Mechanical behavior of closed lamellar membranes as a possible common mechanism for the establishment of developmental shapes.

The mechanical behavior of a closed membrane composed of two layers in contact is described as it is obtained by finding the minimum of the membrane bending energy at constant membrane area, constant difference between the areas of the two layers, and constant enclosed volume. It is shown that the membrane bending energy is a discontinuous function of the volume (v) and difference between the layer areas (delta a) defined relative to the volume and area difference of a sphere with the same membrane area, respectively. However, for different classes of shapes it is possible to obtain regions in the v/delta a diagram within which the shapes change continuously with v and delta a. These regions are shown for the egg, dumbbell and cup shape classes, respectively. The results of the shape analysis are used in the discussion of cell polarity, cytokinesis and gastrulation. Cell polarity is related to the decrease in symmetry during the transition from the radially symmetrical spherical shape to the asymmetrical shapes of the egg class. It is proposed that symmetrical cytokinesis occurs within the dumbbell class and that asymmetrical cytokinesis occurs within the egg class. Gastrulation is described as shape transformations within the class of cup shapes.     

Measurements of mechanical properties of the blastula wall reveal which hypothesized mechanisms of primary invagination are physically plausible in the sea urchin Strongylocentrotus purpuratus.

Computer simulations showed that the elastic modulus of the cell layer relative to the elastic modulus of the extracellular layers predicted the effectiveness of different force-generating mechanisms for sea urchin primary invagination [L. A. Davidson, M. A. R. Koehl, R. Keller, and G. F. Oster (1995) Development 121, 2005-2018]. Here, we measured the composite elastic modulus of the cellular and extracellular matrix layers in the blastula wall of Strongylocentrotus purpuratus embryos at the mesenchyme blastula stage. Combined, these two layers exhibit a viscoelastic response with an initial stiffness ranging from 600 to 2300 Pa. To identify the cellular structures responsible for this stiffness we disrupted these structures and correlated the resulting lesions to changes in the elastic modulus. We treated embryos with cytochalasin D to disrupt the actin-based cytoskeleton, nocodazole to disrupt the microtubule-based cytoskeleton, and a gentle glycine extraction to disrupt the apical extracellular matrix (ECM). Embryos treated less than 60 min in cytochalasin D showed no change in their time-dependent elastic modulus even though F-actin was severely disrupted. Similarly, nocodazole had no effect on the elastic modulus even as the microtubules were severely disrupted. However, glycine extraction resulted in a 40 to 50% decrease in the elastic modulus along with a dramatic reduction in the hyalin protein at the apical ECM, thus implicating the apical ECM as a major mechanical component of the blastula wall. This finding bears on the mechanical plausibility of several models for primary invagination.     

Mechanical Control of Bacterial Cell Shape

In bacteria, cytoskeletal filament bundles such as MreB control the cell morphology and determine whether the cell takes on a spherical or a rod-like shape. Here we use a theoretical model to describe the interplay of cell wall growth, mechanics, and cytoskeletal filaments in shaping the bacterial cell. We predict that growing cells without MreB exhibit an instability that favors rounded cells. MreB can mechanically reinforce the cell wall and prevent the onset of instability. We propose that the overall bacterial shape is determined by a dynamic turnover of cell wall material that is controlled by mechanical stresses in the wall. The model affirms that morphological transformations with and without MreB are reversible, and quantitatively describes the growth of irregular shapes and cells undergoing division. The theory also suggests a unique coupling between mechanics and chemistry that can control organismal shapes in general.     

Phospholipid membrane bending as assessed by the shape sequence of giant oblate phospholipid vesicles

Vesicle shape transformations caused by decreasing the difference between the equilibrium areas of membrane monolayers were studied on phospholipid vesicles with small volume to membrane area ratios. Slow transformations of the vesicle shape were induced by lowering of the concentration of lipid monomers in the solution outside the vesicle. The complete sequence of shapes consisted of a string of pearls, and wormlike, starfish, discocyte and stomatocyte shapes. The transformation from discocyte to stomatocyte vesicle shapes was analyzed theoretically to see whether these observations accord with the area difference elasticity (ADE) model. The membrane shape equation and boundary conditions were derived for axisymmetrical shapes for low volume vesicles, part of whose membranes are in contact. Calculated shapes were arranged into a phase diagram. The theory predicts that the transition between discocyte and stomatocyte shapes is discontinuous for relatively high volumes and continuous for low volumes. The calculated shape sequences matched well with the observed ones. By assuming a linear decrease of the equilibrium area difference with time, the ratio between the nonlocal and local bending constants is in agreement with reported values.     

Phospholipid membrane bending as assessed by the shape sequence of giant oblate phospholipid vesicles

Vesicle shape transformations caused by decreasing the difference between the equilibrium areas of membrane monolayers were studied on phospholipid vesicles with small volume to membrane area ratios. Slow transformations of the vesicle shape were induced by lowering of the concentration of lipid monomers in the solution outside the vesicle. The complete sequence of shapes consisted of a string of pearls, and wormlike, starfish, discocyte and stomatocyte shapes. The transformation from discocyte to stomatocyte vesicle shapes was analyzed theoretically to see whether these observations accord with the area difference elasticity (ADE) model. The membrane shape equation and boundary conditions were derived for axisymmetrical shapes for low volume vesicles, part of whose membranes are in contact. Calculated shapes were arranged into a phase diagram. The theory predicts that the transition between discocyte and stomatocyte shapes is discontinuous for relatively high volumes and continuous for low volumes. The calculated shape sequences matched well with the observed ones. By assuming a linear decrease of the equilibrium area difference with time, the ratio between the nonlocal and local bending constants is in agreement with reported values.     

Spectrin, human erythrocyte shapes, and mechanochemical properties.

Physical studies of human erythrocyte spectrin indicate that isolated spectrin dimers and tetramers in solution are worm-like coils with a persistence length of approximately 20 nm. This finding, the known polyelectrolytic nature of spectrin, and other structural information about spectrin and the membrane skeleton molecular organization have lead us to the hypothesis that the human erythrocyte membrane skeleton constitutes a two-dimensional ionic gel (swollen ionic elastomer). This concept is incorporated in what we refer to as the protein gel-lipid bilayer membrane model. The model accounts quantitatively for red elastic shear modulus and the maximum elastic extension ratio reported for the human erythrocytes membrane. Gel theory further predicts that depending on the environmental conditions, the membrane skeleton modulus of area compression may be small or large relative to the membrane elastic shear modulus. Our analyses show that the ratio between these two parameters affects both the geometry and the stability of the favored cell shapes and that the higher the membrane skeleton compressibility the smaller the values of the gel tension needed to induce cell shape transformations. The main virtue of the protein gel-lipid bilayer membrane model is that it offers a novel theoretical and molecular basis for the various mechanical properties of the membrane skeleton such as the membrane skeleton modulus of area compression and osmotic tension, and the effects of these properties on local membrane skeleton density, cell shape, and shape transformations.     

Fluid-structure Interaction within a Layered Aortic Arch Model

The response of wall stress to the elasticity of each layer in the aorta wall was investigated to understand the role of the different elastic properties of layers in the aortic dissection. The complex mechanical interaction between blood flow and wall dynamics in a three-dimensional arch model of an aorta was studied by means of computational coupled fluid-structure interaction analysis. The results show that stresses in the media layer are highest in three layers and that shear stress is concentrated in the media layer near to the adventitia layer. Hence, the difference in the elastic properties of the layers could be responsible for the pathological state in which a tear splits across the tunica media to near to the tunica adventitia and the dissection spreads along the laminar planes of the media layer where it is near the adventitia layer.     

Bacterial shape

In free-living eubacteria an external shell of peptidoglycan opposes internal hydrostatic pressure and prevents membrane rupture and death. At the same time, this wall imposes on each cell a shape. Because shape is both stable and heritable, as is the ability of many organisms to execute defined morphological transformations, cells must actively choose from among a large repertoire of available shapes. How they do so has been debated for decades, but recently experiment has begun to catch up with theory. Two discoveries are particularly informative. First, specific protein assemblies, nucleated by FtsZ, MreB or Mbl, appear to act as internal scaffolds that influence cell shape, perhaps by correctly localizing synthetic enzymes. Second, defects in cell shape are correlated with the presence of inappropriately placed, metabolically inert patches of peptidoglycan. When combined with what we know about mutants affecting cellular morphology, these observations suggest that bacteria may fabricate specific shapes by directing the synthesis of two kinds of cell wall: a long-lived, rigid framework that defines overall topology, and a metabolically plastic peptidoglycan whose shape is directed by internal scaffolds.     

Crowding-induced membrane remodeling: Interplay of membrane tension,polymer density,architecture

The plasma membrane hosts a wide range of biomolecules, mainly proteins and carbohydrates, that mediate cellular interactions with its environment. The crowding of such biomolecules regulates cellular morphologies and cellular trafficking. Recent discoveries have shown that the structure and density of cell surface polymers and hence the signaling machinery change with the state of the cell, especially in cancer progression. The alterations in membrane-attached glycocalyx and glycosylation of proteins and lipids are common features of cancer cells. The overexpression of glycocalyx polymers, such as mucin and hyaluronan, strongly correlates with cancer metastasis. Here, we present a mesoscale biophysics-based model that accounts for the shape regulation of membranes by crowding of membrane-attached biopolymer-glycocalyx and actin networks. Our computational model is based on the dynamically triangulated Monte Carlo model for membranes and coarse-grained representations of polymer chains. The model allows us to investigate the crowding-induced shape transformations in cell membranes in a tension- and graft polymer density-dependent manner. Our results show that the number of membrane protrusions and their shape depend on membrane tension, with higher membrane tension inducing more tubular protrusions than the vesicular shapes formed at low tension at high surface coverage of polymers. The shape transformations occur above the threshold density predicted by the polymer brush theory, but this threshold also depends on the membrane tension. Increasing the size of the polymer, either by changing the length or by adding side chains, is shown to increase the crowding-induced curvature. The effect of crowding is more prominent for flexible polymers than for semiflexible rigid polymers. We also present an extension of the model that incorporates properties of the actin-like filament networks and demonstrate how tubular structures can be generated by biopolymer crowding on the cytosolic side of cell membranes.     

Six new species of microsporidia of the genus Amblyospora (Microspora: Amblyosporidae) from blood sucking mosquitoes (Diptera: Culicidae) from the west Siberia

Microsporidia of the genus Amblyospora parasiting the adipose body of mosquito larvae of the genus Aedes and Culex has been studied with both light and electron microscopy. Six new species of microsporidia are described based on ultrastructural characteristics of spores and sporogony stages. Amblyospora flavescens sp. n. Mature spores are egg-shaped. The spore wall with three layers, about 165 nm. Exospore is two-membranous. Subexospore is absent. Endospore is electron-translucent. Polaroplast consists of three parts: lamellar, large vesicular, lamellar. The anisofilar polar filament with 10--11 coils (3 1/2 + 2 1/2 + 4-5). Fixed spores are 6.3 +/- 0.1 x 4.24 +/- 0.1 microm. Amblyospora kolarovi sp. n. Mature spores are egg-shaped. The spore wall with three layers, about 265-315 nm. Exospore shapes tucks on the surface of spore. It is two-membranous. Subexospore is quagge, structural. Endospore is electron-translucent. Polaroplast consists of two parts: lamellar and large vesicular. The anisofilar polar filament with 11-13 coils (3 + 8-10). Fixed spores are 5.4-5.6 x 3.5-4.2 microm. Amblyospora orbiculata sp. n. Mature spores are widely egg-shaped. On a back pole there is a small concavity. The spore wall with three layers, about 155 nm. Exospore is shapes tucks on a surface of spore. It is two-membranous. Subexospore is absent. Endospore is electron-translucent. Polaroplast consists of three parts: lamellar, vesicular, lamellar. Polar filament is anisofilar, with 11 1/2 coils (4 1/2 + 1 + 6). Fixed spores are 6.3 +/- 0.1 x x 4.0 +/- 0.1 microm. Amblyospora rugosa sp. n. Mature spores are egg-shaped. On a back pole there is a small concavity. The spore wall with three layers, about 225 nm. Exospore is shapes tucks on a surface of spore. It is two-membranous. Subexospore is quaggy, structural. Endospore is electron-translucent. Polaroplast lamellate. Polar filament is anisofilar, with 17 1/2 coils (3 1/2 + 1 + 13). Fixed spores are 5.3 +/- 0.1 x 3.7 +/- 0.1 microm. Amblyospora undata sp. n. Mature spores are egg-shaped. The spore wall is three-layered, about 220 nm. Exospore is shapes tucks on a surface of spore. It is two-membranous. Subexospore is quaggy, structural. Endospore is electron-translucent. Polaroplast lamellate. The anisofilar polar filament with 8 coils (3 + 5). Fixed spores are 5.0 +/- 0.1 x 3.0 +/- 0.1 microm. Amblyospora urski sp. n. Mature spores have widely oval form. The back pole is concave. The spore wall with three layers, about 280 nm. Exospore is shapes tucks on a surface of spore. It is two-membranous. Subexospore is quaggy, structural. Endospore is electron-translucent. Polaroplast lamellate. Polar filament is anisofilar, with 6 coils (2 + 4). Fixed spores are 4.4 +/- 0.1 x 2.9 +/- 0.1 microm.     

Computer simulation of the hydrostatic skeleton. The physical equivalent, mathematics and application to worm-like forms

The functional principles of a hydrostatic skeleton were combined to obtain a physical model which includes geometry, number and length-tension relationships of the elastic elements in the body wall, internal volume and internal pressure. The model skeleton with pre-set internal volume assumes a certain shape and develops a specific internal pressure in order to minimize the potential energy stored in the elastic elements. This shape is calculated as equilibrium state by using finite element methods and optimization techniques. This model is flexible enough to accommodate different geometries and length-tension-relationships of the elastic elements. Presently, the model is implemented with linear length-tension relationships and certain geometrical restrictions, such as uniform width over the entire animal, and rectangular cross sections; the general case is outlined. First simulations with the "unit-worm" yield stable solutions, i.e. stable shapes for all combinations of parameters tested so far. They define the conditions for bringing all muscles to an optimal operating point. We detected a pressure maximum with increasing volume, assessed the contribution of circular muscles to bending, and determined the shapes of animals with different muscle activations in each body half (Chapman-matrix). We summarize our results by the volume rule and stabilization rule, two simple concepts which predict changes in shape as the result of muscle activation.     

Principles and methods of geometric morphometrics

The basic concepts, notions and methods of geometric morphometrics (GM) are considered. This approach implies multivariate analysis of landmark coordinates located following certain rules on the surface of a morphological object. The aim of GM is to reveal differences between morphological objects by their shapes as such, the "size factor" being excluded. The GM is based on the concept of Kendall's space (KS) defined as a hypersphere with points distributed on its surface. These points are the shapes defined as aligned landmark configurations. KS is a non-Euclidian space, its metrics called Procrustes is defined by landmark configuration of a reference shape relative to which other shapes are aligned and compared. The differences among shapes are measured as Procrustes distances between respective points. For the linear methods of multivariate statistics to be applied to comparison of shapes, the respective points are projected onto the tangent plane (tangent space), the tangent point being defined by the reference. There are two principal methods of shape comparisons in GM: the Procrustes superimposition (a version of the least squares analysis) and thin-plate spline analysis. In the first case, Procrustes residuals are the outcome shape variables which remain after isometric alignment of the shapes being compared. Their summation over all landmarks yields Procrustes distances among these shapes. The Procrustes distances can be used in multivariate analyses just as the Euclidian distances. In the second case, the shapes are fitted to the references by stretching/compressing and shearing until complete identity of their landmark configurations. Eigenvectors of resulting bending energy matrix are defined as new shape variables, principal warps which yield another shape space with the origin defined by the reference. Projections of the shapes being compared onto principal warps yield partial warps, and their covariance matrix decomposition into eigenvectors yields relative warps which are similar to principal components (in particular, they are mutually orthogonal). Both partial and relative warps can be used in many multivariate statistic analyses as quantitative shape variables. Results of thin-plate spline analysis can be represented graphically by transformation grid which displays type, amount and localization of the shape differences. Basis rules of sample composition and landmark positioning to be used in GM are considered. At present, rigid (with minimal degrees of freedom) 2D morphological objects are most suitable for GM applications. It is important to recognize three type of real landmarks, and additionally semi-landmarks and "virtual" landmarks. Some procedures of thin-plate spline analysis are considered exemplified by some study cases, as well as applications of some standard multivariate methods to GM results. They make it possible to evaluate correlation between different shapes, as well as between a shape and some non-shape variables (linear measurements etc); to evaluate the differences among organisms by shape of a morphological structure; to identify landmarks which most accounted for both correlation and differences between the shapes. An annotated list of most popular softwares for GM is provided.     

The human erythrocyte membrane skeleton may be an ionic gel

In the first paper in this series (Stokke et al. Eur Biophys J 1986, 13:203-218) we developed the general theory of the mechanochemical properties and the elastic free energy of the protein gel--lipid bilayer membrane model. Here we report on an extensive numerical analysis of the human erythrocyte shapes and shape transformations predicted by this new cell membrane model. We have calculated the total elastic free energy of deformation of four different cell shape classes: disc-shaped cells, cup-shaped cells, crenated cells, and cells with membrane invaginations. We find that which of these shape classes is favoured depends strongly on the spectrin gel osmotic tension, IIGu, and the surface tensions, IIEu and IIPu, of the extracellular and protoplasmic halves of the membrane lipid bilayer, respectively. For constant ratio IIEu/IIPu greater than O large negative or positive values of IIGu favour respectively the crenated and invaginated cell shape classes. For small absolute values of IIGu, IIEu, and IIPu, biconcave or cup-shaped cells are the stable ones. Our numerical analysis shows that the higher the membrane skeleton compressibility is, the smaller are the values of IIGu needed to induce cell shape transformation. We find that the stable and metastable shapes of discocytes and stomatocytes generally depend both on the shape of the stressfree membrane skeleton and the membrane skeleton compressibility.     

Time-lapse cinemicrographic analysis of superficial cell behavior during and prior to gastrulation in Xenopus laevis

Time-lapse cinemicrography was used to show what changes in the number, size, shape, arrangement and what movements of apices of superficial cells occur during epiboly, extension, convergence and blastopore formation in the blastula or gastrula of Xenopus laevis. Epiboly of the animal region occurs by apical expansion of superficial cells at a nearly constant rate from the midblastula to the midgastrula stage. Egression of deep cells into the superficial layer does not occur. Extension of the dorsal marginal zone begins in the late blastula stage with the rapid spreading of the apices of cells in this region and this continues until the onset of neurulation when rapid shrinkage begins. Extension and convergence of the dorsal marginal zone occurs by a rearrangement in which individual cells exchange neighbors and by a change in the shape of the cell apices. Regional differences in apical expansion are accompanied by differences in rate of anticlinal division of superficial cells such that cells in all sectors of the animal region and the marginal zone show similar patterns of decrease in apparent apical area. Shrinkage of the apices of bottle cells during blastopore formation is described. From this and other studies, a model of the cellular behavior of epiboly, extension and convergence is constructed and several hypotheses as to how these activities might generate the mechanical forces of the gastrulation movements are presented.     

Statistical analysis of sizes and shapes of virus capsids and their resulting elastic properties

From the analysis of sizes of approximately 130 small icosahedral viruses we find that there is a typical structural capsid protein, having a mean diameter of 5 nm and a mean thickness of 3 nm, with more than two thirds of the analyzed capsid proteins having thicknesses between 2 nm and 4 nm. To investigate whether, in addition to the fairly conserved geometry, capsid proteins show similarities in the way they interact with one another, we examined the shapes of the capsids in detail. We classified them numerically according to their similarity to sphere and icosahedron and an interpolating set of shapes in between, all of them obtained from the theory of elasticity of shells. In order to make a unique and straightforward connection between an idealized, numerically calculated shape of an elastic shell and a capsid, we devised a special shape fitting procedure, the outcome of which is the idealized elastic shape fitting the capsid best. Using such a procedure we performed statistical analysis of a series of virus shapes and we found similarities between the capsid elastic properties of even very different viruses. As we explain in the paper, there are both structural and functional reasons for the convergence of protein sizes and capsid elastic properties. Our work presents a specific quantitative scheme to estimate relatedness between different proteins based on the details of the (quaternary) shape they form (capsid). As such, it may provide an information complementary to the one obtained from the studies of other types of protein similarity, such as the overall composition of structural elements, topology of the folded protein backbone, and sequence similarity.     

Morphology of hindwing veins in the shield bug Graphosoma italicum (Heteroptera: Pentatomidae)

Light, fluorescence, and electron microscopy were applied to cross sections and -breakage and whole-mount preparations of the anterior hindwing vein of the shield bug Graphosoma italicum. These analyses were complemented by investigations of the basal part of the forewing Corium and Clavus. The integration of structural, histological, and fluorescence data revealed a complex arrangement of both rigid and elastic structures in the wall of wing veins and provided insights into the constitution of transition zones between rigid and elastic regions. Beneath the exocuticular layers, which are continuous with the dorsal and ventral cuticle of the wing membrane, the lumen of the veins is encompassed by a mesocuticular layer, an internal circular exocuticular layer, and an internal longitudinal endocuticular layer. Separate parallel lumina within the anterior longitudinal vein of the hindwing, arranged side-by-side rostro-caudally, suggest that several veins have fused in the phylogenetic context of vein reduction in the pentatomid hindwing. Gradual structural transition zones and resilin enrichment between sclerotized layers of the vein wall and along the edges of the claval furrow are interpreted as mechanical adaptations to enhance the reliability and durability of the mechanically stressed wing veins.     

Cross-Sectional Shape of Collapsible Tubes

In order to quantify the collapse phenomenon in veins, this paper presents a mathematical analysis of the cross-sectional shape of a flexible tube as its internal pressure varies. Quantitative results are presented in terms of the physical parameters of the tube, such as wall thickness and Young's modulus. It is assumed that the tube is thin walled, that no stretching occurs, that the cross-sectional shape is elliptical when the transmural pressure is zero, and that the longitudinal prestress is zero. The equations were solved on a digital computer which displayed the cross-sectional shapes on an oscilloscope, which were then photographed. A selection of these photographs is presented. Curves are shown which give the cross-sectional area and compliance as functions of transmural pressure. Other curves are shown which are useful for interpolation, and for use in the experimental determination of the physical parameters which may otherwise be difficult or impossible to measure accurately.     

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.