytoskeleton interactions, and cytomechanics.

Quantitative detection of rapid motions in spectrin by NMR

Previous high resolution proton NMR data on human erythrocyte spectrin molecules has indicated the existence of regions exhibiting rapid internal motions within the intact molecules [L. W.-M. Fung, H.-Z. Lu, R. P. Hjelm, jr, M. E. Johnson, FEBS Lett., 197, 234 (1986)]. We have extended the studies by developing quantitative NMR methods to determine the fraction of spectrin protons exhibiting rapid internal motions, in both the isolated molecule and within the spectrin-actin network. Using both one-pulse and spin echo pulse sequences, we find that the fraction of the protons in rapid motion is about 15% of the total protons in the spectrin molecule at 37 degrees C in phosphate buffer with 150 mM NaCl at pH 7.4. Quantitative information on these rapid motions will be important in understanding the structural, mechanical and functional properties of spectrin molecules, as well as in understanding filamentous protein structures in general.

A series expression for the surface area of an ellipsoid and its application to the computation of the surface area of avian erythrocytes

Knowledge of the surface area of cells is necessary for biophysical studies in which the permeability coefficient of a cell-type for a solute is defined. Under the microscope, avian erythrocytes have the appearance of an oval discus usually with a central elongated nucleus. The dimensions of these cells have been obtained for a large range of species over the past century. However, estimates of the surface areas have been obtained using mathematically simplistic models such as flattened circular cylinders. We modelled red cells of the domestic chicken (Gallus domesticus) as ellipsoids using the previously published dimensions to obtain the three semi-axis lengths. Although the mathematical expression for the volume of an ellipsoid is well known, an expression for the surface area is not. We present a general expression for the surface area of an ellipsoid that is a power series involving elliptic integrals and functions; the latter being rapidly evaluated by computer using standard series expressions. Our estimates of area are compared with earlier ones and those obtained by numerical integration of the surface.

Mechanical properties of the red cell membrane skeleton: analysis of axisymmetric deformations

The mechanical properties of erythrocyte membrane composed of a membrane bilayer and membrane skeleton are considered. Two membrane models are described: the model of free boundaries (MFB) and the model of immobilized boundaries (MIB). In MFB, the skeleton is assumed to be attached to the bilayer at a finite number of points, whereas MIB allows the interaction of each spectrin filament with the bilayer along its whole length. For MFB an estimate was made of the mechanical strain generated in the membrane by sucking erythrocytes into a micropipette. The existence of the deformation threshold is demonstrated, below which no mechanical strain, except that of bending, appears in the membrane. Thus only deformations exceeding this threshold result in strain. The relationship between the applied tension and the height of erythrocyte "tongue" sucked into a micropipette was determined. The MIB characteristics correspond to the model of Evans: strains in the membrane are generated at any deformation, however small, i.e. the threshold is equal to zero. A basic feature of this model is quite a different distribution of the skeleton deformations in the membrane. A comparison of the theoretical models and experimental data demonstrated the possibility of either MFB or MIB occurring, depending on the characteristic measurement time.

Relation between the organization of spectrin and of membrane lipids in lymphocytes

In lymphocytes, the cytoskeletal protein spectrin exhibits two organizational states. Because the plasma membrane lipids of lymphocytes also display two organizational states, it was asked whether there is a relation between the organization of spectrin and of membrane lipids. When mouse thymocytes were stained with merocyanine 540 (MC540), a fluorescent lipophilic probe that binds preferentially to loosely packed, disorganized lipid bilayers, some cells fluoresced brightly and some only dimly or not at all. When the same population was stained for spectrin by indirect immunofluorescence, the spectrin in some cells was uniformly distributed, while in others it was concentrated in a unipolar aggregate. Techniques enriching for mature thymocytes selected for cells displaying low MC540 fluorescence and aggregated spectrin, the same characteristics found in peripheral blood lymphocytes. Flow cytometric sorting of thymocytes based on MC540 phenotype simultaneously sorted them by spectrin phenotype. Finally, treatment with agents that alter the distribution of spectrin caused mature lymphocytes to display high MC540 fluorescence and uniform spectrin. Thus, a relation exists between the organizational states of spectrin and of membrane lipids in lymphocytes: aggregated spectrin is found in cells with tightly organized membrane lipids, uniform spectrin in those with loosely organized lipids. Spectrin may thus be involved in modulating membrane lipid organization in lymphocytes as it is in erythrocytes. Since loosely organized lipids may promote adhesion of blood cells to reticuloendothelial cells, spectrin may thereby be involved in transducing an internally generated adhesion signal to the lymphocyte surface.

Model of red blood cell membrane skeleton: electrical and mechanical properties

A theoretical membrane skeleton model of erythrocyte has been developed and successfully applied to interpret electrical and mechanical properties of the red blood cell spectrin-actin network. The model is based on the structure of the membrane skeleton that is comprised of unit cells each containing an actin protofilament and shooting forth a few spectrin heterodimers. The loose ends of the heterodimers of adjacent cells can form bonds with each other giving rise to an integrated network. The number of bonds depends on the temperature. The bond length being excessive (2.6 times the distance between the centers of adjacent cells), the bonds are flexible, and can thus be regarded as entropy springs. The advanced model has been employed to calculate the shear modulus of the membrane skeleton as well as to establish its temperature dependence. In a wide range of temperatures mu(T) is a decreasing function well fitting the experimental data. The relationship between the membrane bilayer-free size of the skeleton and the ionic strength of the solution has been derived to appear in good agreement with the results obtained previously. Experimental data combined with the advanced theory yield the average number of heterodimers per unit cell, m0, as equal to ca. 5; the spectrin heterodimer charge has been estimated.

The molecular basis of erythrocyte shape

Recent discoveries about the molecular organization and physical properties of the mammalian erythrocyte membrane and its associated structural proteins can now be used to explain, and may eventually be used to predict, the shape of the erythrocyte. Such explanations are possible because the relatively few structural proteins of the erythrocyte are regularly distributed over the entire cytoplasmic surface of the cell membrane and because the well-understood topological associations of these proteins seem to be stable in comparison with the time required for the cell to change shape. These simplifications make the erythrocyte the first nonmuscle cell for which it will be possible to extend our knowledge of molecular interactions to the next hierarchical level of organization that deals with shape and shape transformations.

The human erythrocyte membrane skeleton may be an ionic gel. III. Micropipette aspiration of unswollen erythrocytes

We have carried out a theoretical analysis of micropipette aspiration of unswollen erythrocytes using the protein-gel-lipid-bilayer membrane model and taking into account that the modulus of area compression of the membrane skeleton may depend on the environmental conditions. Our analysis shows that the aspiration pressure needed to obtain a certain membrane projection length is strongly dependent on the ratio between the membrane skeleton modulus of area compression and the elastic shear modulus. Our analysis therefore predicts that micropipette aspiration of unswollen erythrocytes may be a sensitive method for detection of changes in this ratio. The analysis thus also shows that micropipette aspiration of unswollen erythrocytes can not be used to determine the membrane shear modulus unless something is known about the membrane skeleton modulus of area compression.

The human erythrocyte membrane skeleton may be an ionic gel. I. Membrane mechanochemical properties

Biochemical and biophysical observations indicate that the erythrocyte membrane skeleton is composed of a swollen network of long, flexible and ionizable macromolecules located at the cytoplasmic surface of the fluid membrane lipid bilayer. We have analyzed the mechanochemical properties of the erythrocyte membrane assuming that the membrane skeleton constitutes an ionic gel (swollen ionic elastomer). Using recently established statistical thermodynamic theory for such gels, our analysis yields mathematical expressions for the mechanochemical properties of erythrocyte membranes that incorporate membrane molecular parameters to an extent not achieved previously. The erythrocyte membrane elastic shear modulus and maximum elastic extension ratio predicted by our membrane model are in quantitative agreement with reported values for these parameters. The gel theory predicts further that the membrane skeleton modulus of area compression, KG, may be small as well as large relative to the membrane elastic shear modulus, G, depending on the environmental conditions. Our analysis shows that the ratio between these two parameters affects both the geometry and the stability of the favoured cell shapes.