Topology of dividing planar tilings: Mitosis and order in epithelial tissues.

We investigate a range of rule-based models of the in-plane structure of growing single-cell-thick epithelia represented by the distribution of frequencies of polygon classes. Within the Markovian framework introduced by Gibson et al. [Nature (London) 442, 1038 (2006)10.1038/nature05014], we discuss various topologically allowed cell division schemes assumed to control the structure of the tissue as well as a phenomenological Gaussian scheme, and we compute the stationary distributions for all of them. Some of the distributions reproduce those seen in tissues characterized by unbiased mitotic events but also in certain tissues with a preferred orientation of the mitotic plane or a cell-rearrangement process such as neighbor exchange. In addition, we propose the asynchronous-division variant of the model, which builds on the Lewis law and on the Aboav-Weaire law as well as on the fact that the dividing cells are larger than the resting cells. This generalization a posteriori validates the original model.

Limiting shapes of confined lipid vesicles.

We theoretically study the shapes of lipid vesicles confined to a spherical cavity, elaborating a framework based on the so-called limiting shapes constructed from geometrically simple structural elements such as double-membrane walls and edges. Partly inspired by numerical results, the proposed non-compartmentalized and compartmentalized limiting shapes are arranged in the bilayer-couple phase diagram which is then compared to its free-vesicle counterpart. We also compute the area-difference-elasticity phase diagram of the limiting shapes and we use it to interpret shape transitions experimentally observed in vesicles confined within another vesicle. The limiting-shape framework may be generalized to theoretically investigate the structure of certain cell organelles such as the mitochondrion.

Structure formation in soft nanocolloids: liquid-drop model.

Using a model where soft nanocolloids such as spherical polymer brushes and star polymers are viewed as compressible liquid drops, we theoretically explore contact interactions between such particles. By numerically minimizing the phenomenological free energy consisting of bulk and surface terms, we find that at small deformations the drop-drop interaction is pairwise additive and described by a power law. We also propose a theory to describe the small-deformation regime, and the agreement is very good at all drop compressibilities. The large-deformation regime, which is dominated by many-body interactions, is marked by a rich phase diagram which includes the face- and body-centered-cubic, σ, A15, and simple hexagonal lattice as well as isostructural and re-entrant transitions. Most of these features are directly related to the non-convex deformation free energy emerging from many-body effects in the partial-faceting regime. The phase diagram, which depends on just two model parameters, contains many of the condensed phases observed in experiments. We also provide statistical-mechanical arguments that relate the two model parameters to the molecular architecture of the polymeric nanocolloids, chain rigidity, and solvent quality. The model represents a generic framework for the overarching features of the phase behavior of polymeric nanocolloids at high compressions.

Phase diagram of elastic spheres.

Experiments show that polymeric nanoparticles often self-assemble into several non-close-packed lattices in addition to the face-centered cubic lattice. Here, we explore theoretically the possibility that the observed phase sequences may be associated with the softness of the particles, which are modeled as elastic spheres interacting upon contact. The spheres are described by two finite-deformation theories of elasticity, the modified Saint-Venant-Kirchhoff model and the neo-Hookean model. We determine the range of indentations where the repulsion between the spheres is pairwise additive and agrees with the Hertz theory. By computing the elastic energies of nine trial crystal lattices at densities far beyond the Hertzian range, we construct the phase diagram and find the face- and body-centered cubic lattices as well as the A15 lattice and the simple hexagonal lattice, with the last two being stable at large densities where the spheres are completely faceted. These results are qualitatively consistent with observations, suggesting that deformability may indeed be viewed as a generic property that determines the phase behavior in nanocolloidal suspensions.

Mosaic two-lengthscale quasicrystals.

Over the past decade, quasicrystalline order has been observed in many soft-matter systems: in dendritic micelles, in star and tetrablock terpolymer melts and in diblock copolymer and surfactant micelles. The formation of quasicrystals from such a broad range of 'soft' macromolecular micelles suggests that they assemble by a generic mechanism rather than being dependent on the specific chemistry of each system. Indeed, micellar softness has been postulated and shown to lead to quasicrystalline order. Here we theoretically explore this link by studying two-dimensional hard disks decorated with step-like square-shoulder repulsion that mimics, for example, the soft alkyl shell around the aromatic core in dendritic micelles. We find a family of quasicrystals with 10-, 12-, 18- and 24-fold bond orientational order which originate from mosaics of equilateral and isosceles triangles formed by particles arranged core-to-core and shoulder-to-shoulder. The pair interaction responsible for these phases highlights the role of local packing geometry in generating quasicrystallinity in soft matter, complementing the principles that lead to quasicrystal formation in hard tetrahedra. Based on simple interparticle potentials, quasicrystalline mosaics may well find use in diverse applications ranging from improved image reproduction to advanced photonic materials.

Composite contact of binary lipid membranes.

Upon adhesion, binary lipid membranes may retain their structural identity but they may also undergo partial fusion promoted by microscopic phase separation of the components. In membranes composed of cylindrical and inverted conical lipids, a partially fused contact zone may consist of inverted micelles sandwiched between two monolayers. We theoretically analyze the elastic properties of such a composite contact zone. We calculate its bending moduli and we show that they may be considerably larger than the moduli of a phase-nonseparated double bilayer. The main mechanism responsible for the enhanced rigidity of the composite contact structure is the increased separation of the two monolayers caused by the presence of water in the inverted micelles.

Many-body contact repulsion of deformable disks.

We use a spring-and-plaquette network model to analyze the repulsion between elastic disks in contact. By studying various 2D geometries, we find that as disks approach the incompressibility limit the many-body effects become dominant and the disk-disk interaction is not pairwise additive. Upon compression, the disks undergo a transition from the localized to the distributed deformation regime accompanied by a steep increase of energy consistent with the onset of a hard core. These results shed new light on the structures formed by deformable objects such as soft nanocolloids.

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.

From lumps to lattices: crystallized clusters made simple.

Using a minimal model based on the continuum theory of a two-dimensional hard-core/square-shoulder ensemble, we reinterpret the main features of cluster mesophases formed by colloids with soft shoulder-like repulsive interactions. We rederive the lattice spacing, the binding energy, and the phase diagram. We also extend the clustering criterion [Likos, C. N. et al. Phys. Rev. E, 2001, 63, 031206; Glaser, M. A. et al. EPL2007, 78, 46004] to include the effect of the hard cores, which precludes the formation of clusters at small densities.

Continuum theory for cluster morphologies of soft colloids.

We introduce a continuum description of the thermodynamics of colloids with a core-corona architecture. In the case of thick coronas, their overlap can be treated approximately by replacing the exact one-particle density distribution by a suitably shaped step profile, which provides a convenient way of modeling the spherical, columnar, lamellar, and inverted cluster morphologies predicted by numerical simulations and the more involved theories. We use the model to study monodisperse particles with the hard-core/square-shoulder pair interaction as the simplest representatives of the core-corona class. We derive approximate analytical expressions for the enthalpies of the cluster morphologies which offer a clear insight into the mechanisms at work, and we calculate the lattice spacing and the cluster size for all morphologies of the phase sequence as well as the phase-transition pressures. By comparing the results with the exact crystalline minimum-enthalpy configurations, we show that the accuracy of the theory increases with shoulder width. We discuss possible extensions of the theory that could account for the finite-temperature effects.

Morphometry and structure of natural random tilings.

A vast range of both living and inanimate planar cellular partitions obeys universal empirical laws describing their structure. To better understand this observation, we analyze the morphometric parameters of a sizeable set of experimental data that includes animal and plant tissues, patterns in desiccated starch slurry, suprafroth in type-I superconductors, soap froths, and geological formations. We characterize the tilings by the distributions of polygon reduced area, a scale-free measure of the roundedness of polygons. These distributions are fairly sharp and seem to belong to the same family. We show that the experimental tilings can be mapped onto the model tilings of equal-area, equal-perimeter polygons obtained by numerical simulations. This suggests that the random two-dimensional patterns can be parametrized by their median reduced area alone.

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.

Field-induced self-assembly of suspended colloidal membranes.

We report experiments that probe the self-assembly of micrometer-size colloids into one-particle-thick, robust, and self-healing membranes. In a magic-angle precessing magnetic field, superparamagnetic spheres experience isotropic pair attraction similar to the van der Waals force between atoms. But the many-body polarization interactions among them steer an ordered aggregation pathway consisting of growth of short chains, cross-linking and network formation, network coarsening, and consolidation of membrane patches. This generic aggregation scenario can be induced in any particles of large enough susceptibility.

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.

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.

Observation of condensed phases of quasiplanar core-softened colloids.

We experimentally study the condensed phases of repelling core-softened spheres in two dimensions. The dipolar pair repulsion between superparamagnetic spheres trapped in a thin cell is induced by a transverse magnetic field and softened by suitably adjusting the cell thickness. We scan a broad density range and we materialize a large part of the theoretically predicted phases in systems of core-softened particles, including expanded and close-packed hexagonal, square, chainlike, stripe or labyrinthine, and honeycomb phase. Further insight into their structure is provided by Monte Carlo simulations.

Colloids on free-standing smectic films.

We study the structure of a free-standing smectic-A film around a micron-size polystyrene colloid adsorbed onto the film. We find that a colloid or a cluster of colloids is surrounded by an optically distinct and radially decorated meniscus ending with a sharp edge. The observed strong and finite-range attraction between the adsorbed colloids is driven by the fusion of menisci. We interpret the structure of the smectic meniscus in terms of a model dominated by the surface free energy and we argue that the characteristic appearance of the meniscus is due to layer undulations.

The foam analogy: from phases to elasticity.

By mapping the interactions of colloidal particles onto the problem of minimizing areas, the physics of foams can be used to understand the phase diagrams of both charged and fuzzy colloids. We extend this analogy to study the elastic properties of such colloidal crystals and consider the face-centered cubic, body-centered cubic and A15 lattices. We discuss two types of soft interparticle potentials corresponding to charged and fuzzy colloids, respectively, and we analyze the dependence of the elastic constants on density as well as on the parameters of the potential. We show that the bulk moduli of the three lattices are generally quite similar, and that the shear moduli of the two non-close-packed lattices are considerably smaller than in the face-centered cubic lattice. We find that in charged colloids, the elastic constants are the largest at a finite screening length, and we discuss a shear instability of the A15 lattice.

Morphology and structure of thin liquid-crystalline films at nematic-isotropic transition.

We review the main features of very thin nematic liquid-crystalline films on solid substrates, focusing on 5CB on oxidized silicon wafers. By discussing the theoretical aspects of the observed structures, we show that the phenomena at work include isotropic capillary condensation and that the coexistence of isotropic and nematic terraces in thin films is a result of the interplay of several mechanisms. Further theoretical as well as experimental efforts are needed to completely understand the wetting behavior of these systems.